TPTP Problem File: SLH0001^1.p

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

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Combinable_Wands/0002_PartialHeapSA/prob_00477_017228__7370296_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1998 ( 384 unt; 719 typ;   0 def)
%            Number of atoms       : 4476 (1186 equ;   0 cnn)
%            Maximal formula atoms :   14 (   3 avg)
%            Number of connectives : 17057 ( 634   ~;  68   |; 374   &;13651   @)
%                                         (   0 <=>;2330  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (   9 avg)
%            Number of types       :  110 ( 109 usr)
%            Number of type conns  : 1853 (1853   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  613 ( 610 usr;  59 con; 0-4 aty)
%            Number of variables   : 4290 (  93   ^;3976   !; 221   ?;4290   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 09:08:42.053
%------------------------------------------------------------------------------
% Could-be-implicit typings (109)
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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,
    set_Pr1932065953672099015od_o_o: $tType ).

thf(ty_n_t__Option__Ooption_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J,type,
    option5747939881698552298on_val: $tType ).

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

thf(ty_n_t__Set__Oset_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J,type,
    set_Pr1004501353197068112on_val: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Mask__Oval_J_J,type,
    set_Pr7730668076155855478ar_val: $tType ).

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,
    produc2934264451710624999od_o_o: $tType ).

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

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_I_Eo_M_Eo_J_Mt__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
    set_Pr7011910401827753159od_o_o: $tType ).

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

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_I_Eo_M_Eo_J_Mt__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    produc4465838814804890855od_o_o: $tType ).

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,
    set_se5158245632345848097at_nat: $tType ).

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

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    set_Pr2507339680178222706t_char: $tType ).

thf(ty_n_t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    option8963830502488799655at_nat: $tType ).

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

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J_J,type,
    set_se5342866143169225671od_o_o: $tType ).

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

thf(ty_n_t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
    option956769437226787789od_o_o: $tType ).

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

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
    set_se491199571785101031od_o_o: $tType ).

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

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

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    set_Product_prod_o_o: $tType ).

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

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

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

thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    product_prod_o_o: $tType ).

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

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

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

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

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

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

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

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

% Explicit typings (610)
thf(sy_c_BNF__Cardinal__Arithmetic_Octwo,type,
    bNF_Cardinal_ctwo: set_Product_prod_o_o ).

thf(sy_c_BNF__Cardinal__Arithmetic_Oczero_001_Eo,type,
    bNF_Cardinal_czero_o: set_Product_prod_o_o ).

thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__of_001_Eo,type,
    bNF_Ca3172636760949973492d_of_o: set_o > set_Product_prod_o_o ).

thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__of_001t__Nat__Onat,type,
    bNF_Ca3793111618940312692of_nat: set_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on_001_Eo,type,
    bNF_Ca8331644756375544342r_on_o: set_o > set_Product_prod_o_o > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal_001_Eo,type,
    bNF_Ca4363140458462388713inal_o: set_o > set_Product_prod_o_o > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal_001t__Nat__Onat,type,
    bNF_Ca5159621140400407487al_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J,type,
    bNF_Ca4299268644204761867on_val: set_Pr7649655576376376530on_val > set_Pr4305388979797279737on_val > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    bNF_Ca3407573850950408152od_o_o: set_se491199571785101031od_o_o > set_Pr1932065953672099015od_o_o > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bNF_Ca8448034394003515314at_nat: set_se7855581050983116737at_nat > set_Pr4329608150637261639at_nat > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Wellorder__Constructions_Obsqr_001_Eo,type,
    bNF_Wellorder_bsqr_o: set_Product_prod_o_o > set_Pr7011910401827753159od_o_o ).

thf(sy_c_BNF__Wellorder__Constructions_Obsqr_001t__Nat__Onat,type,
    bNF_We1732890022510290291qr_nat: set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat ).

thf(sy_c_BNF__Wellorder__Constructions_Obsqr_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J,type,
    bNF_We3507095839090954431on_val: set_Pr4305388979797279737on_val > set_Pr4590028993149660487on_val ).

thf(sy_c_BNF__Wellorder__Constructions_Obsqr_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    bNF_We3935152771354962724od_o_o: set_Pr1932065953672099015od_o_o > set_Pr1013884327308616903od_o_o ).

thf(sy_c_BNF__Wellorder__Constructions_Obsqr_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bNF_We6127972852007642366at_nat: set_Pr4329608150637261639at_nat > set_Pr5564308138774400199at_nat ).

thf(sy_c_BNF__Wellorder__Constructions_OordIso_001_Eo_001_Eo,type,
    bNF_We2654380646378065620so_o_o: set_Pr1932065953672099015od_o_o ).

thf(sy_c_BNF__Wellorder__Constructions_OordIso_001_Eo_001t__Nat__Onat,type,
    bNF_We4116350175183677332_o_nat: set_Pr457366540195662369at_nat ).

thf(sy_c_BNF__Wellorder__Constructions_OordIso_001t__Nat__Onat_001_Eo,type,
    bNF_We7219021536291862642_nat_o: set_Pr8055703017043224813od_o_o ).

thf(sy_c_BNF__Wellorder__Constructions_OordIso_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_We5258908940166488438at_nat: set_Pr4329608150637261639at_nat ).

thf(sy_c_BNF__Wellorder__Constructions_OordLeq_001_Eo_001_Eo,type,
    bNF_We8115293805659342695eq_o_o: set_Pr1932065953672099015od_o_o ).

thf(sy_c_BNF__Wellorder__Constructions_OordLeq_001_Eo_001t__Nat__Onat,type,
    bNF_We3435943820185352641_o_nat: set_Pr457366540195662369at_nat ).

thf(sy_c_BNF__Wellorder__Constructions_OordLeq_001t__Nat__Onat_001_Eo,type,
    bNF_We6538615181293537951_nat_o: set_Pr8055703017043224813od_o_o ).

thf(sy_c_BNF__Wellorder__Constructions_OordLeq_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_We2031159114419172489at_nat: set_Pr4329608150637261639at_nat ).

thf(sy_c_BNF__Wellorder__Constructions_OordLess_001_Eo_001_Eo,type,
    bNF_We2568191115274586186ss_o_o: set_Pr1932065953672099015od_o_o ).

thf(sy_c_BNF__Wellorder__Constructions_OordLess_001_Eo_001t__Nat__Onat,type,
    bNF_We8182288985678559134_o_nat: set_Pr457366540195662369at_nat ).

thf(sy_c_BNF__Wellorder__Constructions_OordLess_001t__Nat__Onat_001_Eo,type,
    bNF_We2061588309931968636_nat_o: set_Pr8055703017043224813od_o_o ).

thf(sy_c_BNF__Wellorder__Constructions_OordLess_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_We3448473429330749420at_nat: set_Pr4329608150637261639at_nat ).

thf(sy_c_BNF__Wellorder__Embedding_Oembed_001_Eo_001_Eo,type,
    bNF_We7100138119184918100ed_o_o: set_Product_prod_o_o > set_Product_prod_o_o > ( $o > $o ) > $o ).

thf(sy_c_BNF__Wellorder__Embedding_Oembed_001_Eo_001t__Nat__Onat,type,
    bNF_We4981740601763737620_o_nat: set_Product_prod_o_o > set_Pr1261947904930325089at_nat > ( $o > nat ) > $o ).

thf(sy_c_BNF__Wellorder__Embedding_Oembed_001t__Nat__Onat_001_Eo,type,
    bNF_We8084411962871922930_nat_o: set_Pr1261947904930325089at_nat > set_Product_prod_o_o > ( nat > $o ) > $o ).

thf(sy_c_BNF__Wellorder__Embedding_Oembed_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_We650101353357441270at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > ( nat > nat ) > $o ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001_Eo,type,
    bNF_We6437023980930612612_rel_o: set_Product_prod_o_o > $o ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Nat__Onat,type,
    bNF_We3818239936649020644el_nat: set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J,type,
    bNF_We4395177084845204272on_val: set_Pr4305388979797279737on_val > $o ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
    bNF_We8581212502267700435od_o_o: set_Pr1013884327308616903od_o_o > $o ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    bNF_We1688423684019388333at_nat: set_Pr4427867301352232263at_nat > $o ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    bNF_We4027175587370390611at_nat: set_Pr5564308138774400199at_nat > $o ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    bNF_We6243063140137587187od_o_o: set_Pr1932065953672099015od_o_o > $o ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bNF_We5602466822260561997at_nat: set_Pr4329608150637261639at_nat > $o ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J,type,
    bNF_We5046398979041125094on_val: set_Pr4305388979797279737on_val > set_Pr7649655576376376530on_val > produc9003781588162623004on_val > $o ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
    bNF_We6886001019485585053od_o_o: set_Pr1013884327308616903od_o_o > set_Pr1932065953672099015od_o_o > produc2934264451710624999od_o_o > $o ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    bNF_We6170040095471856759at_nat: set_Pr4427867301352232263at_nat > set_Pr457366540195662369at_nat > produc732395585841259969at_nat > $o ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    bNF_We2939539520719410205at_nat: set_Pr5564308138774400199at_nat > set_Pr4329608150637261639at_nat > produc3843707927480180839at_nat > $o ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    bNF_We794377122088811965od_o_o: set_Pr1932065953672099015od_o_o > set_se491199571785101031od_o_o > set_Product_prod_o_o > $o ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bNF_We943883530777384727at_nat: set_Pr4329608150637261639at_nat > set_se7855581050983116737at_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001_Eo,type,
    bNF_We3104086342435940326max2_o: set_Product_prod_o_o > $o > $o > $o ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001t__Nat__Onat,type,
    bNF_We2380939065524664066x2_nat: set_Pr1261947904930325089at_nat > nat > nat > nat ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J,type,
    bNF_We3765001366693588814on_val: set_Pr4305388979797279737on_val > produc9003781588162623004on_val > produc9003781588162623004on_val > produc9003781588162623004on_val ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
    bNF_We5307229915971165237od_o_o: set_Pr1013884327308616903od_o_o > produc2934264451710624999od_o_o > produc2934264451710624999od_o_o > produc2934264451710624999od_o_o ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    bNF_We7253295970193759759at_nat: set_Pr4427867301352232263at_nat > produc732395585841259969at_nat > produc732395585841259969at_nat > produc732395585841259969at_nat ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_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,
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thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
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thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_001t__Option__Ooption_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_J,type,
    fun_up273046073882939271on_val: ( produc9003781588162623004on_val > option8887860577033306220on_val ) > produc9003781588162623004on_val > option8887860577033306220on_val > produc9003781588162623004on_val > option8887860577033306220on_val ).

thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Option__Ooption_It__Mask__Oval_J,type,
    fun_up5735975307950539954on_val: ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > option_val > produc1457211279475724562t_char > option_val ).

thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J_001t__Option__Ooption_It__Mask__Oval_J,type,
    fun_up6453461605741198237on_val: ( produc2934264451710624999od_o_o > option_val ) > produc2934264451710624999od_o_o > option_val > produc2934264451710624999od_o_o > option_val ).

thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    fun_up4180510163728908852_state: ( produc2934264451710624999od_o_o > option_state ) > produc2934264451710624999od_o_o > option_state > produc2934264451710624999od_o_o > option_state ).

thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Option__Ooption_It__Mask__Oval_J,type,
    fun_up320048328433459523on_val: ( produc732395585841259969at_nat > option_val ) > produc732395585841259969at_nat > option_val > produc732395585841259969at_nat > option_val ).

thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    fun_up6026250480174712078_state: ( produc732395585841259969at_nat > option_state ) > produc732395585841259969at_nat > option_state > produc732395585841259969at_nat > option_state ).

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__Option__Ooption_It__Mask__Oval_J,type,
    fun_up7479097124709437981on_val: ( produc3843707927480180839at_nat > option_val ) > produc3843707927480180839at_nat > option_val > produc3843707927480180839at_nat > option_val ).

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__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    fun_up3329660993234963380_state: ( produc3843707927480180839at_nat > option_state ) > produc3843707927480180839at_nat > option_state > produc3843707927480180839at_nat > option_state ).

thf(sy_c_Fun_Ofun__upd_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
    fun_up3230021793924660483od_o_o: ( set_Product_prod_o_o > option956769437226787789od_o_o ) > set_Product_prod_o_o > option956769437226787789od_o_o > set_Product_prod_o_o > option956769437226787789od_o_o ).

thf(sy_c_Fun_Ofun__upd_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    fun_up3688893386320429021at_nat: ( set_Product_prod_o_o > option8963830502488799655at_nat ) > set_Product_prod_o_o > option8963830502488799655at_nat > set_Product_prod_o_o > option8963830502488799655at_nat ).

thf(sy_c_Fun_Ofun__upd_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    fun_up1071884767227745027at_nat: ( set_Pr1261947904930325089at_nat > option8963830502488799655at_nat ) > set_Pr1261947904930325089at_nat > option8963830502488799655at_nat > set_Pr1261947904930325089at_nat > option8963830502488799655at_nat ).

thf(sy_c_Fun_Oinj__on_001_Eo_001_Eo,type,
    inj_on_o_o: ( $o > $o ) > set_o > $o ).

thf(sy_c_Fun_Oinj__on_001_Eo_001t__Nat__Onat,type,
    inj_on_o_nat: ( $o > nat ) > set_o > $o ).

thf(sy_c_Fun_Oinj__on_001t__Mask__Oval_001t__Option__Ooption_It__Mask__Oval_J,type,
    inj_on4074814566786472055on_val: ( val > option_val ) > set_val > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001_Eo,type,
    inj_on_nat_o: ( nat > $o ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
    inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__PartialHeapSA__Ostate_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    inj_on3577428053172332983_state: ( state > option_state ) > set_state > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Option__Ooption_It__Mask__Oval_J,type,
    inj_on4474484569326248358on_val: ( produc1457211279475724562t_char > option_val ) > set_Pr2507339680178222706t_char > $o ).

thf(sy_c_Fun__Def_Oreduction__pair_001_Eo,type,
    fun_reduction_pair_o: produc2934264451710624999od_o_o > $o ).

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

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    minus_7052513330764642158od_o_o: set_Product_prod_o_o > set_Product_prod_o_o > set_Product_prod_o_o ).

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

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_J_J,type,
    minus_7211471524202483680on_val: set_Pr4305388979797279737on_val > set_Pr4305388979797279737on_val > set_Pr4305388979797279737on_val ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_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_If_001t__Option__Ooption_It__Mask__Oval_J,type,
    if_option_val: $o > option_val > option_val > option_val ).

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

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__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,
    inf_in3075509010647812879at_nat: set_Pr457366540195662369at_nat > set_Pr457366540195662369at_nat > set_Pr457366540195662369at_nat ).

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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_Eo_J,type,
    sup_sup_set_o: set_o > set_o > set_o ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Mask__Oval_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
    sup_sup_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Mask__Oval_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Mask__Oval_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__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(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_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_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_I_Eo_M_Eo_J_Mt__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__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(sy_c_Orderings_Oord__class_Oless__eq_001t__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(sy_c_Orderings_Oord__class_Oless__eq_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_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_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_PartialHeapSA_Oadd__states,type,
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thf(sy_c_PartialHeapSA_Oadd__states__rel,type,
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thf(sy_c_PartialHeapSA_Ocompatible,type,
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thf(sy_c_PartialHeapSA_Ocompatible__heaps,type,
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thf(sy_c_PartialHeapSA_Ocompatible__options_001t__Mask__Oval,type,
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thf(sy_c_PartialHeapSA_Ocompatible__options_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_PartialHeapSA_Ocompatible__options__rel_001t__Mask__Oval,type,
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thf(sy_c_PartialHeapSA_Ocompatible__options__rel_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_PartialHeapSA_Ocore,type,
    core: state > state ).

thf(sy_c_PartialHeapSA_Oempty__heap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Mask__Oval,type,
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thf(sy_c_PartialHeapSA_Oget__h,type,
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thf(sy_c_PartialHeapSA_Oget__m,type,
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thf(sy_c_PartialHeapSA_Olarger__heap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Mask__Oval,type,
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thf(sy_c_PartialHeapSA_Oplus,type,
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thf(sy_c_PartialHeapSA_Ostable,type,
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thf(sy_c_PartialHeapSA_Ostate_OAbs__state,type,
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thf(sy_c_PartialHeapSA_Ostate_ORep__state,type,
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thf(sy_c_PartialHeapSA_Ounit,type,
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thf(sy_c_PartialHeapSA_Ovalid__heap,type,
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thf(sy_c_PartialHeapSA_Ovalid__state,type,
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thf(sy_c_PartialHeapSA_Ovalid__state__rel,type,
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thf(sy_c_PosRat_Opadd,type,
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thf(sy_c_PosRat_Opinv,type,
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thf(sy_c_PosRat_Oppos,type,
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thf(sy_v__092_060phi_062,type,
    phi: state ).

% Relevant facts (1273)
thf(fact_0_assms,axiom,
    ! [Hl: produc1457211279475724562t_char] :
      ( ( ppos @ ( get_m @ phi @ Hl ) )
      = ( ( get_h @ phi @ Hl )
       != none_val ) ) ).

% assms
thf(fact_1_PartialHeapSA_Ostable__def,axiom,
    ( stable
    = ( ^ [Phi: state] :
        ! [Hl2: produc1457211279475724562t_char] :
          ( ( ppos @ ( get_m @ Phi @ Hl2 ) )
          = ( ( get_h @ Phi @ Hl2 )
           != none_val ) ) ) ) ).

% PartialHeapSA.stable_def
thf(fact_2_larger__heap__anti,axiom,
    ! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val] :
      ( ( larger1532547405891354081ar_val @ A @ B )
     => ( ( larger1532547405891354081ar_val @ B @ A )
       => ( A = B ) ) ) ).

% larger_heap_anti
thf(fact_3_larger__heap__refl,axiom,
    ! [H: produc1457211279475724562t_char > option_val] : ( larger1532547405891354081ar_val @ H @ H ) ).

% larger_heap_refl
thf(fact_4_larger__heap__trans,axiom,
    ! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val,C: produc1457211279475724562t_char > option_val] :
      ( ( larger1532547405891354081ar_val @ A @ B )
     => ( ( larger1532547405891354081ar_val @ B @ C )
       => ( larger1532547405891354081ar_val @ A @ C ) ) ) ).

% larger_heap_trans
thf(fact_5_state__ext,axiom,
    ! [A: state,B: state] :
      ( ( ( get_h @ A )
        = ( get_h @ B ) )
     => ( ( ( get_m @ A )
          = ( get_m @ B ) )
       => ( A = B ) ) ) ).

% state_ext
thf(fact_6_valid__heapI,axiom,
    ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
      ( ! [Hl3: produc1457211279475724562t_char] :
          ( ( ppos @ ( Pi @ Hl3 ) )
         => ( ( H @ Hl3 )
           != none_val ) )
     => ( valid_heap @ Pi @ H ) ) ).

% valid_heapI
thf(fact_7_valid__heap__def,axiom,
    ( valid_heap
    = ( ^ [Pi2: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
        ! [Hl2: produc1457211279475724562t_char] :
          ( ( ppos @ ( Pi2 @ Hl2 ) )
         => ( ( H2 @ Hl2 )
           != none_val ) ) ) ) ).

% valid_heap_def
thf(fact_8_empty__heap__def,axiom,
    ( empty_7512026568161686585ar_val
    = ( ^ [Hl2: produc1457211279475724562t_char] : none_val ) ) ).

% empty_heap_def
thf(fact_9_core__defined_I1_J,axiom,
    ! [Phi2: state] :
      ( ( get_h @ ( core @ Phi2 ) )
      = ( get_h @ Phi2 ) ) ).

% core_defined(1)
thf(fact_10_compatible__options_Osimps_I3_J,axiom,
    ! [Uu: option_state] : ( compat2278460363914054422_state @ Uu @ none_state ) ).

% compatible_options.simps(3)
thf(fact_11_compatible__options_Osimps_I3_J,axiom,
    ! [Uu: option_val] : ( compat7209469420100717861ns_val @ Uu @ none_val ) ).

% compatible_options.simps(3)
thf(fact_12_compatible__options_Osimps_I2_J,axiom,
    ! [Uv: option_state] : ( compat2278460363914054422_state @ none_state @ Uv ) ).

% compatible_options.simps(2)
thf(fact_13_compatible__options_Osimps_I2_J,axiom,
    ! [Uv: option_val] : ( compat7209469420100717861ns_val @ none_val @ Uv ) ).

% compatible_options.simps(2)
thf(fact_14_unit__charact_I1_J,axiom,
    ( ( get_h @ unit )
    = empty_7512026568161686585ar_val ) ).

% unit_charact(1)
thf(fact_15_larger__heap__plus,axiom,
    ! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val,C: produc1457211279475724562t_char > option_val] :
      ( ( larger1532547405891354081ar_val @ A @ B )
     => ( ( larger1532547405891354081ar_val @ A @ C )
       => ( larger1532547405891354081ar_val @ A @ ( map_ad2908576761495157028ar_val @ B @ C ) ) ) ) ).

% larger_heap_plus
thf(fact_16_get__h__m,axiom,
    ( rep_state
    = ( ^ [X: state] : ( produc74620467203698198on_val @ ( get_m @ X ) @ ( get_h @ X ) ) ) ) ).

% get_h_m
thf(fact_17_unit__charact_I2_J,axiom,
    ( ( get_m @ unit )
    = empty_3446695950879338768t_char ) ).

% unit_charact(2)
thf(fact_18_larger__heapI,axiom,
    ! [H: produc1457211279475724562t_char > option_val,H3: produc1457211279475724562t_char > option_val] :
      ( ! [Hl3: produc1457211279475724562t_char,X2: val] :
          ( ( ( H @ Hl3 )
            = ( some_val @ X2 ) )
         => ( ( H3 @ Hl3 )
            = ( some_val @ X2 ) ) )
     => ( larger1532547405891354081ar_val @ H3 @ H ) ) ).

% larger_heapI
thf(fact_19_option_Oinject,axiom,
    ! [X22: state,Y2: state] :
      ( ( ( some_state @ X22 )
        = ( some_state @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% option.inject
thf(fact_20_option_Oinject,axiom,
    ! [X22: val,Y2: val] :
      ( ( ( some_val @ X22 )
        = ( some_val @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% option.inject
thf(fact_21_not__Some__eq,axiom,
    ! [X3: option_state] :
      ( ( ! [Y: state] :
            ( X3
           != ( some_state @ Y ) ) )
      = ( X3 = none_state ) ) ).

% not_Some_eq
thf(fact_22_not__Some__eq,axiom,
    ! [X3: option_val] :
      ( ( ! [Y: val] :
            ( X3
           != ( some_val @ Y ) ) )
      = ( X3 = none_val ) ) ).

% not_Some_eq
thf(fact_23_not__None__eq,axiom,
    ! [X3: option_state] :
      ( ( X3 != none_state )
      = ( ? [Y: state] :
            ( X3
            = ( some_state @ Y ) ) ) ) ).

% not_None_eq
thf(fact_24_not__None__eq,axiom,
    ! [X3: option_val] :
      ( ( X3 != none_val )
      = ( ? [Y: val] :
            ( X3
            = ( some_val @ Y ) ) ) ) ).

% not_None_eq
thf(fact_25_empty__heap__neutral,axiom,
    ! [A: produc1457211279475724562t_char > option_val] :
      ( ( map_ad2908576761495157028ar_val @ A @ empty_7512026568161686585ar_val )
      = A ) ).

% empty_heap_neutral
thf(fact_26_Rep__state__inject,axiom,
    ! [X3: state,Y3: state] :
      ( ( ( rep_state @ X3 )
        = ( rep_state @ Y3 ) )
      = ( X3 = Y3 ) ) ).

% Rep_state_inject
thf(fact_27_valid__state_Ocases,axiom,
    ! [X3: produc9003781588162623004on_val] :
      ~ ! [Pi3: produc1457211279475724562t_char > prat,H4: produc1457211279475724562t_char > option_val] :
          ( X3
         != ( produc74620467203698198on_val @ Pi3 @ H4 ) ) ).

% valid_state.cases
thf(fact_28_compatible__options_Osimps_I1_J,axiom,
    ! [A: state,B: state] :
      ( ( compat2278460363914054422_state @ ( some_state @ A ) @ ( some_state @ B ) )
      = ( A = B ) ) ).

% compatible_options.simps(1)
thf(fact_29_compatible__options_Osimps_I1_J,axiom,
    ! [A: val,B: val] :
      ( ( compat7209469420100717861ns_val @ ( some_val @ A ) @ ( some_val @ B ) )
      = ( A = B ) ) ).

% compatible_options.simps(1)
thf(fact_30_compatible__options_Oelims_I3_J,axiom,
    ! [X3: option_state,Xa: option_state] :
      ( ~ ( compat2278460363914054422_state @ X3 @ Xa )
     => ~ ! [A2: state] :
            ( ( X3
              = ( some_state @ A2 ) )
           => ! [B2: state] :
                ( ( Xa
                  = ( some_state @ B2 ) )
               => ( A2 = B2 ) ) ) ) ).

% compatible_options.elims(3)
thf(fact_31_compatible__options_Oelims_I3_J,axiom,
    ! [X3: option_val,Xa: option_val] :
      ( ~ ( compat7209469420100717861ns_val @ X3 @ Xa )
     => ~ ! [A2: val] :
            ( ( X3
              = ( some_val @ A2 ) )
           => ! [B2: val] :
                ( ( Xa
                  = ( some_val @ B2 ) )
               => ( A2 = B2 ) ) ) ) ).

% compatible_options.elims(3)
thf(fact_32_combine__options__cases,axiom,
    ! [X3: option_state,P: option_state > option_state > $o,Y3: option_state] :
      ( ( ( X3 = none_state )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_state )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A2: state,B2: state] :
              ( ( X3
                = ( some_state @ A2 ) )
             => ( ( Y3
                  = ( some_state @ B2 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_33_combine__options__cases,axiom,
    ! [X3: option_state,P: option_state > option_val > $o,Y3: option_val] :
      ( ( ( X3 = none_state )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_val )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A2: state,B2: val] :
              ( ( X3
                = ( some_state @ A2 ) )
             => ( ( Y3
                  = ( some_val @ B2 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_34_combine__options__cases,axiom,
    ! [X3: option_val,P: option_val > option_state > $o,Y3: option_state] :
      ( ( ( X3 = none_val )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_state )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A2: val,B2: state] :
              ( ( X3
                = ( some_val @ A2 ) )
             => ( ( Y3
                  = ( some_state @ B2 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_35_combine__options__cases,axiom,
    ! [X3: option_val,P: option_val > option_val > $o,Y3: option_val] :
      ( ( ( X3 = none_val )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_val )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A2: val,B2: val] :
              ( ( X3
                = ( some_val @ A2 ) )
             => ( ( Y3
                  = ( some_val @ B2 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_36_split__option__all,axiom,
    ( ( ^ [P2: option_state > $o] :
        ! [X4: option_state] : ( P2 @ X4 ) )
    = ( ^ [P3: option_state > $o] :
          ( ( P3 @ none_state )
          & ! [X: state] : ( P3 @ ( some_state @ X ) ) ) ) ) ).

% split_option_all
thf(fact_37_split__option__all,axiom,
    ( ( ^ [P2: option_val > $o] :
        ! [X4: option_val] : ( P2 @ X4 ) )
    = ( ^ [P3: option_val > $o] :
          ( ( P3 @ none_val )
          & ! [X: val] : ( P3 @ ( some_val @ X ) ) ) ) ) ).

% split_option_all
thf(fact_38_split__option__ex,axiom,
    ( ( ^ [P2: option_state > $o] :
        ? [X4: option_state] : ( P2 @ X4 ) )
    = ( ^ [P3: option_state > $o] :
          ( ( P3 @ none_state )
          | ? [X: state] : ( P3 @ ( some_state @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_39_split__option__ex,axiom,
    ( ( ^ [P2: option_val > $o] :
        ? [X4: option_val] : ( P2 @ X4 ) )
    = ( ^ [P3: option_val > $o] :
          ( ( P3 @ none_val )
          | ? [X: val] : ( P3 @ ( some_val @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_40_option_Oexhaust,axiom,
    ! [Y3: option_state] :
      ( ( Y3 != none_state )
     => ~ ! [X23: state] :
            ( Y3
           != ( some_state @ X23 ) ) ) ).

% option.exhaust
thf(fact_41_option_Oexhaust,axiom,
    ! [Y3: option_val] :
      ( ( Y3 != none_val )
     => ~ ! [X23: val] :
            ( Y3
           != ( some_val @ X23 ) ) ) ).

% option.exhaust
thf(fact_42_option_OdiscI,axiom,
    ! [Option: option_state,X22: state] :
      ( ( Option
        = ( some_state @ X22 ) )
     => ( Option != none_state ) ) ).

% option.discI
thf(fact_43_option_OdiscI,axiom,
    ! [Option: option_val,X22: val] :
      ( ( Option
        = ( some_val @ X22 ) )
     => ( Option != none_val ) ) ).

% option.discI
thf(fact_44_option_Odistinct_I1_J,axiom,
    ! [X22: state] :
      ( none_state
     != ( some_state @ X22 ) ) ).

% option.distinct(1)
thf(fact_45_option_Odistinct_I1_J,axiom,
    ! [X22: val] :
      ( none_val
     != ( some_val @ X22 ) ) ).

% option.distinct(1)
thf(fact_46_compatible__options_Oelims_I1_J,axiom,
    ! [X3: option_state,Xa: option_state,Y3: $o] :
      ( ( ( compat2278460363914054422_state @ X3 @ Xa )
        = Y3 )
     => ( ! [A2: state] :
            ( ( X3
              = ( some_state @ A2 ) )
           => ! [B2: state] :
                ( ( Xa
                  = ( some_state @ B2 ) )
               => ( Y3
                  = ( A2 != B2 ) ) ) )
       => ( ( ( X3 = none_state )
           => ~ Y3 )
         => ~ ( ( Xa = none_state )
             => ~ Y3 ) ) ) ) ).

% compatible_options.elims(1)
thf(fact_47_compatible__options_Oelims_I1_J,axiom,
    ! [X3: option_val,Xa: option_val,Y3: $o] :
      ( ( ( compat7209469420100717861ns_val @ X3 @ Xa )
        = Y3 )
     => ( ! [A2: val] :
            ( ( X3
              = ( some_val @ A2 ) )
           => ! [B2: val] :
                ( ( Xa
                  = ( some_val @ B2 ) )
               => ( Y3
                  = ( A2 != B2 ) ) ) )
       => ( ( ( X3 = none_val )
           => ~ Y3 )
         => ~ ( ( Xa = none_val )
             => ~ Y3 ) ) ) ) ).

% compatible_options.elims(1)
thf(fact_48_compatible__options_Oelims_I2_J,axiom,
    ! [X3: option_state,Xa: option_state] :
      ( ( compat2278460363914054422_state @ X3 @ Xa )
     => ( ! [A2: state] :
            ( ( X3
              = ( some_state @ A2 ) )
           => ! [B2: state] :
                ( ( Xa
                  = ( some_state @ B2 ) )
               => ( A2 != B2 ) ) )
       => ( ( X3 != none_state )
         => ( Xa = none_state ) ) ) ) ).

% compatible_options.elims(2)
thf(fact_49_compatible__options_Oelims_I2_J,axiom,
    ! [X3: option_val,Xa: option_val] :
      ( ( compat7209469420100717861ns_val @ X3 @ Xa )
     => ( ! [A2: val] :
            ( ( X3
              = ( some_val @ A2 ) )
           => ! [B2: val] :
                ( ( Xa
                  = ( some_val @ B2 ) )
               => ( A2 != B2 ) ) )
       => ( ( X3 != none_val )
         => ( Xa = none_val ) ) ) ) ).

% compatible_options.elims(2)
thf(fact_50_core__defined_I2_J,axiom,
    ! [Phi2: state] :
      ( ( get_m @ ( core @ Phi2 ) )
      = empty_3446695950879338768t_char ) ).

% core_defined(2)
thf(fact_51_larger__heap__def,axiom,
    ( larger1532547405891354081ar_val
    = ( ^ [H5: produc1457211279475724562t_char > option_val,H2: produc1457211279475724562t_char > option_val] :
        ! [Hl2: produc1457211279475724562t_char,X: val] :
          ( ( ( H2 @ Hl2 )
            = ( some_val @ X ) )
         => ( ( H5 @ Hl2 )
            = ( some_val @ X ) ) ) ) ) ).

% larger_heap_def
thf(fact_52_map__add__eq__empty__iff,axiom,
    ! [F: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val] :
      ( ( ( map_ad2908576761495157028ar_val @ F @ G )
        = ( ^ [X: produc1457211279475724562t_char] : none_val ) )
      = ( ( F
          = ( ^ [X: produc1457211279475724562t_char] : none_val ) )
        & ( G
          = ( ^ [X: produc1457211279475724562t_char] : none_val ) ) ) ) ).

% map_add_eq_empty_iff
thf(fact_53_map__add__None,axiom,
    ! [M: produc1457211279475724562t_char > option_val,N: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char] :
      ( ( ( map_ad2908576761495157028ar_val @ M @ N @ K )
        = none_val )
      = ( ( ( N @ K )
          = none_val )
        & ( ( M @ K )
          = none_val ) ) ) ).

% map_add_None
thf(fact_54_map__add__find__right,axiom,
    ! [N: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char,Xx: val,M: produc1457211279475724562t_char > option_val] :
      ( ( ( N @ K )
        = ( some_val @ Xx ) )
     => ( ( map_ad2908576761495157028ar_val @ M @ N @ K )
        = ( some_val @ Xx ) ) ) ).

% map_add_find_right
thf(fact_55_map__add__SomeD,axiom,
    ! [M: produc1457211279475724562t_char > option_val,N: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char,X3: val] :
      ( ( ( map_ad2908576761495157028ar_val @ M @ N @ K )
        = ( some_val @ X3 ) )
     => ( ( ( N @ K )
          = ( some_val @ X3 ) )
        | ( ( ( N @ K )
            = none_val )
          & ( ( M @ K )
            = ( some_val @ X3 ) ) ) ) ) ).

% map_add_SomeD
thf(fact_56_map__add__Some__iff,axiom,
    ! [M: produc1457211279475724562t_char > option_val,N: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char,X3: val] :
      ( ( ( map_ad2908576761495157028ar_val @ M @ N @ K )
        = ( some_val @ X3 ) )
      = ( ( ( N @ K )
          = ( some_val @ X3 ) )
        | ( ( ( N @ K )
            = none_val )
          & ( ( M @ K )
            = ( some_val @ X3 ) ) ) ) ) ).

% map_add_Some_iff
thf(fact_57_mem__Collect__eq,axiom,
    ! [A: produc2934264451710624999od_o_o,P: produc2934264451710624999od_o_o > $o] :
      ( ( member444158400953824016od_o_o @ A @ ( collec3302784863738620242od_o_o @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_58_mem__Collect__eq,axiom,
    ! [A: produc732395585841259969at_nat,P: produc732395585841259969at_nat > $o] :
      ( ( member4095101504841534314at_nat @ A @ ( collec1729919521702017196at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_59_mem__Collect__eq,axiom,
    ! [A: produc3843707927480180839at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( member8757157785044589968at_nat @ A @ ( collec6321179662152712658at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_60_Collect__mem__eq,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o] :
      ( ( collec3302784863738620242od_o_o
        @ ^ [X: produc2934264451710624999od_o_o] : ( member444158400953824016od_o_o @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_61_Collect__mem__eq,axiom,
    ! [A3: set_Pr457366540195662369at_nat] :
      ( ( collec1729919521702017196at_nat
        @ ^ [X: produc732395585841259969at_nat] : ( member4095101504841534314at_nat @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_62_Collect__mem__eq,axiom,
    ! [A3: set_Pr4329608150637261639at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_63_unit__def,axiom,
    ( unit
    = ( abs_state @ ( produc74620467203698198on_val @ empty_3446695950879338768t_char @ empty_7512026568161686585ar_val ) ) ) ).

% unit_def
thf(fact_64_map__add__assoc,axiom,
    ! [M1: produc1457211279475724562t_char > option_val,M2: produc1457211279475724562t_char > option_val,M3: produc1457211279475724562t_char > option_val] :
      ( ( map_ad2908576761495157028ar_val @ M1 @ ( map_ad2908576761495157028ar_val @ M2 @ M3 ) )
      = ( map_ad2908576761495157028ar_val @ ( map_ad2908576761495157028ar_val @ M1 @ M2 ) @ M3 ) ) ).

% map_add_assoc
thf(fact_65_core__def,axiom,
    ( core
    = ( ^ [Phi: state] : ( abs_state @ ( produc74620467203698198on_val @ empty_3446695950879338768t_char @ ( get_h @ Phi ) ) ) ) ) ).

% core_def
thf(fact_66_old_Oprod_Oinject,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,A4: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val] :
      ( ( ( produc74620467203698198on_val @ A @ B )
        = ( produc74620467203698198on_val @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_67_old_Oprod_Oinject,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,A4: produc9003781588162623004on_val,B3: produc9003781588162623004on_val] :
      ( ( ( produc4958730763592889809on_val @ A @ B )
        = ( produc4958730763592889809on_val @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_68_old_Oprod_Oinject,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,A4: set_Product_prod_o_o,B3: set_Product_prod_o_o] :
      ( ( ( produc763777882069021527od_o_o @ A @ B )
        = ( produc763777882069021527od_o_o @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_69_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_70_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_71_prod_Oinject,axiom,
    ! [X1: produc1457211279475724562t_char > prat,X22: produc1457211279475724562t_char > option_val,Y1: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char > option_val] :
      ( ( ( produc74620467203698198on_val @ X1 @ X22 )
        = ( produc74620467203698198on_val @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_72_prod_Oinject,axiom,
    ! [X1: produc9003781588162623004on_val,X22: produc9003781588162623004on_val,Y1: produc9003781588162623004on_val,Y2: produc9003781588162623004on_val] :
      ( ( ( produc4958730763592889809on_val @ X1 @ X22 )
        = ( produc4958730763592889809on_val @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_73_prod_Oinject,axiom,
    ! [X1: set_Product_prod_o_o,X22: set_Product_prod_o_o,Y1: set_Product_prod_o_o,Y2: set_Product_prod_o_o] :
      ( ( ( produc763777882069021527od_o_o @ X1 @ X22 )
        = ( produc763777882069021527od_o_o @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_74_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_75_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_76_valid__pre__unit,axiom,
    valid_state @ ( produc74620467203698198on_val @ empty_3446695950879338768t_char @ empty_7512026568161686585ar_val ) ).

% valid_pre_unit
thf(fact_77_add__states_Ocases,axiom,
    ! [X3: produc1317171644805857433on_val] :
      ~ ! [Pi3: produc1457211279475724562t_char > prat,H4: produc1457211279475724562t_char > option_val,Pi4: produc1457211279475724562t_char > prat,H6: produc1457211279475724562t_char > option_val] :
          ( X3
         != ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Pi3 @ H4 ) @ ( produc74620467203698198on_val @ Pi4 @ H6 ) ) ) ).

% add_states.cases
thf(fact_78_compatible__options_Ocases,axiom,
    ! [X3: produc3142500478612311029_state] :
      ( ! [A2: state,B2: state] :
          ( X3
         != ( produc9160152616361873709_state @ ( some_state @ A2 ) @ ( some_state @ B2 ) ) )
     => ( ! [Uv2: option_state] :
            ( X3
           != ( produc9160152616361873709_state @ none_state @ Uv2 ) )
       => ~ ! [Uu2: option_state] :
              ( X3
             != ( produc9160152616361873709_state @ Uu2 @ none_state ) ) ) ) ).

% compatible_options.cases
thf(fact_79_compatible__options_Ocases,axiom,
    ! [X3: produc5122021904355189735on_val] :
      ( ! [A2: val,B2: val] :
          ( X3
         != ( produc904852209897547607on_val @ ( some_val @ A2 ) @ ( some_val @ B2 ) ) )
     => ( ! [Uv2: option_val] :
            ( X3
           != ( produc904852209897547607on_val @ none_val @ Uv2 ) )
       => ~ ! [Uu2: option_val] :
              ( X3
             != ( produc904852209897547607on_val @ Uu2 @ none_val ) ) ) ) ).

% compatible_options.cases
thf(fact_80_Rep__state__inverse,axiom,
    ! [X3: state] :
      ( ( abs_state @ ( rep_state @ X3 ) )
      = X3 ) ).

% Rep_state_inverse
thf(fact_81_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc9003781588162623004on_val] :
      ~ ! [A2: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > option_val] :
          ( Y3
         != ( produc74620467203698198on_val @ A2 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_82_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc1317171644805857433on_val] :
      ~ ! [A2: produc9003781588162623004on_val,B2: produc9003781588162623004on_val] :
          ( Y3
         != ( produc4958730763592889809on_val @ A2 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_83_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc2934264451710624999od_o_o] :
      ~ ! [A2: set_Product_prod_o_o,B2: set_Product_prod_o_o] :
          ( Y3
         != ( produc763777882069021527od_o_o @ A2 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_84_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc732395585841259969at_nat] :
      ~ ! [A2: set_Product_prod_o_o,B2: set_Pr1261947904930325089at_nat] :
          ( Y3
         != ( produc8517790099723286449at_nat @ A2 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_85_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc3843707927480180839at_nat] :
      ~ ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
          ( Y3
         != ( produc2922128104949294807at_nat @ A2 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_86_surj__pair,axiom,
    ! [P4: produc9003781588162623004on_val] :
    ? [X2: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > option_val] :
      ( P4
      = ( produc74620467203698198on_val @ X2 @ Y4 ) ) ).

% surj_pair
thf(fact_87_surj__pair,axiom,
    ! [P4: produc1317171644805857433on_val] :
    ? [X2: produc9003781588162623004on_val,Y4: produc9003781588162623004on_val] :
      ( P4
      = ( produc4958730763592889809on_val @ X2 @ Y4 ) ) ).

% surj_pair
thf(fact_88_surj__pair,axiom,
    ! [P4: produc2934264451710624999od_o_o] :
    ? [X2: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
      ( P4
      = ( produc763777882069021527od_o_o @ X2 @ Y4 ) ) ).

% surj_pair
thf(fact_89_surj__pair,axiom,
    ! [P4: produc732395585841259969at_nat] :
    ? [X2: set_Product_prod_o_o,Y4: set_Pr1261947904930325089at_nat] :
      ( P4
      = ( produc8517790099723286449at_nat @ X2 @ Y4 ) ) ).

% surj_pair
thf(fact_90_surj__pair,axiom,
    ! [P4: produc3843707927480180839at_nat] :
    ? [X2: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
      ( P4
      = ( produc2922128104949294807at_nat @ X2 @ Y4 ) ) ).

% surj_pair
thf(fact_91_prod__cases,axiom,
    ! [P: produc9003781588162623004on_val > $o,P4: produc9003781588162623004on_val] :
      ( ! [A2: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > option_val] : ( P @ ( produc74620467203698198on_val @ A2 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_92_prod__cases,axiom,
    ! [P: produc1317171644805857433on_val > $o,P4: produc1317171644805857433on_val] :
      ( ! [A2: produc9003781588162623004on_val,B2: produc9003781588162623004on_val] : ( P @ ( produc4958730763592889809on_val @ A2 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_93_prod__cases,axiom,
    ! [P: produc2934264451710624999od_o_o > $o,P4: produc2934264451710624999od_o_o] :
      ( ! [A2: set_Product_prod_o_o,B2: set_Product_prod_o_o] : ( P @ ( produc763777882069021527od_o_o @ A2 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_94_prod__cases,axiom,
    ! [P: produc732395585841259969at_nat > $o,P4: produc732395585841259969at_nat] :
      ( ! [A2: set_Product_prod_o_o,B2: set_Pr1261947904930325089at_nat] : ( P @ ( produc8517790099723286449at_nat @ A2 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_95_prod__cases,axiom,
    ! [P: produc3843707927480180839at_nat > $o,P4: produc3843707927480180839at_nat] :
      ( ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( P @ ( produc2922128104949294807at_nat @ A2 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_96_Pair__inject,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,A4: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val] :
      ( ( ( produc74620467203698198on_val @ A @ B )
        = ( produc74620467203698198on_val @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_97_Pair__inject,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,A4: produc9003781588162623004on_val,B3: produc9003781588162623004on_val] :
      ( ( ( produc4958730763592889809on_val @ A @ B )
        = ( produc4958730763592889809on_val @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_98_Pair__inject,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,A4: set_Product_prod_o_o,B3: set_Product_prod_o_o] :
      ( ( ( produc763777882069021527od_o_o @ A @ B )
        = ( produc763777882069021527od_o_o @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_99_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_100_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_101_prod__cases3,axiom,
    ! [Y3: produc1317171644805857433on_val] :
      ~ ! [A2: produc9003781588162623004on_val,B2: produc1457211279475724562t_char > prat,C2: produc1457211279475724562t_char > option_val] :
          ( Y3
         != ( produc4958730763592889809on_val @ A2 @ ( produc74620467203698198on_val @ B2 @ C2 ) ) ) ).

% prod_cases3
thf(fact_102_prod__induct3,axiom,
    ! [P: produc1317171644805857433on_val > $o,X3: produc1317171644805857433on_val] :
      ( ! [A2: produc9003781588162623004on_val,B2: produc1457211279475724562t_char > prat,C2: produc1457211279475724562t_char > option_val] : ( P @ ( produc4958730763592889809on_val @ A2 @ ( produc74620467203698198on_val @ B2 @ C2 ) ) )
     => ( P @ X3 ) ) ).

% prod_induct3
thf(fact_103_valid__stateI,axiom,
    ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
      ( ( valid_mask @ Pi )
     => ( ! [Hl3: produc1457211279475724562t_char] :
            ( ( ppos @ ( Pi @ Hl3 ) )
           => ( ( H @ Hl3 )
             != none_val ) )
       => ( valid_state @ ( produc74620467203698198on_val @ Pi @ H ) ) ) ) ).

% valid_stateI
thf(fact_104_valid__state_Oelims_I3_J,axiom,
    ! [X3: produc9003781588162623004on_val] :
      ( ~ ( valid_state @ X3 )
     => ~ ! [Pi3: produc1457211279475724562t_char > prat,H4: produc1457211279475724562t_char > option_val] :
            ( ( X3
              = ( produc74620467203698198on_val @ Pi3 @ H4 ) )
           => ( ( valid_mask @ Pi3 )
              & ( valid_heap @ Pi3 @ H4 ) ) ) ) ).

% valid_state.elims(3)
thf(fact_105_valid__state_Oelims_I2_J,axiom,
    ! [X3: produc9003781588162623004on_val] :
      ( ( valid_state @ X3 )
     => ~ ! [Pi3: produc1457211279475724562t_char > prat,H4: produc1457211279475724562t_char > option_val] :
            ( ( X3
              = ( produc74620467203698198on_val @ Pi3 @ H4 ) )
           => ~ ( ( valid_mask @ Pi3 )
                & ( valid_heap @ Pi3 @ H4 ) ) ) ) ).

% valid_state.elims(2)
thf(fact_106_valid__state_Oelims_I1_J,axiom,
    ! [X3: produc9003781588162623004on_val,Y3: $o] :
      ( ( ( valid_state @ X3 )
        = Y3 )
     => ~ ! [Pi3: produc1457211279475724562t_char > prat,H4: produc1457211279475724562t_char > option_val] :
            ( ( X3
              = ( produc74620467203698198on_val @ Pi3 @ H4 ) )
           => ( Y3
              = ( ~ ( ( valid_mask @ Pi3 )
                    & ( valid_heap @ Pi3 @ H4 ) ) ) ) ) ) ).

% valid_state.elims(1)
thf(fact_107_valid__state_Osimps,axiom,
    ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
      ( ( valid_state @ ( produc74620467203698198on_val @ Pi @ H ) )
      = ( ( valid_mask @ Pi )
        & ( valid_heap @ Pi @ H ) ) ) ).

% valid_state.simps
thf(fact_108_ssubst__Pair__rhs,axiom,
    ! [R: produc1457211279475724562t_char > prat,S: produc1457211279475724562t_char > option_val,R2: set_Pr7649655576376376530on_val,S2: produc1457211279475724562t_char > option_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_109_ssubst__Pair__rhs,axiom,
    ! [R: produc9003781588162623004on_val,S: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val,S2: produc9003781588162623004on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_110_ssubst__Pair__rhs,axiom,
    ! [R: set_Product_prod_o_o,S: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o,S2: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_111_ssubst__Pair__rhs,axiom,
    ! [R: set_Product_prod_o_o,S: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_112_ssubst__Pair__rhs,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_113_valid__empty,axiom,
    valid_mask @ empty_3446695950879338768t_char ).

% valid_empty
thf(fact_114_valid__mask__full__mask,axiom,
    valid_mask @ full_mask ).

% valid_mask_full_mask
thf(fact_115_same__fstI,axiom,
    ! [P: ( produc1457211279475724562t_char > prat ) > $o,X3: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val,Y3: produc1457211279475724562t_char > option_val,R2: ( produc1457211279475724562t_char > prat ) > set_Pr4698884584343881589on_val] :
      ( ( P @ X3 )
     => ( ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ Y5 @ Y3 ) @ ( R2 @ X3 ) )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ X3 @ Y5 ) @ ( produc74620467203698198on_val @ X3 @ Y3 ) ) @ ( same_f8138205353876899841on_val @ P @ R2 ) ) ) ) ).

% same_fstI
thf(fact_116_same__fstI,axiom,
    ! [P: produc9003781588162623004on_val > $o,X3: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val,R2: produc9003781588162623004on_val > set_Pr4305388979797279737on_val] :
      ( ( P @ X3 )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y5 @ Y3 ) @ ( R2 @ X3 ) )
       => ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ X3 @ Y5 ) @ ( produc4958730763592889809on_val @ X3 @ Y3 ) ) @ ( same_f7521981028318775590on_val @ P @ R2 ) ) ) ) ).

% same_fstI
thf(fact_117_same__fstI,axiom,
    ! [P: set_Product_prod_o_o > $o,X3: set_Product_prod_o_o,Y5: set_Product_prod_o_o,Y3: set_Product_prod_o_o,R2: set_Product_prod_o_o > set_Pr1932065953672099015od_o_o] :
      ( ( P @ X3 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y5 @ Y3 ) @ ( R2 @ X3 ) )
       => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y5 ) @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) ) @ ( same_f884114232143252012od_o_o @ P @ R2 ) ) ) ) ).

% same_fstI
thf(fact_118_same__fstI,axiom,
    ! [P: set_Product_prod_o_o > $o,X3: set_Product_prod_o_o,Y5: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,R2: set_Product_prod_o_o > set_Pr4329608150637261639at_nat] :
      ( ( P @ X3 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ Y3 ) @ ( R2 @ X3 ) )
       => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ X3 @ Y5 ) @ ( produc8517790099723286449at_nat @ X3 @ Y3 ) ) @ ( same_f6937299433074352134at_nat @ P @ R2 ) ) ) ) ).

% same_fstI
thf(fact_119_same__fstI,axiom,
    ! [P: set_Pr1261947904930325089at_nat > $o,X3: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat > set_Pr4329608150637261639at_nat] :
      ( ( P @ X3 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ Y3 ) @ ( R2 @ X3 ) )
       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y5 ) @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) ) @ ( same_f1460074399991292076at_nat @ P @ R2 ) ) ) ) ).

% same_fstI
thf(fact_120_valid__state_Opelims_I1_J,axiom,
    ! [X3: produc9003781588162623004on_val,Y3: $o] :
      ( ( ( valid_state @ X3 )
        = Y3 )
     => ( ( accp_P4405556307444930259on_val @ valid_state_rel @ X3 )
       => ~ ! [Pi3: produc1457211279475724562t_char > prat,H4: produc1457211279475724562t_char > option_val] :
              ( ( X3
                = ( produc74620467203698198on_val @ Pi3 @ H4 ) )
             => ( ( Y3
                  = ( ( valid_mask @ Pi3 )
                    & ( valid_heap @ Pi3 @ H4 ) ) )
               => ~ ( accp_P4405556307444930259on_val @ valid_state_rel @ ( produc74620467203698198on_val @ Pi3 @ H4 ) ) ) ) ) ) ).

% valid_state.pelims(1)
thf(fact_121_valid__state_Opelims_I2_J,axiom,
    ! [X3: produc9003781588162623004on_val] :
      ( ( valid_state @ X3 )
     => ( ( accp_P4405556307444930259on_val @ valid_state_rel @ X3 )
       => ~ ! [Pi3: produc1457211279475724562t_char > prat,H4: produc1457211279475724562t_char > option_val] :
              ( ( X3
                = ( produc74620467203698198on_val @ Pi3 @ H4 ) )
             => ( ( accp_P4405556307444930259on_val @ valid_state_rel @ ( produc74620467203698198on_val @ Pi3 @ H4 ) )
               => ~ ( ( valid_mask @ Pi3 )
                    & ( valid_heap @ Pi3 @ H4 ) ) ) ) ) ) ).

% valid_state.pelims(2)
thf(fact_122_valid__state_Opelims_I3_J,axiom,
    ! [X3: produc9003781588162623004on_val] :
      ( ~ ( valid_state @ X3 )
     => ( ( accp_P4405556307444930259on_val @ valid_state_rel @ X3 )
       => ~ ! [Pi3: produc1457211279475724562t_char > prat,H4: produc1457211279475724562t_char > option_val] :
              ( ( X3
                = ( produc74620467203698198on_val @ Pi3 @ H4 ) )
             => ( ( accp_P4405556307444930259on_val @ valid_state_rel @ ( produc74620467203698198on_val @ Pi3 @ H4 ) )
               => ( ( valid_mask @ Pi3 )
                  & ( valid_heap @ Pi3 @ H4 ) ) ) ) ) ) ).

% valid_state.pelims(3)
thf(fact_123_option_Ocollapse,axiom,
    ! [Option: option_state] :
      ( ( Option != none_state )
     => ( ( some_state @ ( the_state @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_124_option_Ocollapse,axiom,
    ! [Option: option_val] :
      ( ( Option != none_val )
     => ( ( some_val @ ( the_val @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_125_compatible__options_Opelims_I2_J,axiom,
    ! [X3: option_state,Xa: option_state] :
      ( ( compat2278460363914054422_state @ X3 @ Xa )
     => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X3 @ Xa ) )
       => ( ! [A2: state] :
              ( ( X3
                = ( some_state @ A2 ) )
             => ! [B2: state] :
                  ( ( Xa
                    = ( some_state @ B2 ) )
                 => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ ( some_state @ A2 ) @ ( some_state @ B2 ) ) )
                   => ( A2 != B2 ) ) ) )
         => ( ( ( X3 = none_state )
             => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ none_state @ Xa ) ) )
           => ~ ( ( Xa = none_state )
               => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X3 @ none_state ) ) ) ) ) ) ) ).

% compatible_options.pelims(2)
thf(fact_126_compatible__options_Opelims_I2_J,axiom,
    ! [X3: option_val,Xa: option_val] :
      ( ( compat7209469420100717861ns_val @ X3 @ Xa )
     => ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X3 @ Xa ) )
       => ( ! [A2: val] :
              ( ( X3
                = ( some_val @ A2 ) )
             => ! [B2: val] :
                  ( ( Xa
                    = ( some_val @ B2 ) )
                 => ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ ( some_val @ A2 ) @ ( some_val @ B2 ) ) )
                   => ( A2 != B2 ) ) ) )
         => ( ( ( X3 = none_val )
             => ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ none_val @ Xa ) ) )
           => ~ ( ( Xa = none_val )
               => ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X3 @ none_val ) ) ) ) ) ) ) ).

% compatible_options.pelims(2)
thf(fact_127_compatible__options_Opelims_I1_J,axiom,
    ! [X3: option_state,Xa: option_state,Y3: $o] :
      ( ( ( compat2278460363914054422_state @ X3 @ Xa )
        = Y3 )
     => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X3 @ Xa ) )
       => ( ! [A2: state] :
              ( ( X3
                = ( some_state @ A2 ) )
             => ! [B2: state] :
                  ( ( Xa
                    = ( some_state @ B2 ) )
                 => ( ( Y3
                      = ( A2 = B2 ) )
                   => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ ( some_state @ A2 ) @ ( some_state @ B2 ) ) ) ) ) )
         => ( ( ( X3 = none_state )
             => ( Y3
               => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ none_state @ Xa ) ) ) )
           => ~ ( ( Xa = none_state )
               => ( Y3
                 => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X3 @ none_state ) ) ) ) ) ) ) ) ).

% compatible_options.pelims(1)
thf(fact_128_compatible__options_Opelims_I1_J,axiom,
    ! [X3: option_val,Xa: option_val,Y3: $o] :
      ( ( ( compat7209469420100717861ns_val @ X3 @ Xa )
        = Y3 )
     => ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X3 @ Xa ) )
       => ( ! [A2: val] :
              ( ( X3
                = ( some_val @ A2 ) )
             => ! [B2: val] :
                  ( ( Xa
                    = ( some_val @ B2 ) )
                 => ( ( Y3
                      = ( A2 = B2 ) )
                   => ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ ( some_val @ A2 ) @ ( some_val @ B2 ) ) ) ) ) )
         => ( ( ( X3 = none_val )
             => ( Y3
               => ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ none_val @ Xa ) ) ) )
           => ~ ( ( Xa = none_val )
               => ( Y3
                 => ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X3 @ none_val ) ) ) ) ) ) ) ) ).

% compatible_options.pelims(1)
thf(fact_129_valid__state__decompose,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
      ( ( valid_state @ ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ A @ B ) @ H ) )
     => ( valid_state @ ( produc74620467203698198on_val @ A @ H ) ) ) ).

% valid_state_decompose
thf(fact_130_curryI,axiom,
    ! [F: produc9003781588162623004on_val > $o,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val] :
      ( ( F @ ( produc74620467203698198on_val @ A @ B ) )
     => ( produc9184331568460336180_val_o @ F @ A @ B ) ) ).

% curryI
thf(fact_131_curryI,axiom,
    ! [F: produc1317171644805857433on_val > $o,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( F @ ( produc4958730763592889809on_val @ A @ B ) )
     => ( produc4800927984219439315_val_o @ F @ A @ B ) ) ).

% curryI
thf(fact_132_curryI,axiom,
    ! [F: produc2934264451710624999od_o_o > $o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( F @ ( produc763777882069021527od_o_o @ A @ B ) )
     => ( produc4861507100678042957_o_o_o @ F @ A @ B ) ) ).

% curryI
thf(fact_133_curryI,axiom,
    ! [F: produc732395585841259969at_nat > $o,A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat] :
      ( ( F @ ( produc8517790099723286449at_nat @ A @ B ) )
     => ( produc5440495499350717427_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_134_curryI,axiom,
    ! [F: produc3843707927480180839at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( F @ ( produc2922128104949294807at_nat @ A @ B ) )
     => ( produc7820828192191229133_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_135_add__masks__cancellative,axiom,
    ! [B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat,D: produc1457211279475724562t_char > prat] :
      ( ( ( add_ma6586698345353345811t_char @ B @ C )
        = ( add_ma6586698345353345811t_char @ B @ D ) )
     => ( C = D ) ) ).

% add_masks_cancellative
thf(fact_136_add__masks__comm,axiom,
    ( add_ma6586698345353345811t_char
    = ( ^ [A5: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > prat] : ( add_ma6586698345353345811t_char @ B4 @ A5 ) ) ) ).

% add_masks_comm
thf(fact_137_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_138_option_Osel,axiom,
    ! [X22: state] :
      ( ( the_state @ ( some_state @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_139_option_Osel,axiom,
    ! [X22: val] :
      ( ( the_val @ ( some_val @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_140_option_Oexpand,axiom,
    ! [Option: option_val,Option2: option_val] :
      ( ( ( Option = none_val )
        = ( Option2 = none_val ) )
     => ( ( ( Option != none_val )
         => ( ( Option2 != none_val )
           => ( ( the_val @ Option )
              = ( the_val @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_141_option_Oexpand,axiom,
    ! [Option: option_state,Option2: option_state] :
      ( ( ( Option = none_state )
        = ( Option2 = none_state ) )
     => ( ( ( Option != none_state )
         => ( ( Option2 != none_state )
           => ( ( the_state @ Option )
              = ( the_state @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_142_upper__valid__aux,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ A )
     => ( ( A
          = ( add_ma6586698345353345811t_char @ B @ C ) )
       => ( valid_mask @ B ) ) ) ).

% upper_valid_aux
thf(fact_143_upper__valid,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ A )
     => ( ( A
          = ( add_ma6586698345353345811t_char @ B @ C ) )
       => ( ( valid_mask @ B )
          & ( valid_mask @ C ) ) ) ) ).

% upper_valid
thf(fact_144_curryD,axiom,
    ! [F: produc9003781588162623004on_val > $o,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val] :
      ( ( produc9184331568460336180_val_o @ F @ A @ B )
     => ( F @ ( produc74620467203698198on_val @ A @ B ) ) ) ).

% curryD
thf(fact_145_curryD,axiom,
    ! [F: produc1317171644805857433on_val > $o,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( produc4800927984219439315_val_o @ F @ A @ B )
     => ( F @ ( produc4958730763592889809on_val @ A @ B ) ) ) ).

% curryD
thf(fact_146_curryD,axiom,
    ! [F: produc2934264451710624999od_o_o > $o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( produc4861507100678042957_o_o_o @ F @ A @ B )
     => ( F @ ( produc763777882069021527od_o_o @ A @ B ) ) ) ).

% curryD
thf(fact_147_curryD,axiom,
    ! [F: produc732395585841259969at_nat > $o,A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat] :
      ( ( produc5440495499350717427_nat_o @ F @ A @ B )
     => ( F @ ( produc8517790099723286449at_nat @ A @ B ) ) ) ).

% curryD
thf(fact_148_curryD,axiom,
    ! [F: produc3843707927480180839at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( produc7820828192191229133_nat_o @ F @ A @ B )
     => ( F @ ( produc2922128104949294807at_nat @ A @ B ) ) ) ).

% curryD
thf(fact_149_curryE,axiom,
    ! [F: produc9003781588162623004on_val > $o,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val] :
      ( ( produc9184331568460336180_val_o @ F @ A @ B )
     => ( F @ ( produc74620467203698198on_val @ A @ B ) ) ) ).

% curryE
thf(fact_150_curryE,axiom,
    ! [F: produc1317171644805857433on_val > $o,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( produc4800927984219439315_val_o @ F @ A @ B )
     => ( F @ ( produc4958730763592889809on_val @ A @ B ) ) ) ).

% curryE
thf(fact_151_curryE,axiom,
    ! [F: produc2934264451710624999od_o_o > $o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( produc4861507100678042957_o_o_o @ F @ A @ B )
     => ( F @ ( produc763777882069021527od_o_o @ A @ B ) ) ) ).

% curryE
thf(fact_152_curryE,axiom,
    ! [F: produc732395585841259969at_nat > $o,A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat] :
      ( ( produc5440495499350717427_nat_o @ F @ A @ B )
     => ( F @ ( produc8517790099723286449at_nat @ A @ B ) ) ) ).

% curryE
thf(fact_153_curryE,axiom,
    ! [F: produc3843707927480180839at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( produc7820828192191229133_nat_o @ F @ A @ B )
     => ( F @ ( produc2922128104949294807at_nat @ A @ B ) ) ) ).

% curryE
thf(fact_154_minus__empty,axiom,
    ! [Pi: produc1457211279475724562t_char > prat] :
      ( Pi
      = ( add_ma6586698345353345811t_char @ Pi @ empty_3446695950879338768t_char ) ) ).

% minus_empty
thf(fact_155_compatible__options_Opelims_I3_J,axiom,
    ! [X3: option_state,Xa: option_state] :
      ( ~ ( compat2278460363914054422_state @ X3 @ Xa )
     => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X3 @ Xa ) )
       => ~ ! [A2: state] :
              ( ( X3
                = ( some_state @ A2 ) )
             => ! [B2: state] :
                  ( ( Xa
                    = ( some_state @ B2 ) )
                 => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ ( some_state @ A2 ) @ ( some_state @ B2 ) ) )
                   => ( A2 = B2 ) ) ) ) ) ) ).

% compatible_options.pelims(3)
thf(fact_156_compatible__options_Opelims_I3_J,axiom,
    ! [X3: option_val,Xa: option_val] :
      ( ~ ( compat7209469420100717861ns_val @ X3 @ Xa )
     => ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X3 @ Xa ) )
       => ~ ! [A2: val] :
              ( ( X3
                = ( some_val @ A2 ) )
             => ! [B2: val] :
                  ( ( Xa
                    = ( some_val @ B2 ) )
                 => ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ ( some_val @ A2 ) @ ( some_val @ B2 ) ) )
                   => ( A2 = B2 ) ) ) ) ) ) ).

% compatible_options.pelims(3)
thf(fact_157_option_Oexhaust__sel,axiom,
    ! [Option: option_state] :
      ( ( Option != none_state )
     => ( Option
        = ( some_state @ ( the_state @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_158_option_Oexhaust__sel,axiom,
    ! [Option: option_val] :
      ( ( Option != none_val )
     => ( Option
        = ( some_val @ ( the_val @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_159_add__states_Oelims,axiom,
    ! [X3: produc9003781588162623004on_val,Xa: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
      ( ( ( add_states @ X3 @ Xa )
        = Y3 )
     => ~ ! [Pi3: produc1457211279475724562t_char > prat,H4: produc1457211279475724562t_char > option_val] :
            ( ( X3
              = ( produc74620467203698198on_val @ Pi3 @ H4 ) )
           => ! [Pi4: produc1457211279475724562t_char > prat,H6: produc1457211279475724562t_char > option_val] :
                ( ( Xa
                  = ( produc74620467203698198on_val @ Pi4 @ H6 ) )
               => ( Y3
                 != ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ Pi3 @ Pi4 ) @ ( map_ad2908576761495157028ar_val @ H4 @ H6 ) ) ) ) ) ) ).

% add_states.elims
thf(fact_160_add__states_Osimps,axiom,
    ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val,Pi5: produc1457211279475724562t_char > prat,H3: produc1457211279475724562t_char > option_val] :
      ( ( add_states @ ( produc74620467203698198on_val @ Pi @ H ) @ ( produc74620467203698198on_val @ Pi5 @ H3 ) )
      = ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ Pi @ Pi5 ) @ ( map_ad2908576761495157028ar_val @ H @ H3 ) ) ) ).

% add_states.simps
thf(fact_161_valid__larger__mask,axiom,
    ( valid_mask
    = ( greate7680731506090375811t_char @ full_mask ) ) ).

% valid_larger_mask
thf(fact_162_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_163_plus__charact_I1_J,axiom,
    ! [A: state,B: state,X3: state] :
      ( ( ( plus @ A @ B )
        = ( some_state @ X3 ) )
     => ( ( get_m @ X3 )
        = ( add_ma6586698345353345811t_char @ ( get_m @ A ) @ ( get_m @ B ) ) ) ) ).

% plus_charact(1)
thf(fact_164_accp_Ocases,axiom,
    ! [R: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val] :
      ( ( accp_P4405556307444930259on_val @ R @ A )
     => ! [Y6: produc9003781588162623004on_val] :
          ( ( R @ Y6 @ A )
         => ( accp_P4405556307444930259on_val @ R @ Y6 ) ) ) ).

% accp.cases
thf(fact_165_accp_Ocases,axiom,
    ! [R: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,A: produc1317171644805857433on_val] :
      ( ( accp_P6797916352740768290on_val @ R @ A )
     => ! [Y6: produc1317171644805857433on_val] :
          ( ( R @ Y6 @ A )
         => ( accp_P6797916352740768290on_val @ R @ Y6 ) ) ) ).

% accp.cases
thf(fact_166_accp_Osimps,axiom,
    ( accp_P4405556307444930259on_val
    = ( ^ [R3: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A5: produc9003781588162623004on_val] :
        ? [X: produc9003781588162623004on_val] :
          ( ( A5 = X )
          & ! [Y: produc9003781588162623004on_val] :
              ( ( R3 @ Y @ X )
             => ( accp_P4405556307444930259on_val @ R3 @ Y ) ) ) ) ) ).

% accp.simps
thf(fact_167_accp_Osimps,axiom,
    ( accp_P6797916352740768290on_val
    = ( ^ [R3: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,A5: produc1317171644805857433on_val] :
        ? [X: produc1317171644805857433on_val] :
          ( ( A5 = X )
          & ! [Y: produc1317171644805857433on_val] :
              ( ( R3 @ Y @ X )
             => ( accp_P6797916352740768290on_val @ R3 @ Y ) ) ) ) ) ).

% accp.simps
thf(fact_168_greater__mask__properties_I3_J,axiom,
    ! [Pi5: produc1457211279475724562t_char > prat,Pi: produc1457211279475724562t_char > prat] :
      ( ( ( greate7680731506090375811t_char @ Pi5 @ Pi )
        & ( greate7680731506090375811t_char @ Pi @ Pi5 ) )
     => ( Pi = Pi5 ) ) ).

% greater_mask_properties(3)
thf(fact_169_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_170_greater__mask__properties_I1_J,axiom,
    ! [Pi: produc1457211279475724562t_char > prat] : ( greate7680731506090375811t_char @ Pi @ Pi ) ).

% greater_mask_properties(1)
thf(fact_171_commutative,axiom,
    ( plus
    = ( ^ [A5: state,B4: state] : ( plus @ B4 @ A5 ) ) ) ).

% commutative
thf(fact_172_asso1,axiom,
    ! [A: state,B: state,Ab: state,C: state,Bc: state] :
      ( ( ( ( plus @ A @ B )
          = ( some_state @ Ab ) )
        & ( ( plus @ B @ C )
          = ( some_state @ Bc ) ) )
     => ( ( plus @ Ab @ C )
        = ( plus @ A @ Bc ) ) ) ).

% asso1
thf(fact_173_positivity,axiom,
    ! [A: state,B: state,C: state] :
      ( ( ( plus @ A @ B )
        = ( some_state @ C ) )
     => ( ( ( some_state @ C )
          = ( plus @ C @ C ) )
       => ( ( some_state @ A )
          = ( plus @ A @ A ) ) ) ) ).

% positivity
thf(fact_174_asso2,axiom,
    ! [A: state,B: state,Ab: state,C: state] :
      ( ( ( ( plus @ A @ B )
          = ( some_state @ Ab ) )
        & ( ( plus @ B @ C )
          = none_state ) )
     => ( ( plus @ Ab @ C )
        = none_state ) ) ).

% asso2
thf(fact_175_greater__mask__def,axiom,
    ( greate7680731506090375811t_char
    = ( ^ [Pi6: produc1457211279475724562t_char > prat,Pi2: produc1457211279475724562t_char > prat] :
        ? [R3: produc1457211279475724562t_char > prat] :
          ( Pi6
          = ( add_ma6586698345353345811t_char @ Pi2 @ R3 ) ) ) ) ).

% greater_mask_def
thf(fact_176_greater__mask__decomp,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( greate7680731506090375811t_char @ A @ ( add_ma6586698345353345811t_char @ B @ C ) )
     => ? [A1: produc1457211279475724562t_char > prat,A22: produc1457211279475724562t_char > prat] :
          ( ( A
            = ( add_ma6586698345353345811t_char @ A1 @ A22 ) )
          & ( greate7680731506090375811t_char @ A1 @ B )
          & ( greate7680731506090375811t_char @ A22 @ C ) ) ) ).

% greater_mask_decomp
thf(fact_177_const__sum__mask__greater,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat,D: produc1457211279475724562t_char > prat] :
      ( ( ( add_ma6586698345353345811t_char @ A @ B )
        = ( add_ma6586698345353345811t_char @ C @ D ) )
     => ( ( greate7680731506090375811t_char @ A @ C )
       => ( greate7680731506090375811t_char @ D @ B ) ) ) ).

% const_sum_mask_greater
thf(fact_178_plus__charact_I2_J,axiom,
    ! [A: state,B: state,X3: state] :
      ( ( ( plus @ A @ B )
        = ( some_state @ X3 ) )
     => ( ( get_h @ X3 )
        = ( map_ad2908576761495157028ar_val @ ( get_h @ A ) @ ( get_h @ B ) ) ) ) ).

% plus_charact(2)
thf(fact_179_core__is__smaller,axiom,
    ( some_state
    = ( ^ [X: state] : ( plus @ X @ ( core @ X ) ) ) ) ).

% core_is_smaller
thf(fact_180_core__is__pure,axiom,
    ! [X3: state] :
      ( ( some_state @ ( core @ X3 ) )
      = ( plus @ ( core @ X3 ) @ ( core @ X3 ) ) ) ).

% core_is_pure
thf(fact_181_cancellative,axiom,
    ! [A: state,B: state,X3: state,Y3: state] :
      ( ( ( some_state @ A )
        = ( plus @ B @ X3 ) )
     => ( ( ( some_state @ A )
          = ( plus @ B @ Y3 ) )
       => ( ( ( core @ X3 )
            = ( core @ Y3 ) )
         => ( X3 = Y3 ) ) ) ) ).

% cancellative
thf(fact_182_core__sum,axiom,
    ! [C: state,A: state,B: state] :
      ( ( ( some_state @ C )
        = ( plus @ A @ B ) )
     => ( ( some_state @ ( core @ C ) )
        = ( plus @ ( core @ A ) @ ( core @ B ) ) ) ) ).

% core_sum
thf(fact_183_core__max,axiom,
    ! [X3: state,C: state] :
      ( ( ( some_state @ X3 )
        = ( plus @ X3 @ C ) )
     => ? [R4: state] :
          ( ( some_state @ ( core @ X3 ) )
          = ( plus @ C @ R4 ) ) ) ).

% core_max
thf(fact_184_unit__neutral,axiom,
    ( some_state
    = ( ^ [A5: state] : ( plus @ A5 @ unit ) ) ) ).

% unit_neutral
thf(fact_185_accp__induct__rule,axiom,
    ! [R: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val,P: produc9003781588162623004on_val > $o] :
      ( ( accp_P4405556307444930259on_val @ R @ A )
     => ( ! [X2: produc9003781588162623004on_val] :
            ( ( accp_P4405556307444930259on_val @ R @ X2 )
           => ( ! [Y6: produc9003781588162623004on_val] :
                  ( ( R @ Y6 @ X2 )
                 => ( P @ Y6 ) )
             => ( P @ X2 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct_rule
thf(fact_186_accp__induct__rule,axiom,
    ! [R: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,A: produc1317171644805857433on_val,P: produc1317171644805857433on_val > $o] :
      ( ( accp_P6797916352740768290on_val @ R @ A )
     => ( ! [X2: produc1317171644805857433on_val] :
            ( ( accp_P6797916352740768290on_val @ R @ X2 )
           => ( ! [Y6: produc1317171644805857433on_val] :
                  ( ( R @ Y6 @ X2 )
                 => ( P @ Y6 ) )
             => ( P @ X2 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct_rule
thf(fact_187_not__accp__down,axiom,
    ! [R2: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,X3: produc9003781588162623004on_val] :
      ( ~ ( accp_P4405556307444930259on_val @ R2 @ X3 )
     => ~ ! [Z: produc9003781588162623004on_val] :
            ( ( R2 @ Z @ X3 )
           => ( accp_P4405556307444930259on_val @ R2 @ Z ) ) ) ).

% not_accp_down
thf(fact_188_not__accp__down,axiom,
    ! [R2: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,X3: produc1317171644805857433on_val] :
      ( ~ ( accp_P6797916352740768290on_val @ R2 @ X3 )
     => ~ ! [Z: produc1317171644805857433on_val] :
            ( ( R2 @ Z @ X3 )
           => ( accp_P6797916352740768290on_val @ R2 @ Z ) ) ) ).

% not_accp_down
thf(fact_189_accp__downward,axiom,
    ! [R: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,B: produc9003781588162623004on_val,A: produc9003781588162623004on_val] :
      ( ( accp_P4405556307444930259on_val @ R @ B )
     => ( ( R @ A @ B )
       => ( accp_P4405556307444930259on_val @ R @ A ) ) ) ).

% accp_downward
thf(fact_190_accp__downward,axiom,
    ! [R: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,B: produc1317171644805857433on_val,A: produc1317171644805857433on_val] :
      ( ( accp_P6797916352740768290on_val @ R @ B )
     => ( ( R @ A @ B )
       => ( accp_P6797916352740768290on_val @ R @ A ) ) ) ).

% accp_downward
thf(fact_191_accp__induct,axiom,
    ! [R: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val,P: produc9003781588162623004on_val > $o] :
      ( ( accp_P4405556307444930259on_val @ R @ A )
     => ( ! [X2: produc9003781588162623004on_val] :
            ( ( accp_P4405556307444930259on_val @ R @ X2 )
           => ( ! [Y6: produc9003781588162623004on_val] :
                  ( ( R @ Y6 @ X2 )
                 => ( P @ Y6 ) )
             => ( P @ X2 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct
thf(fact_192_accp__induct,axiom,
    ! [R: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,A: produc1317171644805857433on_val,P: produc1317171644805857433on_val > $o] :
      ( ( accp_P6797916352740768290on_val @ R @ A )
     => ( ! [X2: produc1317171644805857433on_val] :
            ( ( accp_P6797916352740768290on_val @ R @ X2 )
           => ( ! [Y6: produc1317171644805857433on_val] :
                  ( ( R @ Y6 @ X2 )
                 => ( P @ Y6 ) )
             => ( P @ X2 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct
thf(fact_193_accpI,axiom,
    ! [R: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,X3: produc9003781588162623004on_val] :
      ( ! [Y4: produc9003781588162623004on_val] :
          ( ( R @ Y4 @ X3 )
         => ( accp_P4405556307444930259on_val @ R @ Y4 ) )
     => ( accp_P4405556307444930259on_val @ R @ X3 ) ) ).

% accpI
thf(fact_194_accpI,axiom,
    ! [R: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,X3: produc1317171644805857433on_val] :
      ( ! [Y4: produc1317171644805857433on_val] :
          ( ( R @ Y4 @ X3 )
         => ( accp_P6797916352740768290on_val @ R @ Y4 ) )
     => ( accp_P6797916352740768290on_val @ R @ X3 ) ) ).

% accpI
thf(fact_195_add__states_Opelims,axiom,
    ! [X3: produc9003781588162623004on_val,Xa: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
      ( ( ( add_states @ X3 @ Xa )
        = Y3 )
     => ( ( accp_P6797916352740768290on_val @ add_states_rel @ ( produc4958730763592889809on_val @ X3 @ Xa ) )
       => ~ ! [Pi3: produc1457211279475724562t_char > prat,H4: produc1457211279475724562t_char > option_val] :
              ( ( X3
                = ( produc74620467203698198on_val @ Pi3 @ H4 ) )
             => ! [Pi4: produc1457211279475724562t_char > prat,H6: produc1457211279475724562t_char > option_val] :
                  ( ( Xa
                    = ( produc74620467203698198on_val @ Pi4 @ H6 ) )
                 => ( ( Y3
                      = ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ Pi3 @ Pi4 ) @ ( map_ad2908576761495157028ar_val @ H4 @ H6 ) ) )
                   => ~ ( accp_P6797916352740768290on_val @ add_states_rel @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Pi3 @ H4 ) @ ( produc74620467203698198on_val @ Pi4 @ H6 ) ) ) ) ) ) ) ) ).

% add_states.pelims
thf(fact_196_plus__def,axiom,
    ( plus
    = ( ^ [A5: state,B4: state] : ( if_option_state @ ( compatible @ ( rep_state @ A5 ) @ ( rep_state @ B4 ) ) @ ( some_state @ ( abs_state @ ( add_states @ ( rep_state @ A5 ) @ ( rep_state @ B4 ) ) ) ) @ none_state ) ) ) ).

% plus_def
thf(fact_197_plus__ab__defined,axiom,
    ! [Phi2: state,Phi3: state] :
      ( ( ( plus @ Phi2 @ Phi3 )
       != none_state )
      = ( ( compatible_heaps @ ( get_h @ Phi2 ) @ ( get_h @ Phi3 ) )
        & ( valid_mask @ ( add_ma6586698345353345811t_char @ ( get_m @ Phi2 ) @ ( get_m @ Phi3 ) ) ) ) ) ).

% plus_ab_defined
thf(fact_198_in__lex__prod,axiom,
    ! [A: produc9003781588162623004on_val,B: set_Product_prod_o_o,A4: produc9003781588162623004on_val,B3: set_Product_prod_o_o,R: set_Pr4305388979797279737on_val,S: set_Pr1932065953672099015od_o_o] :
      ( ( member6243756037710431690od_o_o @ ( produc6284566010912837721od_o_o @ ( produc5870412476076979346od_o_o @ A @ B ) @ ( produc5870412476076979346od_o_o @ A4 @ B3 ) ) @ ( lex_pr4567203704218067578od_o_o @ R @ S ) )
      = ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ A4 ) @ R )
        | ( ( A = A4 )
          & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_199_in__lex__prod,axiom,
    ! [A: produc9003781588162623004on_val,B: set_Pr1261947904930325089at_nat,A4: produc9003781588162623004on_val,B3: set_Pr1261947904930325089at_nat,R: set_Pr4305388979797279737on_val,S: set_Pr4329608150637261639at_nat] :
      ( ( member1987034204009628670at_nat @ ( produc342415708206977933at_nat @ ( produc5399520021890326380at_nat @ A @ B ) @ ( produc5399520021890326380at_nat @ A4 @ B3 ) ) @ ( lex_pr3389479527782218068at_nat @ R @ S ) )
      = ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ A4 ) @ R )
        | ( ( A = A4 )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_200_in__lex__prod,axiom,
    ! [A: set_Product_prod_o_o,B: produc9003781588162623004on_val,A4: set_Product_prod_o_o,B3: produc9003781588162623004on_val,R: set_Pr1932065953672099015od_o_o,S: set_Pr4305388979797279737on_val] :
      ( ( member6502073555826736814on_val @ ( produc4584915227888689725on_val @ ( produc2629970354652812492on_val @ A @ B ) @ ( produc2629970354652812492on_val @ A4 @ B3 ) ) @ ( lex_pr1326761582793900724on_val @ R @ S ) )
      = ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A4 ) @ R )
        | ( ( A = A4 )
          & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_201_in__lex__prod,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: produc9003781588162623004on_val,A4: set_Pr1261947904930325089at_nat,B3: produc9003781588162623004on_val,R: set_Pr4329608150637261639at_nat,S: set_Pr4305388979797279737on_val] :
      ( ( member3753270555006320866on_val @ ( produc4785630383394302321on_val @ ( produc9151609020855878438on_val @ A @ B ) @ ( produc9151609020855878438on_val @ A4 @ B3 ) ) @ ( lex_pr7141568526747770126on_val @ R @ S ) )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A4 ) @ R )
        | ( ( A = A4 )
          & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_202_in__lex__prod,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Product_prod_o_o,A4: set_Pr1261947904930325089at_nat,B3: set_Product_prod_o_o,R: set_Pr4329608150637261639at_nat,S: set_Pr1932065953672099015od_o_o] :
      ( ( member625886800346054288od_o_o @ ( produc4770972565181134295od_o_o @ ( produc2070086378307464573od_o_o @ A @ B ) @ ( produc2070086378307464573od_o_o @ A4 @ B3 ) ) @ ( lex_pr3343786553352869269od_o_o @ R @ S ) )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A4 ) @ R )
        | ( ( A = A4 )
          & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_203_in__lex__prod,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,A4: produc9003781588162623004on_val,B3: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val,S: set_Pr4305388979797279737on_val] :
      ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ A @ B ) @ ( produc4958730763592889809on_val @ A4 @ B3 ) ) @ ( lex_pr8992908938815505897on_val @ R @ S ) )
      = ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ A4 ) @ R )
        | ( ( A = A4 )
          & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_204_in__lex__prod,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,A4: set_Product_prod_o_o,B3: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o,S: set_Pr1932065953672099015od_o_o] :
      ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( produc763777882069021527od_o_o @ A4 @ B3 ) ) @ ( lex_pr2209867684007004015od_o_o @ R @ S ) )
      = ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A4 ) @ R )
        | ( ( A = A4 )
          & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_205_in__lex__prod,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,A4: set_Product_prod_o_o,B3: set_Pr1261947904930325089at_nat,R: set_Pr1932065953672099015od_o_o,S: set_Pr4329608150637261639at_nat] :
      ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ ( produc8517790099723286449at_nat @ A4 @ B3 ) ) @ ( lex_pr568118237913915337at_nat @ R @ S ) )
      = ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A4 ) @ R )
        | ( ( A = A4 )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_206_in__lex__prod,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc2922128104949294807at_nat @ A4 @ B3 ) ) @ ( lex_pr8000699672184251119at_nat @ R @ S ) )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A4 ) @ R )
        | ( ( A = A4 )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_207_in__lex__prod,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,A4: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val,R: set_Pr1685704261205283783r_prat,S: set_Pr4698884584343881589on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ A @ B ) @ ( produc74620467203698198on_val @ A4 @ B3 ) ) @ ( lex_pr8780192377291449342on_val @ R @ S ) )
      = ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ A @ A4 ) @ R )
        | ( ( A = A4 )
          & ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_208_add__masks_Opelims,axiom,
    ! [X3: produc1457211279475724562t_char > prat,Xa: produc1457211279475724562t_char > prat,Xb: produc1457211279475724562t_char,Y3: prat] :
      ( ( ( add_ma6586698345353345811t_char @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P224314924008452035t_char @ add_ma2412268806269088278t_char @ ( produc2957999048406202538t_char @ X3 @ ( produc2591414526589101846t_char @ Xa @ Xb ) ) )
       => ~ ( ( Y3
              = ( padd @ ( X3 @ Xb ) @ ( Xa @ Xb ) ) )
           => ~ ( accp_P224314924008452035t_char @ add_ma2412268806269088278t_char @ ( produc2957999048406202538t_char @ X3 @ ( produc2591414526589101846t_char @ Xa @ Xb ) ) ) ) ) ) ).

% add_masks.pelims
thf(fact_209_mult__greater,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,P4: prat] :
      ( ( greate7680731506090375811t_char @ A @ B )
     => ( greate7680731506090375811t_char @ ( multiply_mask @ P4 @ A ) @ ( multiply_mask @ P4 @ B ) ) ) ).

% mult_greater
thf(fact_210_update__trans,axiom,
    ! [Pi: produc1457211279475724562t_char > prat,Pi5: produc1457211279475724562t_char > prat,Pi7: produc1457211279475724562t_char > prat,Dm: produc1457211279475724562t_char > $o] :
      ( ( greate7680731506090375811t_char @ Pi @ Pi5 )
     => ( ( greate7680731506090375811t_char @ Pi5 @ Pi7 )
       => ( ( update5099441844846592839t_char @ ( update5099441844846592839t_char @ Dm @ Pi @ Pi5 ) @ Pi5 @ Pi7 )
          = ( update5099441844846592839t_char @ Dm @ Pi @ Pi7 ) ) ) ) ).

% update_trans
thf(fact_211_option_Oset__sel,axiom,
    ! [A: option8293164206536526765od_o_o] :
      ( ( A != none_P8254367833036352840od_o_o )
     => ( member444158400953824016od_o_o @ ( the_Pr7254245475272632699od_o_o @ A ) @ ( set_op3337521775309186165od_o_o @ A ) ) ) ).

% option.set_sel
thf(fact_212_option_Oset__sel,axiom,
    ! [A: option7754883786268931591at_nat] :
      ( ( A != none_P5828533823149705634at_nat )
     => ( member4095101504841534314at_nat @ ( the_Pr9097376869753557077at_nat @ A ) @ ( set_op8338180575893423695at_nat @ A ) ) ) ).

% option.set_sel
thf(fact_213_option_Oset__sel,axiom,
    ! [A: option8699557107911914541at_nat] :
      ( ( A != none_P5811896708078067656at_nat )
     => ( member8757157785044589968at_nat @ ( the_Pr6298817542474494203at_nat @ A ) @ ( set_op5866586666149159413at_nat @ A ) ) ) ).

% option.set_sel
thf(fact_214_option_Oset__sel,axiom,
    ! [A: option_val] :
      ( ( A != none_val )
     => ( member_val @ ( the_val @ A ) @ ( set_option_val2 @ A ) ) ) ).

% option.set_sel
thf(fact_215_option_Oset__sel,axiom,
    ! [A: option_state] :
      ( ( A != none_state )
     => ( member_state @ ( the_state @ A ) @ ( set_option_state2 @ A ) ) ) ).

% option.set_sel
thf(fact_216_elem__set,axiom,
    ! [X3: produc2934264451710624999od_o_o,Xo: option8293164206536526765od_o_o] :
      ( ( member444158400953824016od_o_o @ X3 @ ( set_op3337521775309186165od_o_o @ Xo ) )
      = ( Xo
        = ( some_P7068320699308846284od_o_o @ X3 ) ) ) ).

% elem_set
thf(fact_217_elem__set,axiom,
    ! [X3: produc732395585841259969at_nat,Xo: option7754883786268931591at_nat] :
      ( ( member4095101504841534314at_nat @ X3 @ ( set_op8338180575893423695at_nat @ Xo ) )
      = ( Xo
        = ( some_P123625206269196070at_nat @ X3 ) ) ) ).

% elem_set
thf(fact_218_elem__set,axiom,
    ! [X3: produc3843707927480180839at_nat,Xo: option8699557107911914541at_nat] :
      ( ( member8757157785044589968at_nat @ X3 @ ( set_op5866586666149159413at_nat @ Xo ) )
      = ( Xo
        = ( some_P131194445498448716at_nat @ X3 ) ) ) ).

% elem_set
thf(fact_219_elem__set,axiom,
    ! [X3: state,Xo: option_state] :
      ( ( member_state @ X3 @ ( set_option_state2 @ Xo ) )
      = ( Xo
        = ( some_state @ X3 ) ) ) ).

% elem_set
thf(fact_220_elem__set,axiom,
    ! [X3: val,Xo: option_val] :
      ( ( member_val @ X3 @ ( set_option_val2 @ Xo ) )
      = ( Xo
        = ( some_val @ X3 ) ) ) ).

% elem_set
thf(fact_221_compatible__heapsI,axiom,
    ! [H: produc1457211279475724562t_char > option_val,H3: produc1457211279475724562t_char > option_val] :
      ( ! [Hl3: produc1457211279475724562t_char,A2: val,B2: val] :
          ( ( ( H @ Hl3 )
            = ( some_val @ A2 ) )
         => ( ( ( H3 @ Hl3 )
              = ( some_val @ B2 ) )
           => ( A2 = B2 ) ) )
     => ( compatible_heaps @ H @ H3 ) ) ).

% compatible_heapsI
thf(fact_222_compatible__heaps__def,axiom,
    ( compatible_heaps
    = ( ^ [H2: produc1457211279475724562t_char > option_val,H5: produc1457211279475724562t_char > option_val] :
        ! [Hl2: produc1457211279475724562t_char] : ( compat7209469420100717861ns_val @ ( H2 @ Hl2 ) @ ( H5 @ Hl2 ) ) ) ) ).

% compatible_heaps_def
thf(fact_223_larger__compatible__sum__heaps,axiom,
    ! [A: produc1457211279475724562t_char > option_val,X3: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val,Y3: produc1457211279475724562t_char > option_val] :
      ( ( larger1532547405891354081ar_val @ A @ X3 )
     => ( ( larger1532547405891354081ar_val @ B @ Y3 )
       => ( ( compatible_heaps @ A @ B )
         => ( compatible_heaps @ X3 @ Y3 ) ) ) ) ).

% larger_compatible_sum_heaps
thf(fact_224_larger__heap__comp,axiom,
    ! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val,C: produc1457211279475724562t_char > option_val] :
      ( ( larger1532547405891354081ar_val @ A @ B )
     => ( ( compatible_heaps @ A @ C )
       => ( compatible_heaps @ B @ C ) ) ) ).

% larger_heap_comp
thf(fact_225_mult__add__states,axiom,
    ! [A: prat,B: prat,Pi: produc1457211279475724562t_char > prat] :
      ( ( multiply_mask @ ( padd @ A @ B ) @ Pi )
      = ( add_ma6586698345353345811t_char @ ( multiply_mask @ A @ Pi ) @ ( multiply_mask @ B @ Pi ) ) ) ).

% mult_add_states
thf(fact_226_option_Oset__cases,axiom,
    ! [E: produc2934264451710624999od_o_o,A: option8293164206536526765od_o_o] :
      ( ( member444158400953824016od_o_o @ E @ ( set_op3337521775309186165od_o_o @ A ) )
     => ( A
        = ( some_P7068320699308846284od_o_o @ E ) ) ) ).

% option.set_cases
thf(fact_227_option_Oset__cases,axiom,
    ! [E: produc732395585841259969at_nat,A: option7754883786268931591at_nat] :
      ( ( member4095101504841534314at_nat @ E @ ( set_op8338180575893423695at_nat @ A ) )
     => ( A
        = ( some_P123625206269196070at_nat @ E ) ) ) ).

% option.set_cases
thf(fact_228_option_Oset__cases,axiom,
    ! [E: produc3843707927480180839at_nat,A: option8699557107911914541at_nat] :
      ( ( member8757157785044589968at_nat @ E @ ( set_op5866586666149159413at_nat @ A ) )
     => ( A
        = ( some_P131194445498448716at_nat @ E ) ) ) ).

% option.set_cases
thf(fact_229_option_Oset__cases,axiom,
    ! [E: state,A: option_state] :
      ( ( member_state @ E @ ( set_option_state2 @ A ) )
     => ( A
        = ( some_state @ E ) ) ) ).

% option.set_cases
thf(fact_230_option_Oset__cases,axiom,
    ! [E: val,A: option_val] :
      ( ( member_val @ E @ ( set_option_val2 @ A ) )
     => ( A
        = ( some_val @ E ) ) ) ).

% option.set_cases
thf(fact_231_option_Oset__intros,axiom,
    ! [X22: produc2934264451710624999od_o_o] : ( member444158400953824016od_o_o @ X22 @ ( set_op3337521775309186165od_o_o @ ( some_P7068320699308846284od_o_o @ X22 ) ) ) ).

% option.set_intros
thf(fact_232_option_Oset__intros,axiom,
    ! [X22: produc732395585841259969at_nat] : ( member4095101504841534314at_nat @ X22 @ ( set_op8338180575893423695at_nat @ ( some_P123625206269196070at_nat @ X22 ) ) ) ).

% option.set_intros
thf(fact_233_option_Oset__intros,axiom,
    ! [X22: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X22 @ ( set_op5866586666149159413at_nat @ ( some_P131194445498448716at_nat @ X22 ) ) ) ).

% option.set_intros
thf(fact_234_option_Oset__intros,axiom,
    ! [X22: state] : ( member_state @ X22 @ ( set_option_state2 @ ( some_state @ X22 ) ) ) ).

% option.set_intros
thf(fact_235_option_Oset__intros,axiom,
    ! [X22: val] : ( member_val @ X22 @ ( set_option_val2 @ ( some_val @ X22 ) ) ) ).

% option.set_intros
thf(fact_236_ospec,axiom,
    ! [A3: option_state,P: state > $o,X3: state] :
      ( ! [X2: state] :
          ( ( member_state @ X2 @ ( set_option_state2 @ A3 ) )
         => ( P @ X2 ) )
     => ( ( A3
          = ( some_state @ X3 ) )
       => ( P @ X3 ) ) ) ).

% ospec
thf(fact_237_ospec,axiom,
    ! [A3: option_val,P: val > $o,X3: val] :
      ( ! [X2: val] :
          ( ( member_val @ X2 @ ( set_option_val2 @ A3 ) )
         => ( P @ X2 ) )
     => ( ( A3
          = ( some_val @ X3 ) )
       => ( P @ X3 ) ) ) ).

% ospec
thf(fact_238_add__masks_Osimps,axiom,
    ( add_ma6586698345353345811t_char
    = ( ^ [Pi6: produc1457211279475724562t_char > prat,Pi2: produc1457211279475724562t_char > prat,Hl2: produc1457211279475724562t_char] : ( padd @ ( Pi6 @ Hl2 ) @ ( Pi2 @ Hl2 ) ) ) ) ).

% add_masks.simps
thf(fact_239_add__masks_Oelims,axiom,
    ! [X3: produc1457211279475724562t_char > prat,Xa: produc1457211279475724562t_char > prat,Xb: produc1457211279475724562t_char,Y3: prat] :
      ( ( ( add_ma6586698345353345811t_char @ X3 @ Xa @ Xb )
        = Y3 )
     => ( Y3
        = ( padd @ ( X3 @ Xb ) @ ( Xa @ Xb ) ) ) ) ).

% add_masks.elims
thf(fact_240_mult__distr__masks,axiom,
    ! [A: prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( multiply_mask @ A @ ( add_ma6586698345353345811t_char @ B @ C ) )
      = ( add_ma6586698345353345811t_char @ ( multiply_mask @ A @ B ) @ ( multiply_mask @ A @ C ) ) ) ).

% mult_distr_masks
thf(fact_241_larger__heaps__sum__ineq,axiom,
    ! [A4: produc1457211279475724562t_char > option_val,A: produc1457211279475724562t_char > option_val,B3: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val] :
      ( ( larger1532547405891354081ar_val @ A4 @ A )
     => ( ( larger1532547405891354081ar_val @ B3 @ B )
       => ( ( compatible_heaps @ A4 @ B3 )
         => ( larger1532547405891354081ar_val @ ( map_ad2908576761495157028ar_val @ A4 @ B3 ) @ ( map_ad2908576761495157028ar_val @ A @ B ) ) ) ) ) ).

% larger_heaps_sum_ineq
thf(fact_242_compatible__heaps__comm,axiom,
    ! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val] :
      ( ( compatible_heaps @ A @ B )
     => ( ( map_ad2908576761495157028ar_val @ A @ B )
        = ( map_ad2908576761495157028ar_val @ B @ A ) ) ) ).

% compatible_heaps_comm
thf(fact_243_compatible__heaps__sum,axiom,
    ! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val,C: produc1457211279475724562t_char > option_val] :
      ( ( compatible_heaps @ A @ B )
     => ( ( compatible_heaps @ A @ C )
       => ( compatible_heaps @ A @ ( map_ad2908576761495157028ar_val @ B @ C ) ) ) ) ).

% compatible_heaps_sum
thf(fact_244_ppos__add,axiom,
    ! [A: prat,B: prat] :
      ( ( ppos @ A )
     => ( ppos @ ( padd @ A @ B ) ) ) ).

% ppos_add
thf(fact_245_set__empty__eq,axiom,
    ! [Xo: option_val] :
      ( ( ( set_option_val2 @ Xo )
        = bot_bot_set_val )
      = ( Xo = none_val ) ) ).

% set_empty_eq
thf(fact_246_set__empty__eq,axiom,
    ! [Xo: option_state] :
      ( ( ( set_option_state2 @ Xo )
        = bot_bot_set_state )
      = ( Xo = none_state ) ) ).

% set_empty_eq
thf(fact_247_bind__eq__None__conv,axiom,
    ! [A: option_val,F: val > option_val] :
      ( ( ( bind_val_val @ A @ F )
        = none_val )
      = ( ( A = none_val )
        | ( ( F @ ( the_val @ A ) )
          = none_val ) ) ) ).

% bind_eq_None_conv
thf(fact_248_bind__eq__None__conv,axiom,
    ! [A: option_state,F: state > option_val] :
      ( ( ( bind_state_val @ A @ F )
        = none_val )
      = ( ( A = none_state )
        | ( ( F @ ( the_state @ A ) )
          = none_val ) ) ) ).

% bind_eq_None_conv
thf(fact_249_bind__eq__None__conv,axiom,
    ! [A: option_val,F: val > option_state] :
      ( ( ( bind_val_state @ A @ F )
        = none_state )
      = ( ( A = none_val )
        | ( ( F @ ( the_val @ A ) )
          = none_state ) ) ) ).

% bind_eq_None_conv
thf(fact_250_bind__eq__None__conv,axiom,
    ! [A: option_state,F: state > option_state] :
      ( ( ( bind_state_state @ A @ F )
        = none_state )
      = ( ( A = none_state )
        | ( ( F @ ( the_state @ A ) )
          = none_state ) ) ) ).

% bind_eq_None_conv
thf(fact_251_in__graphD,axiom,
    ! [K: produc1457211279475724562t_char > prat,V: produc1457211279475724562t_char > option_val,M: ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ K @ V ) @ ( graph_1288227317702120175on_val @ M ) )
     => ( ( M @ K )
        = ( some_P7669353234731512949on_val @ V ) ) ) ).

% in_graphD
thf(fact_252_in__graphD,axiom,
    ! [K: produc9003781588162623004on_val,V: produc9003781588162623004on_val,M: produc9003781588162623004on_val > option8887860577033306220on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ K @ V ) @ ( graph_6108330199704490680on_val @ M ) )
     => ( ( M @ K )
        = ( some_P2581650809551911671on_val @ V ) ) ) ).

% in_graphD
thf(fact_253_in__graphD,axiom,
    ! [K: set_Product_prod_o_o,V: set_Product_prod_o_o,M: set_Product_prod_o_o > option956769437226787789od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ K @ V ) @ ( graph_5719763352399009470od_o_o @ M ) )
     => ( ( M @ K )
        = ( some_s1360601300120069868od_o_o @ V ) ) ) ).

% in_graphD
thf(fact_254_in__graphD,axiom,
    ! [K: set_Product_prod_o_o,V: set_Pr1261947904930325089at_nat,M: set_Product_prod_o_o > option8963830502488799655at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ K @ V ) @ ( graph_2163970799110208408at_nat @ M ) )
     => ( ( M @ K )
        = ( some_s147305329494351046at_nat @ V ) ) ) ).

% in_graphD
thf(fact_255_in__graphD,axiom,
    ! [K: set_Pr1261947904930325089at_nat,V: set_Pr1261947904930325089at_nat,M: set_Pr1261947904930325089at_nat > option8963830502488799655at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ K @ V ) @ ( graph_2380648924068268862at_nat @ M ) )
     => ( ( M @ K )
        = ( some_s147305329494351046at_nat @ V ) ) ) ).

% in_graphD
thf(fact_256_in__graphI,axiom,
    ! [M: ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val,K: produc1457211279475724562t_char > prat,V: produc1457211279475724562t_char > option_val] :
      ( ( ( M @ K )
        = ( some_P7669353234731512949on_val @ V ) )
     => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ K @ V ) @ ( graph_1288227317702120175on_val @ M ) ) ) ).

% in_graphI
thf(fact_257_in__graphI,axiom,
    ! [M: produc9003781588162623004on_val > option8887860577033306220on_val,K: produc9003781588162623004on_val,V: produc9003781588162623004on_val] :
      ( ( ( M @ K )
        = ( some_P2581650809551911671on_val @ V ) )
     => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ K @ V ) @ ( graph_6108330199704490680on_val @ M ) ) ) ).

% in_graphI
thf(fact_258_in__graphI,axiom,
    ! [M: set_Product_prod_o_o > option956769437226787789od_o_o,K: set_Product_prod_o_o,V: set_Product_prod_o_o] :
      ( ( ( M @ K )
        = ( some_s1360601300120069868od_o_o @ V ) )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ K @ V ) @ ( graph_5719763352399009470od_o_o @ M ) ) ) ).

% in_graphI
thf(fact_259_in__graphI,axiom,
    ! [M: set_Product_prod_o_o > option8963830502488799655at_nat,K: set_Product_prod_o_o,V: set_Pr1261947904930325089at_nat] :
      ( ( ( M @ K )
        = ( some_s147305329494351046at_nat @ V ) )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ K @ V ) @ ( graph_2163970799110208408at_nat @ M ) ) ) ).

% in_graphI
thf(fact_260_in__graphI,axiom,
    ! [M: set_Pr1261947904930325089at_nat > option8963830502488799655at_nat,K: set_Pr1261947904930325089at_nat,V: set_Pr1261947904930325089at_nat] :
      ( ( ( M @ K )
        = ( some_s147305329494351046at_nat @ V ) )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ K @ V ) @ ( graph_2380648924068268862at_nat @ M ) ) ) ).

% in_graphI
thf(fact_261_bind__runit,axiom,
    ! [X3: option_state] :
      ( ( bind_state_state @ X3 @ some_state )
      = X3 ) ).

% bind_runit
thf(fact_262_bind__runit,axiom,
    ! [X3: option_val] :
      ( ( bind_val_val @ X3 @ some_val )
      = X3 ) ).

% bind_runit
thf(fact_263_bind__eq__Some__conv,axiom,
    ! [F: option_state,G: state > option_state,X3: state] :
      ( ( ( bind_state_state @ F @ G )
        = ( some_state @ X3 ) )
      = ( ? [Y: state] :
            ( ( F
              = ( some_state @ Y ) )
            & ( ( G @ Y )
              = ( some_state @ X3 ) ) ) ) ) ).

% bind_eq_Some_conv
thf(fact_264_bind__eq__Some__conv,axiom,
    ! [F: option_val,G: val > option_state,X3: state] :
      ( ( ( bind_val_state @ F @ G )
        = ( some_state @ X3 ) )
      = ( ? [Y: val] :
            ( ( F
              = ( some_val @ Y ) )
            & ( ( G @ Y )
              = ( some_state @ X3 ) ) ) ) ) ).

% bind_eq_Some_conv
thf(fact_265_bind__eq__Some__conv,axiom,
    ! [F: option_state,G: state > option_val,X3: val] :
      ( ( ( bind_state_val @ F @ G )
        = ( some_val @ X3 ) )
      = ( ? [Y: state] :
            ( ( F
              = ( some_state @ Y ) )
            & ( ( G @ Y )
              = ( some_val @ X3 ) ) ) ) ) ).

% bind_eq_Some_conv
thf(fact_266_bind__eq__Some__conv,axiom,
    ! [F: option_val,G: val > option_val,X3: val] :
      ( ( ( bind_val_val @ F @ G )
        = ( some_val @ X3 ) )
      = ( ? [Y: val] :
            ( ( F
              = ( some_val @ Y ) )
            & ( ( G @ Y )
              = ( some_val @ X3 ) ) ) ) ) ).

% bind_eq_Some_conv
thf(fact_267_bind_Obind__lzero,axiom,
    ! [F: val > option_val] :
      ( ( bind_val_val @ none_val @ F )
      = none_val ) ).

% bind.bind_lzero
thf(fact_268_bind_Obind__lzero,axiom,
    ! [F: val > option_state] :
      ( ( bind_val_state @ none_val @ F )
      = none_state ) ).

% bind.bind_lzero
thf(fact_269_bind_Obind__lzero,axiom,
    ! [F: state > option_val] :
      ( ( bind_state_val @ none_state @ F )
      = none_val ) ).

% bind.bind_lzero
thf(fact_270_bind_Obind__lzero,axiom,
    ! [F: state > option_state] :
      ( ( bind_state_state @ none_state @ F )
      = none_state ) ).

% bind.bind_lzero
thf(fact_271_option_Osimps_I14_J,axiom,
    ( ( set_option_val2 @ none_val )
    = bot_bot_set_val ) ).

% option.simps(14)
thf(fact_272_option_Osimps_I14_J,axiom,
    ( ( set_option_state2 @ none_state )
    = bot_bot_set_state ) ).

% option.simps(14)
thf(fact_273_bind__split,axiom,
    ! [P: option_val > $o,M: option_state,F: state > option_val] :
      ( ( P @ ( bind_state_val @ M @ F ) )
      = ( ( ( M = none_state )
         => ( P @ none_val ) )
        & ! [V2: state] :
            ( ( M
              = ( some_state @ V2 ) )
           => ( P @ ( F @ V2 ) ) ) ) ) ).

% bind_split
thf(fact_274_bind__split,axiom,
    ! [P: option_state > $o,M: option_state,F: state > option_state] :
      ( ( P @ ( bind_state_state @ M @ F ) )
      = ( ( ( M = none_state )
         => ( P @ none_state ) )
        & ! [V2: state] :
            ( ( M
              = ( some_state @ V2 ) )
           => ( P @ ( F @ V2 ) ) ) ) ) ).

% bind_split
thf(fact_275_bind__split,axiom,
    ! [P: option_val > $o,M: option_val,F: val > option_val] :
      ( ( P @ ( bind_val_val @ M @ F ) )
      = ( ( ( M = none_val )
         => ( P @ none_val ) )
        & ! [V2: val] :
            ( ( M
              = ( some_val @ V2 ) )
           => ( P @ ( F @ V2 ) ) ) ) ) ).

% bind_split
thf(fact_276_bind__split,axiom,
    ! [P: option_state > $o,M: option_val,F: val > option_state] :
      ( ( P @ ( bind_val_state @ M @ F ) )
      = ( ( ( M = none_val )
         => ( P @ none_state ) )
        & ! [V2: val] :
            ( ( M
              = ( some_val @ V2 ) )
           => ( P @ ( F @ V2 ) ) ) ) ) ).

% bind_split
thf(fact_277_bind__split__asm,axiom,
    ! [P: option_val > $o,M: option_state,F: state > option_val] :
      ( ( P @ ( bind_state_val @ M @ F ) )
      = ( ~ ( ( ( M = none_state )
              & ~ ( P @ none_val ) )
            | ? [X: state] :
                ( ( M
                  = ( some_state @ X ) )
                & ~ ( P @ ( F @ X ) ) ) ) ) ) ).

% bind_split_asm
thf(fact_278_bind__split__asm,axiom,
    ! [P: option_state > $o,M: option_state,F: state > option_state] :
      ( ( P @ ( bind_state_state @ M @ F ) )
      = ( ~ ( ( ( M = none_state )
              & ~ ( P @ none_state ) )
            | ? [X: state] :
                ( ( M
                  = ( some_state @ X ) )
                & ~ ( P @ ( F @ X ) ) ) ) ) ) ).

% bind_split_asm
thf(fact_279_bind__split__asm,axiom,
    ! [P: option_val > $o,M: option_val,F: val > option_val] :
      ( ( P @ ( bind_val_val @ M @ F ) )
      = ( ~ ( ( ( M = none_val )
              & ~ ( P @ none_val ) )
            | ? [X: val] :
                ( ( M
                  = ( some_val @ X ) )
                & ~ ( P @ ( F @ X ) ) ) ) ) ) ).

% bind_split_asm
thf(fact_280_bind__split__asm,axiom,
    ! [P: option_state > $o,M: option_val,F: val > option_state] :
      ( ( P @ ( bind_val_state @ M @ F ) )
      = ( ~ ( ( ( M = none_val )
              & ~ ( P @ none_state ) )
            | ? [X: val] :
                ( ( M
                  = ( some_val @ X ) )
                & ~ ( P @ ( F @ X ) ) ) ) ) ) ).

% bind_split_asm
thf(fact_281_graph__restrictD_I2_J,axiom,
    ! [K: produc1457211279475724562t_char > prat,V: produc1457211279475724562t_char > option_val,M: ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val,A3: set_Pr3433046674199179443r_prat] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ K @ V ) @ ( graph_1288227317702120175on_val @ ( restri7594233908965391805on_val @ M @ A3 ) ) )
     => ( ( M @ K )
        = ( some_P7669353234731512949on_val @ V ) ) ) ).

% graph_restrictD(2)
thf(fact_282_graph__restrictD_I2_J,axiom,
    ! [K: produc9003781588162623004on_val,V: produc9003781588162623004on_val,M: produc9003781588162623004on_val > option8887860577033306220on_val,A3: set_Pr7649655576376376530on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ K @ V ) @ ( graph_6108330199704490680on_val @ ( restri2354629191430751338on_val @ M @ A3 ) ) )
     => ( ( M @ K )
        = ( some_P2581650809551911671on_val @ V ) ) ) ).

% graph_restrictD(2)
thf(fact_283_graph__restrictD_I2_J,axiom,
    ! [K: set_Product_prod_o_o,V: set_Product_prod_o_o,M: set_Product_prod_o_o > option956769437226787789od_o_o,A3: set_se491199571785101031od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ K @ V ) @ ( graph_5719763352399009470od_o_o @ ( restri2278710580326071152od_o_o @ M @ A3 ) ) )
     => ( ( M @ K )
        = ( some_s1360601300120069868od_o_o @ V ) ) ) ).

% graph_restrictD(2)
thf(fact_284_graph__restrictD_I2_J,axiom,
    ! [K: set_Product_prod_o_o,V: set_Pr1261947904930325089at_nat,M: set_Product_prod_o_o > option8963830502488799655at_nat,A3: set_se491199571785101031od_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ K @ V ) @ ( graph_2163970799110208408at_nat @ ( restri2174721201111129930at_nat @ M @ A3 ) ) )
     => ( ( M @ K )
        = ( some_s147305329494351046at_nat @ V ) ) ) ).

% graph_restrictD(2)
thf(fact_285_graph__restrictD_I2_J,axiom,
    ! [K: set_Pr1261947904930325089at_nat,V: set_Pr1261947904930325089at_nat,M: set_Pr1261947904930325089at_nat > option8963830502488799655at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ K @ V ) @ ( graph_2380648924068268862at_nat @ ( restri512981303736921584at_nat @ M @ A3 ) ) )
     => ( ( M @ K )
        = ( some_s147305329494351046at_nat @ V ) ) ) ).

% graph_restrictD(2)
thf(fact_286_option_Osimps_I15_J,axiom,
    ! [X22: state] :
      ( ( set_option_state2 @ ( some_state @ X22 ) )
      = ( insert_state @ X22 @ bot_bot_set_state ) ) ).

% option.simps(15)
thf(fact_287_option_Osimps_I15_J,axiom,
    ! [X22: val] :
      ( ( set_option_val2 @ ( some_val @ X22 ) )
      = ( insert_val @ X22 @ bot_bot_set_val ) ) ).

% option.simps(15)
thf(fact_288_ppos__inv,axiom,
    ! [P4: prat] :
      ( ( ppos @ P4 )
     => ( ppos @ ( pinv @ P4 ) ) ) ).

% ppos_inv
thf(fact_289_graph__restrictD_I1_J,axiom,
    ! [K: produc1457211279475724562t_char > prat,V: produc1457211279475724562t_char > option_val,M: ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val,A3: set_Pr3433046674199179443r_prat] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ K @ V ) @ ( graph_1288227317702120175on_val @ ( restri7594233908965391805on_val @ M @ A3 ) ) )
     => ( member2336342582916103932r_prat @ K @ A3 ) ) ).

% graph_restrictD(1)
thf(fact_290_graph__restrictD_I1_J,axiom,
    ! [K: produc9003781588162623004on_val,V: produc9003781588162623004on_val,M: produc9003781588162623004on_val > option8887860577033306220on_val,A3: set_Pr7649655576376376530on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ K @ V ) @ ( graph_6108330199704490680on_val @ ( restri2354629191430751338on_val @ M @ A3 ) ) )
     => ( member2879030624002546611on_val @ K @ A3 ) ) ).

% graph_restrictD(1)
thf(fact_291_graph__restrictD_I1_J,axiom,
    ! [K: set_Product_prod_o_o,V: set_Product_prod_o_o,M: set_Product_prod_o_o > option956769437226787789od_o_o,A3: set_se491199571785101031od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ K @ V ) @ ( graph_5719763352399009470od_o_o @ ( restri2278710580326071152od_o_o @ M @ A3 ) ) )
     => ( member3157015407008874800od_o_o @ K @ A3 ) ) ).

% graph_restrictD(1)
thf(fact_292_graph__restrictD_I1_J,axiom,
    ! [K: set_Product_prod_o_o,V: set_Pr1261947904930325089at_nat,M: set_Product_prod_o_o > option8963830502488799655at_nat,A3: set_se491199571785101031od_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ K @ V ) @ ( graph_2163970799110208408at_nat @ ( restri2174721201111129930at_nat @ M @ A3 ) ) )
     => ( member3157015407008874800od_o_o @ K @ A3 ) ) ).

% graph_restrictD(1)
thf(fact_293_graph__restrictD_I1_J,axiom,
    ! [K: set_Pr1261947904930325089at_nat,V: set_Pr1261947904930325089at_nat,M: set_Pr1261947904930325089at_nat > option8963830502488799655at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ K @ V ) @ ( graph_2380648924068268862at_nat @ ( restri512981303736921584at_nat @ M @ A3 ) ) )
     => ( member2643936169264416010at_nat @ K @ A3 ) ) ).

% graph_restrictD(1)
thf(fact_294_restrict__out,axiom,
    ! [X3: produc2934264451710624999od_o_o,A3: set_Pr1932065953672099015od_o_o,M: produc2934264451710624999od_o_o > option_val] :
      ( ~ ( member444158400953824016od_o_o @ X3 @ A3 )
     => ( ( restri6212659212090664970_o_val @ M @ A3 @ X3 )
        = none_val ) ) ).

% restrict_out
thf(fact_295_restrict__out,axiom,
    ! [X3: produc732395585841259969at_nat,A3: set_Pr457366540195662369at_nat,M: produc732395585841259969at_nat > option_val] :
      ( ~ ( member4095101504841534314at_nat @ X3 @ A3 )
     => ( ( restri8460040695785523504at_val @ M @ A3 @ X3 )
        = none_val ) ) ).

% restrict_out
thf(fact_296_restrict__out,axiom,
    ! [X3: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,M: produc3843707927480180839at_nat > option_val] :
      ( ~ ( member8757157785044589968at_nat @ X3 @ A3 )
     => ( ( restri1475145662897394570at_val @ M @ A3 @ X3 )
        = none_val ) ) ).

% restrict_out
thf(fact_297_restrict__out,axiom,
    ! [X3: produc2934264451710624999od_o_o,A3: set_Pr1932065953672099015od_o_o,M: produc2934264451710624999od_o_o > option_state] :
      ( ~ ( member444158400953824016od_o_o @ X3 @ A3 )
     => ( ( restri1635145334713054705_state @ M @ A3 @ X3 )
        = none_state ) ) ).

% restrict_out
thf(fact_298_restrict__out,axiom,
    ! [X3: produc732395585841259969at_nat,A3: set_Pr457366540195662369at_nat,M: produc732395585841259969at_nat > option_state] :
      ( ~ ( member4095101504841534314at_nat @ X3 @ A3 )
     => ( ( restri3344998686824062283_state @ M @ A3 @ X3 )
        = none_state ) ) ).

% restrict_out
thf(fact_299_restrict__out,axiom,
    ! [X3: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,M: produc3843707927480180839at_nat > option_state] :
      ( ~ ( member8757157785044589968at_nat @ X3 @ A3 )
     => ( ( restri28699544554891889_state @ M @ A3 @ X3 )
        = none_state ) ) ).

% restrict_out
thf(fact_300_is__none__code_I2_J,axiom,
    ! [X3: state] :
      ~ ( is_none_state @ ( some_state @ X3 ) ) ).

% is_none_code(2)
thf(fact_301_is__none__code_I2_J,axiom,
    ! [X3: val] :
      ~ ( is_none_val @ ( some_val @ X3 ) ) ).

% is_none_code(2)
thf(fact_302_is__none__code_I1_J,axiom,
    is_none_val @ none_val ).

% is_none_code(1)
thf(fact_303_is__none__code_I1_J,axiom,
    is_none_state @ none_state ).

% is_none_code(1)
thf(fact_304_these__insert__None,axiom,
    ! [A3: set_option_val] :
      ( ( these_val @ ( insert_option_val @ none_val @ A3 ) )
      = ( these_val @ A3 ) ) ).

% these_insert_None
thf(fact_305_these__insert__None,axiom,
    ! [A3: set_option_state] :
      ( ( these_state @ ( insert_option_state @ none_state @ A3 ) )
      = ( these_state @ A3 ) ) ).

% these_insert_None
thf(fact_306_these__insert__Some,axiom,
    ! [X3: state,A3: set_option_state] :
      ( ( these_state @ ( insert_option_state @ ( some_state @ X3 ) @ A3 ) )
      = ( insert_state @ X3 @ ( these_state @ A3 ) ) ) ).

% these_insert_Some
thf(fact_307_these__insert__Some,axiom,
    ! [X3: val,A3: set_option_val] :
      ( ( these_val @ ( insert_option_val @ ( some_val @ X3 ) @ A3 ) )
      = ( insert_val @ X3 @ ( these_val @ A3 ) ) ) ).

% these_insert_Some
thf(fact_308_these__not__empty__eq,axiom,
    ! [B5: set_option_val] :
      ( ( ( these_val @ B5 )
       != bot_bot_set_val )
      = ( ( B5 != bot_bo5906790151302476563on_val )
        & ( B5
         != ( insert_option_val @ none_val @ bot_bo5906790151302476563on_val ) ) ) ) ).

% these_not_empty_eq
thf(fact_309_these__not__empty__eq,axiom,
    ! [B5: set_option_state] :
      ( ( ( these_state @ B5 )
       != bot_bot_set_state )
      = ( ( B5 != bot_bo710180891245420500_state )
        & ( B5
         != ( insert_option_state @ none_state @ bot_bo710180891245420500_state ) ) ) ) ).

% these_not_empty_eq
thf(fact_310_these__empty__eq,axiom,
    ! [B5: set_option_val] :
      ( ( ( these_val @ B5 )
        = bot_bot_set_val )
      = ( ( B5 = bot_bo5906790151302476563on_val )
        | ( B5
          = ( insert_option_val @ none_val @ bot_bo5906790151302476563on_val ) ) ) ) ).

% these_empty_eq
thf(fact_311_these__empty__eq,axiom,
    ! [B5: set_option_state] :
      ( ( ( these_state @ B5 )
        = bot_bot_set_state )
      = ( ( B5 = bot_bo710180891245420500_state )
        | ( B5
          = ( insert_option_state @ none_state @ bot_bo710180891245420500_state ) ) ) ) ).

% these_empty_eq
thf(fact_312_restrict__map__def,axiom,
    ( restri6212659212090664970_o_val
    = ( ^ [M4: produc2934264451710624999od_o_o > option_val,A6: set_Pr1932065953672099015od_o_o,X: produc2934264451710624999od_o_o] : ( if_option_val @ ( member444158400953824016od_o_o @ X @ A6 ) @ ( M4 @ X ) @ none_val ) ) ) ).

% restrict_map_def
thf(fact_313_restrict__map__def,axiom,
    ( restri8460040695785523504at_val
    = ( ^ [M4: produc732395585841259969at_nat > option_val,A6: set_Pr457366540195662369at_nat,X: produc732395585841259969at_nat] : ( if_option_val @ ( member4095101504841534314at_nat @ X @ A6 ) @ ( M4 @ X ) @ none_val ) ) ) ).

% restrict_map_def
thf(fact_314_restrict__map__def,axiom,
    ( restri1475145662897394570at_val
    = ( ^ [M4: produc3843707927480180839at_nat > option_val,A6: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] : ( if_option_val @ ( member8757157785044589968at_nat @ X @ A6 ) @ ( M4 @ X ) @ none_val ) ) ) ).

% restrict_map_def
thf(fact_315_restrict__map__def,axiom,
    ( restri1635145334713054705_state
    = ( ^ [M4: produc2934264451710624999od_o_o > option_state,A6: set_Pr1932065953672099015od_o_o,X: produc2934264451710624999od_o_o] : ( if_option_state @ ( member444158400953824016od_o_o @ X @ A6 ) @ ( M4 @ X ) @ none_state ) ) ) ).

% restrict_map_def
thf(fact_316_restrict__map__def,axiom,
    ( restri3344998686824062283_state
    = ( ^ [M4: produc732395585841259969at_nat > option_state,A6: set_Pr457366540195662369at_nat,X: produc732395585841259969at_nat] : ( if_option_state @ ( member4095101504841534314at_nat @ X @ A6 ) @ ( M4 @ X ) @ none_state ) ) ) ).

% restrict_map_def
thf(fact_317_restrict__map__def,axiom,
    ( restri28699544554891889_state
    = ( ^ [M4: produc3843707927480180839at_nat > option_state,A6: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] : ( if_option_state @ ( member8757157785044589968at_nat @ X @ A6 ) @ ( M4 @ X ) @ none_state ) ) ) ).

% restrict_map_def
thf(fact_318_is__none__simps_I2_J,axiom,
    ! [X3: state] :
      ~ ( is_none_state @ ( some_state @ X3 ) ) ).

% is_none_simps(2)
thf(fact_319_is__none__simps_I2_J,axiom,
    ! [X3: val] :
      ~ ( is_none_val @ ( some_val @ X3 ) ) ).

% is_none_simps(2)
thf(fact_320_is__none__simps_I1_J,axiom,
    is_none_val @ none_val ).

% is_none_simps(1)
thf(fact_321_is__none__simps_I1_J,axiom,
    is_none_state @ none_state ).

% is_none_simps(1)
thf(fact_322_Option_Ois__none__def,axiom,
    ( is_none_val
    = ( ^ [X: option_val] : ( X = none_val ) ) ) ).

% Option.is_none_def
thf(fact_323_Option_Ois__none__def,axiom,
    ( is_none_state
    = ( ^ [X: option_state] : ( X = none_state ) ) ) ).

% Option.is_none_def
thf(fact_324_in__these__eq,axiom,
    ! [X3: produc2934264451710624999od_o_o,A3: set_op3605315081486705933od_o_o] :
      ( ( member444158400953824016od_o_o @ X3 @ ( these_2277756779245414180od_o_o @ A3 ) )
      = ( member3623239300863961174od_o_o @ ( some_P7068320699308846284od_o_o @ X3 ) @ A3 ) ) ).

% in_these_eq
thf(fact_325_in__these__eq,axiom,
    ! [X3: produc732395585841259969at_nat,A3: set_op7555165197759865831at_nat] :
      ( ( member4095101504841534314at_nat @ X3 @ ( these_663420305483081086at_nat @ A3 ) )
      = ( member8056002749114821168at_nat @ ( some_P123625206269196070at_nat @ X3 ) @ A3 ) ) ).

% in_these_eq
thf(fact_326_in__these__eq,axiom,
    ! [X3: produc3843707927480180839at_nat,A3: set_op989438649584373901at_nat] :
      ( ( member8757157785044589968at_nat @ X3 @ ( these_1872268174106123684at_nat @ A3 ) )
      = ( member4288423094740844502at_nat @ ( some_P131194445498448716at_nat @ X3 ) @ A3 ) ) ).

% in_these_eq
thf(fact_327_in__these__eq,axiom,
    ! [X3: state,A3: set_option_state] :
      ( ( member_state @ X3 @ ( these_state @ A3 ) )
      = ( member_option_state @ ( some_state @ X3 ) @ A3 ) ) ).

% in_these_eq
thf(fact_328_in__these__eq,axiom,
    ! [X3: val,A3: set_option_val] :
      ( ( member_val @ X3 @ ( these_val @ A3 ) )
      = ( member_option_val @ ( some_val @ X3 ) @ A3 ) ) ).

% in_these_eq
thf(fact_329_relcompEpair,axiom,
    ! [A: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > option_val,R: set_Pr1685704261205283783r_prat,S: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ C ) @ ( relcom2642467956565963995on_val @ R @ S ) )
     => ~ ! [B2: produc1457211279475724562t_char > prat] :
            ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ A @ B2 ) @ R )
           => ~ ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ B2 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_330_relcompEpair,axiom,
    ! [A: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val,S: set_Pr4698884584343881589on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ C ) @ ( relcom3547033359321004872on_val @ R @ S ) )
     => ~ ! [B2: produc1457211279475724562t_char > option_val] :
            ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B2 ) @ R )
           => ~ ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ B2 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_331_relcompEpair,axiom,
    ! [A: produc9003781588162623004on_val,C: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val,S: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ C ) @ ( relcom3664992426205683031on_val @ R @ S ) )
     => ~ ! [B2: produc9003781588162623004on_val] :
            ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B2 ) @ R )
           => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B2 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_332_relcompEpair,axiom,
    ! [A: set_Product_prod_o_o,C: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o,S: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ C ) @ ( relcom8634166700253352710od_o_o @ R @ S ) )
     => ~ ! [B2: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B2 ) @ R )
           => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B2 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_333_relcompEpair,axiom,
    ! [A: set_Product_prod_o_o,C: set_Product_prod_o_o,R: set_Pr457366540195662369at_nat,S: set_Pr8055703017043224813od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ C ) @ ( relcom3302618809389967788od_o_o @ R @ S ) )
     => ~ ! [B2: set_Pr1261947904930325089at_nat] :
            ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B2 ) @ R )
           => ~ ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ B2 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_334_relcompEpair,axiom,
    ! [A: set_Product_prod_o_o,C: set_Pr1261947904930325089at_nat,R: set_Pr1932065953672099015od_o_o,S: set_Pr457366540195662369at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ C ) @ ( relcom526950493951013856at_nat @ R @ S ) )
     => ~ ! [B2: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B2 ) @ R )
           => ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ B2 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_335_relcompEpair,axiom,
    ! [A: set_Product_prod_o_o,C: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ C ) @ ( relcom5236704819349487494at_nat @ R @ S ) )
     => ~ ! [B2: set_Pr1261947904930325089at_nat] :
            ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B2 ) @ R )
           => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_336_relcompEpair,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,R: set_Pr8055703017043224813od_o_o,S: set_Pr457366540195662369at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ C ) @ ( relcom708864787231718074at_nat @ R @ S ) )
     => ~ ! [B2: set_Product_prod_o_o] :
            ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ A @ B2 ) @ R )
           => ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ B2 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_337_relcompEpair,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ C ) @ ( relcom6558574857482195296at_nat @ R @ S ) )
     => ~ ! [B2: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ R )
           => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_338_relcompE,axiom,
    ! [Xz: produc9003781588162623004on_val,R: set_Pr1685704261205283783r_prat,S: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ Xz @ ( relcom2642467956565963995on_val @ R @ S ) )
     => ~ ! [X2: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > prat,Z: produc1457211279475724562t_char > option_val] :
            ( ( Xz
              = ( produc74620467203698198on_val @ X2 @ Z ) )
           => ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X2 @ Y4 ) @ R )
             => ~ ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ Y4 @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_339_relcompE,axiom,
    ! [Xz: produc9003781588162623004on_val,R: set_Pr7649655576376376530on_val,S: set_Pr4698884584343881589on_val] :
      ( ( member2879030624002546611on_val @ Xz @ ( relcom3547033359321004872on_val @ R @ S ) )
     => ~ ! [X2: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > option_val,Z: produc1457211279475724562t_char > option_val] :
            ( ( Xz
              = ( produc74620467203698198on_val @ X2 @ Z ) )
           => ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X2 @ Y4 ) @ R )
             => ~ ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ Y4 @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_340_relcompE,axiom,
    ! [Xz: produc1317171644805857433on_val,R: set_Pr4305388979797279737on_val,S: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ Xz @ ( relcom3664992426205683031on_val @ R @ S ) )
     => ~ ! [X2: produc9003781588162623004on_val,Y4: produc9003781588162623004on_val,Z: produc9003781588162623004on_val] :
            ( ( Xz
              = ( produc4958730763592889809on_val @ X2 @ Z ) )
           => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y4 ) @ R )
             => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y4 @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_341_relcompE,axiom,
    ! [Xz: produc2934264451710624999od_o_o,R: set_Pr1932065953672099015od_o_o,S: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ Xz @ ( relcom8634166700253352710od_o_o @ R @ S ) )
     => ~ ! [X2: set_Product_prod_o_o,Y4: set_Product_prod_o_o,Z: set_Product_prod_o_o] :
            ( ( Xz
              = ( produc763777882069021527od_o_o @ X2 @ Z ) )
           => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y4 ) @ R )
             => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_342_relcompE,axiom,
    ! [Xz: produc2934264451710624999od_o_o,R: set_Pr457366540195662369at_nat,S: set_Pr8055703017043224813od_o_o] :
      ( ( member444158400953824016od_o_o @ Xz @ ( relcom3302618809389967788od_o_o @ R @ S ) )
     => ~ ! [X2: set_Product_prod_o_o,Y4: set_Pr1261947904930325089at_nat,Z: set_Product_prod_o_o] :
            ( ( Xz
              = ( produc763777882069021527od_o_o @ X2 @ Z ) )
           => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X2 @ Y4 ) @ R )
             => ~ ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ Y4 @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_343_relcompE,axiom,
    ! [Xz: produc732395585841259969at_nat,R: set_Pr1932065953672099015od_o_o,S: set_Pr457366540195662369at_nat] :
      ( ( member4095101504841534314at_nat @ Xz @ ( relcom526950493951013856at_nat @ R @ S ) )
     => ~ ! [X2: set_Product_prod_o_o,Y4: set_Product_prod_o_o,Z: set_Pr1261947904930325089at_nat] :
            ( ( Xz
              = ( produc8517790099723286449at_nat @ X2 @ Z ) )
           => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y4 ) @ R )
             => ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ Y4 @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_344_relcompE,axiom,
    ! [Xz: produc732395585841259969at_nat,R: set_Pr457366540195662369at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member4095101504841534314at_nat @ Xz @ ( relcom5236704819349487494at_nat @ R @ S ) )
     => ~ ! [X2: set_Product_prod_o_o,Y4: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
            ( ( Xz
              = ( produc8517790099723286449at_nat @ X2 @ Z ) )
           => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X2 @ Y4 ) @ R )
             => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_345_relcompE,axiom,
    ! [Xz: produc3843707927480180839at_nat,R: set_Pr8055703017043224813od_o_o,S: set_Pr457366540195662369at_nat] :
      ( ( member8757157785044589968at_nat @ Xz @ ( relcom708864787231718074at_nat @ R @ S ) )
     => ~ ! [X2: set_Pr1261947904930325089at_nat,Y4: set_Product_prod_o_o,Z: set_Pr1261947904930325089at_nat] :
            ( ( Xz
              = ( produc2922128104949294807at_nat @ X2 @ Z ) )
           => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ X2 @ Y4 ) @ R )
             => ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ Y4 @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_346_relcompE,axiom,
    ! [Xz: produc3843707927480180839at_nat,R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ Xz @ ( relcom6558574857482195296at_nat @ R @ S ) )
     => ~ ! [X2: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
            ( ( Xz
              = ( produc2922128104949294807at_nat @ X2 @ Z ) )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y4 ) @ R )
             => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_347_relcomp_OrelcompI,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,R: set_Pr1685704261205283783r_prat,C: produc1457211279475724562t_char > option_val,S: set_Pr7649655576376376530on_val] :
      ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ A @ B ) @ R )
     => ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ B @ C ) @ S )
       => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ C ) @ ( relcom2642467956565963995on_val @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_348_relcomp_OrelcompI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Product_prod_o_o,R: set_Pr8055703017043224813od_o_o,C: set_Pr1261947904930325089at_nat,S: set_Pr457366540195662369at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ A @ B ) @ R )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ B @ C ) @ S )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ C ) @ ( relcom708864787231718074at_nat @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_349_relcomp_OrelcompI,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val,C: produc1457211279475724562t_char > option_val,S: set_Pr4698884584343881589on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ R )
     => ( ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ B @ C ) @ S )
       => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ C ) @ ( relcom3547033359321004872on_val @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_350_relcomp_OrelcompI,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val,C: produc9003781588162623004on_val,S: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ C ) @ S )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ C ) @ ( relcom3664992426205683031on_val @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_351_relcomp_OrelcompI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o,C: set_Product_prod_o_o,S: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ C ) @ S )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ C ) @ ( relcom8634166700253352710od_o_o @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_352_relcomp_OrelcompI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o,C: set_Pr1261947904930325089at_nat,S: set_Pr457366540195662369at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ B @ C ) @ S )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ C ) @ ( relcom526950493951013856at_nat @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_353_relcomp_OrelcompI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat,C: set_Product_prod_o_o,S: set_Pr8055703017043224813od_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ B @ C ) @ S )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ C ) @ ( relcom3302618809389967788od_o_o @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_354_relcomp_OrelcompI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat,C: set_Pr1261947904930325089at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ C ) @ S )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ C ) @ ( relcom5236704819349487494at_nat @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_355_relcomp_OrelcompI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,C: set_Pr1261947904930325089at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ C ) @ S )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ C ) @ ( relcom6558574857482195296at_nat @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_356_relcomp_Osimps,axiom,
    ! [A12: produc1457211279475724562t_char > prat,A23: produc1457211279475724562t_char > option_val,R: set_Pr1685704261205283783r_prat,S: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A12 @ A23 ) @ ( relcom2642467956565963995on_val @ R @ S ) )
      = ( ? [A5: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > prat,C3: produc1457211279475724562t_char > option_val] :
            ( ( A12 = A5 )
            & ( A23 = C3 )
            & ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ A5 @ B4 ) @ R )
            & ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ B4 @ C3 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_357_relcomp_Osimps,axiom,
    ! [A12: produc1457211279475724562t_char > prat,A23: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val,S: set_Pr4698884584343881589on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A12 @ A23 ) @ ( relcom3547033359321004872on_val @ R @ S ) )
      = ( ? [A5: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > option_val,C3: produc1457211279475724562t_char > option_val] :
            ( ( A12 = A5 )
            & ( A23 = C3 )
            & ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A5 @ B4 ) @ R )
            & ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ B4 @ C3 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_358_relcomp_Osimps,axiom,
    ! [A12: produc9003781588162623004on_val,A23: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val,S: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A12 @ A23 ) @ ( relcom3664992426205683031on_val @ R @ S ) )
      = ( ? [A5: produc9003781588162623004on_val,B4: produc9003781588162623004on_val,C3: produc9003781588162623004on_val] :
            ( ( A12 = A5 )
            & ( A23 = C3 )
            & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A5 @ B4 ) @ R )
            & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B4 @ C3 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_359_relcomp_Osimps,axiom,
    ! [A12: set_Product_prod_o_o,A23: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o,S: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A12 @ A23 ) @ ( relcom8634166700253352710od_o_o @ R @ S ) )
      = ( ? [A5: set_Product_prod_o_o,B4: set_Product_prod_o_o,C3: set_Product_prod_o_o] :
            ( ( A12 = A5 )
            & ( A23 = C3 )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A5 @ B4 ) @ R )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B4 @ C3 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_360_relcomp_Osimps,axiom,
    ! [A12: set_Product_prod_o_o,A23: set_Product_prod_o_o,R: set_Pr457366540195662369at_nat,S: set_Pr8055703017043224813od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A12 @ A23 ) @ ( relcom3302618809389967788od_o_o @ R @ S ) )
      = ( ? [A5: set_Product_prod_o_o,B4: set_Pr1261947904930325089at_nat,C3: set_Product_prod_o_o] :
            ( ( A12 = A5 )
            & ( A23 = C3 )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A5 @ B4 ) @ R )
            & ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ B4 @ C3 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_361_relcomp_Osimps,axiom,
    ! [A12: set_Product_prod_o_o,A23: set_Pr1261947904930325089at_nat,R: set_Pr1932065953672099015od_o_o,S: set_Pr457366540195662369at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A12 @ A23 ) @ ( relcom526950493951013856at_nat @ R @ S ) )
      = ( ? [A5: set_Product_prod_o_o,B4: set_Product_prod_o_o,C3: set_Pr1261947904930325089at_nat] :
            ( ( A12 = A5 )
            & ( A23 = C3 )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A5 @ B4 ) @ R )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ B4 @ C3 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_362_relcomp_Osimps,axiom,
    ! [A12: set_Product_prod_o_o,A23: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A12 @ A23 ) @ ( relcom5236704819349487494at_nat @ R @ S ) )
      = ( ? [A5: set_Product_prod_o_o,B4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
            ( ( A12 = A5 )
            & ( A23 = C3 )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A5 @ B4 ) @ R )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B4 @ C3 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_363_relcomp_Osimps,axiom,
    ! [A12: set_Pr1261947904930325089at_nat,A23: set_Pr1261947904930325089at_nat,R: set_Pr8055703017043224813od_o_o,S: set_Pr457366540195662369at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A12 @ A23 ) @ ( relcom708864787231718074at_nat @ R @ S ) )
      = ( ? [A5: set_Pr1261947904930325089at_nat,B4: set_Product_prod_o_o,C3: set_Pr1261947904930325089at_nat] :
            ( ( A12 = A5 )
            & ( A23 = C3 )
            & ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ A5 @ B4 ) @ R )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ B4 @ C3 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_364_relcomp_Osimps,axiom,
    ! [A12: set_Pr1261947904930325089at_nat,A23: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A12 @ A23 ) @ ( relcom6558574857482195296at_nat @ R @ S ) )
      = ( ? [A5: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
            ( ( A12 = A5 )
            & ( A23 = C3 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A5 @ B4 ) @ R )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B4 @ C3 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_365_relcomp_Ocases,axiom,
    ! [A12: produc1457211279475724562t_char > prat,A23: produc1457211279475724562t_char > option_val,R: set_Pr1685704261205283783r_prat,S: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A12 @ A23 ) @ ( relcom2642467956565963995on_val @ R @ S ) )
     => ~ ! [B2: produc1457211279475724562t_char > prat] :
            ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ A12 @ B2 ) @ R )
           => ~ ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ B2 @ A23 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_366_relcomp_Ocases,axiom,
    ! [A12: produc1457211279475724562t_char > prat,A23: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val,S: set_Pr4698884584343881589on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A12 @ A23 ) @ ( relcom3547033359321004872on_val @ R @ S ) )
     => ~ ! [B2: produc1457211279475724562t_char > option_val] :
            ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A12 @ B2 ) @ R )
           => ~ ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ B2 @ A23 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_367_relcomp_Ocases,axiom,
    ! [A12: produc9003781588162623004on_val,A23: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val,S: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A12 @ A23 ) @ ( relcom3664992426205683031on_val @ R @ S ) )
     => ~ ! [B2: produc9003781588162623004on_val] :
            ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A12 @ B2 ) @ R )
           => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B2 @ A23 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_368_relcomp_Ocases,axiom,
    ! [A12: set_Product_prod_o_o,A23: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o,S: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A12 @ A23 ) @ ( relcom8634166700253352710od_o_o @ R @ S ) )
     => ~ ! [B2: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A12 @ B2 ) @ R )
           => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B2 @ A23 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_369_relcomp_Ocases,axiom,
    ! [A12: set_Product_prod_o_o,A23: set_Product_prod_o_o,R: set_Pr457366540195662369at_nat,S: set_Pr8055703017043224813od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A12 @ A23 ) @ ( relcom3302618809389967788od_o_o @ R @ S ) )
     => ~ ! [B2: set_Pr1261947904930325089at_nat] :
            ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A12 @ B2 ) @ R )
           => ~ ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ B2 @ A23 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_370_relcomp_Ocases,axiom,
    ! [A12: set_Product_prod_o_o,A23: set_Pr1261947904930325089at_nat,R: set_Pr1932065953672099015od_o_o,S: set_Pr457366540195662369at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A12 @ A23 ) @ ( relcom526950493951013856at_nat @ R @ S ) )
     => ~ ! [B2: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A12 @ B2 ) @ R )
           => ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ B2 @ A23 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_371_relcomp_Ocases,axiom,
    ! [A12: set_Product_prod_o_o,A23: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A12 @ A23 ) @ ( relcom5236704819349487494at_nat @ R @ S ) )
     => ~ ! [B2: set_Pr1261947904930325089at_nat] :
            ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A12 @ B2 ) @ R )
           => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ A23 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_372_relcomp_Ocases,axiom,
    ! [A12: set_Pr1261947904930325089at_nat,A23: set_Pr1261947904930325089at_nat,R: set_Pr8055703017043224813od_o_o,S: set_Pr457366540195662369at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A12 @ A23 ) @ ( relcom708864787231718074at_nat @ R @ S ) )
     => ~ ! [B2: set_Product_prod_o_o] :
            ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ A12 @ B2 ) @ R )
           => ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ B2 @ A23 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_373_relcomp_Ocases,axiom,
    ! [A12: set_Pr1261947904930325089at_nat,A23: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A12 @ A23 ) @ ( relcom6558574857482195296at_nat @ R @ S ) )
     => ~ ! [B2: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A12 @ B2 ) @ R )
           => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ A23 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_374_graph__map__upd,axiom,
    ! [M: ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val,K: produc1457211279475724562t_char > prat,V: produc1457211279475724562t_char > option_val] :
      ( ( graph_1288227317702120175on_val @ ( fun_up1601731230164360576on_val @ M @ K @ ( some_P7669353234731512949on_val @ V ) ) )
      = ( insert3030518898131756108on_val @ ( produc74620467203698198on_val @ K @ V ) @ ( graph_1288227317702120175on_val @ ( fun_up1601731230164360576on_val @ M @ K @ none_P1334921381875262201on_val ) ) ) ) ).

% graph_map_upd
thf(fact_375_graph__map__upd,axiom,
    ! [M: produc9003781588162623004on_val > option8887860577033306220on_val,K: produc9003781588162623004on_val,V: produc9003781588162623004on_val] :
      ( ( graph_6108330199704490680on_val @ ( fun_up273046073882939271on_val @ M @ K @ ( some_P2581650809551911671on_val @ V ) ) )
      = ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ K @ V ) @ ( graph_6108330199704490680on_val @ ( fun_up273046073882939271on_val @ M @ K @ none_P8728355543607141243on_val ) ) ) ) ).

% graph_map_upd
thf(fact_376_graph__map__upd,axiom,
    ! [M: set_Product_prod_o_o > option956769437226787789od_o_o,K: set_Product_prod_o_o,V: set_Product_prod_o_o] :
      ( ( graph_5719763352399009470od_o_o @ ( fun_up3230021793924660483od_o_o @ M @ K @ ( some_s1360601300120069868od_o_o @ V ) ) )
      = ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ K @ V ) @ ( graph_5719763352399009470od_o_o @ ( fun_up3230021793924660483od_o_o @ M @ K @ none_s7246968996079588200od_o_o ) ) ) ) ).

% graph_map_upd
thf(fact_377_graph__map__upd,axiom,
    ! [M: set_Product_prod_o_o > option8963830502488799655at_nat,K: set_Product_prod_o_o,V: set_Pr1261947904930325089at_nat] :
      ( ( graph_2163970799110208408at_nat @ ( fun_up3688893386320429021at_nat @ M @ K @ ( some_s147305329494351046at_nat @ V ) ) )
      = ( insert8109617542356073361at_nat @ ( produc8517790099723286449at_nat @ K @ V ) @ ( graph_2163970799110208408at_nat @ ( fun_up3688893386320429021at_nat @ M @ K @ none_s625347054029921090at_nat ) ) ) ) ).

% graph_map_upd
thf(fact_378_graph__map__upd,axiom,
    ! [M: set_Pr1261947904930325089at_nat > option8963830502488799655at_nat,K: set_Pr1261947904930325089at_nat,V: set_Pr1261947904930325089at_nat] :
      ( ( graph_2380648924068268862at_nat @ ( fun_up1071884767227745027at_nat @ M @ K @ ( some_s147305329494351046at_nat @ V ) ) )
      = ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ K @ V ) @ ( graph_2380648924068268862at_nat @ ( fun_up1071884767227745027at_nat @ M @ K @ none_s625347054029921090at_nat ) ) ) ) ).

% graph_map_upd
thf(fact_379_map__add__upd,axiom,
    ! [F: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val,X3: produc1457211279475724562t_char,Y3: val] :
      ( ( map_ad2908576761495157028ar_val @ F @ ( fun_up5735975307950539954on_val @ G @ X3 @ ( some_val @ Y3 ) ) )
      = ( fun_up5735975307950539954on_val @ ( map_ad2908576761495157028ar_val @ F @ G ) @ X3 @ ( some_val @ Y3 ) ) ) ).

% map_add_upd
thf(fact_380_fun__upd__None__restrict,axiom,
    ! [X3: produc2934264451710624999od_o_o,D2: set_Pr1932065953672099015od_o_o,M: produc2934264451710624999od_o_o > option_val] :
      ( ( ( member444158400953824016od_o_o @ X3 @ D2 )
       => ( ( fun_up6453461605741198237on_val @ ( restri6212659212090664970_o_val @ M @ D2 ) @ X3 @ none_val )
          = ( restri6212659212090664970_o_val @ M @ ( minus_5704166846132356910od_o_o @ D2 @ ( insert3855699742481506999od_o_o @ X3 @ bot_bo498831808395348787od_o_o ) ) ) ) )
      & ( ~ ( member444158400953824016od_o_o @ X3 @ D2 )
       => ( ( fun_up6453461605741198237on_val @ ( restri6212659212090664970_o_val @ M @ D2 ) @ X3 @ none_val )
          = ( restri6212659212090664970_o_val @ M @ D2 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_381_fun__upd__None__restrict,axiom,
    ! [X3: produc732395585841259969at_nat,D2: set_Pr457366540195662369at_nat,M: produc732395585841259969at_nat > option_val] :
      ( ( ( member4095101504841534314at_nat @ X3 @ D2 )
       => ( ( fun_up320048328433459523on_val @ ( restri8460040695785523504at_val @ M @ D2 ) @ X3 @ none_val )
          = ( restri8460040695785523504at_val @ M @ ( minus_1608234471674494216at_nat @ D2 @ ( insert8109617542356073361at_nat @ X3 @ bot_bo1676454896636820109at_nat ) ) ) ) )
      & ( ~ ( member4095101504841534314at_nat @ X3 @ D2 )
       => ( ( fun_up320048328433459523on_val @ ( restri8460040695785523504at_val @ M @ D2 ) @ X3 @ none_val )
          = ( restri8460040695785523504at_val @ M @ D2 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_382_fun__upd__None__restrict,axiom,
    ! [X3: produc3843707927480180839at_nat,D2: set_Pr4329608150637261639at_nat,M: produc3843707927480180839at_nat > option_val] :
      ( ( ( member8757157785044589968at_nat @ X3 @ D2 )
       => ( ( fun_up7479097124709437981on_val @ ( restri1475145662897394570at_val @ M @ D2 ) @ X3 @ none_val )
          = ( restri1475145662897394570at_val @ M @ ( minus_3314409938677909166at_nat @ D2 @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ) )
      & ( ~ ( member8757157785044589968at_nat @ X3 @ D2 )
       => ( ( fun_up7479097124709437981on_val @ ( restri1475145662897394570at_val @ M @ D2 ) @ X3 @ none_val )
          = ( restri1475145662897394570at_val @ M @ D2 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_383_fun__upd__None__restrict,axiom,
    ! [X3: produc2934264451710624999od_o_o,D2: set_Pr1932065953672099015od_o_o,M: produc2934264451710624999od_o_o > option_state] :
      ( ( ( member444158400953824016od_o_o @ X3 @ D2 )
       => ( ( fun_up4180510163728908852_state @ ( restri1635145334713054705_state @ M @ D2 ) @ X3 @ none_state )
          = ( restri1635145334713054705_state @ M @ ( minus_5704166846132356910od_o_o @ D2 @ ( insert3855699742481506999od_o_o @ X3 @ bot_bo498831808395348787od_o_o ) ) ) ) )
      & ( ~ ( member444158400953824016od_o_o @ X3 @ D2 )
       => ( ( fun_up4180510163728908852_state @ ( restri1635145334713054705_state @ M @ D2 ) @ X3 @ none_state )
          = ( restri1635145334713054705_state @ M @ D2 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_384_fun__upd__None__restrict,axiom,
    ! [X3: produc732395585841259969at_nat,D2: set_Pr457366540195662369at_nat,M: produc732395585841259969at_nat > option_state] :
      ( ( ( member4095101504841534314at_nat @ X3 @ D2 )
       => ( ( fun_up6026250480174712078_state @ ( restri3344998686824062283_state @ M @ D2 ) @ X3 @ none_state )
          = ( restri3344998686824062283_state @ M @ ( minus_1608234471674494216at_nat @ D2 @ ( insert8109617542356073361at_nat @ X3 @ bot_bo1676454896636820109at_nat ) ) ) ) )
      & ( ~ ( member4095101504841534314at_nat @ X3 @ D2 )
       => ( ( fun_up6026250480174712078_state @ ( restri3344998686824062283_state @ M @ D2 ) @ X3 @ none_state )
          = ( restri3344998686824062283_state @ M @ D2 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_385_fun__upd__None__restrict,axiom,
    ! [X3: produc3843707927480180839at_nat,D2: set_Pr4329608150637261639at_nat,M: produc3843707927480180839at_nat > option_state] :
      ( ( ( member8757157785044589968at_nat @ X3 @ D2 )
       => ( ( fun_up3329660993234963380_state @ ( restri28699544554891889_state @ M @ D2 ) @ X3 @ none_state )
          = ( restri28699544554891889_state @ M @ ( minus_3314409938677909166at_nat @ D2 @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ) )
      & ( ~ ( member8757157785044589968at_nat @ X3 @ D2 )
       => ( ( fun_up3329660993234963380_state @ ( restri28699544554891889_state @ M @ D2 ) @ X3 @ none_state )
          = ( restri28699544554891889_state @ M @ D2 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_386_Range__insert,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val] :
      ( ( range_8542210083549373368on_val @ ( insert3030518898131756108on_val @ ( produc74620467203698198on_val @ A @ B ) @ R ) )
      = ( insert216038912599248202on_val @ B @ ( range_8542210083549373368on_val @ R ) ) ) ).

% Range_insert
thf(fact_387_Range__insert,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( range_6350497433844811695on_val @ ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R ) )
      = ( insert3030518898131756108on_val @ B @ ( range_6350497433844811695on_val @ R ) ) ) ).

% Range_insert
thf(fact_388_Range__insert,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( range_1318165146934585909od_o_o @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R ) )
      = ( insert3817420688681518039od_o_o @ B @ ( range_1318165146934585909od_o_o @ R ) ) ) ).

% Range_insert
thf(fact_389_Range__insert,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat] :
      ( ( range_8412904782085253007at_nat @ ( insert8109617542356073361at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R ) )
      = ( insert9200635055090092081at_nat @ B @ ( range_8412904782085253007at_nat @ R ) ) ) ).

% Range_insert
thf(fact_390_Range__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( range_880292732526890933at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R ) )
      = ( insert9200635055090092081at_nat @ B @ ( range_880292732526890933at_nat @ R ) ) ) ).

% Range_insert
thf(fact_391_refl__on__singleton,axiom,
    ! [X3: produc9003781588162623004on_val] : ( refl_o6679066329051764061on_val @ ( insert3030518898131756108on_val @ X3 @ bot_bo6722054719108133734on_val ) @ ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ X3 @ X3 ) @ bot_bo402267660617762917on_val ) ) ).

% refl_on_singleton
thf(fact_392_refl__on__singleton,axiom,
    ! [X3: set_Product_prod_o_o] : ( refl_o935183656833700870od_o_o @ ( insert3817420688681518039od_o_o @ X3 @ bot_bo8006819982598807891od_o_o ) @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ X3 @ X3 ) @ bot_bo498831808395348787od_o_o ) ) ).

% refl_on_singleton
thf(fact_393_refl__on__singleton,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] : ( refl_o6955724340487615712at_nat @ ( insert9200635055090092081at_nat @ X3 @ bot_bo3083307316010499117at_nat ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X3 @ X3 ) @ bot_bo228742789529271731at_nat ) ) ).

% refl_on_singleton
thf(fact_394_refl__onD2,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o,X3: produc2934264451710624999od_o_o,Y3: produc2934264451710624999od_o_o] :
      ( ( refl_o8365827894042769382od_o_o @ A3 @ R )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X3 @ Y3 ) @ R )
       => ( member444158400953824016od_o_o @ Y3 @ A3 ) ) ) ).

% refl_onD2
thf(fact_395_refl__onD2,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat,X3: produc732395585841259969at_nat,Y3: produc732395585841259969at_nat] :
      ( ( refl_o3078925143892389184at_nat @ A3 @ R )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X3 @ Y3 ) @ R )
       => ( member4095101504841534314at_nat @ Y3 @ A3 ) ) ) ).

% refl_onD2
thf(fact_396_refl__onD2,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,X3: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A3 @ R )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y3 ) @ R )
       => ( member8757157785044589968at_nat @ Y3 @ A3 ) ) ) ).

% refl_onD2
thf(fact_397_refl__onD2,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
      ( ( refl_o6679066329051764061on_val @ A3 @ R )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R )
       => ( member2879030624002546611on_val @ Y3 @ A3 ) ) ) ).

% refl_onD2
thf(fact_398_refl__onD2,axiom,
    ! [A3: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,X3: set_Product_prod_o_o,Y3: set_Product_prod_o_o] :
      ( ( refl_o935183656833700870od_o_o @ A3 @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R )
       => ( member3157015407008874800od_o_o @ Y3 @ A3 ) ) ) ).

% refl_onD2
thf(fact_399_refl__onD2,axiom,
    ! [A3: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A3 @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R )
       => ( member2643936169264416010at_nat @ Y3 @ A3 ) ) ) ).

% refl_onD2
thf(fact_400_refl__onD1,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o,X3: produc2934264451710624999od_o_o,Y3: produc2934264451710624999od_o_o] :
      ( ( refl_o8365827894042769382od_o_o @ A3 @ R )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X3 @ Y3 ) @ R )
       => ( member444158400953824016od_o_o @ X3 @ A3 ) ) ) ).

% refl_onD1
thf(fact_401_refl__onD1,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat,X3: produc732395585841259969at_nat,Y3: produc732395585841259969at_nat] :
      ( ( refl_o3078925143892389184at_nat @ A3 @ R )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X3 @ Y3 ) @ R )
       => ( member4095101504841534314at_nat @ X3 @ A3 ) ) ) ).

% refl_onD1
thf(fact_402_refl__onD1,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,X3: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A3 @ R )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y3 ) @ R )
       => ( member8757157785044589968at_nat @ X3 @ A3 ) ) ) ).

% refl_onD1
thf(fact_403_refl__onD1,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
      ( ( refl_o6679066329051764061on_val @ A3 @ R )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R )
       => ( member2879030624002546611on_val @ X3 @ A3 ) ) ) ).

% refl_onD1
thf(fact_404_refl__onD1,axiom,
    ! [A3: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,X3: set_Product_prod_o_o,Y3: set_Product_prod_o_o] :
      ( ( refl_o935183656833700870od_o_o @ A3 @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R )
       => ( member3157015407008874800od_o_o @ X3 @ A3 ) ) ) ).

% refl_onD1
thf(fact_405_refl__onD1,axiom,
    ! [A3: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A3 @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R )
       => ( member2643936169264416010at_nat @ X3 @ A3 ) ) ) ).

% refl_onD1
thf(fact_406_refl__onD,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o] :
      ( ( refl_o8365827894042769382od_o_o @ A3 @ R )
     => ( ( member444158400953824016od_o_o @ A @ A3 )
       => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_407_refl__onD,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat] :
      ( ( refl_o3078925143892389184at_nat @ A3 @ R )
     => ( ( member4095101504841534314at_nat @ A @ A3 )
       => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_408_refl__onD,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A3 @ R )
     => ( ( member8757157785044589968at_nat @ A @ A3 )
       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_409_refl__onD,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val] :
      ( ( refl_o6679066329051764061on_val @ A3 @ R )
     => ( ( member2879030624002546611on_val @ A @ A3 )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_410_refl__onD,axiom,
    ! [A3: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o] :
      ( ( refl_o935183656833700870od_o_o @ A3 @ R )
     => ( ( member3157015407008874800od_o_o @ A @ A3 )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_411_refl__onD,axiom,
    ! [A3: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A3 @ R )
     => ( ( member2643936169264416010at_nat @ A @ A3 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_412_Range__iff,axiom,
    ! [A: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val] :
      ( ( member1577473829074448177on_val @ A @ ( range_8542210083549373368on_val @ R ) )
      = ( ? [Y: produc1457211279475724562t_char > prat] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ Y @ A ) @ R ) ) ) ).

% Range_iff
thf(fact_413_Range__iff,axiom,
    ! [A: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member2879030624002546611on_val @ A @ ( range_6350497433844811695on_val @ R ) )
      = ( ? [Y: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y @ A ) @ R ) ) ) ).

% Range_iff
thf(fact_414_Range__iff,axiom,
    ! [A: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( range_1318165146934585909od_o_o @ R ) )
      = ( ? [Y: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ A ) @ R ) ) ) ).

% Range_iff
thf(fact_415_Range__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_8412904782085253007at_nat @ R ) )
      = ( ? [Y: set_Product_prod_o_o] : ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ Y @ A ) @ R ) ) ) ).

% Range_iff
thf(fact_416_Range__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_880292732526890933at_nat @ R ) )
      = ( ? [Y: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ A ) @ R ) ) ) ).

% Range_iff
thf(fact_417_RangeE,axiom,
    ! [B: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val] :
      ( ( member1577473829074448177on_val @ B @ ( range_8542210083549373368on_val @ R ) )
     => ~ ! [A2: produc1457211279475724562t_char > prat] :
            ~ ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A2 @ B ) @ R ) ) ).

% RangeE
thf(fact_418_RangeE,axiom,
    ! [B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member2879030624002546611on_val @ B @ ( range_6350497433844811695on_val @ R ) )
     => ~ ! [A2: produc9003781588162623004on_val] :
            ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A2 @ B ) @ R ) ) ).

% RangeE
thf(fact_419_RangeE,axiom,
    ! [B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ B @ ( range_1318165146934585909od_o_o @ R ) )
     => ~ ! [A2: set_Product_prod_o_o] :
            ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A2 @ B ) @ R ) ) ).

% RangeE
thf(fact_420_RangeE,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat] :
      ( ( member2643936169264416010at_nat @ B @ ( range_8412904782085253007at_nat @ R ) )
     => ~ ! [A2: set_Product_prod_o_o] :
            ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A2 @ B ) @ R ) ) ).

% RangeE
thf(fact_421_RangeE,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ B @ ( range_880292732526890933at_nat @ R ) )
     => ~ ! [A2: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B ) @ R ) ) ).

% RangeE
thf(fact_422_Range_Ointros,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ R )
     => ( member1577473829074448177on_val @ B @ ( range_8542210083549373368on_val @ R ) ) ) ).

% Range.intros
thf(fact_423_Range_Ointros,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
     => ( member2879030624002546611on_val @ B @ ( range_6350497433844811695on_val @ R ) ) ) ).

% Range.intros
thf(fact_424_Range_Ointros,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
     => ( member3157015407008874800od_o_o @ B @ ( range_1318165146934585909od_o_o @ R ) ) ) ).

% Range.intros
thf(fact_425_Range_Ointros,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R )
     => ( member2643936169264416010at_nat @ B @ ( range_8412904782085253007at_nat @ R ) ) ) ).

% Range.intros
thf(fact_426_Range_Ointros,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
     => ( member2643936169264416010at_nat @ B @ ( range_880292732526890933at_nat @ R ) ) ) ).

% Range.intros
thf(fact_427_Range_Osimps,axiom,
    ! [A: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val] :
      ( ( member1577473829074448177on_val @ A @ ( range_8542210083549373368on_val @ R ) )
      = ( ? [A5: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > option_val] :
            ( ( A = B4 )
            & ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A5 @ B4 ) @ R ) ) ) ) ).

% Range.simps
thf(fact_428_Range_Osimps,axiom,
    ! [A: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member2879030624002546611on_val @ A @ ( range_6350497433844811695on_val @ R ) )
      = ( ? [A5: produc9003781588162623004on_val,B4: produc9003781588162623004on_val] :
            ( ( A = B4 )
            & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A5 @ B4 ) @ R ) ) ) ) ).

% Range.simps
thf(fact_429_Range_Osimps,axiom,
    ! [A: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( range_1318165146934585909od_o_o @ R ) )
      = ( ? [A5: set_Product_prod_o_o,B4: set_Product_prod_o_o] :
            ( ( A = B4 )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A5 @ B4 ) @ R ) ) ) ) ).

% Range.simps
thf(fact_430_Range_Osimps,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_8412904782085253007at_nat @ R ) )
      = ( ? [A5: set_Product_prod_o_o,B4: set_Pr1261947904930325089at_nat] :
            ( ( A = B4 )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A5 @ B4 ) @ R ) ) ) ) ).

% Range.simps
thf(fact_431_Range_Osimps,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_880292732526890933at_nat @ R ) )
      = ( ? [A5: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
            ( ( A = B4 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A5 @ B4 ) @ R ) ) ) ) ).

% Range.simps
thf(fact_432_Range_Ocases,axiom,
    ! [A: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val] :
      ( ( member1577473829074448177on_val @ A @ ( range_8542210083549373368on_val @ R ) )
     => ~ ! [A2: produc1457211279475724562t_char > prat] :
            ~ ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A2 @ A ) @ R ) ) ).

% Range.cases
thf(fact_433_Range_Ocases,axiom,
    ! [A: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member2879030624002546611on_val @ A @ ( range_6350497433844811695on_val @ R ) )
     => ~ ! [A2: produc9003781588162623004on_val] :
            ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A2 @ A ) @ R ) ) ).

% Range.cases
thf(fact_434_Range_Ocases,axiom,
    ! [A: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( range_1318165146934585909od_o_o @ R ) )
     => ~ ! [A2: set_Product_prod_o_o] :
            ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A2 @ A ) @ R ) ) ).

% Range.cases
thf(fact_435_Range_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_8412904782085253007at_nat @ R ) )
     => ~ ! [A2: set_Product_prod_o_o] :
            ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A2 @ A ) @ R ) ) ).

% Range.cases
thf(fact_436_Range_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_880292732526890933at_nat @ R ) )
     => ~ ! [A2: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ A ) @ R ) ) ).

% Range.cases
thf(fact_437_refl__on__domain,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( refl_o8365827894042769382od_o_o @ A3 @ R )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R )
       => ( ( member444158400953824016od_o_o @ A @ A3 )
          & ( member444158400953824016od_o_o @ B @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_438_refl__on__domain,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( refl_o3078925143892389184at_nat @ A3 @ R )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R )
       => ( ( member4095101504841534314at_nat @ A @ A3 )
          & ( member4095101504841534314at_nat @ B @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_439_refl__on__domain,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A3 @ R )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R )
       => ( ( member8757157785044589968at_nat @ A @ A3 )
          & ( member8757157785044589968at_nat @ B @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_440_refl__on__domain,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( refl_o6679066329051764061on_val @ A3 @ R )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
       => ( ( member2879030624002546611on_val @ A @ A3 )
          & ( member2879030624002546611on_val @ B @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_441_refl__on__domain,axiom,
    ! [A3: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( refl_o935183656833700870od_o_o @ A3 @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
       => ( ( member3157015407008874800od_o_o @ A @ A3 )
          & ( member3157015407008874800od_o_o @ B @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_442_refl__on__domain,axiom,
    ! [A3: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A3 @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
       => ( ( member2643936169264416010at_nat @ A @ A3 )
          & ( member2643936169264416010at_nat @ B @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_443_linear__order__on__singleton,axiom,
    ! [X3: produc9003781588162623004on_val] : ( order_6061556925426544295on_val @ ( insert3030518898131756108on_val @ X3 @ bot_bo6722054719108133734on_val ) @ ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ X3 @ X3 ) @ bot_bo402267660617762917on_val ) ) ).

% linear_order_on_singleton
thf(fact_444_linear__order__on__singleton,axiom,
    ! [X3: set_Product_prod_o_o] : ( order_2625247082195685180od_o_o @ ( insert3817420688681518039od_o_o @ X3 @ bot_bo8006819982598807891od_o_o ) @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ X3 @ X3 ) @ bot_bo498831808395348787od_o_o ) ) ).

% linear_order_on_singleton
thf(fact_445_linear__order__on__singleton,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] : ( order_5942792088423767318at_nat @ ( insert9200635055090092081at_nat @ X3 @ bot_bo3083307316010499117at_nat ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X3 @ X3 ) @ bot_bo228742789529271731at_nat ) ) ).

% linear_order_on_singleton
thf(fact_446_Domain__insert,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val] :
      ( ( domain6100248181835736161on_val @ ( insert3030518898131756108on_val @ ( produc74620467203698198on_val @ A @ B ) @ R ) )
      = ( insert8113562578550770211r_prat @ A @ ( domain6100248181835736161on_val @ R ) ) ) ).

% Domain_insert
thf(fact_447_Domain__insert,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( domain6215983837468562630on_val @ ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R ) )
      = ( insert3030518898131756108on_val @ A @ ( domain6215983837468562630on_val @ R ) ) ) ).

% Domain_insert
thf(fact_448_Domain__insert,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( domain8908270702770587084od_o_o @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R ) )
      = ( insert3817420688681518039od_o_o @ A @ ( domain8908270702770587084od_o_o @ R ) ) ) ).

% Domain_insert
thf(fact_449_Domain__insert,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat] :
      ( ( domain8494901049294654374at_nat @ ( insert8109617542356073361at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R ) )
      = ( insert3817420688681518039od_o_o @ A @ ( domain8494901049294654374at_nat @ R ) ) ) ).

% Domain_insert
thf(fact_450_Domain__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( domain8802589944165416012at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R ) )
      = ( insert9200635055090092081at_nat @ A @ ( domain8802589944165416012at_nat @ R ) ) ) ).

% Domain_insert
thf(fact_451_Image__singleton__iff,axiom,
    ! [B: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val,A: produc1457211279475724562t_char > prat] :
      ( ( member1577473829074448177on_val @ B @ ( image_4658435527242194650on_val @ R @ ( insert8113562578550770211r_prat @ A @ bot_bo3164680191535668511r_prat ) ) )
      = ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ R ) ) ).

% Image_singleton_iff
thf(fact_452_Image__singleton__iff,axiom,
    ! [B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ B @ ( image_8782378046923076621on_val @ R @ ( insert3030518898131756108on_val @ A @ bot_bo6722054719108133734on_val ) ) )
      = ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R ) ) ).

% Image_singleton_iff
thf(fact_453_Image__singleton__iff,axiom,
    ! [B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o] :
      ( ( member3157015407008874800od_o_o @ B @ ( image_3256184292080368531od_o_o @ R @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) ) )
      = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R ) ) ).

% Image_singleton_iff
thf(fact_454_Image__singleton__iff,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat,A: set_Product_prod_o_o] :
      ( ( member2643936169264416010at_nat @ B @ ( image_2461788823225053677at_nat @ R @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) ) )
      = ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R ) ) ).

% Image_singleton_iff
thf(fact_455_Image__singleton__iff,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ B @ ( image_3892192845344445203at_nat @ R @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R ) ) ).

% Image_singleton_iff
thf(fact_456_map__add__upd__left,axiom,
    ! [M: produc2934264451710624999od_o_o,E2: produc2934264451710624999od_o_o > option_state,E1: produc2934264451710624999od_o_o > option_state,U1: state] :
      ( ~ ( member444158400953824016od_o_o @ M @ ( dom_Pr7503341592092307535_state @ E2 ) )
     => ( ( map_ad8644336254745229100_state @ ( fun_up4180510163728908852_state @ E1 @ M @ ( some_state @ U1 ) ) @ E2 )
        = ( fun_up4180510163728908852_state @ ( map_ad8644336254745229100_state @ E1 @ E2 ) @ M @ ( some_state @ U1 ) ) ) ) ).

% map_add_upd_left
thf(fact_457_map__add__upd__left,axiom,
    ! [M: produc732395585841259969at_nat,E2: produc732395585841259969at_nat > option_state,E1: produc732395585841259969at_nat > option_state,U1: state] :
      ( ~ ( member4095101504841534314at_nat @ M @ ( dom_Pr1362887639694971049_state @ E2 ) )
     => ( ( map_ad490551815367935238_state @ ( fun_up6026250480174712078_state @ E1 @ M @ ( some_state @ U1 ) ) @ E2 )
        = ( fun_up6026250480174712078_state @ ( map_ad490551815367935238_state @ E1 @ E2 ) @ M @ ( some_state @ U1 ) ) ) ) ).

% map_add_upd_left
thf(fact_458_map__add__upd__left,axiom,
    ! [M: produc3843707927480180839at_nat,E2: produc3843707927480180839at_nat > option_state,E1: produc3843707927480180839at_nat > option_state,U1: state] :
      ( ~ ( member8757157785044589968at_nat @ M @ ( dom_Pr2812315296966563535_state @ E2 ) )
     => ( ( map_ad6672435233716844716_state @ ( fun_up3329660993234963380_state @ E1 @ M @ ( some_state @ U1 ) ) @ E2 )
        = ( fun_up3329660993234963380_state @ ( map_ad6672435233716844716_state @ E1 @ E2 ) @ M @ ( some_state @ U1 ) ) ) ) ).

% map_add_upd_left
thf(fact_459_map__add__upd__left,axiom,
    ! [M: produc2934264451710624999od_o_o,E2: produc2934264451710624999od_o_o > option_val,E1: produc2934264451710624999od_o_o > option_val,U1: val] :
      ( ~ ( member444158400953824016od_o_o @ M @ ( dom_Pr6656689653091707692_o_val @ E2 ) )
     => ( ( map_ad2318226471356195983_o_val @ ( fun_up6453461605741198237on_val @ E1 @ M @ ( some_val @ U1 ) ) @ E2 )
        = ( fun_up6453461605741198237on_val @ ( map_ad2318226471356195983_o_val @ E1 @ E2 ) @ M @ ( some_val @ U1 ) ) ) ) ).

% map_add_upd_left
thf(fact_460_map__add__upd__left,axiom,
    ! [M: produc732395585841259969at_nat,E2: produc732395585841259969at_nat > option_val,E1: produc732395585841259969at_nat > option_val,U1: val] :
      ( ~ ( member4095101504841534314at_nat @ M @ ( dom_Pr6138649195375743826at_val @ E2 ) )
     => ( ( map_ad3299894791796522805at_val @ ( fun_up320048328433459523on_val @ E1 @ M @ ( some_val @ U1 ) ) @ E2 )
        = ( fun_up320048328433459523on_val @ ( map_ad3299894791796522805at_val @ E1 @ E2 ) @ M @ ( some_val @ U1 ) ) ) ) ).

% map_add_upd_left
thf(fact_461_map__add__upd__left,axiom,
    ! [M: produc3843707927480180839at_nat,E2: produc3843707927480180839at_nat > option_val,E1: produc3843707927480180839at_nat > option_val,U1: val] :
      ( ~ ( member8757157785044589968at_nat @ M @ ( dom_Pr2146307138176115884at_val @ E2 ) )
     => ( ( map_ad4314939265028211471at_val @ ( fun_up7479097124709437981on_val @ E1 @ M @ ( some_val @ U1 ) ) @ E2 )
        = ( fun_up7479097124709437981on_val @ ( map_ad4314939265028211471at_val @ E1 @ E2 ) @ M @ ( some_val @ U1 ) ) ) ) ).

% map_add_upd_left
thf(fact_462_map__add__upd__left,axiom,
    ! [M: produc1457211279475724562t_char,E2: produc1457211279475724562t_char > option_val,E1: produc1457211279475724562t_char > option_val,U1: val] :
      ( ~ ( member484235747325421115t_char @ M @ ( dom_Pr8989479155164304065ar_val @ E2 ) )
     => ( ( map_ad2908576761495157028ar_val @ ( fun_up5735975307950539954on_val @ E1 @ M @ ( some_val @ U1 ) ) @ E2 )
        = ( fun_up5735975307950539954on_val @ ( map_ad2908576761495157028ar_val @ E1 @ E2 ) @ M @ ( some_val @ U1 ) ) ) ) ).

% map_add_upd_left
thf(fact_463_ImageI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o,A3: set_se491199571785101031od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
     => ( ( member3157015407008874800od_o_o @ A @ A3 )
       => ( member3157015407008874800od_o_o @ B @ ( image_3256184292080368531od_o_o @ R @ A3 ) ) ) ) ).

% ImageI
thf(fact_464_ImageI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat,A3: set_se491199571785101031od_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R )
     => ( ( member3157015407008874800od_o_o @ A @ A3 )
       => ( member2643936169264416010at_nat @ B @ ( image_2461788823225053677at_nat @ R @ A3 ) ) ) ) ).

% ImageI
thf(fact_465_ImageI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
     => ( ( member2643936169264416010at_nat @ A @ A3 )
       => ( member2643936169264416010at_nat @ B @ ( image_3892192845344445203at_nat @ R @ A3 ) ) ) ) ).

% ImageI
thf(fact_466_ImageI,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val,A3: set_Pr3433046674199179443r_prat] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ R )
     => ( ( member2336342582916103932r_prat @ A @ A3 )
       => ( member1577473829074448177on_val @ B @ ( image_4658435527242194650on_val @ R @ A3 ) ) ) ) ).

% ImageI
thf(fact_467_ImageI,axiom,
    ! [A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o,R: set_Pr1013884327308616903od_o_o,A3: set_Pr1932065953672099015od_o_o] :
      ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R )
     => ( ( member444158400953824016od_o_o @ A @ A3 )
       => ( member444158400953824016od_o_o @ B @ ( image_1313870842500862867od_o_o @ R @ A3 ) ) ) ) ).

% ImageI
thf(fact_468_ImageI,axiom,
    ! [A: produc2934264451710624999od_o_o,B: produc732395585841259969at_nat,R: set_Pr6579314390632234657at_nat,A3: set_Pr1932065953672099015od_o_o] :
      ( ( member3940829983276759018at_nat @ ( produc5280317629174639665at_nat @ A @ B ) @ R )
     => ( ( member444158400953824016od_o_o @ A @ A3 )
       => ( member4095101504841534314at_nat @ B @ ( image_8289932575384915565at_nat @ R @ A3 ) ) ) ) ).

% ImageI
thf(fact_469_ImageI,axiom,
    ! [A: produc2934264451710624999od_o_o,B: produc3843707927480180839at_nat,R: set_Pr2596487125543962183at_nat,A3: set_Pr1932065953672099015od_o_o] :
      ( ( member7900332092847368336at_nat @ ( produc2313366179012357847at_nat @ A @ B ) @ R )
     => ( ( member444158400953824016od_o_o @ A @ A3 )
       => ( member8757157785044589968at_nat @ B @ ( image_8716920961790427923at_nat @ R @ A3 ) ) ) ) ).

% ImageI
thf(fact_470_ImageI,axiom,
    ! [A: produc732395585841259969at_nat,B: produc2934264451710624999od_o_o,R: set_Pr7090123087317550701od_o_o,A3: set_Pr457366540195662369at_nat] :
      ( ( member670178137689646518od_o_o @ ( produc7608918879087053565od_o_o @ A @ B ) @ R )
     => ( ( member4095101504841534314at_nat @ A @ A3 )
       => ( member444158400953824016od_o_o @ B @ ( image_1395161788442553657od_o_o @ R @ A3 ) ) ) ) ).

% ImageI
thf(fact_471_ImageI,axiom,
    ! [A: produc732395585841259969at_nat,B: produc732395585841259969at_nat,R: set_Pr4427867301352232263at_nat,A3: set_Pr457366540195662369at_nat] :
      ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R )
     => ( ( member4095101504841534314at_nat @ A @ A3 )
       => ( member4095101504841534314at_nat @ B @ ( image_1966035174115581203at_nat @ R @ A3 ) ) ) ) ).

% ImageI
thf(fact_472_ImageI,axiom,
    ! [A: produc732395585841259969at_nat,B: produc3843707927480180839at_nat,R: set_Pr6877924626643063277at_nat,A3: set_Pr457366540195662369at_nat] :
      ( ( member3999834080665787702at_nat @ ( produc1020012403072254077at_nat @ A @ B ) @ R )
     => ( ( member4095101504841534314at_nat @ A @ A3 )
       => ( member8757157785044589968at_nat @ B @ ( image_8447118000439789241at_nat @ R @ A3 ) ) ) ) ).

% ImageI
thf(fact_473_fun__upd__None__if__notin__dom,axiom,
    ! [K: produc2934264451710624999od_o_o,M: produc2934264451710624999od_o_o > option_val] :
      ( ~ ( member444158400953824016od_o_o @ K @ ( dom_Pr6656689653091707692_o_val @ M ) )
     => ( ( fun_up6453461605741198237on_val @ M @ K @ none_val )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_474_fun__upd__None__if__notin__dom,axiom,
    ! [K: produc732395585841259969at_nat,M: produc732395585841259969at_nat > option_val] :
      ( ~ ( member4095101504841534314at_nat @ K @ ( dom_Pr6138649195375743826at_val @ M ) )
     => ( ( fun_up320048328433459523on_val @ M @ K @ none_val )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_475_fun__upd__None__if__notin__dom,axiom,
    ! [K: produc3843707927480180839at_nat,M: produc3843707927480180839at_nat > option_val] :
      ( ~ ( member8757157785044589968at_nat @ K @ ( dom_Pr2146307138176115884at_val @ M ) )
     => ( ( fun_up7479097124709437981on_val @ M @ K @ none_val )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_476_fun__upd__None__if__notin__dom,axiom,
    ! [K: produc2934264451710624999od_o_o,M: produc2934264451710624999od_o_o > option_state] :
      ( ~ ( member444158400953824016od_o_o @ K @ ( dom_Pr7503341592092307535_state @ M ) )
     => ( ( fun_up4180510163728908852_state @ M @ K @ none_state )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_477_fun__upd__None__if__notin__dom,axiom,
    ! [K: produc732395585841259969at_nat,M: produc732395585841259969at_nat > option_state] :
      ( ~ ( member4095101504841534314at_nat @ K @ ( dom_Pr1362887639694971049_state @ M ) )
     => ( ( fun_up6026250480174712078_state @ M @ K @ none_state )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_478_fun__upd__None__if__notin__dom,axiom,
    ! [K: produc3843707927480180839at_nat,M: produc3843707927480180839at_nat > option_state] :
      ( ~ ( member8757157785044589968at_nat @ K @ ( dom_Pr2812315296966563535_state @ M ) )
     => ( ( fun_up3329660993234963380_state @ M @ K @ none_state )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_479_ImageE,axiom,
    ! [B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o,A3: set_se491199571785101031od_o_o] :
      ( ( member3157015407008874800od_o_o @ B @ ( image_3256184292080368531od_o_o @ R @ A3 ) )
     => ~ ! [X2: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ B ) @ R )
           => ~ ( member3157015407008874800od_o_o @ X2 @ A3 ) ) ) ).

% ImageE
thf(fact_480_ImageE,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat,A3: set_se491199571785101031od_o_o] :
      ( ( member2643936169264416010at_nat @ B @ ( image_2461788823225053677at_nat @ R @ A3 ) )
     => ~ ! [X2: set_Product_prod_o_o] :
            ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X2 @ B ) @ R )
           => ~ ( member3157015407008874800od_o_o @ X2 @ A3 ) ) ) ).

% ImageE
thf(fact_481_ImageE,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( member2643936169264416010at_nat @ B @ ( image_3892192845344445203at_nat @ R @ A3 ) )
     => ~ ! [X2: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ B ) @ R )
           => ~ ( member2643936169264416010at_nat @ X2 @ A3 ) ) ) ).

% ImageE
thf(fact_482_ImageE,axiom,
    ! [B: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val,A3: set_Pr3433046674199179443r_prat] :
      ( ( member1577473829074448177on_val @ B @ ( image_4658435527242194650on_val @ R @ A3 ) )
     => ~ ! [X2: produc1457211279475724562t_char > prat] :
            ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X2 @ B ) @ R )
           => ~ ( member2336342582916103932r_prat @ X2 @ A3 ) ) ) ).

% ImageE
thf(fact_483_ImageE,axiom,
    ! [B: produc2934264451710624999od_o_o,R: set_Pr1013884327308616903od_o_o,A3: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ B @ ( image_1313870842500862867od_o_o @ R @ A3 ) )
     => ~ ! [X2: produc2934264451710624999od_o_o] :
            ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X2 @ B ) @ R )
           => ~ ( member444158400953824016od_o_o @ X2 @ A3 ) ) ) ).

% ImageE
thf(fact_484_ImageE,axiom,
    ! [B: produc2934264451710624999od_o_o,R: set_Pr7090123087317550701od_o_o,A3: set_Pr457366540195662369at_nat] :
      ( ( member444158400953824016od_o_o @ B @ ( image_1395161788442553657od_o_o @ R @ A3 ) )
     => ~ ! [X2: produc732395585841259969at_nat] :
            ( ( member670178137689646518od_o_o @ ( produc7608918879087053565od_o_o @ X2 @ B ) @ R )
           => ~ ( member4095101504841534314at_nat @ X2 @ A3 ) ) ) ).

% ImageE
thf(fact_485_ImageE,axiom,
    ! [B: produc2934264451710624999od_o_o,R: set_Pr4084129990900558663od_o_o,A3: set_Pr4329608150637261639at_nat] :
      ( ( member444158400953824016od_o_o @ B @ ( image_6337422021275444243od_o_o @ R @ A3 ) )
     => ~ ! [X2: produc3843707927480180839at_nat] :
            ( ( member8203642551620930960od_o_o @ ( produc9157239275352149975od_o_o @ X2 @ B ) @ R )
           => ~ ( member8757157785044589968at_nat @ X2 @ A3 ) ) ) ).

% ImageE
thf(fact_486_ImageE,axiom,
    ! [B: produc732395585841259969at_nat,R: set_Pr6579314390632234657at_nat,A3: set_Pr1932065953672099015od_o_o] :
      ( ( member4095101504841534314at_nat @ B @ ( image_8289932575384915565at_nat @ R @ A3 ) )
     => ~ ! [X2: produc2934264451710624999od_o_o] :
            ( ( member3940829983276759018at_nat @ ( produc5280317629174639665at_nat @ X2 @ B ) @ R )
           => ~ ( member444158400953824016od_o_o @ X2 @ A3 ) ) ) ).

% ImageE
thf(fact_487_ImageE,axiom,
    ! [B: produc732395585841259969at_nat,R: set_Pr4427867301352232263at_nat,A3: set_Pr457366540195662369at_nat] :
      ( ( member4095101504841534314at_nat @ B @ ( image_1966035174115581203at_nat @ R @ A3 ) )
     => ~ ! [X2: produc732395585841259969at_nat] :
            ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X2 @ B ) @ R )
           => ~ ( member4095101504841534314at_nat @ X2 @ A3 ) ) ) ).

% ImageE
thf(fact_488_ImageE,axiom,
    ! [B: produc732395585841259969at_nat,R: set_Pr4797530719494571297at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( member4095101504841534314at_nat @ B @ ( image_150793935887408877at_nat @ R @ A3 ) )
     => ~ ! [X2: produc3843707927480180839at_nat] :
            ( ( member3985572510901480042at_nat @ ( produc1947060375374649521at_nat @ X2 @ B ) @ R )
           => ~ ( member8757157785044589968at_nat @ X2 @ A3 ) ) ) ).

% ImageE
thf(fact_489_Image__iff,axiom,
    ! [B: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val,A3: set_Pr3433046674199179443r_prat] :
      ( ( member1577473829074448177on_val @ B @ ( image_4658435527242194650on_val @ R @ A3 ) )
      = ( ? [X: produc1457211279475724562t_char > prat] :
            ( ( member2336342582916103932r_prat @ X @ A3 )
            & ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X @ B ) @ R ) ) ) ) ).

% Image_iff
thf(fact_490_Image__iff,axiom,
    ! [B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val,A3: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ B @ ( image_8782378046923076621on_val @ R @ A3 ) )
      = ( ? [X: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ X @ A3 )
            & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ B ) @ R ) ) ) ) ).

% Image_iff
thf(fact_491_Image__iff,axiom,
    ! [B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o,A3: set_se491199571785101031od_o_o] :
      ( ( member3157015407008874800od_o_o @ B @ ( image_3256184292080368531od_o_o @ R @ A3 ) )
      = ( ? [X: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X @ A3 )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ B ) @ R ) ) ) ) ).

% Image_iff
thf(fact_492_Image__iff,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat,A3: set_se491199571785101031od_o_o] :
      ( ( member2643936169264416010at_nat @ B @ ( image_2461788823225053677at_nat @ R @ A3 ) )
      = ( ? [X: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X @ A3 )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X @ B ) @ R ) ) ) ) ).

% Image_iff
thf(fact_493_Image__iff,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( member2643936169264416010at_nat @ B @ ( image_3892192845344445203at_nat @ R @ A3 ) )
      = ( ? [X: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X @ A3 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ B ) @ R ) ) ) ) ).

% Image_iff
thf(fact_494_rev__ImageI,axiom,
    ! [A: set_Product_prod_o_o,A3: set_se491199571785101031od_o_o,B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ A3 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
       => ( member3157015407008874800od_o_o @ B @ ( image_3256184292080368531od_o_o @ R @ A3 ) ) ) ) ).

% rev_ImageI
thf(fact_495_rev__ImageI,axiom,
    ! [A: set_Product_prod_o_o,A3: set_se491199571785101031od_o_o,B: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat] :
      ( ( member3157015407008874800od_o_o @ A @ A3 )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R )
       => ( member2643936169264416010at_nat @ B @ ( image_2461788823225053677at_nat @ R @ A3 ) ) ) ) ).

% rev_ImageI
thf(fact_496_rev__ImageI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ A3 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
       => ( member2643936169264416010at_nat @ B @ ( image_3892192845344445203at_nat @ R @ A3 ) ) ) ) ).

% rev_ImageI
thf(fact_497_rev__ImageI,axiom,
    ! [A: produc1457211279475724562t_char > prat,A3: set_Pr3433046674199179443r_prat,B: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val] :
      ( ( member2336342582916103932r_prat @ A @ A3 )
     => ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ R )
       => ( member1577473829074448177on_val @ B @ ( image_4658435527242194650on_val @ R @ A3 ) ) ) ) ).

% rev_ImageI
thf(fact_498_rev__ImageI,axiom,
    ! [A: produc2934264451710624999od_o_o,A3: set_Pr1932065953672099015od_o_o,B: produc2934264451710624999od_o_o,R: set_Pr1013884327308616903od_o_o] :
      ( ( member444158400953824016od_o_o @ A @ A3 )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R )
       => ( member444158400953824016od_o_o @ B @ ( image_1313870842500862867od_o_o @ R @ A3 ) ) ) ) ).

% rev_ImageI
thf(fact_499_rev__ImageI,axiom,
    ! [A: produc2934264451710624999od_o_o,A3: set_Pr1932065953672099015od_o_o,B: produc732395585841259969at_nat,R: set_Pr6579314390632234657at_nat] :
      ( ( member444158400953824016od_o_o @ A @ A3 )
     => ( ( member3940829983276759018at_nat @ ( produc5280317629174639665at_nat @ A @ B ) @ R )
       => ( member4095101504841534314at_nat @ B @ ( image_8289932575384915565at_nat @ R @ A3 ) ) ) ) ).

% rev_ImageI
thf(fact_500_rev__ImageI,axiom,
    ! [A: produc2934264451710624999od_o_o,A3: set_Pr1932065953672099015od_o_o,B: produc3843707927480180839at_nat,R: set_Pr2596487125543962183at_nat] :
      ( ( member444158400953824016od_o_o @ A @ A3 )
     => ( ( member7900332092847368336at_nat @ ( produc2313366179012357847at_nat @ A @ B ) @ R )
       => ( member8757157785044589968at_nat @ B @ ( image_8716920961790427923at_nat @ R @ A3 ) ) ) ) ).

% rev_ImageI
thf(fact_501_rev__ImageI,axiom,
    ! [A: produc732395585841259969at_nat,A3: set_Pr457366540195662369at_nat,B: produc2934264451710624999od_o_o,R: set_Pr7090123087317550701od_o_o] :
      ( ( member4095101504841534314at_nat @ A @ A3 )
     => ( ( member670178137689646518od_o_o @ ( produc7608918879087053565od_o_o @ A @ B ) @ R )
       => ( member444158400953824016od_o_o @ B @ ( image_1395161788442553657od_o_o @ R @ A3 ) ) ) ) ).

% rev_ImageI
thf(fact_502_rev__ImageI,axiom,
    ! [A: produc732395585841259969at_nat,A3: set_Pr457366540195662369at_nat,B: produc732395585841259969at_nat,R: set_Pr4427867301352232263at_nat] :
      ( ( member4095101504841534314at_nat @ A @ A3 )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R )
       => ( member4095101504841534314at_nat @ B @ ( image_1966035174115581203at_nat @ R @ A3 ) ) ) ) ).

% rev_ImageI
thf(fact_503_rev__ImageI,axiom,
    ! [A: produc732395585841259969at_nat,A3: set_Pr457366540195662369at_nat,B: produc3843707927480180839at_nat,R: set_Pr6877924626643063277at_nat] :
      ( ( member4095101504841534314at_nat @ A @ A3 )
     => ( ( member3999834080665787702at_nat @ ( produc1020012403072254077at_nat @ A @ B ) @ R )
       => ( member8757157785044589968at_nat @ B @ ( image_8447118000439789241at_nat @ R @ A3 ) ) ) ) ).

% rev_ImageI
thf(fact_504_domI,axiom,
    ! [M: produc2934264451710624999od_o_o > option_state,A: produc2934264451710624999od_o_o,B: state] :
      ( ( ( M @ A )
        = ( some_state @ B ) )
     => ( member444158400953824016od_o_o @ A @ ( dom_Pr7503341592092307535_state @ M ) ) ) ).

% domI
thf(fact_505_domI,axiom,
    ! [M: produc732395585841259969at_nat > option_state,A: produc732395585841259969at_nat,B: state] :
      ( ( ( M @ A )
        = ( some_state @ B ) )
     => ( member4095101504841534314at_nat @ A @ ( dom_Pr1362887639694971049_state @ M ) ) ) ).

% domI
thf(fact_506_domI,axiom,
    ! [M: produc3843707927480180839at_nat > option_state,A: produc3843707927480180839at_nat,B: state] :
      ( ( ( M @ A )
        = ( some_state @ B ) )
     => ( member8757157785044589968at_nat @ A @ ( dom_Pr2812315296966563535_state @ M ) ) ) ).

% domI
thf(fact_507_domI,axiom,
    ! [M: produc2934264451710624999od_o_o > option_val,A: produc2934264451710624999od_o_o,B: val] :
      ( ( ( M @ A )
        = ( some_val @ B ) )
     => ( member444158400953824016od_o_o @ A @ ( dom_Pr6656689653091707692_o_val @ M ) ) ) ).

% domI
thf(fact_508_domI,axiom,
    ! [M: produc732395585841259969at_nat > option_val,A: produc732395585841259969at_nat,B: val] :
      ( ( ( M @ A )
        = ( some_val @ B ) )
     => ( member4095101504841534314at_nat @ A @ ( dom_Pr6138649195375743826at_val @ M ) ) ) ).

% domI
thf(fact_509_domI,axiom,
    ! [M: produc3843707927480180839at_nat > option_val,A: produc3843707927480180839at_nat,B: val] :
      ( ( ( M @ A )
        = ( some_val @ B ) )
     => ( member8757157785044589968at_nat @ A @ ( dom_Pr2146307138176115884at_val @ M ) ) ) ).

% domI
thf(fact_510_domD,axiom,
    ! [A: produc2934264451710624999od_o_o,M: produc2934264451710624999od_o_o > option_state] :
      ( ( member444158400953824016od_o_o @ A @ ( dom_Pr7503341592092307535_state @ M ) )
     => ? [B2: state] :
          ( ( M @ A )
          = ( some_state @ B2 ) ) ) ).

% domD
thf(fact_511_domD,axiom,
    ! [A: produc732395585841259969at_nat,M: produc732395585841259969at_nat > option_state] :
      ( ( member4095101504841534314at_nat @ A @ ( dom_Pr1362887639694971049_state @ M ) )
     => ? [B2: state] :
          ( ( M @ A )
          = ( some_state @ B2 ) ) ) ).

% domD
thf(fact_512_domD,axiom,
    ! [A: produc3843707927480180839at_nat,M: produc3843707927480180839at_nat > option_state] :
      ( ( member8757157785044589968at_nat @ A @ ( dom_Pr2812315296966563535_state @ M ) )
     => ? [B2: state] :
          ( ( M @ A )
          = ( some_state @ B2 ) ) ) ).

% domD
thf(fact_513_domD,axiom,
    ! [A: produc2934264451710624999od_o_o,M: produc2934264451710624999od_o_o > option_val] :
      ( ( member444158400953824016od_o_o @ A @ ( dom_Pr6656689653091707692_o_val @ M ) )
     => ? [B2: val] :
          ( ( M @ A )
          = ( some_val @ B2 ) ) ) ).

% domD
thf(fact_514_domD,axiom,
    ! [A: produc732395585841259969at_nat,M: produc732395585841259969at_nat > option_val] :
      ( ( member4095101504841534314at_nat @ A @ ( dom_Pr6138649195375743826at_val @ M ) )
     => ? [B2: val] :
          ( ( M @ A )
          = ( some_val @ B2 ) ) ) ).

% domD
thf(fact_515_domD,axiom,
    ! [A: produc3843707927480180839at_nat,M: produc3843707927480180839at_nat > option_val] :
      ( ( member8757157785044589968at_nat @ A @ ( dom_Pr2146307138176115884at_val @ M ) )
     => ? [B2: val] :
          ( ( M @ A )
          = ( some_val @ B2 ) ) ) ).

% domD
thf(fact_516_domIff,axiom,
    ! [A: produc2934264451710624999od_o_o,M: produc2934264451710624999od_o_o > option_val] :
      ( ( member444158400953824016od_o_o @ A @ ( dom_Pr6656689653091707692_o_val @ M ) )
      = ( ( M @ A )
       != none_val ) ) ).

% domIff
thf(fact_517_domIff,axiom,
    ! [A: produc732395585841259969at_nat,M: produc732395585841259969at_nat > option_val] :
      ( ( member4095101504841534314at_nat @ A @ ( dom_Pr6138649195375743826at_val @ M ) )
      = ( ( M @ A )
       != none_val ) ) ).

% domIff
thf(fact_518_domIff,axiom,
    ! [A: produc3843707927480180839at_nat,M: produc3843707927480180839at_nat > option_val] :
      ( ( member8757157785044589968at_nat @ A @ ( dom_Pr2146307138176115884at_val @ M ) )
      = ( ( M @ A )
       != none_val ) ) ).

% domIff
thf(fact_519_domIff,axiom,
    ! [A: produc2934264451710624999od_o_o,M: produc2934264451710624999od_o_o > option_state] :
      ( ( member444158400953824016od_o_o @ A @ ( dom_Pr7503341592092307535_state @ M ) )
      = ( ( M @ A )
       != none_state ) ) ).

% domIff
thf(fact_520_domIff,axiom,
    ! [A: produc732395585841259969at_nat,M: produc732395585841259969at_nat > option_state] :
      ( ( member4095101504841534314at_nat @ A @ ( dom_Pr1362887639694971049_state @ M ) )
      = ( ( M @ A )
       != none_state ) ) ).

% domIff
thf(fact_521_domIff,axiom,
    ! [A: produc3843707927480180839at_nat,M: produc3843707927480180839at_nat > option_state] :
      ( ( member8757157785044589968at_nat @ A @ ( dom_Pr2812315296966563535_state @ M ) )
      = ( ( M @ A )
       != none_state ) ) ).

% domIff
thf(fact_522_map__add__dom__app__simps_I1_J,axiom,
    ! [M: produc1457211279475724562t_char,L2: produc1457211279475724562t_char > option_val,L1: produc1457211279475724562t_char > option_val] :
      ( ( member484235747325421115t_char @ M @ ( dom_Pr8989479155164304065ar_val @ L2 ) )
     => ( ( map_ad2908576761495157028ar_val @ L1 @ L2 @ M )
        = ( L2 @ M ) ) ) ).

% map_add_dom_app_simps(1)
thf(fact_523_map__add__dom__app__simps_I2_J,axiom,
    ! [M: produc1457211279475724562t_char,L1: produc1457211279475724562t_char > option_val,L2: produc1457211279475724562t_char > option_val] :
      ( ~ ( member484235747325421115t_char @ M @ ( dom_Pr8989479155164304065ar_val @ L1 ) )
     => ( ( map_ad2908576761495157028ar_val @ L1 @ L2 @ M )
        = ( L2 @ M ) ) ) ).

% map_add_dom_app_simps(2)
thf(fact_524_map__add__dom__app__simps_I3_J,axiom,
    ! [M: produc1457211279475724562t_char,L2: produc1457211279475724562t_char > option_val,L1: produc1457211279475724562t_char > option_val] :
      ( ~ ( member484235747325421115t_char @ M @ ( dom_Pr8989479155164304065ar_val @ L2 ) )
     => ( ( map_ad2908576761495157028ar_val @ L1 @ L2 @ M )
        = ( L1 @ M ) ) ) ).

% map_add_dom_app_simps(3)
thf(fact_525_Domain_Ocases,axiom,
    ! [A: produc1457211279475724562t_char > prat,R: set_Pr7649655576376376530on_val] :
      ( ( member2336342582916103932r_prat @ A @ ( domain6100248181835736161on_val @ R ) )
     => ~ ! [B2: produc1457211279475724562t_char > option_val] :
            ~ ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B2 ) @ R ) ) ).

% Domain.cases
thf(fact_526_Domain_Ocases,axiom,
    ! [A: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member2879030624002546611on_val @ A @ ( domain6215983837468562630on_val @ R ) )
     => ~ ! [B2: produc9003781588162623004on_val] :
            ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B2 ) @ R ) ) ).

% Domain.cases
thf(fact_527_Domain_Ocases,axiom,
    ! [A: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8908270702770587084od_o_o @ R ) )
     => ~ ! [B2: set_Product_prod_o_o] :
            ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B2 ) @ R ) ) ).

% Domain.cases
thf(fact_528_Domain_Ocases,axiom,
    ! [A: set_Product_prod_o_o,R: set_Pr457366540195662369at_nat] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8494901049294654374at_nat @ R ) )
     => ~ ! [B2: set_Pr1261947904930325089at_nat] :
            ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B2 ) @ R ) ) ).

% Domain.cases
thf(fact_529_Domain_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R ) )
     => ~ ! [B2: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ R ) ) ).

% Domain.cases
thf(fact_530_Domain_Osimps,axiom,
    ! [A: produc1457211279475724562t_char > prat,R: set_Pr7649655576376376530on_val] :
      ( ( member2336342582916103932r_prat @ A @ ( domain6100248181835736161on_val @ R ) )
      = ( ? [A5: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > option_val] :
            ( ( A = A5 )
            & ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A5 @ B4 ) @ R ) ) ) ) ).

% Domain.simps
thf(fact_531_Domain_Osimps,axiom,
    ! [A: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member2879030624002546611on_val @ A @ ( domain6215983837468562630on_val @ R ) )
      = ( ? [A5: produc9003781588162623004on_val,B4: produc9003781588162623004on_val] :
            ( ( A = A5 )
            & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A5 @ B4 ) @ R ) ) ) ) ).

% Domain.simps
thf(fact_532_Domain_Osimps,axiom,
    ! [A: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8908270702770587084od_o_o @ R ) )
      = ( ? [A5: set_Product_prod_o_o,B4: set_Product_prod_o_o] :
            ( ( A = A5 )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A5 @ B4 ) @ R ) ) ) ) ).

% Domain.simps
thf(fact_533_Domain_Osimps,axiom,
    ! [A: set_Product_prod_o_o,R: set_Pr457366540195662369at_nat] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8494901049294654374at_nat @ R ) )
      = ( ? [A5: set_Product_prod_o_o,B4: set_Pr1261947904930325089at_nat] :
            ( ( A = A5 )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A5 @ B4 ) @ R ) ) ) ) ).

% Domain.simps
thf(fact_534_Domain_Osimps,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R ) )
      = ( ? [A5: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
            ( ( A = A5 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A5 @ B4 ) @ R ) ) ) ) ).

% Domain.simps
thf(fact_535_Domain_ODomainI,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ R )
     => ( member2336342582916103932r_prat @ A @ ( domain6100248181835736161on_val @ R ) ) ) ).

% Domain.DomainI
thf(fact_536_Domain_ODomainI,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
     => ( member2879030624002546611on_val @ A @ ( domain6215983837468562630on_val @ R ) ) ) ).

% Domain.DomainI
thf(fact_537_Domain_ODomainI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
     => ( member3157015407008874800od_o_o @ A @ ( domain8908270702770587084od_o_o @ R ) ) ) ).

% Domain.DomainI
thf(fact_538_Domain_ODomainI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R: set_Pr457366540195662369at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R )
     => ( member3157015407008874800od_o_o @ A @ ( domain8494901049294654374at_nat @ R ) ) ) ).

% Domain.DomainI
thf(fact_539_Domain_ODomainI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
     => ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R ) ) ) ).

% Domain.DomainI
thf(fact_540_DomainE,axiom,
    ! [A: produc1457211279475724562t_char > prat,R: set_Pr7649655576376376530on_val] :
      ( ( member2336342582916103932r_prat @ A @ ( domain6100248181835736161on_val @ R ) )
     => ~ ! [B2: produc1457211279475724562t_char > option_val] :
            ~ ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B2 ) @ R ) ) ).

% DomainE
thf(fact_541_DomainE,axiom,
    ! [A: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member2879030624002546611on_val @ A @ ( domain6215983837468562630on_val @ R ) )
     => ~ ! [B2: produc9003781588162623004on_val] :
            ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B2 ) @ R ) ) ).

% DomainE
thf(fact_542_DomainE,axiom,
    ! [A: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8908270702770587084od_o_o @ R ) )
     => ~ ! [B2: set_Product_prod_o_o] :
            ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B2 ) @ R ) ) ).

% DomainE
thf(fact_543_DomainE,axiom,
    ! [A: set_Product_prod_o_o,R: set_Pr457366540195662369at_nat] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8494901049294654374at_nat @ R ) )
     => ~ ! [B2: set_Pr1261947904930325089at_nat] :
            ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B2 ) @ R ) ) ).

% DomainE
thf(fact_544_DomainE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R ) )
     => ~ ! [B2: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ R ) ) ).

% DomainE
thf(fact_545_Domain__iff,axiom,
    ! [A: produc1457211279475724562t_char > prat,R: set_Pr7649655576376376530on_val] :
      ( ( member2336342582916103932r_prat @ A @ ( domain6100248181835736161on_val @ R ) )
      = ( ? [Y: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ Y ) @ R ) ) ) ).

% Domain_iff
thf(fact_546_Domain__iff,axiom,
    ! [A: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member2879030624002546611on_val @ A @ ( domain6215983837468562630on_val @ R ) )
      = ( ? [Y: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ Y ) @ R ) ) ) ).

% Domain_iff
thf(fact_547_Domain__iff,axiom,
    ! [A: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8908270702770587084od_o_o @ R ) )
      = ( ? [Y: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ Y ) @ R ) ) ) ).

% Domain_iff
thf(fact_548_Domain__iff,axiom,
    ! [A: set_Product_prod_o_o,R: set_Pr457366540195662369at_nat] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8494901049294654374at_nat @ R ) )
      = ( ? [Y: set_Pr1261947904930325089at_nat] : ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ Y ) @ R ) ) ) ).

% Domain_iff
thf(fact_549_Domain__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R ) )
      = ( ? [Y: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ Y ) @ R ) ) ) ).

% Domain_iff
thf(fact_550_map__le__iff__map__add__commute,axiom,
    ! [F: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val] :
      ( ( map_le3432596473071271956ar_val @ F @ ( map_ad2908576761495157028ar_val @ F @ G ) )
      = ( ( map_ad2908576761495157028ar_val @ F @ G )
        = ( map_ad2908576761495157028ar_val @ G @ F ) ) ) ).

% map_le_iff_map_add_commute
thf(fact_551_map__add__subsumed2,axiom,
    ! [F: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val] :
      ( ( map_le3432596473071271956ar_val @ F @ G )
     => ( ( map_ad2908576761495157028ar_val @ G @ F )
        = G ) ) ).

% map_add_subsumed2
thf(fact_552_map__add__subsumed1,axiom,
    ! [F: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val] :
      ( ( map_le3432596473071271956ar_val @ F @ G )
     => ( ( map_ad2908576761495157028ar_val @ F @ G )
        = G ) ) ).

% map_add_subsumed1
thf(fact_553_map__add__le__mapI,axiom,
    ! [F: produc1457211279475724562t_char > option_val,H: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val] :
      ( ( map_le3432596473071271956ar_val @ F @ H )
     => ( ( map_le3432596473071271956ar_val @ G @ H )
       => ( map_le3432596473071271956ar_val @ ( map_ad2908576761495157028ar_val @ F @ G ) @ H ) ) ) ).

% map_add_le_mapI
thf(fact_554_map__add__le__mapE,axiom,
    ! [F: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val,H: produc1457211279475724562t_char > option_val] :
      ( ( map_le3432596473071271956ar_val @ ( map_ad2908576761495157028ar_val @ F @ G ) @ H )
     => ( map_le3432596473071271956ar_val @ G @ H ) ) ).

% map_add_le_mapE
thf(fact_555_map__le__map__add,axiom,
    ! [F: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val] : ( map_le3432596473071271956ar_val @ F @ ( map_ad2908576761495157028ar_val @ G @ F ) ) ).

% map_le_map_add
thf(fact_556_map__add__comm,axiom,
    ! [M1: produc1457211279475724562t_char > option_val,M2: produc1457211279475724562t_char > option_val] :
      ( ( ( inf_in3544766959038609504t_char @ ( dom_Pr8989479155164304065ar_val @ M1 ) @ ( dom_Pr8989479155164304065ar_val @ M2 ) )
        = bot_bo1414094081888755934t_char )
     => ( ( map_ad2908576761495157028ar_val @ M1 @ M2 )
        = ( map_ad2908576761495157028ar_val @ M2 @ M1 ) ) ) ).

% map_add_comm
thf(fact_557_prod__set__simps_I1_J,axiom,
    ! [X3: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] :
      ( ( basic_7826031515394172198on_val @ ( produc74620467203698198on_val @ X3 @ Y3 ) )
      = ( insert8113562578550770211r_prat @ X3 @ bot_bo3164680191535668511r_prat ) ) ).

% prod_set_simps(1)
thf(fact_558_prod__set__simps_I1_J,axiom,
    ! [X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
      ( ( basic_2381348828333783745on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) )
      = ( insert3030518898131756108on_val @ X3 @ bot_bo6722054719108133734on_val ) ) ).

% prod_set_simps(1)
thf(fact_559_prod__set__simps_I1_J,axiom,
    ! [X3: set_Product_prod_o_o,Y3: set_Product_prod_o_o] :
      ( ( basic_3307925540671692871od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) )
      = ( insert3817420688681518039od_o_o @ X3 @ bot_bo8006819982598807891od_o_o ) ) ).

% prod_set_simps(1)
thf(fact_560_prod__set__simps_I1_J,axiom,
    ! [X3: set_Product_prod_o_o,Y3: set_Pr1261947904930325089at_nat] :
      ( ( basic_3533134110227475617at_nat @ ( produc8517790099723286449at_nat @ X3 @ Y3 ) )
      = ( insert3817420688681518039od_o_o @ X3 @ bot_bo8006819982598807891od_o_o ) ) ).

% prod_set_simps(1)
thf(fact_561_prod__set__simps_I1_J,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( ( basic_3661821009137050567at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) )
      = ( insert9200635055090092081at_nat @ X3 @ bot_bo3083307316010499117at_nat ) ) ).

% prod_set_simps(1)
thf(fact_562_prod__set__simps_I2_J,axiom,
    ! [X3: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] :
      ( ( basic_8841519627501779048on_val @ ( produc74620467203698198on_val @ X3 @ Y3 ) )
      = ( insert216038912599248202on_val @ Y3 @ bot_bo5671083314777306084on_val ) ) ).

% prod_set_simps(2)
thf(fact_563_prod__set__simps_I2_J,axiom,
    ! [X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
      ( ( basic_2713455076908730111on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) )
      = ( insert3030518898131756108on_val @ Y3 @ bot_bo6722054719108133734on_val ) ) ).

% prod_set_simps(2)
thf(fact_564_prod__set__simps_I2_J,axiom,
    ! [X3: set_Product_prod_o_o,Y3: set_Product_prod_o_o] :
      ( ( basic_2174982398181630341od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) )
      = ( insert3817420688681518039od_o_o @ Y3 @ bot_bo8006819982598807891od_o_o ) ) ).

% prod_set_simps(2)
thf(fact_565_prod__set__simps_I2_J,axiom,
    ! [X3: set_Product_prod_o_o,Y3: set_Pr1261947904930325089at_nat] :
      ( ( basic_7607750056555722463at_nat @ ( produc8517790099723286449at_nat @ X3 @ Y3 ) )
      = ( insert9200635055090092081at_nat @ Y3 @ bot_bo3083307316010499117at_nat ) ) ).

% prod_set_simps(2)
thf(fact_566_prod__set__simps_I2_J,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( ( basic_484852565060450053at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) )
      = ( insert9200635055090092081at_nat @ Y3 @ bot_bo3083307316010499117at_nat ) ) ).

% prod_set_simps(2)
thf(fact_567_eq__equiv__class,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o,A3: set_Pr1932065953672099015od_o_o] :
      ( ( ( image_1313870842500862867od_o_o @ R @ ( insert3855699742481506999od_o_o @ A @ bot_bo498831808395348787od_o_o ) )
        = ( image_1313870842500862867od_o_o @ R @ ( insert3855699742481506999od_o_o @ B @ bot_bo498831808395348787od_o_o ) ) )
     => ( ( equiv_3398664326906728093od_o_o @ A3 @ R )
       => ( ( member444158400953824016od_o_o @ B @ A3 )
         => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R ) ) ) ) ).

% eq_equiv_class
thf(fact_568_eq__equiv__class,axiom,
    ! [R: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat,A3: set_Pr457366540195662369at_nat] :
      ( ( ( image_1966035174115581203at_nat @ R @ ( insert8109617542356073361at_nat @ A @ bot_bo1676454896636820109at_nat ) )
        = ( image_1966035174115581203at_nat @ R @ ( insert8109617542356073361at_nat @ B @ bot_bo1676454896636820109at_nat ) ) )
     => ( ( equiv_2520666332425567351at_nat @ A3 @ R )
       => ( ( member4095101504841534314at_nat @ B @ A3 )
         => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R ) ) ) ) ).

% eq_equiv_class
thf(fact_569_eq__equiv__class,axiom,
    ! [R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( ( image_3736809421353964435at_nat @ R @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
        = ( image_3736809421353964435at_nat @ R @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) )
     => ( ( equiv_4443766944827089949at_nat @ A3 @ R )
       => ( ( member8757157785044589968at_nat @ B @ A3 )
         => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R ) ) ) ) ).

% eq_equiv_class
thf(fact_570_eq__equiv__class,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,A3: set_Pr7649655576376376530on_val] :
      ( ( ( image_8782378046923076621on_val @ R @ ( insert3030518898131756108on_val @ A @ bot_bo6722054719108133734on_val ) )
        = ( image_8782378046923076621on_val @ R @ ( insert3030518898131756108on_val @ B @ bot_bo6722054719108133734on_val ) ) )
     => ( ( equiv_7491905333995573990on_val @ A3 @ R )
       => ( ( member2879030624002546611on_val @ B @ A3 )
         => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R ) ) ) ) ).

% eq_equiv_class
thf(fact_571_eq__equiv__class,axiom,
    ! [R: 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 @ R @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) )
        = ( image_3256184292080368531od_o_o @ R @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) )
     => ( ( equiv_4332955795722587581od_o_o @ A3 @ R )
       => ( ( member3157015407008874800od_o_o @ B @ A3 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R ) ) ) ) ).

% eq_equiv_class
thf(fact_572_eq__equiv__class,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( ( image_3892192845344445203at_nat @ R @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) )
        = ( image_3892192845344445203at_nat @ R @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) )
     => ( ( equiv_6458487239543531287at_nat @ A3 @ R )
       => ( ( member2643936169264416010at_nat @ B @ A3 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R ) ) ) ) ).

% eq_equiv_class
thf(fact_573_equiv__class__eq,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( equiv_7491905333995573990on_val @ A3 @ R )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
       => ( ( image_8782378046923076621on_val @ R @ ( insert3030518898131756108on_val @ A @ bot_bo6722054719108133734on_val ) )
          = ( image_8782378046923076621on_val @ R @ ( insert3030518898131756108on_val @ B @ bot_bo6722054719108133734on_val ) ) ) ) ) ).

% equiv_class_eq
thf(fact_574_equiv__class__eq,axiom,
    ! [A3: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
       => ( ( image_3256184292080368531od_o_o @ R @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) )
          = ( image_3256184292080368531od_o_o @ R @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) ) ) ) ).

% equiv_class_eq
thf(fact_575_equiv__class__eq,axiom,
    ! [A3: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
       => ( ( image_3892192845344445203at_nat @ R @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) )
          = ( image_3892192845344445203at_nat @ R @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) ) ) ) ).

% equiv_class_eq
thf(fact_576_dom__map__add,axiom,
    ! [M: produc1457211279475724562t_char > option_val,N: produc1457211279475724562t_char > option_val] :
      ( ( dom_Pr8989479155164304065ar_val @ ( map_ad2908576761495157028ar_val @ M @ N ) )
      = ( sup_su7899656352754981958t_char @ ( dom_Pr8989479155164304065ar_val @ N ) @ ( dom_Pr8989479155164304065ar_val @ M ) ) ) ).

% dom_map_add
thf(fact_577_inj__on__map__add__dom,axiom,
    ! [M: produc1457211279475724562t_char > option_val,M5: produc1457211279475724562t_char > option_val] :
      ( ( inj_on4474484569326248358on_val @ ( map_ad2908576761495157028ar_val @ M @ M5 ) @ ( dom_Pr8989479155164304065ar_val @ M5 ) )
      = ( inj_on4474484569326248358on_val @ M5 @ ( dom_Pr8989479155164304065ar_val @ M5 ) ) ) ).

% inj_on_map_add_dom
thf(fact_578_inj__Some,axiom,
    ! [A3: set_state] : ( inj_on3577428053172332983_state @ some_state @ A3 ) ).

% inj_Some
thf(fact_579_inj__Some,axiom,
    ! [A3: set_val] : ( inj_on4074814566786472055on_val @ some_val @ A3 ) ).

% inj_Some
thf(fact_580_equiv__class__nondisjoint,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o,X3: produc2934264451710624999od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A3 @ R )
     => ( ( member444158400953824016od_o_o @ X3 @ ( inf_in6630186493572433077od_o_o @ ( image_1313870842500862867od_o_o @ R @ ( insert3855699742481506999od_o_o @ A @ bot_bo498831808395348787od_o_o ) ) @ ( image_1313870842500862867od_o_o @ R @ ( insert3855699742481506999od_o_o @ B @ bot_bo498831808395348787od_o_o ) ) ) )
       => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R ) ) ) ).

% equiv_class_nondisjoint
thf(fact_581_equiv__class__nondisjoint,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat,X3: produc732395585841259969at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A3 @ R )
     => ( ( member4095101504841534314at_nat @ X3 @ ( inf_in3075509010647812879at_nat @ ( image_1966035174115581203at_nat @ R @ ( insert8109617542356073361at_nat @ A @ bot_bo1676454896636820109at_nat ) ) @ ( image_1966035174115581203at_nat @ R @ ( insert8109617542356073361at_nat @ B @ bot_bo1676454896636820109at_nat ) ) ) )
       => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R ) ) ) ).

% equiv_class_nondisjoint
thf(fact_582_equiv__class__nondisjoint,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,X3: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A3 @ R )
     => ( ( member8757157785044589968at_nat @ X3 @ ( inf_in7913087082777306421at_nat @ ( image_3736809421353964435at_nat @ R @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) @ ( image_3736809421353964435at_nat @ R @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) )
       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R ) ) ) ).

% equiv_class_nondisjoint
thf(fact_583_equiv__class__nondisjoint,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,X3: produc9003781588162623004on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( equiv_7491905333995573990on_val @ A3 @ R )
     => ( ( member2879030624002546611on_val @ X3 @ ( inf_in1079285664774927716on_val @ ( image_8782378046923076621on_val @ R @ ( insert3030518898131756108on_val @ A @ bot_bo6722054719108133734on_val ) ) @ ( image_8782378046923076621on_val @ R @ ( insert3030518898131756108on_val @ B @ bot_bo6722054719108133734on_val ) ) ) )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R ) ) ) ).

% equiv_class_nondisjoint
thf(fact_584_equiv__class__nondisjoint,axiom,
    ! [A3: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,X3: set_Product_prod_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R )
     => ( ( member3157015407008874800od_o_o @ X3 @ ( inf_in8747346010413679829od_o_o @ ( image_3256184292080368531od_o_o @ R @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) ) @ ( image_3256184292080368531od_o_o @ R @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R ) ) ) ).

% equiv_class_nondisjoint
thf(fact_585_equiv__class__nondisjoint,axiom,
    ! [A3: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,X3: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R )
     => ( ( member2643936169264416010at_nat @ X3 @ ( inf_in767997467307849391at_nat @ ( image_3892192845344445203at_nat @ R @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) ) @ ( image_3892192845344445203at_nat @ R @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R ) ) ) ).

% equiv_class_nondisjoint
thf(fact_586_graph__map__add,axiom,
    ! [M1: produc1457211279475724562t_char > option_val,M2: produc1457211279475724562t_char > option_val] :
      ( ( ( inf_in3544766959038609504t_char @ ( dom_Pr8989479155164304065ar_val @ M1 ) @ ( dom_Pr8989479155164304065ar_val @ M2 ) )
        = bot_bo1414094081888755934t_char )
     => ( ( graph_1733029056113894637ar_val @ ( map_ad2908576761495157028ar_val @ M1 @ M2 ) )
        = ( sup_su5754804325903363146ar_val @ ( graph_1733029056113894637ar_val @ M1 ) @ ( graph_1733029056113894637ar_val @ M2 ) ) ) ) ).

% graph_map_add
thf(fact_587_ran__map__add,axiom,
    ! [M1: produc1457211279475724562t_char > option_val,M2: produc1457211279475724562t_char > option_val] :
      ( ( ( inf_in3544766959038609504t_char @ ( dom_Pr8989479155164304065ar_val @ M1 ) @ ( dom_Pr8989479155164304065ar_val @ M2 ) )
        = bot_bo1414094081888755934t_char )
     => ( ( ran_Pr786515303459910110ar_val @ ( map_ad2908576761495157028ar_val @ M1 @ M2 ) )
        = ( sup_sup_set_val @ ( ran_Pr786515303459910110ar_val @ M1 ) @ ( ran_Pr786515303459910110ar_val @ M2 ) ) ) ) ).

% ran_map_add
thf(fact_588_equiv__class__eq__iff,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o,X3: produc2934264451710624999od_o_o,Y3: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A3 @ R )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X3 @ Y3 ) @ R )
        = ( ( ( image_1313870842500862867od_o_o @ R @ ( insert3855699742481506999od_o_o @ X3 @ bot_bo498831808395348787od_o_o ) )
            = ( image_1313870842500862867od_o_o @ R @ ( insert3855699742481506999od_o_o @ Y3 @ bot_bo498831808395348787od_o_o ) ) )
          & ( member444158400953824016od_o_o @ X3 @ A3 )
          & ( member444158400953824016od_o_o @ Y3 @ A3 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_589_equiv__class__eq__iff,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat,X3: produc732395585841259969at_nat,Y3: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A3 @ R )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X3 @ Y3 ) @ R )
        = ( ( ( image_1966035174115581203at_nat @ R @ ( insert8109617542356073361at_nat @ X3 @ bot_bo1676454896636820109at_nat ) )
            = ( image_1966035174115581203at_nat @ R @ ( insert8109617542356073361at_nat @ Y3 @ bot_bo1676454896636820109at_nat ) ) )
          & ( member4095101504841534314at_nat @ X3 @ A3 )
          & ( member4095101504841534314at_nat @ Y3 @ A3 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_590_equiv__class__eq__iff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,X3: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A3 @ R )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y3 ) @ R )
        = ( ( ( image_3736809421353964435at_nat @ R @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) )
            = ( image_3736809421353964435at_nat @ R @ ( insert9069300056098147895at_nat @ Y3 @ bot_bo228742789529271731at_nat ) ) )
          & ( member8757157785044589968at_nat @ X3 @ A3 )
          & ( member8757157785044589968at_nat @ Y3 @ A3 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_591_equiv__class__eq__iff,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
      ( ( equiv_7491905333995573990on_val @ A3 @ R )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R )
        = ( ( ( image_8782378046923076621on_val @ R @ ( insert3030518898131756108on_val @ X3 @ bot_bo6722054719108133734on_val ) )
            = ( image_8782378046923076621on_val @ R @ ( insert3030518898131756108on_val @ Y3 @ bot_bo6722054719108133734on_val ) ) )
          & ( member2879030624002546611on_val @ X3 @ A3 )
          & ( member2879030624002546611on_val @ Y3 @ A3 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_592_equiv__class__eq__iff,axiom,
    ! [A3: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,X3: set_Product_prod_o_o,Y3: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R )
        = ( ( ( image_3256184292080368531od_o_o @ R @ ( insert3817420688681518039od_o_o @ X3 @ bot_bo8006819982598807891od_o_o ) )
            = ( image_3256184292080368531od_o_o @ R @ ( insert3817420688681518039od_o_o @ Y3 @ bot_bo8006819982598807891od_o_o ) ) )
          & ( member3157015407008874800od_o_o @ X3 @ A3 )
          & ( member3157015407008874800od_o_o @ Y3 @ A3 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_593_equiv__class__eq__iff,axiom,
    ! [A3: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R )
        = ( ( ( image_3892192845344445203at_nat @ R @ ( insert9200635055090092081at_nat @ X3 @ bot_bo3083307316010499117at_nat ) )
            = ( image_3892192845344445203at_nat @ R @ ( insert9200635055090092081at_nat @ Y3 @ bot_bo3083307316010499117at_nat ) ) )
          & ( member2643936169264416010at_nat @ X3 @ A3 )
          & ( member2643936169264416010at_nat @ Y3 @ A3 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_594_eq__equiv__class__iff,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o,X3: produc2934264451710624999od_o_o,Y3: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A3 @ R )
     => ( ( member444158400953824016od_o_o @ X3 @ A3 )
       => ( ( member444158400953824016od_o_o @ Y3 @ A3 )
         => ( ( ( image_1313870842500862867od_o_o @ R @ ( insert3855699742481506999od_o_o @ X3 @ bot_bo498831808395348787od_o_o ) )
              = ( image_1313870842500862867od_o_o @ R @ ( insert3855699742481506999od_o_o @ Y3 @ bot_bo498831808395348787od_o_o ) ) )
            = ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X3 @ Y3 ) @ R ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_595_eq__equiv__class__iff,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat,X3: produc732395585841259969at_nat,Y3: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A3 @ R )
     => ( ( member4095101504841534314at_nat @ X3 @ A3 )
       => ( ( member4095101504841534314at_nat @ Y3 @ A3 )
         => ( ( ( image_1966035174115581203at_nat @ R @ ( insert8109617542356073361at_nat @ X3 @ bot_bo1676454896636820109at_nat ) )
              = ( image_1966035174115581203at_nat @ R @ ( insert8109617542356073361at_nat @ Y3 @ bot_bo1676454896636820109at_nat ) ) )
            = ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X3 @ Y3 ) @ R ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_596_eq__equiv__class__iff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,X3: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A3 @ R )
     => ( ( member8757157785044589968at_nat @ X3 @ A3 )
       => ( ( member8757157785044589968at_nat @ Y3 @ A3 )
         => ( ( ( image_3736809421353964435at_nat @ R @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) )
              = ( image_3736809421353964435at_nat @ R @ ( insert9069300056098147895at_nat @ Y3 @ bot_bo228742789529271731at_nat ) ) )
            = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y3 ) @ R ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_597_eq__equiv__class__iff,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
      ( ( equiv_7491905333995573990on_val @ A3 @ R )
     => ( ( member2879030624002546611on_val @ X3 @ A3 )
       => ( ( member2879030624002546611on_val @ Y3 @ A3 )
         => ( ( ( image_8782378046923076621on_val @ R @ ( insert3030518898131756108on_val @ X3 @ bot_bo6722054719108133734on_val ) )
              = ( image_8782378046923076621on_val @ R @ ( insert3030518898131756108on_val @ Y3 @ bot_bo6722054719108133734on_val ) ) )
            = ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_598_eq__equiv__class__iff,axiom,
    ! [A3: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,X3: set_Product_prod_o_o,Y3: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R )
     => ( ( member3157015407008874800od_o_o @ X3 @ A3 )
       => ( ( member3157015407008874800od_o_o @ Y3 @ A3 )
         => ( ( ( image_3256184292080368531od_o_o @ R @ ( insert3817420688681518039od_o_o @ X3 @ bot_bo8006819982598807891od_o_o ) )
              = ( image_3256184292080368531od_o_o @ R @ ( insert3817420688681518039od_o_o @ Y3 @ bot_bo8006819982598807891od_o_o ) ) )
            = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_599_eq__equiv__class__iff,axiom,
    ! [A3: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R )
     => ( ( member2643936169264416010at_nat @ X3 @ A3 )
       => ( ( member2643936169264416010at_nat @ Y3 @ A3 )
         => ( ( ( image_3892192845344445203at_nat @ R @ ( insert9200635055090092081at_nat @ X3 @ bot_bo3083307316010499117at_nat ) )
              = ( image_3892192845344445203at_nat @ R @ ( insert9200635055090092081at_nat @ Y3 @ bot_bo3083307316010499117at_nat ) ) )
            = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_600_Field__insert,axiom,
    ! [A: $o,B: $o,R: set_Product_prod_o_o] :
      ( ( field_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B ) @ R ) )
      = ( sup_sup_set_o @ ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R ) ) ) ).

% Field_insert
thf(fact_601_Field__insert,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( field_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) )
      = ( sup_sup_set_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R ) ) ) ).

% Field_insert
thf(fact_602_Field__insert,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( field_9010718613293825679on_val @ ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R ) )
      = ( sup_su7506060154676209150on_val @ ( insert3030518898131756108on_val @ A @ ( insert3030518898131756108on_val @ B @ bot_bo6722054719108133734on_val ) ) @ ( field_9010718613293825679on_val @ R ) ) ) ).

% Field_insert
thf(fact_603_Field__insert,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( field_6101175518749917524od_o_o @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R ) )
      = ( sup_su1477959037604056251od_o_o @ ( insert3817420688681518039od_o_o @ A @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) @ ( field_6101175518749917524od_o_o @ R ) ) ) ).

% Field_insert
thf(fact_604_Field__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( field_3145993188331922222at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R ) )
      = ( sup_su3642409539654194069at_nat @ ( insert9200635055090092081at_nat @ A @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) @ ( field_3145993188331922222at_nat @ R ) ) ) ).

% Field_insert
thf(fact_605_disjnt__equiv__class,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( equiv_7491905333995573990on_val @ A3 @ R )
     => ( ( disjnt6704409600597739157on_val @ ( image_8782378046923076621on_val @ R @ ( insert3030518898131756108on_val @ A @ bot_bo6722054719108133734on_val ) ) @ ( image_8782378046923076621on_val @ R @ ( insert3030518898131756108on_val @ B @ bot_bo6722054719108133734on_val ) ) )
        = ( ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R ) ) ) ) ).

% disjnt_equiv_class
thf(fact_606_disjnt__equiv__class,axiom,
    ! [A3: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R )
     => ( ( disjnt556417628418099406od_o_o @ ( image_3256184292080368531od_o_o @ R @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) ) @ ( image_3256184292080368531od_o_o @ R @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) )
        = ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R ) ) ) ) ).

% disjnt_equiv_class
thf(fact_607_disjnt__equiv__class,axiom,
    ! [A3: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R )
     => ( ( disjnt2544082741087298984at_nat @ ( image_3892192845344445203at_nat @ R @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) ) @ ( image_3892192845344445203at_nat @ R @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) )
        = ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R ) ) ) ) ).

% disjnt_equiv_class
thf(fact_608_equiv__class__subset,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( equiv_7491905333995573990on_val @ A3 @ R )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
       => ( ord_le6900622908513426738on_val @ ( image_8782378046923076621on_val @ R @ ( insert3030518898131756108on_val @ A @ bot_bo6722054719108133734on_val ) ) @ ( image_8782378046923076621on_val @ R @ ( insert3030518898131756108on_val @ B @ bot_bo6722054719108133734on_val ) ) ) ) ) ).

% equiv_class_subset
thf(fact_609_equiv__class__subset,axiom,
    ! [A3: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
       => ( ord_le2654036014095304327od_o_o @ ( image_3256184292080368531od_o_o @ R @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) ) @ ( image_3256184292080368531od_o_o @ R @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) ) ) ) ).

% equiv_class_subset
thf(fact_610_equiv__class__subset,axiom,
    ! [A3: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
       => ( ord_le2077887516847798113at_nat @ ( image_3892192845344445203at_nat @ R @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) ) @ ( image_3892192845344445203at_nat @ R @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) ) ) ) ).

% equiv_class_subset
thf(fact_611_subset__equiv__class,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o,B: produc2934264451710624999od_o_o,A: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A3 @ R )
     => ( ( ord_le7586318239878589543od_o_o @ ( image_1313870842500862867od_o_o @ R @ ( insert3855699742481506999od_o_o @ B @ bot_bo498831808395348787od_o_o ) ) @ ( image_1313870842500862867od_o_o @ R @ ( insert3855699742481506999od_o_o @ A @ bot_bo498831808395348787od_o_o ) ) )
       => ( ( member444158400953824016od_o_o @ B @ A3 )
         => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R ) ) ) ) ).

% subset_equiv_class
thf(fact_612_subset__equiv__class,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat,B: produc732395585841259969at_nat,A: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A3 @ R )
     => ( ( ord_le4841768313324958145at_nat @ ( image_1966035174115581203at_nat @ R @ ( insert8109617542356073361at_nat @ B @ bot_bo1676454896636820109at_nat ) ) @ ( image_1966035174115581203at_nat @ R @ ( insert8109617542356073361at_nat @ A @ bot_bo1676454896636820109at_nat ) ) )
       => ( ( member4095101504841534314at_nat @ B @ A3 )
         => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R ) ) ) ) ).

% subset_equiv_class
thf(fact_613_subset__equiv__class,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A3 @ R )
     => ( ( ord_le1268244103169919719at_nat @ ( image_3736809421353964435at_nat @ R @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) @ ( image_3736809421353964435at_nat @ R @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
       => ( ( member8757157785044589968at_nat @ B @ A3 )
         => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R ) ) ) ) ).

% subset_equiv_class
thf(fact_614_subset__equiv__class,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,B: produc9003781588162623004on_val,A: produc9003781588162623004on_val] :
      ( ( equiv_7491905333995573990on_val @ A3 @ R )
     => ( ( ord_le6900622908513426738on_val @ ( image_8782378046923076621on_val @ R @ ( insert3030518898131756108on_val @ B @ bot_bo6722054719108133734on_val ) ) @ ( image_8782378046923076621on_val @ R @ ( insert3030518898131756108on_val @ A @ bot_bo6722054719108133734on_val ) ) )
       => ( ( member2879030624002546611on_val @ B @ A3 )
         => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R ) ) ) ) ).

% subset_equiv_class
thf(fact_615_subset__equiv__class,axiom,
    ! [A3: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,B: set_Product_prod_o_o,A: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R )
     => ( ( ord_le2654036014095304327od_o_o @ ( image_3256184292080368531od_o_o @ R @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) @ ( image_3256184292080368531od_o_o @ R @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) ) )
       => ( ( member3157015407008874800od_o_o @ B @ A3 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R ) ) ) ) ).

% subset_equiv_class
thf(fact_616_subset__equiv__class,axiom,
    ! [A3: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R )
     => ( ( ord_le2077887516847798113at_nat @ ( image_3892192845344445203at_nat @ R @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) @ ( image_3892192845344445203at_nat @ R @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) ) )
       => ( ( member2643936169264416010at_nat @ B @ A3 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R ) ) ) ) ).

% subset_equiv_class
thf(fact_617_FieldI1,axiom,
    ! [I: produc2934264451710624999od_o_o,J: produc2934264451710624999od_o_o,R2: set_Pr1013884327308616903od_o_o] :
      ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I @ J ) @ R2 )
     => ( member444158400953824016od_o_o @ I @ ( field_5121862139100741428od_o_o @ R2 ) ) ) ).

% FieldI1
thf(fact_618_FieldI1,axiom,
    ! [I: produc732395585841259969at_nat,J: produc732395585841259969at_nat,R2: set_Pr4427867301352232263at_nat] :
      ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I @ J ) @ R2 )
     => ( member4095101504841534314at_nat @ I @ ( field_7361900449976055182at_nat @ R2 ) ) ) ).

% FieldI1
thf(fact_619_FieldI1,axiom,
    ! [I: produc3843707927480180839at_nat,J: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I @ J ) @ R2 )
     => ( member8757157785044589968at_nat @ I @ ( field_2862844103579723188at_nat @ R2 ) ) ) ).

% FieldI1
thf(fact_620_FieldI1,axiom,
    ! [I: $o,J: $o,R2: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I @ J ) @ R2 )
     => ( member_o @ I @ ( field_o @ R2 ) ) ) ).

% FieldI1
thf(fact_621_FieldI1,axiom,
    ! [I: nat,J: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R2 )
     => ( member_nat @ I @ ( field_nat @ R2 ) ) ) ).

% FieldI1
thf(fact_622_FieldI1,axiom,
    ! [I: produc9003781588162623004on_val,J: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ I @ J ) @ R2 )
     => ( member2879030624002546611on_val @ I @ ( field_9010718613293825679on_val @ R2 ) ) ) ).

% FieldI1
thf(fact_623_FieldI1,axiom,
    ! [I: set_Product_prod_o_o,J: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I @ J ) @ R2 )
     => ( member3157015407008874800od_o_o @ I @ ( field_6101175518749917524od_o_o @ R2 ) ) ) ).

% FieldI1
thf(fact_624_FieldI1,axiom,
    ! [I: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J ) @ R2 )
     => ( member2643936169264416010at_nat @ I @ ( field_3145993188331922222at_nat @ R2 ) ) ) ).

% FieldI1
thf(fact_625_FieldI2,axiom,
    ! [I: produc2934264451710624999od_o_o,J: produc2934264451710624999od_o_o,R2: set_Pr1013884327308616903od_o_o] :
      ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I @ J ) @ R2 )
     => ( member444158400953824016od_o_o @ J @ ( field_5121862139100741428od_o_o @ R2 ) ) ) ).

% FieldI2
thf(fact_626_FieldI2,axiom,
    ! [I: produc732395585841259969at_nat,J: produc732395585841259969at_nat,R2: set_Pr4427867301352232263at_nat] :
      ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I @ J ) @ R2 )
     => ( member4095101504841534314at_nat @ J @ ( field_7361900449976055182at_nat @ R2 ) ) ) ).

% FieldI2
thf(fact_627_FieldI2,axiom,
    ! [I: produc3843707927480180839at_nat,J: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I @ J ) @ R2 )
     => ( member8757157785044589968at_nat @ J @ ( field_2862844103579723188at_nat @ R2 ) ) ) ).

% FieldI2
thf(fact_628_FieldI2,axiom,
    ! [I: $o,J: $o,R2: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I @ J ) @ R2 )
     => ( member_o @ J @ ( field_o @ R2 ) ) ) ).

% FieldI2
thf(fact_629_FieldI2,axiom,
    ! [I: nat,J: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R2 )
     => ( member_nat @ J @ ( field_nat @ R2 ) ) ) ).

% FieldI2
thf(fact_630_FieldI2,axiom,
    ! [I: produc9003781588162623004on_val,J: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ I @ J ) @ R2 )
     => ( member2879030624002546611on_val @ J @ ( field_9010718613293825679on_val @ R2 ) ) ) ).

% FieldI2
thf(fact_631_FieldI2,axiom,
    ! [I: set_Product_prod_o_o,J: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I @ J ) @ R2 )
     => ( member3157015407008874800od_o_o @ J @ ( field_6101175518749917524od_o_o @ R2 ) ) ) ).

% FieldI2
thf(fact_632_FieldI2,axiom,
    ! [I: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J ) @ R2 )
     => ( member2643936169264416010at_nat @ J @ ( field_3145993188331922222at_nat @ R2 ) ) ) ).

% FieldI2
thf(fact_633_subrelI,axiom,
    ! [R: set_Pr7649655576376376530on_val,S: set_Pr7649655576376376530on_val] :
      ( ! [X2: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > option_val] :
          ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X2 @ Y4 ) @ R )
         => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X2 @ Y4 ) @ S ) )
     => ( ord_le6900622908513426738on_val @ R @ S ) ) ).

% subrelI
thf(fact_634_subrelI,axiom,
    ! [R: set_Pr4305388979797279737on_val,S: set_Pr4305388979797279737on_val] :
      ( ! [X2: produc9003781588162623004on_val,Y4: produc9003781588162623004on_val] :
          ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y4 ) @ R )
         => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y4 ) @ S ) )
     => ( ord_le7224449410216889241on_val @ R @ S ) ) ).

% subrelI
thf(fact_635_subrelI,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,S: set_Pr1932065953672099015od_o_o] :
      ( ! [X2: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
          ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y4 ) @ R )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y4 ) @ S ) )
     => ( ord_le7586318239878589543od_o_o @ R @ S ) ) ).

% subrelI
thf(fact_636_subrelI,axiom,
    ! [R: set_Pr457366540195662369at_nat,S: set_Pr457366540195662369at_nat] :
      ( ! [X2: set_Product_prod_o_o,Y4: set_Pr1261947904930325089at_nat] :
          ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X2 @ Y4 ) @ R )
         => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X2 @ Y4 ) @ S ) )
     => ( ord_le4841768313324958145at_nat @ R @ S ) ) ).

% subrelI
thf(fact_637_subrelI,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y4 ) @ R )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y4 ) @ S ) )
     => ( ord_le1268244103169919719at_nat @ R @ S ) ) ).

% subrelI
thf(fact_638_ssubsetI,axiom,
    ! [A3: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val] :
      ( ! [Pi3: produc1457211279475724562t_char > prat,H4: produc1457211279475724562t_char > option_val] :
          ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ Pi3 @ H4 ) @ A3 )
         => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ Pi3 @ H4 ) @ B5 ) )
     => ( ord_le6900622908513426738on_val @ A3 @ B5 ) ) ).

% ssubsetI
thf(fact_639_ssubsetI,axiom,
    ! [A3: set_Pr4305388979797279737on_val,B5: set_Pr4305388979797279737on_val] :
      ( ! [Pi3: produc9003781588162623004on_val,H4: produc9003781588162623004on_val] :
          ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Pi3 @ H4 ) @ A3 )
         => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Pi3 @ H4 ) @ B5 ) )
     => ( ord_le7224449410216889241on_val @ A3 @ B5 ) ) ).

% ssubsetI
thf(fact_640_ssubsetI,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,B5: set_Pr1932065953672099015od_o_o] :
      ( ! [Pi3: set_Product_prod_o_o,H4: set_Product_prod_o_o] :
          ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Pi3 @ H4 ) @ A3 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Pi3 @ H4 ) @ B5 ) )
     => ( ord_le7586318239878589543od_o_o @ A3 @ B5 ) ) ).

% ssubsetI
thf(fact_641_ssubsetI,axiom,
    ! [A3: set_Pr457366540195662369at_nat,B5: set_Pr457366540195662369at_nat] :
      ( ! [Pi3: set_Product_prod_o_o,H4: set_Pr1261947904930325089at_nat] :
          ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ Pi3 @ H4 ) @ A3 )
         => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ Pi3 @ H4 ) @ B5 ) )
     => ( ord_le4841768313324958145at_nat @ A3 @ B5 ) ) ).

% ssubsetI
thf(fact_642_ssubsetI,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ! [Pi3: set_Pr1261947904930325089at_nat,H4: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Pi3 @ H4 ) @ A3 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Pi3 @ H4 ) @ B5 ) )
     => ( ord_le1268244103169919719at_nat @ A3 @ B5 ) ) ).

% ssubsetI
thf(fact_643_subset__emptyI,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o] :
      ( ! [X2: produc2934264451710624999od_o_o] :
          ~ ( member444158400953824016od_o_o @ X2 @ A3 )
     => ( ord_le7586318239878589543od_o_o @ A3 @ bot_bo498831808395348787od_o_o ) ) ).

% subset_emptyI
thf(fact_644_subset__emptyI,axiom,
    ! [A3: set_Pr457366540195662369at_nat] :
      ( ! [X2: produc732395585841259969at_nat] :
          ~ ( member4095101504841534314at_nat @ X2 @ A3 )
     => ( ord_le4841768313324958145at_nat @ A3 @ bot_bo1676454896636820109at_nat ) ) ).

% subset_emptyI
thf(fact_645_subset__emptyI,axiom,
    ! [A3: set_Pr4329608150637261639at_nat] :
      ( ! [X2: produc3843707927480180839at_nat] :
          ~ ( member8757157785044589968at_nat @ X2 @ A3 )
     => ( ord_le1268244103169919719at_nat @ A3 @ bot_bo228742789529271731at_nat ) ) ).

% subset_emptyI
thf(fact_646_insert__subsetI,axiom,
    ! [X3: produc2934264451710624999od_o_o,A3: set_Pr1932065953672099015od_o_o,X5: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ X3 @ A3 )
     => ( ( ord_le7586318239878589543od_o_o @ X5 @ A3 )
       => ( ord_le7586318239878589543od_o_o @ ( insert3855699742481506999od_o_o @ X3 @ X5 ) @ A3 ) ) ) ).

% insert_subsetI
thf(fact_647_insert__subsetI,axiom,
    ! [X3: produc732395585841259969at_nat,A3: set_Pr457366540195662369at_nat,X5: set_Pr457366540195662369at_nat] :
      ( ( member4095101504841534314at_nat @ X3 @ A3 )
     => ( ( ord_le4841768313324958145at_nat @ X5 @ A3 )
       => ( ord_le4841768313324958145at_nat @ ( insert8109617542356073361at_nat @ X3 @ X5 ) @ A3 ) ) ) ).

% insert_subsetI
thf(fact_648_insert__subsetI,axiom,
    ! [X3: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,X5: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X3 @ A3 )
     => ( ( ord_le1268244103169919719at_nat @ X5 @ A3 )
       => ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ X3 @ X5 ) @ A3 ) ) ) ).

% insert_subsetI
thf(fact_649_subset__Image1__Image1__iff,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( order_3482285145583873534od_o_o @ ( field_5121862139100741428od_o_o @ R ) @ R )
     => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R ) )
       => ( ( member444158400953824016od_o_o @ B @ ( field_5121862139100741428od_o_o @ R ) )
         => ( ( ord_le7586318239878589543od_o_o @ ( image_1313870842500862867od_o_o @ R @ ( insert3855699742481506999od_o_o @ A @ bot_bo498831808395348787od_o_o ) ) @ ( image_1313870842500862867od_o_o @ R @ ( insert3855699742481506999od_o_o @ B @ bot_bo498831808395348787od_o_o ) ) )
            = ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B @ A ) @ R ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_650_subset__Image1__Image1__iff,axiom,
    ! [R: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( order_8058285401901086552at_nat @ ( field_7361900449976055182at_nat @ R ) @ R )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R ) )
       => ( ( member4095101504841534314at_nat @ B @ ( field_7361900449976055182at_nat @ R ) )
         => ( ( ord_le4841768313324958145at_nat @ ( image_1966035174115581203at_nat @ R @ ( insert8109617542356073361at_nat @ A @ bot_bo1676454896636820109at_nat ) ) @ ( image_1966035174115581203at_nat @ R @ ( insert8109617542356073361at_nat @ B @ bot_bo1676454896636820109at_nat ) ) )
            = ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B @ A ) @ R ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_651_subset__Image1__Image1__iff,axiom,
    ! [R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( order_4077073789452270206at_nat @ ( field_2862844103579723188at_nat @ R ) @ R )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R ) )
       => ( ( member8757157785044589968at_nat @ B @ ( field_2862844103579723188at_nat @ R ) )
         => ( ( ord_le1268244103169919719at_nat @ ( image_3736809421353964435at_nat @ R @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) @ ( image_3736809421353964435at_nat @ R @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) )
            = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ A ) @ R ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_652_subset__Image1__Image1__iff,axiom,
    ! [R: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( order_preorder_on_o @ ( field_o @ R ) @ R )
     => ( ( member_o @ A @ ( field_o @ R ) )
       => ( ( member_o @ B @ ( field_o @ R ) )
         => ( ( ord_less_eq_set_o @ ( image_o_o @ R @ ( insert_o @ A @ bot_bot_set_o ) ) @ ( image_o_o @ R @ ( insert_o @ B @ bot_bot_set_o ) ) )
            = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A ) @ R ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_653_subset__Image1__Image1__iff,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( order_4861654808422542329on_nat @ ( field_nat @ R ) @ R )
     => ( ( member_nat @ A @ ( field_nat @ R ) )
       => ( ( member_nat @ B @ ( field_nat @ R ) )
         => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_654_subset__Image1__Image1__iff,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( order_2116471383038404933on_val @ ( field_9010718613293825679on_val @ R ) @ R )
     => ( ( member2879030624002546611on_val @ A @ ( field_9010718613293825679on_val @ R ) )
       => ( ( member2879030624002546611on_val @ B @ ( field_9010718613293825679on_val @ R ) )
         => ( ( ord_le6900622908513426738on_val @ ( image_8782378046923076621on_val @ R @ ( insert3030518898131756108on_val @ A @ bot_bo6722054719108133734on_val ) ) @ ( image_8782378046923076621on_val @ R @ ( insert3030518898131756108on_val @ B @ bot_bo6722054719108133734on_val ) ) )
            = ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ A ) @ R ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_655_subset__Image1__Image1__iff,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( order_6208692358724846110od_o_o @ ( field_6101175518749917524od_o_o @ R ) @ R )
     => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( member3157015407008874800od_o_o @ B @ ( field_6101175518749917524od_o_o @ R ) )
         => ( ( ord_le2654036014095304327od_o_o @ ( image_3256184292080368531od_o_o @ R @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) ) @ ( image_3256184292080368531od_o_o @ R @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) )
            = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ A ) @ R ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_656_subset__Image1__Image1__iff,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( order_7173764778206869240at_nat @ ( field_3145993188331922222at_nat @ R ) @ R )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( member2643936169264416010at_nat @ B @ ( field_3145993188331922222at_nat @ R ) )
         => ( ( ord_le2077887516847798113at_nat @ ( image_3892192845344445203at_nat @ R @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) ) @ ( image_3892192845344445203at_nat @ R @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) )
            = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ A ) @ R ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_657_proj__iff,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
      ( ( equiv_7491905333995573990on_val @ A3 @ R )
     => ( ( ord_le6900622908513426738on_val @ ( insert3030518898131756108on_val @ X3 @ ( insert3030518898131756108on_val @ Y3 @ bot_bo6722054719108133734on_val ) ) @ A3 )
       => ( ( ( equiv_8180404997056654916on_val @ R @ X3 )
            = ( equiv_8180404997056654916on_val @ R @ Y3 ) )
          = ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R ) ) ) ) ).

% proj_iff
thf(fact_658_proj__iff,axiom,
    ! [A3: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,X3: set_Product_prod_o_o,Y3: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R )
     => ( ( ord_le2654036014095304327od_o_o @ ( insert3817420688681518039od_o_o @ X3 @ ( insert3817420688681518039od_o_o @ Y3 @ bot_bo8006819982598807891od_o_o ) ) @ A3 )
       => ( ( ( equiv_6719278489894859850od_o_o @ R @ X3 )
            = ( equiv_6719278489894859850od_o_o @ R @ Y3 ) )
          = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R ) ) ) ) ).

% proj_iff
thf(fact_659_proj__iff,axiom,
    ! [A3: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R )
     => ( ( ord_le2077887516847798113at_nat @ ( insert9200635055090092081at_nat @ X3 @ ( insert9200635055090092081at_nat @ Y3 @ bot_bo3083307316010499117at_nat ) ) @ A3 )
       => ( ( ( equiv_4785085088318667978at_nat @ R @ X3 )
            = ( equiv_4785085088318667978at_nat @ R @ Y3 ) )
          = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R ) ) ) ) ).

% proj_iff
thf(fact_660_subset__Image__Image__iff,axiom,
    ! [R: set_Product_prod_o_o,A3: set_o,B5: set_o] :
      ( ( order_preorder_on_o @ ( field_o @ R ) @ R )
     => ( ( ord_less_eq_set_o @ A3 @ ( field_o @ R ) )
       => ( ( ord_less_eq_set_o @ B5 @ ( field_o @ R ) )
         => ( ( ord_less_eq_set_o @ ( image_o_o @ R @ A3 ) @ ( image_o_o @ R @ B5 ) )
            = ( ! [X: $o] :
                  ( ( member_o @ X @ A3 )
                 => ? [Y: $o] :
                      ( ( member_o @ Y @ B5 )
                      & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y @ X ) @ R ) ) ) ) ) ) ) ) ).

% subset_Image_Image_iff
thf(fact_661_subset__Image__Image__iff,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A3: set_nat,B5: set_nat] :
      ( ( order_4861654808422542329on_nat @ ( field_nat @ R ) @ R )
     => ( ( ord_less_eq_set_nat @ A3 @ ( field_nat @ R ) )
       => ( ( ord_less_eq_set_nat @ B5 @ ( field_nat @ R ) )
         => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R @ A3 ) @ ( image_nat_nat @ R @ B5 ) )
            = ( ! [X: nat] :
                  ( ( member_nat @ X @ A3 )
                 => ? [Y: nat] :
                      ( ( member_nat @ Y @ B5 )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R ) ) ) ) ) ) ) ) ).

% subset_Image_Image_iff
thf(fact_662_subset__Image__Image__iff,axiom,
    ! [R: set_Pr4305388979797279737on_val,A3: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val] :
      ( ( order_2116471383038404933on_val @ ( field_9010718613293825679on_val @ R ) @ R )
     => ( ( ord_le6900622908513426738on_val @ A3 @ ( field_9010718613293825679on_val @ R ) )
       => ( ( ord_le6900622908513426738on_val @ B5 @ ( field_9010718613293825679on_val @ R ) )
         => ( ( ord_le6900622908513426738on_val @ ( image_8782378046923076621on_val @ R @ A3 ) @ ( image_8782378046923076621on_val @ R @ B5 ) )
            = ( ! [X: produc9003781588162623004on_val] :
                  ( ( member2879030624002546611on_val @ X @ A3 )
                 => ? [Y: produc9003781588162623004on_val] :
                      ( ( member2879030624002546611on_val @ Y @ B5 )
                      & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y @ X ) @ R ) ) ) ) ) ) ) ) ).

% subset_Image_Image_iff
thf(fact_663_subset__Image__Image__iff,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A3: set_se491199571785101031od_o_o,B5: set_se491199571785101031od_o_o] :
      ( ( order_6208692358724846110od_o_o @ ( field_6101175518749917524od_o_o @ R ) @ R )
     => ( ( ord_le2654036014095304327od_o_o @ A3 @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( ord_le2654036014095304327od_o_o @ B5 @ ( field_6101175518749917524od_o_o @ R ) )
         => ( ( ord_le2654036014095304327od_o_o @ ( image_3256184292080368531od_o_o @ R @ A3 ) @ ( image_3256184292080368531od_o_o @ R @ B5 ) )
            = ( ! [X: set_Product_prod_o_o] :
                  ( ( member3157015407008874800od_o_o @ X @ A3 )
                 => ? [Y: set_Product_prod_o_o] :
                      ( ( member3157015407008874800od_o_o @ Y @ B5 )
                      & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ X ) @ R ) ) ) ) ) ) ) ) ).

% subset_Image_Image_iff
thf(fact_664_subset__Image__Image__iff,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A3: set_se7855581050983116737at_nat,B5: set_se7855581050983116737at_nat] :
      ( ( order_7173764778206869240at_nat @ ( field_3145993188331922222at_nat @ R ) @ R )
     => ( ( ord_le2077887516847798113at_nat @ A3 @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( ord_le2077887516847798113at_nat @ B5 @ ( field_3145993188331922222at_nat @ R ) )
         => ( ( ord_le2077887516847798113at_nat @ ( image_3892192845344445203at_nat @ R @ A3 ) @ ( image_3892192845344445203at_nat @ R @ B5 ) )
            = ( ! [X: set_Pr1261947904930325089at_nat] :
                  ( ( member2643936169264416010at_nat @ X @ A3 )
                 => ? [Y: set_Pr1261947904930325089at_nat] :
                      ( ( member2643936169264416010at_nat @ Y @ B5 )
                      & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X ) @ R ) ) ) ) ) ) ) ) ).

% subset_Image_Image_iff
thf(fact_665_in__quotient__imp__in__rel,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,X5: set_Pr7649655576376376530on_val,X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
      ( ( equiv_7491905333995573990on_val @ A3 @ R )
     => ( ( member6593899450356729449on_val @ X5 @ ( equiv_6009051154071487125on_val @ A3 @ R ) )
       => ( ( ord_le6900622908513426738on_val @ ( insert3030518898131756108on_val @ X3 @ ( insert3030518898131756108on_val @ Y3 @ bot_bo6722054719108133734on_val ) ) @ X5 )
         => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R ) ) ) ) ).

% in_quotient_imp_in_rel
thf(fact_666_in__quotient__imp__in__rel,axiom,
    ! [A3: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,X5: set_se491199571785101031od_o_o,X3: set_Product_prod_o_o,Y3: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R )
     => ( ( member2239101972932892176od_o_o @ X5 @ ( equiv_3134926782319234254od_o_o @ A3 @ R ) )
       => ( ( ord_le2654036014095304327od_o_o @ ( insert3817420688681518039od_o_o @ X3 @ ( insert3817420688681518039od_o_o @ Y3 @ bot_bo8006819982598807891od_o_o ) ) @ X5 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R ) ) ) ) ).

% in_quotient_imp_in_rel
thf(fact_667_in__quotient__imp__in__rel,axiom,
    ! [A3: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,X5: set_se7855581050983116737at_nat,X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R )
     => ( ( member8313122109188517482at_nat @ X5 @ ( equiv_7923377459618294184at_nat @ A3 @ R ) )
       => ( ( ord_le2077887516847798113at_nat @ ( insert9200635055090092081at_nat @ X3 @ ( insert9200635055090092081at_nat @ Y3 @ bot_bo3083307316010499117at_nat ) ) @ X5 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R ) ) ) ) ).

% in_quotient_imp_in_rel
thf(fact_668_Linear__order__Well__order__iff,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( order_6238756238976269133r_on_o @ ( field_o @ R ) @ R )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
        = ( ! [A6: set_o] :
              ( ( ord_less_eq_set_o @ A6 @ ( field_o @ R ) )
             => ( ( A6 != bot_bot_set_o )
               => ? [X: $o] :
                    ( ( member_o @ X @ A6 )
                    & ! [Y: $o] :
                        ( ( member_o @ Y @ A6 )
                       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_669_Linear__order__Well__order__iff,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R ) @ R )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
        = ( ! [A6: set_nat] :
              ( ( ord_less_eq_set_nat @ A6 @ ( field_nat @ R ) )
             => ( ( A6 != bot_bot_set_nat )
               => ? [X: nat] :
                    ( ( member_nat @ X @ A6 )
                    & ! [Y: nat] :
                        ( ( member_nat @ Y @ A6 )
                       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_670_Linear__order__Well__order__iff,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ( order_6061556925426544295on_val @ ( field_9010718613293825679on_val @ R ) @ R )
     => ( ( order_2579646547231075838on_val @ ( field_9010718613293825679on_val @ R ) @ R )
        = ( ! [A6: set_Pr7649655576376376530on_val] :
              ( ( ord_le6900622908513426738on_val @ A6 @ ( field_9010718613293825679on_val @ R ) )
             => ( ( A6 != bot_bo6722054719108133734on_val )
               => ? [X: produc9003781588162623004on_val] :
                    ( ( member2879030624002546611on_val @ X @ A6 )
                    & ! [Y: produc9003781588162623004on_val] :
                        ( ( member2879030624002546611on_val @ Y @ A6 )
                       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_671_Linear__order__Well__order__iff,axiom,
    ! [R: set_Pr1932065953672099015od_o_o] :
      ( ( order_2625247082195685180od_o_o @ ( field_6101175518749917524od_o_o @ R ) @ R )
     => ( ( order_3767514495197212837od_o_o @ ( field_6101175518749917524od_o_o @ R ) @ R )
        = ( ! [A6: set_se491199571785101031od_o_o] :
              ( ( ord_le2654036014095304327od_o_o @ A6 @ ( field_6101175518749917524od_o_o @ R ) )
             => ( ( A6 != bot_bo8006819982598807891od_o_o )
               => ? [X: set_Product_prod_o_o] :
                    ( ( member3157015407008874800od_o_o @ X @ A6 )
                    & ! [Y: set_Product_prod_o_o] :
                        ( ( member3157015407008874800od_o_o @ Y @ A6 )
                       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_672_Linear__order__Well__order__iff,axiom,
    ! [R: set_Pr4329608150637261639at_nat] :
      ( ( order_5942792088423767318at_nat @ ( field_3145993188331922222at_nat @ R ) @ R )
     => ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R ) @ R )
        = ( ! [A6: set_se7855581050983116737at_nat] :
              ( ( ord_le2077887516847798113at_nat @ A6 @ ( field_3145993188331922222at_nat @ R ) )
             => ( ( A6 != bot_bo3083307316010499117at_nat )
               => ? [X: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ X @ A6 )
                    & ! [Y: set_Pr1261947904930325089at_nat] :
                        ( ( member2643936169264416010at_nat @ Y @ A6 )
                       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_673_underS__incl__iff,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( order_91209724698019100od_o_o @ ( field_5121862139100741428od_o_o @ R ) @ R )
     => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R ) )
       => ( ( member444158400953824016od_o_o @ B @ ( field_5121862139100741428od_o_o @ R ) )
         => ( ( ord_le7586318239878589543od_o_o @ ( order_1929855754104155309od_o_o @ R @ A ) @ ( order_1929855754104155309od_o_o @ R @ B ) )
            = ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_674_underS__incl__iff,axiom,
    ! [R: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( order_9086815480479278966at_nat @ ( field_7361900449976055182at_nat @ R ) @ R )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R ) )
       => ( ( member4095101504841534314at_nat @ B @ ( field_7361900449976055182at_nat @ R ) )
         => ( ( ord_le4841768313324958145at_nat @ ( order_6745690092771398279at_nat @ R @ A ) @ ( order_6745690092771398279at_nat @ R @ B ) )
            = ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_675_underS__incl__iff,axiom,
    ! [R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( order_2711500226769404828at_nat @ ( field_2862844103579723188at_nat @ R ) @ R )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R ) )
       => ( ( member8757157785044589968at_nat @ B @ ( field_2862844103579723188at_nat @ R ) )
         => ( ( ord_le1268244103169919719at_nat @ ( order_3736018788886245933at_nat @ R @ A ) @ ( order_3736018788886245933at_nat @ R @ B ) )
            = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_676_underS__incl__iff,axiom,
    ! [R: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( order_6238756238976269133r_on_o @ ( field_o @ R ) @ R )
     => ( ( member_o @ A @ ( field_o @ R ) )
       => ( ( member_o @ B @ ( field_o @ R ) )
         => ( ( ord_less_eq_set_o @ ( order_underS_o @ R @ A ) @ ( order_underS_o @ R @ B ) )
            = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_677_underS__incl__iff,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R ) @ R )
     => ( ( member_nat @ A @ ( field_nat @ R ) )
       => ( ( member_nat @ B @ ( field_nat @ R ) )
         => ( ( ord_less_eq_set_nat @ ( order_underS_nat @ R @ A ) @ ( order_underS_nat @ R @ B ) )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_678_underS__incl__iff,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( order_6061556925426544295on_val @ ( field_9010718613293825679on_val @ R ) @ R )
     => ( ( member2879030624002546611on_val @ A @ ( field_9010718613293825679on_val @ R ) )
       => ( ( member2879030624002546611on_val @ B @ ( field_9010718613293825679on_val @ R ) )
         => ( ( ord_le6900622908513426738on_val @ ( order_8621009067429368022on_val @ R @ A ) @ ( order_8621009067429368022on_val @ R @ B ) )
            = ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_679_underS__incl__iff,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( order_2625247082195685180od_o_o @ ( field_6101175518749917524od_o_o @ R ) @ R )
     => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( member3157015407008874800od_o_o @ B @ ( field_6101175518749917524od_o_o @ R ) )
         => ( ( ord_le2654036014095304327od_o_o @ ( order_3915675016455977933od_o_o @ R @ A ) @ ( order_3915675016455977933od_o_o @ R @ B ) )
            = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_680_underS__incl__iff,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( order_5942792088423767318at_nat @ ( field_3145993188331922222at_nat @ R ) @ R )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( member2643936169264416010at_nat @ B @ ( field_3145993188331922222at_nat @ R ) )
         => ( ( ord_le2077887516847798113at_nat @ ( order_2644262826035875111at_nat @ R @ A ) @ ( order_2644262826035875111at_nat @ R @ B ) )
            = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_681_well__order__on__domain,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( order_2783320356396525957od_o_o @ A3 @ R )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R )
       => ( ( member444158400953824016od_o_o @ A @ A3 )
          & ( member444158400953824016od_o_o @ B @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_682_well__order__on__domain,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( order_1951249980326167391at_nat @ A3 @ R )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R )
       => ( ( member4095101504841534314at_nat @ A @ A3 )
          & ( member4095101504841534314at_nat @ B @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_683_well__order__on__domain,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( order_2339240034117818117at_nat @ A3 @ R )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R )
       => ( ( member8757157785044589968at_nat @ A @ A3 )
          & ( member8757157785044589968at_nat @ B @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_684_well__order__on__domain,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( order_2579646547231075838on_val @ A3 @ R )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
       => ( ( member2879030624002546611on_val @ A @ A3 )
          & ( member2879030624002546611on_val @ B @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_685_well__order__on__domain,axiom,
    ! [A3: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( order_3767514495197212837od_o_o @ A3 @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
       => ( ( member3157015407008874800od_o_o @ A @ A3 )
          & ( member3157015407008874800od_o_o @ B @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_686_well__order__on__domain,axiom,
    ! [A3: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( order_1968183844437341695at_nat @ A3 @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
       => ( ( member2643936169264416010at_nat @ A @ A3 )
          & ( member2643936169264416010at_nat @ B @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_687_underS__E,axiom,
    ! [I: produc2934264451710624999od_o_o,R2: set_Pr1013884327308616903od_o_o,J: produc2934264451710624999od_o_o] :
      ( ( member444158400953824016od_o_o @ I @ ( order_1929855754104155309od_o_o @ R2 @ J ) )
     => ( ( I != J )
        & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I @ J ) @ R2 ) ) ) ).

% underS_E
thf(fact_688_underS__E,axiom,
    ! [I: produc732395585841259969at_nat,R2: set_Pr4427867301352232263at_nat,J: produc732395585841259969at_nat] :
      ( ( member4095101504841534314at_nat @ I @ ( order_6745690092771398279at_nat @ R2 @ J ) )
     => ( ( I != J )
        & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I @ J ) @ R2 ) ) ) ).

% underS_E
thf(fact_689_underS__E,axiom,
    ! [I: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat,J: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ I @ ( order_3736018788886245933at_nat @ R2 @ J ) )
     => ( ( I != J )
        & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I @ J ) @ R2 ) ) ) ).

% underS_E
thf(fact_690_underS__E,axiom,
    ! [I: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val,J: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ I @ ( order_8621009067429368022on_val @ R2 @ J ) )
     => ( ( I != J )
        & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ I @ J ) @ R2 ) ) ) ).

% underS_E
thf(fact_691_underS__E,axiom,
    ! [I: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o,J: set_Product_prod_o_o] :
      ( ( member3157015407008874800od_o_o @ I @ ( order_3915675016455977933od_o_o @ R2 @ J ) )
     => ( ( I != J )
        & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I @ J ) @ R2 ) ) ) ).

% underS_E
thf(fact_692_underS__E,axiom,
    ! [I: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,J: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ I @ ( order_2644262826035875111at_nat @ R2 @ J ) )
     => ( ( I != J )
        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J ) @ R2 ) ) ) ).

% underS_E
thf(fact_693_underS__I,axiom,
    ! [I: produc2934264451710624999od_o_o,J: produc2934264451710624999od_o_o,R2: set_Pr1013884327308616903od_o_o] :
      ( ( I != J )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I @ J ) @ R2 )
       => ( member444158400953824016od_o_o @ I @ ( order_1929855754104155309od_o_o @ R2 @ J ) ) ) ) ).

% underS_I
thf(fact_694_underS__I,axiom,
    ! [I: produc732395585841259969at_nat,J: produc732395585841259969at_nat,R2: set_Pr4427867301352232263at_nat] :
      ( ( I != J )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I @ J ) @ R2 )
       => ( member4095101504841534314at_nat @ I @ ( order_6745690092771398279at_nat @ R2 @ J ) ) ) ) ).

% underS_I
thf(fact_695_underS__I,axiom,
    ! [I: produc3843707927480180839at_nat,J: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ( I != J )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I @ J ) @ R2 )
       => ( member8757157785044589968at_nat @ I @ ( order_3736018788886245933at_nat @ R2 @ J ) ) ) ) ).

% underS_I
thf(fact_696_underS__I,axiom,
    ! [I: produc9003781588162623004on_val,J: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val] :
      ( ( I != J )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ I @ J ) @ R2 )
       => ( member2879030624002546611on_val @ I @ ( order_8621009067429368022on_val @ R2 @ J ) ) ) ) ).

% underS_I
thf(fact_697_underS__I,axiom,
    ! [I: set_Product_prod_o_o,J: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( I != J )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I @ J ) @ R2 )
       => ( member3157015407008874800od_o_o @ I @ ( order_3915675016455977933od_o_o @ R2 @ J ) ) ) ) ).

% underS_I
thf(fact_698_underS__I,axiom,
    ! [I: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( I != J )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J ) @ R2 )
       => ( member2643936169264416010at_nat @ I @ ( order_2644262826035875111at_nat @ R2 @ J ) ) ) ) ).

% underS_I
thf(fact_699_BNF__Least__Fixpoint_OunderS__Field,axiom,
    ! [I: produc2934264451710624999od_o_o,R2: set_Pr1013884327308616903od_o_o,J: produc2934264451710624999od_o_o] :
      ( ( member444158400953824016od_o_o @ I @ ( order_1929855754104155309od_o_o @ R2 @ J ) )
     => ( member444158400953824016od_o_o @ I @ ( field_5121862139100741428od_o_o @ R2 ) ) ) ).

% BNF_Least_Fixpoint.underS_Field
thf(fact_700_BNF__Least__Fixpoint_OunderS__Field,axiom,
    ! [I: produc732395585841259969at_nat,R2: set_Pr4427867301352232263at_nat,J: produc732395585841259969at_nat] :
      ( ( member4095101504841534314at_nat @ I @ ( order_6745690092771398279at_nat @ R2 @ J ) )
     => ( member4095101504841534314at_nat @ I @ ( field_7361900449976055182at_nat @ R2 ) ) ) ).

% BNF_Least_Fixpoint.underS_Field
thf(fact_701_BNF__Least__Fixpoint_OunderS__Field,axiom,
    ! [I: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat,J: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ I @ ( order_3736018788886245933at_nat @ R2 @ J ) )
     => ( member8757157785044589968at_nat @ I @ ( field_2862844103579723188at_nat @ R2 ) ) ) ).

% BNF_Least_Fixpoint.underS_Field
thf(fact_702_BNF__Least__Fixpoint_OunderS__Field,axiom,
    ! [I: $o,R2: set_Product_prod_o_o,J: $o] :
      ( ( member_o @ I @ ( order_underS_o @ R2 @ J ) )
     => ( member_o @ I @ ( field_o @ R2 ) ) ) ).

% BNF_Least_Fixpoint.underS_Field
thf(fact_703_BNF__Least__Fixpoint_OunderS__Field,axiom,
    ! [I: nat,R2: set_Pr1261947904930325089at_nat,J: nat] :
      ( ( member_nat @ I @ ( order_underS_nat @ R2 @ J ) )
     => ( member_nat @ I @ ( field_nat @ R2 ) ) ) ).

% BNF_Least_Fixpoint.underS_Field
thf(fact_704_quotient__eqI,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o,X5: set_Pr1932065953672099015od_o_o,Y7: set_Pr1932065953672099015od_o_o,X3: produc2934264451710624999od_o_o,Y3: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A3 @ R )
     => ( ( member4387393612980313840od_o_o @ X5 @ ( equiv_1831625524721748142od_o_o @ A3 @ R ) )
       => ( ( member4387393612980313840od_o_o @ Y7 @ ( equiv_1831625524721748142od_o_o @ A3 @ R ) )
         => ( ( member444158400953824016od_o_o @ X3 @ X5 )
           => ( ( member444158400953824016od_o_o @ Y3 @ Y7 )
             => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X3 @ Y3 ) @ R )
               => ( X5 = Y7 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_705_quotient__eqI,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat,X5: set_Pr457366540195662369at_nat,Y7: set_Pr457366540195662369at_nat,X3: produc732395585841259969at_nat,Y3: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A3 @ R )
     => ( ( member3243375828287704010at_nat @ X5 @ ( equiv_2323626691971332616at_nat @ A3 @ R ) )
       => ( ( member3243375828287704010at_nat @ Y7 @ ( equiv_2323626691971332616at_nat @ A3 @ R ) )
         => ( ( member4095101504841534314at_nat @ X3 @ X5 )
           => ( ( member4095101504841534314at_nat @ Y3 @ Y7 )
             => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X3 @ Y3 ) @ R )
               => ( X5 = Y7 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_706_quotient__eqI,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,X5: set_Pr4329608150637261639at_nat,Y7: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A3 @ R )
     => ( ( member6273874620674182256at_nat @ X5 @ ( equiv_8408515162552411438at_nat @ A3 @ R ) )
       => ( ( member6273874620674182256at_nat @ Y7 @ ( equiv_8408515162552411438at_nat @ A3 @ R ) )
         => ( ( member8757157785044589968at_nat @ X3 @ X5 )
           => ( ( member8757157785044589968at_nat @ Y3 @ Y7 )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y3 ) @ R )
               => ( X5 = Y7 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_707_quotient__eqI,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,X5: set_Pr7649655576376376530on_val,Y7: set_Pr7649655576376376530on_val,X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
      ( ( equiv_7491905333995573990on_val @ A3 @ R )
     => ( ( member6593899450356729449on_val @ X5 @ ( equiv_6009051154071487125on_val @ A3 @ R ) )
       => ( ( member6593899450356729449on_val @ Y7 @ ( equiv_6009051154071487125on_val @ A3 @ R ) )
         => ( ( member2879030624002546611on_val @ X3 @ X5 )
           => ( ( member2879030624002546611on_val @ Y3 @ Y7 )
             => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R )
               => ( X5 = Y7 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_708_quotient__eqI,axiom,
    ! [A3: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,X5: set_se491199571785101031od_o_o,Y7: set_se491199571785101031od_o_o,X3: set_Product_prod_o_o,Y3: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R )
     => ( ( member2239101972932892176od_o_o @ X5 @ ( equiv_3134926782319234254od_o_o @ A3 @ R ) )
       => ( ( member2239101972932892176od_o_o @ Y7 @ ( equiv_3134926782319234254od_o_o @ A3 @ R ) )
         => ( ( member3157015407008874800od_o_o @ X3 @ X5 )
           => ( ( member3157015407008874800od_o_o @ Y3 @ Y7 )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R )
               => ( X5 = Y7 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_709_quotient__eqI,axiom,
    ! [A3: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,X5: set_se7855581050983116737at_nat,Y7: set_se7855581050983116737at_nat,X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R )
     => ( ( member8313122109188517482at_nat @ X5 @ ( equiv_7923377459618294184at_nat @ A3 @ R ) )
       => ( ( member8313122109188517482at_nat @ Y7 @ ( equiv_7923377459618294184at_nat @ A3 @ R ) )
         => ( ( member2643936169264416010at_nat @ X3 @ X5 )
           => ( ( member2643936169264416010at_nat @ Y3 @ Y7 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R )
               => ( X5 = Y7 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_710_quotient__eq__iff,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o,X5: set_Pr1932065953672099015od_o_o,Y7: set_Pr1932065953672099015od_o_o,X3: produc2934264451710624999od_o_o,Y3: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A3 @ R )
     => ( ( member4387393612980313840od_o_o @ X5 @ ( equiv_1831625524721748142od_o_o @ A3 @ R ) )
       => ( ( member4387393612980313840od_o_o @ Y7 @ ( equiv_1831625524721748142od_o_o @ A3 @ R ) )
         => ( ( member444158400953824016od_o_o @ X3 @ X5 )
           => ( ( member444158400953824016od_o_o @ Y3 @ Y7 )
             => ( ( X5 = Y7 )
                = ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X3 @ Y3 ) @ R ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_711_quotient__eq__iff,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat,X5: set_Pr457366540195662369at_nat,Y7: set_Pr457366540195662369at_nat,X3: produc732395585841259969at_nat,Y3: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A3 @ R )
     => ( ( member3243375828287704010at_nat @ X5 @ ( equiv_2323626691971332616at_nat @ A3 @ R ) )
       => ( ( member3243375828287704010at_nat @ Y7 @ ( equiv_2323626691971332616at_nat @ A3 @ R ) )
         => ( ( member4095101504841534314at_nat @ X3 @ X5 )
           => ( ( member4095101504841534314at_nat @ Y3 @ Y7 )
             => ( ( X5 = Y7 )
                = ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X3 @ Y3 ) @ R ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_712_quotient__eq__iff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,X5: set_Pr4329608150637261639at_nat,Y7: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A3 @ R )
     => ( ( member6273874620674182256at_nat @ X5 @ ( equiv_8408515162552411438at_nat @ A3 @ R ) )
       => ( ( member6273874620674182256at_nat @ Y7 @ ( equiv_8408515162552411438at_nat @ A3 @ R ) )
         => ( ( member8757157785044589968at_nat @ X3 @ X5 )
           => ( ( member8757157785044589968at_nat @ Y3 @ Y7 )
             => ( ( X5 = Y7 )
                = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y3 ) @ R ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_713_quotient__eq__iff,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,X5: set_Pr7649655576376376530on_val,Y7: set_Pr7649655576376376530on_val,X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
      ( ( equiv_7491905333995573990on_val @ A3 @ R )
     => ( ( member6593899450356729449on_val @ X5 @ ( equiv_6009051154071487125on_val @ A3 @ R ) )
       => ( ( member6593899450356729449on_val @ Y7 @ ( equiv_6009051154071487125on_val @ A3 @ R ) )
         => ( ( member2879030624002546611on_val @ X3 @ X5 )
           => ( ( member2879030624002546611on_val @ Y3 @ Y7 )
             => ( ( X5 = Y7 )
                = ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_714_quotient__eq__iff,axiom,
    ! [A3: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,X5: set_se491199571785101031od_o_o,Y7: set_se491199571785101031od_o_o,X3: set_Product_prod_o_o,Y3: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R )
     => ( ( member2239101972932892176od_o_o @ X5 @ ( equiv_3134926782319234254od_o_o @ A3 @ R ) )
       => ( ( member2239101972932892176od_o_o @ Y7 @ ( equiv_3134926782319234254od_o_o @ A3 @ R ) )
         => ( ( member3157015407008874800od_o_o @ X3 @ X5 )
           => ( ( member3157015407008874800od_o_o @ Y3 @ Y7 )
             => ( ( X5 = Y7 )
                = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_715_quotient__eq__iff,axiom,
    ! [A3: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,X5: set_se7855581050983116737at_nat,Y7: set_se7855581050983116737at_nat,X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R )
     => ( ( member8313122109188517482at_nat @ X5 @ ( equiv_7923377459618294184at_nat @ A3 @ R ) )
       => ( ( member8313122109188517482at_nat @ Y7 @ ( equiv_7923377459618294184at_nat @ A3 @ R ) )
         => ( ( member2643936169264416010at_nat @ X3 @ X5 )
           => ( ( member2643936169264416010at_nat @ Y3 @ Y7 )
             => ( ( X5 = Y7 )
                = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_716_in__quotient__imp__closed,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o,X5: set_Pr1932065953672099015od_o_o,X3: produc2934264451710624999od_o_o,Y3: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A3 @ R )
     => ( ( member4387393612980313840od_o_o @ X5 @ ( equiv_1831625524721748142od_o_o @ A3 @ R ) )
       => ( ( member444158400953824016od_o_o @ X3 @ X5 )
         => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X3 @ Y3 ) @ R )
           => ( member444158400953824016od_o_o @ Y3 @ X5 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_717_in__quotient__imp__closed,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat,X5: set_Pr457366540195662369at_nat,X3: produc732395585841259969at_nat,Y3: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A3 @ R )
     => ( ( member3243375828287704010at_nat @ X5 @ ( equiv_2323626691971332616at_nat @ A3 @ R ) )
       => ( ( member4095101504841534314at_nat @ X3 @ X5 )
         => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X3 @ Y3 ) @ R )
           => ( member4095101504841534314at_nat @ Y3 @ X5 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_718_in__quotient__imp__closed,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,X5: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A3 @ R )
     => ( ( member6273874620674182256at_nat @ X5 @ ( equiv_8408515162552411438at_nat @ A3 @ R ) )
       => ( ( member8757157785044589968at_nat @ X3 @ X5 )
         => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y3 ) @ R )
           => ( member8757157785044589968at_nat @ Y3 @ X5 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_719_in__quotient__imp__closed,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,X5: set_Pr7649655576376376530on_val,X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
      ( ( equiv_7491905333995573990on_val @ A3 @ R )
     => ( ( member6593899450356729449on_val @ X5 @ ( equiv_6009051154071487125on_val @ A3 @ R ) )
       => ( ( member2879030624002546611on_val @ X3 @ X5 )
         => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R )
           => ( member2879030624002546611on_val @ Y3 @ X5 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_720_in__quotient__imp__closed,axiom,
    ! [A3: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,X5: set_se491199571785101031od_o_o,X3: set_Product_prod_o_o,Y3: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R )
     => ( ( member2239101972932892176od_o_o @ X5 @ ( equiv_3134926782319234254od_o_o @ A3 @ R ) )
       => ( ( member3157015407008874800od_o_o @ X3 @ X5 )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R )
           => ( member3157015407008874800od_o_o @ Y3 @ X5 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_721_in__quotient__imp__closed,axiom,
    ! [A3: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,X5: set_se7855581050983116737at_nat,X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R )
     => ( ( member8313122109188517482at_nat @ X5 @ ( equiv_7923377459618294184at_nat @ A3 @ R ) )
       => ( ( member2643936169264416010at_nat @ X3 @ X5 )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R )
           => ( member2643936169264416010at_nat @ Y3 @ X5 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_722_eq__equiv__class__iff2,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o,X3: produc2934264451710624999od_o_o,Y3: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A3 @ R )
     => ( ( member444158400953824016od_o_o @ X3 @ A3 )
       => ( ( member444158400953824016od_o_o @ Y3 @ A3 )
         => ( ( ( equiv_1831625524721748142od_o_o @ ( insert3855699742481506999od_o_o @ X3 @ bot_bo498831808395348787od_o_o ) @ R )
              = ( equiv_1831625524721748142od_o_o @ ( insert3855699742481506999od_o_o @ Y3 @ bot_bo498831808395348787od_o_o ) @ R ) )
            = ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X3 @ Y3 ) @ R ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_723_eq__equiv__class__iff2,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat,X3: produc732395585841259969at_nat,Y3: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A3 @ R )
     => ( ( member4095101504841534314at_nat @ X3 @ A3 )
       => ( ( member4095101504841534314at_nat @ Y3 @ A3 )
         => ( ( ( equiv_2323626691971332616at_nat @ ( insert8109617542356073361at_nat @ X3 @ bot_bo1676454896636820109at_nat ) @ R )
              = ( equiv_2323626691971332616at_nat @ ( insert8109617542356073361at_nat @ Y3 @ bot_bo1676454896636820109at_nat ) @ R ) )
            = ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X3 @ Y3 ) @ R ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_724_eq__equiv__class__iff2,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,X3: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A3 @ R )
     => ( ( member8757157785044589968at_nat @ X3 @ A3 )
       => ( ( member8757157785044589968at_nat @ Y3 @ A3 )
         => ( ( ( equiv_8408515162552411438at_nat @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) @ R )
              = ( equiv_8408515162552411438at_nat @ ( insert9069300056098147895at_nat @ Y3 @ bot_bo228742789529271731at_nat ) @ R ) )
            = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y3 ) @ R ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_725_eq__equiv__class__iff2,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
      ( ( equiv_7491905333995573990on_val @ A3 @ R )
     => ( ( member2879030624002546611on_val @ X3 @ A3 )
       => ( ( member2879030624002546611on_val @ Y3 @ A3 )
         => ( ( ( equiv_6009051154071487125on_val @ ( insert3030518898131756108on_val @ X3 @ bot_bo6722054719108133734on_val ) @ R )
              = ( equiv_6009051154071487125on_val @ ( insert3030518898131756108on_val @ Y3 @ bot_bo6722054719108133734on_val ) @ R ) )
            = ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_726_eq__equiv__class__iff2,axiom,
    ! [A3: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o,X3: set_Product_prod_o_o,Y3: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R )
     => ( ( member3157015407008874800od_o_o @ X3 @ A3 )
       => ( ( member3157015407008874800od_o_o @ Y3 @ A3 )
         => ( ( ( equiv_3134926782319234254od_o_o @ ( insert3817420688681518039od_o_o @ X3 @ bot_bo8006819982598807891od_o_o ) @ R )
              = ( equiv_3134926782319234254od_o_o @ ( insert3817420688681518039od_o_o @ Y3 @ bot_bo8006819982598807891od_o_o ) @ R ) )
            = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_727_eq__equiv__class__iff2,axiom,
    ! [A3: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R )
     => ( ( member2643936169264416010at_nat @ X3 @ A3 )
       => ( ( member2643936169264416010at_nat @ Y3 @ A3 )
         => ( ( ( equiv_7923377459618294184at_nat @ ( insert9200635055090092081at_nat @ X3 @ bot_bo3083307316010499117at_nat ) @ R )
              = ( equiv_7923377459618294184at_nat @ ( insert9200635055090092081at_nat @ Y3 @ bot_bo3083307316010499117at_nat ) @ R ) )
            = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_728_min__ext__compat,axiom,
    ! [R2: set_Pr7011910401827753159od_o_o,S3: set_Pr7011910401827753159od_o_o] :
      ( ( ord_le6026832634849756263od_o_o @ ( relcom4901197468039914022od_o_o @ R2 @ S3 ) @ R2 )
     => ( ord_le7586318239878589543od_o_o @ ( relcom8634166700253352710od_o_o @ ( min_ex2812920868796073424od_o_o @ R2 ) @ ( sup_su4078869789063261851od_o_o @ ( min_ex2812920868796073424od_o_o @ S3 ) @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ bot_bo7073875226086086771od_o_o @ bot_bo7073875226086086771od_o_o ) @ bot_bo498831808395348787od_o_o ) ) ) @ ( min_ex2812920868796073424od_o_o @ R2 ) ) ) ).

% min_ext_compat
thf(fact_729_min__ext__compat,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( relcom7295539661566034944at_nat @ R2 @ S3 ) @ R2 )
     => ( ord_le1268244103169919719at_nat @ ( relcom6558574857482195296at_nat @ ( min_ex6901939911449802026at_nat @ R2 ) @ ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ S3 ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) @ ( min_ex6901939911449802026at_nat @ R2 ) ) ) ).

% min_ext_compat
thf(fact_730_wo__rel_Ocases__Total,axiom,
    ! [R: set_Product_prod_o_o,A: $o,B: $o,Phi4: $o > $o > $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ( ( ord_less_eq_set_o @ ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R ) )
       => ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R )
           => ( Phi4 @ A @ B ) )
         => ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A ) @ R )
             => ( Phi4 @ A @ B ) )
           => ( Phi4 @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_731_wo__rel_Ocases__Total,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat,Phi4: nat > nat > $o] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( ord_less_eq_set_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R ) )
       => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
           => ( Phi4 @ A @ B ) )
         => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R )
             => ( Phi4 @ A @ B ) )
           => ( Phi4 @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_732_wo__rel_Ocases__Total,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,Phi4: produc9003781588162623004on_val > produc9003781588162623004on_val > $o] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( ord_le6900622908513426738on_val @ ( insert3030518898131756108on_val @ A @ ( insert3030518898131756108on_val @ B @ bot_bo6722054719108133734on_val ) ) @ ( field_9010718613293825679on_val @ R ) )
       => ( ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
           => ( Phi4 @ A @ B ) )
         => ( ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ A ) @ R )
             => ( Phi4 @ A @ B ) )
           => ( Phi4 @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_733_wo__rel_Ocases__Total,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o,Phi4: set_Product_prod_o_o > set_Product_prod_o_o > $o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( ord_le2654036014095304327od_o_o @ ( insert3817420688681518039od_o_o @ A @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
           => ( Phi4 @ A @ B ) )
         => ( ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ A ) @ R )
             => ( Phi4 @ A @ B ) )
           => ( Phi4 @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_734_wo__rel_Ocases__Total,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,Phi4: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( ord_le2077887516847798113at_nat @ ( insert9200635055090092081at_nat @ A @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
           => ( Phi4 @ A @ B ) )
         => ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ A ) @ R )
             => ( Phi4 @ A @ B ) )
           => ( Phi4 @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_735_wf__eq__minimal2,axiom,
    ( wf_o
    = ( ^ [R3: set_Product_prod_o_o] :
        ! [A6: set_o] :
          ( ( ( ord_less_eq_set_o @ A6 @ ( field_o @ R3 ) )
            & ( A6 != bot_bot_set_o ) )
         => ? [X: $o] :
              ( ( member_o @ X @ A6 )
              & ! [Y: $o] :
                  ( ( member_o @ Y @ A6 )
                 => ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y @ X ) @ R3 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_736_wf__eq__minimal2,axiom,
    ( wf_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat] :
        ! [A6: set_nat] :
          ( ( ( ord_less_eq_set_nat @ A6 @ ( field_nat @ R3 ) )
            & ( A6 != bot_bot_set_nat ) )
         => ? [X: nat] :
              ( ( member_nat @ X @ A6 )
              & ! [Y: nat] :
                  ( ( member_nat @ Y @ A6 )
                 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R3 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_737_wf__eq__minimal2,axiom,
    ( wf_Pro3300669915054815219on_val
    = ( ^ [R3: set_Pr4305388979797279737on_val] :
        ! [A6: set_Pr7649655576376376530on_val] :
          ( ( ( ord_le6900622908513426738on_val @ A6 @ ( field_9010718613293825679on_val @ R3 ) )
            & ( A6 != bot_bo6722054719108133734on_val ) )
         => ? [X: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ X @ A6 )
              & ! [Y: produc9003781588162623004on_val] :
                  ( ( member2879030624002546611on_val @ Y @ A6 )
                 => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y @ X ) @ R3 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_738_wf__eq__minimal2,axiom,
    ( wf_set2384217916178216176od_o_o
    = ( ^ [R3: set_Pr1932065953672099015od_o_o] :
        ! [A6: set_se491199571785101031od_o_o] :
          ( ( ( ord_le2654036014095304327od_o_o @ A6 @ ( field_6101175518749917524od_o_o @ R3 ) )
            & ( A6 != bot_bo8006819982598807891od_o_o ) )
         => ? [X: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ X @ A6 )
              & ! [Y: set_Product_prod_o_o] :
                  ( ( member3157015407008874800od_o_o @ Y @ A6 )
                 => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ X ) @ R3 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_739_wf__eq__minimal2,axiom,
    ( wf_set7675013139414177994at_nat
    = ( ^ [R3: set_Pr4329608150637261639at_nat] :
        ! [A6: set_se7855581050983116737at_nat] :
          ( ( ( ord_le2077887516847798113at_nat @ A6 @ ( field_3145993188331922222at_nat @ R3 ) )
            & ( A6 != bot_bo3083307316010499117at_nat ) )
         => ? [X: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ X @ A6 )
              & ! [Y: set_Pr1261947904930325089at_nat] :
                  ( ( member2643936169264416010at_nat @ Y @ A6 )
                 => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X ) @ R3 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_740_Linear__order__in__diff__Id,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( order_91209724698019100od_o_o @ ( field_5121862139100741428od_o_o @ R ) @ R )
     => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R ) )
       => ( ( member444158400953824016od_o_o @ B @ ( field_5121862139100741428od_o_o @ R ) )
         => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R )
            = ( ~ ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B @ A ) @ ( minus_5063024033727795502od_o_o @ R @ id_Pro5927853813057311299od_o_o ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_741_Linear__order__in__diff__Id,axiom,
    ! [R: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( order_9086815480479278966at_nat @ ( field_7361900449976055182at_nat @ R ) @ R )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R ) )
       => ( ( member4095101504841534314at_nat @ B @ ( field_7361900449976055182at_nat @ R ) )
         => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R )
            = ( ~ ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B @ A ) @ ( minus_2318593706003940014at_nat @ R @ id_Pro6320881530060322077at_nat ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_742_Linear__order__in__diff__Id,axiom,
    ! [R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( order_2711500226769404828at_nat @ ( field_2862844103579723188at_nat @ R ) @ R )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R ) )
       => ( ( member8757157785044589968at_nat @ B @ ( field_2862844103579723188at_nat @ R ) )
         => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R )
            = ( ~ ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ A ) @ ( minus_8235382265540049710at_nat @ R @ id_Pro5428700290092274115at_nat ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_743_Linear__order__in__diff__Id,axiom,
    ! [R: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( order_6238756238976269133r_on_o @ ( field_o @ R ) @ R )
     => ( ( member_o @ A @ ( field_o @ R ) )
       => ( ( member_o @ B @ ( field_o @ R ) )
         => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R )
            = ( ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A ) @ ( minus_7052513330764642158od_o_o @ R @ id_o ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_744_Linear__order__in__diff__Id,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R ) @ R )
     => ( ( member_nat @ A @ ( field_nat @ R ) )
       => ( ( member_nat @ B @ ( field_nat @ R ) )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
            = ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ ( minus_1356011639430497352at_nat @ R @ id_nat ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_745_Linear__order__in__diff__Id,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( order_6061556925426544295on_val @ ( field_9010718613293825679on_val @ R ) @ R )
     => ( ( member2879030624002546611on_val @ A @ ( field_9010718613293825679on_val @ R ) )
       => ( ( member2879030624002546611on_val @ B @ ( field_9010718613293825679on_val @ R ) )
         => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
            = ( ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ A ) @ ( minus_7211471524202483680on_val @ R @ id_Pro1455714331835775424on_val ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_746_Linear__order__in__diff__Id,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( order_2625247082195685180od_o_o @ ( field_6101175518749917524od_o_o @ R ) @ R )
     => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( member3157015407008874800od_o_o @ B @ ( field_6101175518749917524od_o_o @ R ) )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
            = ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ A ) @ ( minus_5704166846132356910od_o_o @ R @ id_set2647307015711338339od_o_o ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_747_Linear__order__in__diff__Id,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( order_5942792088423767318at_nat @ ( field_3145993188331922222at_nat @ R ) @ R )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( member2643936169264416010at_nat @ B @ ( field_3145993188331922222at_nat @ R ) )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
            = ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ A ) @ ( minus_3314409938677909166at_nat @ R @ id_set1796276052702428605at_nat ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_748_cofinal__def,axiom,
    ( bNF_Ca4363140458462388713inal_o
    = ( ^ [A6: set_o,R3: set_Product_prod_o_o] :
        ! [X: $o] :
          ( ( member_o @ X @ ( field_o @ R3 ) )
         => ? [Y: $o] :
              ( ( member_o @ Y @ A6 )
              & ( X = (~ Y) )
              & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R3 ) ) ) ) ) ).

% cofinal_def
thf(fact_749_cofinal__def,axiom,
    ( bNF_Ca5159621140400407487al_nat
    = ( ^ [A6: set_nat,R3: set_Pr1261947904930325089at_nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ ( field_nat @ R3 ) )
         => ? [Y: nat] :
              ( ( member_nat @ Y @ A6 )
              & ( X != Y )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 ) ) ) ) ) ).

% cofinal_def
thf(fact_750_cofinal__def,axiom,
    ( bNF_Ca4299268644204761867on_val
    = ( ^ [A6: set_Pr7649655576376376530on_val,R3: set_Pr4305388979797279737on_val] :
        ! [X: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ X @ ( field_9010718613293825679on_val @ R3 ) )
         => ? [Y: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ Y @ A6 )
              & ( X != Y )
              & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ R3 ) ) ) ) ) ).

% cofinal_def
thf(fact_751_cofinal__def,axiom,
    ( bNF_Ca3407573850950408152od_o_o
    = ( ^ [A6: set_se491199571785101031od_o_o,R3: set_Pr1932065953672099015od_o_o] :
        ! [X: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X @ ( field_6101175518749917524od_o_o @ R3 ) )
         => ? [Y: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Y @ A6 )
              & ( X != Y )
              & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R3 ) ) ) ) ) ).

% cofinal_def
thf(fact_752_cofinal__def,axiom,
    ( bNF_Ca8448034394003515314at_nat
    = ( ^ [A6: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat] :
        ! [X: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X @ ( field_3145993188331922222at_nat @ R3 ) )
         => ? [Y: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y @ A6 )
              & ( X != Y )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R3 ) ) ) ) ) ).

% cofinal_def
thf(fact_753_pair__in__Id__conv,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ id_Pro1455714331835775424on_val )
      = ( A = B ) ) ).

% pair_in_Id_conv
thf(fact_754_pair__in__Id__conv,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ id_set2647307015711338339od_o_o )
      = ( A = B ) ) ).

% pair_in_Id_conv
thf(fact_755_pair__in__Id__conv,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ id_set1796276052702428605at_nat )
      = ( A = B ) ) ).

% pair_in_Id_conv
thf(fact_756_IdI,axiom,
    ! [A: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ A ) @ id_Pro1455714331835775424on_val ) ).

% IdI
thf(fact_757_IdI,axiom,
    ! [A: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A ) @ id_set2647307015711338339od_o_o ) ).

% IdI
thf(fact_758_IdI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ id_set1796276052702428605at_nat ) ).

% IdI
thf(fact_759_accp__subset,axiom,
    ! [R1: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,R22: produc9003781588162623004on_val > produc9003781588162623004on_val > $o] :
      ( ( ord_le616650860478780142_val_o @ R1 @ R22 )
     => ( ord_le6140313152057869779_val_o @ ( accp_P4405556307444930259on_val @ R22 ) @ ( accp_P4405556307444930259on_val @ R1 ) ) ) ).

% accp_subset
thf(fact_760_accp__subset,axiom,
    ! [R1: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,R22: produc1317171644805857433on_val > produc1317171644805857433on_val > $o] :
      ( ( ord_le2319172193063290574_val_o @ R1 @ R22 )
     => ( ord_le8528861841627966596_val_o @ ( accp_P6797916352740768290on_val @ R22 ) @ ( accp_P6797916352740768290on_val @ R1 ) ) ) ).

% accp_subset
thf(fact_761_wo__rel_Owell__order__induct,axiom,
    ! [R: set_Pr4305388979797279737on_val,P: produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ! [X2: produc9003781588162623004on_val] :
            ( ! [Y6: produc9003781588162623004on_val] :
                ( ( ( Y6 != X2 )
                  & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y6 @ X2 ) @ R ) )
               => ( P @ Y6 ) )
           => ( P @ X2 ) )
       => ( P @ A ) ) ) ).

% wo_rel.well_order_induct
thf(fact_762_wo__rel_Owell__order__induct,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,P: set_Product_prod_o_o > $o,A: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ! [X2: set_Product_prod_o_o] :
            ( ! [Y6: set_Product_prod_o_o] :
                ( ( ( Y6 != X2 )
                  & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y6 @ X2 ) @ R ) )
               => ( P @ Y6 ) )
           => ( P @ X2 ) )
       => ( P @ A ) ) ) ).

% wo_rel.well_order_induct
thf(fact_763_wo__rel_Owell__order__induct,axiom,
    ! [R: set_Pr4329608150637261639at_nat,P: set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ! [X2: set_Pr1261947904930325089at_nat] :
            ( ! [Y6: set_Pr1261947904930325089at_nat] :
                ( ( ( Y6 != X2 )
                  & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y6 @ X2 ) @ R ) )
               => ( P @ Y6 ) )
           => ( P @ X2 ) )
       => ( P @ A ) ) ) ).

% wo_rel.well_order_induct
thf(fact_764_wf__def,axiom,
    ( wf_Pro3300669915054815219on_val
    = ( ^ [R3: set_Pr4305388979797279737on_val] :
        ! [P3: produc9003781588162623004on_val > $o] :
          ( ! [X: produc9003781588162623004on_val] :
              ( ! [Y: produc9003781588162623004on_val] :
                  ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y @ X ) @ R3 )
                 => ( P3 @ Y ) )
             => ( P3 @ X ) )
         => ! [X6: produc9003781588162623004on_val] : ( P3 @ X6 ) ) ) ) ).

% wf_def
thf(fact_765_wf__def,axiom,
    ( wf_set2384217916178216176od_o_o
    = ( ^ [R3: set_Pr1932065953672099015od_o_o] :
        ! [P3: set_Product_prod_o_o > $o] :
          ( ! [X: set_Product_prod_o_o] :
              ( ! [Y: set_Product_prod_o_o] :
                  ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ X ) @ R3 )
                 => ( P3 @ Y ) )
             => ( P3 @ X ) )
         => ! [X6: set_Product_prod_o_o] : ( P3 @ X6 ) ) ) ) ).

% wf_def
thf(fact_766_wf__def,axiom,
    ( wf_set7675013139414177994at_nat
    = ( ^ [R3: set_Pr4329608150637261639at_nat] :
        ! [P3: set_Pr1261947904930325089at_nat > $o] :
          ( ! [X: set_Pr1261947904930325089at_nat] :
              ( ! [Y: set_Pr1261947904930325089at_nat] :
                  ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X ) @ R3 )
                 => ( P3 @ Y ) )
             => ( P3 @ X ) )
         => ! [X6: set_Pr1261947904930325089at_nat] : ( P3 @ X6 ) ) ) ) ).

% wf_def
thf(fact_767_wfE__min,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,X3: produc2934264451710624999od_o_o,Q: set_Pr1932065953672099015od_o_o] :
      ( ( wf_Pro3213319136295691984od_o_o @ R2 )
     => ( ( member444158400953824016od_o_o @ X3 @ Q )
       => ~ ! [Z: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ Z @ Q )
             => ~ ! [Y6: produc2934264451710624999od_o_o] :
                    ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y6 @ Z ) @ R2 )
                   => ~ ( member444158400953824016od_o_o @ Y6 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_768_wfE__min,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,X3: produc732395585841259969at_nat,Q: set_Pr457366540195662369at_nat] :
      ( ( wf_Pro661967360799104810at_nat @ R2 )
     => ( ( member4095101504841534314at_nat @ X3 @ Q )
       => ~ ! [Z: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ Z @ Q )
             => ~ ! [Y6: produc732395585841259969at_nat] :
                    ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y6 @ Z ) @ R2 )
                   => ~ ( member4095101504841534314at_nat @ Y6 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_769_wfE__min,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,X3: produc3843707927480180839at_nat,Q: set_Pr4329608150637261639at_nat] :
      ( ( wf_Pro2357547232388600144at_nat @ R2 )
     => ( ( member8757157785044589968at_nat @ X3 @ Q )
       => ~ ! [Z: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ Z @ Q )
             => ~ ! [Y6: produc3843707927480180839at_nat] :
                    ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y6 @ Z ) @ R2 )
                   => ~ ( member8757157785044589968at_nat @ Y6 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_770_wfE__min,axiom,
    ! [R2: set_Pr4305388979797279737on_val,X3: produc9003781588162623004on_val,Q: set_Pr7649655576376376530on_val] :
      ( ( wf_Pro3300669915054815219on_val @ R2 )
     => ( ( member2879030624002546611on_val @ X3 @ Q )
       => ~ ! [Z: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ Z @ Q )
             => ~ ! [Y6: produc9003781588162623004on_val] :
                    ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y6 @ Z ) @ R2 )
                   => ~ ( member2879030624002546611on_val @ Y6 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_771_wfE__min,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,X3: set_Product_prod_o_o,Q: set_se491199571785101031od_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R2 )
     => ( ( member3157015407008874800od_o_o @ X3 @ Q )
       => ~ ! [Z: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Z @ Q )
             => ~ ! [Y6: set_Product_prod_o_o] :
                    ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y6 @ Z ) @ R2 )
                   => ~ ( member3157015407008874800od_o_o @ Y6 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_772_wfE__min,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,X3: set_Pr1261947904930325089at_nat,Q: set_se7855581050983116737at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R2 )
     => ( ( member2643936169264416010at_nat @ X3 @ Q )
       => ~ ! [Z: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Z @ Q )
             => ~ ! [Y6: set_Pr1261947904930325089at_nat] :
                    ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y6 @ Z ) @ R2 )
                   => ~ ( member2643936169264416010at_nat @ Y6 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_773_wfI__min,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o] :
      ( ! [X2: produc2934264451710624999od_o_o,Q2: set_Pr1932065953672099015od_o_o] :
          ( ( member444158400953824016od_o_o @ X2 @ Q2 )
         => ? [Xa2: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ Xa2 @ Q2 )
              & ! [Y4: produc2934264451710624999od_o_o] :
                  ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y4 @ Xa2 ) @ R2 )
                 => ~ ( member444158400953824016od_o_o @ Y4 @ Q2 ) ) ) )
     => ( wf_Pro3213319136295691984od_o_o @ R2 ) ) ).

% wfI_min
thf(fact_774_wfI__min,axiom,
    ! [R2: set_Pr4427867301352232263at_nat] :
      ( ! [X2: produc732395585841259969at_nat,Q2: set_Pr457366540195662369at_nat] :
          ( ( member4095101504841534314at_nat @ X2 @ Q2 )
         => ? [Xa2: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ Xa2 @ Q2 )
              & ! [Y4: produc732395585841259969at_nat] :
                  ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y4 @ Xa2 ) @ R2 )
                 => ~ ( member4095101504841534314at_nat @ Y4 @ Q2 ) ) ) )
     => ( wf_Pro661967360799104810at_nat @ R2 ) ) ).

% wfI_min
thf(fact_775_wfI__min,axiom,
    ! [R2: set_Pr5564308138774400199at_nat] :
      ( ! [X2: produc3843707927480180839at_nat,Q2: set_Pr4329608150637261639at_nat] :
          ( ( member8757157785044589968at_nat @ X2 @ Q2 )
         => ? [Xa2: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ Xa2 @ Q2 )
              & ! [Y4: produc3843707927480180839at_nat] :
                  ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y4 @ Xa2 ) @ R2 )
                 => ~ ( member8757157785044589968at_nat @ Y4 @ Q2 ) ) ) )
     => ( wf_Pro2357547232388600144at_nat @ R2 ) ) ).

% wfI_min
thf(fact_776_wfI__min,axiom,
    ! [R2: set_Pr4305388979797279737on_val] :
      ( ! [X2: produc9003781588162623004on_val,Q2: set_Pr7649655576376376530on_val] :
          ( ( member2879030624002546611on_val @ X2 @ Q2 )
         => ? [Xa2: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ Xa2 @ Q2 )
              & ! [Y4: produc9003781588162623004on_val] :
                  ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y4 @ Xa2 ) @ R2 )
                 => ~ ( member2879030624002546611on_val @ Y4 @ Q2 ) ) ) )
     => ( wf_Pro3300669915054815219on_val @ R2 ) ) ).

% wfI_min
thf(fact_777_wfI__min,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o] :
      ( ! [X2: set_Product_prod_o_o,Q2: set_se491199571785101031od_o_o] :
          ( ( member3157015407008874800od_o_o @ X2 @ Q2 )
         => ? [Xa2: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Xa2 @ Q2 )
              & ! [Y4: set_Product_prod_o_o] :
                  ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ Xa2 ) @ R2 )
                 => ~ ( member3157015407008874800od_o_o @ Y4 @ Q2 ) ) ) )
     => ( wf_set2384217916178216176od_o_o @ R2 ) ) ).

% wfI_min
thf(fact_778_wfI__min,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Q2: set_se7855581050983116737at_nat] :
          ( ( member2643936169264416010at_nat @ X2 @ Q2 )
         => ? [Xa2: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Xa2 @ Q2 )
              & ! [Y4: set_Pr1261947904930325089at_nat] :
                  ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ Xa2 ) @ R2 )
                 => ~ ( member2643936169264416010at_nat @ Y4 @ Q2 ) ) ) )
     => ( wf_set7675013139414177994at_nat @ R2 ) ) ).

% wfI_min
thf(fact_779_wfUNIVI,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ! [P5: produc9003781588162623004on_val > $o,X2: produc9003781588162623004on_val] :
          ( ! [Xa2: produc9003781588162623004on_val] :
              ( ! [Y4: produc9003781588162623004on_val] :
                  ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y4 @ Xa2 ) @ R )
                 => ( P5 @ Y4 ) )
             => ( P5 @ Xa2 ) )
         => ( P5 @ X2 ) )
     => ( wf_Pro3300669915054815219on_val @ R ) ) ).

% wfUNIVI
thf(fact_780_wfUNIVI,axiom,
    ! [R: set_Pr1932065953672099015od_o_o] :
      ( ! [P5: set_Product_prod_o_o > $o,X2: set_Product_prod_o_o] :
          ( ! [Xa2: set_Product_prod_o_o] :
              ( ! [Y4: set_Product_prod_o_o] :
                  ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ Xa2 ) @ R )
                 => ( P5 @ Y4 ) )
             => ( P5 @ Xa2 ) )
         => ( P5 @ X2 ) )
     => ( wf_set2384217916178216176od_o_o @ R ) ) ).

% wfUNIVI
thf(fact_781_wfUNIVI,axiom,
    ! [R: set_Pr4329608150637261639at_nat] :
      ( ! [P5: set_Pr1261947904930325089at_nat > $o,X2: set_Pr1261947904930325089at_nat] :
          ( ! [Xa2: set_Pr1261947904930325089at_nat] :
              ( ! [Y4: set_Pr1261947904930325089at_nat] :
                  ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ Xa2 ) @ R )
                 => ( P5 @ Y4 ) )
             => ( P5 @ Xa2 ) )
         => ( P5 @ X2 ) )
     => ( wf_set7675013139414177994at_nat @ R ) ) ).

% wfUNIVI
thf(fact_782_wf__asym,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,X3: produc9003781588162623004on_val] :
      ( ( wf_Pro3300669915054815219on_val @ R )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ X3 ) @ R )
       => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ A ) @ R ) ) ) ).

% wf_asym
thf(fact_783_wf__asym,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,X3: set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ X3 ) @ R )
       => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ A ) @ R ) ) ) ).

% wf_asym
thf(fact_784_wf__asym,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,X3: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ X3 ) @ R )
       => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ A ) @ R ) ) ) ).

% wf_asym
thf(fact_785_wf__induct,axiom,
    ! [R: set_Pr4305388979797279737on_val,P: produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val] :
      ( ( wf_Pro3300669915054815219on_val @ R )
     => ( ! [X2: produc9003781588162623004on_val] :
            ( ! [Y6: produc9003781588162623004on_val] :
                ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y6 @ X2 ) @ R )
               => ( P @ Y6 ) )
           => ( P @ X2 ) )
       => ( P @ A ) ) ) ).

% wf_induct
thf(fact_786_wf__induct,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,P: set_Product_prod_o_o > $o,A: set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R )
     => ( ! [X2: set_Product_prod_o_o] :
            ( ! [Y6: set_Product_prod_o_o] :
                ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y6 @ X2 ) @ R )
               => ( P @ Y6 ) )
           => ( P @ X2 ) )
       => ( P @ A ) ) ) ).

% wf_induct
thf(fact_787_wf__induct,axiom,
    ! [R: set_Pr4329608150637261639at_nat,P: set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R )
     => ( ! [X2: set_Pr1261947904930325089at_nat] :
            ( ! [Y6: set_Pr1261947904930325089at_nat] :
                ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y6 @ X2 ) @ R )
               => ( P @ Y6 ) )
           => ( P @ X2 ) )
       => ( P @ A ) ) ) ).

% wf_induct
thf(fact_788_wf__irrefl,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val] :
      ( ( wf_Pro3300669915054815219on_val @ R )
     => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ A ) @ R ) ) ).

% wf_irrefl
thf(fact_789_wf__irrefl,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R )
     => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A ) @ R ) ) ).

% wf_irrefl
thf(fact_790_wf__irrefl,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R )
     => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ R ) ) ).

% wf_irrefl
thf(fact_791_wf__not__sym,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,X3: produc9003781588162623004on_val] :
      ( ( wf_Pro3300669915054815219on_val @ R )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ X3 ) @ R )
       => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ A ) @ R ) ) ) ).

% wf_not_sym
thf(fact_792_wf__not__sym,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,X3: set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ X3 ) @ R )
       => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ A ) @ R ) ) ) ).

% wf_not_sym
thf(fact_793_wf__not__sym,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,X3: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ X3 ) @ R )
       => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ A ) @ R ) ) ) ).

% wf_not_sym
thf(fact_794_wf__not__refl,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val] :
      ( ( wf_Pro3300669915054815219on_val @ R )
     => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ A ) @ R ) ) ).

% wf_not_refl
thf(fact_795_wf__not__refl,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R )
     => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A ) @ R ) ) ).

% wf_not_refl
thf(fact_796_wf__not__refl,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R )
     => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ R ) ) ).

% wf_not_refl
thf(fact_797_wf__eq__minimal,axiom,
    ( wf_Pro3213319136295691984od_o_o
    = ( ^ [R3: set_Pr1013884327308616903od_o_o] :
        ! [Q3: set_Pr1932065953672099015od_o_o] :
          ( ? [X: produc2934264451710624999od_o_o] : ( member444158400953824016od_o_o @ X @ Q3 )
         => ? [X: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ X @ Q3 )
              & ! [Y: produc2934264451710624999od_o_o] :
                  ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y @ X ) @ R3 )
                 => ~ ( member444158400953824016od_o_o @ Y @ Q3 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_798_wf__eq__minimal,axiom,
    ( wf_Pro661967360799104810at_nat
    = ( ^ [R3: set_Pr4427867301352232263at_nat] :
        ! [Q3: set_Pr457366540195662369at_nat] :
          ( ? [X: produc732395585841259969at_nat] : ( member4095101504841534314at_nat @ X @ Q3 )
         => ? [X: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ X @ Q3 )
              & ! [Y: produc732395585841259969at_nat] :
                  ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y @ X ) @ R3 )
                 => ~ ( member4095101504841534314at_nat @ Y @ Q3 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_799_wf__eq__minimal,axiom,
    ( wf_Pro2357547232388600144at_nat
    = ( ^ [R3: set_Pr5564308138774400199at_nat] :
        ! [Q3: set_Pr4329608150637261639at_nat] :
          ( ? [X: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X @ Q3 )
         => ? [X: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X @ Q3 )
              & ! [Y: produc3843707927480180839at_nat] :
                  ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y @ X ) @ R3 )
                 => ~ ( member8757157785044589968at_nat @ Y @ Q3 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_800_wf__eq__minimal,axiom,
    ( wf_Pro3300669915054815219on_val
    = ( ^ [R3: set_Pr4305388979797279737on_val] :
        ! [Q3: set_Pr7649655576376376530on_val] :
          ( ? [X: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X @ Q3 )
         => ? [X: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ X @ Q3 )
              & ! [Y: produc9003781588162623004on_val] :
                  ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y @ X ) @ R3 )
                 => ~ ( member2879030624002546611on_val @ Y @ Q3 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_801_wf__eq__minimal,axiom,
    ( wf_set2384217916178216176od_o_o
    = ( ^ [R3: set_Pr1932065953672099015od_o_o] :
        ! [Q3: set_se491199571785101031od_o_o] :
          ( ? [X: set_Product_prod_o_o] : ( member3157015407008874800od_o_o @ X @ Q3 )
         => ? [X: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ X @ Q3 )
              & ! [Y: set_Product_prod_o_o] :
                  ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ X ) @ R3 )
                 => ~ ( member3157015407008874800od_o_o @ Y @ Q3 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_802_wf__eq__minimal,axiom,
    ( wf_set7675013139414177994at_nat
    = ( ^ [R3: set_Pr4329608150637261639at_nat] :
        ! [Q3: set_se7855581050983116737at_nat] :
          ( ? [X: set_Pr1261947904930325089at_nat] : ( member2643936169264416010at_nat @ X @ Q3 )
         => ? [X: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ X @ Q3 )
              & ! [Y: set_Pr1261947904930325089at_nat] :
                  ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X ) @ R3 )
                 => ~ ( member2643936169264416010at_nat @ Y @ Q3 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_803_wf__induct__rule,axiom,
    ! [R: set_Pr4305388979797279737on_val,P: produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val] :
      ( ( wf_Pro3300669915054815219on_val @ R )
     => ( ! [X2: produc9003781588162623004on_val] :
            ( ! [Y6: produc9003781588162623004on_val] :
                ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y6 @ X2 ) @ R )
               => ( P @ Y6 ) )
           => ( P @ X2 ) )
       => ( P @ A ) ) ) ).

% wf_induct_rule
thf(fact_804_wf__induct__rule,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,P: set_Product_prod_o_o > $o,A: set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R )
     => ( ! [X2: set_Product_prod_o_o] :
            ( ! [Y6: set_Product_prod_o_o] :
                ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y6 @ X2 ) @ R )
               => ( P @ Y6 ) )
           => ( P @ X2 ) )
       => ( P @ A ) ) ) ).

% wf_induct_rule
thf(fact_805_wf__induct__rule,axiom,
    ! [R: set_Pr4329608150637261639at_nat,P: set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R )
     => ( ! [X2: set_Pr1261947904930325089at_nat] :
            ( ! [Y6: set_Pr1261947904930325089at_nat] :
                ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y6 @ X2 ) @ R )
               => ( P @ Y6 ) )
           => ( P @ X2 ) )
       => ( P @ A ) ) ) ).

% wf_induct_rule
thf(fact_806_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr4305388979797279737on_val,R: set_Pr4305388979797279737on_val,F: produc9003781588162623004on_val > produc9003781588162623004on_val] :
      ( ( wf_Pro3300669915054815219on_val @ S )
     => ( ! [X2: produc9003781588162623004on_val,Y4: produc9003781588162623004on_val] :
            ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y4 ) @ R )
           => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( F @ X2 ) @ ( F @ Y4 ) ) @ S ) )
       => ( wf_Pro3300669915054815219on_val @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_807_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr1932065953672099015od_o_o,R: set_Pr4305388979797279737on_val,F: produc9003781588162623004on_val > set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ S )
     => ( ! [X2: produc9003781588162623004on_val,Y4: produc9003781588162623004on_val] :
            ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y4 ) @ R )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( F @ X2 ) @ ( F @ Y4 ) ) @ S ) )
       => ( wf_Pro3300669915054815219on_val @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_808_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr4329608150637261639at_nat,R: set_Pr4305388979797279737on_val,F: produc9003781588162623004on_val > set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ S )
     => ( ! [X2: produc9003781588162623004on_val,Y4: produc9003781588162623004on_val] :
            ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y4 ) @ R )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) @ S ) )
       => ( wf_Pro3300669915054815219on_val @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_809_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr4305388979797279737on_val,R: set_Pr1932065953672099015od_o_o,F: set_Product_prod_o_o > produc9003781588162623004on_val] :
      ( ( wf_Pro3300669915054815219on_val @ S )
     => ( ! [X2: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y4 ) @ R )
           => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( F @ X2 ) @ ( F @ Y4 ) ) @ S ) )
       => ( wf_set2384217916178216176od_o_o @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_810_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr1932065953672099015od_o_o,R: set_Pr1932065953672099015od_o_o,F: set_Product_prod_o_o > set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ S )
     => ( ! [X2: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y4 ) @ R )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( F @ X2 ) @ ( F @ Y4 ) ) @ S ) )
       => ( wf_set2384217916178216176od_o_o @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_811_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr4329608150637261639at_nat,R: set_Pr1932065953672099015od_o_o,F: set_Product_prod_o_o > set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ S )
     => ( ! [X2: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y4 ) @ R )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) @ S ) )
       => ( wf_set2384217916178216176od_o_o @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_812_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr4305388979797279737on_val,R: set_Pr4329608150637261639at_nat,F: set_Pr1261947904930325089at_nat > produc9003781588162623004on_val] :
      ( ( wf_Pro3300669915054815219on_val @ S )
     => ( ! [X2: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y4 ) @ R )
           => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( F @ X2 ) @ ( F @ Y4 ) ) @ S ) )
       => ( wf_set7675013139414177994at_nat @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_813_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr1932065953672099015od_o_o,R: set_Pr4329608150637261639at_nat,F: set_Pr1261947904930325089at_nat > set_Product_prod_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ S )
     => ( ! [X2: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y4 ) @ R )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( F @ X2 ) @ ( F @ Y4 ) ) @ S ) )
       => ( wf_set7675013139414177994at_nat @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_814_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr4329608150637261639at_nat,R: set_Pr4329608150637261639at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ( wf_set7675013139414177994at_nat @ S )
     => ( ! [X2: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y4 ) @ R )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) @ S ) )
       => ( wf_set7675013139414177994at_nat @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_815_accp__subset__induct,axiom,
    ! [D2: produc9003781588162623004on_val > $o,R2: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,X3: produc9003781588162623004on_val,P: produc9003781588162623004on_val > $o] :
      ( ( ord_le6140313152057869779_val_o @ D2 @ ( accp_P4405556307444930259on_val @ R2 ) )
     => ( ! [X2: produc9003781588162623004on_val,Z: produc9003781588162623004on_val] :
            ( ( D2 @ X2 )
           => ( ( R2 @ Z @ X2 )
             => ( D2 @ Z ) ) )
       => ( ( D2 @ X3 )
         => ( ! [X2: produc9003781588162623004on_val] :
                ( ( D2 @ X2 )
               => ( ! [Z2: produc9003781588162623004on_val] :
                      ( ( R2 @ Z2 @ X2 )
                     => ( P @ Z2 ) )
                 => ( P @ X2 ) ) )
           => ( P @ X3 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_816_accp__subset__induct,axiom,
    ! [D2: produc1317171644805857433on_val > $o,R2: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,X3: produc1317171644805857433on_val,P: produc1317171644805857433on_val > $o] :
      ( ( ord_le8528861841627966596_val_o @ D2 @ ( accp_P6797916352740768290on_val @ R2 ) )
     => ( ! [X2: produc1317171644805857433on_val,Z: produc1317171644805857433on_val] :
            ( ( D2 @ X2 )
           => ( ( R2 @ Z @ X2 )
             => ( D2 @ Z ) ) )
       => ( ( D2 @ X3 )
         => ( ! [X2: produc1317171644805857433on_val] :
                ( ( D2 @ X2 )
               => ( ! [Z2: produc1317171644805857433on_val] :
                      ( ( R2 @ Z2 @ X2 )
                     => ( P @ Z2 ) )
                 => ( P @ X2 ) ) )
           => ( P @ X3 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_817_IdE,axiom,
    ! [P4: produc1317171644805857433on_val] :
      ( ( member6505152944228695362on_val @ P4 @ id_Pro1455714331835775424on_val )
     => ~ ! [X2: produc9003781588162623004on_val] :
            ( P4
           != ( produc4958730763592889809on_val @ X2 @ X2 ) ) ) ).

% IdE
thf(fact_818_IdE,axiom,
    ! [P4: produc2934264451710624999od_o_o] :
      ( ( member444158400953824016od_o_o @ P4 @ id_set2647307015711338339od_o_o )
     => ~ ! [X2: set_Product_prod_o_o] :
            ( P4
           != ( produc763777882069021527od_o_o @ X2 @ X2 ) ) ) ).

% IdE
thf(fact_819_IdE,axiom,
    ! [P4: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ P4 @ id_set1796276052702428605at_nat )
     => ~ ! [X2: set_Pr1261947904930325089at_nat] :
            ( P4
           != ( produc2922128104949294807at_nat @ X2 @ X2 ) ) ) ).

% IdE
thf(fact_820_wo__rel_Oin__notinI,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,J: produc2934264451710624999od_o_o,I: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R )
     => ( ( ~ ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ J @ I ) @ R )
          | ( J = I ) )
       => ( ( member444158400953824016od_o_o @ I @ ( field_5121862139100741428od_o_o @ R ) )
         => ( ( member444158400953824016od_o_o @ J @ ( field_5121862139100741428od_o_o @ R ) )
           => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I @ J ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_821_wo__rel_Oin__notinI,axiom,
    ! [R: set_Pr4427867301352232263at_nat,J: produc732395585841259969at_nat,I: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R )
     => ( ( ~ ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ J @ I ) @ R )
          | ( J = I ) )
       => ( ( member4095101504841534314at_nat @ I @ ( field_7361900449976055182at_nat @ R ) )
         => ( ( member4095101504841534314at_nat @ J @ ( field_7361900449976055182at_nat @ R ) )
           => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I @ J ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_822_wo__rel_Oin__notinI,axiom,
    ! [R: set_Pr5564308138774400199at_nat,J: produc3843707927480180839at_nat,I: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R )
     => ( ( ~ ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ J @ I ) @ R )
          | ( J = I ) )
       => ( ( member8757157785044589968at_nat @ I @ ( field_2862844103579723188at_nat @ R ) )
         => ( ( member8757157785044589968at_nat @ J @ ( field_2862844103579723188at_nat @ R ) )
           => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I @ J ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_823_wo__rel_Oin__notinI,axiom,
    ! [R: set_Product_prod_o_o,J: $o,I: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ( ( ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ J @ I ) @ R )
          | ( J = I ) )
       => ( ( member_o @ I @ ( field_o @ R ) )
         => ( ( member_o @ J @ ( field_o @ R ) )
           => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I @ J ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_824_wo__rel_Oin__notinI,axiom,
    ! [R: set_Pr1261947904930325089at_nat,J: nat,I: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ J @ I ) @ R )
          | ( J = I ) )
       => ( ( member_nat @ I @ ( field_nat @ R ) )
         => ( ( member_nat @ J @ ( field_nat @ R ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_825_wo__rel_Oin__notinI,axiom,
    ! [R: set_Pr4305388979797279737on_val,J: produc9003781588162623004on_val,I: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ J @ I ) @ R )
          | ( J = I ) )
       => ( ( member2879030624002546611on_val @ I @ ( field_9010718613293825679on_val @ R ) )
         => ( ( member2879030624002546611on_val @ J @ ( field_9010718613293825679on_val @ R ) )
           => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ I @ J ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_826_wo__rel_Oin__notinI,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,J: set_Product_prod_o_o,I: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ J @ I ) @ R )
          | ( J = I ) )
       => ( ( member3157015407008874800od_o_o @ I @ ( field_6101175518749917524od_o_o @ R ) )
         => ( ( member3157015407008874800od_o_o @ J @ ( field_6101175518749917524od_o_o @ R ) )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I @ J ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_827_wo__rel_Oin__notinI,axiom,
    ! [R: set_Pr4329608150637261639at_nat,J: set_Pr1261947904930325089at_nat,I: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ J @ I ) @ R )
          | ( J = I ) )
       => ( ( member2643936169264416010at_nat @ I @ ( field_3145993188331922222at_nat @ R ) )
         => ( ( member2643936169264416010at_nat @ J @ ( field_3145993188331922222at_nat @ R ) )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_828_wo__rel_OTOTALS,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ! [X7: $o] :
          ( ( member_o @ X7 @ ( field_o @ R ) )
         => ! [Xa2: $o] :
              ( ( member_o @ Xa2 @ ( field_o @ R ) )
             => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X7 @ Xa2 ) @ R )
                | ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Xa2 @ X7 ) @ R ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_829_wo__rel_OTOTALS,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ! [X7: nat] :
          ( ( member_nat @ X7 @ ( field_nat @ R ) )
         => ! [Xa2: nat] :
              ( ( member_nat @ Xa2 @ ( field_nat @ R ) )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X7 @ Xa2 ) @ R )
                | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa2 @ X7 ) @ R ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_830_wo__rel_OTOTALS,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ! [X7: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ X7 @ ( field_9010718613293825679on_val @ R ) )
         => ! [Xa2: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ Xa2 @ ( field_9010718613293825679on_val @ R ) )
             => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X7 @ Xa2 ) @ R )
                | ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Xa2 @ X7 ) @ R ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_831_wo__rel_OTOTALS,axiom,
    ! [R: set_Pr1932065953672099015od_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ! [X7: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X7 @ ( field_6101175518749917524od_o_o @ R ) )
         => ! [Xa2: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Xa2 @ ( field_6101175518749917524od_o_o @ R ) )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X7 @ Xa2 ) @ R )
                | ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Xa2 @ X7 ) @ R ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_832_wo__rel_OTOTALS,axiom,
    ! [R: set_Pr4329608150637261639at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ! [X7: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X7 @ ( field_3145993188331922222at_nat @ R ) )
         => ! [Xa2: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Xa2 @ ( field_3145993188331922222at_nat @ R ) )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ Xa2 ) @ R )
                | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Xa2 @ X7 ) @ R ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_833_wfE__min_H,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,Q: set_Pr1932065953672099015od_o_o] :
      ( ( wf_Pro3213319136295691984od_o_o @ R2 )
     => ( ( Q != bot_bo498831808395348787od_o_o )
       => ~ ! [Z: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ Z @ Q )
             => ~ ! [Y6: produc2934264451710624999od_o_o] :
                    ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y6 @ Z ) @ R2 )
                   => ~ ( member444158400953824016od_o_o @ Y6 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_834_wfE__min_H,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,Q: set_Pr457366540195662369at_nat] :
      ( ( wf_Pro661967360799104810at_nat @ R2 )
     => ( ( Q != bot_bo1676454896636820109at_nat )
       => ~ ! [Z: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ Z @ Q )
             => ~ ! [Y6: produc732395585841259969at_nat] :
                    ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y6 @ Z ) @ R2 )
                   => ~ ( member4095101504841534314at_nat @ Y6 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_835_wfE__min_H,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,Q: set_Pr4329608150637261639at_nat] :
      ( ( wf_Pro2357547232388600144at_nat @ R2 )
     => ( ( Q != bot_bo228742789529271731at_nat )
       => ~ ! [Z: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ Z @ Q )
             => ~ ! [Y6: produc3843707927480180839at_nat] :
                    ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y6 @ Z ) @ R2 )
                   => ~ ( member8757157785044589968at_nat @ Y6 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_836_wfE__min_H,axiom,
    ! [R2: set_Pr4305388979797279737on_val,Q: set_Pr7649655576376376530on_val] :
      ( ( wf_Pro3300669915054815219on_val @ R2 )
     => ( ( Q != bot_bo6722054719108133734on_val )
       => ~ ! [Z: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ Z @ Q )
             => ~ ! [Y6: produc9003781588162623004on_val] :
                    ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y6 @ Z ) @ R2 )
                   => ~ ( member2879030624002546611on_val @ Y6 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_837_wfE__min_H,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,Q: set_se491199571785101031od_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ R2 )
     => ( ( Q != bot_bo8006819982598807891od_o_o )
       => ~ ! [Z: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Z @ Q )
             => ~ ! [Y6: set_Product_prod_o_o] :
                    ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y6 @ Z ) @ R2 )
                   => ~ ( member3157015407008874800od_o_o @ Y6 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_838_wfE__min_H,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,Q: set_se7855581050983116737at_nat] :
      ( ( wf_set7675013139414177994at_nat @ R2 )
     => ( ( Q != bot_bo3083307316010499117at_nat )
       => ~ ! [Z: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Z @ Q )
             => ~ ! [Y6: set_Pr1261947904930325089at_nat] :
                    ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y6 @ Z ) @ R2 )
                   => ~ ( member2643936169264416010at_nat @ Y6 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_839_wo__rel_Ocases__Total3,axiom,
    ! [R: set_Product_prod_o_o,A: $o,B: $o,Phi4: $o > $o > $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ( ( ord_less_eq_set_o @ ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R ) )
       => ( ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( minus_7052513330764642158od_o_o @ R @ id_o ) )
              | ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A ) @ ( minus_7052513330764642158od_o_o @ R @ id_o ) ) )
           => ( Phi4 @ A @ B ) )
         => ( ( ( A = B )
             => ( Phi4 @ A @ B ) )
           => ( Phi4 @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_840_wo__rel_Ocases__Total3,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat,Phi4: nat > nat > $o] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( ord_less_eq_set_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R ) )
       => ( ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( minus_1356011639430497352at_nat @ R @ id_nat ) )
              | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ ( minus_1356011639430497352at_nat @ R @ id_nat ) ) )
           => ( Phi4 @ A @ B ) )
         => ( ( ( A = B )
             => ( Phi4 @ A @ B ) )
           => ( Phi4 @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_841_wo__rel_Ocases__Total3,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,Phi4: produc9003781588162623004on_val > produc9003781588162623004on_val > $o] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( ord_le6900622908513426738on_val @ ( insert3030518898131756108on_val @ A @ ( insert3030518898131756108on_val @ B @ bot_bo6722054719108133734on_val ) ) @ ( field_9010718613293825679on_val @ R ) )
       => ( ( ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ ( minus_7211471524202483680on_val @ R @ id_Pro1455714331835775424on_val ) )
              | ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ A ) @ ( minus_7211471524202483680on_val @ R @ id_Pro1455714331835775424on_val ) ) )
           => ( Phi4 @ A @ B ) )
         => ( ( ( A = B )
             => ( Phi4 @ A @ B ) )
           => ( Phi4 @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_842_wo__rel_Ocases__Total3,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o,Phi4: set_Product_prod_o_o > set_Product_prod_o_o > $o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( ord_le2654036014095304327od_o_o @ ( insert3817420688681518039od_o_o @ A @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( minus_5704166846132356910od_o_o @ R @ id_set2647307015711338339od_o_o ) )
              | ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ A ) @ ( minus_5704166846132356910od_o_o @ R @ id_set2647307015711338339od_o_o ) ) )
           => ( Phi4 @ A @ B ) )
         => ( ( ( A = B )
             => ( Phi4 @ A @ B ) )
           => ( Phi4 @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_843_wo__rel_Ocases__Total3,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,Phi4: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( ord_le2077887516847798113at_nat @ ( insert9200635055090092081at_nat @ A @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( minus_3314409938677909166at_nat @ R @ id_set1796276052702428605at_nat ) )
              | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ A ) @ ( minus_3314409938677909166at_nat @ R @ id_set1796276052702428605at_nat ) ) )
           => ( Phi4 @ A @ B ) )
         => ( ( ( A = B )
             => ( Phi4 @ A @ B ) )
           => ( Phi4 @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_844_well__order__induct__imp,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,P: produc2934264451710624999od_o_o > $o,A: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R )
     => ( ! [X2: produc2934264451710624999od_o_o] :
            ( ! [Y6: produc2934264451710624999od_o_o] :
                ( ( ( Y6 != X2 )
                  & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y6 @ X2 ) @ R ) )
               => ( ( member444158400953824016od_o_o @ Y6 @ ( field_5121862139100741428od_o_o @ R ) )
                 => ( P @ Y6 ) ) )
           => ( ( member444158400953824016od_o_o @ X2 @ ( field_5121862139100741428od_o_o @ R ) )
             => ( P @ X2 ) ) )
       => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R ) )
         => ( P @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_845_well__order__induct__imp,axiom,
    ! [R: set_Pr4427867301352232263at_nat,P: produc732395585841259969at_nat > $o,A: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R )
     => ( ! [X2: produc732395585841259969at_nat] :
            ( ! [Y6: produc732395585841259969at_nat] :
                ( ( ( Y6 != X2 )
                  & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y6 @ X2 ) @ R ) )
               => ( ( member4095101504841534314at_nat @ Y6 @ ( field_7361900449976055182at_nat @ R ) )
                 => ( P @ Y6 ) ) )
           => ( ( member4095101504841534314at_nat @ X2 @ ( field_7361900449976055182at_nat @ R ) )
             => ( P @ X2 ) ) )
       => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R ) )
         => ( P @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_846_well__order__induct__imp,axiom,
    ! [R: set_Pr5564308138774400199at_nat,P: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R )
     => ( ! [X2: produc3843707927480180839at_nat] :
            ( ! [Y6: produc3843707927480180839at_nat] :
                ( ( ( Y6 != X2 )
                  & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y6 @ X2 ) @ R ) )
               => ( ( member8757157785044589968at_nat @ Y6 @ ( field_2862844103579723188at_nat @ R ) )
                 => ( P @ Y6 ) ) )
           => ( ( member8757157785044589968at_nat @ X2 @ ( field_2862844103579723188at_nat @ R ) )
             => ( P @ X2 ) ) )
       => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R ) )
         => ( P @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_847_well__order__induct__imp,axiom,
    ! [R: set_Product_prod_o_o,P: $o > $o,A: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ( ! [X2: $o] :
            ( ! [Y6: $o] :
                ( ( ( Y6 != X2 )
                  & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y6 @ X2 ) @ R ) )
               => ( ( member_o @ Y6 @ ( field_o @ R ) )
                 => ( P @ Y6 ) ) )
           => ( ( member_o @ X2 @ ( field_o @ R ) )
             => ( P @ X2 ) ) )
       => ( ( member_o @ A @ ( field_o @ R ) )
         => ( P @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_848_well__order__induct__imp,axiom,
    ! [R: set_Pr1261947904930325089at_nat,P: nat > $o,A: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ! [X2: nat] :
            ( ! [Y6: nat] :
                ( ( ( Y6 != X2 )
                  & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y6 @ X2 ) @ R ) )
               => ( ( member_nat @ Y6 @ ( field_nat @ R ) )
                 => ( P @ Y6 ) ) )
           => ( ( member_nat @ X2 @ ( field_nat @ R ) )
             => ( P @ X2 ) ) )
       => ( ( member_nat @ A @ ( field_nat @ R ) )
         => ( P @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_849_well__order__induct__imp,axiom,
    ! [R: set_Pr4305388979797279737on_val,P: produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ! [X2: produc9003781588162623004on_val] :
            ( ! [Y6: produc9003781588162623004on_val] :
                ( ( ( Y6 != X2 )
                  & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y6 @ X2 ) @ R ) )
               => ( ( member2879030624002546611on_val @ Y6 @ ( field_9010718613293825679on_val @ R ) )
                 => ( P @ Y6 ) ) )
           => ( ( member2879030624002546611on_val @ X2 @ ( field_9010718613293825679on_val @ R ) )
             => ( P @ X2 ) ) )
       => ( ( member2879030624002546611on_val @ A @ ( field_9010718613293825679on_val @ R ) )
         => ( P @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_850_well__order__induct__imp,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,P: set_Product_prod_o_o > $o,A: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ! [X2: set_Product_prod_o_o] :
            ( ! [Y6: set_Product_prod_o_o] :
                ( ( ( Y6 != X2 )
                  & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y6 @ X2 ) @ R ) )
               => ( ( member3157015407008874800od_o_o @ Y6 @ ( field_6101175518749917524od_o_o @ R ) )
                 => ( P @ Y6 ) ) )
           => ( ( member3157015407008874800od_o_o @ X2 @ ( field_6101175518749917524od_o_o @ R ) )
             => ( P @ X2 ) ) )
       => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R ) )
         => ( P @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_851_well__order__induct__imp,axiom,
    ! [R: set_Pr4329608150637261639at_nat,P: set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ! [X2: set_Pr1261947904930325089at_nat] :
            ( ! [Y6: set_Pr1261947904930325089at_nat] :
                ( ( ( Y6 != X2 )
                  & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y6 @ X2 ) @ R ) )
               => ( ( member2643936169264416010at_nat @ Y6 @ ( field_3145993188331922222at_nat @ R ) )
                 => ( P @ Y6 ) ) )
           => ( ( member2643936169264416010at_nat @ X2 @ ( field_3145993188331922222at_nat @ R ) )
             => ( P @ X2 ) ) )
       => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R ) )
         => ( P @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_852_Linear__order__wf__diff__Id,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( order_6238756238976269133r_on_o @ ( field_o @ R ) @ R )
     => ( ( wf_o @ ( minus_7052513330764642158od_o_o @ R @ id_o ) )
        = ( ! [A6: set_o] :
              ( ( ord_less_eq_set_o @ A6 @ ( field_o @ R ) )
             => ( ( A6 != bot_bot_set_o )
               => ? [X: $o] :
                    ( ( member_o @ X @ A6 )
                    & ! [Y: $o] :
                        ( ( member_o @ Y @ A6 )
                       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_853_Linear__order__wf__diff__Id,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R ) @ R )
     => ( ( wf_nat @ ( minus_1356011639430497352at_nat @ R @ id_nat ) )
        = ( ! [A6: set_nat] :
              ( ( ord_less_eq_set_nat @ A6 @ ( field_nat @ R ) )
             => ( ( A6 != bot_bot_set_nat )
               => ? [X: nat] :
                    ( ( member_nat @ X @ A6 )
                    & ! [Y: nat] :
                        ( ( member_nat @ Y @ A6 )
                       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_854_Linear__order__wf__diff__Id,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ( order_6061556925426544295on_val @ ( field_9010718613293825679on_val @ R ) @ R )
     => ( ( wf_Pro3300669915054815219on_val @ ( minus_7211471524202483680on_val @ R @ id_Pro1455714331835775424on_val ) )
        = ( ! [A6: set_Pr7649655576376376530on_val] :
              ( ( ord_le6900622908513426738on_val @ A6 @ ( field_9010718613293825679on_val @ R ) )
             => ( ( A6 != bot_bo6722054719108133734on_val )
               => ? [X: produc9003781588162623004on_val] :
                    ( ( member2879030624002546611on_val @ X @ A6 )
                    & ! [Y: produc9003781588162623004on_val] :
                        ( ( member2879030624002546611on_val @ Y @ A6 )
                       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_855_Linear__order__wf__diff__Id,axiom,
    ! [R: set_Pr1932065953672099015od_o_o] :
      ( ( order_2625247082195685180od_o_o @ ( field_6101175518749917524od_o_o @ R ) @ R )
     => ( ( wf_set2384217916178216176od_o_o @ ( minus_5704166846132356910od_o_o @ R @ id_set2647307015711338339od_o_o ) )
        = ( ! [A6: set_se491199571785101031od_o_o] :
              ( ( ord_le2654036014095304327od_o_o @ A6 @ ( field_6101175518749917524od_o_o @ R ) )
             => ( ( A6 != bot_bo8006819982598807891od_o_o )
               => ? [X: set_Product_prod_o_o] :
                    ( ( member3157015407008874800od_o_o @ X @ A6 )
                    & ! [Y: set_Product_prod_o_o] :
                        ( ( member3157015407008874800od_o_o @ Y @ A6 )
                       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_856_Linear__order__wf__diff__Id,axiom,
    ! [R: set_Pr4329608150637261639at_nat] :
      ( ( order_5942792088423767318at_nat @ ( field_3145993188331922222at_nat @ R ) @ R )
     => ( ( wf_set7675013139414177994at_nat @ ( minus_3314409938677909166at_nat @ R @ id_set1796276052702428605at_nat ) )
        = ( ! [A6: set_se7855581050983116737at_nat] :
              ( ( ord_le2077887516847798113at_nat @ A6 @ ( field_3145993188331922222at_nat @ R ) )
             => ( ( A6 != bot_bo3083307316010499117at_nat )
               => ? [X: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ X @ A6 )
                    & ! [Y: set_Pr1261947904930325089at_nat] :
                        ( ( member2643936169264416010at_nat @ Y @ A6 )
                       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_857_max__ext__compat,axiom,
    ! [R2: set_Pr7011910401827753159od_o_o,S3: set_Pr7011910401827753159od_o_o] :
      ( ( ord_le6026832634849756263od_o_o @ ( relcom4901197468039914022od_o_o @ R2 @ S3 ) @ R2 )
     => ( ord_le7586318239878589543od_o_o @ ( relcom8634166700253352710od_o_o @ ( max_ex7572203241849361634od_o_o @ R2 ) @ ( sup_su4078869789063261851od_o_o @ ( max_ex7572203241849361634od_o_o @ S3 ) @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ bot_bo7073875226086086771od_o_o @ bot_bo7073875226086086771od_o_o ) @ bot_bo498831808395348787od_o_o ) ) ) @ ( max_ex7572203241849361634od_o_o @ R2 ) ) ) ).

% max_ext_compat
thf(fact_858_max__ext__compat,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( relcom7295539661566034944at_nat @ R2 @ S3 ) @ R2 )
     => ( ord_le1268244103169919719at_nat @ ( relcom6558574857482195296at_nat @ ( max_ex8135407076693332796at_nat @ R2 ) @ ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ S3 ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) @ ( max_ex8135407076693332796at_nat @ R2 ) ) ) ).

% max_ext_compat
thf(fact_859_reduction__pairI,axiom,
    ! [R2: set_Product_prod_o_o,S3: set_Product_prod_o_o] :
      ( ( wf_o @ R2 )
     => ( ( ord_le192195778054999463od_o_o @ ( relcomp_o_o_o @ R2 @ S3 ) @ R2 )
       => ( fun_reduction_pair_o @ ( produc763777882069021527od_o_o @ R2 @ S3 ) ) ) ) ).

% reduction_pairI
thf(fact_860_reduction__pairI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R2 )
     => ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R2 @ S3 ) @ R2 )
       => ( fun_re1951171424197210636ir_nat @ ( produc2922128104949294807at_nat @ R2 @ S3 ) ) ) ) ).

% reduction_pairI
thf(fact_861_Total__subset__Id,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( total_on_o @ ( field_o @ R ) @ R )
     => ( ( ord_le192195778054999463od_o_o @ R @ id_o )
       => ( ( R = bot_bo7073875226086086771od_o_o )
          | ? [A2: $o] :
              ( R
              = ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A2 @ A2 ) @ bot_bo7073875226086086771od_o_o ) ) ) ) ) ).

% Total_subset_Id
thf(fact_862_Total__subset__Id,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( total_on_nat @ ( field_nat @ R ) @ R )
     => ( ( ord_le3146513528884898305at_nat @ R @ id_nat )
       => ( ( R = bot_bo2099793752762293965at_nat )
          | ? [A2: nat] :
              ( R
              = ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% Total_subset_Id
thf(fact_863_Total__subset__Id,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ( total_4784279244336621976on_val @ ( field_9010718613293825679on_val @ R ) @ R )
     => ( ( ord_le7224449410216889241on_val @ R @ id_Pro1455714331835775424on_val )
       => ( ( R = bot_bo402267660617762917on_val )
          | ? [A2: produc9003781588162623004on_val] :
              ( R
              = ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ A2 @ A2 ) @ bot_bo402267660617762917on_val ) ) ) ) ) ).

% Total_subset_Id
thf(fact_864_Total__subset__Id,axiom,
    ! [R: set_Pr1932065953672099015od_o_o] :
      ( ( total_5784220159268842891od_o_o @ ( field_6101175518749917524od_o_o @ R ) @ R )
     => ( ( ord_le7586318239878589543od_o_o @ R @ id_set2647307015711338339od_o_o )
       => ( ( R = bot_bo498831808395348787od_o_o )
          | ? [A2: set_Product_prod_o_o] :
              ( R
              = ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ A2 @ A2 ) @ bot_bo498831808395348787od_o_o ) ) ) ) ) ).

% Total_subset_Id
thf(fact_865_Total__subset__Id,axiom,
    ! [R: set_Pr4329608150637261639at_nat] :
      ( ( total_7212417256332265445at_nat @ ( field_3145993188331922222at_nat @ R ) @ R )
     => ( ( ord_le1268244103169919719at_nat @ R @ id_set1796276052702428605at_nat )
       => ( ( R = bot_bo228742789529271731at_nat )
          | ? [A2: set_Pr1261947904930325089at_nat] :
              ( R
              = ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A2 @ A2 ) @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% Total_subset_Id
thf(fact_866_wo__rel_Oequals__minim,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,B5: set_Pr1932065953672099015od_o_o,A: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R )
     => ( ( ord_le7586318239878589543od_o_o @ B5 @ ( field_5121862139100741428od_o_o @ R ) )
       => ( ( member444158400953824016od_o_o @ A @ B5 )
         => ( ! [B2: produc2934264451710624999od_o_o] :
                ( ( member444158400953824016od_o_o @ B2 @ B5 )
               => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B2 ) @ R ) )
           => ( A
              = ( bNF_We4719373430050929331od_o_o @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_867_wo__rel_Oequals__minim,axiom,
    ! [R: set_Pr4427867301352232263at_nat,B5: set_Pr457366540195662369at_nat,A: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R )
     => ( ( ord_le4841768313324958145at_nat @ B5 @ ( field_7361900449976055182at_nat @ R ) )
       => ( ( member4095101504841534314at_nat @ A @ B5 )
         => ( ! [B2: produc732395585841259969at_nat] :
                ( ( member4095101504841534314at_nat @ B2 @ B5 )
               => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B2 ) @ R ) )
           => ( A
              = ( bNF_We6249471513304245645at_nat @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_868_wo__rel_Oequals__minim,axiom,
    ! [R: set_Pr5564308138774400199at_nat,B5: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R )
     => ( ( ord_le1268244103169919719at_nat @ B5 @ ( field_2862844103579723188at_nat @ R ) )
       => ( ( member8757157785044589968at_nat @ A @ B5 )
         => ( ! [B2: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ B2 @ B5 )
               => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B2 ) @ R ) )
           => ( A
              = ( bNF_We7473068394052389427at_nat @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_869_wo__rel_Oequals__minim,axiom,
    ! [R: set_Product_prod_o_o,B5: set_o,A: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ( ( ord_less_eq_set_o @ B5 @ ( field_o @ R ) )
       => ( ( member_o @ A @ B5 )
         => ( ! [B2: $o] :
                ( ( member_o @ B2 @ B5 )
               => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B2 ) @ R ) )
           => ( A
              = ( bNF_We537455116209922916inim_o @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_870_wo__rel_Oequals__minim,axiom,
    ! [R: set_Pr1261947904930325089at_nat,B5: set_nat,A: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( ord_less_eq_set_nat @ B5 @ ( field_nat @ R ) )
       => ( ( member_nat @ A @ B5 )
         => ( ! [B2: nat] :
                ( ( member_nat @ B2 @ B5 )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ R ) )
           => ( A
              = ( bNF_We3275124272252294916im_nat @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_871_wo__rel_Oequals__minim,axiom,
    ! [R: set_Pr4305388979797279737on_val,B5: set_Pr7649655576376376530on_val,A: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( ord_le6900622908513426738on_val @ B5 @ ( field_9010718613293825679on_val @ R ) )
       => ( ( member2879030624002546611on_val @ A @ B5 )
         => ( ! [B2: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ B2 @ B5 )
               => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B2 ) @ R ) )
           => ( A
              = ( bNF_We316632424347587920on_val @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_872_wo__rel_Oequals__minim,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,B5: set_se491199571785101031od_o_o,A: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( ord_le2654036014095304327od_o_o @ B5 @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( member3157015407008874800od_o_o @ A @ B5 )
         => ( ! [B2: set_Product_prod_o_o] :
                ( ( member3157015407008874800od_o_o @ B2 @ B5 )
               => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B2 ) @ R ) )
           => ( A
              = ( bNF_We6151898823597705171od_o_o @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_873_wo__rel_Oequals__minim,axiom,
    ! [R: set_Pr4329608150637261639at_nat,B5: set_se7855581050983116737at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( ord_le2077887516847798113at_nat @ B5 @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( member2643936169264416010at_nat @ A @ B5 )
         => ( ! [B2: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ B2 @ B5 )
               => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ R ) )
           => ( A
              = ( bNF_We6832956061294468653at_nat @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_874_wo__rel_Ominim__least,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,B5: set_Pr1932065953672099015od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R )
     => ( ( ord_le7586318239878589543od_o_o @ B5 @ ( field_5121862139100741428od_o_o @ R ) )
       => ( ( member444158400953824016od_o_o @ B @ B5 )
         => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( bNF_We4719373430050929331od_o_o @ R @ B5 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_875_wo__rel_Ominim__least,axiom,
    ! [R: set_Pr4427867301352232263at_nat,B5: set_Pr457366540195662369at_nat,B: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R )
     => ( ( ord_le4841768313324958145at_nat @ B5 @ ( field_7361900449976055182at_nat @ R ) )
       => ( ( member4095101504841534314at_nat @ B @ B5 )
         => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( bNF_We6249471513304245645at_nat @ R @ B5 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_876_wo__rel_Ominim__least,axiom,
    ! [R: set_Pr5564308138774400199at_nat,B5: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R )
     => ( ( ord_le1268244103169919719at_nat @ B5 @ ( field_2862844103579723188at_nat @ R ) )
       => ( ( member8757157785044589968at_nat @ B @ B5 )
         => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( bNF_We7473068394052389427at_nat @ R @ B5 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_877_wo__rel_Ominim__least,axiom,
    ! [R: set_Product_prod_o_o,B5: set_o,B: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ( ( ord_less_eq_set_o @ B5 @ ( field_o @ R ) )
       => ( ( member_o @ B @ B5 )
         => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( bNF_We537455116209922916inim_o @ R @ B5 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_878_wo__rel_Ominim__least,axiom,
    ! [R: set_Pr1261947904930325089at_nat,B5: set_nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( ord_less_eq_set_nat @ B5 @ ( field_nat @ R ) )
       => ( ( member_nat @ B @ B5 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( bNF_We3275124272252294916im_nat @ R @ B5 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_879_wo__rel_Ominim__least,axiom,
    ! [R: set_Pr4305388979797279737on_val,B5: set_Pr7649655576376376530on_val,B: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( ord_le6900622908513426738on_val @ B5 @ ( field_9010718613293825679on_val @ R ) )
       => ( ( member2879030624002546611on_val @ B @ B5 )
         => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( bNF_We316632424347587920on_val @ R @ B5 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_880_wo__rel_Ominim__least,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,B5: set_se491199571785101031od_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( ord_le2654036014095304327od_o_o @ B5 @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( member3157015407008874800od_o_o @ B @ B5 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_We6151898823597705171od_o_o @ R @ B5 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_881_wo__rel_Ominim__least,axiom,
    ! [R: set_Pr4329608150637261639at_nat,B5: set_se7855581050983116737at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( ord_le2077887516847798113at_nat @ B5 @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( member2643936169264416010at_nat @ B @ B5 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_We6832956061294468653at_nat @ R @ B5 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_882_IdD,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ id_Pro1455714331835775424on_val )
     => ( A = B ) ) ).

% IdD
thf(fact_883_IdD,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ id_set2647307015711338339od_o_o )
     => ( A = B ) ) ).

% IdD
thf(fact_884_IdD,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ id_set1796276052702428605at_nat )
     => ( A = B ) ) ).

% IdD
thf(fact_885_total__on__def,axiom,
    ( total_4784279244336621976on_val
    = ( ^ [A6: set_Pr7649655576376376530on_val,R3: set_Pr4305388979797279737on_val] :
        ! [X: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ X @ A6 )
         => ! [Y: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ Y @ A6 )
             => ( ( X != Y )
               => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ R3 )
                  | ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y @ X ) @ R3 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_886_total__on__def,axiom,
    ( total_5784220159268842891od_o_o
    = ( ^ [A6: set_se491199571785101031od_o_o,R3: set_Pr1932065953672099015od_o_o] :
        ! [X: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X @ A6 )
         => ! [Y: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Y @ A6 )
             => ( ( X != Y )
               => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R3 )
                  | ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ X ) @ R3 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_887_total__on__def,axiom,
    ( total_7212417256332265445at_nat
    = ( ^ [A6: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat] :
        ! [X: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X @ A6 )
         => ! [Y: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y @ A6 )
             => ( ( X != Y )
               => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R3 )
                  | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X ) @ R3 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_888_total__onI,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o] :
      ( ! [X2: produc2934264451710624999od_o_o,Y4: produc2934264451710624999od_o_o] :
          ( ( member444158400953824016od_o_o @ X2 @ A3 )
         => ( ( member444158400953824016od_o_o @ Y4 @ A3 )
           => ( ( X2 != Y4 )
             => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X2 @ Y4 ) @ R )
                | ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y4 @ X2 ) @ R ) ) ) ) )
     => ( total_3055748755304958059od_o_o @ A3 @ R ) ) ).

% total_onI
thf(fact_889_total__onI,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat] :
      ( ! [X2: produc732395585841259969at_nat,Y4: produc732395585841259969at_nat] :
          ( ( member4095101504841534314at_nat @ X2 @ A3 )
         => ( ( member4095101504841534314at_nat @ Y4 @ A3 )
           => ( ( X2 != Y4 )
             => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X2 @ Y4 ) @ R )
                | ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y4 @ X2 ) @ R ) ) ) ) )
     => ( total_8996415722781341509at_nat @ A3 @ R ) ) ).

% total_onI
thf(fact_890_total__onI,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat] :
      ( ! [X2: produc3843707927480180839at_nat,Y4: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X2 @ A3 )
         => ( ( member8757157785044589968at_nat @ Y4 @ A3 )
           => ( ( X2 != Y4 )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ Y4 ) @ R )
                | ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y4 @ X2 ) @ R ) ) ) ) )
     => ( total_1513685709984341995at_nat @ A3 @ R ) ) ).

% total_onI
thf(fact_891_total__onI,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val] :
      ( ! [X2: produc9003781588162623004on_val,Y4: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ X2 @ A3 )
         => ( ( member2879030624002546611on_val @ Y4 @ A3 )
           => ( ( X2 != Y4 )
             => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y4 ) @ R )
                | ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y4 @ X2 ) @ R ) ) ) ) )
     => ( total_4784279244336621976on_val @ A3 @ R ) ) ).

% total_onI
thf(fact_892_total__onI,axiom,
    ! [A3: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ! [X2: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X2 @ A3 )
         => ( ( member3157015407008874800od_o_o @ Y4 @ A3 )
           => ( ( X2 != Y4 )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y4 ) @ R )
                | ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ X2 ) @ R ) ) ) ) )
     => ( total_5784220159268842891od_o_o @ A3 @ R ) ) ).

% total_onI
thf(fact_893_total__onI,axiom,
    ! [A3: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X2 @ A3 )
         => ( ( member2643936169264416010at_nat @ Y4 @ A3 )
           => ( ( X2 != Y4 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y4 ) @ R )
                | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ X2 ) @ R ) ) ) ) )
     => ( total_7212417256332265445at_nat @ A3 @ R ) ) ).

% total_onI
thf(fact_894_max__ext__additive,axiom,
    ! [A3: set_Product_prod_o_o,B5: set_Product_prod_o_o,R2: set_Pr7011910401827753159od_o_o,C4: set_Product_prod_o_o,D2: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A3 @ B5 ) @ ( max_ex7572203241849361634od_o_o @ R2 ) )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ C4 @ D2 ) @ ( max_ex7572203241849361634od_o_o @ R2 ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( sup_su5769328420594410459od_o_o @ A3 @ C4 ) @ ( sup_su5769328420594410459od_o_o @ B5 @ D2 ) ) @ ( max_ex7572203241849361634od_o_o @ R2 ) ) ) ) ).

% max_ext_additive
thf(fact_895_max__ext__additive,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,C4: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A3 @ B5 ) @ ( max_ex8135407076693332796at_nat @ R2 ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ C4 @ D2 ) @ ( max_ex8135407076693332796at_nat @ R2 ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ C4 ) @ ( sup_su6327502436637775413at_nat @ B5 @ D2 ) ) @ ( max_ex8135407076693332796at_nat @ R2 ) ) ) ) ).

% max_ext_additive
thf(fact_896_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R )
     => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R ) )
       => ( ( member444158400953824016od_o_o @ B @ ( field_5121862139100741428od_o_o @ R ) )
         => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ ( bNF_We5307229915971165237od_o_o @ R @ A @ B ) ) @ R )
            & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B @ ( bNF_We5307229915971165237od_o_o @ R @ A @ B ) ) @ R )
            & ( member444158400953824016od_o_o @ ( bNF_We5307229915971165237od_o_o @ R @ A @ B ) @ ( insert3855699742481506999od_o_o @ A @ ( insert3855699742481506999od_o_o @ B @ bot_bo498831808395348787od_o_o ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_897_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R ) )
       => ( ( member4095101504841534314at_nat @ B @ ( field_7361900449976055182at_nat @ R ) )
         => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ ( bNF_We7253295970193759759at_nat @ R @ A @ B ) ) @ R )
            & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B @ ( bNF_We7253295970193759759at_nat @ R @ A @ B ) ) @ R )
            & ( member4095101504841534314at_nat @ ( bNF_We7253295970193759759at_nat @ R @ A @ B ) @ ( insert8109617542356073361at_nat @ A @ ( insert8109617542356073361at_nat @ B @ bot_bo1676454896636820109at_nat ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_898_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R ) )
       => ( ( member8757157785044589968at_nat @ B @ ( field_2862844103579723188at_nat @ R ) )
         => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ ( bNF_We1807050579049552309at_nat @ R @ A @ B ) ) @ R )
            & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ ( bNF_We1807050579049552309at_nat @ R @ A @ B ) ) @ R )
            & ( member8757157785044589968at_nat @ ( bNF_We1807050579049552309at_nat @ R @ A @ B ) @ ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_899_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ( ( member_o @ A @ ( field_o @ R ) )
       => ( ( member_o @ B @ ( field_o @ R ) )
         => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ ( bNF_We3104086342435940326max2_o @ R @ A @ B ) ) @ R )
            & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ ( bNF_We3104086342435940326max2_o @ R @ A @ B ) ) @ R )
            & ( member_o @ ( bNF_We3104086342435940326max2_o @ R @ A @ B ) @ ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_900_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( member_nat @ A @ ( field_nat @ R ) )
       => ( ( member_nat @ B @ ( field_nat @ R ) )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ ( bNF_We2380939065524664066x2_nat @ R @ A @ B ) ) @ R )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ ( bNF_We2380939065524664066x2_nat @ R @ A @ B ) ) @ R )
            & ( member_nat @ ( bNF_We2380939065524664066x2_nat @ R @ A @ B ) @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_901_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( member2879030624002546611on_val @ A @ ( field_9010718613293825679on_val @ R ) )
       => ( ( member2879030624002546611on_val @ B @ ( field_9010718613293825679on_val @ R ) )
         => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ ( bNF_We3765001366693588814on_val @ R @ A @ B ) ) @ R )
            & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ ( bNF_We3765001366693588814on_val @ R @ A @ B ) ) @ R )
            & ( member2879030624002546611on_val @ ( bNF_We3765001366693588814on_val @ R @ A @ B ) @ ( insert3030518898131756108on_val @ A @ ( insert3030518898131756108on_val @ B @ bot_bo6722054719108133734on_val ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_902_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( member3157015407008874800od_o_o @ B @ ( field_6101175518749917524od_o_o @ R ) )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ ( bNF_We9141258785624724309od_o_o @ R @ A @ B ) ) @ R )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ ( bNF_We9141258785624724309od_o_o @ R @ A @ B ) ) @ R )
            & ( member3157015407008874800od_o_o @ ( bNF_We9141258785624724309od_o_o @ R @ A @ B ) @ ( insert3817420688681518039od_o_o @ A @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_903_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( member2643936169264416010at_nat @ B @ ( field_3145993188331922222at_nat @ R ) )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ ( bNF_We6094461251688579247at_nat @ R @ A @ B ) ) @ R )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ ( bNF_We6094461251688579247at_nat @ R @ A @ B ) ) @ R )
            & ( member2643936169264416010at_nat @ ( bNF_We6094461251688579247at_nat @ R @ A @ B ) @ ( insert9200635055090092081at_nat @ A @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_904_wo__rel_Omax2__greater,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R )
     => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R ) )
       => ( ( member444158400953824016od_o_o @ B @ ( field_5121862139100741428od_o_o @ R ) )
         => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ ( bNF_We5307229915971165237od_o_o @ R @ A @ B ) ) @ R )
            & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B @ ( bNF_We5307229915971165237od_o_o @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_905_wo__rel_Omax2__greater,axiom,
    ! [R: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R ) )
       => ( ( member4095101504841534314at_nat @ B @ ( field_7361900449976055182at_nat @ R ) )
         => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ ( bNF_We7253295970193759759at_nat @ R @ A @ B ) ) @ R )
            & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B @ ( bNF_We7253295970193759759at_nat @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_906_wo__rel_Omax2__greater,axiom,
    ! [R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R ) )
       => ( ( member8757157785044589968at_nat @ B @ ( field_2862844103579723188at_nat @ R ) )
         => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ ( bNF_We1807050579049552309at_nat @ R @ A @ B ) ) @ R )
            & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ ( bNF_We1807050579049552309at_nat @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_907_wo__rel_Omax2__greater,axiom,
    ! [R: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ( ( member_o @ A @ ( field_o @ R ) )
       => ( ( member_o @ B @ ( field_o @ R ) )
         => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ ( bNF_We3104086342435940326max2_o @ R @ A @ B ) ) @ R )
            & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ ( bNF_We3104086342435940326max2_o @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_908_wo__rel_Omax2__greater,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( member_nat @ A @ ( field_nat @ R ) )
       => ( ( member_nat @ B @ ( field_nat @ R ) )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ ( bNF_We2380939065524664066x2_nat @ R @ A @ B ) ) @ R )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ ( bNF_We2380939065524664066x2_nat @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_909_wo__rel_Omax2__greater,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( member2879030624002546611on_val @ A @ ( field_9010718613293825679on_val @ R ) )
       => ( ( member2879030624002546611on_val @ B @ ( field_9010718613293825679on_val @ R ) )
         => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ ( bNF_We3765001366693588814on_val @ R @ A @ B ) ) @ R )
            & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ ( bNF_We3765001366693588814on_val @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_910_wo__rel_Omax2__greater,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( member3157015407008874800od_o_o @ B @ ( field_6101175518749917524od_o_o @ R ) )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ ( bNF_We9141258785624724309od_o_o @ R @ A @ B ) ) @ R )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ ( bNF_We9141258785624724309od_o_o @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_911_wo__rel_Omax2__greater,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( member2643936169264416010at_nat @ B @ ( field_3145993188331922222at_nat @ R ) )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ ( bNF_We6094461251688579247at_nat @ R @ A @ B ) ) @ R )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ ( bNF_We6094461251688579247at_nat @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_912_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R )
     => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R ) )
       => ( ( member444158400953824016od_o_o @ B @ ( field_5121862139100741428od_o_o @ R ) )
         => ( ( ( bNF_We5307229915971165237od_o_o @ R @ A @ B )
              = B )
            = ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_913_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R ) )
       => ( ( member4095101504841534314at_nat @ B @ ( field_7361900449976055182at_nat @ R ) )
         => ( ( ( bNF_We7253295970193759759at_nat @ R @ A @ B )
              = B )
            = ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_914_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R ) )
       => ( ( member8757157785044589968at_nat @ B @ ( field_2862844103579723188at_nat @ R ) )
         => ( ( ( bNF_We1807050579049552309at_nat @ R @ A @ B )
              = B )
            = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_915_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ( ( member_o @ A @ ( field_o @ R ) )
       => ( ( member_o @ B @ ( field_o @ R ) )
         => ( ( ( bNF_We3104086342435940326max2_o @ R @ A @ B )
              = B )
            = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_916_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( member_nat @ A @ ( field_nat @ R ) )
       => ( ( member_nat @ B @ ( field_nat @ R ) )
         => ( ( ( bNF_We2380939065524664066x2_nat @ R @ A @ B )
              = B )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_917_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( member2879030624002546611on_val @ A @ ( field_9010718613293825679on_val @ R ) )
       => ( ( member2879030624002546611on_val @ B @ ( field_9010718613293825679on_val @ R ) )
         => ( ( ( bNF_We3765001366693588814on_val @ R @ A @ B )
              = B )
            = ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_918_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( member3157015407008874800od_o_o @ B @ ( field_6101175518749917524od_o_o @ R ) )
         => ( ( ( bNF_We9141258785624724309od_o_o @ R @ A @ B )
              = B )
            = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_919_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( member2643936169264416010at_nat @ B @ ( field_3145993188331922222at_nat @ R ) )
         => ( ( ( bNF_We6094461251688579247at_nat @ R @ A @ B )
              = B )
            = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_920_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R )
     => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R ) )
       => ( ( member444158400953824016od_o_o @ B @ ( field_5121862139100741428od_o_o @ R ) )
         => ( ( ( bNF_We5307229915971165237od_o_o @ R @ A @ B )
              = A )
            = ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_921_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R ) )
       => ( ( member4095101504841534314at_nat @ B @ ( field_7361900449976055182at_nat @ R ) )
         => ( ( ( bNF_We7253295970193759759at_nat @ R @ A @ B )
              = A )
            = ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_922_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R ) )
       => ( ( member8757157785044589968at_nat @ B @ ( field_2862844103579723188at_nat @ R ) )
         => ( ( ( bNF_We1807050579049552309at_nat @ R @ A @ B )
              = A )
            = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_923_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ( ( member_o @ A @ ( field_o @ R ) )
       => ( ( member_o @ B @ ( field_o @ R ) )
         => ( ( ( bNF_We3104086342435940326max2_o @ R @ A @ B )
              = A )
            = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_924_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( member_nat @ A @ ( field_nat @ R ) )
       => ( ( member_nat @ B @ ( field_nat @ R ) )
         => ( ( ( bNF_We2380939065524664066x2_nat @ R @ A @ B )
              = A )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_925_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( member2879030624002546611on_val @ A @ ( field_9010718613293825679on_val @ R ) )
       => ( ( member2879030624002546611on_val @ B @ ( field_9010718613293825679on_val @ R ) )
         => ( ( ( bNF_We3765001366693588814on_val @ R @ A @ B )
              = A )
            = ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_926_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( member3157015407008874800od_o_o @ B @ ( field_6101175518749917524od_o_o @ R ) )
         => ( ( ( bNF_We9141258785624724309od_o_o @ R @ A @ B )
              = A )
            = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_927_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( member2643936169264416010at_nat @ B @ ( field_3145993188331922222at_nat @ R ) )
         => ( ( ( bNF_We6094461251688579247at_nat @ R @ A @ B )
              = A )
            = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_928_wo__rel_Omax2__def,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
         => ( ( bNF_We3765001366693588814on_val @ R @ A @ B )
            = B ) )
        & ( ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
         => ( ( bNF_We3765001366693588814on_val @ R @ A @ B )
            = A ) ) ) ) ).

% wo_rel.max2_def
thf(fact_929_wo__rel_Omax2__def,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
         => ( ( bNF_We9141258785624724309od_o_o @ R @ A @ B )
            = B ) )
        & ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
         => ( ( bNF_We9141258785624724309od_o_o @ R @ A @ B )
            = A ) ) ) ) ).

% wo_rel.max2_def
thf(fact_930_wo__rel_Omax2__def,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
         => ( ( bNF_We6094461251688579247at_nat @ R @ A @ B )
            = B ) )
        & ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
         => ( ( bNF_We6094461251688579247at_nat @ R @ A @ B )
            = A ) ) ) ) ).

% wo_rel.max2_def
thf(fact_931_bsqr__max2,axiom,
    ! [R: set_Product_prod_o_o,A12: $o,A23: $o,B1: $o,B22: $o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ ( product_Pair_o_o @ A12 @ A23 ) @ ( product_Pair_o_o @ B1 @ B22 ) ) @ ( bNF_Wellorder_bsqr_o @ R ) )
       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( bNF_We3104086342435940326max2_o @ R @ A12 @ A23 ) @ ( bNF_We3104086342435940326max2_o @ R @ B1 @ B22 ) ) @ R ) ) ) ).

% bsqr_max2
thf(fact_932_bsqr__max2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A12: nat,A23: nat,B1: nat,B22: nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A12 @ A23 ) @ ( product_Pair_nat_nat @ B1 @ B22 ) ) @ ( bNF_We1732890022510290291qr_nat @ R ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( bNF_We2380939065524664066x2_nat @ R @ A12 @ A23 ) @ ( bNF_We2380939065524664066x2_nat @ R @ B1 @ B22 ) ) @ R ) ) ) ).

% bsqr_max2
thf(fact_933_bsqr__max2,axiom,
    ! [R: set_Pr4305388979797279737on_val,A12: produc9003781588162623004on_val,A23: produc9003781588162623004on_val,B1: produc9003781588162623004on_val,B22: produc9003781588162623004on_val] :
      ( ( order_2579646547231075838on_val @ ( field_9010718613293825679on_val @ R ) @ R )
     => ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ A12 @ A23 ) @ ( produc4958730763592889809on_val @ B1 @ B22 ) ) @ ( bNF_We3507095839090954431on_val @ R ) )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( bNF_We3765001366693588814on_val @ R @ A12 @ A23 ) @ ( bNF_We3765001366693588814on_val @ R @ B1 @ B22 ) ) @ R ) ) ) ).

% bsqr_max2
thf(fact_934_bsqr__max2,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A12: set_Product_prod_o_o,A23: set_Product_prod_o_o,B1: set_Product_prod_o_o,B22: set_Product_prod_o_o] :
      ( ( order_3767514495197212837od_o_o @ ( field_6101175518749917524od_o_o @ R ) @ R )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ A12 @ A23 ) @ ( produc763777882069021527od_o_o @ B1 @ B22 ) ) @ ( bNF_We3935152771354962724od_o_o @ R ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_We9141258785624724309od_o_o @ R @ A12 @ A23 ) @ ( bNF_We9141258785624724309od_o_o @ R @ B1 @ B22 ) ) @ R ) ) ) ).

% bsqr_max2
thf(fact_935_bsqr__max2,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A12: set_Pr1261947904930325089at_nat,A23: set_Pr1261947904930325089at_nat,B1: set_Pr1261947904930325089at_nat,B22: set_Pr1261947904930325089at_nat] :
      ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R ) @ R )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ A12 @ A23 ) @ ( produc2922128104949294807at_nat @ B1 @ B22 ) ) @ ( bNF_We6127972852007642366at_nat @ R ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_We6094461251688579247at_nat @ R @ A12 @ A23 ) @ ( bNF_We6094461251688579247at_nat @ R @ B1 @ B22 ) ) @ R ) ) ) ).

% bsqr_max2
thf(fact_936_wo__rel_OisMinim__def,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,A3: set_Pr1932065953672099015od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R )
     => ( ( bNF_We6886001019485585053od_o_o @ R @ A3 @ B )
        = ( ( member444158400953824016od_o_o @ B @ A3 )
          & ! [X: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ X @ A3 )
             => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B @ X ) @ R ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_937_wo__rel_OisMinim__def,axiom,
    ! [R: set_Pr4427867301352232263at_nat,A3: set_Pr457366540195662369at_nat,B: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R )
     => ( ( bNF_We6170040095471856759at_nat @ R @ A3 @ B )
        = ( ( member4095101504841534314at_nat @ B @ A3 )
          & ! [X: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ X @ A3 )
             => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B @ X ) @ R ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_938_wo__rel_OisMinim__def,axiom,
    ! [R: set_Pr5564308138774400199at_nat,A3: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R )
     => ( ( bNF_We2939539520719410205at_nat @ R @ A3 @ B )
        = ( ( member8757157785044589968at_nat @ B @ A3 )
          & ! [X: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X @ A3 )
             => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ X ) @ R ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_939_wo__rel_OisMinim__def,axiom,
    ! [R: set_Pr4305388979797279737on_val,A3: set_Pr7649655576376376530on_val,B: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( bNF_We5046398979041125094on_val @ R @ A3 @ B )
        = ( ( member2879030624002546611on_val @ B @ A3 )
          & ! [X: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ X @ A3 )
             => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ X ) @ R ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_940_wo__rel_OisMinim__def,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A3: set_se491199571785101031od_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( bNF_We794377122088811965od_o_o @ R @ A3 @ B )
        = ( ( member3157015407008874800od_o_o @ B @ A3 )
          & ! [X: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ X @ A3 )
             => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ X ) @ R ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_941_wo__rel_OisMinim__def,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A3: set_se7855581050983116737at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( bNF_We943883530777384727at_nat @ R @ A3 @ B )
        = ( ( member2643936169264416010at_nat @ B @ A3 )
          & ! [X: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ X @ A3 )
             => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ X ) @ R ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_942_max__ext_Ocases,axiom,
    ! [A12: set_Pr1932065953672099015od_o_o,A23: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o] :
      ( ( member3740103928775426832od_o_o @ ( produc2282539314773629783od_o_o @ A12 @ A23 ) @ ( max_ex9141845325704764834od_o_o @ R2 ) )
     => ~ ( ( finite820925532130942480od_o_o @ A12 )
         => ( ( finite820925532130942480od_o_o @ A23 )
           => ( ( A23 != bot_bo498831808395348787od_o_o )
             => ~ ! [X7: produc2934264451710624999od_o_o] :
                    ( ( member444158400953824016od_o_o @ X7 @ A12 )
                   => ? [Xa3: produc2934264451710624999od_o_o] :
                        ( ( member444158400953824016od_o_o @ Xa3 @ A23 )
                        & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X7 @ Xa3 ) @ R2 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_943_max__ext_Ocases,axiom,
    ! [A12: set_Pr457366540195662369at_nat,A23: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat] :
      ( ( member7768518508883498896at_nat @ ( produc4325049062531119319at_nat @ A12 @ A23 ) @ ( max_ex1111939896555521276at_nat @ R2 ) )
     => ~ ( ( finite8072506237346428522at_nat @ A12 )
         => ( ( finite8072506237346428522at_nat @ A23 )
           => ( ( A23 != bot_bo1676454896636820109at_nat )
             => ~ ! [X7: produc732395585841259969at_nat] :
                    ( ( member4095101504841534314at_nat @ X7 @ A12 )
                   => ? [Xa3: produc732395585841259969at_nat] :
                        ( ( member4095101504841534314at_nat @ Xa3 @ A23 )
                        & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X7 @ Xa3 ) @ R2 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_944_max__ext_Ocases,axiom,
    ! [A12: set_Pr4329608150637261639at_nat,A23: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A12 @ A23 ) @ ( max_ex8833738776070432290at_nat @ R2 ) )
     => ~ ( ( finite4343798906461161616at_nat @ A12 )
         => ( ( finite4343798906461161616at_nat @ A23 )
           => ( ( A23 != bot_bo228742789529271731at_nat )
             => ~ ! [X7: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ X7 @ A12 )
                   => ? [Xa3: produc3843707927480180839at_nat] :
                        ( ( member8757157785044589968at_nat @ Xa3 @ A23 )
                        & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X7 @ Xa3 ) @ R2 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_945_max__ext_Ocases,axiom,
    ! [A12: set_Pr7649655576376376530on_val,A23: set_Pr7649655576376376530on_val,R2: set_Pr4305388979797279737on_val] :
      ( ( member806545433280400814on_val @ ( produc113594951777976637on_val @ A12 @ A23 ) @ ( max_ex7543004642404436129on_val @ R2 ) )
     => ~ ( ( finite8324237431073730739on_val @ A12 )
         => ( ( finite8324237431073730739on_val @ A23 )
           => ( ( A23 != bot_bo6722054719108133734on_val )
             => ~ ! [X7: produc9003781588162623004on_val] :
                    ( ( member2879030624002546611on_val @ X7 @ A12 )
                   => ? [Xa3: produc9003781588162623004on_val] :
                        ( ( member2879030624002546611on_val @ Xa3 @ A23 )
                        & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X7 @ Xa3 ) @ R2 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_946_max__ext_Ocases,axiom,
    ! [A12: set_se491199571785101031od_o_o,A23: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member9169340693537962768od_o_o @ ( produc4478624491032480599od_o_o @ A12 @ A23 ) @ ( max_ex5401630698724402626od_o_o @ R2 ) )
     => ~ ( ( finite9011636195296603184od_o_o @ A12 )
         => ( ( finite9011636195296603184od_o_o @ A23 )
           => ( ( A23 != bot_bo8006819982598807891od_o_o )
             => ~ ! [X7: set_Product_prod_o_o] :
                    ( ( member3157015407008874800od_o_o @ X7 @ A12 )
                   => ? [Xa3: set_Product_prod_o_o] :
                        ( ( member3157015407008874800od_o_o @ Xa3 @ A23 )
                        & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X7 @ Xa3 ) @ R2 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_947_max__ext_Ocases,axiom,
    ! [A12: set_se7855581050983116737at_nat,A23: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ A12 @ A23 ) @ ( max_ex4535816423174879388at_nat @ R2 ) )
     => ~ ( ( finite9047747110432174090at_nat @ A12 )
         => ( ( finite9047747110432174090at_nat @ A23 )
           => ( ( A23 != bot_bo3083307316010499117at_nat )
             => ~ ! [X7: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ X7 @ A12 )
                   => ? [Xa3: set_Pr1261947904930325089at_nat] :
                        ( ( member2643936169264416010at_nat @ Xa3 @ A23 )
                        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ Xa3 ) @ R2 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_948_max__ext_Ocases,axiom,
    ! [A12: set_Product_prod_o_o,A23: set_Product_prod_o_o,R2: set_Pr7011910401827753159od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A12 @ A23 ) @ ( max_ex7572203241849361634od_o_o @ R2 ) )
     => ~ ( ( finite6120865539452801872od_o_o @ A12 )
         => ( ( finite6120865539452801872od_o_o @ A23 )
           => ( ( A23 != bot_bo7073875226086086771od_o_o )
             => ~ ! [X7: product_prod_o_o] :
                    ( ( member7466972457876170832od_o_o @ X7 @ A12 )
                   => ? [Xa3: product_prod_o_o] :
                        ( ( member7466972457876170832od_o_o @ Xa3 @ A23 )
                        & ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X7 @ Xa3 ) @ R2 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_949_max__ext_Ocases,axiom,
    ! [A12: set_Pr1261947904930325089at_nat,A23: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A12 @ A23 ) @ ( max_ex8135407076693332796at_nat @ R2 ) )
     => ~ ( ( finite6177210948735845034at_nat @ A12 )
         => ( ( finite6177210948735845034at_nat @ A23 )
           => ( ( A23 != bot_bo2099793752762293965at_nat )
             => ~ ! [X7: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X7 @ A12 )
                   => ? [Xa3: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ Xa3 @ A23 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X7 @ Xa3 ) @ R2 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_950_max__ext_Osimps,axiom,
    ! [A12: set_Pr1932065953672099015od_o_o,A23: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o] :
      ( ( member3740103928775426832od_o_o @ ( produc2282539314773629783od_o_o @ A12 @ A23 ) @ ( max_ex9141845325704764834od_o_o @ R2 ) )
      = ( ( finite820925532130942480od_o_o @ A12 )
        & ( finite820925532130942480od_o_o @ A23 )
        & ( A23 != bot_bo498831808395348787od_o_o )
        & ! [X: produc2934264451710624999od_o_o] :
            ( ( member444158400953824016od_o_o @ X @ A12 )
           => ? [Y: produc2934264451710624999od_o_o] :
                ( ( member444158400953824016od_o_o @ Y @ A23 )
                & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Y ) @ R2 ) ) ) ) ) ).

% max_ext.simps
thf(fact_951_max__ext_Osimps,axiom,
    ! [A12: set_Pr457366540195662369at_nat,A23: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat] :
      ( ( member7768518508883498896at_nat @ ( produc4325049062531119319at_nat @ A12 @ A23 ) @ ( max_ex1111939896555521276at_nat @ R2 ) )
      = ( ( finite8072506237346428522at_nat @ A12 )
        & ( finite8072506237346428522at_nat @ A23 )
        & ( A23 != bot_bo1676454896636820109at_nat )
        & ! [X: produc732395585841259969at_nat] :
            ( ( member4095101504841534314at_nat @ X @ A12 )
           => ? [Y: produc732395585841259969at_nat] :
                ( ( member4095101504841534314at_nat @ Y @ A23 )
                & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Y ) @ R2 ) ) ) ) ) ).

% max_ext.simps
thf(fact_952_max__ext_Osimps,axiom,
    ! [A12: set_Pr4329608150637261639at_nat,A23: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A12 @ A23 ) @ ( max_ex8833738776070432290at_nat @ R2 ) )
      = ( ( finite4343798906461161616at_nat @ A12 )
        & ( finite4343798906461161616at_nat @ A23 )
        & ( A23 != bot_bo228742789529271731at_nat )
        & ! [X: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X @ A12 )
           => ? [Y: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ Y @ A23 )
                & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R2 ) ) ) ) ) ).

% max_ext.simps
thf(fact_953_max__ext_Osimps,axiom,
    ! [A12: set_Pr7649655576376376530on_val,A23: set_Pr7649655576376376530on_val,R2: set_Pr4305388979797279737on_val] :
      ( ( member806545433280400814on_val @ ( produc113594951777976637on_val @ A12 @ A23 ) @ ( max_ex7543004642404436129on_val @ R2 ) )
      = ( ( finite8324237431073730739on_val @ A12 )
        & ( finite8324237431073730739on_val @ A23 )
        & ( A23 != bot_bo6722054719108133734on_val )
        & ! [X: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ X @ A12 )
           => ? [Y: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ Y @ A23 )
                & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ R2 ) ) ) ) ) ).

% max_ext.simps
thf(fact_954_max__ext_Osimps,axiom,
    ! [A12: set_se491199571785101031od_o_o,A23: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member9169340693537962768od_o_o @ ( produc4478624491032480599od_o_o @ A12 @ A23 ) @ ( max_ex5401630698724402626od_o_o @ R2 ) )
      = ( ( finite9011636195296603184od_o_o @ A12 )
        & ( finite9011636195296603184od_o_o @ A23 )
        & ( A23 != bot_bo8006819982598807891od_o_o )
        & ! [X: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X @ A12 )
           => ? [Y: set_Product_prod_o_o] :
                ( ( member3157015407008874800od_o_o @ Y @ A23 )
                & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R2 ) ) ) ) ) ).

% max_ext.simps
thf(fact_955_max__ext_Osimps,axiom,
    ! [A12: set_se7855581050983116737at_nat,A23: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ A12 @ A23 ) @ ( max_ex4535816423174879388at_nat @ R2 ) )
      = ( ( finite9047747110432174090at_nat @ A12 )
        & ( finite9047747110432174090at_nat @ A23 )
        & ( A23 != bot_bo3083307316010499117at_nat )
        & ! [X: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X @ A12 )
           => ? [Y: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Y @ A23 )
                & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 ) ) ) ) ) ).

% max_ext.simps
thf(fact_956_max__ext_Osimps,axiom,
    ! [A12: set_Product_prod_o_o,A23: set_Product_prod_o_o,R2: set_Pr7011910401827753159od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A12 @ A23 ) @ ( max_ex7572203241849361634od_o_o @ R2 ) )
      = ( ( finite6120865539452801872od_o_o @ A12 )
        & ( finite6120865539452801872od_o_o @ A23 )
        & ( A23 != bot_bo7073875226086086771od_o_o )
        & ! [X: product_prod_o_o] :
            ( ( member7466972457876170832od_o_o @ X @ A12 )
           => ? [Y: product_prod_o_o] :
                ( ( member7466972457876170832od_o_o @ Y @ A23 )
                & ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X @ Y ) @ R2 ) ) ) ) ) ).

% max_ext.simps
thf(fact_957_max__ext_Osimps,axiom,
    ! [A12: set_Pr1261947904930325089at_nat,A23: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A12 @ A23 ) @ ( max_ex8135407076693332796at_nat @ R2 ) )
      = ( ( finite6177210948735845034at_nat @ A12 )
        & ( finite6177210948735845034at_nat @ A23 )
        & ( A23 != bot_bo2099793752762293965at_nat )
        & ! [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ A12 )
           => ? [Y: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y @ A23 )
                & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 ) ) ) ) ) ).

% max_ext.simps
thf(fact_958_max__ext_Omax__extI,axiom,
    ! [X5: set_Pr1932065953672099015od_o_o,Y7: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o] :
      ( ( finite820925532130942480od_o_o @ X5 )
     => ( ( finite820925532130942480od_o_o @ Y7 )
       => ( ( Y7 != bot_bo498831808395348787od_o_o )
         => ( ! [X2: produc2934264451710624999od_o_o] :
                ( ( member444158400953824016od_o_o @ X2 @ X5 )
               => ? [Xa2: produc2934264451710624999od_o_o] :
                    ( ( member444158400953824016od_o_o @ Xa2 @ Y7 )
                    & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X2 @ Xa2 ) @ R2 ) ) )
           => ( member3740103928775426832od_o_o @ ( produc2282539314773629783od_o_o @ X5 @ Y7 ) @ ( max_ex9141845325704764834od_o_o @ R2 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_959_max__ext_Omax__extI,axiom,
    ! [X5: set_Pr457366540195662369at_nat,Y7: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat] :
      ( ( finite8072506237346428522at_nat @ X5 )
     => ( ( finite8072506237346428522at_nat @ Y7 )
       => ( ( Y7 != bot_bo1676454896636820109at_nat )
         => ( ! [X2: produc732395585841259969at_nat] :
                ( ( member4095101504841534314at_nat @ X2 @ X5 )
               => ? [Xa2: produc732395585841259969at_nat] :
                    ( ( member4095101504841534314at_nat @ Xa2 @ Y7 )
                    & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X2 @ Xa2 ) @ R2 ) ) )
           => ( member7768518508883498896at_nat @ ( produc4325049062531119319at_nat @ X5 @ Y7 ) @ ( max_ex1111939896555521276at_nat @ R2 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_960_max__ext_Omax__extI,axiom,
    ! [X5: set_Pr4329608150637261639at_nat,Y7: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ( finite4343798906461161616at_nat @ X5 )
     => ( ( finite4343798906461161616at_nat @ Y7 )
       => ( ( Y7 != bot_bo228742789529271731at_nat )
         => ( ! [X2: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X2 @ X5 )
               => ? [Xa2: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ Xa2 @ Y7 )
                    & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ Xa2 ) @ R2 ) ) )
           => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y7 ) @ ( max_ex8833738776070432290at_nat @ R2 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_961_max__ext_Omax__extI,axiom,
    ! [X5: set_Product_prod_o_o,Y7: set_Product_prod_o_o,R2: set_Pr7011910401827753159od_o_o] :
      ( ( finite6120865539452801872od_o_o @ X5 )
     => ( ( finite6120865539452801872od_o_o @ Y7 )
       => ( ( Y7 != bot_bo7073875226086086771od_o_o )
         => ( ! [X2: product_prod_o_o] :
                ( ( member7466972457876170832od_o_o @ X2 @ X5 )
               => ? [Xa2: product_prod_o_o] :
                    ( ( member7466972457876170832od_o_o @ Xa2 @ Y7 )
                    & ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X2 @ Xa2 ) @ R2 ) ) )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X5 @ Y7 ) @ ( max_ex7572203241849361634od_o_o @ R2 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_962_max__ext_Omax__extI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( finite6177210948735845034at_nat @ X5 )
     => ( ( finite6177210948735845034at_nat @ Y7 )
       => ( ( Y7 != bot_bo2099793752762293965at_nat )
         => ( ! [X2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X2 @ X5 )
               => ? [Xa2: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Xa2 @ Y7 )
                    & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Xa2 ) @ R2 ) ) )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y7 ) @ ( max_ex8135407076693332796at_nat @ R2 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_963_max__ext_Omax__extI,axiom,
    ! [X5: set_Pr7649655576376376530on_val,Y7: set_Pr7649655576376376530on_val,R2: set_Pr4305388979797279737on_val] :
      ( ( finite8324237431073730739on_val @ X5 )
     => ( ( finite8324237431073730739on_val @ Y7 )
       => ( ( Y7 != bot_bo6722054719108133734on_val )
         => ( ! [X2: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ X2 @ X5 )
               => ? [Xa2: produc9003781588162623004on_val] :
                    ( ( member2879030624002546611on_val @ Xa2 @ Y7 )
                    & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Xa2 ) @ R2 ) ) )
           => ( member806545433280400814on_val @ ( produc113594951777976637on_val @ X5 @ Y7 ) @ ( max_ex7543004642404436129on_val @ R2 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_964_max__ext_Omax__extI,axiom,
    ! [X5: set_se491199571785101031od_o_o,Y7: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( finite9011636195296603184od_o_o @ X5 )
     => ( ( finite9011636195296603184od_o_o @ Y7 )
       => ( ( Y7 != bot_bo8006819982598807891od_o_o )
         => ( ! [X2: set_Product_prod_o_o] :
                ( ( member3157015407008874800od_o_o @ X2 @ X5 )
               => ? [Xa2: set_Product_prod_o_o] :
                    ( ( member3157015407008874800od_o_o @ Xa2 @ Y7 )
                    & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Xa2 ) @ R2 ) ) )
           => ( member9169340693537962768od_o_o @ ( produc4478624491032480599od_o_o @ X5 @ Y7 ) @ ( max_ex5401630698724402626od_o_o @ R2 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_965_max__ext_Omax__extI,axiom,
    ! [X5: set_se7855581050983116737at_nat,Y7: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( finite9047747110432174090at_nat @ X5 )
     => ( ( finite9047747110432174090at_nat @ Y7 )
       => ( ( Y7 != bot_bo3083307316010499117at_nat )
         => ( ! [X2: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ X2 @ X5 )
               => ? [Xa2: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ Xa2 @ Y7 )
                    & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Xa2 ) @ R2 ) ) )
           => ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ X5 @ Y7 ) @ ( max_ex4535816423174879388at_nat @ R2 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_966_finite__ordLess__infinite,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 )
       => ( ( finite_finite_nat @ ( field_nat @ R ) )
         => ( ~ ( finite_finite_o @ ( field_o @ R5 ) )
           => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R5 ) @ bNF_We2061588309931968636_nat_o ) ) ) ) ) ).

% finite_ordLess_infinite
thf(fact_967_finite__ordLess__infinite,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 )
       => ( ( finite_finite_o @ ( field_o @ R ) )
         => ( ~ ( finite_finite_o @ ( field_o @ R5 ) )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ) ) ).

% finite_ordLess_infinite
thf(fact_968_finite__ordLess__infinite,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( finite_finite_nat @ ( field_nat @ R ) )
         => ( ~ ( finite_finite_nat @ ( field_nat @ R5 ) )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We3448473429330749420at_nat ) ) ) ) ) ).

% finite_ordLess_infinite
thf(fact_969_finite__ordLess__infinite,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( finite_finite_o @ ( field_o @ R ) )
         => ( ~ ( finite_finite_nat @ ( field_nat @ R5 ) )
           => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat ) ) ) ) ) ).

% finite_ordLess_infinite
thf(fact_970_initial__segment__of__Diff,axiom,
    ! [P4: set_Product_prod_o_o,Q4: set_Product_prod_o_o,S: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P4 @ Q4 ) @ init_seg_of_o )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( minus_7052513330764642158od_o_o @ P4 @ S ) @ ( minus_7052513330764642158od_o_o @ Q4 @ S ) ) @ init_seg_of_o ) ) ).

% initial_segment_of_Diff
thf(fact_971_initial__segment__of__Diff,axiom,
    ! [P4: set_Pr1261947904930325089at_nat,Q4: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P4 @ Q4 ) @ init_seg_of_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( minus_1356011639430497352at_nat @ P4 @ S ) @ ( minus_1356011639430497352at_nat @ Q4 @ S ) ) @ init_seg_of_nat ) ) ).

% initial_segment_of_Diff
thf(fact_972_wo__rel_Osuc__greater,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,B5: set_Pr1932065953672099015od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R )
     => ( ( ord_le7586318239878589543od_o_o @ B5 @ ( field_5121862139100741428od_o_o @ R ) )
       => ( ( ( order_909087248044410034od_o_o @ R @ B5 )
           != bot_bo498831808395348787od_o_o )
         => ( ( member444158400953824016od_o_o @ B @ B5 )
           => ( ( ( bNF_We7386194254115604808od_o_o @ R @ B5 )
               != B )
              & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B @ ( bNF_We7386194254115604808od_o_o @ R @ B5 ) ) @ R ) ) ) ) ) ) ).

% wo_rel.suc_greater
thf(fact_973_wo__rel_Osuc__greater,axiom,
    ! [R: set_Pr4427867301352232263at_nat,B5: set_Pr457366540195662369at_nat,B: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R )
     => ( ( ord_le4841768313324958145at_nat @ B5 @ ( field_7361900449976055182at_nat @ R ) )
       => ( ( ( order_3539579856470909964at_nat @ R @ B5 )
           != bot_bo1676454896636820109at_nat )
         => ( ( member4095101504841534314at_nat @ B @ B5 )
           => ( ( ( bNF_We5419384366696408482at_nat @ R @ B5 )
               != B )
              & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B @ ( bNF_We5419384366696408482at_nat @ R @ B5 ) ) @ R ) ) ) ) ) ) ).

% wo_rel.suc_greater
thf(fact_974_wo__rel_Osuc__greater,axiom,
    ! [R: set_Pr5564308138774400199at_nat,B5: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R )
     => ( ( ord_le1268244103169919719at_nat @ B5 @ ( field_2862844103579723188at_nat @ R ) )
       => ( ( ( order_5334716200017444146at_nat @ R @ B5 )
           != bot_bo228742789529271731at_nat )
         => ( ( member8757157785044589968at_nat @ B @ B5 )
           => ( ( ( bNF_We2397293392365222856at_nat @ R @ B5 )
               != B )
              & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ ( bNF_We2397293392365222856at_nat @ R @ B5 ) ) @ R ) ) ) ) ) ) ).

% wo_rel.suc_greater
thf(fact_975_wo__rel_Osuc__greater,axiom,
    ! [R: set_Product_prod_o_o,B5: set_o,B: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ( ( ord_less_eq_set_o @ B5 @ ( field_o @ R ) )
       => ( ( ( order_AboveS_o @ R @ B5 )
           != bot_bot_set_o )
         => ( ( member_o @ B @ B5 )
           => ( ( ( bNF_We8594000642646999417_suc_o @ R @ B5 )
               != B )
              & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ ( bNF_We8594000642646999417_suc_o @ R @ B5 ) ) @ R ) ) ) ) ) ) ).

% wo_rel.suc_greater
thf(fact_976_wo__rel_Osuc__greater,axiom,
    ! [R: set_Pr1261947904930325089at_nat,B5: set_nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( ord_less_eq_set_nat @ B5 @ ( field_nat @ R ) )
       => ( ( ( order_AboveS_nat @ R @ B5 )
           != bot_bot_set_nat )
         => ( ( member_nat @ B @ B5 )
           => ( ( ( bNF_We6243639482952820783uc_nat @ R @ B5 )
               != B )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ ( bNF_We6243639482952820783uc_nat @ R @ B5 ) ) @ R ) ) ) ) ) ) ).

% wo_rel.suc_greater
thf(fact_977_wo__rel_Osuc__greater,axiom,
    ! [R: set_Pr4305388979797279737on_val,B5: set_Pr7649655576376376530on_val,B: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( ord_le6900622908513426738on_val @ B5 @ ( field_9010718613293825679on_val @ R ) )
       => ( ( ( order_3343947986673401233on_val @ R @ B5 )
           != bot_bo6722054719108133734on_val )
         => ( ( member2879030624002546611on_val @ B @ B5 )
           => ( ( ( bNF_We7054565829724620667on_val @ R @ B5 )
               != B )
              & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ ( bNF_We7054565829724620667on_val @ R @ B5 ) ) @ R ) ) ) ) ) ) ).

% wo_rel.suc_greater
thf(fact_978_wo__rel_Osuc__greater,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,B5: set_se491199571785101031od_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( ord_le2654036014095304327od_o_o @ B5 @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( ( order_5655281162628242642od_o_o @ R @ B5 )
           != bot_bo8006819982598807891od_o_o )
         => ( ( member3157015407008874800od_o_o @ B @ B5 )
           => ( ( ( bNF_We1257987224342916968od_o_o @ R @ B5 )
               != B )
              & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ ( bNF_We1257987224342916968od_o_o @ R @ B5 ) ) @ R ) ) ) ) ) ) ).

% wo_rel.suc_greater
thf(fact_979_wo__rel_Osuc__greater,axiom,
    ! [R: set_Pr4329608150637261639at_nat,B5: set_se7855581050983116737at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( ord_le2077887516847798113at_nat @ B5 @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( ( order_9095727932502173612at_nat @ R @ B5 )
           != bot_bo3083307316010499117at_nat )
         => ( ( member2643936169264416010at_nat @ B @ B5 )
           => ( ( ( bNF_We3445218718054152002at_nat @ R @ B5 )
               != B )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ ( bNF_We3445218718054152002at_nat @ R @ B5 ) ) @ R ) ) ) ) ) ) ).

% wo_rel.suc_greater
thf(fact_980_wf__insert,axiom,
    ! [Y3: produc9003781588162623004on_val,X3: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( wf_Pro3300669915054815219on_val @ ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ Y3 @ X3 ) @ R ) )
      = ( ( wf_Pro3300669915054815219on_val @ R )
        & ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ ( transi3365842532184619097on_val @ R ) ) ) ) ).

% wf_insert
thf(fact_981_wf__insert,axiom,
    ! [Y3: set_Product_prod_o_o,X3: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( wf_set2384217916178216176od_o_o @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ X3 ) @ R ) )
      = ( ( wf_set2384217916178216176od_o_o @ R )
        & ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ ( transi8975321201238539274od_o_o @ R ) ) ) ) ).

% wf_insert
thf(fact_982_wf__insert,axiom,
    ! [Y3: set_Pr1261947904930325089at_nat,X3: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( wf_set7675013139414177994at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X3 ) @ R ) )
      = ( ( wf_set7675013139414177994at_nat @ R )
        & ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ ( transi6170703685534862052at_nat @ R ) ) ) ) ).

% wf_insert
thf(fact_983_ordLess__irreflexive,axiom,
    ! [R: set_Product_prod_o_o] :
      ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R ) @ bNF_We2568191115274586186ss_o_o ) ).

% ordLess_irreflexive
thf(fact_984_ordLess__irreflexive,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R ) @ bNF_We3448473429330749420at_nat ) ).

% ordLess_irreflexive
thf(fact_985_ordLess__transitive,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R6 ) @ bNF_We2568191115274586186ss_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R6 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLess_transitive
thf(fact_986_ordLess__transitive,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We3448473429330749420at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We3448473429330749420at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R6 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLess_transitive
thf(fact_987_ordLess__transitive,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R5 ) @ bNF_We2061588309931968636_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We8182288985678559134_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R6 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLess_transitive
thf(fact_988_ordLess__transitive,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We8182288985678559134_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R6 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLess_transitive
thf(fact_989_ordLess__transitive,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R6 ) @ bNF_We2061588309931968636_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R6 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLess_transitive
thf(fact_990_ordLess__transitive,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We3448473429330749420at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R6 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLess_transitive
thf(fact_991_wo__rel_Osuc__least__AboveS,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B5: set_Pr1932065953672099015od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R )
     => ( ( member444158400953824016od_o_o @ A @ ( order_909087248044410034od_o_o @ R @ B5 ) )
       => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( bNF_We7386194254115604808od_o_o @ R @ B5 ) @ A ) @ R ) ) ) ).

% wo_rel.suc_least_AboveS
thf(fact_992_wo__rel_Osuc__least__AboveS,axiom,
    ! [R: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B5: set_Pr457366540195662369at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R )
     => ( ( member4095101504841534314at_nat @ A @ ( order_3539579856470909964at_nat @ R @ B5 ) )
       => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( bNF_We5419384366696408482at_nat @ R @ B5 ) @ A ) @ R ) ) ) ).

% wo_rel.suc_least_AboveS
thf(fact_993_wo__rel_Osuc__least__AboveS,axiom,
    ! [R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R )
     => ( ( member8757157785044589968at_nat @ A @ ( order_5334716200017444146at_nat @ R @ B5 ) )
       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( bNF_We2397293392365222856at_nat @ R @ B5 ) @ A ) @ R ) ) ) ).

% wo_rel.suc_least_AboveS
thf(fact_994_wo__rel_Osuc__least__AboveS,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B5: set_Pr7649655576376376530on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( member2879030624002546611on_val @ A @ ( order_3343947986673401233on_val @ R @ B5 ) )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( bNF_We7054565829724620667on_val @ R @ B5 ) @ A ) @ R ) ) ) ).

% wo_rel.suc_least_AboveS
thf(fact_995_wo__rel_Osuc__least__AboveS,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B5: set_se491199571785101031od_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( member3157015407008874800od_o_o @ A @ ( order_5655281162628242642od_o_o @ R @ B5 ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_We1257987224342916968od_o_o @ R @ B5 ) @ A ) @ R ) ) ) ).

% wo_rel.suc_least_AboveS
thf(fact_996_wo__rel_Osuc__least__AboveS,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B5: set_se7855581050983116737at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( member2643936169264416010at_nat @ A @ ( order_9095727932502173612at_nat @ R @ B5 ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_We3445218718054152002at_nat @ R @ B5 ) @ A ) @ R ) ) ) ).

% wo_rel.suc_least_AboveS
thf(fact_997_refl__on__init__seg__of,axiom,
    ! [R: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R ) @ init_seg_of_o ) ).

% refl_on_init_seg_of
thf(fact_998_refl__on__init__seg__of,axiom,
    ! [R: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R ) @ init_seg_of_nat ) ).

% refl_on_init_seg_of
thf(fact_999_antisym__init__seg__of,axiom,
    ! [R: set_Product_prod_o_o,S: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ S ) @ init_seg_of_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ S @ R ) @ init_seg_of_o )
       => ( R = S ) ) ) ).

% antisym_init_seg_of
thf(fact_1000_antisym__init__seg__of,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ S ) @ init_seg_of_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ S @ R ) @ init_seg_of_nat )
       => ( R = S ) ) ) ).

% antisym_init_seg_of
thf(fact_1001_trans__init__seg__of,axiom,
    ! [R: set_Product_prod_o_o,S: set_Product_prod_o_o,T: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ S ) @ init_seg_of_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ S @ T ) @ init_seg_of_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ T ) @ init_seg_of_o ) ) ) ).

% trans_init_seg_of
thf(fact_1002_trans__init__seg__of,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ S ) @ init_seg_of_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ S @ T ) @ init_seg_of_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ T ) @ init_seg_of_nat ) ) ) ).

% trans_init_seg_of
thf(fact_1003_wo__rel_Oequals__suc__AboveS,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,B5: set_Pr1932065953672099015od_o_o,A: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R )
     => ( ( ord_le7586318239878589543od_o_o @ B5 @ ( field_5121862139100741428od_o_o @ R ) )
       => ( ( member444158400953824016od_o_o @ A @ ( order_909087248044410034od_o_o @ R @ B5 ) )
         => ( ! [A7: produc2934264451710624999od_o_o] :
                ( ( member444158400953824016od_o_o @ A7 @ ( order_909087248044410034od_o_o @ R @ B5 ) )
               => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ A7 ) @ R ) )
           => ( A
              = ( bNF_We7386194254115604808od_o_o @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
thf(fact_1004_wo__rel_Oequals__suc__AboveS,axiom,
    ! [R: set_Pr4427867301352232263at_nat,B5: set_Pr457366540195662369at_nat,A: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R )
     => ( ( ord_le4841768313324958145at_nat @ B5 @ ( field_7361900449976055182at_nat @ R ) )
       => ( ( member4095101504841534314at_nat @ A @ ( order_3539579856470909964at_nat @ R @ B5 ) )
         => ( ! [A7: produc732395585841259969at_nat] :
                ( ( member4095101504841534314at_nat @ A7 @ ( order_3539579856470909964at_nat @ R @ B5 ) )
               => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ A7 ) @ R ) )
           => ( A
              = ( bNF_We5419384366696408482at_nat @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
thf(fact_1005_wo__rel_Oequals__suc__AboveS,axiom,
    ! [R: set_Pr5564308138774400199at_nat,B5: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R )
     => ( ( ord_le1268244103169919719at_nat @ B5 @ ( field_2862844103579723188at_nat @ R ) )
       => ( ( member8757157785044589968at_nat @ A @ ( order_5334716200017444146at_nat @ R @ B5 ) )
         => ( ! [A7: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ A7 @ ( order_5334716200017444146at_nat @ R @ B5 ) )
               => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ A7 ) @ R ) )
           => ( A
              = ( bNF_We2397293392365222856at_nat @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
thf(fact_1006_wo__rel_Oequals__suc__AboveS,axiom,
    ! [R: set_Product_prod_o_o,B5: set_o,A: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R )
     => ( ( ord_less_eq_set_o @ B5 @ ( field_o @ R ) )
       => ( ( member_o @ A @ ( order_AboveS_o @ R @ B5 ) )
         => ( ! [A7: $o] :
                ( ( member_o @ A7 @ ( order_AboveS_o @ R @ B5 ) )
               => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ A7 ) @ R ) )
           => ( A
              = ( bNF_We8594000642646999417_suc_o @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
thf(fact_1007_wo__rel_Oequals__suc__AboveS,axiom,
    ! [R: set_Pr1261947904930325089at_nat,B5: set_nat,A: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( ord_less_eq_set_nat @ B5 @ ( field_nat @ R ) )
       => ( ( member_nat @ A @ ( order_AboveS_nat @ R @ B5 ) )
         => ( ! [A7: nat] :
                ( ( member_nat @ A7 @ ( order_AboveS_nat @ R @ B5 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A7 ) @ R ) )
           => ( A
              = ( bNF_We6243639482952820783uc_nat @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
thf(fact_1008_wo__rel_Oequals__suc__AboveS,axiom,
    ! [R: set_Pr4305388979797279737on_val,B5: set_Pr7649655576376376530on_val,A: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( ord_le6900622908513426738on_val @ B5 @ ( field_9010718613293825679on_val @ R ) )
       => ( ( member2879030624002546611on_val @ A @ ( order_3343947986673401233on_val @ R @ B5 ) )
         => ( ! [A7: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ A7 @ ( order_3343947986673401233on_val @ R @ B5 ) )
               => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ A7 ) @ R ) )
           => ( A
              = ( bNF_We7054565829724620667on_val @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
thf(fact_1009_wo__rel_Oequals__suc__AboveS,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,B5: set_se491199571785101031od_o_o,A: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( ord_le2654036014095304327od_o_o @ B5 @ ( field_6101175518749917524od_o_o @ R ) )
       => ( ( member3157015407008874800od_o_o @ A @ ( order_5655281162628242642od_o_o @ R @ B5 ) )
         => ( ! [A7: set_Product_prod_o_o] :
                ( ( member3157015407008874800od_o_o @ A7 @ ( order_5655281162628242642od_o_o @ R @ B5 ) )
               => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A7 ) @ R ) )
           => ( A
              = ( bNF_We1257987224342916968od_o_o @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
thf(fact_1010_wo__rel_Oequals__suc__AboveS,axiom,
    ! [R: set_Pr4329608150637261639at_nat,B5: set_se7855581050983116737at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( ord_le2077887516847798113at_nat @ B5 @ ( field_3145993188331922222at_nat @ R ) )
       => ( ( member2643936169264416010at_nat @ A @ ( order_9095727932502173612at_nat @ R @ B5 ) )
         => ( ! [A7: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ A7 @ ( order_9095727932502173612at_nat @ R @ B5 ) )
               => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A7 ) @ R ) )
           => ( A
              = ( bNF_We3445218718054152002at_nat @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
thf(fact_1011_in__finite__psubset,axiom,
    ! [A3: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A3 @ B5 ) @ finite1053609180499918954od_o_o )
      = ( ( ord_le3253702748690112691od_o_o @ A3 @ B5 )
        & ( finite6120865539452801872od_o_o @ B5 ) ) ) ).

% in_finite_psubset
thf(fact_1012_in__finite__psubset,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A3 @ B5 ) @ finite469560695537375940at_nat )
      = ( ( ord_le7866589430770878221at_nat @ A3 @ B5 )
        & ( finite6177210948735845034at_nat @ B5 ) ) ) ).

% in_finite_psubset
thf(fact_1013_Not__Domain__rtrancl,axiom,
    ! [X3: produc2934264451710624999od_o_o,R2: set_Pr1013884327308616903od_o_o,Y3: produc2934264451710624999od_o_o] :
      ( ~ ( member444158400953824016od_o_o @ X3 @ ( domain2187190646820045772od_o_o @ R2 ) )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X3 @ Y3 ) @ ( transi3610599525415160810od_o_o @ R2 ) )
        = ( X3 = Y3 ) ) ) ).

% Not_Domain_rtrancl
thf(fact_1014_Not__Domain__rtrancl,axiom,
    ! [X3: produc732395585841259969at_nat,R2: set_Pr4427867301352232263at_nat,Y3: produc732395585841259969at_nat] :
      ( ~ ( member4095101504841534314at_nat @ X3 @ ( domain8659765193365367884at_nat @ R2 ) )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X3 @ Y3 ) @ ( transi4536970581772660548at_nat @ R2 ) )
        = ( X3 = Y3 ) ) ) ).

% Not_Domain_rtrancl
thf(fact_1015_Not__Domain__rtrancl,axiom,
    ! [X3: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat,Y3: produc3843707927480180839at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X3 @ ( domain1102310642530566604at_nat @ R2 ) )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y3 ) @ ( transi5049001397978532970at_nat @ R2 ) )
        = ( X3 = Y3 ) ) ) ).

% Not_Domain_rtrancl
thf(fact_1016_Not__Domain__rtrancl,axiom,
    ! [X3: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val,Y3: produc9003781588162623004on_val] :
      ( ~ ( member2879030624002546611on_val @ X3 @ ( domain6215983837468562630on_val @ R2 ) )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ ( transi3365842532184619097on_val @ R2 ) )
        = ( X3 = Y3 ) ) ) ).

% Not_Domain_rtrancl
thf(fact_1017_Not__Domain__rtrancl,axiom,
    ! [X3: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o,Y3: set_Product_prod_o_o] :
      ( ~ ( member3157015407008874800od_o_o @ X3 @ ( domain8908270702770587084od_o_o @ R2 ) )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ ( transi8975321201238539274od_o_o @ R2 ) )
        = ( X3 = Y3 ) ) ) ).

% Not_Domain_rtrancl
thf(fact_1018_Not__Domain__rtrancl,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( ~ ( member2643936169264416010at_nat @ X3 @ ( domain8802589944165416012at_nat @ R2 ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ ( transi6170703685534862052at_nat @ R2 ) )
        = ( X3 = Y3 ) ) ) ).

% Not_Domain_rtrancl
thf(fact_1019_rtrancl__Un__separatorE,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,P: set_Pr4305388979797279737on_val,Q: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ ( transi3365842532184619097on_val @ ( sup_su3890347010539237325on_val @ P @ Q ) ) )
     => ( ! [X2: produc9003781588162623004on_val,Y4: produc9003781588162623004on_val] :
            ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ X2 ) @ ( transi3365842532184619097on_val @ P ) )
           => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y4 ) @ Q )
             => ( X2 = Y4 ) ) )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ ( transi3365842532184619097on_val @ P ) ) ) ) ).

% rtrancl_Un_separatorE
thf(fact_1020_rtrancl__Un__separatorE,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,P: set_Pr1932065953672099015od_o_o,Q: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( transi8975321201238539274od_o_o @ ( sup_su4078869789063261851od_o_o @ P @ Q ) ) )
     => ( ! [X2: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ X2 ) @ ( transi8975321201238539274od_o_o @ P ) )
           => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y4 ) @ Q )
             => ( X2 = Y4 ) ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( transi8975321201238539274od_o_o @ P ) ) ) ) ).

% rtrancl_Un_separatorE
thf(fact_1021_rtrancl__Un__separatorE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,P: set_Pr4329608150637261639at_nat,Q: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( transi6170703685534862052at_nat @ ( sup_su5525570899277871387at_nat @ P @ Q ) ) )
     => ( ! [X2: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ X2 ) @ ( transi6170703685534862052at_nat @ P ) )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y4 ) @ Q )
             => ( X2 = Y4 ) ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( transi6170703685534862052at_nat @ P ) ) ) ) ).

% rtrancl_Un_separatorE
thf(fact_1022_rtrancl__Un__separator__converseE,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,P: set_Pr4305388979797279737on_val,Q: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ ( transi3365842532184619097on_val @ ( sup_su3890347010539237325on_val @ P @ Q ) ) )
     => ( ! [X2: produc9003781588162623004on_val,Y4: produc9003781588162623004on_val] :
            ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ B ) @ ( transi3365842532184619097on_val @ P ) )
           => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y4 @ X2 ) @ Q )
             => ( Y4 = X2 ) ) )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ ( transi3365842532184619097on_val @ P ) ) ) ) ).

% rtrancl_Un_separator_converseE
thf(fact_1023_rtrancl__Un__separator__converseE,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,P: set_Pr1932065953672099015od_o_o,Q: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( transi8975321201238539274od_o_o @ ( sup_su4078869789063261851od_o_o @ P @ Q ) ) )
     => ( ! [X2: set_Product_prod_o_o,Y4: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ B ) @ ( transi8975321201238539274od_o_o @ P ) )
           => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ X2 ) @ Q )
             => ( Y4 = X2 ) ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( transi8975321201238539274od_o_o @ P ) ) ) ) ).

% rtrancl_Un_separator_converseE
thf(fact_1024_rtrancl__Un__separator__converseE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,P: set_Pr4329608150637261639at_nat,Q: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( transi6170703685534862052at_nat @ ( sup_su5525570899277871387at_nat @ P @ Q ) ) )
     => ( ! [X2: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ B ) @ ( transi6170703685534862052at_nat @ P ) )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ X2 ) @ Q )
             => ( Y4 = X2 ) ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( transi6170703685534862052at_nat @ P ) ) ) ) ).

% rtrancl_Un_separator_converseE
thf(fact_1025_converse__rtrancl__induct2,axiom,
    ! [Ax: produc9003781588162623004on_val,Ay: produc9003781588162623004on_val,Bx: produc9003781588162623004on_val,By: produc9003781588162623004on_val,R: set_Pr4590028993149660487on_val,P: produc9003781588162623004on_val > produc9003781588162623004on_val > $o] :
      ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ Ax @ Ay ) @ ( produc4958730763592889809on_val @ Bx @ By ) ) @ ( transi725908587811456796on_val @ R ) )
     => ( ( P @ Bx @ By )
       => ( ! [A2: produc9003781588162623004on_val,B2: produc9003781588162623004on_val,Aa: produc9003781588162623004on_val,Ba: produc9003781588162623004on_val] :
              ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ A2 @ B2 ) @ ( produc4958730763592889809on_val @ Aa @ Ba ) ) @ R )
             => ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ Aa @ Ba ) @ ( produc4958730763592889809on_val @ Bx @ By ) ) @ ( transi725908587811456796on_val @ R ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A2 @ B2 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_1026_converse__rtrancl__induct2,axiom,
    ! [Ax: set_Product_prod_o_o,Ay: set_Product_prod_o_o,Bx: set_Product_prod_o_o,By: set_Product_prod_o_o,R: set_Pr1013884327308616903od_o_o,P: set_Product_prod_o_o > set_Product_prod_o_o > $o] :
      ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ Ax @ Ay ) @ ( produc763777882069021527od_o_o @ Bx @ By ) ) @ ( transi3610599525415160810od_o_o @ R ) )
     => ( ( P @ Bx @ By )
       => ( ! [A2: set_Product_prod_o_o,B2: set_Product_prod_o_o,Aa: set_Product_prod_o_o,Ba: set_Product_prod_o_o] :
              ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ A2 @ B2 ) @ ( produc763777882069021527od_o_o @ Aa @ Ba ) ) @ R )
             => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ Aa @ Ba ) @ ( produc763777882069021527od_o_o @ Bx @ By ) ) @ ( transi3610599525415160810od_o_o @ R ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A2 @ B2 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_1027_converse__rtrancl__induct2,axiom,
    ! [Ax: set_Product_prod_o_o,Ay: set_Pr1261947904930325089at_nat,Bx: set_Product_prod_o_o,By: set_Pr1261947904930325089at_nat,R: set_Pr4427867301352232263at_nat,P: set_Product_prod_o_o > set_Pr1261947904930325089at_nat > $o] :
      ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ Ax @ Ay ) @ ( produc8517790099723286449at_nat @ Bx @ By ) ) @ ( transi4536970581772660548at_nat @ R ) )
     => ( ( P @ Bx @ By )
       => ( ! [A2: set_Product_prod_o_o,B2: set_Pr1261947904930325089at_nat,Aa: set_Product_prod_o_o,Ba: set_Pr1261947904930325089at_nat] :
              ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ A2 @ B2 ) @ ( produc8517790099723286449at_nat @ Aa @ Ba ) ) @ R )
             => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ Aa @ Ba ) @ ( produc8517790099723286449at_nat @ Bx @ By ) ) @ ( transi4536970581772660548at_nat @ R ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A2 @ B2 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_1028_converse__rtrancl__induct2,axiom,
    ! [Ax: set_Pr1261947904930325089at_nat,Ay: set_Pr1261947904930325089at_nat,Bx: set_Pr1261947904930325089at_nat,By: set_Pr1261947904930325089at_nat,R: set_Pr5564308138774400199at_nat,P: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ Ax @ Ay ) @ ( produc2922128104949294807at_nat @ Bx @ By ) ) @ ( transi5049001397978532970at_nat @ R ) )
     => ( ( P @ Bx @ By )
       => ( ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,Aa: set_Pr1261947904930325089at_nat,Ba: set_Pr1261947904930325089at_nat] :
              ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ ( produc2922128104949294807at_nat @ Aa @ Ba ) ) @ R )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ Aa @ Ba ) @ ( produc2922128104949294807at_nat @ Bx @ By ) ) @ ( transi5049001397978532970at_nat @ R ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A2 @ B2 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_1029_converse__rtrancl__induct2,axiom,
    ! [Ax: produc1457211279475724562t_char > prat,Ay: produc1457211279475724562t_char > option_val,Bx: produc1457211279475724562t_char > prat,By: produc1457211279475724562t_char > option_val,R: set_Pr4305388979797279737on_val,P: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Ax @ Ay ) @ ( produc74620467203698198on_val @ Bx @ By ) ) @ ( transi3365842532184619097on_val @ R ) )
     => ( ( P @ Bx @ By )
       => ( ! [A2: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > option_val,Aa: produc1457211279475724562t_char > prat,Ba: produc1457211279475724562t_char > option_val] :
              ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ A2 @ B2 ) @ ( produc74620467203698198on_val @ Aa @ Ba ) ) @ R )
             => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Aa @ Ba ) @ ( produc74620467203698198on_val @ Bx @ By ) ) @ ( transi3365842532184619097on_val @ R ) )
               => ( ( P @ Aa @ Ba )
                 => ( P @ A2 @ B2 ) ) ) )
         => ( P @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_1030_converse__rtrancl__into__rtrancl,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val,C: produc9003781588162623004on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ C ) @ ( transi3365842532184619097on_val @ R ) )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ C ) @ ( transi3365842532184619097on_val @ R ) ) ) ) ).

% converse_rtrancl_into_rtrancl
thf(fact_1031_converse__rtrancl__into__rtrancl,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o,C: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ C ) @ ( transi8975321201238539274od_o_o @ R ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ C ) @ ( transi8975321201238539274od_o_o @ R ) ) ) ) ).

% converse_rtrancl_into_rtrancl
thf(fact_1032_converse__rtrancl__into__rtrancl,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ C ) @ ( transi6170703685534862052at_nat @ R ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ C ) @ ( transi6170703685534862052at_nat @ R ) ) ) ) ).

% converse_rtrancl_into_rtrancl
thf(fact_1033_converse__rtrancl__induct,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val,P: produc9003781588162623004on_val > $o] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ ( transi3365842532184619097on_val @ R ) )
     => ( ( P @ B )
       => ( ! [Y4: produc9003781588162623004on_val,Z: produc9003781588162623004on_val] :
              ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y4 @ Z ) @ R )
             => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Z @ B ) @ ( transi3365842532184619097on_val @ R ) )
               => ( ( P @ Z )
                 => ( P @ Y4 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_rtrancl_induct
thf(fact_1034_converse__rtrancl__induct,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o,P: set_Product_prod_o_o > $o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( transi8975321201238539274od_o_o @ R ) )
     => ( ( P @ B )
       => ( ! [Y4: set_Product_prod_o_o,Z: set_Product_prod_o_o] :
              ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ Z ) @ R )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Z @ B ) @ ( transi8975321201238539274od_o_o @ R ) )
               => ( ( P @ Z )
                 => ( P @ Y4 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_rtrancl_induct
thf(fact_1035_converse__rtrancl__induct,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( transi6170703685534862052at_nat @ R ) )
     => ( ( P @ B )
       => ( ! [Y4: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
              ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ Z ) @ R )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Z @ B ) @ ( transi6170703685534862052at_nat @ R ) )
               => ( ( P @ Z )
                 => ( P @ Y4 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_rtrancl_induct
thf(fact_1036_converse__rtranclE,axiom,
    ! [X3: produc9003781588162623004on_val,Z3: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Z3 ) @ ( transi3365842532184619097on_val @ R ) )
     => ( ( X3 != Z3 )
       => ~ ! [Y4: produc9003781588162623004on_val] :
              ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y4 ) @ R )
             => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y4 @ Z3 ) @ ( transi3365842532184619097on_val @ R ) ) ) ) ) ).

% converse_rtranclE
thf(fact_1037_converse__rtranclE,axiom,
    ! [X3: set_Product_prod_o_o,Z3: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Z3 ) @ ( transi8975321201238539274od_o_o @ R ) )
     => ( ( X3 != Z3 )
       => ~ ! [Y4: set_Product_prod_o_o] :
              ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y4 ) @ R )
             => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ Z3 ) @ ( transi8975321201238539274od_o_o @ R ) ) ) ) ) ).

% converse_rtranclE
thf(fact_1038_converse__rtranclE,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Z3 ) @ ( transi6170703685534862052at_nat @ R ) )
     => ( ( X3 != Z3 )
       => ~ ! [Y4: set_Pr1261947904930325089at_nat] :
              ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y4 ) @ R )
             => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ Z3 ) @ ( transi6170703685534862052at_nat @ R ) ) ) ) ) ).

% converse_rtranclE
thf(fact_1039_rtrancl__induct,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val,P: produc9003781588162623004on_val > $o] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ ( transi3365842532184619097on_val @ R ) )
     => ( ( P @ A )
       => ( ! [Y4: produc9003781588162623004on_val,Z: produc9003781588162623004on_val] :
              ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ Y4 ) @ ( transi3365842532184619097on_val @ R ) )
             => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y4 @ Z ) @ R )
               => ( ( P @ Y4 )
                 => ( P @ Z ) ) ) )
         => ( P @ B ) ) ) ) ).

% rtrancl_induct
thf(fact_1040_rtrancl__induct,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o,P: set_Product_prod_o_o > $o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( transi8975321201238539274od_o_o @ R ) )
     => ( ( P @ A )
       => ( ! [Y4: set_Product_prod_o_o,Z: set_Product_prod_o_o] :
              ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ Y4 ) @ ( transi8975321201238539274od_o_o @ R ) )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ Z ) @ R )
               => ( ( P @ Y4 )
                 => ( P @ Z ) ) ) )
         => ( P @ B ) ) ) ) ).

% rtrancl_induct
thf(fact_1041_rtrancl__induct,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( transi6170703685534862052at_nat @ R ) )
     => ( ( P @ A )
       => ( ! [Y4: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
              ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ Y4 ) @ ( transi6170703685534862052at_nat @ R ) )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ Z ) @ R )
               => ( ( P @ Y4 )
                 => ( P @ Z ) ) ) )
         => ( P @ B ) ) ) ) ).

% rtrancl_induct
thf(fact_1042_rtrancl__trans,axiom,
    ! [X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val,Z3: produc9003781588162623004on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ ( transi3365842532184619097on_val @ R ) )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y3 @ Z3 ) @ ( transi3365842532184619097on_val @ R ) )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Z3 ) @ ( transi3365842532184619097on_val @ R ) ) ) ) ).

% rtrancl_trans
thf(fact_1043_rtrancl__trans,axiom,
    ! [X3: set_Product_prod_o_o,Y3: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o,Z3: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ ( transi8975321201238539274od_o_o @ R ) )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ Z3 ) @ ( transi8975321201238539274od_o_o @ R ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Z3 ) @ ( transi8975321201238539274od_o_o @ R ) ) ) ) ).

% rtrancl_trans
thf(fact_1044_rtrancl__trans,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ ( transi6170703685534862052at_nat @ R ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ Z3 ) @ ( transi6170703685534862052at_nat @ R ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Z3 ) @ ( transi6170703685534862052at_nat @ R ) ) ) ) ).

% rtrancl_trans
thf(fact_1045_rtranclE,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ ( transi3365842532184619097on_val @ R ) )
     => ( ( A != B )
       => ~ ! [Y4: produc9003781588162623004on_val] :
              ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ Y4 ) @ ( transi3365842532184619097on_val @ R ) )
             => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y4 @ B ) @ R ) ) ) ) ).

% rtranclE
thf(fact_1046_rtranclE,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( transi8975321201238539274od_o_o @ R ) )
     => ( ( A != B )
       => ~ ! [Y4: set_Product_prod_o_o] :
              ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ Y4 ) @ ( transi8975321201238539274od_o_o @ R ) )
             => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ B ) @ R ) ) ) ) ).

% rtranclE
thf(fact_1047_rtranclE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( transi6170703685534862052at_nat @ R ) )
     => ( ( A != B )
       => ~ ! [Y4: set_Pr1261947904930325089at_nat] :
              ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ Y4 ) @ ( transi6170703685534862052at_nat @ R ) )
             => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ B ) @ R ) ) ) ) ).

% rtranclE
thf(fact_1048_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val,C: produc9003781588162623004on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ ( transi3365842532184619097on_val @ R ) )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ C ) @ R )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ C ) @ ( transi3365842532184619097on_val @ R ) ) ) ) ).

% rtrancl.rtrancl_into_rtrancl
thf(fact_1049_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o,C: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( transi8975321201238539274od_o_o @ R ) )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ C ) @ R )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ C ) @ ( transi8975321201238539274od_o_o @ R ) ) ) ) ).

% rtrancl.rtrancl_into_rtrancl
thf(fact_1050_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( transi6170703685534862052at_nat @ R ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ C ) @ R )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ C ) @ ( transi6170703685534862052at_nat @ R ) ) ) ) ).

% rtrancl.rtrancl_into_rtrancl
thf(fact_1051_rtrancl_Ortrancl__refl,axiom,
    ! [A: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ A ) @ ( transi3365842532184619097on_val @ R ) ) ).

% rtrancl.rtrancl_refl
thf(fact_1052_rtrancl_Ortrancl__refl,axiom,
    ! [A: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A ) @ ( transi8975321201238539274od_o_o @ R ) ) ).

% rtrancl.rtrancl_refl
thf(fact_1053_rtrancl_Ortrancl__refl,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ ( transi6170703685534862052at_nat @ R ) ) ).

% rtrancl.rtrancl_refl
thf(fact_1054_rtrancl_Osimps,axiom,
    ! [A12: produc9003781588162623004on_val,A23: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A12 @ A23 ) @ ( transi3365842532184619097on_val @ R ) )
      = ( ? [A5: produc9003781588162623004on_val] :
            ( ( A12 = A5 )
            & ( A23 = A5 ) )
        | ? [A5: produc9003781588162623004on_val,B4: produc9003781588162623004on_val,C3: produc9003781588162623004on_val] :
            ( ( A12 = A5 )
            & ( A23 = C3 )
            & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A5 @ B4 ) @ ( transi3365842532184619097on_val @ R ) )
            & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B4 @ C3 ) @ R ) ) ) ) ).

% rtrancl.simps
thf(fact_1055_rtrancl_Osimps,axiom,
    ! [A12: set_Product_prod_o_o,A23: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A12 @ A23 ) @ ( transi8975321201238539274od_o_o @ R ) )
      = ( ? [A5: set_Product_prod_o_o] :
            ( ( A12 = A5 )
            & ( A23 = A5 ) )
        | ? [A5: set_Product_prod_o_o,B4: set_Product_prod_o_o,C3: set_Product_prod_o_o] :
            ( ( A12 = A5 )
            & ( A23 = C3 )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A5 @ B4 ) @ ( transi8975321201238539274od_o_o @ R ) )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B4 @ C3 ) @ R ) ) ) ) ).

% rtrancl.simps
thf(fact_1056_rtrancl_Osimps,axiom,
    ! [A12: set_Pr1261947904930325089at_nat,A23: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A12 @ A23 ) @ ( transi6170703685534862052at_nat @ R ) )
      = ( ? [A5: set_Pr1261947904930325089at_nat] :
            ( ( A12 = A5 )
            & ( A23 = A5 ) )
        | ? [A5: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
            ( ( A12 = A5 )
            & ( A23 = C3 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A5 @ B4 ) @ ( transi6170703685534862052at_nat @ R ) )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B4 @ C3 ) @ R ) ) ) ) ).

% rtrancl.simps
thf(fact_1057_rtrancl_Ocases,axiom,
    ! [A12: produc9003781588162623004on_val,A23: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A12 @ A23 ) @ ( transi3365842532184619097on_val @ R ) )
     => ( ( A23 != A12 )
       => ~ ! [B2: produc9003781588162623004on_val] :
              ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A12 @ B2 ) @ ( transi3365842532184619097on_val @ R ) )
             => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B2 @ A23 ) @ R ) ) ) ) ).

% rtrancl.cases
thf(fact_1058_rtrancl_Ocases,axiom,
    ! [A12: set_Product_prod_o_o,A23: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A12 @ A23 ) @ ( transi8975321201238539274od_o_o @ R ) )
     => ( ( A23 != A12 )
       => ~ ! [B2: set_Product_prod_o_o] :
              ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A12 @ B2 ) @ ( transi8975321201238539274od_o_o @ R ) )
             => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B2 @ A23 ) @ R ) ) ) ) ).

% rtrancl.cases
thf(fact_1059_rtrancl_Ocases,axiom,
    ! [A12: set_Pr1261947904930325089at_nat,A23: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A12 @ A23 ) @ ( transi6170703685534862052at_nat @ R ) )
     => ( ( A23 != A12 )
       => ~ ! [B2: set_Pr1261947904930325089at_nat] :
              ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A12 @ B2 ) @ ( transi6170703685534862052at_nat @ R ) )
             => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ A23 ) @ R ) ) ) ) ).

% rtrancl.cases
thf(fact_1060_rtrancl__induct2,axiom,
    ! [Ax: produc9003781588162623004on_val,Ay: produc9003781588162623004on_val,Bx: produc9003781588162623004on_val,By: produc9003781588162623004on_val,R: set_Pr4590028993149660487on_val,P: produc9003781588162623004on_val > produc9003781588162623004on_val > $o] :
      ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ Ax @ Ay ) @ ( produc4958730763592889809on_val @ Bx @ By ) ) @ ( transi725908587811456796on_val @ R ) )
     => ( ( P @ Ax @ Ay )
       => ( ! [A2: produc9003781588162623004on_val,B2: produc9003781588162623004on_val,Aa: produc9003781588162623004on_val,Ba: produc9003781588162623004on_val] :
              ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ Ax @ Ay ) @ ( produc4958730763592889809on_val @ A2 @ B2 ) ) @ ( transi725908587811456796on_val @ R ) )
             => ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ A2 @ B2 ) @ ( produc4958730763592889809on_val @ Aa @ Ba ) ) @ R )
               => ( ( P @ A2 @ B2 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_1061_rtrancl__induct2,axiom,
    ! [Ax: set_Product_prod_o_o,Ay: set_Product_prod_o_o,Bx: set_Product_prod_o_o,By: set_Product_prod_o_o,R: set_Pr1013884327308616903od_o_o,P: set_Product_prod_o_o > set_Product_prod_o_o > $o] :
      ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ Ax @ Ay ) @ ( produc763777882069021527od_o_o @ Bx @ By ) ) @ ( transi3610599525415160810od_o_o @ R ) )
     => ( ( P @ Ax @ Ay )
       => ( ! [A2: set_Product_prod_o_o,B2: set_Product_prod_o_o,Aa: set_Product_prod_o_o,Ba: set_Product_prod_o_o] :
              ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ Ax @ Ay ) @ ( produc763777882069021527od_o_o @ A2 @ B2 ) ) @ ( transi3610599525415160810od_o_o @ R ) )
             => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ A2 @ B2 ) @ ( produc763777882069021527od_o_o @ Aa @ Ba ) ) @ R )
               => ( ( P @ A2 @ B2 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_1062_rtrancl__induct2,axiom,
    ! [Ax: set_Product_prod_o_o,Ay: set_Pr1261947904930325089at_nat,Bx: set_Product_prod_o_o,By: set_Pr1261947904930325089at_nat,R: set_Pr4427867301352232263at_nat,P: set_Product_prod_o_o > set_Pr1261947904930325089at_nat > $o] :
      ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ Ax @ Ay ) @ ( produc8517790099723286449at_nat @ Bx @ By ) ) @ ( transi4536970581772660548at_nat @ R ) )
     => ( ( P @ Ax @ Ay )
       => ( ! [A2: set_Product_prod_o_o,B2: set_Pr1261947904930325089at_nat,Aa: set_Product_prod_o_o,Ba: set_Pr1261947904930325089at_nat] :
              ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ Ax @ Ay ) @ ( produc8517790099723286449at_nat @ A2 @ B2 ) ) @ ( transi4536970581772660548at_nat @ R ) )
             => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ A2 @ B2 ) @ ( produc8517790099723286449at_nat @ Aa @ Ba ) ) @ R )
               => ( ( P @ A2 @ B2 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_1063_rtrancl__induct2,axiom,
    ! [Ax: set_Pr1261947904930325089at_nat,Ay: set_Pr1261947904930325089at_nat,Bx: set_Pr1261947904930325089at_nat,By: set_Pr1261947904930325089at_nat,R: set_Pr5564308138774400199at_nat,P: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ Ax @ Ay ) @ ( produc2922128104949294807at_nat @ Bx @ By ) ) @ ( transi5049001397978532970at_nat @ R ) )
     => ( ( P @ Ax @ Ay )
       => ( ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,Aa: set_Pr1261947904930325089at_nat,Ba: set_Pr1261947904930325089at_nat] :
              ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ Ax @ Ay ) @ ( produc2922128104949294807at_nat @ A2 @ B2 ) ) @ ( transi5049001397978532970at_nat @ R ) )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ ( produc2922128104949294807at_nat @ Aa @ Ba ) ) @ R )
               => ( ( P @ A2 @ B2 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_1064_rtrancl__induct2,axiom,
    ! [Ax: produc1457211279475724562t_char > prat,Ay: produc1457211279475724562t_char > option_val,Bx: produc1457211279475724562t_char > prat,By: produc1457211279475724562t_char > option_val,R: set_Pr4305388979797279737on_val,P: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Ax @ Ay ) @ ( produc74620467203698198on_val @ Bx @ By ) ) @ ( transi3365842532184619097on_val @ R ) )
     => ( ( P @ Ax @ Ay )
       => ( ! [A2: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > option_val,Aa: produc1457211279475724562t_char > prat,Ba: produc1457211279475724562t_char > option_val] :
              ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Ax @ Ay ) @ ( produc74620467203698198on_val @ A2 @ B2 ) ) @ ( transi3365842532184619097on_val @ R ) )
             => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ A2 @ B2 ) @ ( produc74620467203698198on_val @ Aa @ Ba ) ) @ R )
               => ( ( P @ A2 @ B2 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_1065_converse__rtranclE2,axiom,
    ! [Xa: produc9003781588162623004on_val,Xb: produc9003781588162623004on_val,Za: produc9003781588162623004on_val,Zb: produc9003781588162623004on_val,R: set_Pr4590028993149660487on_val] :
      ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ Xa @ Xb ) @ ( produc4958730763592889809on_val @ Za @ Zb ) ) @ ( transi725908587811456796on_val @ R ) )
     => ( ( ( produc4958730763592889809on_val @ Xa @ Xb )
         != ( produc4958730763592889809on_val @ Za @ Zb ) )
       => ~ ! [A2: produc9003781588162623004on_val,B2: produc9003781588162623004on_val] :
              ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ Xa @ Xb ) @ ( produc4958730763592889809on_val @ A2 @ B2 ) ) @ R )
             => ~ ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ A2 @ B2 ) @ ( produc4958730763592889809on_val @ Za @ Zb ) ) @ ( transi725908587811456796on_val @ R ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_1066_converse__rtranclE2,axiom,
    ! [Xa: set_Product_prod_o_o,Xb: set_Product_prod_o_o,Za: set_Product_prod_o_o,Zb: set_Product_prod_o_o,R: set_Pr1013884327308616903od_o_o] :
      ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ Xa @ Xb ) @ ( produc763777882069021527od_o_o @ Za @ Zb ) ) @ ( transi3610599525415160810od_o_o @ R ) )
     => ( ( ( produc763777882069021527od_o_o @ Xa @ Xb )
         != ( produc763777882069021527od_o_o @ Za @ Zb ) )
       => ~ ! [A2: set_Product_prod_o_o,B2: set_Product_prod_o_o] :
              ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ Xa @ Xb ) @ ( produc763777882069021527od_o_o @ A2 @ B2 ) ) @ R )
             => ~ ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ ( produc763777882069021527od_o_o @ A2 @ B2 ) @ ( produc763777882069021527od_o_o @ Za @ Zb ) ) @ ( transi3610599525415160810od_o_o @ R ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_1067_converse__rtranclE2,axiom,
    ! [Xa: set_Product_prod_o_o,Xb: set_Pr1261947904930325089at_nat,Za: set_Product_prod_o_o,Zb: set_Pr1261947904930325089at_nat,R: set_Pr4427867301352232263at_nat] :
      ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ Xa @ Xb ) @ ( produc8517790099723286449at_nat @ Za @ Zb ) ) @ ( transi4536970581772660548at_nat @ R ) )
     => ( ( ( produc8517790099723286449at_nat @ Xa @ Xb )
         != ( produc8517790099723286449at_nat @ Za @ Zb ) )
       => ~ ! [A2: set_Product_prod_o_o,B2: set_Pr1261947904930325089at_nat] :
              ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ Xa @ Xb ) @ ( produc8517790099723286449at_nat @ A2 @ B2 ) ) @ R )
             => ~ ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ A2 @ B2 ) @ ( produc8517790099723286449at_nat @ Za @ Zb ) ) @ ( transi4536970581772660548at_nat @ R ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_1068_converse__rtranclE2,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Xb: set_Pr1261947904930325089at_nat,Za: set_Pr1261947904930325089at_nat,Zb: set_Pr1261947904930325089at_nat,R: set_Pr5564308138774400199at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ Xa @ Xb ) @ ( produc2922128104949294807at_nat @ Za @ Zb ) ) @ ( transi5049001397978532970at_nat @ R ) )
     => ( ( ( produc2922128104949294807at_nat @ Xa @ Xb )
         != ( produc2922128104949294807at_nat @ Za @ Zb ) )
       => ~ ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
              ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ Xa @ Xb ) @ ( produc2922128104949294807at_nat @ A2 @ B2 ) ) @ R )
             => ~ ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ ( produc2922128104949294807at_nat @ Za @ Zb ) ) @ ( transi5049001397978532970at_nat @ R ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_1069_converse__rtranclE2,axiom,
    ! [Xa: produc1457211279475724562t_char > prat,Xb: produc1457211279475724562t_char > option_val,Za: produc1457211279475724562t_char > prat,Zb: produc1457211279475724562t_char > option_val,R: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Xa @ Xb ) @ ( produc74620467203698198on_val @ Za @ Zb ) ) @ ( transi3365842532184619097on_val @ R ) )
     => ( ( ( produc74620467203698198on_val @ Xa @ Xb )
         != ( produc74620467203698198on_val @ Za @ Zb ) )
       => ~ ! [A2: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > option_val] :
              ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Xa @ Xb ) @ ( produc74620467203698198on_val @ A2 @ B2 ) ) @ R )
             => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ A2 @ B2 ) @ ( produc74620467203698198on_val @ Za @ Zb ) ) @ ( transi3365842532184619097on_val @ R ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_1070_wo__rel_Osuc__ofilter__in,axiom,
    ! [R: set_Pr1013884327308616903od_o_o,A3: set_Pr1932065953672099015od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R )
     => ( ( order_543634532542674813od_o_o @ R @ A3 )
       => ( ( ( order_909087248044410034od_o_o @ R @ A3 )
           != bot_bo498831808395348787od_o_o )
         => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B @ ( bNF_We7386194254115604808od_o_o @ R @ A3 ) ) @ R )
           => ( ( B
               != ( bNF_We7386194254115604808od_o_o @ R @ A3 ) )
             => ( member444158400953824016od_o_o @ B @ A3 ) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
thf(fact_1071_wo__rel_Osuc__ofilter__in,axiom,
    ! [R: set_Pr4427867301352232263at_nat,A3: set_Pr457366540195662369at_nat,B: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R )
     => ( ( order_3205064739578148183at_nat @ R @ A3 )
       => ( ( ( order_3539579856470909964at_nat @ R @ A3 )
           != bot_bo1676454896636820109at_nat )
         => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B @ ( bNF_We5419384366696408482at_nat @ R @ A3 ) ) @ R )
           => ( ( B
               != ( bNF_We5419384366696408482at_nat @ R @ A3 ) )
             => ( member4095101504841534314at_nat @ B @ A3 ) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
thf(fact_1072_wo__rel_Osuc__ofilter__in,axiom,
    ! [R: set_Pr5564308138774400199at_nat,A3: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R )
     => ( ( order_3142731893281451261at_nat @ R @ A3 )
       => ( ( ( order_5334716200017444146at_nat @ R @ A3 )
           != bot_bo228742789529271731at_nat )
         => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ ( bNF_We2397293392365222856at_nat @ R @ A3 ) ) @ R )
           => ( ( B
               != ( bNF_We2397293392365222856at_nat @ R @ A3 ) )
             => ( member8757157785044589968at_nat @ B @ A3 ) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
thf(fact_1073_wo__rel_Osuc__ofilter__in,axiom,
    ! [R: set_Pr4305388979797279737on_val,A3: set_Pr7649655576376376530on_val,B: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( order_6652311712475968006on_val @ R @ A3 )
       => ( ( ( order_3343947986673401233on_val @ R @ A3 )
           != bot_bo6722054719108133734on_val )
         => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ ( bNF_We7054565829724620667on_val @ R @ A3 ) ) @ R )
           => ( ( B
               != ( bNF_We7054565829724620667on_val @ R @ A3 ) )
             => ( member2879030624002546611on_val @ B @ A3 ) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
thf(fact_1074_wo__rel_Osuc__ofilter__in,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,A3: set_se491199571785101031od_o_o,B: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R )
     => ( ( order_136921456694023837od_o_o @ R @ A3 )
       => ( ( ( order_5655281162628242642od_o_o @ R @ A3 )
           != bot_bo8006819982598807891od_o_o )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ ( bNF_We1257987224342916968od_o_o @ R @ A3 ) ) @ R )
           => ( ( B
               != ( bNF_We1257987224342916968od_o_o @ R @ A3 ) )
             => ( member3157015407008874800od_o_o @ B @ A3 ) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
thf(fact_1075_wo__rel_Osuc__ofilter__in,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A3: set_se7855581050983116737at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R )
     => ( ( order_6860593127430111223at_nat @ R @ A3 )
       => ( ( ( order_9095727932502173612at_nat @ R @ A3 )
           != bot_bo3083307316010499117at_nat )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ ( bNF_We3445218718054152002at_nat @ R @ A3 ) ) @ R )
           => ( ( B
               != ( bNF_We3445218718054152002at_nat @ R @ A3 ) )
             => ( member2643936169264416010at_nat @ B @ A3 ) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
thf(fact_1076_acyclic__insert,axiom,
    ! [Y3: produc9003781588162623004on_val,X3: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( transi5235682513840182669on_val @ ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ Y3 @ X3 ) @ R ) )
      = ( ( transi5235682513840182669on_val @ R )
        & ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ ( transi3365842532184619097on_val @ R ) ) ) ) ).

% acyclic_insert
thf(fact_1077_acyclic__insert,axiom,
    ! [Y3: set_Product_prod_o_o,X3: set_Product_prod_o_o,R: set_Pr1932065953672099015od_o_o] :
      ( ( transi6732886088652640214od_o_o @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ X3 ) @ R ) )
      = ( ( transi6732886088652640214od_o_o @ R )
        & ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ ( transi8975321201238539274od_o_o @ R ) ) ) ) ).

% acyclic_insert
thf(fact_1078_acyclic__insert,axiom,
    ! [Y3: set_Pr1261947904930325089at_nat,X3: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( transi5940216143853766832at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X3 ) @ R ) )
      = ( ( transi5940216143853766832at_nat @ R )
        & ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ ( transi6170703685534862052at_nat @ R ) ) ) ) ).

% acyclic_insert
thf(fact_1079_ordLess__iff,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R5 ) @ bNF_We2061588309931968636_nat_o )
      = ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
        & ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 )
        & ~ ? [X6: $o > nat] : ( bNF_We4981740601763737620_o_nat @ R5 @ R @ X6 ) ) ) ).

% ordLess_iff
thf(fact_1080_ordLess__iff,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o )
      = ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
        & ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 )
        & ~ ? [X6: $o > $o] : ( bNF_We7100138119184918100ed_o_o @ R5 @ R @ X6 ) ) ) ).

% ordLess_iff
thf(fact_1081_ordLess__iff,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We3448473429330749420at_nat )
      = ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
        & ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
        & ~ ? [X6: nat > nat] : ( bNF_We650101353357441270at_nat @ R5 @ R @ X6 ) ) ) ).

% ordLess_iff
thf(fact_1082_ordLess__iff,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat )
      = ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
        & ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
        & ~ ? [X6: nat > $o] : ( bNF_We8084411962871922930_nat_o @ R5 @ R @ X6 ) ) ) ).

% ordLess_iff
thf(fact_1083_not__ordLess__iff__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( ~ ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R ) @ bNF_We2061588309931968636_nat_o ) )
          = ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We3435943820185352641_o_nat ) ) ) ) ).

% not_ordLess_iff_ordLeq
thf(fact_1084_not__ordLess__iff__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 )
       => ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R ) @ bNF_We2568191115274586186ss_o_o ) )
          = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ).

% not_ordLess_iff_ordLeq
thf(fact_1085_not__ordLess__iff__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R ) @ bNF_We3448473429330749420at_nat ) )
          = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% not_ordLess_iff_ordLeq
thf(fact_1086_not__ordLess__iff__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 )
       => ( ( ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R ) @ bNF_We8182288985678559134_o_nat ) )
          = ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R5 ) @ bNF_We6538615181293537951_nat_o ) ) ) ) ).

% not_ordLess_iff_ordLeq
thf(fact_1087_not__ordLeq__iff__ordLess,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 )
       => ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R ) @ bNF_We8115293805659342695eq_o_o ) )
          = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ) ).

% not_ordLeq_iff_ordLess
thf(fact_1088_not__ordLeq__iff__ordLess,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 )
       => ( ( ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R ) @ bNF_We3435943820185352641_o_nat ) )
          = ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R5 ) @ bNF_We2061588309931968636_nat_o ) ) ) ) ).

% not_ordLeq_iff_ordLess
thf(fact_1089_not__ordLeq__iff__ordLess,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R ) @ bNF_We2031159114419172489at_nat ) )
          = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We3448473429330749420at_nat ) ) ) ) ).

% not_ordLeq_iff_ordLess
thf(fact_1090_not__ordLeq__iff__ordLess,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( ~ ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R ) @ bNF_We6538615181293537951_nat_o ) )
          = ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat ) ) ) ) ).

% not_ordLeq_iff_ordLess
thf(fact_1091_ordLeq__transitive,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R5 ) @ bNF_We6538615181293537951_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We3435943820185352641_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R6 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordLeq_transitive
thf(fact_1092_ordLeq__transitive,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R6 ) @ bNF_We8115293805659342695eq_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R6 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% ordLeq_transitive
thf(fact_1093_ordLeq__transitive,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We3435943820185352641_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R6 ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% ordLeq_transitive
thf(fact_1094_ordLeq__transitive,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R6 ) @ bNF_We6538615181293537951_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R6 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% ordLeq_transitive
thf(fact_1095_ordLeq__transitive,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We2031159114419172489at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R6 ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% ordLeq_transitive
thf(fact_1096_ordLeq__transitive,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We2031159114419172489at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We2031159114419172489at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R6 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordLeq_transitive
thf(fact_1097_not__ordLeq__ordLess,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We8115293805659342695eq_o_o )
     => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R ) @ bNF_We2568191115274586186ss_o_o ) ) ).

% not_ordLeq_ordLess
thf(fact_1098_not__ordLeq__ordLess,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We3435943820185352641_o_nat )
     => ~ ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R ) @ bNF_We2061588309931968636_nat_o ) ) ).

% not_ordLeq_ordLess
thf(fact_1099_not__ordLeq__ordLess,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We2031159114419172489at_nat )
     => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R ) @ bNF_We3448473429330749420at_nat ) ) ).

% not_ordLeq_ordLess
thf(fact_1100_not__ordLeq__ordLess,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R5 ) @ bNF_We6538615181293537951_nat_o )
     => ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R ) @ bNF_We8182288985678559134_o_nat ) ) ).

% not_ordLeq_ordLess
thf(fact_1101_not__ordLess__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R5 ) @ bNF_We2061588309931968636_nat_o )
     => ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R ) @ bNF_We3435943820185352641_o_nat ) ) ).

% not_ordLess_ordLeq
thf(fact_1102_not__ordLess__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o )
     => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% not_ordLess_ordLeq
thf(fact_1103_not__ordLess__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We3448473429330749420at_nat )
     => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R ) @ bNF_We2031159114419172489at_nat ) ) ).

% not_ordLess_ordLeq
thf(fact_1104_not__ordLess__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat )
     => ~ ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R ) @ bNF_We6538615181293537951_nat_o ) ) ).

% not_ordLess_ordLeq
thf(fact_1105_ordLess__imp__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% ordLess_imp_ordLeq
thf(fact_1106_ordLess__imp__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We3448473429330749420at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We2031159114419172489at_nat ) ) ).

% ordLess_imp_ordLeq
thf(fact_1107_ordLess__imp__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We3435943820185352641_o_nat ) ) ).

% ordLess_imp_ordLeq
thf(fact_1108_ordLeq__ordLess__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R6 ) @ bNF_We2568191115274586186ss_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R6 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLeq_ordLess_trans
thf(fact_1109_ordLeq__ordLess__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R6 ) @ bNF_We2061588309931968636_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R6 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLeq_ordLess_trans
thf(fact_1110_ordLeq__ordLess__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We2031159114419172489at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We3448473429330749420at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R6 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLeq_ordLess_trans
thf(fact_1111_ordLeq__ordLess__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We3448473429330749420at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R6 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLeq_ordLess_trans
thf(fact_1112_ordLeq__ordLess__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R5 ) @ bNF_We6538615181293537951_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We8182288985678559134_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R6 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLeq_ordLess_trans
thf(fact_1113_ordLeq__ordLess__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We8182288985678559134_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R6 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLeq_ordLess_trans
thf(fact_1114_ordLess__ordLeq__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R5 ) @ bNF_We2061588309931968636_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We3435943820185352641_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R6 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLess_ordLeq_trans
thf(fact_1115_ordLess__ordLeq__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R6 ) @ bNF_We8115293805659342695eq_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R6 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLess_ordLeq_trans
thf(fact_1116_ordLess__ordLeq__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We3448473429330749420at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We2031159114419172489at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R6 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLess_ordLeq_trans
thf(fact_1117_ordLess__ordLeq__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We3435943820185352641_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R6 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLess_ordLeq_trans
thf(fact_1118_ordLess__ordLeq__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R6 ) @ bNF_We6538615181293537951_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R6 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLess_ordLeq_trans
thf(fact_1119_ordLess__ordLeq__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We2031159114419172489at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R6 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLess_ordLeq_trans
thf(fact_1120_ordLess__not__embed,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o )
     => ~ ? [X_1: $o > $o] : ( bNF_We7100138119184918100ed_o_o @ R5 @ R @ X_1 ) ) ).

% ordLess_not_embed
thf(fact_1121_ordLess__not__embed,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We3448473429330749420at_nat )
     => ~ ? [X_1: nat > nat] : ( bNF_We650101353357441270at_nat @ R5 @ R @ X_1 ) ) ).

% ordLess_not_embed
thf(fact_1122_ordLess__not__embed,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat )
     => ~ ? [X_1: nat > $o] : ( bNF_We8084411962871922930_nat_o @ R5 @ R @ X_1 ) ) ).

% ordLess_not_embed
thf(fact_1123_ordLeq__total,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R5 ) @ bNF_We6538615181293537951_nat_o )
          | ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R ) @ bNF_We3435943820185352641_o_nat ) ) ) ) ).

% ordLeq_total
thf(fact_1124_ordLeq__total,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We8115293805659342695eq_o_o )
          | ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ).

% ordLeq_total
thf(fact_1125_ordLeq__total,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We3435943820185352641_o_nat )
          | ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R ) @ bNF_We6538615181293537951_nat_o ) ) ) ) ).

% ordLeq_total
thf(fact_1126_ordLeq__total,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We2031159114419172489at_nat )
          | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% ordLeq_total
thf(fact_1127_ordLeq__reflexive,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% ordLeq_reflexive
thf(fact_1128_ordLeq__reflexive,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R ) @ bNF_We2031159114419172489at_nat ) ) ).

% ordLeq_reflexive
thf(fact_1129_ordLeq__Well__order__simp,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R5 ) @ bNF_We6538615181293537951_nat_o )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
        & ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 ) ) ) ).

% ordLeq_Well_order_simp
thf(fact_1130_ordLeq__Well__order__simp,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
        & ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 ) ) ) ).

% ordLeq_Well_order_simp
thf(fact_1131_ordLeq__Well__order__simp,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
        & ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 ) ) ) ).

% ordLeq_Well_order_simp
thf(fact_1132_ordLeq__Well__order__simp,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We2031159114419172489at_nat )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
        & ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 ) ) ) ).

% ordLeq_Well_order_simp
thf(fact_1133_exists__minim__Well__order,axiom,
    ! [R2: set_se491199571785101031od_o_o] :
      ( ( R2 != bot_bo8006819982598807891od_o_o )
     => ( ! [X2: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X2 @ R2 )
           => ( order_2615775362358406454r_on_o @ ( field_o @ X2 ) @ X2 ) )
       => ? [X2: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X2 @ R2 )
            & ! [Xa2: set_Product_prod_o_o] :
                ( ( member3157015407008874800od_o_o @ Xa2 @ R2 )
               => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Xa2 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ) ).

% exists_minim_Well_order
thf(fact_1134_exists__minim__Well__order,axiom,
    ! [R2: set_se7855581050983116737at_nat] :
      ( ( R2 != bot_bo3083307316010499117at_nat )
     => ( ! [X2: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X2 @ R2 )
           => ( order_2888998067076097458on_nat @ ( field_nat @ X2 ) @ X2 ) )
       => ? [X2: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X2 @ R2 )
            & ! [Xa2: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Xa2 @ R2 )
               => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Xa2 ) @ bNF_We2031159114419172489at_nat ) ) ) ) ) ).

% exists_minim_Well_order
thf(fact_1135_ordLess__or__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 )
       => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R5 ) @ bNF_We2061588309931968636_nat_o )
          | ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R ) @ bNF_We3435943820185352641_o_nat ) ) ) ) ).

% ordLess_or_ordLeq
thf(fact_1136_ordLess__or__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R5 ) @ R5 )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o )
          | ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ).

% ordLess_or_ordLeq
thf(fact_1137_ordLess__or__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We3448473429330749420at_nat )
          | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% ordLess_or_ordLeq
thf(fact_1138_ordLess__or__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat )
          | ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R ) @ bNF_We6538615181293537951_nat_o ) ) ) ) ).

% ordLess_or_ordLeq
thf(fact_1139_BNF__Wellorder__Constructions_OordLess__Field,axiom,
    ! [R12: set_Pr1261947904930325089at_nat,R23: set_Product_prod_o_o,F: nat > $o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R12 @ R23 ) @ bNF_We2061588309931968636_nat_o )
     => ( ( bNF_We8084411962871922930_nat_o @ R12 @ R23 @ F )
       => ( ( image_nat_o @ F @ ( field_nat @ R12 ) )
         != ( field_o @ R23 ) ) ) ) ).

% BNF_Wellorder_Constructions.ordLess_Field
thf(fact_1140_BNF__Wellorder__Constructions_OordLess__Field,axiom,
    ! [R12: set_Product_prod_o_o,R23: set_Product_prod_o_o,F: $o > $o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R12 @ R23 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( bNF_We7100138119184918100ed_o_o @ R12 @ R23 @ F )
       => ( ( image_o_o2 @ F @ ( field_o @ R12 ) )
         != ( field_o @ R23 ) ) ) ) ).

% BNF_Wellorder_Constructions.ordLess_Field
thf(fact_1141_BNF__Wellorder__Constructions_OordLess__Field,axiom,
    ! [R12: set_Pr1261947904930325089at_nat,R23: set_Pr1261947904930325089at_nat,F: nat > nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R12 @ R23 ) @ bNF_We3448473429330749420at_nat )
     => ( ( bNF_We650101353357441270at_nat @ R12 @ R23 @ F )
       => ( ( image_nat_nat2 @ F @ ( field_nat @ R12 ) )
         != ( field_nat @ R23 ) ) ) ) ).

% BNF_Wellorder_Constructions.ordLess_Field
thf(fact_1142_BNF__Wellorder__Constructions_OordLess__Field,axiom,
    ! [R12: set_Product_prod_o_o,R23: set_Pr1261947904930325089at_nat,F: $o > nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R12 @ R23 ) @ bNF_We8182288985678559134_o_nat )
     => ( ( bNF_We4981740601763737620_o_nat @ R12 @ R23 @ F )
       => ( ( image_o_nat @ F @ ( field_o @ R12 ) )
         != ( field_nat @ R23 ) ) ) ) ).

% BNF_Wellorder_Constructions.ordLess_Field
thf(fact_1143_internalize__ordLess,axiom,
    ! [R5: set_Product_prod_o_o,R: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R ) @ bNF_We2568191115274586186ss_o_o )
      = ( ? [P6: set_Product_prod_o_o] :
            ( ( ord_less_set_o @ ( field_o @ P6 ) @ ( field_o @ R ) )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ P6 ) @ bNF_We2654380646378065620so_o_o )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P6 @ R ) @ bNF_We2568191115274586186ss_o_o ) ) ) ) ).

% internalize_ordLess
thf(fact_1144_internalize__ordLess,axiom,
    ! [R5: set_Pr1261947904930325089at_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R ) @ bNF_We3448473429330749420at_nat )
      = ( ? [P6: set_Pr1261947904930325089at_nat] :
            ( ( ord_less_set_nat @ ( field_nat @ P6 ) @ ( field_nat @ R ) )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ P6 ) @ bNF_We5258908940166488438at_nat )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P6 @ R ) @ bNF_We3448473429330749420at_nat ) ) ) ) ).

% internalize_ordLess
thf(fact_1145_internalize__ordLess,axiom,
    ! [R5: set_Product_prod_o_o,R: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R ) @ bNF_We8182288985678559134_o_nat )
      = ( ? [P6: set_Pr1261947904930325089at_nat] :
            ( ( ord_less_set_nat @ ( field_nat @ P6 ) @ ( field_nat @ R ) )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ P6 ) @ bNF_We4116350175183677332_o_nat )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P6 @ R ) @ bNF_We3448473429330749420at_nat ) ) ) ) ).

% internalize_ordLess
thf(fact_1146_card__of__Field__ordLess,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R ) ) @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% card_of_Field_ordLess
thf(fact_1147_card__of__Field__ordLess,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R ) ) @ R ) @ bNF_We2031159114419172489at_nat ) ) ).

% card_of_Field_ordLess
thf(fact_1148_card__of__Un__ordLess__infinite,axiom,
    ! [C4: set_o,A3: set_o,B5: set_o] :
      ( ~ ( finite_finite_o @ C4 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3172636760949973492d_of_o @ C4 ) ) @ bNF_We2568191115274586186ss_o_o )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B5 ) @ ( bNF_Ca3172636760949973492d_of_o @ C4 ) ) @ bNF_We2568191115274586186ss_o_o )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( sup_sup_set_o @ A3 @ B5 ) ) @ ( bNF_Ca3172636760949973492d_of_o @ C4 ) ) @ bNF_We2568191115274586186ss_o_o ) ) ) ) ).

% card_of_Un_ordLess_infinite
thf(fact_1149_card__of__Un__ordLess__infinite,axiom,
    ! [C4: set_nat,A3: set_nat,B5: set_nat] :
      ( ~ ( finite_finite_nat @ C4 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ C4 ) ) @ bNF_We3448473429330749420at_nat )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B5 ) @ ( bNF_Ca3793111618940312692of_nat @ C4 ) ) @ bNF_We3448473429330749420at_nat )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( sup_sup_set_nat @ A3 @ B5 ) ) @ ( bNF_Ca3793111618940312692of_nat @ C4 ) ) @ bNF_We3448473429330749420at_nat ) ) ) ) ).

% card_of_Un_ordLess_infinite
thf(fact_1150_card__of__Un__ordLess__infinite,axiom,
    ! [C4: set_nat,A3: set_o,B5: set_o] :
      ( ~ ( finite_finite_nat @ C4 )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ C4 ) ) @ bNF_We8182288985678559134_o_nat )
       => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B5 ) @ ( bNF_Ca3793111618940312692of_nat @ C4 ) ) @ bNF_We8182288985678559134_o_nat )
         => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ ( sup_sup_set_o @ A3 @ B5 ) ) @ ( bNF_Ca3793111618940312692of_nat @ C4 ) ) @ bNF_We8182288985678559134_o_nat ) ) ) ) ).

% card_of_Un_ordLess_infinite
thf(fact_1151_internalize__ordLeq,axiom,
    ! [R5: set_Product_prod_o_o,R: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R ) @ bNF_We3435943820185352641_o_nat )
      = ( ? [P6: set_Pr1261947904930325089at_nat] :
            ( ( ord_less_eq_set_nat @ ( field_nat @ P6 ) @ ( field_nat @ R ) )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ P6 ) @ bNF_We4116350175183677332_o_nat )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P6 @ R ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% internalize_ordLeq
thf(fact_1152_internalize__ordLeq,axiom,
    ! [R5: set_Product_prod_o_o,R: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R ) @ bNF_We8115293805659342695eq_o_o )
      = ( ? [P6: set_Product_prod_o_o] :
            ( ( ord_less_eq_set_o @ ( field_o @ P6 ) @ ( field_o @ R ) )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ P6 ) @ bNF_We2654380646378065620so_o_o )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ P6 @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ).

% internalize_ordLeq
thf(fact_1153_internalize__ordLeq,axiom,
    ! [R5: set_Pr1261947904930325089at_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R ) @ bNF_We2031159114419172489at_nat )
      = ( ? [P6: set_Pr1261947904930325089at_nat] :
            ( ( ord_less_eq_set_nat @ ( field_nat @ P6 ) @ ( field_nat @ R ) )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ P6 ) @ bNF_We5258908940166488438at_nat )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ P6 @ R ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% internalize_ordLeq
thf(fact_1154_image__map__upd,axiom,
    ! [X3: produc2934264451710624999od_o_o,A3: set_Pr1932065953672099015od_o_o,M: produc2934264451710624999od_o_o > option_state,Y3: state] :
      ( ~ ( member444158400953824016od_o_o @ X3 @ A3 )
     => ( ( image_4422564924380504236_state @ ( fun_up4180510163728908852_state @ M @ X3 @ ( some_state @ Y3 ) ) @ A3 )
        = ( image_4422564924380504236_state @ M @ A3 ) ) ) ).

% image_map_upd
thf(fact_1155_image__map__upd,axiom,
    ! [X3: produc732395585841259969at_nat,A3: set_Pr457366540195662369at_nat,M: produc732395585841259969at_nat > option_state,Y3: state] :
      ( ~ ( member4095101504841534314at_nat @ X3 @ A3 )
     => ( ( image_773556728021010310_state @ ( fun_up6026250480174712078_state @ M @ X3 @ ( some_state @ Y3 ) ) @ A3 )
        = ( image_773556728021010310_state @ M @ A3 ) ) ) ).

% image_map_upd
thf(fact_1156_image__map__upd,axiom,
    ! [X3: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,M: produc3843707927480180839at_nat > option_state,Y3: state] :
      ( ~ ( member8757157785044589968at_nat @ X3 @ A3 )
     => ( ( image_3396607859836187180_state @ ( fun_up3329660993234963380_state @ M @ X3 @ ( some_state @ Y3 ) ) @ A3 )
        = ( image_3396607859836187180_state @ M @ A3 ) ) ) ).

% image_map_upd
thf(fact_1157_image__map__upd,axiom,
    ! [X3: produc2934264451710624999od_o_o,A3: set_Pr1932065953672099015od_o_o,M: produc2934264451710624999od_o_o > option_val,Y3: val] :
      ( ~ ( member444158400953824016od_o_o @ X3 @ A3 )
     => ( ( image_4505417531320963877on_val @ ( fun_up6453461605741198237on_val @ M @ X3 @ ( some_val @ Y3 ) ) @ A3 )
        = ( image_4505417531320963877on_val @ M @ A3 ) ) ) ).

% image_map_upd
thf(fact_1158_image__map__upd,axiom,
    ! [X3: produc732395585841259969at_nat,A3: set_Pr457366540195662369at_nat,M: produc732395585841259969at_nat > option_val,Y3: val] :
      ( ~ ( member4095101504841534314at_nat @ X3 @ A3 )
     => ( ( image_1232738520333227211on_val @ ( fun_up320048328433459523on_val @ M @ X3 @ ( some_val @ Y3 ) ) @ A3 )
        = ( image_1232738520333227211on_val @ M @ A3 ) ) ) ).

% image_map_upd
thf(fact_1159_image__map__upd,axiom,
    ! [X3: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,M: produc3843707927480180839at_nat > option_val,Y3: val] :
      ( ~ ( member8757157785044589968at_nat @ X3 @ A3 )
     => ( ( image_3891918652046008741on_val @ ( fun_up7479097124709437981on_val @ M @ X3 @ ( some_val @ Y3 ) ) @ A3 )
        = ( image_3891918652046008741on_val @ M @ A3 ) ) ) ).

% image_map_upd
thf(fact_1160_these__image__Some__eq,axiom,
    ! [A3: set_state] :
      ( ( these_state @ ( image_6076465424260689483_state @ some_state @ A3 ) )
      = A3 ) ).

% these_image_Some_eq
thf(fact_1161_these__image__Some__eq,axiom,
    ! [A3: set_val] :
      ( ( these_val @ ( image_val_option_val @ some_val @ A3 ) )
      = A3 ) ).

% these_image_Some_eq
thf(fact_1162_card__of__image,axiom,
    ! [F: $o > $o,A3: set_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( image_o_o2 @ F @ A3 ) ) @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) ) @ bNF_We8115293805659342695eq_o_o ) ).

% card_of_image
thf(fact_1163_card__of__image,axiom,
    ! [F: nat > $o,A3: set_nat] : ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ ( image_nat_o @ F @ A3 ) ) @ ( bNF_Ca3793111618940312692of_nat @ A3 ) ) @ bNF_We3435943820185352641_o_nat ) ).

% card_of_image
thf(fact_1164_card__of__image,axiom,
    ! [F: nat > nat,A3: set_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( image_nat_nat2 @ F @ A3 ) ) @ ( bNF_Ca3793111618940312692of_nat @ A3 ) ) @ bNF_We2031159114419172489at_nat ) ).

% card_of_image
thf(fact_1165_card__of__ordIso__finite,axiom,
    ! [A3: set_o,B5: set_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3172636760949973492d_of_o @ B5 ) ) @ bNF_We2654380646378065620so_o_o )
     => ( ( finite_finite_o @ A3 )
        = ( finite_finite_o @ B5 ) ) ) ).

% card_of_ordIso_finite
thf(fact_1166_card__of__ordIso__finite,axiom,
    ! [A3: set_o,B5: set_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ B5 ) ) @ bNF_We4116350175183677332_o_nat )
     => ( ( finite_finite_o @ A3 )
        = ( finite_finite_nat @ B5 ) ) ) ).

% card_of_ordIso_finite
thf(fact_1167_card__of__ordIso__finite,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ B5 ) ) @ bNF_We5258908940166488438at_nat )
     => ( ( finite_finite_nat @ A3 )
        = ( finite_finite_nat @ B5 ) ) ) ).

% card_of_ordIso_finite
thf(fact_1168_ordIso__transitive,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R6 ) @ bNF_We7219021536291862642_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R6 ) @ bNF_We2654380646378065620so_o_o ) ) ) ).

% ordIso_transitive
thf(fact_1169_ordIso__transitive,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R5 ) @ bNF_We7219021536291862642_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We4116350175183677332_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R6 ) @ bNF_We5258908940166488438at_nat ) ) ) ).

% ordIso_transitive
thf(fact_1170_ordIso__transitive,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We5258908940166488438at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R6 ) @ bNF_We4116350175183677332_o_nat ) ) ) ).

% ordIso_transitive
thf(fact_1171_ordIso__transitive,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We4116350175183677332_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R6 ) @ bNF_We4116350175183677332_o_nat ) ) ) ).

% ordIso_transitive
thf(fact_1172_ordIso__transitive,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R6 ) @ bNF_We2654380646378065620so_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R6 ) @ bNF_We2654380646378065620so_o_o ) ) ) ).

% ordIso_transitive
thf(fact_1173_ordIso__transitive,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We5258908940166488438at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We5258908940166488438at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R6 ) @ bNF_We5258908940166488438at_nat ) ) ) ).

% ordIso_transitive
thf(fact_1174_ordIso__symmetric,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R5 ) @ bNF_We7219021536291862642_nat_o )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R ) @ bNF_We4116350175183677332_o_nat ) ) ).

% ordIso_symmetric
thf(fact_1175_ordIso__symmetric,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We4116350175183677332_o_nat )
     => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R ) @ bNF_We7219021536291862642_nat_o ) ) ).

% ordIso_symmetric
thf(fact_1176_ordIso__symmetric,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2654380646378065620so_o_o )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R ) @ bNF_We2654380646378065620so_o_o ) ) ).

% ordIso_symmetric
thf(fact_1177_ordIso__symmetric,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We5258908940166488438at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R ) @ bNF_We5258908940166488438at_nat ) ) ).

% ordIso_symmetric
thf(fact_1178_card__of__cong,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2654380646378065620so_o_o )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R ) ) @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R5 ) ) ) @ bNF_We2654380646378065620so_o_o ) ) ).

% card_of_cong
thf(fact_1179_card__of__cong,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We4116350175183677332_o_nat )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R ) ) @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R5 ) ) ) @ bNF_We4116350175183677332_o_nat ) ) ).

% card_of_cong
thf(fact_1180_card__of__cong,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R5 ) @ bNF_We7219021536291862642_nat_o )
     => ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R ) ) @ ( bNF_Ca3172636760949973492d_of_o @ ( field_o @ R5 ) ) ) @ bNF_We7219021536291862642_nat_o ) ) ).

% card_of_cong
thf(fact_1181_card__of__cong,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We5258908940166488438at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R ) ) @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ R5 ) ) ) @ bNF_We5258908940166488438at_nat ) ) ).

% card_of_cong
thf(fact_1182_card__of__refl,axiom,
    ! [A3: set_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) ) @ bNF_We2654380646378065620so_o_o ) ).

% card_of_refl
thf(fact_1183_card__of__refl,axiom,
    ! [A3: set_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ A3 ) ) @ bNF_We5258908940166488438at_nat ) ).

% card_of_refl
thf(fact_1184_card__of__empty__ordIso,axiom,
    member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) ) @ bNF_We2654380646378065620so_o_o ).

% card_of_empty_ordIso
thf(fact_1185_card__of__empty__ordIso,axiom,
    member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) ) @ bNF_We4116350175183677332_o_nat ).

% card_of_empty_ordIso
thf(fact_1186_card__of__empty__ordIso,axiom,
    member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) ) @ bNF_We5258908940166488438at_nat ).

% card_of_empty_ordIso
thf(fact_1187_card__of__empty2,axiom,
    ! [A3: set_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) ) @ bNF_We2654380646378065620so_o_o )
     => ( A3 = bot_bot_set_o ) ) ).

% card_of_empty2
thf(fact_1188_card__of__empty2,axiom,
    ! [A3: set_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) ) @ bNF_We4116350175183677332_o_nat )
     => ( A3 = bot_bot_set_o ) ) ).

% card_of_empty2
thf(fact_1189_card__of__empty2,axiom,
    ! [A3: set_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) ) @ bNF_We5258908940166488438at_nat )
     => ( A3 = bot_bot_set_nat ) ) ).

% card_of_empty2
thf(fact_1190_card__of__ordLeq2,axiom,
    ! [A3: set_o,B5: set_o] :
      ( ( A3 != bot_bot_set_o )
     => ( ( ? [G2: $o > $o] :
              ( ( image_o_o2 @ G2 @ B5 )
              = A3 ) )
        = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3172636760949973492d_of_o @ B5 ) ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% card_of_ordLeq2
thf(fact_1191_card__of__ordLeq2,axiom,
    ! [A3: set_o,B5: set_nat] :
      ( ( A3 != bot_bot_set_o )
     => ( ( ? [G2: nat > $o] :
              ( ( image_nat_o @ G2 @ B5 )
              = A3 ) )
        = ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ B5 ) ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% card_of_ordLeq2
thf(fact_1192_card__of__ordLeq2,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( ? [G2: nat > nat] :
              ( ( image_nat_nat2 @ G2 @ B5 )
              = A3 ) )
        = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ B5 ) ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% card_of_ordLeq2
thf(fact_1193_surj__imp__ordLeq,axiom,
    ! [B5: set_o,F: $o > $o,A3: set_o] :
      ( ( ord_less_eq_set_o @ B5 @ ( image_o_o2 @ F @ A3 ) )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B5 ) @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% surj_imp_ordLeq
thf(fact_1194_surj__imp__ordLeq,axiom,
    ! [B5: set_o,F: nat > $o,A3: set_nat] :
      ( ( ord_less_eq_set_o @ B5 @ ( image_nat_o @ F @ A3 ) )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B5 ) @ ( bNF_Ca3793111618940312692of_nat @ A3 ) ) @ bNF_We3435943820185352641_o_nat ) ) ).

% surj_imp_ordLeq
thf(fact_1195_surj__imp__ordLeq,axiom,
    ! [B5: set_nat,F: nat > nat,A3: set_nat] :
      ( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat2 @ F @ A3 ) )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B5 ) @ ( bNF_Ca3793111618940312692of_nat @ A3 ) ) @ bNF_We2031159114419172489at_nat ) ) ).

% surj_imp_ordLeq
thf(fact_1196_internalize__card__of__ordLeq2,axiom,
    ! [A3: set_o,C4: set_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3172636760949973492d_of_o @ C4 ) ) @ bNF_We8115293805659342695eq_o_o )
      = ( ? [B6: set_o] :
            ( ( ord_less_eq_set_o @ B6 @ C4 )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) ) @ bNF_We2654380646378065620so_o_o )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) @ ( bNF_Ca3172636760949973492d_of_o @ C4 ) ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ).

% internalize_card_of_ordLeq2
thf(fact_1197_internalize__card__of__ordLeq2,axiom,
    ! [A3: set_o,C4: set_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ C4 ) ) @ bNF_We3435943820185352641_o_nat )
      = ( ? [B6: set_nat] :
            ( ( ord_less_eq_set_nat @ B6 @ C4 )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ B6 ) ) @ bNF_We4116350175183677332_o_nat )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B6 ) @ ( bNF_Ca3793111618940312692of_nat @ C4 ) ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% internalize_card_of_ordLeq2
thf(fact_1198_internalize__card__of__ordLeq2,axiom,
    ! [A3: set_nat,C4: set_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A3 ) @ ( bNF_Ca3172636760949973492d_of_o @ C4 ) ) @ bNF_We6538615181293537951_nat_o )
      = ( ? [B6: set_o] :
            ( ( ord_less_eq_set_o @ B6 @ C4 )
            & ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A3 ) @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) ) @ bNF_We7219021536291862642_nat_o )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) @ ( bNF_Ca3172636760949973492d_of_o @ C4 ) ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ).

% internalize_card_of_ordLeq2
thf(fact_1199_internalize__card__of__ordLeq2,axiom,
    ! [A3: set_nat,C4: set_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ C4 ) ) @ bNF_We2031159114419172489at_nat )
      = ( ? [B6: set_nat] :
            ( ( ord_less_eq_set_nat @ B6 @ C4 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ B6 ) ) @ bNF_We5258908940166488438at_nat )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B6 ) @ ( bNF_Ca3793111618940312692of_nat @ C4 ) ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% internalize_card_of_ordLeq2
thf(fact_1200_card__of__ordLess2,axiom,
    ! [B5: set_o,A3: set_o] :
      ( ( B5 != bot_bot_set_o )
     => ( ( ~ ? [F2: $o > $o] :
                ( ( image_o_o2 @ F2 @ A3 )
                = B5 ) )
        = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3172636760949973492d_of_o @ B5 ) ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% card_of_ordLess2
thf(fact_1201_card__of__ordLess2,axiom,
    ! [B5: set_nat,A3: set_nat] :
      ( ( B5 != bot_bot_set_nat )
     => ( ( ~ ? [F2: nat > nat] :
                ( ( image_nat_nat2 @ F2 @ A3 )
                = B5 ) )
        = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ B5 ) ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% card_of_ordLess2
thf(fact_1202_card__of__ordLess2,axiom,
    ! [B5: set_nat,A3: set_o] :
      ( ( B5 != bot_bot_set_nat )
     => ( ( ~ ? [F2: $o > nat] :
                ( ( image_o_nat @ F2 @ A3 )
                = B5 ) )
        = ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ B5 ) ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% card_of_ordLess2
thf(fact_1203_internalize__card__of__ordLeq,axiom,
    ! [A3: set_o,R: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ R ) @ bNF_We8115293805659342695eq_o_o )
      = ( ? [B6: set_o] :
            ( ( ord_less_eq_set_o @ B6 @ ( field_o @ R ) )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) ) @ bNF_We2654380646378065620so_o_o )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ).

% internalize_card_of_ordLeq
thf(fact_1204_internalize__card__of__ordLeq,axiom,
    ! [A3: set_o,R: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ R ) @ bNF_We3435943820185352641_o_nat )
      = ( ? [B6: set_nat] :
            ( ( ord_less_eq_set_nat @ B6 @ ( field_nat @ R ) )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ B6 ) ) @ bNF_We4116350175183677332_o_nat )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B6 ) @ R ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% internalize_card_of_ordLeq
thf(fact_1205_internalize__card__of__ordLeq,axiom,
    ! [A3: set_nat,R: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A3 ) @ R ) @ bNF_We6538615181293537951_nat_o )
      = ( ? [B6: set_o] :
            ( ( ord_less_eq_set_o @ B6 @ ( field_o @ R ) )
            & ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A3 ) @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) ) @ bNF_We7219021536291862642_nat_o )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ) ) ).

% internalize_card_of_ordLeq
thf(fact_1206_internalize__card__of__ordLeq,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A3 ) @ R ) @ bNF_We2031159114419172489at_nat )
      = ( ? [B6: set_nat] :
            ( ( ord_less_eq_set_nat @ B6 @ ( field_nat @ R ) )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ B6 ) ) @ bNF_We5258908940166488438at_nat )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B6 ) @ R ) @ bNF_We2031159114419172489at_nat ) ) ) ) ).

% internalize_card_of_ordLeq
thf(fact_1207_card__of__ordLeq,axiom,
    ! [A3: set_o,B5: set_o] :
      ( ( ? [F2: $o > $o] :
            ( ( inj_on_o_o @ F2 @ A3 )
            & ( ord_less_eq_set_o @ ( image_o_o2 @ F2 @ A3 ) @ B5 ) ) )
      = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3172636760949973492d_of_o @ B5 ) ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% card_of_ordLeq
thf(fact_1208_card__of__ordLeq,axiom,
    ! [A3: set_o,B5: set_nat] :
      ( ( ? [F2: $o > nat] :
            ( ( inj_on_o_nat @ F2 @ A3 )
            & ( ord_less_eq_set_nat @ ( image_o_nat @ F2 @ A3 ) @ B5 ) ) )
      = ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ B5 ) ) @ bNF_We3435943820185352641_o_nat ) ) ).

% card_of_ordLeq
thf(fact_1209_card__of__ordLeq,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( ( ? [F2: nat > nat] :
            ( ( inj_on_nat_nat @ F2 @ A3 )
            & ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F2 @ A3 ) @ B5 ) ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ B5 ) ) @ bNF_We2031159114419172489at_nat ) ) ).

% card_of_ordLeq
thf(fact_1210_card__of__ordLess,axiom,
    ! [A3: set_o,B5: set_o] :
      ( ( ~ ? [F2: $o > $o] :
              ( ( inj_on_o_o @ F2 @ A3 )
              & ( ord_less_eq_set_o @ ( image_o_o2 @ F2 @ A3 ) @ B5 ) ) )
      = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B5 ) @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) ) @ bNF_We2568191115274586186ss_o_o ) ) ).

% card_of_ordLess
thf(fact_1211_card__of__ordLess,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( ( ~ ? [F2: nat > nat] :
              ( ( inj_on_nat_nat @ F2 @ A3 )
              & ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F2 @ A3 ) @ B5 ) ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B5 ) @ ( bNF_Ca3793111618940312692of_nat @ A3 ) ) @ bNF_We3448473429330749420at_nat ) ) ).

% card_of_ordLess
thf(fact_1212_card__of__ordLess,axiom,
    ! [A3: set_nat,B5: set_o] :
      ( ( ~ ? [F2: nat > $o] :
              ( ( inj_on_nat_o @ F2 @ A3 )
              & ( ord_less_eq_set_o @ ( image_nat_o @ F2 @ A3 ) @ B5 ) ) )
      = ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ B5 ) @ ( bNF_Ca3793111618940312692of_nat @ A3 ) ) @ bNF_We8182288985678559134_o_nat ) ) ).

% card_of_ordLess
thf(fact_1213_internalize__card__of__ordLess,axiom,
    ! [A3: set_o,R: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ R ) @ bNF_We2568191115274586186ss_o_o )
      = ( ? [B6: set_o] :
            ( ( ord_less_set_o @ B6 @ ( field_o @ R ) )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) ) @ bNF_We2654380646378065620so_o_o )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) @ R ) @ bNF_We2568191115274586186ss_o_o ) ) ) ) ).

% internalize_card_of_ordLess
thf(fact_1214_internalize__card__of__ordLess,axiom,
    ! [A3: set_nat,R: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A3 ) @ R ) @ bNF_We2061588309931968636_nat_o )
      = ( ? [B6: set_o] :
            ( ( ord_less_set_o @ B6 @ ( field_o @ R ) )
            & ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ ( bNF_Ca3793111618940312692of_nat @ A3 ) @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) ) @ bNF_We7219021536291862642_nat_o )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ B6 ) @ R ) @ bNF_We2568191115274586186ss_o_o ) ) ) ) ).

% internalize_card_of_ordLess
thf(fact_1215_internalize__card__of__ordLess,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A3 ) @ R ) @ bNF_We3448473429330749420at_nat )
      = ( ? [B6: set_nat] :
            ( ( ord_less_set_nat @ B6 @ ( field_nat @ R ) )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ B6 ) ) @ bNF_We5258908940166488438at_nat )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B6 ) @ R ) @ bNF_We3448473429330749420at_nat ) ) ) ) ).

% internalize_card_of_ordLess
thf(fact_1216_internalize__card__of__ordLess,axiom,
    ! [A3: set_o,R: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ R ) @ bNF_We8182288985678559134_o_nat )
      = ( ? [B6: set_nat] :
            ( ( ord_less_set_nat @ B6 @ ( field_nat @ R ) )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ B6 ) ) @ bNF_We4116350175183677332_o_nat )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ B6 ) @ R ) @ bNF_We3448473429330749420at_nat ) ) ) ) ).

% internalize_card_of_ordLess
thf(fact_1217_None__notin__image__Some,axiom,
    ! [A3: set_state] :
      ~ ( member_option_state @ none_state @ ( image_6076465424260689483_state @ some_state @ A3 ) ) ).

% None_notin_image_Some
thf(fact_1218_None__notin__image__Some,axiom,
    ! [A3: set_val] :
      ~ ( member_option_val @ none_val @ ( image_val_option_val @ some_val @ A3 ) ) ).

% None_notin_image_Some
thf(fact_1219_ordLeq__ordIso__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R5 ) @ bNF_We6538615181293537951_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We4116350175183677332_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R6 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordLeq_ordIso_trans
thf(fact_1220_ordLeq__ordIso__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We4116350175183677332_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R6 ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% ordLeq_ordIso_trans
thf(fact_1221_ordLeq__ordIso__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R6 ) @ bNF_We7219021536291862642_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R6 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% ordLeq_ordIso_trans
thf(fact_1222_ordLeq__ordIso__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R6 ) @ bNF_We2654380646378065620so_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R6 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% ordLeq_ordIso_trans
thf(fact_1223_ordLeq__ordIso__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We3435943820185352641_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We5258908940166488438at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R6 ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% ordLeq_ordIso_trans
thf(fact_1224_ordLeq__ordIso__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We2031159114419172489at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We5258908940166488438at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R6 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordLeq_ordIso_trans
thf(fact_1225_ordIso__ordLeq__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R5 ) @ bNF_We7219021536291862642_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We3435943820185352641_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R6 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordIso_ordLeq_trans
thf(fact_1226_ordIso__ordLeq__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R6 ) @ bNF_We6538615181293537951_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R6 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% ordIso_ordLeq_trans
thf(fact_1227_ordIso__ordLeq__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We2031159114419172489at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R6 ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% ordIso_ordLeq_trans
thf(fact_1228_ordIso__ordLeq__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R6 ) @ bNF_We8115293805659342695eq_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R6 ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% ordIso_ordLeq_trans
thf(fact_1229_ordIso__ordLeq__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We3435943820185352641_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R6 ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% ordIso_ordLeq_trans
thf(fact_1230_ordIso__ordLeq__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We5258908940166488438at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We2031159114419172489at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R6 ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordIso_ordLeq_trans
thf(fact_1231_ordIso__imp__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We4116350175183677332_o_nat )
     => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We3435943820185352641_o_nat ) ) ).

% ordIso_imp_ordLeq
thf(fact_1232_ordIso__imp__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2654380646378065620so_o_o )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We8115293805659342695eq_o_o ) ) ).

% ordIso_imp_ordLeq
thf(fact_1233_ordIso__imp__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We5258908940166488438at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We2031159114419172489at_nat ) ) ).

% ordIso_imp_ordLeq
thf(fact_1234_ordIso__iff__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R5 ) @ bNF_We7219021536291862642_nat_o )
      = ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R5 ) @ bNF_We6538615181293537951_nat_o )
        & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R ) @ bNF_We3435943820185352641_o_nat ) ) ) ).

% ordIso_iff_ordLeq
thf(fact_1235_ordIso__iff__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We4116350175183677332_o_nat )
      = ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We3435943820185352641_o_nat )
        & ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R ) @ bNF_We6538615181293537951_nat_o ) ) ) ).

% ordIso_iff_ordLeq
thf(fact_1236_ordIso__iff__ordLeq,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2654380646378065620so_o_o )
      = ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We8115293805659342695eq_o_o )
        & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R ) @ bNF_We8115293805659342695eq_o_o ) ) ) ).

% ordIso_iff_ordLeq
thf(fact_1237_ordIso__iff__ordLeq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We5258908940166488438at_nat )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We2031159114419172489at_nat )
        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R ) @ bNF_We2031159114419172489at_nat ) ) ) ).

% ordIso_iff_ordLeq
thf(fact_1238_ordLess__ordIso__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R5 ) @ bNF_We2061588309931968636_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We4116350175183677332_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R6 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLess_ordIso_trans
thf(fact_1239_ordLess__ordIso__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R6 ) @ bNF_We2654380646378065620so_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R6 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLess_ordIso_trans
thf(fact_1240_ordLess__ordIso__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We3448473429330749420at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We5258908940166488438at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R6 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordLess_ordIso_trans
thf(fact_1241_ordLess__ordIso__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We4116350175183677332_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R6 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLess_ordIso_trans
thf(fact_1242_ordLess__ordIso__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R6 ) @ bNF_We7219021536291862642_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R6 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordLess_ordIso_trans
thf(fact_1243_ordLess__ordIso__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We5258908940166488438at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R6 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordLess_ordIso_trans
thf(fact_1244_ordIso__ordLess__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Product_prod_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R5 @ R6 ) @ bNF_We2061588309931968636_nat_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R6 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordIso_ordLess_trans
thf(fact_1245_ordIso__ordLess__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R5 @ R6 ) @ bNF_We2568191115274586186ss_o_o )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R6 ) @ bNF_We2568191115274586186ss_o_o ) ) ) ).

% ordIso_ordLess_trans
thf(fact_1246_ordIso__ordLess__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We5258908940166488438at_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We3448473429330749420at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R6 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordIso_ordLess_trans
thf(fact_1247_ordIso__ordLess__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We4116350175183677332_o_nat )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R5 @ R6 ) @ bNF_We3448473429330749420at_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R6 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordIso_ordLess_trans
thf(fact_1248_ordIso__ordLess__trans,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ R @ R5 ) @ bNF_We7219021536291862642_nat_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We8182288985678559134_o_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R6 ) @ bNF_We3448473429330749420at_nat ) ) ) ).

% ordIso_ordLess_trans
thf(fact_1249_ordIso__ordLess__trans,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o,R6: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2654380646378065620so_o_o )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R5 @ R6 ) @ bNF_We8182288985678559134_o_nat )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R6 ) @ bNF_We8182288985678559134_o_nat ) ) ) ).

% ordIso_ordLess_trans
thf(fact_1250_not__ordLess__ordIso,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2568191115274586186ss_o_o )
     => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R @ R5 ) @ bNF_We2654380646378065620so_o_o ) ) ).

% not_ordLess_ordIso
thf(fact_1251_not__ordLess__ordIso,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We3448473429330749420at_nat )
     => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ R5 ) @ bNF_We5258908940166488438at_nat ) ) ).

% not_ordLess_ordIso
thf(fact_1252_not__ordLess__ordIso,axiom,
    ! [R: set_Product_prod_o_o,R5: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We8182288985678559134_o_nat )
     => ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R @ R5 ) @ bNF_We4116350175183677332_o_nat ) ) ).

% not_ordLess_ordIso
thf(fact_1253_wf__bounded__measure,axiom,
    ! [R: set_Pr4305388979797279737on_val,Ub: produc9003781588162623004on_val > nat,F: produc9003781588162623004on_val > nat] :
      ( ! [A2: produc9003781588162623004on_val,B2: produc9003781588162623004on_val] :
          ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B2 @ A2 ) @ R )
         => ( ( ord_less_eq_nat @ ( Ub @ B2 ) @ ( Ub @ A2 ) )
            & ( ord_less_eq_nat @ ( F @ B2 ) @ ( Ub @ A2 ) )
            & ( ord_less_nat @ ( F @ A2 ) @ ( F @ B2 ) ) ) )
     => ( wf_Pro3300669915054815219on_val @ R ) ) ).

% wf_bounded_measure
thf(fact_1254_wf__bounded__measure,axiom,
    ! [R: set_Pr1932065953672099015od_o_o,Ub: set_Product_prod_o_o > nat,F: set_Product_prod_o_o > nat] :
      ( ! [A2: set_Product_prod_o_o,B2: set_Product_prod_o_o] :
          ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B2 @ A2 ) @ R )
         => ( ( ord_less_eq_nat @ ( Ub @ B2 ) @ ( Ub @ A2 ) )
            & ( ord_less_eq_nat @ ( F @ B2 ) @ ( Ub @ A2 ) )
            & ( ord_less_nat @ ( F @ A2 ) @ ( F @ B2 ) ) ) )
     => ( wf_set2384217916178216176od_o_o @ R ) ) ).

% wf_bounded_measure
thf(fact_1255_wf__bounded__measure,axiom,
    ! [R: set_Pr4329608150637261639at_nat,Ub: set_Pr1261947904930325089at_nat > nat,F: set_Pr1261947904930325089at_nat > nat] :
      ( ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ A2 ) @ R )
         => ( ( ord_less_eq_nat @ ( Ub @ B2 ) @ ( Ub @ A2 ) )
            & ( ord_less_eq_nat @ ( F @ B2 ) @ ( Ub @ A2 ) )
            & ( ord_less_nat @ ( F @ A2 ) @ ( F @ B2 ) ) ) )
     => ( wf_set7675013139414177994at_nat @ R ) ) ).

% wf_bounded_measure
thf(fact_1256_card__of__empty3,axiom,
    ! [A3: set_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) ) @ bNF_We8115293805659342695eq_o_o )
     => ( A3 = bot_bot_set_o ) ) ).

% card_of_empty3
thf(fact_1257_card__of__empty3,axiom,
    ! [A3: set_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) ) @ bNF_We3435943820185352641_o_nat )
     => ( A3 = bot_bot_set_o ) ) ).

% card_of_empty3
thf(fact_1258_card__of__empty3,axiom,
    ! [A3: set_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) ) @ bNF_We2031159114419172489at_nat )
     => ( A3 = bot_bot_set_nat ) ) ).

% card_of_empty3
thf(fact_1259_card__of__empty,axiom,
    ! [A3: set_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) ) @ bNF_We8115293805659342695eq_o_o ) ).

% card_of_empty
thf(fact_1260_card__of__empty,axiom,
    ! [A3: set_nat] : ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ bot_bot_set_o ) @ ( bNF_Ca3793111618940312692of_nat @ A3 ) ) @ bNF_We3435943820185352641_o_nat ) ).

% card_of_empty
thf(fact_1261_card__of__empty,axiom,
    ! [A3: set_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ bot_bot_set_nat ) @ ( bNF_Ca3793111618940312692of_nat @ A3 ) ) @ bNF_We2031159114419172489at_nat ) ).

% card_of_empty
thf(fact_1262_card__of__ordLeq__finite,axiom,
    ! [A3: set_o,B5: set_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3172636760949973492d_of_o @ B5 ) ) @ bNF_We8115293805659342695eq_o_o )
     => ( ( finite_finite_o @ B5 )
       => ( finite_finite_o @ A3 ) ) ) ).

% card_of_ordLeq_finite
thf(fact_1263_card__of__ordLeq__finite,axiom,
    ! [A3: set_o,B5: set_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ ( bNF_Ca3172636760949973492d_of_o @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ B5 ) ) @ bNF_We3435943820185352641_o_nat )
     => ( ( finite_finite_nat @ B5 )
       => ( finite_finite_o @ A3 ) ) ) ).

% card_of_ordLeq_finite
thf(fact_1264_card__of__ordLeq__finite,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ B5 ) ) @ bNF_We2031159114419172489at_nat )
     => ( ( finite_finite_nat @ B5 )
       => ( finite_finite_nat @ A3 ) ) ) ).

% card_of_ordLeq_finite
thf(fact_1265_card__of__ordLeq__infinite,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ A3 ) @ ( bNF_Ca3793111618940312692of_nat @ B5 ) ) @ bNF_We2031159114419172489at_nat )
     => ( ~ ( finite_finite_nat @ A3 )
       => ~ ( finite_finite_nat @ B5 ) ) ) ).

% card_of_ordLeq_infinite
thf(fact_1266_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_1267_ctwo__ordLess__natLeq,axiom,
    member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ bNF_Cardinal_ctwo @ bNF_Ca8665028551170535155natLeq ) @ bNF_We8182288985678559134_o_nat ).

% ctwo_ordLess_natLeq
thf(fact_1268_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_1269_card__of__nat,axiom,
    member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ top_top_set_nat ) @ bNF_Ca8665028551170535155natLeq ) @ bNF_We5258908940166488438at_nat ).

% card_of_nat
thf(fact_1270_get__pre_I2_J,axiom,
    ( get_m
    = ( ^ [X: state] : ( produc5239619271231976746on_val @ ( rep_state @ X ) ) ) ) ).

% get_pre(2)
thf(fact_1271_get__m_Oelims,axiom,
    ! [X3: state,Y3: produc1457211279475724562t_char > prat] :
      ( ( ( get_m @ X3 )
        = Y3 )
     => ( Y3
        = ( produc5239619271231976746on_val @ ( rep_state @ X3 ) ) ) ) ).

% get_m.elims
thf(fact_1272_get__m_Osimps,axiom,
    ( get_m
    = ( ^ [A5: state] : ( produc5239619271231976746on_val @ ( rep_state @ A5 ) ) ) ) ).

% get_m.simps

% Helper facts (5)
thf(help_If_2_1_If_001t__Option__Ooption_It__Mask__Oval_J_T,axiom,
    ! [X3: option_val,Y3: option_val] :
      ( ( if_option_val @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Mask__Oval_J_T,axiom,
    ! [X3: option_val,Y3: option_val] :
      ( ( if_option_val @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_3_1_If_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_T,axiom,
    ! [X3: option_state,Y3: option_state] :
      ( ( if_option_state @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_T,axiom,
    ! [X3: option_state,Y3: option_state] :
      ( ( if_option_state @ $true @ X3 @ Y3 )
      = X3 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    stable @ phi ).

%------------------------------------------------------------------------------