%------------------------------------------------------------------------------
% 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    : Rewrite_Properties_Reduction/0017_Replace_Constant/prob_00068_003190__13856038_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1755 ( 470 unt; 498 typ;   0 def)
%            Number of atoms       : 3974 (1432 equ;   0 cnn)
%            Maximal formula atoms :   20 (   3 avg)
%            Number of connectives : 17051 ( 729   ~;  72   |; 326   &;13806   @)
%                                         (   0 <=>;2118  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   9 avg)
%            Number of types       :   87 (  86 usr)
%            Number of type conns  : 1136 (1136   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  415 ( 412 usr;  51 con; 0-4 aty)
%            Number of variables   : 4608 (  60   ^;4319   !; 229   ?;4608   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:18:33.078
%------------------------------------------------------------------------------
% Could-be-implicit typings (86)
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thf(sy_c_List_Olenlex_001t__Nat__Onat,type,
    lenlex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_List_Olenlex_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    lenlex_term_a_b: set_Pr4386577575007340137rm_a_b > set_Pr8564414093027780873rm_a_b ).

thf(sy_c_List_Olex_001t__Nat__Onat,type,
    lex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_List_Olex_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    lex_term_a_b: set_Pr4386577575007340137rm_a_b > set_Pr8564414093027780873rm_a_b ).

thf(sy_c_List_Olexord_001t__Nat__Onat,type,
    lexord_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_List_Olexord_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    lexord_term_a_b: set_Pr4386577575007340137rm_a_b > set_Pr8564414093027780873rm_a_b ).

thf(sy_c_List_Olist_OCons_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    cons_nat_nat: ( nat > nat ) > list_nat_nat > list_nat_nat ).

thf(sy_c_List_Olist_OCons_001_062_Itf__b_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    cons_b_term_a_b: ( b > term_a_b ) > list_b_term_a_b > list_b_term_a_b ).

thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
    cons_list_nat: list_nat > list_list_nat > list_list_nat ).

thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    cons_list_term_a_b: list_term_a_b > list_list_term_a_b > list_list_term_a_b ).

thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
    cons_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    cons_P3536305108106557631rm_a_b: produc357393685978478089rm_a_b > list_P8875379029341186191rm_a_b > list_P8875379029341186191rm_a_b ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    cons_P5205166803686508359_a_nat: product_prod_a_nat > list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat ).

thf(sy_c_List_Olist_OCons_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    cons_term_a_b: term_a_b > list_term_a_b > list_term_a_b ).

thf(sy_c_List_Olist_OCons_001tf__a,type,
    cons_a: a > list_a > list_a ).

thf(sy_c_List_Olist_OCons_001tf__b,type,
    cons_b: b > list_b > list_b ).

thf(sy_c_List_Olist_ONil_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    nil_nat_nat: list_nat_nat ).

thf(sy_c_List_Olist_ONil_001_062_Itf__b_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    nil_b_term_a_b: list_b_term_a_b ).

thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
    nil_list_nat: list_list_nat ).

thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    nil_list_term_a_b: list_list_term_a_b ).

thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    nil_Pr6942174756412032271rm_a_b: list_P8875379029341186191rm_a_b ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    nil_Pr7402525243500994295_a_nat: list_P3592885314253461005_a_nat ).

thf(sy_c_List_Olist_ONil_001t__Set__Oset_It__Nat__Onat_J,type,
    nil_set_nat: list_set_nat ).

thf(sy_c_List_Olist_ONil_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    nil_set_term_a_b: list_set_term_a_b ).

thf(sy_c_List_Olist_ONil_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    nil_term_a_b: list_term_a_b ).

thf(sy_c_List_Olist_ONil_001tf__a,type,
    nil_a: list_a ).

thf(sy_c_List_Olist_ONil_001tf__b,type,
    nil_b: list_b ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
    map_nat_nat: ( nat > nat ) > list_nat > list_nat ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    map_nat_term_a_b: ( nat > term_a_b ) > list_nat > list_term_a_b ).

thf(sy_c_List_Olist_Omap_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
    map_term_a_b_nat: ( term_a_b > nat ) > list_term_a_b > list_nat ).

thf(sy_c_List_Olist_Omap_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    map_te2328734355998148206rm_a_b: ( term_a_b > term_a_b ) > list_term_a_b > list_term_a_b ).

thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
    list_ex1_nat: ( nat > $o ) > list_nat > $o ).

thf(sy_c_List_Olist__ex1_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    list_ex1_term_a_b: ( term_a_b > $o ) > list_term_a_b > $o ).

thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
    list_update_nat: list_nat > nat > nat > list_nat ).

thf(sy_c_List_Olist__update_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    list_update_term_a_b: list_term_a_b > nat > term_a_b > list_term_a_b ).

thf(sy_c_List_Olistrel1_001t__Nat__Onat,type,
    listrel1_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_List_Olistrel1_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    listrel1_term_a_b: set_Pr4386577575007340137rm_a_b > set_Pr8564414093027780873rm_a_b ).

thf(sy_c_List_Olistrel_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    listrel_nat_nat_nat: set_Pr9093778441882193744at_nat > set_Pr5771542735269860976st_nat ).

thf(sy_c_List_Olistrel_001t__Nat__Onat_001t__Nat__Onat,type,
    listrel_nat_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_List_Olistrel_001t__Nat__Onat_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    listrel_nat_term_a_b: set_Pr4549835640365387557rm_a_b > set_Pr2653742144321748037rm_a_b ).

thf(sy_c_List_Olistrel_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
    listrel_term_a_b_nat: set_Pr6708357783813671845_b_nat > set_Pr5972536988916469445st_nat ).

thf(sy_c_List_Olistrel_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    listre1194016999521814427rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr8564414093027780873rm_a_b ).

thf(sy_c_List_Olistrel_001tf__a_001t__Nat__Onat,type,
    listrel_a_nat: set_Pr4934435412358123699_a_nat > set_Pr5046312416420021961st_nat ).

thf(sy_c_List_Olists_001t__List__Olist_It__Nat__Onat_J,type,
    lists_list_nat: set_list_nat > set_list_list_nat ).

thf(sy_c_List_Olists_001t__Nat__Onat,type,
    lists_nat: set_nat > set_list_nat ).

thf(sy_c_List_Olists_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    lists_3936190290964281341rm_a_b: set_Pr4386577575007340137rm_a_b > set_li7311725489111463791rm_a_b ).

thf(sy_c_List_Olists_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    lists_3983152277583228297_a_nat: set_Pr4934435412358123699_a_nat > set_li7590439355475177155_a_nat ).

thf(sy_c_List_Olists_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    lists_term_a_b: set_term_a_b > set_list_term_a_b ).

thf(sy_c_List_Olists_001tf__b,type,
    lists_b: set_b > set_list_b ).

thf(sy_c_List_Olistset_001t__Nat__Onat,type,
    listset_nat: list_set_nat > set_list_nat ).

thf(sy_c_List_Olistset_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    listset_term_a_b: list_set_term_a_b > set_list_term_a_b ).

thf(sy_c_List_On__lists_001t__Nat__Onat,type,
    n_lists_nat: nat > list_nat > list_list_nat ).

thf(sy_c_List_On__lists_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    n_lists_term_a_b: nat > list_term_a_b > list_list_term_a_b ).

thf(sy_c_List_Onth_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    nth_nat_nat: list_nat_nat > nat > nat > nat ).

thf(sy_c_List_Onth_001t__Nat__Onat,type,
    nth_nat: list_nat > nat > nat ).

thf(sy_c_List_Onth_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    nth_term_a_b: list_term_a_b > nat > term_a_b ).

thf(sy_c_List_Onth_001tf__a,type,
    nth_a: list_a > nat > a ).

thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
    product_lists_nat: list_list_nat > list_list_nat ).

thf(sy_c_List_Oproduct__lists_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    produc17669015410068453rm_a_b: list_list_term_a_b > list_list_term_a_b ).

thf(sy_c_Missing__List_Oadjust__idx__rev,type,
    missin3815256168798769645dx_rev: nat > nat > nat ).

thf(sy_c_Missing__List_Oconcat__lists_001t__Nat__Onat,type,
    missin4567272213201432058ts_nat: list_list_nat > list_list_nat ).

thf(sy_c_Missing__List_Oconcat__lists_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    missin8060632096978918206rm_a_b: list_list_term_a_b > list_list_term_a_b ).

thf(sy_c_Nat_OSuc,type,
    suc: nat > nat ).

thf(sy_c_Nat_Ocompow_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    compow4057154403645558940rm_a_b: nat > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s8208510060688613859at_nat: list_nat_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
    size_size_list_nat: list_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    size_s8906293707977694520rm_a_b: list_term_a_b > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
    size_size_list_a: list_a > nat ).

thf(sy_c_Option_Ooption_ONone_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
    none_Sum_sum_a_b: option_Sum_sum_a_b ).

thf(sy_c_Order__Relation_Olinear__order__on_001t__List__Olist_It__Nat__Onat_J,type,
    order_2931667645265552619st_nat: set_list_nat > set_Pr3451248702717554689st_nat > $o ).

thf(sy_c_Order__Relation_Olinear__order__on_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    order_681589137112254398rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr2972776593051762503rm_a_b > $o ).

thf(sy_c_Order__Relation_Olinear__order__on_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    order_6388386362885710600_a_nat: set_Pr4934435412358123699_a_nat > set_Pr1811044260758604347_a_nat > $o ).

thf(sy_c_Order__Relation_Olinear__order__on_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    order_5388802246213473311rm_a_b: set_term_a_b > set_Pr4386577575007340137rm_a_b > $o ).

thf(sy_c_Order__Relation_Olinear__order__on_001tf__b,type,
    order_8768733634509060148r_on_b: set_b > set_Product_prod_b_b > $o ).

thf(sy_c_Order__Relation_Owell__order__on_001t__List__Olist_It__Nat__Onat_J,type,
    order_3262761615943460802st_nat: set_list_nat > set_Pr3451248702717554689st_nat > $o ).

thf(sy_c_Order__Relation_Owell__order__on_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    order_8637870607158948775rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr2972776593051762503rm_a_b > $o ).

thf(sy_c_Order__Relation_Owell__order__on_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    order_7753855512070164063_a_nat: set_Pr4934435412358123699_a_nat > set_Pr1811044260758604347_a_nat > $o ).

thf(sy_c_Order__Relation_Owell__order__on_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    order_5719896216891381494rm_a_b: set_term_a_b > set_Pr4386577575007340137rm_a_b > $o ).

thf(sy_c_Order__Relation_Owell__order__on_001tf__b,type,
    order_6972113574731384221r_on_b: set_b > set_Product_prod_b_b > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    bot_bot_set_list_nat: set_list_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    bot_bo8941492048910247406rm_a_b: set_list_term_a_b ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    bot_bo197521221353338581rm_a_b: set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    bot_bo9049108969261143879_a_nat: set_Pr4934435412358123699_a_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    bot_bot_set_term_a_b: set_term_a_b ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
    bot_bot_set_b: set_b ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: nat > nat > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    ord_le3678578370064672496at_nat: set_Pr9093778441882193744at_nat > set_Pr9093778441882193744at_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    ord_le118470702582115849rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    ord_le8666007276011122963_a_nat: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    ord_le2705286416250468010rm_a_b: set_term_a_b > set_term_a_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
    ord_less_eq_set_b: set_b > set_b > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    top_top_set_list_nat: set_list_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    top_to1314267278846557113rm_a_b: set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    top_to3353692345378799459_a_nat: set_Pr4934435412358123699_a_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    top_top_set_term_a_b: set_term_a_b ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__b_J,type,
    top_top_set_b: set_b ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc4727192421694094319st_nat: ( nat > nat > $o ) > list_nat > produc254973753779126261st_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc3127733452865184594st_nat: ( nat > nat > $o ) > produc1828647624359046049st_nat > produc4787317212837456354st_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Term__Oterm_Itf__a_Mtf__b_J_M_Eo_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    produc2750092164434624090rm_a_b: ( nat > term_a_b > $o ) > produc9043357390500214885rm_a_b > produc8734217775477389290rm_a_b ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_Eo_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc8587622027977423880st_nat: ( nat > $o ) > list_nat > produc4226810134323546766st_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc5770335208449155351at_nat: ( nat > nat ) > ( nat > nat ) > produc1932156733058919263at_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    produc72220940542539688at_nat: ( nat > nat ) > nat > produc8199716216217303280at_nat ).

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    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    member7514656309480873070rm_a_b: produc1234881154892807749rm_a_b > set_Pr4549835640365387557rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
    member8962352052110095674_nat_a: product_prod_nat_a > set_Pr4193341848836149977_nat_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    member3795392313164512464at_nat: produc492020038885415591at_nat > set_Pr7047708467701517959at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    member8417600551952982416rm_a_b: produc4523971788519308903rm_a_b > set_Pr2972776593051762503rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    member9062615507155100804_a_nat: produc4708774622424448987_a_nat > set_Pr1811044260758604347_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
    member9000184225777080558_b_nat: produc2720409071189015237_b_nat > set_Pr6708357783813671845_b_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    member5869715511025134514rm_a_b: produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member5463269792551873667at_nat: produc551447040318622060at_nat > set_Pr6841066879934740386at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    member5724188588386418708_a_nat: product_prod_a_nat > set_Pr4934435412358123699_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    member7862447936710763792od_b_b: product_prod_b_b > set_Product_prod_b_b > $o ).

thf(sy_c_member_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    member_term_a_b: term_a_b > set_term_a_b > $o ).

thf(sy_c_member_001tf__b,type,
    member_b: b > set_b > $o ).

thf(sy_v_C____,type,
    c: subterm_and_ctxt_a_b ).

thf(sy_v__092_060R_062,type,
    r: set_Pr4386577575007340137rm_a_b ).

thf(sy_v__092_060sigma_062____,type,
    sigma: b > term_a_b ).

thf(sy_v_c,type,
    c2: a ).

thf(sy_v_l____,type,
    l: term_a_b ).

thf(sy_v_p,type,
    p: list_nat ).

thf(sy_v_q_H____,type,
    q: list_nat ).

thf(sy_v_q____,type,
    q2: list_nat ).

thf(sy_v_r____,type,
    r2: term_a_b ).

thf(sy_v_s,type,
    s: term_a_b ).

thf(sy_v_t,type,
    t: term_a_b ).

thf(sy_v_u,type,
    u: term_a_b ).

thf(sy_v_v____,type,
    v: list_nat ).

thf(sy_v_x____,type,
    x: b ).

% Relevant facts (1251)
thf(fact_0_varposs_I1_J,axiom,
    member_list_nat @ v @ ( terms_varposs_a_b @ l ) ).

% varposs(1)
thf(fact_1_varposs__r_I1_J,axiom,
    member_list_nat @ q @ ( terms_varposs_a_b @ r2 ) ).

% varposs_r(1)
thf(fact_2_varposs_I2_J,axiom,
    term_p3503116865373065078eq_nat @ v @ q2 ).

% varposs(2)
thf(fact_3_varposs_I3_J,axiom,
    ( ( term_subt_at_a_b @ l @ v )
    = ( var_b_a @ x ) ) ).

% varposs(3)
thf(fact_4_varposs__r_I2_J,axiom,
    ( ( term_subt_at_a_b @ r2 @ q )
    = ( var_b_a @ x ) ) ).

% varposs_r(2)
thf(fact_5_poss,axiom,
    member_list_nat @ q2 @ ( term_poss_a_b @ ( subst_7999470309526761004_a_b_b @ l @ sigma ) ) ).

% poss
thf(fact_6_local_Oconst,axiom,
    ( ( term_subt_at_a_b @ ( subst_7999470309526761004_a_b_b @ l @ sigma ) @ q2 )
    = ( fun_a_b @ c2 @ nil_term_a_b ) ) ).

% local.const
thf(fact_7__092_060open_062q_A_N_092_060_094sub_062p_Av_A_092_060in_062_Aposs_A_Il_A_092_060cdot_062_A_092_060sigma_062_A_124___Av_J_092_060close_062,axiom,
    member_list_nat @ ( term_pos_diff_nat @ q2 @ v ) @ ( term_poss_a_b @ ( term_subt_at_a_b @ ( subst_7999470309526761004_a_b_b @ l @ sigma ) @ v ) ) ).

% \<open>q -\<^sub>p v \<in> poss (l \<cdot> \<sigma> |_ v)\<close>
thf(fact_8_below,axiom,
    term_p3503116865373065078eq_nat @ ( term_hole_pos_a_b @ c ) @ p ).

% below
thf(fact_9__092_060open_062q_H_A_064_Aq_A_N_092_060_094sub_062p_Av_A_092_060in_062_Aposs_A_Ir_A_092_060cdot_062_A_092_060sigma_062_J_A_092_060Longrightarrow_062_Ar_A_092_060cdot_062_A_092_060sigma_062_A_124___A_Iq_H_A_064_Aq_A_N_092_060_094sub_062p_Av_J_A_061_Ar_A_092_060cdot_062_A_092_060sigma_062_A_124___Aq_H_A_124___A_Iq_A_N_092_060_094sub_062p_Av_J_092_060close_062,axiom,
    ( ( member_list_nat @ ( append_nat @ q @ ( term_pos_diff_nat @ q2 @ v ) ) @ ( term_poss_a_b @ ( subst_7999470309526761004_a_b_b @ r2 @ sigma ) ) )
   => ( ( term_subt_at_a_b @ ( subst_7999470309526761004_a_b_b @ r2 @ sigma ) @ ( append_nat @ q @ ( term_pos_diff_nat @ q2 @ v ) ) )
      = ( term_subt_at_a_b @ ( term_subt_at_a_b @ ( subst_7999470309526761004_a_b_b @ r2 @ sigma ) @ q ) @ ( term_pos_diff_nat @ q2 @ v ) ) ) ) ).

% \<open>q' @ q -\<^sub>p v \<in> poss (r \<cdot> \<sigma>) \<Longrightarrow> r \<cdot> \<sigma> |_ (q' @ q -\<^sub>p v) = r \<cdot> \<sigma> |_ q' |_ (q -\<^sub>p v)\<close>
thf(fact_10__092_060open_062_092_060exists_062r_O_Ar_A_092_060in_062_Avarposs_Al_A_092_060and_062_Ar_A_092_060le_062_092_060_094sub_062p_Aq_092_060close_062,axiom,
    ? [R: list_nat] :
      ( ( member_list_nat @ R @ ( terms_varposs_a_b @ l ) )
      & ( term_p3503116865373065078eq_nat @ R @ q2 ) ) ).

% \<open>\<exists>r. r \<in> varposs l \<and> r \<le>\<^sub>p q\<close>
thf(fact_11_occ,axiom,
    member_list_nat @ p @ ( terms_7168686267159881682rm_a_b @ ( fun_a_b @ c2 @ nil_term_a_b ) @ s ) ).

% occ
thf(fact_12_pos__diff__append__itself,axiom,
    ! [P: list_nat,Q: list_nat] :
      ( ( term_pos_diff_nat @ ( append_nat @ P @ Q ) @ P )
      = Q ) ).

% pos_diff_append_itself
thf(fact_13_position__diff__Nil,axiom,
    ! [Q: list_term_a_b] :
      ( ( term_p798503758663136087rm_a_b @ Q @ nil_term_a_b )
      = Q ) ).

% position_diff_Nil
thf(fact_14_position__diff__Nil,axiom,
    ! [Q: list_nat] :
      ( ( term_pos_diff_nat @ Q @ nil_nat )
      = Q ) ).

% position_diff_Nil
thf(fact_15_pos__diff__itself,axiom,
    ! [P: list_term_a_b] :
      ( ( term_p798503758663136087rm_a_b @ P @ P )
      = nil_term_a_b ) ).

% pos_diff_itself
thf(fact_16_pos__diff__itself,axiom,
    ! [P: list_nat] :
      ( ( term_pos_diff_nat @ P @ P )
      = nil_nat ) ).

% pos_diff_itself
thf(fact_17_append_Oright__neutral,axiom,
    ! [A: list_term_a_b] :
      ( ( append_term_a_b @ A @ nil_term_a_b )
      = A ) ).

% append.right_neutral
thf(fact_18_append_Oright__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ A @ nil_nat )
      = A ) ).

% append.right_neutral
thf(fact_19_append__Nil2,axiom,
    ! [Xs: list_term_a_b] :
      ( ( append_term_a_b @ Xs @ nil_term_a_b )
      = Xs ) ).

% append_Nil2
thf(fact_20_append__Nil2,axiom,
    ! [Xs: list_nat] :
      ( ( append_nat @ Xs @ nil_nat )
      = Xs ) ).

% append_Nil2
thf(fact_21_append__self__conv,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( ( append_term_a_b @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_term_a_b ) ) ).

% append_self_conv
thf(fact_22_append__self__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_nat ) ) ).

% append_self_conv
thf(fact_23_self__append__conv,axiom,
    ! [Y: list_term_a_b,Ys: list_term_a_b] :
      ( ( Y
        = ( append_term_a_b @ Y @ Ys ) )
      = ( Ys = nil_term_a_b ) ) ).

% self_append_conv
thf(fact_24_self__append__conv,axiom,
    ! [Y: list_nat,Ys: list_nat] :
      ( ( Y
        = ( append_nat @ Y @ Ys ) )
      = ( Ys = nil_nat ) ) ).

% self_append_conv
thf(fact_25_append__self__conv2,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( ( append_term_a_b @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_term_a_b ) ) ).

% append_self_conv2
thf(fact_26_append__self__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_nat ) ) ).

% append_self_conv2
thf(fact_27_self__append__conv2,axiom,
    ! [Y: list_term_a_b,Xs: list_term_a_b] :
      ( ( Y
        = ( append_term_a_b @ Xs @ Y ) )
      = ( Xs = nil_term_a_b ) ) ).

% self_append_conv2
thf(fact_28_self__append__conv2,axiom,
    ! [Y: list_nat,Xs: list_nat] :
      ( ( Y
        = ( append_nat @ Xs @ Y ) )
      = ( Xs = nil_nat ) ) ).

% self_append_conv2
thf(fact_29_Nil__is__append__conv,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( nil_term_a_b
        = ( append_term_a_b @ Xs @ Ys ) )
      = ( ( Xs = nil_term_a_b )
        & ( Ys = nil_term_a_b ) ) ) ).

% Nil_is_append_conv
thf(fact_30_Nil__is__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( nil_nat
        = ( append_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_31_append__is__Nil__conv,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( ( append_term_a_b @ Xs @ Ys )
        = nil_term_a_b )
      = ( ( Xs = nil_term_a_b )
        & ( Ys = nil_term_a_b ) ) ) ).

% append_is_Nil_conv
thf(fact_32_append__is__Nil__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = nil_nat )
      = ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_33_same__append__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_34_append__same__eq,axiom,
    ! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Ys @ Xs )
        = ( append_nat @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_35_append__assoc,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
      = ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_36_append_Oassoc,axiom,
    ! [A: list_nat,B: list_nat,C: list_nat] :
      ( ( append_nat @ ( append_nat @ A @ B ) @ C )
      = ( append_nat @ A @ ( append_nat @ B @ C ) ) ) ).

% append.assoc
thf(fact_37_position__less__Nil__is__bot2,axiom,
    ! [P: list_term_a_b] :
      ( ( term_p8391561492822560442rm_a_b @ P @ nil_term_a_b )
      = ( P = nil_term_a_b ) ) ).

% position_less_Nil_is_bot2
thf(fact_38_position__less__Nil__is__bot2,axiom,
    ! [P: list_nat] :
      ( ( term_p3503116865373065078eq_nat @ P @ nil_nat )
      = ( P = nil_nat ) ) ).

% position_less_Nil_is_bot2
thf(fact_39__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062q_H_O_A_092_060lbrakk_062q_H_A_092_060in_062_Avarposs_Ar_059_Ar_A_124___Aq_H_A_061_AVar_Ax_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Q2: list_nat] :
        ( ( member_list_nat @ Q2 @ ( terms_varposs_a_b @ r2 ) )
       => ( ( term_subt_at_a_b @ r2 @ Q2 )
         != ( var_b_a @ x ) ) ) ).

% \<open>\<And>thesis. (\<And>q'. \<lbrakk>q' \<in> varposs r; r |_ q' = Var x\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_40__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_Av_O_A_092_060lbrakk_062v_A_092_060in_062_Avarposs_Al_059_Av_A_092_060le_062_092_060_094sub_062p_Aq_059_Al_A_124___Av_A_061_AVar_Ax_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [X: b,V: list_nat] :
        ( ( member_list_nat @ V @ ( terms_varposs_a_b @ l ) )
       => ( ( term_p3503116865373065078eq_nat @ V @ q2 )
         => ( ( term_subt_at_a_b @ l @ V )
           != ( var_b_a @ X ) ) ) ) ).

% \<open>\<And>thesis. (\<And>x v. \<lbrakk>v \<in> varposs l; v \<le>\<^sub>p q; l |_ v = Var x\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_41_pos__les__eq__append__diff,axiom,
    ! [P: list_nat,Q: list_nat] :
      ( ( term_p3503116865373065078eq_nat @ P @ Q )
     => ( ( append_nat @ P @ ( term_pos_diff_nat @ Q @ P ) )
        = Q ) ) ).

% pos_les_eq_append_diff
thf(fact_42_poss__of__term__possI,axiom,
    ! [P: list_nat,S: term_a_b,U: term_a_b] :
      ( ( member_list_nat @ P @ ( term_poss_a_b @ S ) )
     => ( ( ( term_subt_at_a_b @ S @ P )
          = U )
       => ( member_list_nat @ P @ ( terms_7168686267159881682rm_a_b @ U @ S ) ) ) ) ).

% poss_of_term_possI
thf(fact_43__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062q_O_A_092_060lbrakk_062p_A_061_Ahole__pos_AC_A_064_Aq_059_Aq_A_092_060in_062_Aposs_A_Il_A_092_060cdot_062_A_092_060sigma_062_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Q3: list_nat] :
        ( ( p
          = ( append_nat @ ( term_hole_pos_a_b @ c ) @ Q3 ) )
       => ~ ( member_list_nat @ Q3 @ ( term_poss_a_b @ ( subst_7999470309526761004_a_b_b @ l @ sigma ) ) ) ) ).

% \<open>\<And>thesis. (\<And>q. \<lbrakk>p = hole_pos C @ q; q \<in> poss (l \<cdot> \<sigma>)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_44_True,axiom,
    member_b @ x @ ( vars_term_a_b @ r2 ) ).

% True
thf(fact_45__092_060open_062p_A_061_Ahole__pos_AC_A_064_Aq_092_060close_062,axiom,
    ( p
    = ( append_nat @ ( term_hole_pos_a_b @ c ) @ q2 ) ) ).

% \<open>p = hole_pos C @ q\<close>
thf(fact_46_subt__at_Osimps_I1_J,axiom,
    ! [S: term_a_b] :
      ( ( term_subt_at_a_b @ S @ nil_nat )
      = S ) ).

% subt_at.simps(1)
thf(fact_47_Nil__in__poss,axiom,
    ! [T: term_a_b] : ( member_list_nat @ nil_nat @ ( term_poss_a_b @ T ) ) ).

% Nil_in_poss
thf(fact_48_position__less__refl,axiom,
    ! [P: list_nat] : ( term_p3503116865373065078eq_nat @ P @ P ) ).

% position_less_refl
thf(fact_49_varposs__imp__poss,axiom,
    ! [P: list_nat,S: term_a_b] :
      ( ( member_list_nat @ P @ ( terms_varposs_a_b @ S ) )
     => ( member_list_nat @ P @ ( term_poss_a_b @ S ) ) ) ).

% varposs_imp_poss
thf(fact_50_subst__subt__at__dist,axiom,
    ! [P: list_nat,S: term_a_b,Sigma: b > term_a_b] :
      ( ( member_list_nat @ P @ ( term_poss_a_b @ S ) )
     => ( ( term_subt_at_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ P )
        = ( subst_7999470309526761004_a_b_b @ ( term_subt_at_a_b @ S @ P ) @ Sigma ) ) ) ).

% subst_subt_at_dist
thf(fact_51_poss__pos__diffI,axiom,
    ! [P: list_nat,Q: list_nat,S: term_a_b] :
      ( ( term_p3503116865373065078eq_nat @ P @ Q )
     => ( ( member_list_nat @ Q @ ( term_poss_a_b @ S ) )
       => ( member_list_nat @ ( term_pos_diff_nat @ Q @ P ) @ ( term_poss_a_b @ ( term_subt_at_a_b @ S @ P ) ) ) ) ) ).

% poss_pos_diffI
thf(fact_52_subt__at__append__dist,axiom,
    ! [P: list_nat,Q: list_nat,S: term_a_b] :
      ( ( member_list_nat @ ( append_nat @ P @ Q ) @ ( term_poss_a_b @ S ) )
     => ( ( term_subt_at_a_b @ S @ ( append_nat @ P @ Q ) )
        = ( term_subt_at_a_b @ ( term_subt_at_a_b @ S @ P ) @ Q ) ) ) ).

% subt_at_append_dist
thf(fact_53_poss__append__poss,axiom,
    ! [P: list_nat,Q: list_nat,T: term_a_b] :
      ( ( member_list_nat @ ( append_nat @ P @ Q ) @ ( term_poss_a_b @ T ) )
      = ( ( member_list_nat @ P @ ( term_poss_a_b @ T ) )
        & ( member_list_nat @ Q @ ( term_poss_a_b @ ( term_subt_at_a_b @ T @ P ) ) ) ) ) ).

% poss_append_poss
thf(fact_54_poss__of__termE,axiom,
    ! [P: list_nat,U: term_a_b,S: term_a_b] :
      ( ( member_list_nat @ P @ ( terms_7168686267159881682rm_a_b @ U @ S ) )
     => ~ ( ( member_list_nat @ P @ ( term_poss_a_b @ S ) )
         => ( ( term_subt_at_a_b @ S @ P )
           != U ) ) ) ).

% poss_of_termE
thf(fact_55_linear__term_Ocases,axiom,
    ! [X2: term_a_b] :
      ( ! [Uu: b] :
          ( X2
         != ( var_b_a @ Uu ) )
     => ~ ! [Uv: a,Ts: list_term_a_b] :
            ( X2
           != ( fun_a_b @ Uv @ Ts ) ) ) ).

% linear_term.cases
thf(fact_56_mem__Collect__eq,axiom,
    ! [A: list_nat,P2: list_nat > $o] :
      ( ( member_list_nat @ A @ ( collect_list_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_57_mem__Collect__eq,axiom,
    ! [A: b,P2: b > $o] :
      ( ( member_b @ A @ ( collect_b @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_58_mem__Collect__eq,axiom,
    ! [A: produc357393685978478089rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
      ( ( member5869715511025134514rm_a_b @ A @ ( collec99177395925924084rm_a_b @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_59_mem__Collect__eq,axiom,
    ! [A: product_prod_a_nat,P2: product_prod_a_nat > $o] :
      ( ( member5724188588386418708_a_nat @ A @ ( collec4464134535221767506_a_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_60_Collect__mem__eq,axiom,
    ! [A2: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_61_Collect__mem__eq,axiom,
    ! [A2: set_b] :
      ( ( collect_b
        @ ^ [X3: b] : ( member_b @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_62_Collect__mem__eq,axiom,
    ! [A2: set_Pr4386577575007340137rm_a_b] :
      ( ( collec99177395925924084rm_a_b
        @ ^ [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_63_Collect__mem__eq,axiom,
    ! [A2: set_Pr4934435412358123699_a_nat] :
      ( ( collec4464134535221767506_a_nat
        @ ^ [X3: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_64_funas__term_Ocases,axiom,
    ! [X2: term_a_b] :
      ( ! [X: b] :
          ( X2
         != ( var_b_a @ X ) )
     => ~ ! [F: a,Ts: list_term_a_b] :
            ( X2
           != ( fun_a_b @ F @ Ts ) ) ) ).

% funas_term.cases
thf(fact_65_position__less__Nil__is__bot,axiom,
    ! [P: list_term_a_b] : ( term_p8391561492822560442rm_a_b @ nil_term_a_b @ P ) ).

% position_less_Nil_is_bot
thf(fact_66_position__less__Nil__is__bot,axiom,
    ! [P: list_nat] : ( term_p3503116865373065078eq_nat @ nil_nat @ P ) ).

% position_less_Nil_is_bot
thf(fact_67_less__eq__poss__append__itself,axiom,
    ! [P: list_nat,Q: list_nat] : ( term_p3503116865373065078eq_nat @ P @ ( append_nat @ P @ Q ) ) ).

% less_eq_poss_append_itself
thf(fact_68_position__less__eq__induct,axiom,
    ! [P: list_nat,Q: list_nat,P2: list_nat > list_nat > $o] :
      ( ( term_p3503116865373065078eq_nat @ P @ Q )
     => ( ! [P3: list_nat] : ( P2 @ P3 @ P3 )
       => ( ! [P3: list_nat,Q3: list_nat,R: list_nat] :
              ( ( term_p3503116865373065078eq_nat @ P3 @ Q3 )
             => ( ( P2 @ P3 @ Q3 )
               => ( P2 @ P3 @ ( append_nat @ Q3 @ R ) ) ) )
         => ( P2 @ P @ Q ) ) ) ) ).

% position_less_eq_induct
thf(fact_69_position__less__eq__def,axiom,
    ( term_p3503116865373065078eq_nat
    = ( ^ [P4: list_nat,Q4: list_nat] :
        ? [R2: list_nat] :
          ( ( append_nat @ P4 @ R2 )
          = Q4 ) ) ) ).

% position_less_eq_def
thf(fact_70_append__eq__append__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts2: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Zs @ Ts2 ) )
      = ( ? [Us: list_nat] :
            ( ( ( Xs
                = ( append_nat @ Zs @ Us ) )
              & ( ( append_nat @ Us @ Ys )
                = Ts2 ) )
            | ( ( ( append_nat @ Xs @ Us )
                = Zs )
              & ( Ys
                = ( append_nat @ Us @ Ts2 ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_71_append__eq__appendI,axiom,
    ! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us2: list_nat] :
      ( ( ( append_nat @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append_nat @ Xs1 @ Us2 ) )
       => ( ( append_nat @ Xs @ Ys )
          = ( append_nat @ Zs @ Us2 ) ) ) ) ).

% append_eq_appendI
thf(fact_72_eq__Nil__appendI,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_term_a_b @ nil_term_a_b @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_73_eq__Nil__appendI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_nat @ nil_nat @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_74_append_Oleft__neutral,axiom,
    ! [A: list_term_a_b] :
      ( ( append_term_a_b @ nil_term_a_b @ A )
      = A ) ).

% append.left_neutral
thf(fact_75_append_Oleft__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ nil_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_76_append__Nil,axiom,
    ! [Ys: list_term_a_b] :
      ( ( append_term_a_b @ nil_term_a_b @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_77_append__Nil,axiom,
    ! [Ys: list_nat] :
      ( ( append_nat @ nil_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_78__092_060open_062_092_060And_062thesis_O_A_092_060lbrakk_062p_A_092_060bottom_062_Ahole__pos_AC_A_092_060Longrightarrow_062_Athesis_059_Ahole__pos_AC_A_092_060le_062_092_060_094sub_062p_Ap_A_092_060Longrightarrow_062_Athesis_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ( ~ ( term_p5017330785391824242ar_nat @ p @ ( term_hole_pos_a_b @ c ) )
   => ( term_p3503116865373065078eq_nat @ ( term_hole_pos_a_b @ c ) @ p ) ) ).

% \<open>\<And>thesis. \<lbrakk>p \<bottom> hole_pos C \<Longrightarrow> thesis; hole_pos C \<le>\<^sub>p p \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis\<close>
thf(fact_79_subst__apply__term__empty,axiom,
    ! [T: term_a_b] :
      ( ( subst_7999470309526761004_a_b_b @ T @ var_b_a )
      = T ) ).

% subst_apply_term_empty
thf(fact_80_lin_I2_J,axiom,
    terms_5523818207328828897rm_a_b @ r2 ).

% lin(2)
thf(fact_81_lin_I1_J,axiom,
    terms_5523818207328828897rm_a_b @ l ).

% lin(1)
thf(fact_82_t_I2_J,axiom,
    ( t
    = ( subter2376574525758040790rm_a_b @ c @ ( subst_7999470309526761004_a_b_b @ r2 @ sigma ) ) ) ).

% t(2)
thf(fact_83_t_I1_J,axiom,
    ( s
    = ( subter2376574525758040790rm_a_b @ c @ ( subst_7999470309526761004_a_b_b @ l @ sigma ) ) ) ).

% t(1)
thf(fact_84_term_Oinject_I2_J,axiom,
    ! [X21: a,X22: list_term_a_b,Y21: a,Y22: list_term_a_b] :
      ( ( ( fun_a_b @ X21 @ X22 )
        = ( fun_a_b @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% term.inject(2)
thf(fact_85_term_Oinject_I1_J,axiom,
    ! [X1: b,Y1: b] :
      ( ( ( var_b_a @ X1 )
        = ( var_b_a @ Y1 ) )
      = ( X1 = Y1 ) ) ).

% term.inject(1)
thf(fact_86_rule,axiom,
    member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ l @ r2 ) @ r ).

% rule
thf(fact_87_subst__apply__term_Osimps_I1_J,axiom,
    ! [X2: b,Sigma: b > term_a_b] :
      ( ( subst_7999470309526761004_a_b_b @ ( var_b_a @ X2 ) @ Sigma )
      = ( Sigma @ X2 ) ) ).

% subst_apply_term.simps(1)
thf(fact_88_subst__apply__eq__Var,axiom,
    ! [S: term_a_b,Sigma: b > term_a_b,X2: b] :
      ( ( ( subst_7999470309526761004_a_b_b @ S @ Sigma )
        = ( var_b_a @ X2 ) )
     => ~ ! [Y2: b] :
            ( ( S
              = ( var_b_a @ Y2 ) )
           => ( ( Sigma @ Y2 )
             != ( var_b_a @ X2 ) ) ) ) ).

% subst_apply_eq_Var
thf(fact_89_par__not__refl,axiom,
    ! [P: list_nat] :
      ~ ( term_p5017330785391824242ar_nat @ P @ P ) ).

% par_not_refl
thf(fact_90_Nil__not__par_I2_J,axiom,
    ! [P: list_term_a_b] :
      ~ ( term_p7407996180858101430rm_a_b @ P @ nil_term_a_b ) ).

% Nil_not_par(2)
thf(fact_91_Nil__not__par_I2_J,axiom,
    ! [P: list_nat] :
      ~ ( term_p5017330785391824242ar_nat @ P @ nil_nat ) ).

% Nil_not_par(2)
thf(fact_92_Nil__not__par_I1_J,axiom,
    ! [P: list_term_a_b] :
      ~ ( term_p7407996180858101430rm_a_b @ nil_term_a_b @ P ) ).

% Nil_not_par(1)
thf(fact_93_Nil__not__par_I1_J,axiom,
    ! [P: list_nat] :
      ~ ( term_p5017330785391824242ar_nat @ nil_nat @ P ) ).

% Nil_not_par(1)
thf(fact_94__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062C_Al_Ar_A_092_060sigma_062_O_A_092_060lbrakk_062s_A_061_AC_092_060langle_062l_A_092_060cdot_062_A_092_060sigma_062_092_060rangle_062_059_At_A_061_AC_092_060langle_062r_A_092_060cdot_062_A_092_060sigma_062_092_060rangle_062_059_A_Il_M_Ar_J_A_092_060in_062_A_092_060R_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [C2: subterm_and_ctxt_a_b,L: term_a_b,R: term_a_b,Sigma2: b > term_a_b] :
        ( ( s
          = ( subter2376574525758040790rm_a_b @ C2 @ ( subst_7999470309526761004_a_b_b @ L @ Sigma2 ) ) )
       => ( ( t
            = ( subter2376574525758040790rm_a_b @ C2 @ ( subst_7999470309526761004_a_b_b @ R @ Sigma2 ) ) )
         => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ r ) ) ) ).

% \<open>\<And>thesis. (\<And>C l r \<sigma>. \<lbrakk>s = C\<langle>l \<cdot> \<sigma>\<rangle>; t = C\<langle>r \<cdot> \<sigma>\<rangle>; (l, r) \<in> \<R>\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_95_ctxt__apply__subt__at__hole__pos,axiom,
    ! [C3: subterm_and_ctxt_a_b,S: term_a_b] :
      ( ( term_subt_at_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ S ) @ ( term_hole_pos_a_b @ C3 ) )
      = S ) ).

% ctxt_apply_subt_at_hole_pos
thf(fact_96_poss__ctxt__apply,axiom,
    ! [C3: subterm_and_ctxt_a_b,P: list_nat,S: term_a_b] :
      ( ( member_list_nat @ ( append_nat @ ( term_hole_pos_a_b @ C3 ) @ P ) @ ( term_poss_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ S ) ) )
      = ( member_list_nat @ P @ ( term_poss_a_b @ S ) ) ) ).

% poss_ctxt_apply
thf(fact_97_ctxt__apply__term__subt__at__hole__pos,axiom,
    ! [C3: subterm_and_ctxt_a_b,S: term_a_b,Q: list_nat] :
      ( ( term_subt_at_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ S ) @ ( append_nat @ ( term_hole_pos_a_b @ C3 ) @ Q ) )
      = ( term_subt_at_a_b @ S @ Q ) ) ).

% ctxt_apply_term_subt_at_hole_pos
thf(fact_98_local_Ostep,axiom,
    member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ s @ t ) @ ( rstep_a_b @ r ) ).

% local.step
thf(fact_99_term_Oset__intros_I3_J,axiom,
    ! [X1: b] : ( member_b @ X1 @ ( vars_term_a_b @ ( var_b_a @ X1 ) ) ) ).

% term.set_intros(3)
thf(fact_100_term__subst__eq__conv,axiom,
    ! [T: term_a_b,Sigma: b > term_a_b,Tau: b > term_a_b] :
      ( ( ( subst_7999470309526761004_a_b_b @ T @ Sigma )
        = ( subst_7999470309526761004_a_b_b @ T @ Tau ) )
      = ( ! [X3: b] :
            ( ( member_b @ X3 @ ( vars_term_a_b @ T ) )
           => ( ( Sigma @ X3 )
              = ( Tau @ X3 ) ) ) ) ) ).

% term_subst_eq_conv
thf(fact_101_term__subst__eq__rev,axiom,
    ! [T: term_a_b,Sigma: b > term_a_b,Tau: b > term_a_b] :
      ( ( ( subst_7999470309526761004_a_b_b @ T @ Sigma )
        = ( subst_7999470309526761004_a_b_b @ T @ Tau ) )
     => ! [X4: b] :
          ( ( member_b @ X4 @ ( vars_term_a_b @ T ) )
         => ( ( Sigma @ X4 )
            = ( Tau @ X4 ) ) ) ) ).

% term_subst_eq_rev
thf(fact_102_term__subst__eq,axiom,
    ! [T: term_a_b,Sigma: b > term_a_b,Tau: b > term_a_b] :
      ( ! [X: b] :
          ( ( member_b @ X @ ( vars_term_a_b @ T ) )
         => ( ( Sigma @ X )
            = ( Tau @ X ) ) )
     => ( ( subst_7999470309526761004_a_b_b @ T @ Sigma )
        = ( subst_7999470309526761004_a_b_b @ T @ Tau ) ) ) ).

% term_subst_eq
thf(fact_103_linear__term_Osimps_I1_J,axiom,
    ! [Uu2: b] : ( terms_5523818207328828897rm_a_b @ ( var_b_a @ Uu2 ) ) ).

% linear_term.simps(1)
thf(fact_104_hole__pos__in__ctxt__apply,axiom,
    ! [C3: subterm_and_ctxt_a_b,U: term_a_b] : ( member_list_nat @ ( term_hole_pos_a_b @ C3 ) @ ( term_poss_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ U ) ) ) ).

% hole_pos_in_ctxt_apply
thf(fact_105_position__par__def,axiom,
    ( term_p5017330785391824242ar_nat
    = ( ^ [P4: list_nat,Q4: list_nat] :
          ( ~ ( term_p3503116865373065078eq_nat @ P4 @ Q4 )
          & ~ ( term_p3503116865373065078eq_nat @ Q4 @ P4 ) ) ) ) ).

% position_par_def
thf(fact_106_subt__at__vars__term,axiom,
    ! [P: list_nat,S: term_a_b,X2: b] :
      ( ( member_list_nat @ P @ ( term_poss_a_b @ S ) )
     => ( ( ( term_subt_at_a_b @ S @ P )
          = ( var_b_a @ X2 ) )
       => ( member_b @ X2 @ ( vars_term_a_b @ S ) ) ) ) ).

% subt_at_vars_term
thf(fact_107_vars__term__varposs__iff,axiom,
    ! [X2: b,S: term_a_b] :
      ( ( member_b @ X2 @ ( vars_term_a_b @ S ) )
      = ( ? [X3: list_nat] :
            ( ( member_list_nat @ X3 @ ( terms_varposs_a_b @ S ) )
            & ( ( term_subt_at_a_b @ S @ X3 )
              = ( var_b_a @ X2 ) ) ) ) ) ).

% vars_term_varposs_iff
thf(fact_108_poss__of__term__const__ctxt__apply,axiom,
    ! [P: list_nat,C: a,C3: subterm_and_ctxt_a_b,S: term_a_b] :
      ( ( member_list_nat @ P @ ( terms_7168686267159881682rm_a_b @ ( fun_a_b @ C @ nil_term_a_b ) @ ( subter2376574525758040790rm_a_b @ C3 @ S ) ) )
     => ( ( term_p5017330785391824242ar_nat @ P @ ( term_hole_pos_a_b @ C3 ) )
        | ( term_p3503116865373065078eq_nat @ ( term_hole_pos_a_b @ C3 ) @ P ) ) ) ).

% poss_of_term_const_ctxt_apply
thf(fact_109_subst__term__eqI,axiom,
    ! [Sigma: b > term_a_b,Tau: b > term_a_b] :
      ( ! [T2: term_a_b] :
          ( ( subst_7999470309526761004_a_b_b @ T2 @ Sigma )
          = ( subst_7999470309526761004_a_b_b @ T2 @ Tau ) )
     => ( Sigma = Tau ) ) ).

% subst_term_eqI
thf(fact_110_term_Odistinct_I1_J,axiom,
    ! [X1: b,X21: a,X22: list_term_a_b] :
      ( ( var_b_a @ X1 )
     != ( fun_a_b @ X21 @ X22 ) ) ).

% term.distinct(1)
thf(fact_111_Term_Oterm_Oexhaust,axiom,
    ! [Y: term_a_b] :
      ( ! [X12: b] :
          ( Y
         != ( var_b_a @ X12 ) )
     => ~ ! [X212: a,X222: list_term_a_b] :
            ( Y
           != ( fun_a_b @ X212 @ X222 ) ) ) ).

% Term.term.exhaust
thf(fact_112_possc__subt__at__ctxt__apply,axiom,
    ! [P: list_nat,C3: subterm_and_ctxt_a_b,S: term_a_b,T: term_a_b] :
      ( ( member_list_nat @ P @ ( terms_possc_a_b @ C3 ) )
     => ( ( term_p5017330785391824242ar_nat @ P @ ( term_hole_pos_a_b @ C3 ) )
       => ( ( term_subt_at_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ S ) @ P )
          = ( term_subt_at_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ T ) @ P ) ) ) ) ).

% possc_subt_at_ctxt_apply
thf(fact_113_calculation,axiom,
    member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( term_r6860082780075436317at_a_b @ s @ p @ u ) @ ( term_r6860082780075436317at_a_b @ t @ ( append_nat @ ( term_hole_pos_a_b @ c ) @ ( append_nat @ q @ ( term_pos_diff_nat @ q2 @ v ) ) ) @ u ) ) @ ( rstep_a_b @ r ) ).

% calculation
thf(fact_114_ctxt__eq,axiom,
    ! [C3: subterm_and_ctxt_a_b,S: term_a_b,T: term_a_b] :
      ( ( ( subter2376574525758040790rm_a_b @ C3 @ S )
        = ( subter2376574525758040790rm_a_b @ C3 @ T ) )
      = ( S = T ) ) ).

% ctxt_eq
thf(fact_115_old_Oprod_Oinject,axiom,
    ! [A: term_a_b,B: term_a_b,A3: term_a_b,B2: term_a_b] :
      ( ( ( produc7020197800436672577rm_a_b @ A @ B )
        = ( produc7020197800436672577rm_a_b @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_116_old_Oprod_Oinject,axiom,
    ! [A: a,B: nat,A3: a,B2: nat] :
      ( ( ( product_Pair_a_nat @ A @ B )
        = ( product_Pair_a_nat @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_117_old_Oprod_Oinject,axiom,
    ! [A: nat > nat,B: nat,A3: nat > nat,B2: nat] :
      ( ( ( produc72220940542539688at_nat @ A @ B )
        = ( produc72220940542539688at_nat @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_118_prod_Oinject,axiom,
    ! [X1: term_a_b,X23: term_a_b,Y1: term_a_b,Y23: term_a_b] :
      ( ( ( produc7020197800436672577rm_a_b @ X1 @ X23 )
        = ( produc7020197800436672577rm_a_b @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_119_prod_Oinject,axiom,
    ! [X1: a,X23: nat,Y1: a,Y23: nat] :
      ( ( ( product_Pair_a_nat @ X1 @ X23 )
        = ( product_Pair_a_nat @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_120_prod_Oinject,axiom,
    ! [X1: nat > nat,X23: nat,Y1: nat > nat,Y23: nat] :
      ( ( ( produc72220940542539688at_nat @ X1 @ X23 )
        = ( produc72220940542539688at_nat @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_121_gterm__of__term_Ocases,axiom,
    ! [X2: term_a_b] :
      ( ! [F: a,Ts: list_term_a_b] :
          ( X2
         != ( fun_a_b @ F @ Ts ) )
     => ~ ! [V: b] :
            ( X2
           != ( var_b_a @ V ) ) ) ).

% gterm_of_term.cases
thf(fact_122_hole__pos__replace__term__at,axiom,
    ! [C3: subterm_and_ctxt_a_b,P: list_nat,S: term_a_b,U: term_a_b] :
      ( ( term_p3503116865373065078eq_nat @ ( term_hole_pos_a_b @ C3 ) @ P )
     => ( ( term_r6860082780075436317at_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ S ) @ P @ U )
        = ( subter2376574525758040790rm_a_b @ C3 @ ( term_r6860082780075436317at_a_b @ S @ ( term_pos_diff_nat @ P @ ( term_hole_pos_a_b @ C3 ) ) @ U ) ) ) ) ).

% hole_pos_replace_term_at
thf(fact_123_fun__at__hole__pos__ctxt__apply,axiom,
    ! [C3: subterm_and_ctxt_a_b,T: term_a_b] :
      ( ( term_fun_at_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ T ) @ ( term_hole_pos_a_b @ C3 ) )
      = ( term_fun_at_a_b @ T @ nil_nat ) ) ).

% fun_at_hole_pos_ctxt_apply
thf(fact_124_subt__at__subterm,axiom,
    ! [P: list_nat,T: term_a_b] :
      ( ( member_list_nat @ P @ ( term_poss_a_b @ T ) )
     => ( ( P != nil_nat )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ ( term_subt_at_a_b @ T @ P ) ) @ subterm_and_supt_a_b ) ) ) ).

% subt_at_subterm
thf(fact_125_subst__ident,axiom,
    ! [X2: b,T: term_a_b,U: term_a_b] :
      ( ~ ( member_b @ X2 @ ( vars_term_a_b @ T ) )
     => ( ( subst_7999470309526761004_a_b_b @ T @ ( subst_b_a @ X2 @ U ) )
        = T ) ) ).

% subst_ident
thf(fact_126_replace__term__at__same__pos,axiom,
    ! [S: term_a_b,P: list_nat,U: term_a_b,T: term_a_b] :
      ( ( term_r6860082780075436317at_a_b @ ( term_r6860082780075436317at_a_b @ S @ P @ U ) @ P @ T )
      = ( term_r6860082780075436317at_a_b @ S @ P @ T ) ) ).

% replace_term_at_same_pos
thf(fact_127_replace__term__at__not__poss,axiom,
    ! [P: list_nat,S: term_a_b,T: term_a_b] :
      ( ~ ( member_list_nat @ P @ ( term_poss_a_b @ S ) )
     => ( ( term_r6860082780075436317at_a_b @ S @ P @ T )
        = S ) ) ).

% replace_term_at_not_poss
thf(fact_128_replace__term__at__subt__at__id,axiom,
    ! [S: term_a_b,P: list_nat] :
      ( ( term_r6860082780075436317at_a_b @ S @ P @ ( term_subt_at_a_b @ S @ P ) )
      = S ) ).

% replace_term_at_subt_at_id
thf(fact_129_subst__simps_I2_J,axiom,
    ! [X2: b] :
      ( ( subst_b_a @ X2 @ ( var_b_a @ X2 ) )
      = var_b_a ) ).

% subst_simps(2)
thf(fact_130_replace__term__at__above,axiom,
    ! [P: list_nat,Q: list_nat,S: term_a_b,T: term_a_b,U: term_a_b] :
      ( ( term_p3503116865373065078eq_nat @ P @ Q )
     => ( ( term_r6860082780075436317at_a_b @ ( term_r6860082780075436317at_a_b @ S @ Q @ T ) @ P @ U )
        = ( term_r6860082780075436317at_a_b @ S @ P @ U ) ) ) ).

% replace_term_at_above
thf(fact_131_subst__apply__left__idemp,axiom,
    ! [Sigma: b > term_a_b,X2: b,T: term_a_b,S: term_a_b] :
      ( ( ( Sigma @ X2 )
        = ( subst_7999470309526761004_a_b_b @ T @ Sigma ) )
     => ( ( subst_7999470309526761004_a_b_b @ ( subst_7999470309526761004_a_b_b @ S @ ( subst_b_a @ X2 @ T ) ) @ Sigma )
        = ( subst_7999470309526761004_a_b_b @ S @ Sigma ) ) ) ).

% subst_apply_left_idemp
thf(fact_132_supt__subst,axiom,
    ! [S: term_a_b,T: term_a_b,Sigma: b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ subterm_and_supt_a_b ) ) ).

% supt_subst
thf(fact_133_replace__at__hole__pos,axiom,
    ! [C3: subterm_and_ctxt_a_b,S: term_a_b,T: term_a_b] :
      ( ( term_r6860082780075436317at_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ S ) @ ( term_hole_pos_a_b @ C3 ) @ T )
      = ( subter2376574525758040790rm_a_b @ C3 @ T ) ) ).

% replace_at_hole_pos
thf(fact_134_replace__subterm__at__itself,axiom,
    ! [S: term_a_b,P: list_nat,Q: list_nat,T: term_a_b] :
      ( ( term_r6860082780075436317at_a_b @ S @ P @ ( term_r6860082780075436317at_a_b @ ( term_subt_at_a_b @ S @ P ) @ Q @ T ) )
      = ( term_r6860082780075436317at_a_b @ S @ ( append_nat @ P @ Q ) @ T ) ) ).

% replace_subterm_at_itself
thf(fact_135_replace__term__at__below,axiom,
    ! [P: list_nat,Q: list_nat,S: term_a_b,T: term_a_b,U: term_a_b] :
      ( ( ( P != Q )
        & ( term_p3503116865373065078eq_nat @ P @ Q ) )
     => ( ( term_r6860082780075436317at_a_b @ ( term_r6860082780075436317at_a_b @ S @ P @ T ) @ Q @ U )
        = ( term_r6860082780075436317at_a_b @ S @ P @ ( term_r6860082780075436317at_a_b @ T @ ( term_pos_diff_nat @ Q @ P ) @ U ) ) ) ) ).

% replace_term_at_below
thf(fact_136_ctxt__apply__term__replace__term__hole__pos,axiom,
    ! [C3: subterm_and_ctxt_a_b,S: term_a_b,Q: list_nat,U: term_a_b] :
      ( ( term_r6860082780075436317at_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ S ) @ ( append_nat @ ( term_hole_pos_a_b @ C3 ) @ Q ) @ U )
      = ( subter2376574525758040790rm_a_b @ C3 @ ( term_r6860082780075436317at_a_b @ S @ Q @ U ) ) ) ).

% ctxt_apply_term_replace_term_hole_pos
thf(fact_137_subterm_Oless__asym,axiom,
    ! [Y: term_a_b,X2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ subterm_and_supt_a_b ) ) ).

% subterm.less_asym
thf(fact_138_subterm_Oless__asym_H,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b ) ) ).

% subterm.less_asym'
thf(fact_139_subterm_Oless__trans,axiom,
    ! [Y: term_a_b,X2: term_a_b,Z: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ Y ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ X2 ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.less_trans
thf(fact_140_subterm_Oorder_Oasym,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b ) ) ).

% subterm.order.asym
thf(fact_141_subterm_Oless__irrefl,axiom,
    ! [X2: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ X2 ) @ subterm_and_supt_a_b ) ).

% subterm.less_irrefl
thf(fact_142_subterm_Oless__imp__neq,axiom,
    ! [Y: term_a_b,X2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subterm_and_supt_a_b )
     => ( X2 != Y ) ) ).

% subterm.less_imp_neq
thf(fact_143_subterm_Oless__not__sym,axiom,
    ! [Y: term_a_b,X2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ subterm_and_supt_a_b ) ) ).

% subterm.less_not_sym
thf(fact_144_subterm_Oless__imp__triv,axiom,
    ! [Y: term_a_b,X2: term_a_b,P2: $o] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ subterm_and_supt_a_b )
       => P2 ) ) ).

% subterm.less_imp_triv
thf(fact_145_subterm_Odual__order_Oasym,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b ) ) ).

% subterm.dual_order.asym
thf(fact_146_subterm_Oless__imp__not__eq,axiom,
    ! [Y: term_a_b,X2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subterm_and_supt_a_b )
     => ( X2 != Y ) ) ).

% subterm.less_imp_not_eq
thf(fact_147_subterm_Oless__imp__not__eq2,axiom,
    ! [Y: term_a_b,X2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subterm_and_supt_a_b )
     => ( Y != X2 ) ) ).

% subterm.less_imp_not_eq2
thf(fact_148_subterm_Odual__order_Oirrefl,axiom,
    ! [A: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ subterm_and_supt_a_b ) ).

% subterm.dual_order.irrefl
thf(fact_149_subterm_Oless__imp__not__less,axiom,
    ! [Y: term_a_b,X2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ subterm_and_supt_a_b ) ) ).

% subterm.less_imp_not_less
thf(fact_150_subterm_Oord__eq__less__trans,axiom,
    ! [A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( A = B )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.ord_eq_less_trans
thf(fact_151_subterm_Oord__less__eq__trans,axiom,
    ! [B: term_a_b,A: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ( ( B = C )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.ord_less_eq_trans
thf(fact_152_subterm_Oorder_Ostrict__trans,axiom,
    ! [B: term_a_b,A: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.order.strict_trans
thf(fact_153_subterm_Odual__order_Ostrict__trans,axiom,
    ! [A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.dual_order.strict_trans
thf(fact_154_subterm_Oorder_Ostrict__implies__not__eq,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ( A != B ) ) ).

% subterm.order.strict_implies_not_eq
thf(fact_155_subterm_Odual__order_Ostrict__implies__not__eq,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
     => ( A != B ) ) ).

% subterm.dual_order.strict_implies_not_eq
thf(fact_156_supt__neqD,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ( S != T ) ) ).

% supt_neqD
thf(fact_157_supt__trans,axiom,
    ! [S: term_a_b,T: term_a_b,U: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ U ) @ subterm_and_supt_a_b ) ) ) ).

% supt_trans
thf(fact_158_supt__irrefl,axiom,
    ! [T: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ T ) @ subterm_and_supt_a_b ) ).

% supt_irrefl
thf(fact_159_supt__not__sym,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ subterm_and_supt_a_b ) ) ).

% supt_not_sym
thf(fact_160_supt__not__refl,axiom,
    ! [T: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ T ) @ subterm_and_supt_a_b ) ).

% supt_not_refl
thf(fact_161_subterm__induct,axiom,
    ! [P2: term_a_b > $o,T: term_a_b] :
      ( ! [T2: term_a_b] :
          ( ! [S2: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ S2 ) @ subterm_and_supt_a_b )
             => ( P2 @ S2 ) )
         => ( P2 @ T2 ) )
     => ( P2 @ T ) ) ).

% subterm_induct
thf(fact_162_replace__term__at_Osimps_I1_J,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( term_r6860082780075436317at_a_b @ S @ nil_nat @ T )
      = T ) ).

% replace_term_at.simps(1)
thf(fact_163_pos__replace__at__pres,axiom,
    ! [P: list_nat,S: term_a_b,T: term_a_b] :
      ( ( member_list_nat @ P @ ( term_poss_a_b @ S ) )
     => ( member_list_nat @ P @ ( term_poss_a_b @ ( term_r6860082780075436317at_a_b @ S @ P @ T ) ) ) ) ).

% pos_replace_at_pres
thf(fact_164_supt__var,axiom,
    ! [X2: b,U: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( var_b_a @ X2 ) @ U ) @ subterm_and_supt_a_b ) ).

% supt_var
thf(fact_165_Var__supt,axiom,
    ! [X2: b,T: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( var_b_a @ X2 ) @ T ) @ subterm_and_supt_a_b ) ).

% Var_supt
thf(fact_166_parallel__replace__term__commute,axiom,
    ! [P: list_nat,Q: list_nat,S: term_a_b,T: term_a_b,U: term_a_b] :
      ( ( term_p5017330785391824242ar_nat @ P @ Q )
     => ( ( term_r6860082780075436317at_a_b @ ( term_r6860082780075436317at_a_b @ S @ P @ T ) @ Q @ U )
        = ( term_r6860082780075436317at_a_b @ ( term_r6860082780075436317at_a_b @ S @ Q @ U ) @ P @ T ) ) ) ).

% parallel_replace_term_commute
thf(fact_167_eq__term__by__poss__fun__at,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( ( term_poss_a_b @ S )
        = ( term_poss_a_b @ T ) )
     => ( ! [P3: list_nat] :
            ( ( member_list_nat @ P3 @ ( term_poss_a_b @ S ) )
           => ( ( term_fun_at_a_b @ S @ P3 )
              = ( term_fun_at_a_b @ T @ P3 ) ) )
       => ( S = T ) ) ) ).

% eq_term_by_poss_fun_at
thf(fact_168_supt__const,axiom,
    ! [F2: a,U: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( fun_a_b @ F2 @ nil_term_a_b ) @ U ) @ subterm_and_supt_a_b ) ).

% supt_const
thf(fact_169_poss__of__term__replace__term__at,axiom,
    ! [P: list_nat,S: term_a_b,U: term_a_b] :
      ( ( member_list_nat @ P @ ( term_poss_a_b @ S ) )
     => ( member_list_nat @ P @ ( terms_7168686267159881682rm_a_b @ U @ ( term_r6860082780075436317at_a_b @ S @ P @ U ) ) ) ) ).

% poss_of_term_replace_term_at
thf(fact_170_par__pos__replace__pres,axiom,
    ! [P: list_nat,S: term_a_b,Q: list_nat,T: term_a_b] :
      ( ( member_list_nat @ P @ ( term_poss_a_b @ S ) )
     => ( ( term_p5017330785391824242ar_nat @ P @ Q )
       => ( member_list_nat @ P @ ( term_poss_a_b @ ( term_r6860082780075436317at_a_b @ S @ Q @ T ) ) ) ) ) ).

% par_pos_replace_pres
thf(fact_171_old_Oprod_Oexhaust,axiom,
    ! [Y: produc357393685978478089rm_a_b] :
      ~ ! [A4: term_a_b,B3: term_a_b] :
          ( Y
         != ( produc7020197800436672577rm_a_b @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_172_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_a_nat] :
      ~ ! [A4: a,B3: nat] :
          ( Y
         != ( product_Pair_a_nat @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_173_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8199716216217303280at_nat] :
      ~ ! [A4: nat > nat,B3: nat] :
          ( Y
         != ( produc72220940542539688at_nat @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_174_surj__pair,axiom,
    ! [P: produc357393685978478089rm_a_b] :
    ? [X: term_a_b,Y2: term_a_b] :
      ( P
      = ( produc7020197800436672577rm_a_b @ X @ Y2 ) ) ).

% surj_pair
thf(fact_175_surj__pair,axiom,
    ! [P: product_prod_a_nat] :
    ? [X: a,Y2: nat] :
      ( P
      = ( product_Pair_a_nat @ X @ Y2 ) ) ).

% surj_pair
thf(fact_176_surj__pair,axiom,
    ! [P: produc8199716216217303280at_nat] :
    ? [X: nat > nat,Y2: nat] :
      ( P
      = ( produc72220940542539688at_nat @ X @ Y2 ) ) ).

% surj_pair
thf(fact_177_prod__cases,axiom,
    ! [P2: produc357393685978478089rm_a_b > $o,P: produc357393685978478089rm_a_b] :
      ( ! [A4: term_a_b,B3: term_a_b] : ( P2 @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_178_prod__cases,axiom,
    ! [P2: product_prod_a_nat > $o,P: product_prod_a_nat] :
      ( ! [A4: a,B3: nat] : ( P2 @ ( product_Pair_a_nat @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_179_prod__cases,axiom,
    ! [P2: produc8199716216217303280at_nat > $o,P: produc8199716216217303280at_nat] :
      ( ! [A4: nat > nat,B3: nat] : ( P2 @ ( produc72220940542539688at_nat @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_180_Pair__inject,axiom,
    ! [A: term_a_b,B: term_a_b,A3: term_a_b,B2: term_a_b] :
      ( ( ( produc7020197800436672577rm_a_b @ A @ B )
        = ( produc7020197800436672577rm_a_b @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_181_Pair__inject,axiom,
    ! [A: a,B: nat,A3: a,B2: nat] :
      ( ( ( product_Pair_a_nat @ A @ B )
        = ( product_Pair_a_nat @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_182_Pair__inject,axiom,
    ! [A: nat > nat,B: nat,A3: nat > nat,B2: nat] :
      ( ( ( produc72220940542539688at_nat @ A @ B )
        = ( produc72220940542539688at_nat @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_183_term__to__sig_Ocases,axiom,
    ! [X2: produc5279506192219892694rm_a_b] :
      ( ! [F3: set_Pr4934435412358123699_a_nat,V: b,X: b] :
          ( X2
         != ( produc8030969961714872974rm_a_b @ F3 @ ( produc1437816968797971900rm_a_b @ V @ ( var_b_a @ X ) ) ) )
     => ~ ! [F3: set_Pr4934435412358123699_a_nat,V: b,F: a,Ts: list_term_a_b] :
            ( X2
           != ( produc8030969961714872974rm_a_b @ F3 @ ( produc1437816968797971900rm_a_b @ V @ ( fun_a_b @ F @ Ts ) ) ) ) ) ).

% term_to_sig.cases
thf(fact_184_par__pos__replace__term__at,axiom,
    ! [P: list_nat,S: term_a_b,Q: list_nat,T: term_a_b] :
      ( ( member_list_nat @ P @ ( term_poss_a_b @ S ) )
     => ( ( term_p5017330785391824242ar_nat @ P @ Q )
       => ( ( term_subt_at_a_b @ ( term_r6860082780075436317at_a_b @ S @ Q @ T ) @ P )
          = ( term_subt_at_a_b @ S @ P ) ) ) ) ).

% par_pos_replace_term_at
thf(fact_185_greater__eq__subt__at__replace,axiom,
    ! [P: list_nat,S: term_a_b,Q: list_nat,T: term_a_b] :
      ( ( member_list_nat @ P @ ( term_poss_a_b @ S ) )
     => ( ( term_p3503116865373065078eq_nat @ Q @ P )
       => ( ( term_subt_at_a_b @ ( term_r6860082780075436317at_a_b @ S @ Q @ T ) @ P )
          = ( term_subt_at_a_b @ T @ ( term_pos_diff_nat @ P @ Q ) ) ) ) ) ).

% greater_eq_subt_at_replace
thf(fact_186_less__eq__subt__at__replace,axiom,
    ! [P: list_nat,S: term_a_b,Q: list_nat,T: term_a_b] :
      ( ( member_list_nat @ P @ ( term_poss_a_b @ S ) )
     => ( ( term_p3503116865373065078eq_nat @ P @ Q )
       => ( ( term_subt_at_a_b @ ( term_r6860082780075436317at_a_b @ S @ Q @ T ) @ P )
          = ( term_r6860082780075436317at_a_b @ ( term_subt_at_a_b @ S @ P ) @ ( term_pos_diff_nat @ Q @ P ) @ T ) ) ) ) ).

% less_eq_subt_at_replace
thf(fact_187_rstepI,axiom,
    ! [L2: term_a_b,R3: term_a_b,R4: set_Pr4386577575007340137rm_a_b,S: term_a_b,C3: subterm_and_ctxt_a_b,Sigma: b > term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R3 ) @ R4 )
     => ( ( S
          = ( subter2376574525758040790rm_a_b @ C3 @ ( subst_7999470309526761004_a_b_b @ L2 @ Sigma ) ) )
       => ( ( T
            = ( subter2376574525758040790rm_a_b @ C3 @ ( subst_7999470309526761004_a_b_b @ R3 @ Sigma ) ) )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R4 ) ) ) ) ) ).

% rstepI
thf(fact_188_rstep__ctxtI,axiom,
    ! [S: term_a_b,T: term_a_b,R4: set_Pr4386577575007340137rm_a_b,C3: subterm_and_ctxt_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R4 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ S ) @ ( subter2376574525758040790rm_a_b @ C3 @ T ) ) @ ( rstep_a_b @ R4 ) ) ) ).

% rstep_ctxtI
thf(fact_189_rstep__substI,axiom,
    ! [S: term_a_b,T: term_a_b,R4: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R4 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( rstep_a_b @ R4 ) ) ) ).

% rstep_substI
thf(fact_190_pos__replace__to__rstep,axiom,
    ! [P: list_nat,S: term_a_b,L2: term_a_b,R3: term_a_b,R4: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b,T: term_a_b] :
      ( ( member_list_nat @ P @ ( term_poss_a_b @ S ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R3 ) @ R4 )
       => ( ( ( term_subt_at_a_b @ S @ P )
            = ( subst_7999470309526761004_a_b_b @ L2 @ Sigma ) )
         => ( ( T
              = ( term_r6860082780075436317at_a_b @ S @ P @ ( subst_7999470309526761004_a_b_b @ R3 @ Sigma ) ) )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R4 ) ) ) ) ) ) ).

% pos_replace_to_rstep
thf(fact_191_rstep__to__pos__replace,axiom,
    ! [S: term_a_b,T: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R4 ) )
     => ? [P3: list_nat,L: term_a_b,R: term_a_b,Sigma2: b > term_a_b] :
          ( ( member_list_nat @ P3 @ ( term_poss_a_b @ S ) )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R4 )
          & ( ( term_subt_at_a_b @ S @ P3 )
            = ( subst_7999470309526761004_a_b_b @ L @ Sigma2 ) )
          & ( T
            = ( term_r6860082780075436317at_a_b @ S @ P3 @ ( subst_7999470309526761004_a_b_b @ R @ Sigma2 ) ) ) ) ) ).

% rstep_to_pos_replace
thf(fact_192_rstep__ruleI,axiom,
    ! [L2: term_a_b,R3: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R3 ) @ R4 )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R3 ) @ ( rstep_a_b @ R4 ) ) ) ).

% rstep_ruleI
thf(fact_193_rstep__imp__C__s__r_H,axiom,
    ! [S: term_a_b,T: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R4 ) )
     => ~ ! [C2: subterm_and_ctxt_a_b,L: term_a_b,R: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R4 )
           => ! [Sigma2: b > term_a_b] :
                ( ( S
                  = ( subter2376574525758040790rm_a_b @ C2 @ ( subst_7999470309526761004_a_b_b @ L @ Sigma2 ) ) )
               => ( T
                 != ( subter2376574525758040790rm_a_b @ C2 @ ( subst_7999470309526761004_a_b_b @ R @ Sigma2 ) ) ) ) ) ) ).

% rstep_imp_C_s_r'
thf(fact_194_rstep__imp__C__s__r,axiom,
    ! [S: term_a_b,T: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R4 ) )
     => ? [C2: subterm_and_ctxt_a_b,Sigma2: b > term_a_b,L: term_a_b,R: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R4 )
          & ( S
            = ( subter2376574525758040790rm_a_b @ C2 @ ( subst_7999470309526761004_a_b_b @ L @ Sigma2 ) ) )
          & ( T
            = ( subter2376574525758040790rm_a_b @ C2 @ ( subst_7999470309526761004_a_b_b @ R @ Sigma2 ) ) ) ) ) ).

% rstep_imp_C_s_r
thf(fact_195_fresh,axiom,
    ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ c2 @ zero_zero_nat ) @ ( terms_7988297476397195622_a_b_b @ r ) ) ).

% fresh
thf(fact_196_nt__lhs,axiom,
    ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ c2 @ zero_zero_nat ) @ ( term_funas_term_a_b @ l ) ) ).

% nt_lhs
thf(fact_197_par__hole__pos__replace__term__context__at,axiom,
    ! [P: list_nat,C3: subterm_and_ctxt_a_b,S: term_a_b,U: term_a_b] :
      ( ( term_p5017330785391824242ar_nat @ P @ ( term_hole_pos_a_b @ C3 ) )
     => ( ( term_r6860082780075436317at_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ S ) @ P @ U )
        = ( subter2376574525758040790rm_a_b @ ( terms_4774307173741787698at_a_b @ C3 @ P @ U ) @ S ) ) ) ).

% par_hole_pos_replace_term_context_at
thf(fact_198_depth_Osimps_I2_J,axiom,
    ! [F2: a] :
      ( ( term_depth_a_b @ ( fun_a_b @ F2 @ nil_term_a_b ) )
      = zero_zero_nat ) ).

% depth.simps(2)
thf(fact_199_eq__ctxt__at__pos__by__poss,axiom,
    ! [P: list_nat,S: term_a_b,T: term_a_b] :
      ( ( member_list_nat @ P @ ( term_poss_a_b @ S ) )
     => ( ( member_list_nat @ P @ ( term_poss_a_b @ T ) )
       => ( ! [Q3: list_nat] :
              ( ~ ( term_p3503116865373065078eq_nat @ P @ Q3 )
             => ( ( member_list_nat @ Q3 @ ( term_poss_a_b @ S ) )
                = ( member_list_nat @ Q3 @ ( term_poss_a_b @ T ) ) ) )
         => ( ! [Q3: list_nat] :
                ( ( member_list_nat @ Q3 @ ( term_poss_a_b @ S ) )
               => ( ~ ( term_p3503116865373065078eq_nat @ P @ Q3 )
                 => ( ( term_fun_at_a_b @ S @ Q3 )
                    = ( term_fun_at_a_b @ T @ Q3 ) ) ) )
           => ( ( term_ctxt_at_pos_a_b @ S @ P )
              = ( term_ctxt_at_pos_a_b @ T @ P ) ) ) ) ) ) ).

% eq_ctxt_at_pos_by_poss
thf(fact_200_ctxt__of__pos__term__apply__replace__at__ident,axiom,
    ! [P: list_nat,S: term_a_b,T: term_a_b] :
      ( ( member_list_nat @ P @ ( term_poss_a_b @ S ) )
     => ( ( subter2376574525758040790rm_a_b @ ( term_ctxt_at_pos_a_b @ S @ P ) @ T )
        = ( term_r6860082780075436317at_a_b @ S @ P @ T ) ) ) ).

% ctxt_of_pos_term_apply_replace_at_ident
thf(fact_201_ctxt__at__pos__subt__at__pos,axiom,
    ! [P: list_nat,T: term_a_b,U: term_a_b] :
      ( ( member_list_nat @ P @ ( term_poss_a_b @ T ) )
     => ( ( term_subt_at_a_b @ ( subter2376574525758040790rm_a_b @ ( term_ctxt_at_pos_a_b @ T @ P ) @ U ) @ P )
        = U ) ) ).

% ctxt_at_pos_subt_at_pos
thf(fact_202_ctxt__at__pos__subt__at__id,axiom,
    ! [P: list_nat,T: term_a_b] :
      ( ( member_list_nat @ P @ ( term_poss_a_b @ T ) )
     => ( ( subter2376574525758040790rm_a_b @ ( term_ctxt_at_pos_a_b @ T @ P ) @ ( term_subt_at_a_b @ T @ P ) )
        = T ) ) ).

% ctxt_at_pos_subt_at_id
thf(fact_203_ctxt__at__pos__hole__pos,axiom,
    ! [C3: subterm_and_ctxt_a_b,S: term_a_b] :
      ( ( term_ctxt_at_pos_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ S ) @ ( term_hole_pos_a_b @ C3 ) )
      = C3 ) ).

% ctxt_at_pos_hole_pos
thf(fact_204_zero__reorient,axiom,
    ! [X2: nat] :
      ( ( zero_zero_nat = X2 )
      = ( X2 = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_205_subst__at__ctxt__at__eq__termD,axiom,
    ! [S: term_a_b,T: term_a_b,P: list_nat] :
      ( ( S = T )
     => ( ( member_list_nat @ P @ ( term_poss_a_b @ T ) )
       => ( ( ( term_subt_at_a_b @ S @ P )
            = ( term_subt_at_a_b @ T @ P ) )
          & ( ( term_ctxt_at_pos_a_b @ S @ P )
            = ( term_ctxt_at_pos_a_b @ T @ P ) ) ) ) ) ).

% subst_at_ctxt_at_eq_termD
thf(fact_206_subst__at__ctxt__at__eq__termI,axiom,
    ! [P: list_nat,S: term_a_b,T: term_a_b] :
      ( ( member_list_nat @ P @ ( term_poss_a_b @ S ) )
     => ( ( member_list_nat @ P @ ( term_poss_a_b @ T ) )
       => ( ( ( term_subt_at_a_b @ S @ P )
            = ( term_subt_at_a_b @ T @ P ) )
         => ( ( ( term_ctxt_at_pos_a_b @ S @ P )
              = ( term_ctxt_at_pos_a_b @ T @ P ) )
           => ( S = T ) ) ) ) ) ).

% subst_at_ctxt_at_eq_termI
thf(fact_207_depth_Osimps_I1_J,axiom,
    ! [X2: b] :
      ( ( term_depth_a_b @ ( var_b_a @ X2 ) )
      = zero_zero_nat ) ).

% depth.simps(1)
thf(fact_208_constT__nfunas__term__poss__of__term__empty,axiom,
    ! [C: a,T: term_a_b] :
      ( ( ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ C @ zero_zero_nat ) @ ( term_funas_term_a_b @ T ) ) )
      = ( ( terms_7168686267159881682rm_a_b @ ( fun_a_b @ C @ nil_term_a_b ) @ T )
        = bot_bot_set_list_nat ) ) ).

% constT_nfunas_term_poss_of_term_empty
thf(fact_209_funas__term__const__subst__conv,axiom,
    ! [C: a,L2: term_a_b] :
      ( ( ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ C @ zero_zero_nat ) @ ( term_funas_term_a_b @ L2 ) ) )
      = ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ ( fun_a_b @ C @ nil_term_a_b ) ) @ subter523971068842742411eq_a_b ) ) ) ).

% funas_term_const_subst_conv
thf(fact_210_funas__term__subt__at,axiom,
    ! [F2: a,N: nat,T: term_a_b] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F2 @ N ) @ ( term_funas_term_a_b @ T ) )
     => ? [P3: list_nat,Ts: list_term_a_b] :
          ( ( member_list_nat @ P3 @ ( term_poss_a_b @ T ) )
          & ( ( term_subt_at_a_b @ T @ P3 )
            = ( fun_a_b @ F2 @ Ts ) )
          & ( ( size_s8906293707977694520rm_a_b @ Ts )
            = N ) ) ) ).

% funas_term_subt_at
thf(fact_211_restrict__subst__domain__Var,axiom,
    ! [V2: set_b] :
      ( ( restri22458263168500592in_b_a @ V2 @ var_b_a )
      = var_b_a ) ).

% restrict_subst_domain_Var
thf(fact_212_rename__subst__domain__range__Var__lhs,axiom,
    ! [Sigma: b > term_a_b] :
      ( ( rename5388498879213883313_b_a_b @ var_b_a @ Sigma )
      = Sigma ) ).

% rename_subst_domain_range_Var_lhs
thf(fact_213_rename__subst__domain__range__Var__rhs,axiom,
    ! [Rho: b > term_a_b] :
      ( ( rename5388498879213883313_b_a_b @ Rho @ var_b_a )
      = var_b_a ) ).

% rename_subst_domain_range_Var_rhs
thf(fact_214_rstep_Ocases,axiom,
    ! [A1: term_a_b,A22: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rstep_a_b @ R4 ) )
     => ~ ! [S3: term_a_b,T2: term_a_b,C2: subterm_and_ctxt_a_b] :
            ( ( A1
              = ( subter2376574525758040790rm_a_b @ C2 @ S3 ) )
           => ( ( A22
                = ( subter2376574525758040790rm_a_b @ C2 @ T2 ) )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S3 @ T2 ) @ ( rrstep_a_b @ R4 ) ) ) ) ) ).

% rstep.cases
thf(fact_215_append__eq__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat,Us2: list_nat,Vs: list_nat] :
      ( ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        | ( ( size_size_list_nat @ Us2 )
          = ( size_size_list_nat @ Vs ) ) )
     => ( ( ( append_nat @ Xs @ Us2 )
          = ( append_nat @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us2 = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_216_rrstep__basicI,axiom,
    ! [L2: term_a_b,R3: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R3 ) @ R4 )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R3 ) @ ( rrstep_a_b @ R4 ) ) ) ).

% rrstep_basicI
thf(fact_217_length__0__conv,axiom,
    ! [Xs: list_term_a_b] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_term_a_b ) ) ).

% length_0_conv
thf(fact_218_length__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_nat ) ) ).

% length_0_conv
thf(fact_219_suptI,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( A != B )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b ) ) ) ).

% suptI
thf(fact_220_supteq__subst,axiom,
    ! [S: term_a_b,T: term_a_b,Sigma: b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ subter523971068842742411eq_a_b ) ) ).

% supteq_subst
thf(fact_221_ctxt__supteq,axiom,
    ! [S: term_a_b,C3: subterm_and_ctxt_a_b,T: term_a_b] :
      ( ( S
        = ( subter2376574525758040790rm_a_b @ C3 @ T ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b ) ) ).

% ctxt_supteq
thf(fact_222_restrict__subst__domain__empty,axiom,
    ! [Sigma: b > term_a_b] :
      ( ( restri22458263168500592in_b_a @ bot_bot_set_b @ Sigma )
      = var_b_a ) ).

% restrict_subst_domain_empty
thf(fact_223_subt__at__imp__supteq,axiom,
    ! [P: list_nat,S: term_a_b] :
      ( ( member_list_nat @ P @ ( term_poss_a_b @ S ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ ( term_subt_at_a_b @ S @ P ) ) @ subter523971068842742411eq_a_b ) ) ).

% subt_at_imp_supteq
thf(fact_224_subt__at__subterm__eq,axiom,
    ! [P: list_nat,T: term_a_b] :
      ( ( member_list_nat @ P @ ( term_poss_a_b @ T ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ ( term_subt_at_a_b @ T @ P ) ) @ subter523971068842742411eq_a_b ) ) ).

% subt_at_subterm_eq
thf(fact_225_supteq__antisym,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ subter523971068842742411eq_a_b )
       => ( S = T ) ) ) ).

% supteq_antisym
thf(fact_226_supteq__trans,axiom,
    ! [S: term_a_b,T: term_a_b,U: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ U ) @ subter523971068842742411eq_a_b ) ) ) ).

% supteq_trans
thf(fact_227_eq__supteq,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( S = T )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b ) ) ).

% eq_supteq
thf(fact_228_subterm_Odual__order_Oantisym,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
       => ( A = B ) ) ) ).

% subterm.dual_order.antisym
thf(fact_229_subterm_Odual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: term_a_b,Z2: term_a_b] : ( Y3 = Z2 ) )
    = ( ^ [A5: term_a_b,B4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B4 ) @ subter523971068842742411eq_a_b )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ A5 ) @ subter523971068842742411eq_a_b ) ) ) ) ).

% subterm.dual_order.eq_iff
thf(fact_230_subterm_Odual__order_Otrans,axiom,
    ! [A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.dual_order.trans
thf(fact_231_subterm_Oord__le__eq__trans,axiom,
    ! [B: term_a_b,A: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
     => ( ( B = C )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.ord_le_eq_trans
thf(fact_232_subterm_Oord__eq__le__trans,axiom,
    ! [A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( A = B )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.ord_eq_le_trans
thf(fact_233_subterm_Odual__order_Orefl,axiom,
    ! [A: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ subter523971068842742411eq_a_b ) ).

% subterm.dual_order.refl
thf(fact_234_subterm_Oorder__antisym,axiom,
    ! [Y: term_a_b,X2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ subter523971068842742411eq_a_b )
       => ( X2 = Y ) ) ) ).

% subterm.order_antisym
thf(fact_235_subterm_Oorder_Oantisym,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
       => ( A = B ) ) ) ).

% subterm.order.antisym
thf(fact_236_subterm_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: term_a_b,Z2: term_a_b] : ( Y3 = Z2 ) )
    = ( ^ [X3: term_a_b,Y4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ X3 ) @ subter523971068842742411eq_a_b )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ subter523971068842742411eq_a_b ) ) ) ) ).

% subterm.order_eq_iff
thf(fact_237_subterm_Oorder_Oeq__iff,axiom,
    ( ( ^ [Y3: term_a_b,Z2: term_a_b] : ( Y3 = Z2 ) )
    = ( ^ [A5: term_a_b,B4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ A5 ) @ subter523971068842742411eq_a_b )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B4 ) @ subter523971068842742411eq_a_b ) ) ) ) ).

% subterm.order.eq_iff
thf(fact_238_subterm_Oantisym__conv,axiom,
    ! [X2: term_a_b,Y: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subter523971068842742411eq_a_b )
        = ( X2 = Y ) ) ) ).

% subterm.antisym_conv
thf(fact_239_subterm_Oorder__trans,axiom,
    ! [Y: term_a_b,X2: term_a_b,Z: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ Y ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ X2 ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.order_trans
thf(fact_240_subterm_Oorder_Otrans,axiom,
    ! [B: term_a_b,A: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.order.trans
thf(fact_241_subterm_Oorder__refl,axiom,
    ! [X2: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ X2 ) @ subter523971068842742411eq_a_b ) ).

% subterm.order_refl
thf(fact_242_subterm_Oeq__refl,axiom,
    ! [X2: term_a_b,Y: term_a_b] :
      ( ( X2 = Y )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subter523971068842742411eq_a_b ) ) ).

% subterm.eq_refl
thf(fact_243_supteq_Orefl,axiom,
    ! [T: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ T ) @ subter523971068842742411eq_a_b ) ).

% supteq.refl
thf(fact_244_list_Osize_I3_J,axiom,
    ( ( size_s8906293707977694520rm_a_b @ nil_term_a_b )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_245_list_Osize_I3_J,axiom,
    ( ( size_size_list_nat @ nil_nat )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_246_subterm_OleD,axiom,
    ! [X2: term_a_b,Y: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ subter523971068842742411eq_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subterm_and_supt_a_b ) ) ).

% subterm.leD
thf(fact_247_subterm_Ole__less,axiom,
    ! [Y: term_a_b,X2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subter523971068842742411eq_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subterm_and_supt_a_b )
        | ( X2 = Y ) ) ) ).

% subterm.le_less
thf(fact_248_subterm_Oless__le,axiom,
    ! [Y: term_a_b,X2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subter523971068842742411eq_a_b )
        & ( X2 != Y ) ) ) ).

% subterm.less_le
thf(fact_249_subterm_Onless__le,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b ) )
      = ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
        | ( A = B ) ) ) ).

% subterm.nless_le
thf(fact_250_subterm_Oless__imp__le,axiom,
    ! [Y: term_a_b,X2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subterm_and_supt_a_b )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subter523971068842742411eq_a_b ) ) ).

% subterm.less_imp_le
thf(fact_251_subterm_Ole__neq__trans,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
     => ( ( A != B )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.le_neq_trans
thf(fact_252_subterm_Oantisym__conv1,axiom,
    ! [Y: term_a_b,X2: term_a_b] :
      ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subter523971068842742411eq_a_b )
        = ( X2 = Y ) ) ) ).

% subterm.antisym_conv1
thf(fact_253_subterm_Oantisym__conv2,axiom,
    ! [Y: term_a_b,X2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subter523971068842742411eq_a_b )
     => ( ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subterm_and_supt_a_b ) )
        = ( X2 = Y ) ) ) ).

% subterm.antisym_conv2
thf(fact_254_subterm_Ole__less__trans,axiom,
    ! [Y: term_a_b,X2: term_a_b,Z: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ Y ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ X2 ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.le_less_trans
thf(fact_255_subterm_Oless__le__trans,axiom,
    ! [Y: term_a_b,X2: term_a_b,Z: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ Y ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ X2 ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.less_le_trans
thf(fact_256_subterm_Oless__le__not__le,axiom,
    ! [Y: term_a_b,X2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subter523971068842742411eq_a_b )
        & ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.less_le_not_le
thf(fact_257_subterm_Ole__imp__less__or__eq,axiom,
    ! [Y: term_a_b,X2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ subterm_and_supt_a_b )
        | ( X2 = Y ) ) ) ).

% subterm.le_imp_less_or_eq
thf(fact_258_subterm_Oorder_Ostrict__trans1,axiom,
    ! [B: term_a_b,A: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.order.strict_trans1
thf(fact_259_subterm_Oorder_Ostrict__trans2,axiom,
    ! [B: term_a_b,A: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.order.strict_trans2
thf(fact_260_subterm_Oorder_Ostrict__iff__not,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
        & ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.order.strict_iff_not
thf(fact_261_subterm_Oorder_Oorder__iff__strict,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
        | ( A = B ) ) ) ).

% subterm.order.order_iff_strict
thf(fact_262_subterm_Oorder_Ostrict__iff__order,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
        & ( A != B ) ) ) ).

% subterm.order.strict_iff_order
thf(fact_263_subterm_Odual__order_Ostrict__trans1,axiom,
    ! [A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.dual_order.strict_trans1
thf(fact_264_subterm_Odual__order_Ostrict__trans2,axiom,
    ! [A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.dual_order.strict_trans2
thf(fact_265_subterm_Odual__order_Ostrict__iff__not,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
        & ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.dual_order.strict_iff_not
thf(fact_266_subterm_Oorder_Ostrict__implies__order,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b ) ) ).

% subterm.order.strict_implies_order
thf(fact_267_subterm_Odual__order_Oorder__iff__strict,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
        | ( A = B ) ) ) ).

% subterm.dual_order.order_iff_strict
thf(fact_268_subterm_Odual__order_Ostrict__iff__order,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
        & ( A != B ) ) ) ).

% subterm.dual_order.strict_iff_order
thf(fact_269_subterm_Odual__order_Ostrict__implies__order,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b ) ) ).

% subterm.dual_order.strict_implies_order
thf(fact_270_subterm_Oorder_Onot__eq__order__implies__strict,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( A != B )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.order.not_eq_order_implies_strict
thf(fact_271_suptE,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
       => ( S != T ) ) ) ).

% suptE
thf(fact_272_supt__imp__supteq,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b ) ) ).

% supt_imp_supteq
thf(fact_273_supteq__not__supt,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
     => ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
       => ( S = T ) ) ) ).

% supteq_not_supt
thf(fact_274_supt__supteq__conv,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
        & ( S != T ) ) ) ).

% supt_supteq_conv
thf(fact_275_supteq__supt__conv,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
        | ( S = T ) ) ) ).

% supteq_supt_conv
thf(fact_276_supt__supteq__trans,axiom,
    ! [S: term_a_b,T: term_a_b,U: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ U ) @ subterm_and_supt_a_b ) ) ) ).

% supt_supteq_trans
thf(fact_277_supteq__supt__trans,axiom,
    ! [S: term_a_b,T: term_a_b,U: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ U ) @ subterm_and_supt_a_b ) ) ) ).

% supteq_supt_trans
thf(fact_278_supt__supteq__not__supteq,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
        & ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ subter523971068842742411eq_a_b ) ) ) ).

% supt_supteq_not_supteq
thf(fact_279_supteq__var__imp__eq,axiom,
    ! [X2: b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( var_b_a @ X2 ) @ T ) @ subter523971068842742411eq_a_b )
      = ( T
        = ( var_b_a @ X2 ) ) ) ).

% supteq_var_imp_eq
thf(fact_280_supteq__ctxt__conv,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
      = ( ? [C4: subterm_and_ctxt_a_b] :
            ( S
            = ( subter2376574525758040790rm_a_b @ C4 @ T ) ) ) ) ).

% supteq_ctxt_conv
thf(fact_281_ctxt__imp__supteq,axiom,
    ! [C3: subterm_and_ctxt_a_b,T: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ T ) @ T ) @ subter523971068842742411eq_a_b ) ).

% ctxt_imp_supteq
thf(fact_282_supteq__ctxtE,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
     => ~ ! [C2: subterm_and_ctxt_a_b] :
            ( S
           != ( subter2376574525758040790rm_a_b @ C2 @ T ) ) ) ).

% supteq_ctxtE
thf(fact_283_restrict__subst__domain__def,axiom,
    ( restri22458263168500592in_b_a
    = ( ^ [V3: set_b,Sigma3: b > term_a_b,X3: b] : ( if_term_a_b @ ( member_b @ X3 @ V3 ) @ ( Sigma3 @ X3 ) @ ( var_b_a @ X3 ) ) ) ) ).

% restrict_subst_domain_def
thf(fact_284_rrstep_Ocases,axiom,
    ! [A1: term_a_b,A22: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rrstep_a_b @ R4 ) )
     => ~ ! [L: term_a_b,R: term_a_b,Sigma2: b > term_a_b] :
            ( ( A1
              = ( subst_7999470309526761004_a_b_b @ L @ Sigma2 ) )
           => ( ( A22
                = ( subst_7999470309526761004_a_b_b @ R @ Sigma2 ) )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R4 ) ) ) ) ).

% rrstep.cases
thf(fact_285_rrstep_Osimps,axiom,
    ! [A1: term_a_b,A22: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rrstep_a_b @ R4 ) )
      = ( ? [L3: term_a_b,R2: term_a_b,Sigma3: b > term_a_b] :
            ( ( A1
              = ( subst_7999470309526761004_a_b_b @ L3 @ Sigma3 ) )
            & ( A22
              = ( subst_7999470309526761004_a_b_b @ R2 @ Sigma3 ) )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L3 @ R2 ) @ R4 ) ) ) ) ).

% rrstep.simps
thf(fact_286_rrstep_Ointros,axiom,
    ! [L2: term_a_b,R3: term_a_b,R4: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R3 ) @ R4 )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ L2 @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ R3 @ Sigma ) ) @ ( rrstep_a_b @ R4 ) ) ) ).

% rrstep.intros
thf(fact_287_rrstep__subst,axiom,
    ! [S: term_a_b,T: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R4 ) )
     => ~ ! [L: term_a_b,R: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R4 )
           => ! [Sigma2: b > term_a_b] :
                ( ( S
                  = ( subst_7999470309526761004_a_b_b @ L @ Sigma2 ) )
               => ( T
                 != ( subst_7999470309526761004_a_b_b @ R @ Sigma2 ) ) ) ) ) ).

% rrstep_subst
thf(fact_288_vars__term__supteq,axiom,
    ! [X2: b,T: term_a_b] :
      ( ( member_b @ X2 @ ( vars_term_a_b @ T ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ ( var_b_a @ X2 ) ) @ subter523971068842742411eq_a_b ) ) ).

% vars_term_supteq
thf(fact_289_subt__at__imp__supteq_H,axiom,
    ! [P: list_nat,S: term_a_b,T: term_a_b] :
      ( ( member_list_nat @ P @ ( term_poss_a_b @ S ) )
     => ( ( ( term_subt_at_a_b @ S @ P )
          = T )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b ) ) ) ).

% subt_at_imp_supteq'
thf(fact_290_poss__of__term__poss__emptyD,axiom,
    ! [U: term_a_b,S: term_a_b,P: list_nat] :
      ( ( ( terms_7168686267159881682rm_a_b @ U @ S )
        = bot_bot_set_list_nat )
     => ( ( member_list_nat @ P @ ( term_poss_a_b @ S ) )
       => ( ( term_subt_at_a_b @ S @ P )
         != U ) ) ) ).

% poss_of_term_poss_emptyD
thf(fact_291_rstep_Ointros,axiom,
    ! [S: term_a_b,T: term_a_b,R4: set_Pr4386577575007340137rm_a_b,C3: subterm_and_ctxt_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R4 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ S ) @ ( subter2376574525758040790rm_a_b @ C3 @ T ) ) @ ( rstep_a_b @ R4 ) ) ) ).

% rstep.intros
thf(fact_292_rstep_Osimps,axiom,
    ! [A1: term_a_b,A22: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rstep_a_b @ R4 ) )
      = ( ? [S4: term_a_b,T3: term_a_b,C4: subterm_and_ctxt_a_b] :
            ( ( A1
              = ( subter2376574525758040790rm_a_b @ C4 @ S4 ) )
            & ( A22
              = ( subter2376574525758040790rm_a_b @ C4 @ T3 ) )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T3 ) @ ( rrstep_a_b @ R4 ) ) ) ) ) ).

% rstep.simps
thf(fact_293_varposs__Var,axiom,
    ! [X2: b] :
      ( ( terms_varposs_a_b @ ( var_b_a @ X2 ) )
      = ( insert_list_nat @ nil_nat @ bot_bot_set_list_nat ) ) ).

% varposs_Var
thf(fact_294_poss_Osimps_I1_J,axiom,
    ! [X2: b] :
      ( ( term_poss_a_b @ ( var_b_a @ X2 ) )
      = ( insert_list_nat @ nil_nat @ bot_bot_set_list_nat ) ) ).

% poss.simps(1)
thf(fact_295_range__vars__Var,axiom,
    ( ( range_vars_b_a @ var_b_a )
    = bot_bot_set_b ) ).

% range_vars_Var
thf(fact_296_list__ex1__simps_I1_J,axiom,
    ! [P2: term_a_b > $o] :
      ~ ( list_ex1_term_a_b @ P2 @ nil_term_a_b ) ).

% list_ex1_simps(1)
thf(fact_297_list__ex1__simps_I1_J,axiom,
    ! [P2: nat > $o] :
      ~ ( list_ex1_nat @ P2 @ nil_nat ) ).

% list_ex1_simps(1)
thf(fact_298_funas__term__subterm__atI,axiom,
    ! [P: list_nat,S: term_a_b,F4: set_Pr4934435412358123699_a_nat] :
      ( ( member_list_nat @ P @ ( term_poss_a_b @ S ) )
     => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F4 )
       => ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( term_subt_at_a_b @ S @ P ) ) @ F4 ) ) ) ).

% funas_term_subterm_atI
thf(fact_299_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_300_zero__le,axiom,
    ! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).

% zero_le
thf(fact_301_mem__range__varsI,axiom,
    ! [Sigma: b > term_a_b,X2: b,Y: b] :
      ( ( ( Sigma @ X2 )
        = ( var_b_a @ Y ) )
     => ( ( X2 != Y )
       => ( member_b @ Y @ ( range_vars_b_a @ Sigma ) ) ) ) ).

% mem_range_varsI
thf(fact_302_vars__term__subset__subst__eq,axiom,
    ! [T: term_a_b,S: term_a_b,Sigma: b > term_a_b,Tau: b > term_a_b] :
      ( ( ord_less_eq_set_b @ ( vars_term_a_b @ T ) @ ( vars_term_a_b @ S ) )
     => ( ( ( subst_7999470309526761004_a_b_b @ S @ Sigma )
          = ( subst_7999470309526761004_a_b_b @ S @ Tau ) )
       => ( ( subst_7999470309526761004_a_b_b @ T @ Sigma )
          = ( subst_7999470309526761004_a_b_b @ T @ Tau ) ) ) ) ).

% vars_term_subset_subst_eq
thf(fact_303_funas__term_Osimps_I1_J,axiom,
    ! [X2: b] :
      ( ( term_funas_term_a_b @ ( var_b_a @ X2 ) )
      = bot_bo9049108969261143879_a_nat ) ).

% funas_term.simps(1)
thf(fact_304_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_length_term_a_b @ N @ nil_term_a_b )
      = N ) ).

% gen_length_code(1)
thf(fact_305_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_length_nat @ N @ nil_nat )
      = N ) ).

% gen_length_code(1)
thf(fact_306_Term_Oterm_Osimps_I17_J,axiom,
    ! [X1: b] :
      ( ( vars_term_a_b @ ( var_b_a @ X1 ) )
      = ( insert_b @ X1 @ bot_bot_set_b ) ) ).

% Term.term.simps(17)
thf(fact_307_supteq__imp__vars__term__subset,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
     => ( ord_less_eq_set_b @ ( vars_term_a_b @ T ) @ ( vars_term_a_b @ S ) ) ) ).

% supteq_imp_vars_term_subset
thf(fact_308_funas__term__replace__at__lower,axiom,
    ! [P: list_nat,S: term_a_b,T: term_a_b] :
      ( ( member_list_nat @ P @ ( term_poss_a_b @ S ) )
     => ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ ( term_funas_term_a_b @ ( term_r6860082780075436317at_a_b @ S @ P @ T ) ) ) ) ).

% funas_term_replace_at_lower
thf(fact_309_subst__apply__term__restrict__subst__domain,axiom,
    ! [T: term_a_b,V2: set_b,Sigma: b > term_a_b] :
      ( ( ord_less_eq_set_b @ ( vars_term_a_b @ T ) @ V2 )
     => ( ( subst_7999470309526761004_a_b_b @ T @ ( restri22458263168500592in_b_a @ V2 @ Sigma ) )
        = ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) ) ).

% subst_apply_term_restrict_subst_domain
thf(fact_310_rstep__term__to__sig__l,axiom,
    ! [S: term_a_b,T: term_a_b,R4: set_Pr4386577575007340137rm_a_b,F4: set_Pr4934435412358123699_a_nat,V4: b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R4 ) )
     => ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R4 ) @ F4 )
       => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F4 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( terms_8519481630511763164ig_a_b @ F4 @ V4 @ S ) @ T ) @ ( rstep_a_b @ R4 ) ) ) ) ) ).

% rstep_term_to_sig_l
thf(fact_311_rstep__term__to__sig__r,axiom,
    ! [S: term_a_b,T: term_a_b,R4: set_Pr4386577575007340137rm_a_b,F4: set_Pr4934435412358123699_a_nat,V4: b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R4 ) )
     => ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R4 ) @ F4 )
       => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F4 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ ( terms_8519481630511763164ig_a_b @ F4 @ V4 @ T ) ) @ ( rstep_a_b @ R4 ) ) ) ) ) ).

% rstep_term_to_sig_r
thf(fact_312_listset_Osimps_I1_J,axiom,
    ( ( listset_term_a_b @ nil_set_term_a_b )
    = ( insert_list_term_a_b @ nil_term_a_b @ bot_bo8941492048910247406rm_a_b ) ) ).

% listset.simps(1)
thf(fact_313_listset_Osimps_I1_J,axiom,
    ( ( listset_nat @ nil_set_nat )
    = ( insert_list_nat @ nil_nat @ bot_bot_set_list_nat ) ) ).

% listset.simps(1)
thf(fact_314_subrelI,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ! [X: term_a_b,Y2: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ R3 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ S ) )
     => ( ord_le118470702582115849rm_a_b @ R3 @ S ) ) ).

% subrelI
thf(fact_315_subrelI,axiom,
    ! [R3: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
      ( ! [X: a,Y2: nat] :
          ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y2 ) @ R3 )
         => ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y2 ) @ S ) )
     => ( ord_le8666007276011122963_a_nat @ R3 @ S ) ) ).

% subrelI
thf(fact_316_subrelI,axiom,
    ! [R3: set_Pr9093778441882193744at_nat,S: set_Pr9093778441882193744at_nat] :
      ( ! [X: nat > nat,Y2: nat] :
          ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y2 ) @ R3 )
         => ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y2 ) @ S ) )
     => ( ord_le3678578370064672496at_nat @ R3 @ S ) ) ).

% subrelI
thf(fact_317_rstep__srstepI,axiom,
    ! [R4: set_Pr4386577575007340137rm_a_b,F4: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R4 ) @ F4 )
     => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F4 )
       => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F4 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R4 ) )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) ) ) ) ) ).

% rstep_srstepI
thf(fact_318_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_319_term__to__sig__id,axiom,
    ! [T: term_a_b,F4: set_Pr4934435412358123699_a_nat,V4: b] :
      ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F4 )
     => ( ( terms_8519481630511763164ig_a_b @ F4 @ V4 @ T )
        = T ) ) ).

% term_to_sig_id
thf(fact_320_sig__stepI,axiom,
    ! [S: term_a_b,F4: set_Pr4934435412358123699_a_nat,T: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F4 )
     => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F4 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ R4 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F4 @ R4 ) ) ) ) ) ).

% sig_stepI
thf(fact_321_srstep__monp,axiom,
    ! [F4: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ F4 @ G )
     => ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) @ ( sig_step_a_b @ G @ ( rstep_a_b @ R4 ) ) ) ) ).

% srstep_monp
thf(fact_322_term__to__sig_Osimps_I1_J,axiom,
    ! [F4: set_Pr4934435412358123699_a_nat,V4: b,X2: b] :
      ( ( terms_8519481630511763164ig_a_b @ F4 @ V4 @ ( var_b_a @ X2 ) )
      = ( var_b_a @ X2 ) ) ).

% term_to_sig.simps(1)
thf(fact_323_subst__poss__mono,axiom,
    ! [S: term_a_b,Sigma: b > term_a_b] : ( ord_le6045566169113846134st_nat @ ( term_poss_a_b @ S ) @ ( term_poss_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) ) ) ).

% subst_poss_mono
thf(fact_324_rrstep__rstep__mono,axiom,
    ! [R4: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( rrstep_a_b @ R4 ) @ ( rstep_a_b @ R4 ) ) ).

% rrstep_rstep_mono
thf(fact_325_fuans__term__term__to__sig,axiom,
    ! [F4: set_Pr4934435412358123699_a_nat,V4: b,T: term_a_b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( terms_8519481630511763164ig_a_b @ F4 @ V4 @ T ) ) @ F4 ) ).

% fuans_term_term_to_sig
thf(fact_326_sig__stepE,axiom,
    ! [S: term_a_b,T: term_a_b,F4: set_Pr4934435412358123699_a_nat,R4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F4 @ R4 ) )
     => ~ ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ R4 )
         => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F4 )
           => ~ ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F4 ) ) ) ) ).

% sig_stepE
thf(fact_327_srstepD,axiom,
    ! [S: term_a_b,T: term_a_b,F4: set_Pr4934435412358123699_a_nat,R4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R4 ) )
        & ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F4 )
        & ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F4 ) ) ) ).

% srstepD
thf(fact_328_srrstepD,axiom,
    ! [S: term_a_b,T: term_a_b,F4: set_Pr4934435412358123699_a_nat,R4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F4 @ ( rrstep_a_b @ R4 ) ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R4 ) )
        & ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F4 )
        & ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F4 ) ) ) ).

% srrstepD
thf(fact_329_srstep__subst__closed,axiom,
    ! [S: term_a_b,T: term_a_b,F4: set_Pr4934435412358123699_a_nat,R4: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) )
     => ( ! [X: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X ) ) @ F4 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) ) ) ).

% srstep_subst_closed
thf(fact_330_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_331_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_332_rsteps__eq__srsteps__eqI,axiom,
    ! [R4: set_Pr4386577575007340137rm_a_b,F4: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R4 ) @ F4 )
     => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F4 )
       => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F4 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ R4 ) ) )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) ) ) ) ) ) ).

% rsteps_eq_srsteps_eqI
thf(fact_333_rsteps__srstepsI,axiom,
    ! [R4: set_Pr4386577575007340137rm_a_b,F4: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R4 ) @ F4 )
     => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F4 )
       => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F4 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R4 ) ) )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) ) ) ) ) ) ).

% rsteps_srstepsI
thf(fact_334_rstep__trancl__sig__step__l,axiom,
    ! [S: term_a_b,T: term_a_b,R4: set_Pr4386577575007340137rm_a_b,F4: set_Pr4934435412358123699_a_nat,V4: b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R4 ) ) )
     => ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R4 ) @ F4 )
       => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F4 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( terms_8519481630511763164ig_a_b @ F4 @ V4 @ S ) @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) ) ) ) ) ).

% rstep_trancl_sig_step_l
thf(fact_335_rstep__trancl__sig__step__r,axiom,
    ! [S: term_a_b,T: term_a_b,R4: set_Pr4386577575007340137rm_a_b,F4: set_Pr4934435412358123699_a_nat,V4: b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R4 ) ) )
     => ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R4 ) @ F4 )
       => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F4 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ ( terms_8519481630511763164ig_a_b @ F4 @ V4 @ T ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) ) ) ) ) ).

% rstep_trancl_sig_step_r
thf(fact_336_srsteps__monp,axiom,
    ! [F4: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ F4 @ G )
     => ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ G @ ( rstep_a_b @ R4 ) ) ) ) ) ).

% srsteps_monp
thf(fact_337_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_338_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_339_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_340_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_341_srsteps__eq__monp,axiom,
    ! [F4: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ F4 @ G )
     => ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ G @ ( rstep_a_b @ R4 ) ) ) ) ) ).

% srsteps_eq_monp
thf(fact_342_sig__step__rsteps__dist,axiom,
    ! [R4: set_Pr4386577575007340137rm_a_b,F4: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R4 ) @ F4 )
     => ( ( sig_step_a_b @ F4 @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R4 ) ) )
        = ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) ) ) ).

% sig_step_rsteps_dist
thf(fact_343_srstepsD,axiom,
    ! [S: term_a_b,T: term_a_b,F4: set_Pr4934435412358123699_a_nat,R4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R4 ) ) )
        & ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F4 )
        & ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F4 ) ) ) ).

% srstepsD
thf(fact_344_srsteps__subst__closed,axiom,
    ! [S: term_a_b,T: term_a_b,F4: set_Pr4934435412358123699_a_nat,R4: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) )
     => ( ! [X: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X ) ) @ F4 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) ) ) ) ).

% srsteps_subst_closed
thf(fact_345_srsteps__eq__subst__closed,axiom,
    ! [S: term_a_b,T: term_a_b,F4: set_Pr4934435412358123699_a_nat,R4: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) )
     => ( ! [X: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X ) ) @ F4 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) ) ) ) ).

% srsteps_eq_subst_closed
thf(fact_346_tranclD,axiom,
    ! [X2: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( transi7922773638565587891rm_a_b @ R5 ) )
     => ? [Z3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z3 ) @ R5 )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ Y ) @ ( transi7742714808557438673rm_a_b @ R5 ) ) ) ) ).

% tranclD
thf(fact_347_rtranclD,axiom,
    ! [A: term_a_b,B: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
     => ( ( A = B )
        | ( ( A != B )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R5 ) ) ) ) ) ).

% rtranclD
thf(fact_348_tranclD2,axiom,
    ! [X2: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( transi7922773638565587891rm_a_b @ R5 ) )
     => ? [Z3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z3 ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ Y ) @ R5 ) ) ) ).

% tranclD2
thf(fact_349_trancl__into__rtrancl,axiom,
    ! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ).

% trancl_into_rtrancl
thf(fact_350_rtrancl__eq__or__trancl,axiom,
    ! [X2: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
      = ( ( X2 = Y )
        | ( ( X2 != Y )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( transi7922773638565587891rm_a_b @ R5 ) ) ) ) ) ).

% rtrancl_eq_or_trancl
thf(fact_351_rtrancl__into__trancl1,axiom,
    ! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ R3 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ).

% rtrancl_into_trancl1
thf(fact_352_converse__rtrancl__induct2,axiom,
    ! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R3: set_Pr2972776593051762503rm_a_b,P2: term_a_b > term_a_b > $o] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A4: term_a_b,B3: term_a_b,Aa: term_a_b,Ba: term_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R3 )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
               => ( ( P2 @ Aa @ Ba )
                 => ( P2 @ A4 @ B3 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_353_converse__rtrancl__induct2,axiom,
    ! [Ax: a,Ay: nat,Bx: a,By: nat,R3: set_Pr1811044260758604347_a_nat,P2: a > nat > $o] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ Bx @ By ) ) @ ( transi2726145917338391738_a_nat @ R3 ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A4: a,B3: nat,Aa: a,Ba: nat] :
              ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ A4 @ B3 ) @ ( product_Pair_a_nat @ Aa @ Ba ) ) @ R3 )
             => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Aa @ Ba ) @ ( product_Pair_a_nat @ Bx @ By ) ) @ ( transi2726145917338391738_a_nat @ R3 ) )
               => ( ( P2 @ Aa @ Ba )
                 => ( P2 @ A4 @ B3 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_354_converse__rtrancl__induct2,axiom,
    ! [Ax: nat > nat,Ay: nat,Bx: nat > nat,By: nat,R3: set_Pr7047708467701517959at_nat,P2: ( nat > nat ) > nat > $o] :
      ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ Bx @ By ) ) @ ( transi3422749762518297843at_nat @ R3 ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A4: nat > nat,B3: nat,Aa: nat > nat,Ba: nat] :
              ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ A4 @ B3 ) @ ( produc72220940542539688at_nat @ Aa @ Ba ) ) @ R3 )
             => ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Aa @ Ba ) @ ( produc72220940542539688at_nat @ Bx @ By ) ) @ ( transi3422749762518297843at_nat @ R3 ) )
               => ( ( P2 @ Aa @ Ba )
                 => ( P2 @ A4 @ B3 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_355_converse__rtranclE2,axiom,
    ! [Xa: term_a_b,Xb: term_a_b,Za: term_a_b,Zb: term_a_b,R3: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa @ Xb ) @ ( produc7020197800436672577rm_a_b @ Za @ Zb ) ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
     => ( ( ( produc7020197800436672577rm_a_b @ Xa @ Xb )
         != ( produc7020197800436672577rm_a_b @ Za @ Zb ) )
       => ~ ! [A4: term_a_b,B3: term_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa @ Xb ) @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) ) @ R3 )
             => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Za @ Zb ) ) @ ( transi2615809358984392588rm_a_b @ R3 ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_356_converse__rtranclE2,axiom,
    ! [Xa: a,Xb: nat,Za: a,Zb: nat,R3: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Xa @ Xb ) @ ( product_Pair_a_nat @ Za @ Zb ) ) @ ( transi2726145917338391738_a_nat @ R3 ) )
     => ( ( ( product_Pair_a_nat @ Xa @ Xb )
         != ( product_Pair_a_nat @ Za @ Zb ) )
       => ~ ! [A4: a,B3: nat] :
              ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Xa @ Xb ) @ ( product_Pair_a_nat @ A4 @ B3 ) ) @ R3 )
             => ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ A4 @ B3 ) @ ( product_Pair_a_nat @ Za @ Zb ) ) @ ( transi2726145917338391738_a_nat @ R3 ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_357_converse__rtranclE2,axiom,
    ! [Xa: nat > nat,Xb: nat,Za: nat > nat,Zb: nat,R3: set_Pr7047708467701517959at_nat] :
      ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Xa @ Xb ) @ ( produc72220940542539688at_nat @ Za @ Zb ) ) @ ( transi3422749762518297843at_nat @ R3 ) )
     => ( ( ( produc72220940542539688at_nat @ Xa @ Xb )
         != ( produc72220940542539688at_nat @ Za @ Zb ) )
       => ~ ! [A4: nat > nat,B3: nat] :
              ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Xa @ Xb ) @ ( produc72220940542539688at_nat @ A4 @ B3 ) ) @ R3 )
             => ~ ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ A4 @ B3 ) @ ( produc72220940542539688at_nat @ Za @ Zb ) ) @ ( transi3422749762518297843at_nat @ R3 ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_358_rtrancl__induct2,axiom,
    ! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R3: set_Pr2972776593051762503rm_a_b,P2: term_a_b > term_a_b > $o] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A4: term_a_b,B3: term_a_b,Aa: term_a_b,Ba: term_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R3 )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_359_rtrancl__induct2,axiom,
    ! [Ax: a,Ay: nat,Bx: a,By: nat,R3: set_Pr1811044260758604347_a_nat,P2: a > nat > $o] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ Bx @ By ) ) @ ( transi2726145917338391738_a_nat @ R3 ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A4: a,B3: nat,Aa: a,Ba: nat] :
              ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ A4 @ B3 ) ) @ ( transi2726145917338391738_a_nat @ R3 ) )
             => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ A4 @ B3 ) @ ( product_Pair_a_nat @ Aa @ Ba ) ) @ R3 )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_360_rtrancl__induct2,axiom,
    ! [Ax: nat > nat,Ay: nat,Bx: nat > nat,By: nat,R3: set_Pr7047708467701517959at_nat,P2: ( nat > nat ) > nat > $o] :
      ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ Bx @ By ) ) @ ( transi3422749762518297843at_nat @ R3 ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A4: nat > nat,B3: nat,Aa: nat > nat,Ba: nat] :
              ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ A4 @ B3 ) ) @ ( transi3422749762518297843at_nat @ R3 ) )
             => ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ A4 @ B3 ) @ ( produc72220940542539688at_nat @ Aa @ Ba ) ) @ R3 )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_361_trancl__induct2,axiom,
    ! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R3: set_Pr2972776593051762503rm_a_b,P2: term_a_b > term_a_b > $o] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
     => ( ! [A4: term_a_b,B3: term_a_b] :
            ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) ) @ R3 )
           => ( P2 @ A4 @ B3 ) )
       => ( ! [A4: term_a_b,B3: term_a_b,Aa: term_a_b,Ba: term_a_b] :
              ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R3 )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_362_trancl__induct2,axiom,
    ! [Ax: a,Ay: nat,Bx: a,By: nat,R3: set_Pr1811044260758604347_a_nat,P2: a > nat > $o] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ Bx @ By ) ) @ ( transi6805586507763294364_a_nat @ R3 ) )
     => ( ! [A4: a,B3: nat] :
            ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ A4 @ B3 ) ) @ R3 )
           => ( P2 @ A4 @ B3 ) )
       => ( ! [A4: a,B3: nat,Aa: a,Ba: nat] :
              ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ A4 @ B3 ) ) @ ( transi6805586507763294364_a_nat @ R3 ) )
             => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ A4 @ B3 ) @ ( product_Pair_a_nat @ Aa @ Ba ) ) @ R3 )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_363_trancl__induct2,axiom,
    ! [Ax: nat > nat,Ay: nat,Bx: nat > nat,By: nat,R3: set_Pr7047708467701517959at_nat,P2: ( nat > nat ) > nat > $o] :
      ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ Bx @ By ) ) @ ( transi3146596849628481425at_nat @ R3 ) )
     => ( ! [A4: nat > nat,B3: nat] :
            ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ A4 @ B3 ) ) @ R3 )
           => ( P2 @ A4 @ B3 ) )
       => ( ! [A4: nat > nat,B3: nat,Aa: nat > nat,Ba: nat] :
              ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ A4 @ B3 ) ) @ ( transi3146596849628481425at_nat @ R3 ) )
             => ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ A4 @ B3 ) @ ( produc72220940542539688at_nat @ Aa @ Ba ) ) @ R3 )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_364_rtrancl_Ocases,axiom,
    ! [A1: term_a_b,A22: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
     => ( ( A22 != A1 )
       => ~ ! [B3: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B3 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A22 ) @ R3 ) ) ) ) ).

% rtrancl.cases
thf(fact_365_rtrancl_Osimps,axiom,
    ! [A1: term_a_b,A22: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
      = ( ? [A5: term_a_b] :
            ( ( A1 = A5 )
            & ( A22 = A5 ) )
        | ? [A5: term_a_b,B4: term_a_b,C5: term_a_b] :
            ( ( A1 = A5 )
            & ( A22 = C5 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B4 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C5 ) @ R3 ) ) ) ) ).

% rtrancl.simps
thf(fact_366_rtrancl_Ortrancl__refl,axiom,
    ! [A: term_a_b,R3: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ).

% rtrancl.rtrancl_refl
thf(fact_367_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ R3 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ) ).

% rtrancl.rtrancl_into_rtrancl
thf(fact_368_rtranclE,axiom,
    ! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
     => ( ( A != B )
       => ~ ! [Y2: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y2 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ B ) @ R3 ) ) ) ) ).

% rtranclE
thf(fact_369_rtrancl__trans,axiom,
    ! [X2: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b,Z: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ) ).

% rtrancl_trans
thf(fact_370_rtrancl__induct,axiom,
    ! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
     => ( ( P2 @ A )
       => ( ! [Y2: term_a_b,Z3: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y2 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z3 ) @ R3 )
               => ( ( P2 @ Y2 )
                 => ( P2 @ Z3 ) ) ) )
         => ( P2 @ B ) ) ) ) ).

% rtrancl_induct
thf(fact_371_converse__rtranclE,axiom,
    ! [X2: term_a_b,Z: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
     => ( ( X2 != Z )
       => ~ ! [Y2: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y2 ) @ R3 )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ) ) ).

% converse_rtranclE
thf(fact_372_converse__rtrancl__induct,axiom,
    ! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
     => ( ( P2 @ B )
       => ( ! [Y2: term_a_b,Z3: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z3 ) @ R3 )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ B ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
               => ( ( P2 @ Z3 )
                 => ( P2 @ Y2 ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_rtrancl_induct
thf(fact_373_converse__rtrancl__into__rtrancl,axiom,
    ! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ) ).

% converse_rtrancl_into_rtrancl
thf(fact_374_trancl_Ocases,axiom,
    ! [A1: term_a_b,A22: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
     => ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ R3 )
       => ~ ! [B3: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B3 ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A22 ) @ R3 ) ) ) ) ).

% trancl.cases
thf(fact_375_trancl_Osimps,axiom,
    ! [A1: term_a_b,A22: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
      = ( ? [A5: term_a_b,B4: term_a_b] :
            ( ( A1 = A5 )
            & ( A22 = B4 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B4 ) @ R3 ) )
        | ? [A5: term_a_b,B4: term_a_b,C5: term_a_b] :
            ( ( A1 = A5 )
            & ( A22 = C5 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B4 ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C5 ) @ R3 ) ) ) ) ).

% trancl.simps
thf(fact_376_trancl_Or__into__trancl,axiom,
    ! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ).

% trancl.r_into_trancl
thf(fact_377_tranclE,axiom,
    ! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
     => ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
       => ~ ! [C6: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C6 ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C6 @ B ) @ R3 ) ) ) ) ).

% tranclE
thf(fact_378_trancl__trans,axiom,
    ! [X2: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b,Z: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ).

% trancl_trans
thf(fact_379_trancl__induct,axiom,
    ! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
     => ( ! [Y2: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y2 ) @ R3 )
           => ( P2 @ Y2 ) )
       => ( ! [Y2: term_a_b,Z3: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y2 ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z3 ) @ R3 )
               => ( ( P2 @ Y2 )
                 => ( P2 @ Z3 ) ) ) )
         => ( P2 @ B ) ) ) ) ).

% trancl_induct
thf(fact_380_r__r__into__trancl,axiom,
    ! [A: term_a_b,B: term_a_b,R5: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R5 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ R5 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7922773638565587891rm_a_b @ R5 ) ) ) ) ).

% r_r_into_trancl
thf(fact_381_converse__tranclE,axiom,
    ! [X2: term_a_b,Z: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
     => ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z ) @ R3 )
       => ~ ! [Y2: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y2 ) @ R3 )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ) ).

% converse_tranclE
thf(fact_382_irrefl__trancl__rD,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,X2: term_a_b,Y: term_a_b] :
      ( ! [X: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ R3 )
       => ( X2 != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_383_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ R3 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_384_trancl__into__trancl2,axiom,
    ! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ).

% trancl_into_trancl2
thf(fact_385_trancl__trans__induct,axiom,
    ! [X2: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b,P2: term_a_b > term_a_b > $o] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
     => ( ! [X: term_a_b,Y2: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ R3 )
           => ( P2 @ X @ Y2 ) )
       => ( ! [X: term_a_b,Y2: term_a_b,Z3: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
             => ( ( P2 @ X @ Y2 )
               => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z3 ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
                 => ( ( P2 @ Y2 @ Z3 )
                   => ( P2 @ X @ Z3 ) ) ) ) )
         => ( P2 @ X2 @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_386_converse__trancl__induct,axiom,
    ! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
     => ( ! [Y2: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ B ) @ R3 )
           => ( P2 @ Y2 ) )
       => ( ! [Y2: term_a_b,Z3: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z3 ) @ R3 )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ B ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
               => ( ( P2 @ Z3 )
                 => ( P2 @ Y2 ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_387_trancl__rtrancl__trancl,axiom,
    ! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ).

% trancl_rtrancl_trancl
thf(fact_388_rtrancl__trancl__trancl,axiom,
    ! [X2: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b,Z: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ).

% rtrancl_trancl_trancl
thf(fact_389_rtrancl__into__trancl2,axiom,
    ! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ).

% rtrancl_into_trancl2
thf(fact_390_rsteps__eq__relcomp__srsteps__eq__relcompI,axiom,
    ! [R4: set_Pr4386577575007340137rm_a_b,F4: set_Pr4934435412358123699_a_nat,S5: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R4 ) @ F4 )
     => ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ S5 ) @ F4 )
       => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F4 )
         => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F4 )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ R4 ) ) @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ S5 ) ) ) )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ S5 ) ) ) ) ) ) ) ) ) ) ).

% rsteps_eq_relcomp_srsteps_eq_relcompI
thf(fact_391_srsteps__ctxt__closed,axiom,
    ! [C3: subterm_and_ctxt_a_b,F4: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ ( term_funas_ctxt_a_b @ C3 ) @ F4 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ S ) @ ( subter2376574525758040790rm_a_b @ C3 @ T ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) ) ) ) ).

% srsteps_ctxt_closed
thf(fact_392_srsteps__eq__ctxt__closed,axiom,
    ! [C3: subterm_and_ctxt_a_b,F4: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ ( term_funas_ctxt_a_b @ C3 ) @ F4 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ S ) @ ( subter2376574525758040790rm_a_b @ C3 @ T ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) ) ) ) ).

% srsteps_eq_ctxt_closed
thf(fact_393_srsteps__eq__subst__relcomp__closed,axiom,
    ! [S: term_a_b,T: term_a_b,F4: set_Pr4934435412358123699_a_nat,R4: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ S5 ) ) ) ) )
     => ( ! [X: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X ) ) @ F4 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ S5 ) ) ) ) ) ) ) ).

% srsteps_eq_subst_relcomp_closed
thf(fact_394_srstep__ctxt__closed,axiom,
    ! [C3: subterm_and_ctxt_a_b,F4: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ ( term_funas_ctxt_a_b @ C3 ) @ F4 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ S ) @ ( subter2376574525758040790rm_a_b @ C3 @ T ) ) @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) ) ) ).

% srstep_ctxt_closed
thf(fact_395_relcompEpair,axiom,
    ! [A: term_a_b,C: term_a_b,R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( relcom370159955682700863rm_a_b @ R3 @ S ) )
     => ~ ! [B3: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ R3 )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_396_relcompEpair,axiom,
    ! [A: a,C: nat,R3: set_Product_prod_a_a,S: set_Pr4934435412358123699_a_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ C ) @ ( relcomp_a_a_nat @ R3 @ S ) )
     => ~ ! [B3: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B3 ) @ R3 )
           => ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_397_relcompEpair,axiom,
    ! [A: a,C: nat,R3: set_Pr6841066879934740386at_nat,S: set_Pr9093778441882193744at_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ C ) @ ( relcom8552328690651882190at_nat @ R3 @ S ) )
     => ~ ! [B3: nat > nat] :
            ( ( member5463269792551873667at_nat @ ( produc7514922136058452582at_nat @ A @ B3 ) @ R3 )
           => ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_398_relcompEpair,axiom,
    ! [A: a,C: nat,R3: set_Pr4934435412358123699_a_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ C ) @ ( relcomp_a_nat_nat @ R3 @ S ) )
     => ~ ! [B3: nat] :
            ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ B3 ) @ R3 )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_399_relcompEpair,axiom,
    ! [A: nat > nat,C: nat,R3: set_Pr3254233319718257578_nat_a,S: set_Pr4934435412358123699_a_nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ C ) @ ( relcom1163327185954966926_a_nat @ R3 @ S ) )
     => ~ ! [B3: a] :
            ( ( member7754445317155421323_nat_a @ ( produc3941937540969104678_nat_a @ A @ B3 ) @ R3 )
           => ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_400_relcompEpair,axiom,
    ! [A: nat > nat,C: nat,R3: set_Pr7682762132356531903at_nat,S: set_Pr9093778441882193744at_nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ C ) @ ( relcom9054258243621454929at_nat @ R3 @ S ) )
     => ~ ! [B3: nat > nat] :
            ( ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A @ B3 ) @ R3 )
           => ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_401_relcompEpair,axiom,
    ! [A: nat > nat,C: nat,R3: set_Pr9093778441882193744at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ C ) @ ( relcom1859588284133128290at_nat @ R3 @ S ) )
     => ~ ! [B3: nat] :
            ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ B3 ) @ R3 )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_402_relcompE,axiom,
    ! [Xz: produc357393685978478089rm_a_b,R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ Xz @ ( relcom370159955682700863rm_a_b @ R3 @ S ) )
     => ~ ! [X: term_a_b,Y2: term_a_b,Z3: term_a_b] :
            ( ( Xz
              = ( produc7020197800436672577rm_a_b @ X @ Z3 ) )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ R3 )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z3 ) @ S ) ) ) ) ).

% relcompE
thf(fact_403_relcompE,axiom,
    ! [Xz: product_prod_a_nat,R3: set_Product_prod_a_a,S: set_Pr4934435412358123699_a_nat] :
      ( ( member5724188588386418708_a_nat @ Xz @ ( relcomp_a_a_nat @ R3 @ S ) )
     => ~ ! [X: a,Y2: a,Z3: nat] :
            ( ( Xz
              = ( product_Pair_a_nat @ X @ Z3 ) )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y2 ) @ R3 )
             => ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ Y2 @ Z3 ) @ S ) ) ) ) ).

% relcompE
thf(fact_404_relcompE,axiom,
    ! [Xz: product_prod_a_nat,R3: set_Pr6841066879934740386at_nat,S: set_Pr9093778441882193744at_nat] :
      ( ( member5724188588386418708_a_nat @ Xz @ ( relcom8552328690651882190at_nat @ R3 @ S ) )
     => ~ ! [X: a,Y2: nat > nat,Z3: nat] :
            ( ( Xz
              = ( product_Pair_a_nat @ X @ Z3 ) )
           => ( ( member5463269792551873667at_nat @ ( produc7514922136058452582at_nat @ X @ Y2 ) @ R3 )
             => ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ Y2 @ Z3 ) @ S ) ) ) ) ).

% relcompE
thf(fact_405_relcompE,axiom,
    ! [Xz: product_prod_a_nat,R3: set_Pr4934435412358123699_a_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member5724188588386418708_a_nat @ Xz @ ( relcomp_a_nat_nat @ R3 @ S ) )
     => ~ ! [X: a,Y2: nat,Z3: nat] :
            ( ( Xz
              = ( product_Pair_a_nat @ X @ Z3 ) )
           => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y2 ) @ R3 )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ S ) ) ) ) ).

% relcompE
thf(fact_406_relcompE,axiom,
    ! [Xz: produc8199716216217303280at_nat,R3: set_Pr3254233319718257578_nat_a,S: set_Pr4934435412358123699_a_nat] :
      ( ( member7226740684066999833at_nat @ Xz @ ( relcom1163327185954966926_a_nat @ R3 @ S ) )
     => ~ ! [X: nat > nat,Y2: a,Z3: nat] :
            ( ( Xz
              = ( produc72220940542539688at_nat @ X @ Z3 ) )
           => ( ( member7754445317155421323_nat_a @ ( produc3941937540969104678_nat_a @ X @ Y2 ) @ R3 )
             => ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ Y2 @ Z3 ) @ S ) ) ) ) ).

% relcompE
thf(fact_407_relcompE,axiom,
    ! [Xz: produc8199716216217303280at_nat,R3: set_Pr7682762132356531903at_nat,S: set_Pr9093778441882193744at_nat] :
      ( ( member7226740684066999833at_nat @ Xz @ ( relcom9054258243621454929at_nat @ R3 @ S ) )
     => ~ ! [X: nat > nat,Y2: nat > nat,Z3: nat] :
            ( ( Xz
              = ( produc72220940542539688at_nat @ X @ Z3 ) )
           => ( ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ X @ Y2 ) @ R3 )
             => ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ Y2 @ Z3 ) @ S ) ) ) ) ).

% relcompE
thf(fact_408_relcompE,axiom,
    ! [Xz: produc8199716216217303280at_nat,R3: set_Pr9093778441882193744at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member7226740684066999833at_nat @ Xz @ ( relcom1859588284133128290at_nat @ R3 @ S ) )
     => ~ ! [X: nat > nat,Y2: nat,Z3: nat] :
            ( ( Xz
              = ( produc72220940542539688at_nat @ X @ Z3 ) )
           => ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y2 ) @ R3 )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ S ) ) ) ) ).

% relcompE
thf(fact_409_relcomp_OrelcompI,axiom,
    ! [A: a,B: a,R3: set_Product_prod_a_a,C: nat,S: set_Pr4934435412358123699_a_nat] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R3 )
     => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B @ C ) @ S )
       => ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ C ) @ ( relcomp_a_a_nat @ R3 @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_410_relcomp_OrelcompI,axiom,
    ! [A: nat > nat,B: a,R3: set_Pr3254233319718257578_nat_a,C: nat,S: set_Pr4934435412358123699_a_nat] :
      ( ( member7754445317155421323_nat_a @ ( produc3941937540969104678_nat_a @ A @ B ) @ R3 )
     => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B @ C ) @ S )
       => ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ C ) @ ( relcom1163327185954966926_a_nat @ R3 @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_411_relcomp_OrelcompI,axiom,
    ! [A: a,B: nat > nat,R3: set_Pr6841066879934740386at_nat,C: nat,S: set_Pr9093778441882193744at_nat] :
      ( ( member5463269792551873667at_nat @ ( produc7514922136058452582at_nat @ A @ B ) @ R3 )
     => ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B @ C ) @ S )
       => ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ C ) @ ( relcom8552328690651882190at_nat @ R3 @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_412_relcomp_OrelcompI,axiom,
    ! [A: nat > nat,B: nat > nat,R3: set_Pr7682762132356531903at_nat,C: nat,S: set_Pr9093778441882193744at_nat] :
      ( ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A @ B ) @ R3 )
     => ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B @ C ) @ S )
       => ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ C ) @ ( relcom9054258243621454929at_nat @ R3 @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_413_relcomp_OrelcompI,axiom,
    ! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,C: term_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ S )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( relcom370159955682700863rm_a_b @ R3 @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_414_relcomp_OrelcompI,axiom,
    ! [A: a,B: nat,R3: set_Pr4934435412358123699_a_nat,C: nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ B ) @ R3 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ S )
       => ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ C ) @ ( relcomp_a_nat_nat @ R3 @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_415_relcomp_OrelcompI,axiom,
    ! [A: nat > nat,B: nat,R3: set_Pr9093778441882193744at_nat,C: nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ B ) @ R3 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ S )
       => ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ C ) @ ( relcom1859588284133128290at_nat @ R3 @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_416_relcomp_Osimps,axiom,
    ! [A1: term_a_b,A22: term_a_b,R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( relcom370159955682700863rm_a_b @ R3 @ S ) )
      = ( ? [A5: term_a_b,B4: term_a_b,C5: term_a_b] :
            ( ( A1 = A5 )
            & ( A22 = C5 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B4 ) @ R3 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C5 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_417_relcomp_Osimps,axiom,
    ! [A1: a,A22: nat,R3: set_Product_prod_a_a,S: set_Pr4934435412358123699_a_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcomp_a_a_nat @ R3 @ S ) )
      = ( ? [A5: a,B4: a,C5: nat] :
            ( ( A1 = A5 )
            & ( A22 = C5 )
            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A5 @ B4 ) @ R3 )
            & ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B4 @ C5 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_418_relcomp_Osimps,axiom,
    ! [A1: a,A22: nat,R3: set_Pr6841066879934740386at_nat,S: set_Pr9093778441882193744at_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcom8552328690651882190at_nat @ R3 @ S ) )
      = ( ? [A5: a,B4: nat > nat,C5: nat] :
            ( ( A1 = A5 )
            & ( A22 = C5 )
            & ( member5463269792551873667at_nat @ ( produc7514922136058452582at_nat @ A5 @ B4 ) @ R3 )
            & ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B4 @ C5 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_419_relcomp_Osimps,axiom,
    ! [A1: a,A22: nat,R3: set_Pr4934435412358123699_a_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcomp_a_nat_nat @ R3 @ S ) )
      = ( ? [A5: a,B4: nat,C5: nat] :
            ( ( A1 = A5 )
            & ( A22 = C5 )
            & ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A5 @ B4 ) @ R3 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C5 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_420_relcomp_Osimps,axiom,
    ! [A1: nat > nat,A22: nat,R3: set_Pr3254233319718257578_nat_a,S: set_Pr4934435412358123699_a_nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A1 @ A22 ) @ ( relcom1163327185954966926_a_nat @ R3 @ S ) )
      = ( ? [A5: nat > nat,B4: a,C5: nat] :
            ( ( A1 = A5 )
            & ( A22 = C5 )
            & ( member7754445317155421323_nat_a @ ( produc3941937540969104678_nat_a @ A5 @ B4 ) @ R3 )
            & ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B4 @ C5 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_421_relcomp_Osimps,axiom,
    ! [A1: nat > nat,A22: nat,R3: set_Pr7682762132356531903at_nat,S: set_Pr9093778441882193744at_nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A1 @ A22 ) @ ( relcom9054258243621454929at_nat @ R3 @ S ) )
      = ( ? [A5: nat > nat,B4: nat > nat,C5: nat] :
            ( ( A1 = A5 )
            & ( A22 = C5 )
            & ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A5 @ B4 ) @ R3 )
            & ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B4 @ C5 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_422_relcomp_Osimps,axiom,
    ! [A1: nat > nat,A22: nat,R3: set_Pr9093778441882193744at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A1 @ A22 ) @ ( relcom1859588284133128290at_nat @ R3 @ S ) )
      = ( ? [A5: nat > nat,B4: nat,C5: nat] :
            ( ( A1 = A5 )
            & ( A22 = C5 )
            & ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A5 @ B4 ) @ R3 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C5 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_423_relcomp_Ocases,axiom,
    ! [A1: term_a_b,A22: term_a_b,R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( relcom370159955682700863rm_a_b @ R3 @ S ) )
     => ~ ! [B3: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B3 ) @ R3 )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_424_relcomp_Ocases,axiom,
    ! [A1: a,A22: nat,R3: set_Product_prod_a_a,S: set_Pr4934435412358123699_a_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcomp_a_a_nat @ R3 @ S ) )
     => ~ ! [B3: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ B3 ) @ R3 )
           => ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_425_relcomp_Ocases,axiom,
    ! [A1: a,A22: nat,R3: set_Pr6841066879934740386at_nat,S: set_Pr9093778441882193744at_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcom8552328690651882190at_nat @ R3 @ S ) )
     => ~ ! [B3: nat > nat] :
            ( ( member5463269792551873667at_nat @ ( produc7514922136058452582at_nat @ A1 @ B3 ) @ R3 )
           => ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_426_relcomp_Ocases,axiom,
    ! [A1: a,A22: nat,R3: set_Pr4934435412358123699_a_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcomp_a_nat_nat @ R3 @ S ) )
     => ~ ! [B3: nat] :
            ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ B3 ) @ R3 )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_427_relcomp_Ocases,axiom,
    ! [A1: nat > nat,A22: nat,R3: set_Pr3254233319718257578_nat_a,S: set_Pr4934435412358123699_a_nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A1 @ A22 ) @ ( relcom1163327185954966926_a_nat @ R3 @ S ) )
     => ~ ! [B3: a] :
            ( ( member7754445317155421323_nat_a @ ( produc3941937540969104678_nat_a @ A1 @ B3 ) @ R3 )
           => ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_428_relcomp_Ocases,axiom,
    ! [A1: nat > nat,A22: nat,R3: set_Pr7682762132356531903at_nat,S: set_Pr9093778441882193744at_nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A1 @ A22 ) @ ( relcom9054258243621454929at_nat @ R3 @ S ) )
     => ~ ! [B3: nat > nat] :
            ( ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A1 @ B3 ) @ R3 )
           => ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_429_relcomp_Ocases,axiom,
    ! [A1: nat > nat,A22: nat,R3: set_Pr9093778441882193744at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A1 @ A22 ) @ ( relcom1859588284133128290at_nat @ R3 @ S ) )
     => ~ ! [B3: nat] :
            ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A1 @ B3 ) @ R3 )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_430_srsteps__with__root__step__def,axiom,
    ( srstep7844470518422762656ep_a_b
    = ( ^ [F5: set_Pr4934435412358123699_a_nat,R6: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F5 @ ( rstep_a_b @ R6 ) ) ) @ ( relcom370159955682700863rm_a_b @ ( sig_step_a_b @ F5 @ ( rrstep_a_b @ R6 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F5 @ ( rstep_a_b @ R6 ) ) ) ) ) ) ) ).

% srsteps_with_root_step_def
thf(fact_431_non__strict__into__strict,axiom,
    ! [NS: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S6 ) @ S6 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ NS ) @ S6 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ S6 ) ) ) ).

% non_strict_into_strict
thf(fact_432_non__strict__into__strict,axiom,
    ! [NS: set_Product_prod_a_a,S6: set_Pr4934435412358123699_a_nat,S: a,T: nat] :
      ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ NS @ S6 ) @ S6 )
     => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T ) @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ NS ) @ S6 ) )
       => ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T ) @ S6 ) ) ) ).

% non_strict_into_strict
thf(fact_433_non__strict__into__strict,axiom,
    ! [NS: set_Pr7682762132356531903at_nat,S6: set_Pr9093778441882193744at_nat,S: nat > nat,T: nat] :
      ( ( ord_le3678578370064672496at_nat @ ( relcom9054258243621454929at_nat @ NS @ S6 ) @ S6 )
     => ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ S @ T ) @ ( relcom9054258243621454929at_nat @ ( transi3031656534272781564at_nat @ NS ) @ S6 ) )
       => ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ S @ T ) @ S6 ) ) ) ).

% non_strict_into_strict
thf(fact_434_steps__map,axiom,
    ! [P2: term_a_b > $o,Q5: set_Pr4386577575007340137rm_a_b > $o,F2: term_a_b > term_a_b,G2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,T: term_a_b,R5: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,U: term_a_b] :
      ( ! [T2: term_a_b,U2: term_a_b,R7: set_Pr4386577575007340137rm_a_b] :
          ( ( P2 @ T2 )
         => ( ( Q5 @ R7 )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ U2 ) @ R7 )
             => ( ( P2 @ U2 )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T2 ) @ ( F2 @ U2 ) ) @ ( G2 @ R7 ) ) ) ) ) )
     => ( ( P2 @ T )
       => ( ( Q5 @ R5 )
         => ( ( Q5 @ S6 )
           => ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi7742714808557438673rm_a_b @ ( G2 @ R5 ) ) ) )
              & ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R5 ) @ ( relcom370159955682700863rm_a_b @ S6 @ ( transi7742714808557438673rm_a_b @ R5 ) ) ) )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( G2 @ R5 ) ) @ ( relcom370159955682700863rm_a_b @ ( G2 @ S6 ) @ ( transi7742714808557438673rm_a_b @ ( G2 @ R5 ) ) ) ) ) ) ) ) ) ) ) ).

% steps_map
thf(fact_435_sig__steps__join__ctxt__closed,axiom,
    ! [C3: subterm_and_ctxt_a_b,F4: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le8666007276011122963_a_nat @ ( term_funas_ctxt_a_b @ C3 ) @ F4 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( abstra4096080454567261402rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ S ) @ ( subter2376574525758040790rm_a_b @ C3 @ T ) ) @ ( abstra4096080454567261402rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) ) ) ) ).

% sig_steps_join_ctxt_closed
thf(fact_436_trancl__map,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b,S: set_Pr4386577575007340137rm_a_b,X2: term_a_b,Y: term_a_b] :
      ( ! [X: term_a_b,Y2: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ R3 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X ) @ ( F2 @ Y2 ) ) @ S ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X2 ) @ ( F2 @ Y ) ) @ ( transi7922773638565587891rm_a_b @ S ) ) ) ) ).

% trancl_map
thf(fact_437_relcomp3__I,axiom,
    ! [T: nat,U: a,A2: set_Pr4193341848836149977_nat_a,S: a,B5: set_Pr4934435412358123699_a_nat,V4: nat] :
      ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ T @ U ) @ A2 )
     => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T ) @ B5 )
       => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ U @ V4 ) @ B5 )
         => ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ V4 ) @ ( relcomp_a_nat_nat @ B5 @ ( relcomp_nat_a_nat @ A2 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_438_relcomp3__I,axiom,
    ! [T: nat,U: nat > nat,A2: set_Pr6370437063884598352at_nat,S: nat > nat,B5: set_Pr9093778441882193744at_nat,V4: nat] :
      ( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ T @ U ) @ A2 )
     => ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ S @ T ) @ B5 )
       => ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ U @ V4 ) @ B5 )
         => ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ S @ V4 ) @ ( relcom1859588284133128290at_nat @ B5 @ ( relcom6405311860805859042at_nat @ A2 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_439_relcomp3__I,axiom,
    ! [T: term_a_b,U: term_a_b,A2: set_Pr4386577575007340137rm_a_b,S: term_a_b,B5: set_Pr4386577575007340137rm_a_b,V4: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ A2 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ B5 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ V4 ) @ B5 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ V4 ) @ ( relcom370159955682700863rm_a_b @ B5 @ ( relcom370159955682700863rm_a_b @ A2 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_440_relcomp3__I,axiom,
    ! [T: a,U: nat,A2: set_Pr4934435412358123699_a_nat,S: nat,B5: set_Pr4193341848836149977_nat_a,V4: a] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ T @ U ) @ A2 )
     => ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ S @ T ) @ B5 )
       => ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ U @ V4 ) @ B5 )
         => ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ S @ V4 ) @ ( relcomp_nat_a_a @ B5 @ ( relcomp_a_nat_a @ A2 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_441_relcomp3__I,axiom,
    ! [T: nat > nat,U: nat,A2: set_Pr9093778441882193744at_nat,S: nat,B5: set_Pr6370437063884598352at_nat,V4: nat > nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ T @ U ) @ A2 )
     => ( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ S @ T ) @ B5 )
       => ( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ U @ V4 ) @ B5 )
         => ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ S @ V4 ) @ ( relcom896424912281157969at_nat @ B5 @ ( relcom7598182128343433937at_nat @ A2 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_442_joinI__right,axiom,
    ! [B: term_a_b,A: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A2 ) ) ) ).

% joinI_right
thf(fact_443_joinI__left,axiom,
    ! [A: term_a_b,B: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A2 ) ) ) ).

% joinI_left
thf(fact_444_joinI,axiom,
    ! [A: term_a_b,C: term_a_b,A2: set_Pr4386577575007340137rm_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A2 ) ) ) ) ).

% joinI
thf(fact_445_join__sym,axiom,
    ! [S: term_a_b,T: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( abstra4096080454567261402rm_a_b @ A2 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ ( abstra4096080454567261402rm_a_b @ A2 ) ) ) ).

% join_sym
thf(fact_446_rtrancl__join__join,axiom,
    ! [A: term_a_b,B: term_a_b,A2: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( abstra4096080454567261402rm_a_b @ A2 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( abstra4096080454567261402rm_a_b @ A2 ) ) ) ) ).

% rtrancl_join_join
thf(fact_447_join__rtrancl__join,axiom,
    ! [A: term_a_b,B: term_a_b,A2: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A2 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( abstra4096080454567261402rm_a_b @ A2 ) ) ) ) ).

% join_rtrancl_join
thf(fact_448_joinE,axiom,
    ! [A: term_a_b,B: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A2 ) )
     => ~ ! [C6: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C6 ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C6 ) @ ( transi7742714808557438673rm_a_b @ A2 ) ) ) ) ).

% joinE
thf(fact_449_joinD,axiom,
    ! [A: term_a_b,B: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A2 ) )
     => ? [C6: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C6 ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C6 ) @ ( transi7742714808557438673rm_a_b @ A2 ) ) ) ) ).

% joinD
thf(fact_450_CR__on__singletonI,axiom,
    ! [A: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ! [B3: term_a_b,C6: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C6 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C6 ) @ ( abstra4096080454567261402rm_a_b @ R3 ) ) ) )
     => ( abstra8448919418672941150rm_a_b @ R3 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) ) ).

% CR_on_singletonI
thf(fact_451_meetI,axiom,
    ! [A: term_a_b,B: term_a_b,A2: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( abstra1093061187967292567rm_a_b @ A2 ) ) ) ) ).

% meetI
thf(fact_452_subst__subst__domain,axiom,
    ! [T: term_a_b,X2: b] :
      ( ( ( T
          = ( var_b_a @ X2 ) )
       => ( ( subst_domain_b_a @ ( subst_b_a @ X2 @ T ) )
          = bot_bot_set_b ) )
      & ( ( T
         != ( var_b_a @ X2 ) )
       => ( ( subst_domain_b_a @ ( subst_b_a @ X2 @ T ) )
          = ( insert_b @ X2 @ bot_bot_set_b ) ) ) ) ).

% subst_subst_domain
thf(fact_453_subst__subst__range,axiom,
    ! [T: term_a_b,X2: b] :
      ( ( ( T
          = ( var_b_a @ X2 ) )
       => ( ( subst_range_b_a @ ( subst_b_a @ X2 @ T ) )
          = bot_bot_set_term_a_b ) )
      & ( ( T
         != ( var_b_a @ X2 ) )
       => ( ( subst_range_b_a @ ( subst_b_a @ X2 @ T ) )
          = ( insert_term_a_b @ T @ bot_bot_set_term_a_b ) ) ) ) ).

% subst_subst_range
thf(fact_454_subst__domain__Var,axiom,
    ( ( subst_domain_b_a @ var_b_a )
    = bot_bot_set_b ) ).

% subst_domain_Var
thf(fact_455_subst__range__Var,axiom,
    ( ( subst_range_b_a @ var_b_a )
    = bot_bot_set_term_a_b ) ).

% subst_range_Var
thf(fact_456_CR__onI,axiom,
    ! [A2: set_list_nat,R3: set_Pr3451248702717554689st_nat] :
      ( ! [A4: list_nat,B3: list_nat,C6: list_nat] :
          ( ( member_list_nat @ A4 @ A2 )
         => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A4 @ B3 ) @ ( transi5285580207609517981st_nat @ R3 ) )
           => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A4 @ C6 ) @ ( transi5285580207609517981st_nat @ R3 ) )
             => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B3 @ C6 ) @ ( abstra1638945853619340710st_nat @ R3 ) ) ) ) )
     => ( abstra5991784817725020458st_nat @ R3 @ A2 ) ) ).

% CR_onI
thf(fact_457_CR__onI,axiom,
    ! [A2: set_b,R3: set_Product_prod_b_b] :
      ( ! [A4: b,B3: b,C6: b] :
          ( ( member_b @ A4 @ A2 )
         => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A4 @ B3 ) @ ( transitive_rtrancl_b @ R3 ) )
           => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A4 @ C6 ) @ ( transitive_rtrancl_b @ R3 ) )
             => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B3 @ C6 ) @ ( abstract_join_b @ R3 ) ) ) ) )
     => ( abstract_CR_on_b @ R3 @ A2 ) ) ).

% CR_onI
thf(fact_458_CR__onI,axiom,
    ! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
      ( ! [A4: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,C6: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ A4 @ A2 )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B3 ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
           => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ C6 ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
             => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ C6 ) @ ( abstra4889682940153293507rm_a_b @ R3 ) ) ) ) )
     => ( abstra7359694574113289279rm_a_b @ R3 @ A2 ) ) ).

% CR_onI
thf(fact_459_CR__onI,axiom,
    ! [A2: set_Pr4934435412358123699_a_nat,R3: set_Pr1811044260758604347_a_nat] :
      ( ! [A4: product_prod_a_nat,B3: product_prod_a_nat,C6: product_prod_a_nat] :
          ( ( member5724188588386418708_a_nat @ A4 @ A2 )
         => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A4 @ B3 ) @ ( transi2726145917338391738_a_nat @ R3 ) )
           => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A4 @ C6 ) @ ( transi2726145917338391738_a_nat @ R3 ) )
             => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B3 @ C6 ) @ ( abstra881512052393830979_a_nat @ R3 ) ) ) ) )
     => ( abstra7964079670522473415_a_nat @ R3 @ A2 ) ) ).

% CR_onI
thf(fact_460_CR__onI,axiom,
    ! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ! [A4: term_a_b,B3: term_a_b,C6: term_a_b] :
          ( ( member_term_a_b @ A4 @ A2 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ C6 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C6 ) @ ( abstra4096080454567261402rm_a_b @ R3 ) ) ) ) )
     => ( abstra8448919418672941150rm_a_b @ R3 @ A2 ) ) ).

% CR_onI
thf(fact_461_CR__on__def,axiom,
    ( abstra8448919418672941150rm_a_b
    = ( ^ [R2: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
        ! [X3: term_a_b] :
          ( ( member_term_a_b @ X3 @ A6 )
         => ! [B4: term_a_b,C5: term_a_b] :
              ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ B4 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ C5 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C5 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) ) ) ).

% CR_on_def
thf(fact_462_CR__onE,axiom,
    ! [R3: set_Pr3451248702717554689st_nat,A2: set_list_nat,A: list_nat,B: list_nat,C: list_nat] :
      ( ( abstra5991784817725020458st_nat @ R3 @ A2 )
     => ( ( member_list_nat @ A @ A2 )
       => ( ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ C ) @ ( abstra1638945853619340710st_nat @ R3 ) )
         => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( transi5285580207609517981st_nat @ R3 ) )
           => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ C ) @ ( transi5285580207609517981st_nat @ R3 ) ) ) ) ) ) ).

% CR_onE
thf(fact_463_CR__onE,axiom,
    ! [R3: set_Product_prod_b_b,A2: set_b,A: b,B: b,C: b] :
      ( ( abstract_CR_on_b @ R3 @ A2 )
     => ( ( member_b @ A @ A2 )
       => ( ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ C ) @ ( abstract_join_b @ R3 ) )
         => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( transitive_rtrancl_b @ R3 ) )
           => ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ C ) @ ( transitive_rtrancl_b @ R3 ) ) ) ) ) ) ).

% CR_onE
thf(fact_464_CR__onE,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( abstra7359694574113289279rm_a_b @ R3 @ A2 )
     => ( ( member5869715511025134514rm_a_b @ A @ A2 )
       => ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ ( abstra4889682940153293507rm_a_b @ R3 ) )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
           => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( transi2615809358984392588rm_a_b @ R3 ) ) ) ) ) ) ).

% CR_onE
thf(fact_465_CR__onE,axiom,
    ! [R3: set_Pr1811044260758604347_a_nat,A2: set_Pr4934435412358123699_a_nat,A: product_prod_a_nat,B: product_prod_a_nat,C: product_prod_a_nat] :
      ( ( abstra7964079670522473415_a_nat @ R3 @ A2 )
     => ( ( member5724188588386418708_a_nat @ A @ A2 )
       => ( ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ C ) @ ( abstra881512052393830979_a_nat @ R3 ) )
         => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( transi2726145917338391738_a_nat @ R3 ) )
           => ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ C ) @ ( transi2726145917338391738_a_nat @ R3 ) ) ) ) ) ) ).

% CR_onE
thf(fact_466_CR__onE,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b,A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( abstra8448919418672941150rm_a_b @ R3 @ A2 )
     => ( ( member_term_a_b @ A @ A2 )
       => ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( abstra4096080454567261402rm_a_b @ R3 ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ) ) ) ).

% CR_onE
thf(fact_467_CR__onD,axiom,
    ! [R3: set_Pr3451248702717554689st_nat,A2: set_list_nat,A: list_nat,B: list_nat,C: list_nat] :
      ( ( abstra5991784817725020458st_nat @ R3 @ A2 )
     => ( ( member_list_nat @ A @ A2 )
       => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( transi5285580207609517981st_nat @ R3 ) )
         => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ C ) @ ( transi5285580207609517981st_nat @ R3 ) )
           => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ C ) @ ( abstra1638945853619340710st_nat @ R3 ) ) ) ) ) ) ).

% CR_onD
thf(fact_468_CR__onD,axiom,
    ! [R3: set_Product_prod_b_b,A2: set_b,A: b,B: b,C: b] :
      ( ( abstract_CR_on_b @ R3 @ A2 )
     => ( ( member_b @ A @ A2 )
       => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( transitive_rtrancl_b @ R3 ) )
         => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ C ) @ ( transitive_rtrancl_b @ R3 ) )
           => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ C ) @ ( abstract_join_b @ R3 ) ) ) ) ) ) ).

% CR_onD
thf(fact_469_CR__onD,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( abstra7359694574113289279rm_a_b @ R3 @ A2 )
     => ( ( member5869715511025134514rm_a_b @ A @ A2 )
       => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
           => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ ( abstra4889682940153293507rm_a_b @ R3 ) ) ) ) ) ) ).

% CR_onD
thf(fact_470_CR__onD,axiom,
    ! [R3: set_Pr1811044260758604347_a_nat,A2: set_Pr4934435412358123699_a_nat,A: product_prod_a_nat,B: product_prod_a_nat,C: product_prod_a_nat] :
      ( ( abstra7964079670522473415_a_nat @ R3 @ A2 )
     => ( ( member5724188588386418708_a_nat @ A @ A2 )
       => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( transi2726145917338391738_a_nat @ R3 ) )
         => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ C ) @ ( transi2726145917338391738_a_nat @ R3 ) )
           => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ C ) @ ( abstra881512052393830979_a_nat @ R3 ) ) ) ) ) ) ).

% CR_onD
thf(fact_471_CR__onD,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b,A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( abstra8448919418672941150rm_a_b @ R3 @ A2 )
     => ( ( member_term_a_b @ A @ A2 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( abstra4096080454567261402rm_a_b @ R3 ) ) ) ) ) ) ).

% CR_onD
thf(fact_472_range__varsE,axiom,
    ! [X2: b,Sigma: b > term_a_b] :
      ( ( member_b @ X2 @ ( range_vars_b_a @ Sigma ) )
     => ~ ! [T2: term_a_b] :
            ( ( member_b @ X2 @ ( vars_term_a_b @ T2 ) )
           => ~ ( member_term_a_b @ T2 @ ( subst_range_b_a @ Sigma ) ) ) ) ).

% range_varsE
thf(fact_473_meetD,axiom,
    ! [B: term_a_b,C: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( abstra1093061187967292567rm_a_b @ A2 ) )
     => ? [A4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ C ) @ ( transi7742714808557438673rm_a_b @ A2 ) ) ) ) ).

% meetD
thf(fact_474_meetE,axiom,
    ! [B: term_a_b,C: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( abstra1093061187967292567rm_a_b @ A2 ) )
     => ~ ! [A4: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ C ) @ ( transi7742714808557438673rm_a_b @ A2 ) ) ) ) ).

% meetE
thf(fact_475_term__to__sig__ctxt__apply,axiom,
    ! [F4: set_Pr4934435412358123699_a_nat,C3: subterm_and_ctxt_a_b,V4: b,S: term_a_b] :
      ( ( terms_8374513854926927137th_a_b @ F4 @ C3 )
     => ( ( terms_8519481630511763164ig_a_b @ F4 @ V4 @ ( subter2376574525758040790rm_a_b @ C3 @ S ) )
        = ( subter2376574525758040790rm_a_b @ ( terms_3799363064701517935xt_a_b @ F4 @ V4 @ C3 ) @ ( terms_8519481630511763164ig_a_b @ F4 @ V4 @ S ) ) ) ) ).

% term_to_sig_ctxt_apply
thf(fact_476_WCR__onI,axiom,
    ! [A2: set_list_nat,R3: set_Pr3451248702717554689st_nat] :
      ( ! [A4: list_nat,B3: list_nat,C6: list_nat] :
          ( ( member_list_nat @ A4 @ A2 )
         => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A4 @ B3 ) @ R3 )
           => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A4 @ C6 ) @ R3 )
             => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B3 @ C6 ) @ ( abstra1638945853619340710st_nat @ R3 ) ) ) ) )
     => ( abstra2569714506102940299st_nat @ R3 @ A2 ) ) ).

% WCR_onI
thf(fact_477_WCR__onI,axiom,
    ! [A2: set_b,R3: set_Product_prod_b_b] :
      ( ! [A4: b,B3: b,C6: b] :
          ( ( member_b @ A4 @ A2 )
         => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A4 @ B3 ) @ R3 )
           => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A4 @ C6 ) @ R3 )
             => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B3 @ C6 ) @ ( abstract_join_b @ R3 ) ) ) ) )
     => ( abstract_WCR_on_b @ R3 @ A2 ) ) ).

% WCR_onI
thf(fact_478_WCR__onI,axiom,
    ! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
      ( ! [A4: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,C6: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ A4 @ A2 )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B3 ) @ R3 )
           => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ C6 ) @ R3 )
             => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ C6 ) @ ( abstra4889682940153293507rm_a_b @ R3 ) ) ) ) )
     => ( abstra875157153669124638rm_a_b @ R3 @ A2 ) ) ).

% WCR_onI
thf(fact_479_WCR__onI,axiom,
    ! [A2: set_Pr4934435412358123699_a_nat,R3: set_Pr1811044260758604347_a_nat] :
      ( ! [A4: product_prod_a_nat,B3: product_prod_a_nat,C6: product_prod_a_nat] :
          ( ( member5724188588386418708_a_nat @ A4 @ A2 )
         => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A4 @ B3 ) @ R3 )
           => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A4 @ C6 ) @ R3 )
             => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B3 @ C6 ) @ ( abstra881512052393830979_a_nat @ R3 ) ) ) ) )
     => ( abstra6648775111202918568_a_nat @ R3 @ A2 ) ) ).

% WCR_onI
thf(fact_480_WCR__onI,axiom,
    ! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ! [A4: term_a_b,B3: term_a_b,C6: term_a_b] :
          ( ( member_term_a_b @ A4 @ A2 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ R3 )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ C6 ) @ R3 )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C6 ) @ ( abstra4096080454567261402rm_a_b @ R3 ) ) ) ) )
     => ( abstra5026849107050860991rm_a_b @ R3 @ A2 ) ) ).

% WCR_onI
thf(fact_481_subst__apply__term__subst__apply__term__eq__subst__apply__term__lhs,axiom,
    ! [Sigma: b > term_a_b,Delta: b > term_a_b,T: term_a_b] :
      ( ( ( inf_inf_set_b @ ( range_vars_b_a @ Sigma ) @ ( subst_domain_b_a @ Delta ) )
        = bot_bot_set_b )
     => ( ( ( inf_inf_set_b @ ( vars_term_a_b @ T ) @ ( subst_domain_b_a @ Delta ) )
          = bot_bot_set_b )
       => ( ( subst_7999470309526761004_a_b_b @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) @ Delta )
          = ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) ) ) ).

% subst_apply_term_subst_apply_term_eq_subst_apply_term_lhs
thf(fact_482_lists__empty,axiom,
    ( ( lists_term_a_b @ bot_bot_set_term_a_b )
    = ( insert_list_term_a_b @ nil_term_a_b @ bot_bo8941492048910247406rm_a_b ) ) ).

% lists_empty
thf(fact_483_lists__empty,axiom,
    ( ( lists_nat @ bot_bot_set_nat )
    = ( insert_list_nat @ nil_nat @ bot_bot_set_list_nat ) ) ).

% lists_empty
thf(fact_484_comp__rtrancl__trancl,axiom,
    ! [R5: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R5 @ S6 ) @ S6 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R5 @ S6 ) ) @ S6 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ S6 ) ) ) ) ).

% comp_rtrancl_trancl
thf(fact_485_append__in__lists__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat,A2: set_nat] :
      ( ( member_list_nat @ ( append_nat @ Xs @ Ys ) @ ( lists_nat @ A2 ) )
      = ( ( member_list_nat @ Xs @ ( lists_nat @ A2 ) )
        & ( member_list_nat @ Ys @ ( lists_nat @ A2 ) ) ) ) ).

% append_in_lists_conv
thf(fact_486_funas__ctxt__apply,axiom,
    ! [C3: subterm_and_ctxt_a_b,T: term_a_b] :
      ( ( term_funas_term_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ T ) )
      = ( sup_su459911885395995103_a_nat @ ( term_funas_ctxt_a_b @ C3 ) @ ( term_funas_term_a_b @ T ) ) ) ).

% funas_ctxt_apply
thf(fact_487_lists__IntI,axiom,
    ! [L2: list_nat,A2: set_nat,B5: set_nat] :
      ( ( member_list_nat @ L2 @ ( lists_nat @ A2 ) )
     => ( ( member_list_nat @ L2 @ ( lists_nat @ B5 ) )
       => ( member_list_nat @ L2 @ ( lists_nat @ ( inf_inf_set_nat @ A2 @ B5 ) ) ) ) ) ).

% lists_IntI
thf(fact_488_lists_ONil,axiom,
    ! [A2: set_term_a_b] : ( member_list_term_a_b @ nil_term_a_b @ ( lists_term_a_b @ A2 ) ) ).

% lists.Nil
thf(fact_489_lists_ONil,axiom,
    ! [A2: set_nat] : ( member_list_nat @ nil_nat @ ( lists_nat @ A2 ) ) ).

% lists.Nil
thf(fact_490_rtrancl__Un__separator__converseE,axiom,
    ! [A: term_a_b,B: term_a_b,P2: set_Pr4386577575007340137rm_a_b,Q5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P2 @ Q5 ) ) )
     => ( ! [X: term_a_b,Y2: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ B ) @ ( transi7742714808557438673rm_a_b @ P2 ) )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ X ) @ Q5 )
             => ( Y2 = X ) ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ P2 ) ) ) ) ).

% rtrancl_Un_separator_converseE
thf(fact_491_rtrancl__Un__separatorE,axiom,
    ! [A: term_a_b,B: term_a_b,P2: set_Pr4386577575007340137rm_a_b,Q5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P2 @ Q5 ) ) )
     => ( ! [X: term_a_b,Y2: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ X ) @ ( transi7742714808557438673rm_a_b @ P2 ) )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ Q5 )
             => ( X = Y2 ) ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ P2 ) ) ) ) ).

% rtrancl_Un_separatorE
thf(fact_492_right__comp__S,axiom,
    ! [X2: term_a_b,Y: term_a_b,S6: set_Pr4386577575007340137rm_a_b,NS: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( relcom370159955682700863rm_a_b @ S6 @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S6 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S6 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( relcom370159955682700863rm_a_b @ S6 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S6 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).

% right_comp_S
thf(fact_493_first__step__O,axiom,
    ! [C3: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,X2: term_a_b,Y: term_a_b] :
      ( ( C3
        = ( sup_su6776935440552674877rm_a_b @ A2 @ B5 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ C3 ) @ B5 ) )
       => ? [Y2: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y2 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A2 ) @ B5 ) ) ) ) ).

% first_step_O
thf(fact_494_first__step,axiom,
    ! [C3: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,X2: term_a_b,Y: term_a_b,Z: term_a_b] :
      ( ( C3
        = ( sup_su6776935440552674877rm_a_b @ A2 @ B5 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( transi7742714808557438673rm_a_b @ C3 ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ B5 )
         => ? [Y2: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y2 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A2 ) @ B5 ) ) ) ) ) ).

% first_step
thf(fact_495_firstStep,axiom,
    ! [L4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,R5: set_Pr4386577575007340137rm_a_b,X2: term_a_b,Y: term_a_b] :
      ( ( L4
        = ( sup_su6776935440552674877rm_a_b @ S6 @ R5 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( transi7742714808557438673rm_a_b @ L4 ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
          | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R5 ) @ ( relcom370159955682700863rm_a_b @ S6 @ ( transi7742714808557438673rm_a_b @ L4 ) ) ) ) ) ) ) ).

% firstStep
thf(fact_496_vars__term__disjoint__imp__unifier,axiom,
    ! [S: term_a_b,T: term_a_b,Sigma: b > term_a_b,Tau: b > term_a_b] :
      ( ( ( inf_inf_set_b @ ( vars_term_a_b @ S ) @ ( vars_term_a_b @ T ) )
        = bot_bot_set_b )
     => ( ( ( subst_7999470309526761004_a_b_b @ S @ Sigma )
          = ( subst_7999470309526761004_a_b_b @ T @ Tau ) )
       => ? [Mu: b > term_a_b] :
            ( ( subst_7999470309526761004_a_b_b @ S @ Mu )
            = ( subst_7999470309526761004_a_b_b @ T @ Mu ) ) ) ) ).

% vars_term_disjoint_imp_unifier
thf(fact_497_funas__term__replace__at__upper,axiom,
    ! [S: term_a_b,P: list_nat,T: term_a_b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( term_r6860082780075436317at_a_b @ S @ P @ T ) ) @ ( sup_su459911885395995103_a_nat @ ( term_funas_term_a_b @ S ) @ ( term_funas_term_a_b @ T ) ) ) ).

% funas_term_replace_at_upper
thf(fact_498_WCR__onD,axiom,
    ! [R3: set_Pr3451248702717554689st_nat,A2: set_list_nat,A: list_nat,B: list_nat,C: list_nat] :
      ( ( abstra2569714506102940299st_nat @ R3 @ A2 )
     => ( ( member_list_nat @ A @ A2 )
       => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ R3 )
         => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ C ) @ R3 )
           => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ C ) @ ( abstra1638945853619340710st_nat @ R3 ) ) ) ) ) ) ).

% WCR_onD
thf(fact_499_WCR__onD,axiom,
    ! [R3: set_Product_prod_b_b,A2: set_b,A: b,B: b,C: b] :
      ( ( abstract_WCR_on_b @ R3 @ A2 )
     => ( ( member_b @ A @ A2 )
       => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ R3 )
         => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ C ) @ R3 )
           => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ C ) @ ( abstract_join_b @ R3 ) ) ) ) ) ) ).

% WCR_onD
thf(fact_500_WCR__onD,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( abstra875157153669124638rm_a_b @ R3 @ A2 )
     => ( ( member5869715511025134514rm_a_b @ A @ A2 )
       => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ R3 )
           => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ ( abstra4889682940153293507rm_a_b @ R3 ) ) ) ) ) ) ).

% WCR_onD
thf(fact_501_WCR__onD,axiom,
    ! [R3: set_Pr1811044260758604347_a_nat,A2: set_Pr4934435412358123699_a_nat,A: product_prod_a_nat,B: product_prod_a_nat,C: product_prod_a_nat] :
      ( ( abstra6648775111202918568_a_nat @ R3 @ A2 )
     => ( ( member5724188588386418708_a_nat @ A @ A2 )
       => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ R3 )
         => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ C ) @ R3 )
           => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ C ) @ ( abstra881512052393830979_a_nat @ R3 ) ) ) ) ) ) ).

% WCR_onD
thf(fact_502_WCR__onD,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b,A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( abstra5026849107050860991rm_a_b @ R3 @ A2 )
     => ( ( member_term_a_b @ A @ A2 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ R3 )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( abstra4096080454567261402rm_a_b @ R3 ) ) ) ) ) ) ).

% WCR_onD
thf(fact_503_WCR__onE,axiom,
    ! [R3: set_Pr3451248702717554689st_nat,A2: set_list_nat,A: list_nat,B: list_nat,C: list_nat] :
      ( ( abstra2569714506102940299st_nat @ R3 @ A2 )
     => ( ( member_list_nat @ A @ A2 )
       => ( ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ C ) @ ( abstra1638945853619340710st_nat @ R3 ) )
         => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ R3 )
           => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ C ) @ R3 ) ) ) ) ) ).

% WCR_onE
thf(fact_504_WCR__onE,axiom,
    ! [R3: set_Product_prod_b_b,A2: set_b,A: b,B: b,C: b] :
      ( ( abstract_WCR_on_b @ R3 @ A2 )
     => ( ( member_b @ A @ A2 )
       => ( ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ C ) @ ( abstract_join_b @ R3 ) )
         => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ R3 )
           => ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ C ) @ R3 ) ) ) ) ) ).

% WCR_onE
thf(fact_505_WCR__onE,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( abstra875157153669124638rm_a_b @ R3 @ A2 )
     => ( ( member5869715511025134514rm_a_b @ A @ A2 )
       => ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ ( abstra4889682940153293507rm_a_b @ R3 ) )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 )
           => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ R3 ) ) ) ) ) ).

% WCR_onE
thf(fact_506_WCR__onE,axiom,
    ! [R3: set_Pr1811044260758604347_a_nat,A2: set_Pr4934435412358123699_a_nat,A: product_prod_a_nat,B: product_prod_a_nat,C: product_prod_a_nat] :
      ( ( abstra6648775111202918568_a_nat @ R3 @ A2 )
     => ( ( member5724188588386418708_a_nat @ A @ A2 )
       => ( ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ C ) @ ( abstra881512052393830979_a_nat @ R3 ) )
         => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ R3 )
           => ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ C ) @ R3 ) ) ) ) ) ).

% WCR_onE
thf(fact_507_WCR__onE,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b,A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( abstra5026849107050860991rm_a_b @ R3 @ A2 )
     => ( ( member_term_a_b @ A @ A2 )
       => ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( abstra4096080454567261402rm_a_b @ R3 ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ R3 ) ) ) ) ) ).

% WCR_onE
thf(fact_508_WCR__on__def,axiom,
    ( abstra5026849107050860991rm_a_b
    = ( ^ [R2: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
        ! [X3: term_a_b] :
          ( ( member_term_a_b @ X3 @ A6 )
         => ! [B4: term_a_b,C5: term_a_b] :
              ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ B4 ) @ R2 )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ C5 ) @ R2 ) )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C5 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) ) ) ).

% WCR_on_def
thf(fact_509_compatible__rtrancl__split,axiom,
    ! [NS: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,X2: term_a_b,Y: term_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S6 ) @ S6 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S6 ) ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S6 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S6 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).

% compatible_rtrancl_split
thf(fact_510_subst__apply__term__ident,axiom,
    ! [T: term_a_b,Sigma: b > term_a_b] :
      ( ( ( inf_inf_set_b @ ( vars_term_a_b @ T ) @ ( subst_domain_b_a @ Sigma ) )
        = bot_bot_set_b )
     => ( ( subst_7999470309526761004_a_b_b @ T @ Sigma )
        = T ) ) ).

% subst_apply_term_ident
thf(fact_511_term__to__sig__ctxt__apply_H,axiom,
    ! [F4: set_Pr4934435412358123699_a_nat,C3: subterm_and_ctxt_a_b,V4: b,S: term_a_b] :
      ( ~ ( terms_8374513854926927137th_a_b @ F4 @ C3 )
     => ( ( terms_8519481630511763164ig_a_b @ F4 @ V4 @ ( subter2376574525758040790rm_a_b @ C3 @ S ) )
        = ( terms_130083692264552600xt_a_b @ F4 @ V4 @ C3 ) ) ) ).

% term_to_sig_ctxt_apply'
thf(fact_512_sig__step__rsteps__eq__dist,axiom,
    ! [R4: set_Pr4386577575007340137rm_a_b,F4: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R4 ) @ F4 )
     => ( ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F4 @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R4 ) ) ) @ id_term_a_b )
        = ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) ) ) ).

% sig_step_rsteps_eq_dist
thf(fact_513_vars__term__subst__apply__term__subset,axiom,
    ! [T: term_a_b,Sigma: b > term_a_b] : ( ord_less_eq_set_b @ ( vars_term_a_b @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( sup_sup_set_b @ ( minus_minus_set_b @ ( vars_term_a_b @ T ) @ ( subst_domain_b_a @ Sigma ) ) @ ( range_vars_b_a @ Sigma ) ) ) ).

% vars_term_subst_apply_term_subset
thf(fact_514_vars__term__ctxt__apply,axiom,
    ! [C3: subterm_and_ctxt_a_b,T: term_a_b] :
      ( ( vars_term_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ T ) )
      = ( sup_sup_set_b @ ( subter6041326469635110383xt_a_b @ C3 ) @ ( vars_term_a_b @ T ) ) ) ).

% vars_term_ctxt_apply
thf(fact_515_term__to__sig_Osimps_I2_J,axiom,
    ! [F2: a,Ts2: list_term_a_b,F4: set_Pr4934435412358123699_a_nat,V4: b] :
      ( ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F2 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) ) @ F4 )
       => ( ( terms_8519481630511763164ig_a_b @ F4 @ V4 @ ( fun_a_b @ F2 @ Ts2 ) )
          = ( fun_a_b @ F2 @ ( map_te2328734355998148206rm_a_b @ ( terms_8519481630511763164ig_a_b @ F4 @ V4 ) @ Ts2 ) ) ) )
      & ( ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F2 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) ) @ F4 )
       => ( ( terms_8519481630511763164ig_a_b @ F4 @ V4 @ ( fun_a_b @ F2 @ Ts2 ) )
          = ( var_b_a @ V4 ) ) ) ) ).

% term_to_sig.simps(2)
thf(fact_516_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ A )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_517_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_518_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_519_map__is__Nil__conv,axiom,
    ! [F2: term_a_b > term_a_b,Xs: list_term_a_b] :
      ( ( ( map_te2328734355998148206rm_a_b @ F2 @ Xs )
        = nil_term_a_b )
      = ( Xs = nil_term_a_b ) ) ).

% map_is_Nil_conv
thf(fact_520_map__is__Nil__conv,axiom,
    ! [F2: nat > term_a_b,Xs: list_nat] :
      ( ( ( map_nat_term_a_b @ F2 @ Xs )
        = nil_term_a_b )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_521_map__is__Nil__conv,axiom,
    ! [F2: term_a_b > nat,Xs: list_term_a_b] :
      ( ( ( map_term_a_b_nat @ F2 @ Xs )
        = nil_nat )
      = ( Xs = nil_term_a_b ) ) ).

% map_is_Nil_conv
thf(fact_522_map__is__Nil__conv,axiom,
    ! [F2: nat > nat,Xs: list_nat] :
      ( ( ( map_nat_nat @ F2 @ Xs )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_523_Nil__is__map__conv,axiom,
    ! [F2: term_a_b > term_a_b,Xs: list_term_a_b] :
      ( ( nil_term_a_b
        = ( map_te2328734355998148206rm_a_b @ F2 @ Xs ) )
      = ( Xs = nil_term_a_b ) ) ).

% Nil_is_map_conv
thf(fact_524_Nil__is__map__conv,axiom,
    ! [F2: nat > term_a_b,Xs: list_nat] :
      ( ( nil_term_a_b
        = ( map_nat_term_a_b @ F2 @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_525_Nil__is__map__conv,axiom,
    ! [F2: term_a_b > nat,Xs: list_term_a_b] :
      ( ( nil_nat
        = ( map_term_a_b_nat @ F2 @ Xs ) )
      = ( Xs = nil_term_a_b ) ) ).

% Nil_is_map_conv
thf(fact_526_Nil__is__map__conv,axiom,
    ! [F2: nat > nat,Xs: list_nat] :
      ( ( nil_nat
        = ( map_nat_nat @ F2 @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_527_list_Omap__disc__iff,axiom,
    ! [F2: term_a_b > term_a_b,A: list_term_a_b] :
      ( ( ( map_te2328734355998148206rm_a_b @ F2 @ A )
        = nil_term_a_b )
      = ( A = nil_term_a_b ) ) ).

% list.map_disc_iff
thf(fact_528_list_Omap__disc__iff,axiom,
    ! [F2: nat > term_a_b,A: list_nat] :
      ( ( ( map_nat_term_a_b @ F2 @ A )
        = nil_term_a_b )
      = ( A = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_529_list_Omap__disc__iff,axiom,
    ! [F2: term_a_b > nat,A: list_term_a_b] :
      ( ( ( map_term_a_b_nat @ F2 @ A )
        = nil_nat )
      = ( A = nil_term_a_b ) ) ).

% list.map_disc_iff
thf(fact_530_list_Omap__disc__iff,axiom,
    ! [F2: nat > nat,A: list_nat] :
      ( ( ( map_nat_nat @ F2 @ A )
        = nil_nat )
      = ( A = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_531_map__append,axiom,
    ! [F2: nat > nat,Xs: list_nat,Ys: list_nat] :
      ( ( map_nat_nat @ F2 @ ( append_nat @ Xs @ Ys ) )
      = ( append_nat @ ( map_nat_nat @ F2 @ Xs ) @ ( map_nat_nat @ F2 @ Ys ) ) ) ).

% map_append
thf(fact_532_IdI,axiom,
    ! [A: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ id_term_a_b ) ).

% IdI
thf(fact_533_pair__in__Id__conv,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ id_term_a_b )
      = ( A = B ) ) ).

% pair_in_Id_conv
thf(fact_534_diff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_535_append__eq__map__conv,axiom,
    ! [Ys: list_nat,Zs: list_nat,F2: nat > nat,Xs: list_nat] :
      ( ( ( append_nat @ Ys @ Zs )
        = ( map_nat_nat @ F2 @ Xs ) )
      = ( ? [Us: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us @ Vs2 ) )
            & ( Ys
              = ( map_nat_nat @ F2 @ Us ) )
            & ( Zs
              = ( map_nat_nat @ F2 @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_536_map__eq__append__conv,axiom,
    ! [F2: nat > nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( map_nat_nat @ F2 @ Xs )
        = ( append_nat @ Ys @ Zs ) )
      = ( ? [Us: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us @ Vs2 ) )
            & ( Ys
              = ( map_nat_nat @ F2 @ Us ) )
            & ( Zs
              = ( map_nat_nat @ F2 @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_537_list_Osimps_I8_J,axiom,
    ! [F2: term_a_b > term_a_b] :
      ( ( map_te2328734355998148206rm_a_b @ F2 @ nil_term_a_b )
      = nil_term_a_b ) ).

% list.simps(8)
thf(fact_538_list_Osimps_I8_J,axiom,
    ! [F2: term_a_b > nat] :
      ( ( map_term_a_b_nat @ F2 @ nil_term_a_b )
      = nil_nat ) ).

% list.simps(8)
thf(fact_539_list_Osimps_I8_J,axiom,
    ! [F2: nat > term_a_b] :
      ( ( map_nat_term_a_b @ F2 @ nil_nat )
      = nil_term_a_b ) ).

% list.simps(8)
thf(fact_540_list_Osimps_I8_J,axiom,
    ! [F2: nat > nat] :
      ( ( map_nat_nat @ F2 @ nil_nat )
      = nil_nat ) ).

% list.simps(8)
thf(fact_541_IdE,axiom,
    ! [P: produc357393685978478089rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ P @ id_term_a_b )
     => ~ ! [X: term_a_b] :
            ( P
           != ( produc7020197800436672577rm_a_b @ X @ X ) ) ) ).

% IdE
thf(fact_542_rtrancl__diff__decomp,axiom,
    ! [X2: term_a_b,Y: term_a_b,A2: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( minus_5192120951422937424rm_a_b @ ( transi7742714808557438673rm_a_b @ A2 ) @ ( transi7742714808557438673rm_a_b @ B5 ) ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A2 ) @ ( relcom370159955682700863rm_a_b @ ( minus_5192120951422937424rm_a_b @ A2 @ B5 ) @ ( transi7742714808557438673rm_a_b @ A2 ) ) ) ) ) ).

% rtrancl_diff_decomp
thf(fact_543_term__to__sig_Oelims,axiom,
    ! [X2: set_Pr4934435412358123699_a_nat,Xa: b,Xb: term_a_b,Y: term_a_b] :
      ( ( ( terms_8519481630511763164ig_a_b @ X2 @ Xa @ Xb )
        = Y )
     => ( ! [X: b] :
            ( ( Xb
              = ( var_b_a @ X ) )
           => ( Y
             != ( var_b_a @ X ) ) )
       => ~ ! [F: a,Ts: list_term_a_b] :
              ( ( Xb
                = ( fun_a_b @ F @ Ts ) )
             => ~ ( ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F @ ( size_s8906293707977694520rm_a_b @ Ts ) ) @ X2 )
                   => ( Y
                      = ( fun_a_b @ F @ ( map_te2328734355998148206rm_a_b @ ( terms_8519481630511763164ig_a_b @ X2 @ Xa ) @ Ts ) ) ) )
                  & ( ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F @ ( size_s8906293707977694520rm_a_b @ Ts ) ) @ X2 )
                   => ( Y
                      = ( var_b_a @ Xa ) ) ) ) ) ) ) ).

% term_to_sig.elims
thf(fact_544_strongly__confluent__on__E11,axiom,
    ! [R3: set_Pr3451248702717554689st_nat,A2: set_list_nat,X2: list_nat,Y: list_nat,Z: list_nat] :
      ( ( abstra3206962040680252735st_nat @ R3 @ A2 )
     => ( ( member_list_nat @ X2 @ A2 )
       => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X2 @ Y ) @ R3 )
         => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X2 @ Z ) @ R3 )
           => ? [U2: list_nat] :
                ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y @ U2 ) @ ( transi5285580207609517981st_nat @ R3 ) )
                & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Z @ U2 ) @ ( sup_su5841606568262889429st_nat @ R3 @ id_list_nat ) ) ) ) ) ) ) ).

% strongly_confluent_on_E11
thf(fact_545_strongly__confluent__on__E11,axiom,
    ! [R3: set_Product_prod_b_b,A2: set_b,X2: b,Y: b,Z: b] :
      ( ( abstra4021214168631544544t_on_b @ R3 @ A2 )
     => ( ( member_b @ X2 @ A2 )
       => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X2 @ Y ) @ R3 )
         => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X2 @ Z ) @ R3 )
           => ? [U2: b] :
                ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y @ U2 ) @ ( transitive_rtrancl_b @ R3 ) )
                & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Z @ U2 ) @ ( sup_su2483643821041016987od_b_b @ R3 @ id_b ) ) ) ) ) ) ) ).

% strongly_confluent_on_E11
thf(fact_546_strongly__confluent__on__E11,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b,X2: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
      ( ( abstra837640964651051114rm_a_b @ R3 @ A2 )
     => ( ( member5869715511025134514rm_a_b @ X2 @ A2 )
       => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ Y ) @ R3 )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ Z ) @ R3 )
           => ? [U2: produc357393685978478089rm_a_b] :
                ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ U2 ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
                & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z @ U2 ) @ ( sup_su1845815533690797339rm_a_b @ R3 @ id_Pro3488709451141449061rm_a_b ) ) ) ) ) ) ) ).

% strongly_confluent_on_E11
thf(fact_547_strongly__confluent__on__E11,axiom,
    ! [R3: set_Pr1811044260758604347_a_nat,A2: set_Pr4934435412358123699_a_nat,X2: product_prod_a_nat,Y: product_prod_a_nat,Z: product_prod_a_nat] :
      ( ( abstra5315942894939654492_a_nat @ R3 @ A2 )
     => ( ( member5724188588386418708_a_nat @ X2 @ A2 )
       => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X2 @ Y ) @ R3 )
         => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X2 @ Z ) @ R3 )
           => ? [U2: product_prod_a_nat] :
                ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y @ U2 ) @ ( transi2726145917338391738_a_nat @ R3 ) )
                & ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Z @ U2 ) @ ( sup_su958340139570419215_a_nat @ R3 @ id_Pro5207055338379199009_a_nat ) ) ) ) ) ) ) ).

% strongly_confluent_on_E11
thf(fact_548_strongly__confluent__on__E11,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b,X2: term_a_b,Y: term_a_b,Z: term_a_b] :
      ( ( abstra5664096641628173427rm_a_b @ R3 @ A2 )
     => ( ( member_term_a_b @ X2 @ A2 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ R3 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z ) @ R3 )
           => ? [U2: term_a_b] :
                ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ U2 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ U2 ) @ ( sup_su6776935440552674877rm_a_b @ R3 @ id_term_a_b ) ) ) ) ) ) ) ).

% strongly_confluent_on_E11
thf(fact_549_strongly__confluent__on__def,axiom,
    ( abstra5664096641628173427rm_a_b
    = ( ^ [R2: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
        ! [X3: term_a_b] :
          ( ( member_term_a_b @ X3 @ A6 )
         => ! [Y4: term_a_b,Z4: term_a_b] :
              ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R2 )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z4 ) @ R2 ) )
             => ? [U3: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ U3 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
                  & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z4 @ U3 ) @ ( sup_su6776935440552674877rm_a_b @ R2 @ id_term_a_b ) ) ) ) ) ) ) ).

% strongly_confluent_on_def
thf(fact_550_term__to__sig_Opelims,axiom,
    ! [X2: set_Pr4934435412358123699_a_nat,Xa: b,Xb: term_a_b,Y: term_a_b] :
      ( ( ( terms_8519481630511763164ig_a_b @ X2 @ Xa @ Xb )
        = Y )
     => ( ( accp_P3366084933459579103rm_a_b @ terms_8023737178250165727el_a_b @ ( produc8030969961714872974rm_a_b @ X2 @ ( produc1437816968797971900rm_a_b @ Xa @ Xb ) ) )
       => ( ! [X: b] :
              ( ( Xb
                = ( var_b_a @ X ) )
             => ( ( Y
                  = ( var_b_a @ X ) )
               => ~ ( accp_P3366084933459579103rm_a_b @ terms_8023737178250165727el_a_b @ ( produc8030969961714872974rm_a_b @ X2 @ ( produc1437816968797971900rm_a_b @ Xa @ ( var_b_a @ X ) ) ) ) ) )
         => ~ ! [F: a,Ts: list_term_a_b] :
                ( ( Xb
                  = ( fun_a_b @ F @ Ts ) )
               => ( ( ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F @ ( size_s8906293707977694520rm_a_b @ Ts ) ) @ X2 )
                     => ( Y
                        = ( fun_a_b @ F @ ( map_te2328734355998148206rm_a_b @ ( terms_8519481630511763164ig_a_b @ X2 @ Xa ) @ Ts ) ) ) )
                    & ( ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F @ ( size_s8906293707977694520rm_a_b @ Ts ) ) @ X2 )
                     => ( Y
                        = ( var_b_a @ Xa ) ) ) )
                 => ~ ( accp_P3366084933459579103rm_a_b @ terms_8023737178250165727el_a_b @ ( produc8030969961714872974rm_a_b @ X2 @ ( produc1437816968797971900rm_a_b @ Xa @ ( fun_a_b @ F @ Ts ) ) ) ) ) ) ) ) ) ).

% term_to_sig.pelims
thf(fact_551_IdD,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ id_term_a_b )
     => ( A = B ) ) ).

% IdD
thf(fact_552_varposs__ground__replace__at,axiom,
    ! [P: list_nat,S: term_a_b,U: term_a_b] :
      ( ( member_list_nat @ P @ ( terms_varposs_a_b @ S ) )
     => ( ( term_ground_a_b @ U )
       => ( ( terms_varposs_a_b @ ( term_r6860082780075436317at_a_b @ S @ P @ U ) )
          = ( minus_7954133019191499631st_nat @ ( terms_varposs_a_b @ S ) @ ( insert_list_nat @ P @ bot_bot_set_list_nat ) ) ) ) ) ).

% varposs_ground_replace_at
thf(fact_553_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_554_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_555_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_556_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% diff_is_0_eq
thf(fact_557_ground__vars__term__empty,axiom,
    ! [T: term_a_b] :
      ( ( term_ground_a_b @ T )
     => ( ( vars_term_a_b @ T )
        = bot_bot_set_b ) ) ).

% ground_vars_term_empty
thf(fact_558_ground__substI,axiom,
    ! [S: term_a_b,Sigma: b > term_a_b] :
      ( ! [X: b] :
          ( ( member_b @ X @ ( vars_term_a_b @ S ) )
         => ( term_ground_a_b @ ( Sigma @ X ) ) )
     => ( term_ground_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) ) ) ).

% ground_substI
thf(fact_559_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M )
          = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
thf(fact_560_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_561_ground_Osimps_I1_J,axiom,
    ! [X2: b] :
      ~ ( term_ground_a_b @ ( var_b_a @ X2 ) ) ).

% ground.simps(1)
thf(fact_562_ground__subst__apply,axiom,
    ! [T: term_a_b,Sigma: b > term_a_b] :
      ( ( term_ground_a_b @ T )
     => ( ( subst_7999470309526761004_a_b_b @ T @ Sigma )
        = T ) ) ).

% ground_subst_apply
thf(fact_563_gound__linear,axiom,
    ! [T: term_a_b] :
      ( ( term_ground_a_b @ T )
     => ( terms_5523818207328828897rm_a_b @ T ) ) ).

% gound_linear
thf(fact_564_subterm__eq__pres__ground,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( term_ground_a_b @ S )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
       => ( term_ground_a_b @ T ) ) ) ).

% subterm_eq_pres_ground
thf(fact_565_vars__term__empty__ground,axiom,
    ! [S: term_a_b] :
      ( ( ( vars_term_a_b @ S )
        = bot_bot_set_b )
     => ( term_ground_a_b @ S ) ) ).

% vars_term_empty_ground
thf(fact_566_ground__substD,axiom,
    ! [L2: term_a_b,Sigma: b > term_a_b,X2: b] :
      ( ( term_ground_a_b @ ( subst_7999470309526761004_a_b_b @ L2 @ Sigma ) )
     => ( ( member_b @ X2 @ ( vars_term_a_b @ L2 ) )
       => ( term_ground_a_b @ ( Sigma @ X2 ) ) ) ) ).

% ground_substD
thf(fact_567_varposs__empty__gound,axiom,
    ! [S: term_a_b] :
      ( ( ( terms_varposs_a_b @ S )
        = bot_bot_set_list_nat )
      = ( term_ground_a_b @ S ) ) ).

% varposs_empty_gound
thf(fact_568_strongly__confluentI,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b] :
      ( ! [X: term_a_b,Y2: term_a_b,Z3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ R3 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z3 ) @ R3 )
           => ? [U4: term_a_b] :
                ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ U4 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ U4 ) @ ( sup_su6776935440552674877rm_a_b @ R3 @ id_term_a_b ) ) ) ) )
     => ( abstra5664096641628173427rm_a_b @ R3 @ top_top_set_term_a_b ) ) ).

% strongly_confluentI
thf(fact_569_Field__insert,axiom,
    ! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( field_term_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 ) )
      = ( sup_sup_set_term_a_b @ ( insert_term_a_b @ A @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R3 ) ) ) ).

% Field_insert
thf(fact_570_subst__monoid__mult_Omult_Oright__neutral,axiom,
    ! [A: b > term_a_b] :
      ( ( subst_1216682049913447528_a_b_b @ A @ var_b_a )
      = A ) ).

% subst_monoid_mult.mult.right_neutral
thf(fact_571_subst__monoid__mult_Omult_Oleft__neutral,axiom,
    ! [A: b > term_a_b] :
      ( ( subst_1216682049913447528_a_b_b @ var_b_a @ A )
      = A ) ).

% subst_monoid_mult.mult.left_neutral
thf(fact_572_subst__subst__compose,axiom,
    ! [T: term_a_b,Sigma: b > term_a_b,Tau: b > term_a_b] :
      ( ( subst_7999470309526761004_a_b_b @ T @ ( subst_1216682049913447528_a_b_b @ Sigma @ Tau ) )
      = ( subst_7999470309526761004_a_b_b @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) @ Tau ) ) ).

% subst_subst_compose
thf(fact_573_subst__compose__left__idemp,axiom,
    ! [Sigma: b > term_a_b,X2: b,T: term_a_b] :
      ( ( ( Sigma @ X2 )
        = ( subst_7999470309526761004_a_b_b @ T @ Sigma ) )
     => ( ( subst_1216682049913447528_a_b_b @ ( subst_b_a @ X2 @ T ) @ Sigma )
        = Sigma ) ) ).

% subst_compose_left_idemp
thf(fact_574_subst__self__idemp,axiom,
    ! [X2: b,T: term_a_b] :
      ( ~ ( member_b @ X2 @ ( vars_term_a_b @ T ) )
     => ( ( subst_1216682049913447528_a_b_b @ ( subst_b_a @ X2 @ T ) @ ( subst_b_a @ X2 @ T ) )
        = ( subst_b_a @ X2 @ T ) ) ) ).

% subst_self_idemp
thf(fact_575_FieldI2,axiom,
    ! [I: list_nat,J: list_nat,R5: set_Pr3451248702717554689st_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ I @ J ) @ R5 )
     => ( member_list_nat @ J @ ( field_list_nat @ R5 ) ) ) ).

% FieldI2
thf(fact_576_FieldI2,axiom,
    ! [I: b,J: b,R5: set_Product_prod_b_b] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ I @ J ) @ R5 )
     => ( member_b @ J @ ( field_b @ R5 ) ) ) ).

% FieldI2
thf(fact_577_FieldI2,axiom,
    ! [I: produc357393685978478089rm_a_b,J: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I @ J ) @ R5 )
     => ( member5869715511025134514rm_a_b @ J @ ( field_6884932134483023318rm_a_b @ R5 ) ) ) ).

% FieldI2
thf(fact_578_FieldI2,axiom,
    ! [I: product_prod_a_nat,J: product_prod_a_nat,R5: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ I @ J ) @ R5 )
     => ( member5724188588386418708_a_nat @ J @ ( field_8954927560578634480_a_nat @ R5 ) ) ) ).

% FieldI2
thf(fact_579_FieldI2,axiom,
    ! [I: term_a_b,J: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R5 )
     => ( member_term_a_b @ J @ ( field_term_a_b @ R5 ) ) ) ).

% FieldI2
thf(fact_580_FieldI1,axiom,
    ! [I: list_nat,J: list_nat,R5: set_Pr3451248702717554689st_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ I @ J ) @ R5 )
     => ( member_list_nat @ I @ ( field_list_nat @ R5 ) ) ) ).

% FieldI1
thf(fact_581_FieldI1,axiom,
    ! [I: b,J: b,R5: set_Product_prod_b_b] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ I @ J ) @ R5 )
     => ( member_b @ I @ ( field_b @ R5 ) ) ) ).

% FieldI1
thf(fact_582_FieldI1,axiom,
    ! [I: produc357393685978478089rm_a_b,J: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I @ J ) @ R5 )
     => ( member5869715511025134514rm_a_b @ I @ ( field_6884932134483023318rm_a_b @ R5 ) ) ) ).

% FieldI1
thf(fact_583_FieldI1,axiom,
    ! [I: product_prod_a_nat,J: product_prod_a_nat,R5: set_Pr1811044260758604347_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ I @ J ) @ R5 )
     => ( member5724188588386418708_a_nat @ I @ ( field_8954927560578634480_a_nat @ R5 ) ) ) ).

% FieldI1
thf(fact_584_FieldI1,axiom,
    ! [I: term_a_b,J: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R5 )
     => ( member_term_a_b @ I @ ( field_term_a_b @ R5 ) ) ) ).

% FieldI1
thf(fact_585_CR__divergence__imp__join,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,X2: term_a_b,Y: term_a_b,Z: term_a_b] :
      ( ( abstra8448919418672941150rm_a_b @ R3 @ top_top_set_term_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( abstra4096080454567261402rm_a_b @ R3 ) ) ) ) ) ).

% CR_divergence_imp_join
thf(fact_586_partially__localize__CR,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra8448919418672941150rm_a_b @ R3 @ top_top_set_term_a_b )
      = ( ! [X3: term_a_b,Y4: term_a_b,Z4: term_a_b] :
            ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R3 )
              & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z4 ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Z4 ) @ ( abstra4096080454567261402rm_a_b @ R3 ) ) ) ) ) ).

% partially_localize_CR
thf(fact_587_CR__join__right__I,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,X2: term_a_b,Y: term_a_b,Z: term_a_b] :
      ( ( abstra8448919418672941150rm_a_b @ R3 @ top_top_set_term_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( abstra4096080454567261402rm_a_b @ R3 ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z ) @ ( abstra4096080454567261402rm_a_b @ R3 ) ) ) ) ) ).

% CR_join_right_I
thf(fact_588_CR__join__left__I,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,X2: term_a_b,Y: term_a_b,Z: term_a_b] :
      ( ( abstra8448919418672941150rm_a_b @ R3 @ top_top_set_term_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z ) @ ( abstra4096080454567261402rm_a_b @ R3 ) )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( abstra4096080454567261402rm_a_b @ R3 ) ) ) ) ) ).

% CR_join_left_I
thf(fact_589_strongly__confluent__E1n,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,X2: term_a_b,Y: term_a_b,Z: term_a_b,N: nat] :
      ( ( abstra5664096641628173427rm_a_b @ R3 @ top_top_set_term_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( sup_su6776935440552674877rm_a_b @ R3 @ id_term_a_b ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) )
         => ? [U2: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ U2 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
              & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ U2 ) @ ( sup_su6776935440552674877rm_a_b @ R3 @ id_term_a_b ) ) ) ) ) ) ).

% strongly_confluent_E1n
thf(fact_590_acyclic__insert,axiom,
    ! [Y: term_a_b,X2: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( transi5314701259734769157rm_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ R3 ) )
      = ( ( transi5314701259734769157rm_a_b @ R3 )
        & ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ) ).

% acyclic_insert
thf(fact_591_wo__rel_Ocases__Total3,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b,Phi: term_a_b > term_a_b > $o] :
      ( ( bNF_We251836928773027112rm_a_b @ R3 )
     => ( ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ A @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R3 ) )
       => ( ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( minus_5192120951422937424rm_a_b @ R3 @ id_term_a_b ) )
              | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( minus_5192120951422937424rm_a_b @ R3 @ id_term_a_b ) ) )
           => ( Phi @ A @ B ) )
         => ( ( ( A = B )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_592_refl__on__singleton,axiom,
    ! [X2: term_a_b] : ( refl_on_term_a_b @ ( insert_term_a_b @ X2 @ bot_bot_set_term_a_b ) @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ X2 ) @ bot_bo197521221353338581rm_a_b ) ) ).

% refl_on_singleton
thf(fact_593_wo__rel_Oin__notinI,axiom,
    ! [R3: set_Pr3451248702717554689st_nat,J: list_nat,I: list_nat] :
      ( ( bNF_We7018074364679882228st_nat @ R3 )
     => ( ( ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ J @ I ) @ R3 )
          | ( J = I ) )
       => ( ( member_list_nat @ I @ ( field_list_nat @ R3 ) )
         => ( ( member_list_nat @ J @ ( field_list_nat @ R3 ) )
           => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ I @ J ) @ R3 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_594_wo__rel_Oin__notinI,axiom,
    ! [R3: set_Product_prod_b_b,J: b,I: b] :
      ( ( bNF_We1162827675446709995_rel_b @ R3 )
     => ( ( ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ J @ I ) @ R3 )
          | ( J = I ) )
       => ( ( member_b @ I @ ( field_b @ R3 ) )
         => ( ( member_b @ J @ ( field_b @ R3 ) )
           => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ I @ J ) @ R3 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_595_wo__rel_Oin__notinI,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b,J: produc357393685978478089rm_a_b,I: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R3 )
     => ( ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ J @ I ) @ R3 )
          | ( J = I ) )
       => ( ( member5869715511025134514rm_a_b @ I @ ( field_6884932134483023318rm_a_b @ R3 ) )
         => ( ( member5869715511025134514rm_a_b @ J @ ( field_6884932134483023318rm_a_b @ R3 ) )
           => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I @ J ) @ R3 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_596_wo__rel_Oin__notinI,axiom,
    ! [R3: set_Pr1811044260758604347_a_nat,J: product_prod_a_nat,I: product_prod_a_nat] :
      ( ( bNF_We4298404315170990993_a_nat @ R3 )
     => ( ( ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ J @ I ) @ R3 )
          | ( J = I ) )
       => ( ( member5724188588386418708_a_nat @ I @ ( field_8954927560578634480_a_nat @ R3 ) )
         => ( ( member5724188588386418708_a_nat @ J @ ( field_8954927560578634480_a_nat @ R3 ) )
           => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ I @ J ) @ R3 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_597_wo__rel_Oin__notinI,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,J: term_a_b,I: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R3 )
     => ( ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ J @ I ) @ R3 )
          | ( J = I ) )
       => ( ( member_term_a_b @ I @ ( field_term_a_b @ R3 ) )
         => ( ( member_term_a_b @ J @ ( field_term_a_b @ R3 ) )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R3 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_598_wo__rel_OTOTALS,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R3 )
     => ! [X4: term_a_b] :
          ( ( member_term_a_b @ X4 @ ( field_term_a_b @ R3 ) )
         => ! [Xa2: term_a_b] :
              ( ( member_term_a_b @ Xa2 @ ( field_term_a_b @ R3 ) )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Xa2 ) @ R3 )
                | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa2 @ X4 ) @ R3 ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_599_refl__onD,axiom,
    ! [A2: set_list_nat,R3: set_Pr3451248702717554689st_nat,A: list_nat] :
      ( ( refl_on_list_nat @ A2 @ R3 )
     => ( ( member_list_nat @ A @ A2 )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ A ) @ R3 ) ) ) ).

% refl_onD
thf(fact_600_refl__onD,axiom,
    ! [A2: set_b,R3: set_Product_prod_b_b,A: b] :
      ( ( refl_on_b @ A2 @ R3 )
     => ( ( member_b @ A @ A2 )
       => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ A ) @ R3 ) ) ) ).

% refl_onD
thf(fact_601_refl__onD,axiom,
    ! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b] :
      ( ( refl_o5344404488616963464rm_a_b @ A2 @ R3 )
     => ( ( member5869715511025134514rm_a_b @ A @ A2 )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ A ) @ R3 ) ) ) ).

% refl_onD
thf(fact_602_refl__onD,axiom,
    ! [A2: set_Pr4934435412358123699_a_nat,R3: set_Pr1811044260758604347_a_nat,A: product_prod_a_nat] :
      ( ( refl_o8237544196271505854_a_nat @ A2 @ R3 )
     => ( ( member5724188588386418708_a_nat @ A @ A2 )
       => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ A ) @ R3 ) ) ) ).

% refl_onD
thf(fact_603_refl__onD,axiom,
    ! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
      ( ( refl_on_term_a_b @ A2 @ R3 )
     => ( ( member_term_a_b @ A @ A2 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R3 ) ) ) ).

% refl_onD
thf(fact_604_refl__onD1,axiom,
    ! [A2: set_list_nat,R3: set_Pr3451248702717554689st_nat,X2: list_nat,Y: list_nat] :
      ( ( refl_on_list_nat @ A2 @ R3 )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X2 @ Y ) @ R3 )
       => ( member_list_nat @ X2 @ A2 ) ) ) ).

% refl_onD1
thf(fact_605_refl__onD1,axiom,
    ! [A2: set_b,R3: set_Product_prod_b_b,X2: b,Y: b] :
      ( ( refl_on_b @ A2 @ R3 )
     => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X2 @ Y ) @ R3 )
       => ( member_b @ X2 @ A2 ) ) ) ).

% refl_onD1
thf(fact_606_refl__onD1,axiom,
    ! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b,X2: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
      ( ( refl_o5344404488616963464rm_a_b @ A2 @ R3 )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ Y ) @ R3 )
       => ( member5869715511025134514rm_a_b @ X2 @ A2 ) ) ) ).

% refl_onD1
thf(fact_607_refl__onD1,axiom,
    ! [A2: set_Pr4934435412358123699_a_nat,R3: set_Pr1811044260758604347_a_nat,X2: product_prod_a_nat,Y: product_prod_a_nat] :
      ( ( refl_o8237544196271505854_a_nat @ A2 @ R3 )
     => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X2 @ Y ) @ R3 )
       => ( member5724188588386418708_a_nat @ X2 @ A2 ) ) ) ).

% refl_onD1
thf(fact_608_refl__onD1,axiom,
    ! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,X2: term_a_b,Y: term_a_b] :
      ( ( refl_on_term_a_b @ A2 @ R3 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ R3 )
       => ( member_term_a_b @ X2 @ A2 ) ) ) ).

% refl_onD1
thf(fact_609_refl__onD2,axiom,
    ! [A2: set_list_nat,R3: set_Pr3451248702717554689st_nat,X2: list_nat,Y: list_nat] :
      ( ( refl_on_list_nat @ A2 @ R3 )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X2 @ Y ) @ R3 )
       => ( member_list_nat @ Y @ A2 ) ) ) ).

% refl_onD2
thf(fact_610_refl__onD2,axiom,
    ! [A2: set_b,R3: set_Product_prod_b_b,X2: b,Y: b] :
      ( ( refl_on_b @ A2 @ R3 )
     => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X2 @ Y ) @ R3 )
       => ( member_b @ Y @ A2 ) ) ) ).

% refl_onD2
thf(fact_611_refl__onD2,axiom,
    ! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b,X2: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
      ( ( refl_o5344404488616963464rm_a_b @ A2 @ R3 )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ Y ) @ R3 )
       => ( member5869715511025134514rm_a_b @ Y @ A2 ) ) ) ).

% refl_onD2
thf(fact_612_refl__onD2,axiom,
    ! [A2: set_Pr4934435412358123699_a_nat,R3: set_Pr1811044260758604347_a_nat,X2: product_prod_a_nat,Y: product_prod_a_nat] :
      ( ( refl_o8237544196271505854_a_nat @ A2 @ R3 )
     => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X2 @ Y ) @ R3 )
       => ( member5724188588386418708_a_nat @ Y @ A2 ) ) ) ).

% refl_onD2
thf(fact_613_refl__onD2,axiom,
    ! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,X2: term_a_b,Y: term_a_b] :
      ( ( refl_on_term_a_b @ A2 @ R3 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ R3 )
       => ( member_term_a_b @ Y @ A2 ) ) ) ).

% refl_onD2
thf(fact_614_wo__rel_Owell__order__induct,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o,A: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R3 )
     => ( ! [X: term_a_b] :
            ( ! [Y5: term_a_b] :
                ( ( ( Y5 != X )
                  & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X ) @ R3 ) )
               => ( P2 @ Y5 ) )
           => ( P2 @ X ) )
       => ( P2 @ A ) ) ) ).

% wo_rel.well_order_induct
thf(fact_615_relpow__Suc__D2_H,axiom,
    ! [N: nat,R5: set_Pr4386577575007340137rm_a_b,X4: term_a_b,Y5: term_a_b,Z5: term_a_b] :
      ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
        & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ Z5 ) @ R5 ) )
     => ? [W: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ W ) @ R5 )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ W @ Z5 ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) ) ) ) ).

% relpow_Suc_D2'
thf(fact_616_relpow__image,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b,R8: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b,N: nat] :
      ( ! [S3: term_a_b,T2: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S3 @ T2 ) @ R3 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ S3 ) @ ( F2 @ T2 ) ) @ R8 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ S ) @ ( F2 @ T ) ) @ ( compow4057154403645558940rm_a_b @ N @ R8 ) ) ) ) ).

% relpow_image
thf(fact_617_relpow__0__I,axiom,
    ! [X2: term_a_b,R5: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ X2 ) @ ( compow4057154403645558940rm_a_b @ zero_zero_nat @ R5 ) ) ).

% relpow_0_I
thf(fact_618_relpow__0__E,axiom,
    ! [X2: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( compow4057154403645558940rm_a_b @ zero_zero_nat @ R5 ) )
     => ( X2 = Y ) ) ).

% relpow_0_E
thf(fact_619_relpow__refl__mono,axiom,
    ! [Rel: set_Pr4386577575007340137rm_a_b,M: nat,N: nat,A: term_a_b,B: term_a_b] :
      ( ! [X: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ Rel )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( compow4057154403645558940rm_a_b @ M @ Rel ) )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( compow4057154403645558940rm_a_b @ N @ Rel ) ) ) ) ) ).

% relpow_refl_mono
thf(fact_620_rtrancl__len__E,axiom,
    ! [X2: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
     => ~ ! [N2: nat] :
            ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( compow4057154403645558940rm_a_b @ N2 @ R3 ) ) ) ).

% rtrancl_len_E
thf(fact_621_reflI,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b] :
      ( ! [X: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R3 )
     => ( refl_on_term_a_b @ top_top_set_term_a_b @ R3 ) ) ).

% reflI
thf(fact_622_reflD,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
      ( ( refl_on_term_a_b @ top_top_set_term_a_b @ R3 )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R3 ) ) ).

% reflD
thf(fact_623_acyclic__def,axiom,
    ( transi5314701259734769157rm_a_b
    = ( ^ [R2: set_Pr4386577575007340137rm_a_b] :
        ! [X3: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).

% acyclic_def
thf(fact_624_acyclicI,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b] :
      ( ! [X: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
     => ( transi5314701259734769157rm_a_b @ R3 ) ) ).

% acyclicI
thf(fact_625_wo__rel_Ocases__Total,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b,Phi: term_a_b > term_a_b > $o] :
      ( ( bNF_We251836928773027112rm_a_b @ R3 )
     => ( ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ A @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R3 ) )
       => ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
           => ( Phi @ A @ B ) )
         => ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R3 )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_626_trancl__steps__relpow,axiom,
    ! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,X2: term_a_b,Y: term_a_b,N: nat] :
      ( ( ord_le118470702582115849rm_a_b @ A @ ( transi7922773638565587891rm_a_b @ B ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( compow4057154403645558940rm_a_b @ N @ A ) )
       => ? [M2: nat] :
            ( ( ord_less_eq_nat @ N @ M2 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( compow4057154403645558940rm_a_b @ M2 @ B ) ) ) ) ) ).

% trancl_steps_relpow
thf(fact_627_subterm_OacyclicI__order,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b] :
      ( ! [A4: term_a_b,B3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ R3 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ A4 ) @ ( F2 @ B3 ) ) @ subterm_and_supt_a_b ) )
     => ( transi5314701259734769157rm_a_b @ R3 ) ) ).

% subterm.acyclicI_order
thf(fact_628_well__order__induct__imp,axiom,
    ! [R3: set_Pr3451248702717554689st_nat,P2: list_nat > $o,A: list_nat] :
      ( ( bNF_We7018074364679882228st_nat @ R3 )
     => ( ! [X: list_nat] :
            ( ! [Y5: list_nat] :
                ( ( ( Y5 != X )
                  & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y5 @ X ) @ R3 ) )
               => ( ( member_list_nat @ Y5 @ ( field_list_nat @ R3 ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member_list_nat @ X @ ( field_list_nat @ R3 ) )
             => ( P2 @ X ) ) )
       => ( ( member_list_nat @ A @ ( field_list_nat @ R3 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_629_well__order__induct__imp,axiom,
    ! [R3: set_Product_prod_b_b,P2: b > $o,A: b] :
      ( ( bNF_We1162827675446709995_rel_b @ R3 )
     => ( ! [X: b] :
            ( ! [Y5: b] :
                ( ( ( Y5 != X )
                  & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y5 @ X ) @ R3 ) )
               => ( ( member_b @ Y5 @ ( field_b @ R3 ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member_b @ X @ ( field_b @ R3 ) )
             => ( P2 @ X ) ) )
       => ( ( member_b @ A @ ( field_b @ R3 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_630_well__order__induct__imp,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b,P2: produc357393685978478089rm_a_b > $o,A: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R3 )
     => ( ! [X: produc357393685978478089rm_a_b] :
            ( ! [Y5: produc357393685978478089rm_a_b] :
                ( ( ( Y5 != X )
                  & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y5 @ X ) @ R3 ) )
               => ( ( member5869715511025134514rm_a_b @ Y5 @ ( field_6884932134483023318rm_a_b @ R3 ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member5869715511025134514rm_a_b @ X @ ( field_6884932134483023318rm_a_b @ R3 ) )
             => ( P2 @ X ) ) )
       => ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R3 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_631_well__order__induct__imp,axiom,
    ! [R3: set_Pr1811044260758604347_a_nat,P2: product_prod_a_nat > $o,A: product_prod_a_nat] :
      ( ( bNF_We4298404315170990993_a_nat @ R3 )
     => ( ! [X: product_prod_a_nat] :
            ( ! [Y5: product_prod_a_nat] :
                ( ( ( Y5 != X )
                  & ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y5 @ X ) @ R3 ) )
               => ( ( member5724188588386418708_a_nat @ Y5 @ ( field_8954927560578634480_a_nat @ R3 ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member5724188588386418708_a_nat @ X @ ( field_8954927560578634480_a_nat @ R3 ) )
             => ( P2 @ X ) ) )
       => ( ( member5724188588386418708_a_nat @ A @ ( field_8954927560578634480_a_nat @ R3 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_632_well__order__induct__imp,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o,A: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R3 )
     => ( ! [X: term_a_b] :
            ( ! [Y5: term_a_b] :
                ( ( ( Y5 != X )
                  & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X ) @ R3 ) )
               => ( ( member_term_a_b @ Y5 @ ( field_term_a_b @ R3 ) )
                 => ( P2 @ Y5 ) ) )
           => ( ( member_term_a_b @ X @ ( field_term_a_b @ R3 ) )
             => ( P2 @ X ) ) )
       => ( ( member_term_a_b @ A @ ( field_term_a_b @ R3 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_633_wo__rel_Oequals__minim,axiom,
    ! [R3: set_Pr3451248702717554689st_nat,B5: set_list_nat,A: list_nat] :
      ( ( bNF_We7018074364679882228st_nat @ R3 )
     => ( ( ord_le6045566169113846134st_nat @ B5 @ ( field_list_nat @ R3 ) )
       => ( ( member_list_nat @ A @ B5 )
         => ( ! [B3: list_nat] :
                ( ( member_list_nat @ B3 @ B5 )
               => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B3 ) @ R3 ) )
           => ( A
              = ( bNF_We3801768462575224852st_nat @ R3 @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_634_wo__rel_Oequals__minim,axiom,
    ! [R3: set_Product_prod_b_b,B5: set_b,A: b] :
      ( ( bNF_We1162827675446709995_rel_b @ R3 )
     => ( ( ord_less_eq_set_b @ B5 @ ( field_b @ R3 ) )
       => ( ( member_b @ A @ B5 )
         => ( ! [B3: b] :
                ( ( member_b @ B3 @ B5 )
               => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B3 ) @ R3 ) )
           => ( A
              = ( bNF_We5615626441682584779inim_b @ R3 @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_635_wo__rel_Oequals__minim,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b,B5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R3 )
     => ( ( ord_le118470702582115849rm_a_b @ B5 @ ( field_6884932134483023318rm_a_b @ R3 ) )
       => ( ( member5869715511025134514rm_a_b @ A @ B5 )
         => ( ! [B3: produc357393685978478089rm_a_b] :
                ( ( member5869715511025134514rm_a_b @ B3 @ B5 )
               => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B3 ) @ R3 ) )
           => ( A
              = ( bNF_We1338740440619677141rm_a_b @ R3 @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_636_wo__rel_Oequals__minim,axiom,
    ! [R3: set_Pr1811044260758604347_a_nat,B5: set_Pr4934435412358123699_a_nat,A: product_prod_a_nat] :
      ( ( bNF_We4298404315170990993_a_nat @ R3 )
     => ( ( ord_le8666007276011122963_a_nat @ B5 @ ( field_8954927560578634480_a_nat @ R3 ) )
       => ( ( member5724188588386418708_a_nat @ A @ B5 )
         => ( ! [B3: product_prod_a_nat] :
                ( ( member5724188588386418708_a_nat @ B3 @ B5 )
               => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B3 ) @ R3 ) )
           => ( A
              = ( bNF_We3895015349946260913_a_nat @ R3 @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_637_wo__rel_Oequals__minim,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,B5: set_term_a_b,A: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R3 )
     => ( ( ord_le2705286416250468010rm_a_b @ B5 @ ( field_term_a_b @ R3 ) )
       => ( ( member_term_a_b @ A @ B5 )
         => ( ! [B3: term_a_b] :
                ( ( member_term_a_b @ B3 @ B5 )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ R3 ) )
           => ( A
              = ( bNF_We6258903063523145544rm_a_b @ R3 @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_638_wo__rel_Ominim__least,axiom,
    ! [R3: set_Pr3451248702717554689st_nat,B5: set_list_nat,B: list_nat] :
      ( ( bNF_We7018074364679882228st_nat @ R3 )
     => ( ( ord_le6045566169113846134st_nat @ B5 @ ( field_list_nat @ R3 ) )
       => ( ( member_list_nat @ B @ B5 )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( bNF_We3801768462575224852st_nat @ R3 @ B5 ) @ B ) @ R3 ) ) ) ) ).

% wo_rel.minim_least
thf(fact_639_wo__rel_Ominim__least,axiom,
    ! [R3: set_Product_prod_b_b,B5: set_b,B: b] :
      ( ( bNF_We1162827675446709995_rel_b @ R3 )
     => ( ( ord_less_eq_set_b @ B5 @ ( field_b @ R3 ) )
       => ( ( member_b @ B @ B5 )
         => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( bNF_We5615626441682584779inim_b @ R3 @ B5 ) @ B ) @ R3 ) ) ) ) ).

% wo_rel.minim_least
thf(fact_640_wo__rel_Ominim__least,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b,B5: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R3 )
     => ( ( ord_le118470702582115849rm_a_b @ B5 @ ( field_6884932134483023318rm_a_b @ R3 ) )
       => ( ( member5869715511025134514rm_a_b @ B @ B5 )
         => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( bNF_We1338740440619677141rm_a_b @ R3 @ B5 ) @ B ) @ R3 ) ) ) ) ).

% wo_rel.minim_least
thf(fact_641_wo__rel_Ominim__least,axiom,
    ! [R3: set_Pr1811044260758604347_a_nat,B5: set_Pr4934435412358123699_a_nat,B: product_prod_a_nat] :
      ( ( bNF_We4298404315170990993_a_nat @ R3 )
     => ( ( ord_le8666007276011122963_a_nat @ B5 @ ( field_8954927560578634480_a_nat @ R3 ) )
       => ( ( member5724188588386418708_a_nat @ B @ B5 )
         => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( bNF_We3895015349946260913_a_nat @ R3 @ B5 ) @ B ) @ R3 ) ) ) ) ).

% wo_rel.minim_least
thf(fact_642_wo__rel_Ominim__least,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,B5: set_term_a_b,B: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R3 )
     => ( ( ord_le2705286416250468010rm_a_b @ B5 @ ( field_term_a_b @ R3 ) )
       => ( ( member_term_a_b @ B @ B5 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( bNF_We6258903063523145544rm_a_b @ R3 @ B5 ) @ B ) @ R3 ) ) ) ) ).

% wo_rel.minim_least
thf(fact_643_wo__rel_Omax2__greater__among,axiom,
    ! [R3: set_Pr3451248702717554689st_nat,A: list_nat,B: list_nat] :
      ( ( bNF_We7018074364679882228st_nat @ R3 )
     => ( ( member_list_nat @ A @ ( field_list_nat @ R3 ) )
       => ( ( member_list_nat @ B @ ( field_list_nat @ R3 ) )
         => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ ( bNF_We2161624777392673042st_nat @ R3 @ A @ B ) ) @ R3 )
            & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ ( bNF_We2161624777392673042st_nat @ R3 @ A @ B ) ) @ R3 )
            & ( member_list_nat @ ( bNF_We2161624777392673042st_nat @ R3 @ A @ B ) @ ( insert_list_nat @ A @ ( insert_list_nat @ B @ bot_bot_set_list_nat ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_644_wo__rel_Omax2__greater__among,axiom,
    ! [R3: set_Product_prod_b_b,A: b,B: b] :
      ( ( bNF_We1162827675446709995_rel_b @ R3 )
     => ( ( member_b @ A @ ( field_b @ R3 ) )
       => ( ( member_b @ B @ ( field_b @ R3 ) )
         => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ ( bNF_We3763454674811381837max2_b @ R3 @ A @ B ) ) @ R3 )
            & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ ( bNF_We3763454674811381837max2_b @ R3 @ A @ B ) ) @ R3 )
            & ( member_b @ ( bNF_We3763454674811381837max2_b @ R3 @ A @ B ) @ ( insert_b @ A @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_645_wo__rel_Omax2__greater__among,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R3 )
     => ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R3 ) )
       => ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R3 ) )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ ( bNF_We1369222903935127127rm_a_b @ R3 @ A @ B ) ) @ R3 )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ ( bNF_We1369222903935127127rm_a_b @ R3 @ A @ B ) ) @ R3 )
            & ( member5869715511025134514rm_a_b @ ( bNF_We1369222903935127127rm_a_b @ R3 @ A @ B ) @ ( insert7009541432154983385rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_646_wo__rel_Omax2__greater__among,axiom,
    ! [R3: set_Pr1811044260758604347_a_nat,A: product_prod_a_nat,B: product_prod_a_nat] :
      ( ( bNF_We4298404315170990993_a_nat @ R3 )
     => ( ( member5724188588386418708_a_nat @ A @ ( field_8954927560578634480_a_nat @ R3 ) )
       => ( ( member5724188588386418708_a_nat @ B @ ( field_8954927560578634480_a_nat @ R3 ) )
         => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ ( bNF_We7882469860449324975_a_nat @ R3 @ A @ B ) ) @ R3 )
            & ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ ( bNF_We7882469860449324975_a_nat @ R3 @ A @ B ) ) @ R3 )
            & ( member5724188588386418708_a_nat @ ( bNF_We7882469860449324975_a_nat @ R3 @ A @ B ) @ ( insert8054603423593749677_a_nat @ A @ ( insert8054603423593749677_a_nat @ B @ bot_bo9049108969261143879_a_nat ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_647_wo__rel_Omax2__greater__among,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R3 )
     => ( ( member_term_a_b @ A @ ( field_term_a_b @ R3 ) )
       => ( ( member_term_a_b @ B @ ( field_term_a_b @ R3 ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ ( bNF_We4618759378340593734rm_a_b @ R3 @ A @ B ) ) @ R3 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ ( bNF_We4618759378340593734rm_a_b @ R3 @ A @ B ) ) @ R3 )
            & ( member_term_a_b @ ( bNF_We4618759378340593734rm_a_b @ R3 @ A @ B ) @ ( insert_term_a_b @ A @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_648_ssubst__Pair__rhs,axiom,
    ! [R3: term_a_b,S: term_a_b,R5: set_Pr4386577575007340137rm_a_b,S7: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R3 @ S ) @ R5 )
     => ( ( S7 = S )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R3 @ S7 ) @ R5 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_649_ssubst__Pair__rhs,axiom,
    ! [R3: a,S: nat,R5: set_Pr4934435412358123699_a_nat,S7: nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ R3 @ S ) @ R5 )
     => ( ( S7 = S )
       => ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ R3 @ S7 ) @ R5 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_650_ssubst__Pair__rhs,axiom,
    ! [R3: nat > nat,S: nat,R5: set_Pr9093778441882193744at_nat,S7: nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ R3 @ S ) @ R5 )
     => ( ( S7 = S )
       => ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ R3 @ S7 ) @ R5 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_651_wo__rel_Omax2__def,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R3 )
     => ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
         => ( ( bNF_We4618759378340593734rm_a_b @ R3 @ A @ B )
            = B ) )
        & ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
         => ( ( bNF_We4618759378340593734rm_a_b @ R3 @ A @ B )
            = A ) ) ) ) ).

% wo_rel.max2_def
thf(fact_652_wo__rel_Omax2__equals1,axiom,
    ! [R3: set_Pr3451248702717554689st_nat,A: list_nat,B: list_nat] :
      ( ( bNF_We7018074364679882228st_nat @ R3 )
     => ( ( member_list_nat @ A @ ( field_list_nat @ R3 ) )
       => ( ( member_list_nat @ B @ ( field_list_nat @ R3 ) )
         => ( ( ( bNF_We2161624777392673042st_nat @ R3 @ A @ B )
              = A )
            = ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ A ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_653_wo__rel_Omax2__equals1,axiom,
    ! [R3: set_Product_prod_b_b,A: b,B: b] :
      ( ( bNF_We1162827675446709995_rel_b @ R3 )
     => ( ( member_b @ A @ ( field_b @ R3 ) )
       => ( ( member_b @ B @ ( field_b @ R3 ) )
         => ( ( ( bNF_We3763454674811381837max2_b @ R3 @ A @ B )
              = A )
            = ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ A ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_654_wo__rel_Omax2__equals1,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R3 )
     => ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R3 ) )
       => ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R3 ) )
         => ( ( ( bNF_We1369222903935127127rm_a_b @ R3 @ A @ B )
              = A )
            = ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_655_wo__rel_Omax2__equals1,axiom,
    ! [R3: set_Pr1811044260758604347_a_nat,A: product_prod_a_nat,B: product_prod_a_nat] :
      ( ( bNF_We4298404315170990993_a_nat @ R3 )
     => ( ( member5724188588386418708_a_nat @ A @ ( field_8954927560578634480_a_nat @ R3 ) )
       => ( ( member5724188588386418708_a_nat @ B @ ( field_8954927560578634480_a_nat @ R3 ) )
         => ( ( ( bNF_We7882469860449324975_a_nat @ R3 @ A @ B )
              = A )
            = ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ A ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_656_wo__rel_Omax2__equals1,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R3 )
     => ( ( member_term_a_b @ A @ ( field_term_a_b @ R3 ) )
       => ( ( member_term_a_b @ B @ ( field_term_a_b @ R3 ) )
         => ( ( ( bNF_We4618759378340593734rm_a_b @ R3 @ A @ B )
              = A )
            = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_657_wo__rel_Omax2__equals2,axiom,
    ! [R3: set_Pr3451248702717554689st_nat,A: list_nat,B: list_nat] :
      ( ( bNF_We7018074364679882228st_nat @ R3 )
     => ( ( member_list_nat @ A @ ( field_list_nat @ R3 ) )
       => ( ( member_list_nat @ B @ ( field_list_nat @ R3 ) )
         => ( ( ( bNF_We2161624777392673042st_nat @ R3 @ A @ B )
              = B )
            = ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_658_wo__rel_Omax2__equals2,axiom,
    ! [R3: set_Product_prod_b_b,A: b,B: b] :
      ( ( bNF_We1162827675446709995_rel_b @ R3 )
     => ( ( member_b @ A @ ( field_b @ R3 ) )
       => ( ( member_b @ B @ ( field_b @ R3 ) )
         => ( ( ( bNF_We3763454674811381837max2_b @ R3 @ A @ B )
              = B )
            = ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_659_wo__rel_Omax2__equals2,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R3 )
     => ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R3 ) )
       => ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R3 ) )
         => ( ( ( bNF_We1369222903935127127rm_a_b @ R3 @ A @ B )
              = B )
            = ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_660_wo__rel_Omax2__equals2,axiom,
    ! [R3: set_Pr1811044260758604347_a_nat,A: product_prod_a_nat,B: product_prod_a_nat] :
      ( ( bNF_We4298404315170990993_a_nat @ R3 )
     => ( ( member5724188588386418708_a_nat @ A @ ( field_8954927560578634480_a_nat @ R3 ) )
       => ( ( member5724188588386418708_a_nat @ B @ ( field_8954927560578634480_a_nat @ R3 ) )
         => ( ( ( bNF_We7882469860449324975_a_nat @ R3 @ A @ B )
              = B )
            = ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_661_wo__rel_Omax2__equals2,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R3 )
     => ( ( member_term_a_b @ A @ ( field_term_a_b @ R3 ) )
       => ( ( member_term_a_b @ B @ ( field_term_a_b @ R3 ) )
         => ( ( ( bNF_We4618759378340593734rm_a_b @ R3 @ A @ B )
              = B )
            = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_662_wo__rel_Omax2__greater,axiom,
    ! [R3: set_Pr3451248702717554689st_nat,A: list_nat,B: list_nat] :
      ( ( bNF_We7018074364679882228st_nat @ R3 )
     => ( ( member_list_nat @ A @ ( field_list_nat @ R3 ) )
       => ( ( member_list_nat @ B @ ( field_list_nat @ R3 ) )
         => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ ( bNF_We2161624777392673042st_nat @ R3 @ A @ B ) ) @ R3 )
            & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ ( bNF_We2161624777392673042st_nat @ R3 @ A @ B ) ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_663_wo__rel_Omax2__greater,axiom,
    ! [R3: set_Product_prod_b_b,A: b,B: b] :
      ( ( bNF_We1162827675446709995_rel_b @ R3 )
     => ( ( member_b @ A @ ( field_b @ R3 ) )
       => ( ( member_b @ B @ ( field_b @ R3 ) )
         => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ ( bNF_We3763454674811381837max2_b @ R3 @ A @ B ) ) @ R3 )
            & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ ( bNF_We3763454674811381837max2_b @ R3 @ A @ B ) ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_664_wo__rel_Omax2__greater,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( bNF_We1688910813892497397rm_a_b @ R3 )
     => ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R3 ) )
       => ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R3 ) )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ ( bNF_We1369222903935127127rm_a_b @ R3 @ A @ B ) ) @ R3 )
            & ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ ( bNF_We1369222903935127127rm_a_b @ R3 @ A @ B ) ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_665_wo__rel_Omax2__greater,axiom,
    ! [R3: set_Pr1811044260758604347_a_nat,A: product_prod_a_nat,B: product_prod_a_nat] :
      ( ( bNF_We4298404315170990993_a_nat @ R3 )
     => ( ( member5724188588386418708_a_nat @ A @ ( field_8954927560578634480_a_nat @ R3 ) )
       => ( ( member5724188588386418708_a_nat @ B @ ( field_8954927560578634480_a_nat @ R3 ) )
         => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ ( bNF_We7882469860449324975_a_nat @ R3 @ A @ B ) ) @ R3 )
            & ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ ( bNF_We7882469860449324975_a_nat @ R3 @ A @ B ) ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_666_wo__rel_Omax2__greater,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( bNF_We251836928773027112rm_a_b @ R3 )
     => ( ( member_term_a_b @ A @ ( field_term_a_b @ R3 ) )
       => ( ( member_term_a_b @ B @ ( field_term_a_b @ R3 ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ ( bNF_We4618759378340593734rm_a_b @ R3 @ A @ B ) ) @ R3 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ ( bNF_We4618759378340593734rm_a_b @ R3 @ A @ B ) ) @ R3 ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_667_refl__on__domain,axiom,
    ! [A2: set_list_nat,R3: set_Pr3451248702717554689st_nat,A: list_nat,B: list_nat] :
      ( ( refl_on_list_nat @ A2 @ R3 )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ R3 )
       => ( ( member_list_nat @ A @ A2 )
          & ( member_list_nat @ B @ A2 ) ) ) ) ).

% refl_on_domain
thf(fact_668_refl__on__domain,axiom,
    ! [A2: set_b,R3: set_Product_prod_b_b,A: b,B: b] :
      ( ( refl_on_b @ A2 @ R3 )
     => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ R3 )
       => ( ( member_b @ A @ A2 )
          & ( member_b @ B @ A2 ) ) ) ) ).

% refl_on_domain
thf(fact_669_refl__on__domain,axiom,
    ! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( refl_o5344404488616963464rm_a_b @ A2 @ R3 )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 )
       => ( ( member5869715511025134514rm_a_b @ A @ A2 )
          & ( member5869715511025134514rm_a_b @ B @ A2 ) ) ) ) ).

% refl_on_domain
thf(fact_670_refl__on__domain,axiom,
    ! [A2: set_Pr4934435412358123699_a_nat,R3: set_Pr1811044260758604347_a_nat,A: product_prod_a_nat,B: product_prod_a_nat] :
      ( ( refl_o8237544196271505854_a_nat @ A2 @ R3 )
     => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ R3 )
       => ( ( member5724188588386418708_a_nat @ A @ A2 )
          & ( member5724188588386418708_a_nat @ B @ A2 ) ) ) ) ).

% refl_on_domain
thf(fact_671_refl__on__domain,axiom,
    ! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( refl_on_term_a_b @ A2 @ R3 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
       => ( ( member_term_a_b @ A @ A2 )
          & ( member_term_a_b @ B @ A2 ) ) ) ) ).

% refl_on_domain
thf(fact_672_Total__subset__Id,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b] :
      ( ( total_on_term_a_b @ ( field_term_a_b @ R3 ) @ R3 )
     => ( ( ord_le118470702582115849rm_a_b @ R3 @ id_term_a_b )
       => ( ( R3 = bot_bo197521221353338581rm_a_b )
          | ? [A4: term_a_b] :
              ( R3
              = ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ A4 ) @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ).

% Total_subset_Id
thf(fact_673_subst__monoid__mult_Oprod__list_ONil,axiom,
    ( ( groups5454103906155650226rm_a_b @ var_b_a @ subst_1216682049913447528_a_b_b @ nil_b_term_a_b )
    = var_b_a ) ).

% subst_monoid_mult.prod_list.Nil
thf(fact_674_subst__monoid__mult_Oprod__list_Oappend,axiom,
    ! [Xs: list_b_term_a_b,Ys: list_b_term_a_b] :
      ( ( groups5454103906155650226rm_a_b @ var_b_a @ subst_1216682049913447528_a_b_b @ ( append_b_term_a_b @ Xs @ Ys ) )
      = ( subst_1216682049913447528_a_b_b @ ( groups5454103906155650226rm_a_b @ var_b_a @ subst_1216682049913447528_a_b_b @ Xs ) @ ( groups5454103906155650226rm_a_b @ var_b_a @ subst_1216682049913447528_a_b_b @ Ys ) ) ) ).

% subst_monoid_mult.prod_list.append
thf(fact_675_total__onI,axiom,
    ! [A2: set_list_nat,R3: set_Pr3451248702717554689st_nat] :
      ( ! [X: list_nat,Y2: list_nat] :
          ( ( member_list_nat @ X @ A2 )
         => ( ( member_list_nat @ Y2 @ A2 )
           => ( ( X != Y2 )
             => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y2 ) @ R3 )
                | ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y2 @ X ) @ R3 ) ) ) ) )
     => ( total_on_list_nat @ A2 @ R3 ) ) ).

% total_onI
thf(fact_676_total__onI,axiom,
    ! [A2: set_b,R3: set_Product_prod_b_b] :
      ( ! [X: b,Y2: b] :
          ( ( member_b @ X @ A2 )
         => ( ( member_b @ Y2 @ A2 )
           => ( ( X != Y2 )
             => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y2 ) @ R3 )
                | ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y2 @ X ) @ R3 ) ) ) ) )
     => ( total_on_b @ A2 @ R3 ) ) ).

% total_onI
thf(fact_677_total__onI,axiom,
    ! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
      ( ! [X: produc357393685978478089rm_a_b,Y2: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X @ A2 )
         => ( ( member5869715511025134514rm_a_b @ Y2 @ A2 )
           => ( ( X != Y2 )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y2 ) @ R3 )
                | ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ X ) @ R3 ) ) ) ) )
     => ( total_4010809703584253837rm_a_b @ A2 @ R3 ) ) ).

% total_onI
thf(fact_678_total__onI,axiom,
    ! [A2: set_Pr4934435412358123699_a_nat,R3: set_Pr1811044260758604347_a_nat] :
      ( ! [X: product_prod_a_nat,Y2: product_prod_a_nat] :
          ( ( member5724188588386418708_a_nat @ X @ A2 )
         => ( ( member5724188588386418708_a_nat @ Y2 @ A2 )
           => ( ( X != Y2 )
             => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y2 ) @ R3 )
                | ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y2 @ X ) @ R3 ) ) ) ) )
     => ( total_9110697612923228665_a_nat @ A2 @ R3 ) ) ).

% total_onI
thf(fact_679_total__onI,axiom,
    ! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ! [X: term_a_b,Y2: term_a_b] :
          ( ( member_term_a_b @ X @ A2 )
         => ( ( member_term_a_b @ Y2 @ A2 )
           => ( ( X != Y2 )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ R3 )
                | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ X ) @ R3 ) ) ) ) )
     => ( total_on_term_a_b @ A2 @ R3 ) ) ).

% total_onI
thf(fact_680_total__on__def,axiom,
    ( total_on_term_a_b
    = ( ^ [A6: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
        ! [X3: term_a_b] :
          ( ( member_term_a_b @ X3 @ A6 )
         => ! [Y4: term_a_b] :
              ( ( member_term_a_b @ Y4 @ A6 )
             => ( ( X3 != Y4 )
               => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R2 )
                  | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ X3 ) @ R2 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_681_totalI,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b] :
      ( ! [X: term_a_b,Y2: term_a_b] :
          ( ( X != Y2 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ R3 )
            | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ X ) @ R3 ) ) )
     => ( total_on_term_a_b @ top_top_set_term_a_b @ R3 ) ) ).

% totalI
thf(fact_682_linear__order__on__singleton,axiom,
    ! [X2: term_a_b] : ( order_5388802246213473311rm_a_b @ ( insert_term_a_b @ X2 @ bot_bot_set_term_a_b ) @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ X2 ) @ bot_bo197521221353338581rm_a_b ) ) ).

% linear_order_on_singleton
thf(fact_683_Linear__order__in__diff__Id,axiom,
    ! [R3: set_Pr3451248702717554689st_nat,A: list_nat,B: list_nat] :
      ( ( order_2931667645265552619st_nat @ ( field_list_nat @ R3 ) @ R3 )
     => ( ( member_list_nat @ A @ ( field_list_nat @ R3 ) )
       => ( ( member_list_nat @ B @ ( field_list_nat @ R3 ) )
         => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ R3 )
            = ( ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ A ) @ ( minus_4256792079133151976st_nat @ R3 @ id_list_nat ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_684_Linear__order__in__diff__Id,axiom,
    ! [R3: set_Product_prod_b_b,A: b,B: b] :
      ( ( order_8768733634509060148r_on_b @ ( field_b @ R3 ) @ R3 )
     => ( ( member_b @ A @ ( field_b @ R3 ) )
       => ( ( member_b @ B @ ( field_b @ R3 ) )
         => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ R3 )
            = ( ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ A ) @ ( minus_6252421959248544046od_b_b @ R3 @ id_b ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_685_Linear__order__in__diff__Id,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( order_681589137112254398rm_a_b @ ( field_6884932134483023318rm_a_b @ R3 ) @ R3 )
     => ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R3 ) )
       => ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R3 ) )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 )
            = ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ ( minus_45395518312058030rm_a_b @ R3 @ id_Pro3488709451141449061rm_a_b ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_686_Linear__order__in__diff__Id,axiom,
    ! [R3: set_Pr1811044260758604347_a_nat,A: product_prod_a_nat,B: product_prod_a_nat] :
      ( ( order_6388386362885710600_a_nat @ ( field_8954927560578634480_a_nat @ R3 ) @ R3 )
     => ( ( member5724188588386418708_a_nat @ A @ ( field_8954927560578634480_a_nat @ R3 ) )
       => ( ( member5724188588386418708_a_nat @ B @ ( field_8954927560578634480_a_nat @ R3 ) )
         => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ R3 )
            = ( ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ A ) @ ( minus_1341117088723627554_a_nat @ R3 @ id_Pro5207055338379199009_a_nat ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_687_Linear__order__in__diff__Id,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R3 ) @ R3 )
     => ( ( member_term_a_b @ A @ ( field_term_a_b @ R3 ) )
       => ( ( member_term_a_b @ B @ ( field_term_a_b @ R3 ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
            = ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( minus_5192120951422937424rm_a_b @ R3 @ id_term_a_b ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_688_subst__monoid__mult_Oprod__list__def,axiom,
    ( ( groups5454103906155650226rm_a_b @ var_b_a @ subst_1216682049913447528_a_b_b )
    = ( groups7935052335273062087rm_a_b @ subst_1216682049913447528_a_b_b @ var_b_a ) ) ).

% subst_monoid_mult.prod_list_def
thf(fact_689_Nil__lenlex__iff1,axiom,
    ! [Ns: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ nil_term_a_b @ Ns ) @ ( lenlex_term_a_b @ R3 ) )
      = ( Ns != nil_term_a_b ) ) ).

% Nil_lenlex_iff1
thf(fact_690_Nil__lenlex__iff1,axiom,
    ! [Ns: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ns ) @ ( lenlex_nat @ R3 ) )
      = ( Ns != nil_nat ) ) ).

% Nil_lenlex_iff1
thf(fact_691_lenlex__irreflexive,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b] :
      ( ! [X: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R3 )
     => ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Xs ) @ ( lenlex_term_a_b @ R3 ) ) ) ).

% lenlex_irreflexive
thf(fact_692_Nil__lenlex__iff2,axiom,
    ! [Ns: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ns @ nil_term_a_b ) @ ( lenlex_term_a_b @ R3 ) ) ).

% Nil_lenlex_iff2
thf(fact_693_Nil__lenlex__iff2,axiom,
    ! [Ns: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ns @ nil_nat ) @ ( lenlex_nat @ R3 ) ) ).

% Nil_lenlex_iff2
thf(fact_694_lenlex__append1,axiom,
    ! [Us2: list_nat,Xs: list_nat,R5: set_Pr1261947904930325089at_nat,Vs: list_nat,Ys: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us2 @ Xs ) @ ( lenlex_nat @ R5 ) )
     => ( ( ( size_size_list_nat @ Vs )
          = ( size_size_list_nat @ Ys ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Us2 @ Vs ) @ ( append_nat @ Xs @ Ys ) ) @ ( lenlex_nat @ R5 ) ) ) ) ).

% lenlex_append1
thf(fact_695_Linear__order__wf__diff__Id,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b] :
      ( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R3 ) @ R3 )
     => ( ( wf_term_a_b @ ( minus_5192120951422937424rm_a_b @ R3 @ id_term_a_b ) )
        = ( ! [A6: set_term_a_b] :
              ( ( ord_le2705286416250468010rm_a_b @ A6 @ ( field_term_a_b @ R3 ) )
             => ( ( A6 != bot_bot_set_term_a_b )
               => ? [X3: term_a_b] :
                    ( ( member_term_a_b @ X3 @ A6 )
                    & ! [Y4: term_a_b] :
                        ( ( member_term_a_b @ Y4 @ A6 )
                       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R3 ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_696_lenlex__append2,axiom,
    ! [R5: set_Pr1261947904930325089at_nat,Us2: list_nat,Xs: list_nat,Ys: list_nat] :
      ( ( irrefl_on_nat @ top_top_set_nat @ R5 )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Us2 @ Xs ) @ ( append_nat @ Us2 @ Ys ) ) @ ( lenlex_nat @ R5 ) )
        = ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lenlex_nat @ R5 ) ) ) ) ).

% lenlex_append2
thf(fact_697_Linear__order__Well__order__iff,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b] :
      ( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R3 ) @ R3 )
     => ( ( order_5719896216891381494rm_a_b @ ( field_term_a_b @ R3 ) @ R3 )
        = ( ! [A6: set_term_a_b] :
              ( ( ord_le2705286416250468010rm_a_b @ A6 @ ( field_term_a_b @ R3 ) )
             => ( ( A6 != bot_bot_set_term_a_b )
               => ? [X3: term_a_b] :
                    ( ( member_term_a_b @ X3 @ A6 )
                    & ! [Y4: term_a_b] :
                        ( ( member_term_a_b @ Y4 @ A6 )
                       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R3 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_698_lexord__same__pref__if__irrefl,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( irrefl_on_nat @ top_top_set_nat @ R3 )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lexord_nat @ R3 ) )
        = ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lexord_nat @ R3 ) ) ) ) ).

% lexord_same_pref_if_irrefl
thf(fact_699_irrefl__onD,axiom,
    ! [A2: set_list_nat,R3: set_Pr3451248702717554689st_nat,A: list_nat] :
      ( ( irrefl_on_list_nat @ A2 @ R3 )
     => ( ( member_list_nat @ A @ A2 )
       => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ A ) @ R3 ) ) ) ).

% irrefl_onD
thf(fact_700_irrefl__onD,axiom,
    ! [A2: set_b,R3: set_Product_prod_b_b,A: b] :
      ( ( irrefl_on_b @ A2 @ R3 )
     => ( ( member_b @ A @ A2 )
       => ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ A ) @ R3 ) ) ) ).

% irrefl_onD
thf(fact_701_irrefl__onD,axiom,
    ! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b] :
      ( ( irrefl6620156600162922239rm_a_b @ A2 @ R3 )
     => ( ( member5869715511025134514rm_a_b @ A @ A2 )
       => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ A ) @ R3 ) ) ) ).

% irrefl_onD
thf(fact_702_irrefl__onD,axiom,
    ! [A2: set_Pr4934435412358123699_a_nat,R3: set_Pr1811044260758604347_a_nat,A: product_prod_a_nat] :
      ( ( irrefl177138428007194887_a_nat @ A2 @ R3 )
     => ( ( member5724188588386418708_a_nat @ A @ A2 )
       => ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ A ) @ R3 ) ) ) ).

% irrefl_onD
thf(fact_703_irrefl__onD,axiom,
    ! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
      ( ( irrefl_on_term_a_b @ A2 @ R3 )
     => ( ( member_term_a_b @ A @ A2 )
       => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R3 ) ) ) ).

% irrefl_onD
thf(fact_704_irrefl__onI,axiom,
    ! [A2: set_list_nat,R3: set_Pr3451248702717554689st_nat] :
      ( ! [A4: list_nat] :
          ( ( member_list_nat @ A4 @ A2 )
         => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A4 @ A4 ) @ R3 ) )
     => ( irrefl_on_list_nat @ A2 @ R3 ) ) ).

% irrefl_onI
thf(fact_705_irrefl__onI,axiom,
    ! [A2: set_b,R3: set_Product_prod_b_b] :
      ( ! [A4: b] :
          ( ( member_b @ A4 @ A2 )
         => ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A4 @ A4 ) @ R3 ) )
     => ( irrefl_on_b @ A2 @ R3 ) ) ).

% irrefl_onI
thf(fact_706_irrefl__onI,axiom,
    ! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
      ( ! [A4: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ A4 @ A2 )
         => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ A4 ) @ R3 ) )
     => ( irrefl6620156600162922239rm_a_b @ A2 @ R3 ) ) ).

% irrefl_onI
thf(fact_707_irrefl__onI,axiom,
    ! [A2: set_Pr4934435412358123699_a_nat,R3: set_Pr1811044260758604347_a_nat] :
      ( ! [A4: product_prod_a_nat] :
          ( ( member5724188588386418708_a_nat @ A4 @ A2 )
         => ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A4 @ A4 ) @ R3 ) )
     => ( irrefl177138428007194887_a_nat @ A2 @ R3 ) ) ).

% irrefl_onI
thf(fact_708_irrefl__onI,axiom,
    ! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ! [A4: term_a_b] :
          ( ( member_term_a_b @ A4 @ A2 )
         => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ A4 ) @ R3 ) )
     => ( irrefl_on_term_a_b @ A2 @ R3 ) ) ).

% irrefl_onI
thf(fact_709_irrefl__on__def,axiom,
    ( irrefl_on_term_a_b
    = ( ^ [A6: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
        ! [X3: term_a_b] :
          ( ( member_term_a_b @ X3 @ A6 )
         => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ R2 ) ) ) ) ).

% irrefl_on_def
thf(fact_710_lexord__irreflexive,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b] :
      ( ! [X: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R3 )
     => ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Xs ) @ ( lexord_term_a_b @ R3 ) ) ) ).

% lexord_irreflexive
thf(fact_711_lexord__linear,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,X2: list_term_a_b,Y: list_term_a_b] :
      ( ! [A4: term_a_b,B3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ R3 )
          | ( A4 = B3 )
          | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A4 ) @ R3 ) )
     => ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X2 @ Y ) @ ( lexord_term_a_b @ R3 ) )
        | ( X2 = Y )
        | ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Y @ X2 ) @ ( lexord_term_a_b @ R3 ) ) ) ) ).

% lexord_linear
thf(fact_712_lexord__Nil__right,axiom,
    ! [X2: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X2 @ nil_term_a_b ) @ ( lexord_term_a_b @ R3 ) ) ).

% lexord_Nil_right
thf(fact_713_lexord__Nil__right,axiom,
    ! [X2: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X2 @ nil_nat ) @ ( lexord_nat @ R3 ) ) ).

% lexord_Nil_right
thf(fact_714_lexord__append__leftI,axiom,
    ! [U: list_nat,V4: list_nat,R3: set_Pr1261947904930325089at_nat,X2: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ V4 ) @ ( lexord_nat @ R3 ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ X2 @ U ) @ ( append_nat @ X2 @ V4 ) ) @ ( lexord_nat @ R3 ) ) ) ).

% lexord_append_leftI
thf(fact_715_well__order__on__domain,axiom,
    ! [A2: set_list_nat,R3: set_Pr3451248702717554689st_nat,A: list_nat,B: list_nat] :
      ( ( order_3262761615943460802st_nat @ A2 @ R3 )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ R3 )
       => ( ( member_list_nat @ A @ A2 )
          & ( member_list_nat @ B @ A2 ) ) ) ) ).

% well_order_on_domain
thf(fact_716_well__order__on__domain,axiom,
    ! [A2: set_b,R3: set_Product_prod_b_b,A: b,B: b] :
      ( ( order_6972113574731384221r_on_b @ A2 @ R3 )
     => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ R3 )
       => ( ( member_b @ A @ A2 )
          & ( member_b @ B @ A2 ) ) ) ) ).

% well_order_on_domain
thf(fact_717_well__order__on__domain,axiom,
    ! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( order_8637870607158948775rm_a_b @ A2 @ R3 )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 )
       => ( ( member5869715511025134514rm_a_b @ A @ A2 )
          & ( member5869715511025134514rm_a_b @ B @ A2 ) ) ) ) ).

% well_order_on_domain
thf(fact_718_well__order__on__domain,axiom,
    ! [A2: set_Pr4934435412358123699_a_nat,R3: set_Pr1811044260758604347_a_nat,A: product_prod_a_nat,B: product_prod_a_nat] :
      ( ( order_7753855512070164063_a_nat @ A2 @ R3 )
     => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ R3 )
       => ( ( member5724188588386418708_a_nat @ A @ A2 )
          & ( member5724188588386418708_a_nat @ B @ A2 ) ) ) ) ).

% well_order_on_domain
thf(fact_719_well__order__on__domain,axiom,
    ! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( order_5719896216891381494rm_a_b @ A2 @ R3 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
       => ( ( member_term_a_b @ A @ A2 )
          & ( member_term_a_b @ B @ A2 ) ) ) ) ).

% well_order_on_domain
thf(fact_720_lexord__append__leftD,axiom,
    ! [X2: list_nat,U: list_nat,V4: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ X2 @ U ) @ ( append_nat @ X2 @ V4 ) ) @ ( lexord_nat @ R3 ) )
     => ( ! [A4: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A4 ) @ R3 )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ V4 ) @ ( lexord_nat @ R3 ) ) ) ) ).

% lexord_append_leftD
thf(fact_721_lexord__append__leftD,axiom,
    ! [X2: list_term_a_b,U: list_term_a_b,V4: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ X2 @ U ) @ ( append_term_a_b @ X2 @ V4 ) ) @ ( lexord_term_a_b @ R3 ) )
     => ( ! [A4: term_a_b] :
            ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ A4 ) @ R3 )
       => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ U @ V4 ) @ ( lexord_term_a_b @ R3 ) ) ) ) ).

% lexord_append_leftD
thf(fact_722_lexord__sufE,axiom,
    ! [Xs: list_nat,Zs: list_nat,Ys: list_nat,Qs: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Zs ) @ ( append_nat @ Ys @ Qs ) ) @ ( lexord_nat @ R3 ) )
     => ( ( Xs != Ys )
       => ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) )
         => ( ( ( size_size_list_nat @ Zs )
              = ( size_size_list_nat @ Qs ) )
           => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lexord_nat @ R3 ) ) ) ) ) ) ).

% lexord_sufE
thf(fact_723_irreflI,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b] :
      ( ! [A4: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ A4 ) @ R3 )
     => ( irrefl_on_term_a_b @ top_top_set_term_a_b @ R3 ) ) ).

% irreflI
thf(fact_724_irreflD,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,X2: term_a_b] :
      ( ( irrefl_on_term_a_b @ top_top_set_term_a_b @ R3 )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ X2 ) @ R3 ) ) ).

% irreflD
thf(fact_725_lexord__sufI,axiom,
    ! [U: list_nat,W2: list_nat,R3: set_Pr1261947904930325089at_nat,V4: list_nat,Z: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ W2 ) @ ( lexord_nat @ R3 ) )
     => ( ( ord_less_eq_nat @ ( size_size_list_nat @ W2 ) @ ( size_size_list_nat @ U ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ U @ V4 ) @ ( append_nat @ W2 @ Z ) ) @ ( lexord_nat @ R3 ) ) ) ) ).

% lexord_sufI
thf(fact_726_wf__eq__minimal2,axiom,
    ( wf_term_a_b
    = ( ^ [R2: set_Pr4386577575007340137rm_a_b] :
        ! [A6: set_term_a_b] :
          ( ( ( ord_le2705286416250468010rm_a_b @ A6 @ ( field_term_a_b @ R2 ) )
            & ( A6 != bot_bot_set_term_a_b ) )
         => ? [X3: term_a_b] :
              ( ( member_term_a_b @ X3 @ A6 )
              & ! [Y4: term_a_b] :
                  ( ( member_term_a_b @ Y4 @ A6 )
                 => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ X3 ) @ R2 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_727_wf__insert,axiom,
    ! [Y: term_a_b,X2: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( wf_term_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X2 ) @ R3 ) )
      = ( ( wf_term_a_b @ R3 )
        & ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ) ).

% wf_insert
thf(fact_728_wfE__min_H,axiom,
    ! [R5: set_Pr3451248702717554689st_nat,Q5: set_list_nat] :
      ( ( wf_list_nat @ R5 )
     => ( ( Q5 != bot_bot_set_list_nat )
       => ~ ! [Z3: list_nat] :
              ( ( member_list_nat @ Z3 @ Q5 )
             => ~ ! [Y5: list_nat] :
                    ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y5 @ Z3 ) @ R5 )
                   => ~ ( member_list_nat @ Y5 @ Q5 ) ) ) ) ) ).

% wfE_min'
thf(fact_729_wfE__min_H,axiom,
    ! [R5: set_Product_prod_b_b,Q5: set_b] :
      ( ( wf_b @ R5 )
     => ( ( Q5 != bot_bot_set_b )
       => ~ ! [Z3: b] :
              ( ( member_b @ Z3 @ Q5 )
             => ~ ! [Y5: b] :
                    ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y5 @ Z3 ) @ R5 )
                   => ~ ( member_b @ Y5 @ Q5 ) ) ) ) ) ).

% wfE_min'
thf(fact_730_wfE__min_H,axiom,
    ! [R5: set_Pr2972776593051762503rm_a_b,Q5: set_Pr4386577575007340137rm_a_b] :
      ( ( wf_Pro2335863617654816626rm_a_b @ R5 )
     => ( ( Q5 != bot_bo197521221353338581rm_a_b )
       => ~ ! [Z3: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ Z3 @ Q5 )
             => ~ ! [Y5: produc357393685978478089rm_a_b] :
                    ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y5 @ Z3 ) @ R5 )
                   => ~ ( member5869715511025134514rm_a_b @ Y5 @ Q5 ) ) ) ) ) ).

% wfE_min'
thf(fact_731_wfE__min_H,axiom,
    ! [R5: set_Pr1811044260758604347_a_nat,Q5: set_Pr4934435412358123699_a_nat] :
      ( ( wf_Pro8109884346892000340_a_nat @ R5 )
     => ( ( Q5 != bot_bo9049108969261143879_a_nat )
       => ~ ! [Z3: product_prod_a_nat] :
              ( ( member5724188588386418708_a_nat @ Z3 @ Q5 )
             => ~ ! [Y5: product_prod_a_nat] :
                    ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y5 @ Z3 ) @ R5 )
                   => ~ ( member5724188588386418708_a_nat @ Y5 @ Q5 ) ) ) ) ) ).

% wfE_min'
thf(fact_732_wfE__min_H,axiom,
    ! [R5: set_Pr4386577575007340137rm_a_b,Q5: set_term_a_b] :
      ( ( wf_term_a_b @ R5 )
     => ( ( Q5 != bot_bot_set_term_a_b )
       => ~ ! [Z3: term_a_b] :
              ( ( member_term_a_b @ Z3 @ Q5 )
             => ~ ! [Y5: term_a_b] :
                    ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ Z3 ) @ R5 )
                   => ~ ( member_term_a_b @ Y5 @ Q5 ) ) ) ) ) ).

% wfE_min'
thf(fact_733_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr4386577575007340137rm_a_b,R3: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b] :
      ( ( wf_term_a_b @ S )
     => ( ! [X: term_a_b,Y2: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ R3 )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X ) @ ( F2 @ Y2 ) ) @ S ) )
       => ( wf_term_a_b @ R3 ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_734_wf__def,axiom,
    ( wf_term_a_b
    = ( ^ [R2: set_Pr4386577575007340137rm_a_b] :
        ! [P5: term_a_b > $o] :
          ( ! [X3: term_a_b] :
              ( ! [Y4: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ X3 ) @ R2 )
                 => ( P5 @ Y4 ) )
             => ( P5 @ X3 ) )
         => ! [X5: term_a_b] : ( P5 @ X5 ) ) ) ) ).

% wf_def
thf(fact_735_wfE__min,axiom,
    ! [R5: set_Pr3451248702717554689st_nat,X2: list_nat,Q5: set_list_nat] :
      ( ( wf_list_nat @ R5 )
     => ( ( member_list_nat @ X2 @ Q5 )
       => ~ ! [Z3: list_nat] :
              ( ( member_list_nat @ Z3 @ Q5 )
             => ~ ! [Y5: list_nat] :
                    ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y5 @ Z3 ) @ R5 )
                   => ~ ( member_list_nat @ Y5 @ Q5 ) ) ) ) ) ).

% wfE_min
thf(fact_736_wfE__min,axiom,
    ! [R5: set_Product_prod_b_b,X2: b,Q5: set_b] :
      ( ( wf_b @ R5 )
     => ( ( member_b @ X2 @ Q5 )
       => ~ ! [Z3: b] :
              ( ( member_b @ Z3 @ Q5 )
             => ~ ! [Y5: b] :
                    ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y5 @ Z3 ) @ R5 )
                   => ~ ( member_b @ Y5 @ Q5 ) ) ) ) ) ).

% wfE_min
thf(fact_737_wfE__min,axiom,
    ! [R5: set_Pr2972776593051762503rm_a_b,X2: produc357393685978478089rm_a_b,Q5: set_Pr4386577575007340137rm_a_b] :
      ( ( wf_Pro2335863617654816626rm_a_b @ R5 )
     => ( ( member5869715511025134514rm_a_b @ X2 @ Q5 )
       => ~ ! [Z3: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ Z3 @ Q5 )
             => ~ ! [Y5: produc357393685978478089rm_a_b] :
                    ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y5 @ Z3 ) @ R5 )
                   => ~ ( member5869715511025134514rm_a_b @ Y5 @ Q5 ) ) ) ) ) ).

% wfE_min
thf(fact_738_wfE__min,axiom,
    ! [R5: set_Pr1811044260758604347_a_nat,X2: product_prod_a_nat,Q5: set_Pr4934435412358123699_a_nat] :
      ( ( wf_Pro8109884346892000340_a_nat @ R5 )
     => ( ( member5724188588386418708_a_nat @ X2 @ Q5 )
       => ~ ! [Z3: product_prod_a_nat] :
              ( ( member5724188588386418708_a_nat @ Z3 @ Q5 )
             => ~ ! [Y5: product_prod_a_nat] :
                    ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y5 @ Z3 ) @ R5 )
                   => ~ ( member5724188588386418708_a_nat @ Y5 @ Q5 ) ) ) ) ) ).

% wfE_min
thf(fact_739_wfE__min,axiom,
    ! [R5: set_Pr4386577575007340137rm_a_b,X2: term_a_b,Q5: set_term_a_b] :
      ( ( wf_term_a_b @ R5 )
     => ( ( member_term_a_b @ X2 @ Q5 )
       => ~ ! [Z3: term_a_b] :
              ( ( member_term_a_b @ Z3 @ Q5 )
             => ~ ! [Y5: term_a_b] :
                    ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ Z3 ) @ R5 )
                   => ~ ( member_term_a_b @ Y5 @ Q5 ) ) ) ) ) ).

% wfE_min
thf(fact_740_wfI__min,axiom,
    ! [R5: set_Pr3451248702717554689st_nat] :
      ( ! [X: list_nat,Q6: set_list_nat] :
          ( ( member_list_nat @ X @ Q6 )
         => ? [Xa2: list_nat] :
              ( ( member_list_nat @ Xa2 @ Q6 )
              & ! [Y2: list_nat] :
                  ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y2 @ Xa2 ) @ R5 )
                 => ~ ( member_list_nat @ Y2 @ Q6 ) ) ) )
     => ( wf_list_nat @ R5 ) ) ).

% wfI_min
thf(fact_741_wfI__min,axiom,
    ! [R5: set_Product_prod_b_b] :
      ( ! [X: b,Q6: set_b] :
          ( ( member_b @ X @ Q6 )
         => ? [Xa2: b] :
              ( ( member_b @ Xa2 @ Q6 )
              & ! [Y2: b] :
                  ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y2 @ Xa2 ) @ R5 )
                 => ~ ( member_b @ Y2 @ Q6 ) ) ) )
     => ( wf_b @ R5 ) ) ).

% wfI_min
thf(fact_742_wfI__min,axiom,
    ! [R5: set_Pr2972776593051762503rm_a_b] :
      ( ! [X: produc357393685978478089rm_a_b,Q6: set_Pr4386577575007340137rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X @ Q6 )
         => ? [Xa2: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ Xa2 @ Q6 )
              & ! [Y2: produc357393685978478089rm_a_b] :
                  ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ Xa2 ) @ R5 )
                 => ~ ( member5869715511025134514rm_a_b @ Y2 @ Q6 ) ) ) )
     => ( wf_Pro2335863617654816626rm_a_b @ R5 ) ) ).

% wfI_min
thf(fact_743_wfI__min,axiom,
    ! [R5: set_Pr1811044260758604347_a_nat] :
      ( ! [X: product_prod_a_nat,Q6: set_Pr4934435412358123699_a_nat] :
          ( ( member5724188588386418708_a_nat @ X @ Q6 )
         => ? [Xa2: product_prod_a_nat] :
              ( ( member5724188588386418708_a_nat @ Xa2 @ Q6 )
              & ! [Y2: product_prod_a_nat] :
                  ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y2 @ Xa2 ) @ R5 )
                 => ~ ( member5724188588386418708_a_nat @ Y2 @ Q6 ) ) ) )
     => ( wf_Pro8109884346892000340_a_nat @ R5 ) ) ).

% wfI_min
thf(fact_744_wfI__min,axiom,
    ! [R5: set_Pr4386577575007340137rm_a_b] :
      ( ! [X: term_a_b,Q6: set_term_a_b] :
          ( ( member_term_a_b @ X @ Q6 )
         => ? [Xa2: term_a_b] :
              ( ( member_term_a_b @ Xa2 @ Q6 )
              & ! [Y2: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Xa2 ) @ R5 )
                 => ~ ( member_term_a_b @ Y2 @ Q6 ) ) ) )
     => ( wf_term_a_b @ R5 ) ) ).

% wfI_min
thf(fact_745_wfUNIVI,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b] :
      ( ! [P6: term_a_b > $o,X: term_a_b] :
          ( ! [Xa2: term_a_b] :
              ( ! [Y2: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Xa2 ) @ R3 )
                 => ( P6 @ Y2 ) )
             => ( P6 @ Xa2 ) )
         => ( P6 @ X ) )
     => ( wf_term_a_b @ R3 ) ) ).

% wfUNIVI
thf(fact_746_wf__asym,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,X2: term_a_b] :
      ( ( wf_term_a_b @ R3 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ X2 ) @ R3 )
       => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ A ) @ R3 ) ) ) ).

% wf_asym
thf(fact_747_wf__induct,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o,A: term_a_b] :
      ( ( wf_term_a_b @ R3 )
     => ( ! [X: term_a_b] :
            ( ! [Y5: term_a_b] :
                ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X ) @ R3 )
               => ( P2 @ Y5 ) )
           => ( P2 @ X ) )
       => ( P2 @ A ) ) ) ).

% wf_induct
thf(fact_748_wf__irrefl,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
      ( ( wf_term_a_b @ R3 )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R3 ) ) ).

% wf_irrefl
thf(fact_749_wf__not__sym,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,X2: term_a_b] :
      ( ( wf_term_a_b @ R3 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ X2 ) @ R3 )
       => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ A ) @ R3 ) ) ) ).

% wf_not_sym
thf(fact_750_wf__not__refl,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
      ( ( wf_term_a_b @ R3 )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R3 ) ) ).

% wf_not_refl
thf(fact_751_wf__eq__minimal,axiom,
    ( wf_list_nat
    = ( ^ [R2: set_Pr3451248702717554689st_nat] :
        ! [Q7: set_list_nat] :
          ( ? [X3: list_nat] : ( member_list_nat @ X3 @ Q7 )
         => ? [X3: list_nat] :
              ( ( member_list_nat @ X3 @ Q7 )
              & ! [Y4: list_nat] :
                  ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y4 @ X3 ) @ R2 )
                 => ~ ( member_list_nat @ Y4 @ Q7 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_752_wf__eq__minimal,axiom,
    ( wf_b
    = ( ^ [R2: set_Product_prod_b_b] :
        ! [Q7: set_b] :
          ( ? [X3: b] : ( member_b @ X3 @ Q7 )
         => ? [X3: b] :
              ( ( member_b @ X3 @ Q7 )
              & ! [Y4: b] :
                  ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y4 @ X3 ) @ R2 )
                 => ~ ( member_b @ Y4 @ Q7 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_753_wf__eq__minimal,axiom,
    ( wf_Pro2335863617654816626rm_a_b
    = ( ^ [R2: set_Pr2972776593051762503rm_a_b] :
        ! [Q7: set_Pr4386577575007340137rm_a_b] :
          ( ? [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ Q7 )
         => ? [X3: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ X3 @ Q7 )
              & ! [Y4: produc357393685978478089rm_a_b] :
                  ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y4 @ X3 ) @ R2 )
                 => ~ ( member5869715511025134514rm_a_b @ Y4 @ Q7 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_754_wf__eq__minimal,axiom,
    ( wf_Pro8109884346892000340_a_nat
    = ( ^ [R2: set_Pr1811044260758604347_a_nat] :
        ! [Q7: set_Pr4934435412358123699_a_nat] :
          ( ? [X3: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X3 @ Q7 )
         => ? [X3: product_prod_a_nat] :
              ( ( member5724188588386418708_a_nat @ X3 @ Q7 )
              & ! [Y4: product_prod_a_nat] :
                  ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y4 @ X3 ) @ R2 )
                 => ~ ( member5724188588386418708_a_nat @ Y4 @ Q7 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_755_wf__eq__minimal,axiom,
    ( wf_term_a_b
    = ( ^ [R2: set_Pr4386577575007340137rm_a_b] :
        ! [Q7: set_term_a_b] :
          ( ? [X3: term_a_b] : ( member_term_a_b @ X3 @ Q7 )
         => ? [X3: term_a_b] :
              ( ( member_term_a_b @ X3 @ Q7 )
              & ! [Y4: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ X3 ) @ R2 )
                 => ~ ( member_term_a_b @ Y4 @ Q7 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_756_wf__induct__rule,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o,A: term_a_b] :
      ( ( wf_term_a_b @ R3 )
     => ( ! [X: term_a_b] :
            ( ! [Y5: term_a_b] :
                ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X ) @ R3 )
               => ( P2 @ Y5 ) )
           => ( P2 @ X ) )
       => ( P2 @ A ) ) ) ).

% wf_induct_rule
thf(fact_757_Nil2__notin__lex,axiom,
    ! [Xs: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ nil_term_a_b ) @ ( lex_term_a_b @ R3 ) ) ).

% Nil2_notin_lex
thf(fact_758_Nil2__notin__lex,axiom,
    ! [Xs: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( lex_nat @ R3 ) ) ).

% Nil2_notin_lex
thf(fact_759_Nil__notin__lex,axiom,
    ! [Ys: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ nil_term_a_b @ Ys ) @ ( lex_term_a_b @ R3 ) ) ).

% Nil_notin_lex
thf(fact_760_Nil__notin__lex,axiom,
    ! [Ys: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys ) @ ( lex_nat @ R3 ) ) ).

% Nil_notin_lex
thf(fact_761_lex__append__leftI,axiom,
    ! [Ys: list_nat,Zs: list_nat,R3: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R3 ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R3 ) ) ) ).

% lex_append_leftI
thf(fact_762_lex__append__leftD,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ! [X: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R3 )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R3 ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R3 ) ) ) ) ).

% lex_append_leftD
thf(fact_763_lex__append__leftD,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b,Ys: list_term_a_b,Zs: list_term_a_b] :
      ( ! [X: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R3 )
     => ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ Xs @ Ys ) @ ( append_term_a_b @ Xs @ Zs ) ) @ ( lex_term_a_b @ R3 ) )
       => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ys @ Zs ) @ ( lex_term_a_b @ R3 ) ) ) ) ).

% lex_append_leftD
thf(fact_764_lex__append__left__iff,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ! [X: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R3 )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R3 ) )
        = ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R3 ) ) ) ) ).

% lex_append_left_iff
thf(fact_765_lex__append__left__iff,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b,Ys: list_term_a_b,Zs: list_term_a_b] :
      ( ! [X: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R3 )
     => ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ Xs @ Ys ) @ ( append_term_a_b @ Xs @ Zs ) ) @ ( lex_term_a_b @ R3 ) )
        = ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ys @ Zs ) @ ( lex_term_a_b @ R3 ) ) ) ) ).

% lex_append_left_iff
thf(fact_766_lex__append__rightI,axiom,
    ! [Xs: list_nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat,Vs: list_nat,Us2: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R3 ) )
     => ( ( ( size_size_list_nat @ Vs )
          = ( size_size_list_nat @ Us2 ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Us2 ) @ ( append_nat @ Ys @ Vs ) ) @ ( lex_nat @ R3 ) ) ) ) ).

% lex_append_rightI
thf(fact_767_bsqr__max2,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A1: term_a_b,A22: term_a_b,B1: term_a_b,B22: term_a_b] :
      ( ( order_5719896216891381494rm_a_b @ ( field_term_a_b @ R3 ) @ R3 )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( produc7020197800436672577rm_a_b @ B1 @ B22 ) ) @ ( bNF_We8522098035112346167rm_a_b @ R3 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( bNF_We4618759378340593734rm_a_b @ R3 @ A1 @ A22 ) @ ( bNF_We4618759378340593734rm_a_b @ R3 @ B1 @ B22 ) ) @ R3 ) ) ) ).

% bsqr_max2
thf(fact_768_Cons__in__lex,axiom,
    ! [X2: nat,Xs: list_nat,Y: nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( lex_nat @ R3 ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R3 )
          & ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) ) )
        | ( ( X2 = Y )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R3 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_769_Cons__in__lex,axiom,
    ! [X2: term_a_b,Xs: list_term_a_b,Y: term_a_b,Ys: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X2 @ Xs ) @ ( cons_term_a_b @ Y @ Ys ) ) @ ( lex_term_a_b @ R3 ) )
      = ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ R3 )
          & ( ( size_s8906293707977694520rm_a_b @ Xs )
            = ( size_s8906293707977694520rm_a_b @ Ys ) ) )
        | ( ( X2 = Y )
          & ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( lex_term_a_b @ R3 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_770_in__lex__prod,axiom,
    ! [A: term_a_b,B: term_a_b,A3: term_a_b,B2: term_a_b,R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) ) @ ( lex_pr6163557265797435481rm_a_b @ R3 @ S ) )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A3 ) @ R3 )
        | ( ( A = A3 )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_771_in__lex__prod,axiom,
    ! [A: a,B: nat,A3: a,B2: nat,R3: set_Product_prod_a_a,S: set_Pr1261947904930325089at_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ A @ B ) @ ( product_Pair_a_nat @ A3 @ B2 ) ) @ ( lex_prod_a_nat @ R3 @ S ) )
      = ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A3 ) @ R3 )
        | ( ( A = A3 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_772_in__lex__prod,axiom,
    ! [A: nat > nat,B: nat,A3: nat > nat,B2: nat,R3: set_Pr7682762132356531903at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ A @ B ) @ ( produc72220940542539688at_nat @ A3 @ B2 ) ) @ ( lex_prod_nat_nat_nat @ R3 @ S ) )
      = ( ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A @ A3 ) @ R3 )
        | ( ( A = A3 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_773_list_Oinject,axiom,
    ! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
      ( ( ( cons_nat @ X21 @ X22 )
        = ( cons_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_774_Cons__in__lists__iff,axiom,
    ! [X2: list_nat,Xs: list_list_nat,A2: set_list_nat] :
      ( ( member_list_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ ( lists_list_nat @ A2 ) )
      = ( ( member_list_nat @ X2 @ A2 )
        & ( member_list_list_nat @ Xs @ ( lists_list_nat @ A2 ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_775_Cons__in__lists__iff,axiom,
    ! [X2: b,Xs: list_b,A2: set_b] :
      ( ( member_list_b @ ( cons_b @ X2 @ Xs ) @ ( lists_b @ A2 ) )
      = ( ( member_b @ X2 @ A2 )
        & ( member_list_b @ Xs @ ( lists_b @ A2 ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_776_Cons__in__lists__iff,axiom,
    ! [X2: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
      ( ( member7646937072540562872rm_a_b @ ( cons_P3536305108106557631rm_a_b @ X2 @ Xs ) @ ( lists_3936190290964281341rm_a_b @ A2 ) )
      = ( ( member5869715511025134514rm_a_b @ X2 @ A2 )
        & ( member7646937072540562872rm_a_b @ Xs @ ( lists_3936190290964281341rm_a_b @ A2 ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_777_Cons__in__lists__iff,axiom,
    ! [X2: product_prod_a_nat,Xs: list_P3592885314253461005_a_nat,A2: set_Pr4934435412358123699_a_nat] :
      ( ( member4041938288099225636_a_nat @ ( cons_P5205166803686508359_a_nat @ X2 @ Xs ) @ ( lists_3983152277583228297_a_nat @ A2 ) )
      = ( ( member5724188588386418708_a_nat @ X2 @ A2 )
        & ( member4041938288099225636_a_nat @ Xs @ ( lists_3983152277583228297_a_nat @ A2 ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_778_Cons__in__lists__iff,axiom,
    ! [X2: nat,Xs: list_nat,A2: set_nat] :
      ( ( member_list_nat @ ( cons_nat @ X2 @ Xs ) @ ( lists_nat @ A2 ) )
      = ( ( member_nat @ X2 @ A2 )
        & ( member_list_nat @ Xs @ ( lists_nat @ A2 ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_779_position__less__eq__Cons,axiom,
    ! [I: nat,Ps: list_nat,J: nat,Qs: list_nat] :
      ( ( term_p3503116865373065078eq_nat @ ( cons_nat @ I @ Ps ) @ ( cons_nat @ J @ Qs ) )
      = ( ( I = J )
        & ( term_p3503116865373065078eq_nat @ Ps @ Qs ) ) ) ).

% position_less_eq_Cons
thf(fact_780_position__diff__Cons,axiom,
    ! [I: nat,Ps: list_nat,Qs: list_nat] :
      ( ( term_pos_diff_nat @ ( cons_nat @ I @ Ps ) @ ( cons_nat @ I @ Qs ) )
      = ( term_pos_diff_nat @ Ps @ Qs ) ) ).

% position_diff_Cons
thf(fact_781_append1__eq__conv,axiom,
    ! [Xs: list_term_a_b,X2: term_a_b,Ys: list_term_a_b,Y: term_a_b] :
      ( ( ( append_term_a_b @ Xs @ ( cons_term_a_b @ X2 @ nil_term_a_b ) )
        = ( append_term_a_b @ Ys @ ( cons_term_a_b @ Y @ nil_term_a_b ) ) )
      = ( ( Xs = Ys )
        & ( X2 = Y ) ) ) ).

% append1_eq_conv
thf(fact_782_append1__eq__conv,axiom,
    ! [Xs: list_nat,X2: nat,Ys: list_nat,Y: nat] :
      ( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) )
        = ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
      = ( ( Xs = Ys )
        & ( X2 = Y ) ) ) ).

% append1_eq_conv
thf(fact_783_subst__monoid__mult_Oprod__list_OCons,axiom,
    ! [X2: b > term_a_b,Xs: list_b_term_a_b] :
      ( ( groups5454103906155650226rm_a_b @ var_b_a @ subst_1216682049913447528_a_b_b @ ( cons_b_term_a_b @ X2 @ Xs ) )
      = ( subst_1216682049913447528_a_b_b @ X2 @ ( groups5454103906155650226rm_a_b @ var_b_a @ subst_1216682049913447528_a_b_b @ Xs ) ) ) ).

% subst_monoid_mult.prod_list.Cons
thf(fact_784_lexord__cons__cons,axiom,
    ! [A: nat,X2: list_nat,B: nat,Y: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ A @ X2 ) @ ( cons_nat @ B @ Y ) ) @ ( lexord_nat @ R3 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
        | ( ( A = B )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X2 @ Y ) @ ( lexord_nat @ R3 ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_785_lexord__cons__cons,axiom,
    ! [A: term_a_b,X2: list_term_a_b,B: term_a_b,Y: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ A @ X2 ) @ ( cons_term_a_b @ B @ Y ) ) @ ( lexord_term_a_b @ R3 ) )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
        | ( ( A = B )
          & ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X2 @ Y ) @ ( lexord_term_a_b @ R3 ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_786_lexord__Nil__left,axiom,
    ! [Y: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ nil_term_a_b @ Y ) @ ( lexord_term_a_b @ R3 ) )
      = ( ? [A5: term_a_b,X3: list_term_a_b] :
            ( Y
            = ( cons_term_a_b @ A5 @ X3 ) ) ) ) ).

% lexord_Nil_left
thf(fact_787_lexord__Nil__left,axiom,
    ! [Y: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Y ) @ ( lexord_nat @ R3 ) )
      = ( ? [A5: nat,X3: list_nat] :
            ( Y
            = ( cons_nat @ A5 @ X3 ) ) ) ) ).

% lexord_Nil_left
thf(fact_788_successively_Ocases,axiom,
    ! [X2: produc8847469151730319409rm_a_b] :
      ( ! [P6: term_a_b > term_a_b > $o] :
          ( X2
         != ( produc5931190720334929579rm_a_b @ P6 @ nil_term_a_b ) )
     => ( ! [P6: term_a_b > term_a_b > $o,X: term_a_b] :
            ( X2
           != ( produc5931190720334929579rm_a_b @ P6 @ ( cons_term_a_b @ X @ nil_term_a_b ) ) )
       => ~ ! [P6: term_a_b > term_a_b > $o,X: term_a_b,Y2: term_a_b,Xs2: list_term_a_b] :
              ( X2
             != ( produc5931190720334929579rm_a_b @ P6 @ ( cons_term_a_b @ X @ ( cons_term_a_b @ Y2 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_789_successively_Ocases,axiom,
    ! [X2: produc254973753779126261st_nat] :
      ( ! [P6: nat > nat > $o] :
          ( X2
         != ( produc4727192421694094319st_nat @ P6 @ nil_nat ) )
     => ( ! [P6: nat > nat > $o,X: nat] :
            ( X2
           != ( produc4727192421694094319st_nat @ P6 @ ( cons_nat @ X @ nil_nat ) ) )
       => ~ ! [P6: nat > nat > $o,X: nat,Y2: nat,Xs2: list_nat] :
              ( X2
             != ( produc4727192421694094319st_nat @ P6 @ ( cons_nat @ X @ ( cons_nat @ Y2 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_790_list__nonempty__induct,axiom,
    ! [Xs: list_term_a_b,P2: list_term_a_b > $o] :
      ( ( Xs != nil_term_a_b )
     => ( ! [X: term_a_b] : ( P2 @ ( cons_term_a_b @ X @ nil_term_a_b ) )
       => ( ! [X: term_a_b,Xs2: list_term_a_b] :
              ( ( Xs2 != nil_term_a_b )
             => ( ( P2 @ Xs2 )
               => ( P2 @ ( cons_term_a_b @ X @ Xs2 ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_791_list__nonempty__induct,axiom,
    ! [Xs: list_nat,P2: list_nat > $o] :
      ( ( Xs != nil_nat )
     => ( ! [X: nat] : ( P2 @ ( cons_nat @ X @ nil_nat ) )
       => ( ! [X: nat,Xs2: list_nat] :
              ( ( Xs2 != nil_nat )
             => ( ( P2 @ Xs2 )
               => ( P2 @ ( cons_nat @ X @ Xs2 ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_792_list__induct2_H,axiom,
    ! [P2: list_term_a_b > list_term_a_b > $o,Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( P2 @ nil_term_a_b @ nil_term_a_b )
     => ( ! [X: term_a_b,Xs2: list_term_a_b] : ( P2 @ ( cons_term_a_b @ X @ Xs2 ) @ nil_term_a_b )
       => ( ! [Y2: term_a_b,Ys2: list_term_a_b] : ( P2 @ nil_term_a_b @ ( cons_term_a_b @ Y2 @ Ys2 ) )
         => ( ! [X: term_a_b,Xs2: list_term_a_b,Y2: term_a_b,Ys2: list_term_a_b] :
                ( ( P2 @ Xs2 @ Ys2 )
               => ( P2 @ ( cons_term_a_b @ X @ Xs2 ) @ ( cons_term_a_b @ Y2 @ Ys2 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_793_list__induct2_H,axiom,
    ! [P2: list_term_a_b > list_nat > $o,Xs: list_term_a_b,Ys: list_nat] :
      ( ( P2 @ nil_term_a_b @ nil_nat )
     => ( ! [X: term_a_b,Xs2: list_term_a_b] : ( P2 @ ( cons_term_a_b @ X @ Xs2 ) @ nil_nat )
       => ( ! [Y2: nat,Ys2: list_nat] : ( P2 @ nil_term_a_b @ ( cons_nat @ Y2 @ Ys2 ) )
         => ( ! [X: term_a_b,Xs2: list_term_a_b,Y2: nat,Ys2: list_nat] :
                ( ( P2 @ Xs2 @ Ys2 )
               => ( P2 @ ( cons_term_a_b @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_794_list__induct2_H,axiom,
    ! [P2: list_nat > list_term_a_b > $o,Xs: list_nat,Ys: list_term_a_b] :
      ( ( P2 @ nil_nat @ nil_term_a_b )
     => ( ! [X: nat,Xs2: list_nat] : ( P2 @ ( cons_nat @ X @ Xs2 ) @ nil_term_a_b )
       => ( ! [Y2: term_a_b,Ys2: list_term_a_b] : ( P2 @ nil_nat @ ( cons_term_a_b @ Y2 @ Ys2 ) )
         => ( ! [X: nat,Xs2: list_nat,Y2: term_a_b,Ys2: list_term_a_b] :
                ( ( P2 @ Xs2 @ Ys2 )
               => ( P2 @ ( cons_nat @ X @ Xs2 ) @ ( cons_term_a_b @ Y2 @ Ys2 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_795_list__induct2_H,axiom,
    ! [P2: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
      ( ( P2 @ nil_nat @ nil_nat )
     => ( ! [X: nat,Xs2: list_nat] : ( P2 @ ( cons_nat @ X @ Xs2 ) @ nil_nat )
       => ( ! [Y2: nat,Ys2: list_nat] : ( P2 @ nil_nat @ ( cons_nat @ Y2 @ Ys2 ) )
         => ( ! [X: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat] :
                ( ( P2 @ Xs2 @ Ys2 )
               => ( P2 @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_796_neq__Nil__conv,axiom,
    ! [Xs: list_term_a_b] :
      ( ( Xs != nil_term_a_b )
      = ( ? [Y4: term_a_b,Ys3: list_term_a_b] :
            ( Xs
            = ( cons_term_a_b @ Y4 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_797_neq__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
      = ( ? [Y4: nat,Ys3: list_nat] :
            ( Xs
            = ( cons_nat @ Y4 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_798_remdups__adj_Ocases,axiom,
    ! [X2: list_term_a_b] :
      ( ( X2 != nil_term_a_b )
     => ( ! [X: term_a_b] :
            ( X2
           != ( cons_term_a_b @ X @ nil_term_a_b ) )
       => ~ ! [X: term_a_b,Y2: term_a_b,Xs2: list_term_a_b] :
              ( X2
             != ( cons_term_a_b @ X @ ( cons_term_a_b @ Y2 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_799_remdups__adj_Ocases,axiom,
    ! [X2: list_nat] :
      ( ( X2 != nil_nat )
     => ( ! [X: nat] :
            ( X2
           != ( cons_nat @ X @ nil_nat ) )
       => ~ ! [X: nat,Y2: nat,Xs2: list_nat] :
              ( X2
             != ( cons_nat @ X @ ( cons_nat @ Y2 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_800_List_Omin__list_Ocases,axiom,
    ! [X2: list_nat] :
      ( ! [X: nat,Xs2: list_nat] :
          ( X2
         != ( cons_nat @ X @ Xs2 ) )
     => ( X2 = nil_nat ) ) ).

% List.min_list.cases
thf(fact_801_list_Oexhaust,axiom,
    ! [Y: list_term_a_b] :
      ( ( Y != nil_term_a_b )
     => ~ ! [X212: term_a_b,X222: list_term_a_b] :
            ( Y
           != ( cons_term_a_b @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_802_list_Oexhaust,axiom,
    ! [Y: list_nat] :
      ( ( Y != nil_nat )
     => ~ ! [X212: nat,X222: list_nat] :
            ( Y
           != ( cons_nat @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_803_list_OdiscI,axiom,
    ! [List: list_term_a_b,X21: term_a_b,X22: list_term_a_b] :
      ( ( List
        = ( cons_term_a_b @ X21 @ X22 ) )
     => ( List != nil_term_a_b ) ) ).

% list.discI
thf(fact_804_list_OdiscI,axiom,
    ! [List: list_nat,X21: nat,X22: list_nat] :
      ( ( List
        = ( cons_nat @ X21 @ X22 ) )
     => ( List != nil_nat ) ) ).

% list.discI
thf(fact_805_list_Odistinct_I1_J,axiom,
    ! [X21: term_a_b,X22: list_term_a_b] :
      ( nil_term_a_b
     != ( cons_term_a_b @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_806_list_Odistinct_I1_J,axiom,
    ! [X21: nat,X22: list_nat] :
      ( nil_nat
     != ( cons_nat @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_807_transpose_Ocases,axiom,
    ! [X2: list_list_term_a_b] :
      ( ( X2 != nil_list_term_a_b )
     => ( ! [Xss: list_list_term_a_b] :
            ( X2
           != ( cons_list_term_a_b @ nil_term_a_b @ Xss ) )
       => ~ ! [X: term_a_b,Xs2: list_term_a_b,Xss: list_list_term_a_b] :
              ( X2
             != ( cons_list_term_a_b @ ( cons_term_a_b @ X @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_808_transpose_Ocases,axiom,
    ! [X2: list_list_nat] :
      ( ( X2 != nil_list_nat )
     => ( ! [Xss: list_list_nat] :
            ( X2
           != ( cons_list_nat @ nil_nat @ Xss ) )
       => ~ ! [X: nat,Xs2: list_nat,Xss: list_list_nat] :
              ( X2
             != ( cons_list_nat @ ( cons_nat @ X @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_809_Cons__eq__appendI,axiom,
    ! [X2: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs: list_nat] :
      ( ( ( cons_nat @ X2 @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_nat @ Xs1 @ Zs ) )
       => ( ( cons_nat @ X2 @ Xs )
          = ( append_nat @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_810_append__Cons,axiom,
    ! [X2: nat,Xs: list_nat,Ys: list_nat] :
      ( ( append_nat @ ( cons_nat @ X2 @ Xs ) @ Ys )
      = ( cons_nat @ X2 @ ( append_nat @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_811_par__Cons__iff,axiom,
    ! [I: nat,Ps: list_nat,J: nat,Qs: list_nat] :
      ( ( term_p5017330785391824242ar_nat @ ( cons_nat @ I @ Ps ) @ ( cons_nat @ J @ Qs ) )
      = ( ( I != J )
        | ( term_p5017330785391824242ar_nat @ Ps @ Qs ) ) ) ).

% par_Cons_iff
thf(fact_812_par__pos__prefix,axiom,
    ! [I: nat,P: list_nat,Q: list_nat] :
      ( ( term_p5017330785391824242ar_nat @ ( cons_nat @ I @ P ) @ ( cons_nat @ I @ Q ) )
     => ( term_p5017330785391824242ar_nat @ P @ Q ) ) ).

% par_pos_prefix
thf(fact_813_map__eq__Cons__conv,axiom,
    ! [F2: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F2 @ Xs )
        = ( cons_nat @ Y @ Ys ) )
      = ( ? [Z4: nat,Zs2: list_nat] :
            ( ( Xs
              = ( cons_nat @ Z4 @ Zs2 ) )
            & ( ( F2 @ Z4 )
              = Y )
            & ( ( map_nat_nat @ F2 @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_814_Cons__eq__map__conv,axiom,
    ! [X2: nat,Xs: list_nat,F2: nat > nat,Ys: list_nat] :
      ( ( ( cons_nat @ X2 @ Xs )
        = ( map_nat_nat @ F2 @ Ys ) )
      = ( ? [Z4: nat,Zs2: list_nat] :
            ( ( Ys
              = ( cons_nat @ Z4 @ Zs2 ) )
            & ( X2
              = ( F2 @ Z4 ) )
            & ( Xs
              = ( map_nat_nat @ F2 @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_815_map__eq__Cons__D,axiom,
    ! [F2: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F2 @ Xs )
        = ( cons_nat @ Y @ Ys ) )
     => ? [Z3: nat,Zs3: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z3 @ Zs3 ) )
          & ( ( F2 @ Z3 )
            = Y )
          & ( ( map_nat_nat @ F2 @ Zs3 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_816_Cons__eq__map__D,axiom,
    ! [X2: nat,Xs: list_nat,F2: nat > nat,Ys: list_nat] :
      ( ( ( cons_nat @ X2 @ Xs )
        = ( map_nat_nat @ F2 @ Ys ) )
     => ? [Z3: nat,Zs3: list_nat] :
          ( ( Ys
            = ( cons_nat @ Z3 @ Zs3 ) )
          & ( X2
            = ( F2 @ Z3 ) )
          & ( Xs
            = ( map_nat_nat @ F2 @ Zs3 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_817_list_Osimps_I9_J,axiom,
    ! [F2: nat > nat,X21: nat,X22: list_nat] :
      ( ( map_nat_nat @ F2 @ ( cons_nat @ X21 @ X22 ) )
      = ( cons_nat @ ( F2 @ X21 ) @ ( map_nat_nat @ F2 @ X22 ) ) ) ).

% list.simps(9)
thf(fact_818_lists_OCons,axiom,
    ! [A: list_nat,A2: set_list_nat,L2: list_list_nat] :
      ( ( member_list_nat @ A @ A2 )
     => ( ( member_list_list_nat @ L2 @ ( lists_list_nat @ A2 ) )
       => ( member_list_list_nat @ ( cons_list_nat @ A @ L2 ) @ ( lists_list_nat @ A2 ) ) ) ) ).

% lists.Cons
thf(fact_819_lists_OCons,axiom,
    ! [A: b,A2: set_b,L2: list_b] :
      ( ( member_b @ A @ A2 )
     => ( ( member_list_b @ L2 @ ( lists_b @ A2 ) )
       => ( member_list_b @ ( cons_b @ A @ L2 ) @ ( lists_b @ A2 ) ) ) ) ).

% lists.Cons
thf(fact_820_lists_OCons,axiom,
    ! [A: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,L2: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ A2 )
     => ( ( member7646937072540562872rm_a_b @ L2 @ ( lists_3936190290964281341rm_a_b @ A2 ) )
       => ( member7646937072540562872rm_a_b @ ( cons_P3536305108106557631rm_a_b @ A @ L2 ) @ ( lists_3936190290964281341rm_a_b @ A2 ) ) ) ) ).

% lists.Cons
thf(fact_821_lists_OCons,axiom,
    ! [A: product_prod_a_nat,A2: set_Pr4934435412358123699_a_nat,L2: list_P3592885314253461005_a_nat] :
      ( ( member5724188588386418708_a_nat @ A @ A2 )
     => ( ( member4041938288099225636_a_nat @ L2 @ ( lists_3983152277583228297_a_nat @ A2 ) )
       => ( member4041938288099225636_a_nat @ ( cons_P5205166803686508359_a_nat @ A @ L2 ) @ ( lists_3983152277583228297_a_nat @ A2 ) ) ) ) ).

% lists.Cons
thf(fact_822_lists_OCons,axiom,
    ! [A: nat,A2: set_nat,L2: list_nat] :
      ( ( member_nat @ A @ A2 )
     => ( ( member_list_nat @ L2 @ ( lists_nat @ A2 ) )
       => ( member_list_nat @ ( cons_nat @ A @ L2 ) @ ( lists_nat @ A2 ) ) ) ) ).

% lists.Cons
thf(fact_823_listsE,axiom,
    ! [X2: list_nat,L2: list_list_nat,A2: set_list_nat] :
      ( ( member_list_list_nat @ ( cons_list_nat @ X2 @ L2 ) @ ( lists_list_nat @ A2 ) )
     => ~ ( ( member_list_nat @ X2 @ A2 )
         => ~ ( member_list_list_nat @ L2 @ ( lists_list_nat @ A2 ) ) ) ) ).

% listsE
thf(fact_824_listsE,axiom,
    ! [X2: b,L2: list_b,A2: set_b] :
      ( ( member_list_b @ ( cons_b @ X2 @ L2 ) @ ( lists_b @ A2 ) )
     => ~ ( ( member_b @ X2 @ A2 )
         => ~ ( member_list_b @ L2 @ ( lists_b @ A2 ) ) ) ) ).

% listsE
thf(fact_825_listsE,axiom,
    ! [X2: produc357393685978478089rm_a_b,L2: list_P8875379029341186191rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
      ( ( member7646937072540562872rm_a_b @ ( cons_P3536305108106557631rm_a_b @ X2 @ L2 ) @ ( lists_3936190290964281341rm_a_b @ A2 ) )
     => ~ ( ( member5869715511025134514rm_a_b @ X2 @ A2 )
         => ~ ( member7646937072540562872rm_a_b @ L2 @ ( lists_3936190290964281341rm_a_b @ A2 ) ) ) ) ).

% listsE
thf(fact_826_listsE,axiom,
    ! [X2: product_prod_a_nat,L2: list_P3592885314253461005_a_nat,A2: set_Pr4934435412358123699_a_nat] :
      ( ( member4041938288099225636_a_nat @ ( cons_P5205166803686508359_a_nat @ X2 @ L2 ) @ ( lists_3983152277583228297_a_nat @ A2 ) )
     => ~ ( ( member5724188588386418708_a_nat @ X2 @ A2 )
         => ~ ( member4041938288099225636_a_nat @ L2 @ ( lists_3983152277583228297_a_nat @ A2 ) ) ) ) ).

% listsE
thf(fact_827_listsE,axiom,
    ! [X2: nat,L2: list_nat,A2: set_nat] :
      ( ( member_list_nat @ ( cons_nat @ X2 @ L2 ) @ ( lists_nat @ A2 ) )
     => ~ ( ( member_nat @ X2 @ A2 )
         => ~ ( member_list_nat @ L2 @ ( lists_nat @ A2 ) ) ) ) ).

% listsE
thf(fact_828_not__Cons__self2,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( cons_nat @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_829_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ws: list_nat,P2: list_nat > list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P2 @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
           => ( ! [X: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat,Z3: nat,Zs3: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_nat @ Ys2 ) )
                 => ( ( ( size_size_list_nat @ Ys2 )
                      = ( size_size_list_nat @ Zs3 ) )
                   => ( ( ( size_size_list_nat @ Zs3 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P2 @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
                       => ( P2 @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_830_list__induct4,axiom,
    ! [Xs: list_term_a_b,Ys: list_nat,Zs: list_nat,Ws: list_nat,P2: list_term_a_b > list_nat > list_nat > list_nat > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P2 @ nil_term_a_b @ nil_nat @ nil_nat @ nil_nat )
           => ( ! [X: term_a_b,Xs2: list_term_a_b,Y2: nat,Ys2: list_nat,Z3: nat,Zs3: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
                    = ( size_size_list_nat @ Ys2 ) )
                 => ( ( ( size_size_list_nat @ Ys2 )
                      = ( size_size_list_nat @ Zs3 ) )
                   => ( ( ( size_size_list_nat @ Zs3 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P2 @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
                       => ( P2 @ ( cons_term_a_b @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_831_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_term_a_b,Zs: list_nat,Ws: list_nat,P2: list_nat > list_term_a_b > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P2 @ nil_nat @ nil_term_a_b @ nil_nat @ nil_nat )
           => ( ! [X: nat,Xs2: list_nat,Y2: term_a_b,Ys2: list_term_a_b,Z3: nat,Zs3: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_s8906293707977694520rm_a_b @ Ys2 ) )
                 => ( ( ( size_s8906293707977694520rm_a_b @ Ys2 )
                      = ( size_size_list_nat @ Zs3 ) )
                   => ( ( ( size_size_list_nat @ Zs3 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P2 @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
                       => ( P2 @ ( cons_nat @ X @ Xs2 ) @ ( cons_term_a_b @ Y2 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_832_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_term_a_b,Ws: list_nat,P2: list_nat > list_nat > list_term_a_b > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s8906293707977694520rm_a_b @ Zs ) )
       => ( ( ( size_s8906293707977694520rm_a_b @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P2 @ nil_nat @ nil_nat @ nil_term_a_b @ nil_nat )
           => ( ! [X: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat,Z3: term_a_b,Zs3: list_term_a_b,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_nat @ Ys2 ) )
                 => ( ( ( size_size_list_nat @ Ys2 )
                      = ( size_s8906293707977694520rm_a_b @ Zs3 ) )
                   => ( ( ( size_s8906293707977694520rm_a_b @ Zs3 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P2 @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
                       => ( P2 @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_term_a_b @ Z3 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_833_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ws: list_term_a_b,P2: list_nat > list_nat > list_nat > list_term_a_b > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_s8906293707977694520rm_a_b @ Ws ) )
         => ( ( P2 @ nil_nat @ nil_nat @ nil_nat @ nil_term_a_b )
           => ( ! [X: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat,Z3: nat,Zs3: list_nat,W: term_a_b,Ws2: list_term_a_b] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_nat @ Ys2 ) )
                 => ( ( ( size_size_list_nat @ Ys2 )
                      = ( size_size_list_nat @ Zs3 ) )
                   => ( ( ( size_size_list_nat @ Zs3 )
                        = ( size_s8906293707977694520rm_a_b @ Ws2 ) )
                     => ( ( P2 @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
                       => ( P2 @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs3 ) @ ( cons_term_a_b @ W @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_834_list__induct4,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b,Zs: list_nat,Ws: list_nat,P2: list_term_a_b > list_term_a_b > list_nat > list_nat > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P2 @ nil_term_a_b @ nil_term_a_b @ nil_nat @ nil_nat )
           => ( ! [X: term_a_b,Xs2: list_term_a_b,Y2: term_a_b,Ys2: list_term_a_b,Z3: nat,Zs3: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
                    = ( size_s8906293707977694520rm_a_b @ Ys2 ) )
                 => ( ( ( size_s8906293707977694520rm_a_b @ Ys2 )
                      = ( size_size_list_nat @ Zs3 ) )
                   => ( ( ( size_size_list_nat @ Zs3 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P2 @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
                       => ( P2 @ ( cons_term_a_b @ X @ Xs2 ) @ ( cons_term_a_b @ Y2 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_835_list__induct4,axiom,
    ! [Xs: list_term_a_b,Ys: list_nat,Zs: list_term_a_b,Ws: list_nat,P2: list_term_a_b > list_nat > list_term_a_b > list_nat > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s8906293707977694520rm_a_b @ Zs ) )
       => ( ( ( size_s8906293707977694520rm_a_b @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P2 @ nil_term_a_b @ nil_nat @ nil_term_a_b @ nil_nat )
           => ( ! [X: term_a_b,Xs2: list_term_a_b,Y2: nat,Ys2: list_nat,Z3: term_a_b,Zs3: list_term_a_b,W: nat,Ws2: list_nat] :
                  ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
                    = ( size_size_list_nat @ Ys2 ) )
                 => ( ( ( size_size_list_nat @ Ys2 )
                      = ( size_s8906293707977694520rm_a_b @ Zs3 ) )
                   => ( ( ( size_s8906293707977694520rm_a_b @ Zs3 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P2 @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
                       => ( P2 @ ( cons_term_a_b @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_term_a_b @ Z3 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_836_list__induct4,axiom,
    ! [Xs: list_term_a_b,Ys: list_nat,Zs: list_nat,Ws: list_term_a_b,P2: list_term_a_b > list_nat > list_nat > list_term_a_b > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_s8906293707977694520rm_a_b @ Ws ) )
         => ( ( P2 @ nil_term_a_b @ nil_nat @ nil_nat @ nil_term_a_b )
           => ( ! [X: term_a_b,Xs2: list_term_a_b,Y2: nat,Ys2: list_nat,Z3: nat,Zs3: list_nat,W: term_a_b,Ws2: list_term_a_b] :
                  ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
                    = ( size_size_list_nat @ Ys2 ) )
                 => ( ( ( size_size_list_nat @ Ys2 )
                      = ( size_size_list_nat @ Zs3 ) )
                   => ( ( ( size_size_list_nat @ Zs3 )
                        = ( size_s8906293707977694520rm_a_b @ Ws2 ) )
                     => ( ( P2 @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
                       => ( P2 @ ( cons_term_a_b @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs3 ) @ ( cons_term_a_b @ W @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_837_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_term_a_b,Zs: list_term_a_b,Ws: list_nat,P2: list_nat > list_term_a_b > list_term_a_b > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Ys )
          = ( size_s8906293707977694520rm_a_b @ Zs ) )
       => ( ( ( size_s8906293707977694520rm_a_b @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P2 @ nil_nat @ nil_term_a_b @ nil_term_a_b @ nil_nat )
           => ( ! [X: nat,Xs2: list_nat,Y2: term_a_b,Ys2: list_term_a_b,Z3: term_a_b,Zs3: list_term_a_b,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_s8906293707977694520rm_a_b @ Ys2 ) )
                 => ( ( ( size_s8906293707977694520rm_a_b @ Ys2 )
                      = ( size_s8906293707977694520rm_a_b @ Zs3 ) )
                   => ( ( ( size_s8906293707977694520rm_a_b @ Zs3 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P2 @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
                       => ( P2 @ ( cons_nat @ X @ Xs2 ) @ ( cons_term_a_b @ Y2 @ Ys2 ) @ ( cons_term_a_b @ Z3 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_838_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_term_a_b,Zs: list_nat,Ws: list_term_a_b,P2: list_nat > list_term_a_b > list_nat > list_term_a_b > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_s8906293707977694520rm_a_b @ Ws ) )
         => ( ( P2 @ nil_nat @ nil_term_a_b @ nil_nat @ nil_term_a_b )
           => ( ! [X: nat,Xs2: list_nat,Y2: term_a_b,Ys2: list_term_a_b,Z3: nat,Zs3: list_nat,W: term_a_b,Ws2: list_term_a_b] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_s8906293707977694520rm_a_b @ Ys2 ) )
                 => ( ( ( size_s8906293707977694520rm_a_b @ Ys2 )
                      = ( size_size_list_nat @ Zs3 ) )
                   => ( ( ( size_size_list_nat @ Zs3 )
                        = ( size_s8906293707977694520rm_a_b @ Ws2 ) )
                     => ( ( P2 @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
                       => ( P2 @ ( cons_nat @ X @ Xs2 ) @ ( cons_term_a_b @ Y2 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs3 ) @ ( cons_term_a_b @ W @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_839_list__induct3,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b,Zs: list_term_a_b,P2: list_term_a_b > list_term_a_b > list_term_a_b > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Ys )
          = ( size_s8906293707977694520rm_a_b @ Zs ) )
       => ( ( P2 @ nil_term_a_b @ nil_term_a_b @ nil_term_a_b )
         => ( ! [X: term_a_b,Xs2: list_term_a_b,Y2: term_a_b,Ys2: list_term_a_b,Z3: term_a_b,Zs3: list_term_a_b] :
                ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
                  = ( size_s8906293707977694520rm_a_b @ Ys2 ) )
               => ( ( ( size_s8906293707977694520rm_a_b @ Ys2 )
                    = ( size_s8906293707977694520rm_a_b @ Zs3 ) )
                 => ( ( P2 @ Xs2 @ Ys2 @ Zs3 )
                   => ( P2 @ ( cons_term_a_b @ X @ Xs2 ) @ ( cons_term_a_b @ Y2 @ Ys2 ) @ ( cons_term_a_b @ Z3 @ Zs3 ) ) ) ) )
           => ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_840_list__induct3,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b,Zs: list_nat,P2: list_term_a_b > list_term_a_b > list_nat > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P2 @ nil_term_a_b @ nil_term_a_b @ nil_nat )
         => ( ! [X: term_a_b,Xs2: list_term_a_b,Y2: term_a_b,Ys2: list_term_a_b,Z3: nat,Zs3: list_nat] :
                ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
                  = ( size_s8906293707977694520rm_a_b @ Ys2 ) )
               => ( ( ( size_s8906293707977694520rm_a_b @ Ys2 )
                    = ( size_size_list_nat @ Zs3 ) )
                 => ( ( P2 @ Xs2 @ Ys2 @ Zs3 )
                   => ( P2 @ ( cons_term_a_b @ X @ Xs2 ) @ ( cons_term_a_b @ Y2 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs3 ) ) ) ) )
           => ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_841_list__induct3,axiom,
    ! [Xs: list_term_a_b,Ys: list_nat,Zs: list_term_a_b,P2: list_term_a_b > list_nat > list_term_a_b > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s8906293707977694520rm_a_b @ Zs ) )
       => ( ( P2 @ nil_term_a_b @ nil_nat @ nil_term_a_b )
         => ( ! [X: term_a_b,Xs2: list_term_a_b,Y2: nat,Ys2: list_nat,Z3: term_a_b,Zs3: list_term_a_b] :
                ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
                  = ( size_size_list_nat @ Ys2 ) )
               => ( ( ( size_size_list_nat @ Ys2 )
                    = ( size_s8906293707977694520rm_a_b @ Zs3 ) )
                 => ( ( P2 @ Xs2 @ Ys2 @ Zs3 )
                   => ( P2 @ ( cons_term_a_b @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_term_a_b @ Z3 @ Zs3 ) ) ) ) )
           => ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_842_list__induct3,axiom,
    ! [Xs: list_term_a_b,Ys: list_nat,Zs: list_nat,P2: list_term_a_b > list_nat > list_nat > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P2 @ nil_term_a_b @ nil_nat @ nil_nat )
         => ( ! [X: term_a_b,Xs2: list_term_a_b,Y2: nat,Ys2: list_nat,Z3: nat,Zs3: list_nat] :
                ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
                  = ( size_size_list_nat @ Ys2 ) )
               => ( ( ( size_size_list_nat @ Ys2 )
                    = ( size_size_list_nat @ Zs3 ) )
                 => ( ( P2 @ Xs2 @ Ys2 @ Zs3 )
                   => ( P2 @ ( cons_term_a_b @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs3 ) ) ) ) )
           => ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_843_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_term_a_b,Zs: list_term_a_b,P2: list_nat > list_term_a_b > list_term_a_b > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Ys )
          = ( size_s8906293707977694520rm_a_b @ Zs ) )
       => ( ( P2 @ nil_nat @ nil_term_a_b @ nil_term_a_b )
         => ( ! [X: nat,Xs2: list_nat,Y2: term_a_b,Ys2: list_term_a_b,Z3: term_a_b,Zs3: list_term_a_b] :
                ( ( ( size_size_list_nat @ Xs2 )
                  = ( size_s8906293707977694520rm_a_b @ Ys2 ) )
               => ( ( ( size_s8906293707977694520rm_a_b @ Ys2 )
                    = ( size_s8906293707977694520rm_a_b @ Zs3 ) )
                 => ( ( P2 @ Xs2 @ Ys2 @ Zs3 )
                   => ( P2 @ ( cons_nat @ X @ Xs2 ) @ ( cons_term_a_b @ Y2 @ Ys2 ) @ ( cons_term_a_b @ Z3 @ Zs3 ) ) ) ) )
           => ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_844_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_term_a_b,Zs: list_nat,P2: list_nat > list_term_a_b > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P2 @ nil_nat @ nil_term_a_b @ nil_nat )
         => ( ! [X: nat,Xs2: list_nat,Y2: term_a_b,Ys2: list_term_a_b,Z3: nat,Zs3: list_nat] :
                ( ( ( size_size_list_nat @ Xs2 )
                  = ( size_s8906293707977694520rm_a_b @ Ys2 ) )
               => ( ( ( size_s8906293707977694520rm_a_b @ Ys2 )
                    = ( size_size_list_nat @ Zs3 ) )
                 => ( ( P2 @ Xs2 @ Ys2 @ Zs3 )
                   => ( P2 @ ( cons_nat @ X @ Xs2 ) @ ( cons_term_a_b @ Y2 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs3 ) ) ) ) )
           => ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_845_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_term_a_b,P2: list_nat > list_nat > list_term_a_b > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s8906293707977694520rm_a_b @ Zs ) )
       => ( ( P2 @ nil_nat @ nil_nat @ nil_term_a_b )
         => ( ! [X: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat,Z3: term_a_b,Zs3: list_term_a_b] :
                ( ( ( size_size_list_nat @ Xs2 )
                  = ( size_size_list_nat @ Ys2 ) )
               => ( ( ( size_size_list_nat @ Ys2 )
                    = ( size_s8906293707977694520rm_a_b @ Zs3 ) )
                 => ( ( P2 @ Xs2 @ Ys2 @ Zs3 )
                   => ( P2 @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_term_a_b @ Z3 @ Zs3 ) ) ) ) )
           => ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_846_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,P2: list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P2 @ nil_nat @ nil_nat @ nil_nat )
         => ( ! [X: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat,Z3: nat,Zs3: list_nat] :
                ( ( ( size_size_list_nat @ Xs2 )
                  = ( size_size_list_nat @ Ys2 ) )
               => ( ( ( size_size_list_nat @ Ys2 )
                    = ( size_size_list_nat @ Zs3 ) )
                 => ( ( P2 @ Xs2 @ Ys2 @ Zs3 )
                   => ( P2 @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs3 ) ) ) ) )
           => ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_847_list__induct2,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b,P2: list_term_a_b > list_term_a_b > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( P2 @ nil_term_a_b @ nil_term_a_b )
       => ( ! [X: term_a_b,Xs2: list_term_a_b,Y2: term_a_b,Ys2: list_term_a_b] :
              ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
                = ( size_s8906293707977694520rm_a_b @ Ys2 ) )
             => ( ( P2 @ Xs2 @ Ys2 )
               => ( P2 @ ( cons_term_a_b @ X @ Xs2 ) @ ( cons_term_a_b @ Y2 @ Ys2 ) ) ) )
         => ( P2 @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_848_list__induct2,axiom,
    ! [Xs: list_term_a_b,Ys: list_nat,P2: list_term_a_b > list_nat > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P2 @ nil_term_a_b @ nil_nat )
       => ( ! [X: term_a_b,Xs2: list_term_a_b,Y2: nat,Ys2: list_nat] :
              ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
                = ( size_size_list_nat @ Ys2 ) )
             => ( ( P2 @ Xs2 @ Ys2 )
               => ( P2 @ ( cons_term_a_b @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) ) )
         => ( P2 @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_849_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_term_a_b,P2: list_nat > list_term_a_b > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( P2 @ nil_nat @ nil_term_a_b )
       => ( ! [X: nat,Xs2: list_nat,Y2: term_a_b,Ys2: list_term_a_b] :
              ( ( ( size_size_list_nat @ Xs2 )
                = ( size_s8906293707977694520rm_a_b @ Ys2 ) )
             => ( ( P2 @ Xs2 @ Ys2 )
               => ( P2 @ ( cons_nat @ X @ Xs2 ) @ ( cons_term_a_b @ Y2 @ Ys2 ) ) ) )
         => ( P2 @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_850_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_nat,P2: list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P2 @ nil_nat @ nil_nat )
       => ( ! [X: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat] :
              ( ( ( size_size_list_nat @ Xs2 )
                = ( size_size_list_nat @ Ys2 ) )
             => ( ( P2 @ Xs2 @ Ys2 )
               => ( P2 @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) ) )
         => ( P2 @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_851_impossible__Cons,axiom,
    ! [Xs: list_nat,Ys: list_nat,X2: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
     => ( Xs
       != ( cons_nat @ X2 @ Ys ) ) ) ).

% impossible_Cons
thf(fact_852_rev__nonempty__induct,axiom,
    ! [Xs: list_term_a_b,P2: list_term_a_b > $o] :
      ( ( Xs != nil_term_a_b )
     => ( ! [X: term_a_b] : ( P2 @ ( cons_term_a_b @ X @ nil_term_a_b ) )
       => ( ! [X: term_a_b,Xs2: list_term_a_b] :
              ( ( Xs2 != nil_term_a_b )
             => ( ( P2 @ Xs2 )
               => ( P2 @ ( append_term_a_b @ Xs2 @ ( cons_term_a_b @ X @ nil_term_a_b ) ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_853_rev__nonempty__induct,axiom,
    ! [Xs: list_nat,P2: list_nat > $o] :
      ( ( Xs != nil_nat )
     => ( ! [X: nat] : ( P2 @ ( cons_nat @ X @ nil_nat ) )
       => ( ! [X: nat,Xs2: list_nat] :
              ( ( Xs2 != nil_nat )
             => ( ( P2 @ Xs2 )
               => ( P2 @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_854_append__eq__Cons__conv,axiom,
    ! [Ys: list_term_a_b,Zs: list_term_a_b,X2: term_a_b,Xs: list_term_a_b] :
      ( ( ( append_term_a_b @ Ys @ Zs )
        = ( cons_term_a_b @ X2 @ Xs ) )
      = ( ( ( Ys = nil_term_a_b )
          & ( Zs
            = ( cons_term_a_b @ X2 @ Xs ) ) )
        | ? [Ys4: list_term_a_b] :
            ( ( Ys
              = ( cons_term_a_b @ X2 @ Ys4 ) )
            & ( ( append_term_a_b @ Ys4 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_855_append__eq__Cons__conv,axiom,
    ! [Ys: list_nat,Zs: list_nat,X2: nat,Xs: list_nat] :
      ( ( ( append_nat @ Ys @ Zs )
        = ( cons_nat @ X2 @ Xs ) )
      = ( ( ( Ys = nil_nat )
          & ( Zs
            = ( cons_nat @ X2 @ Xs ) ) )
        | ? [Ys4: list_nat] :
            ( ( Ys
              = ( cons_nat @ X2 @ Ys4 ) )
            & ( ( append_nat @ Ys4 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_856_Cons__eq__append__conv,axiom,
    ! [X2: term_a_b,Xs: list_term_a_b,Ys: list_term_a_b,Zs: list_term_a_b] :
      ( ( ( cons_term_a_b @ X2 @ Xs )
        = ( append_term_a_b @ Ys @ Zs ) )
      = ( ( ( Ys = nil_term_a_b )
          & ( ( cons_term_a_b @ X2 @ Xs )
            = Zs ) )
        | ? [Ys4: list_term_a_b] :
            ( ( ( cons_term_a_b @ X2 @ Ys4 )
              = Ys )
            & ( Xs
              = ( append_term_a_b @ Ys4 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_857_Cons__eq__append__conv,axiom,
    ! [X2: nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( cons_nat @ X2 @ Xs )
        = ( append_nat @ Ys @ Zs ) )
      = ( ( ( Ys = nil_nat )
          & ( ( cons_nat @ X2 @ Xs )
            = Zs ) )
        | ? [Ys4: list_nat] :
            ( ( ( cons_nat @ X2 @ Ys4 )
              = Ys )
            & ( Xs
              = ( append_nat @ Ys4 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_858_rev__exhaust,axiom,
    ! [Xs: list_term_a_b] :
      ( ( Xs != nil_term_a_b )
     => ~ ! [Ys2: list_term_a_b,Y2: term_a_b] :
            ( Xs
           != ( append_term_a_b @ Ys2 @ ( cons_term_a_b @ Y2 @ nil_term_a_b ) ) ) ) ).

% rev_exhaust
thf(fact_859_rev__exhaust,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ~ ! [Ys2: list_nat,Y2: nat] :
            ( Xs
           != ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) ).

% rev_exhaust
thf(fact_860_rev__induct,axiom,
    ! [P2: list_term_a_b > $o,Xs: list_term_a_b] :
      ( ( P2 @ nil_term_a_b )
     => ( ! [X: term_a_b,Xs2: list_term_a_b] :
            ( ( P2 @ Xs2 )
           => ( P2 @ ( append_term_a_b @ Xs2 @ ( cons_term_a_b @ X @ nil_term_a_b ) ) ) )
       => ( P2 @ Xs ) ) ) ).

% rev_induct
thf(fact_861_rev__induct,axiom,
    ! [P2: list_nat > $o,Xs: list_nat] :
      ( ( P2 @ nil_nat )
     => ( ! [X: nat,Xs2: list_nat] :
            ( ( P2 @ Xs2 )
           => ( P2 @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) ) )
       => ( P2 @ Xs ) ) ) ).

% rev_induct
thf(fact_862_shuffles_Ocases,axiom,
    ! [X2: produc51424535725745577rm_a_b] :
      ( ! [Ys2: list_term_a_b] :
          ( X2
         != ( produc4885699992713594593rm_a_b @ nil_term_a_b @ Ys2 ) )
     => ( ! [Xs2: list_term_a_b] :
            ( X2
           != ( produc4885699992713594593rm_a_b @ Xs2 @ nil_term_a_b ) )
       => ~ ! [X: term_a_b,Xs2: list_term_a_b,Y2: term_a_b,Ys2: list_term_a_b] :
              ( X2
             != ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X @ Xs2 ) @ ( cons_term_a_b @ Y2 @ Ys2 ) ) ) ) ) ).

% shuffles.cases
thf(fact_863_shuffles_Ocases,axiom,
    ! [X2: produc1828647624359046049st_nat] :
      ( ! [Ys2: list_nat] :
          ( X2
         != ( produc2694037385005941721st_nat @ nil_nat @ Ys2 ) )
     => ( ! [Xs2: list_nat] :
            ( X2
           != ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) )
       => ~ ! [X: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat] :
              ( X2
             != ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) ) ) ) ).

% shuffles.cases
thf(fact_864_remove__prefix_Ocases,axiom,
    ! [X2: produc51424535725745577rm_a_b] :
      ( ! [X: term_a_b,Xs2: list_term_a_b,Y2: term_a_b,Ys2: list_term_a_b] :
          ( X2
         != ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X @ Xs2 ) @ ( cons_term_a_b @ Y2 @ Ys2 ) ) )
     => ( ! [Ys2: list_term_a_b] :
            ( X2
           != ( produc4885699992713594593rm_a_b @ nil_term_a_b @ Ys2 ) )
       => ~ ! [V: term_a_b,Va: list_term_a_b] :
              ( X2
             != ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ V @ Va ) @ nil_term_a_b ) ) ) ) ).

% remove_prefix.cases
thf(fact_865_remove__prefix_Ocases,axiom,
    ! [X2: produc1828647624359046049st_nat] :
      ( ! [X: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat] :
          ( X2
         != ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
     => ( ! [Ys2: list_nat] :
            ( X2
           != ( produc2694037385005941721st_nat @ nil_nat @ Ys2 ) )
       => ~ ! [V: nat,Va: list_nat] :
              ( X2
             != ( produc2694037385005941721st_nat @ ( cons_nat @ V @ Va ) @ nil_nat ) ) ) ) ).

% remove_prefix.cases
thf(fact_866_lists_Ocases,axiom,
    ! [A: list_list_nat,A2: set_list_nat] :
      ( ( member_list_list_nat @ A @ ( lists_list_nat @ A2 ) )
     => ( ( A != nil_list_nat )
       => ~ ! [A4: list_nat,L: list_list_nat] :
              ( ( A
                = ( cons_list_nat @ A4 @ L ) )
             => ( ( member_list_nat @ A4 @ A2 )
               => ~ ( member_list_list_nat @ L @ ( lists_list_nat @ A2 ) ) ) ) ) ) ).

% lists.cases
thf(fact_867_lists_Ocases,axiom,
    ! [A: list_b,A2: set_b] :
      ( ( member_list_b @ A @ ( lists_b @ A2 ) )
     => ( ( A != nil_b )
       => ~ ! [A4: b,L: list_b] :
              ( ( A
                = ( cons_b @ A4 @ L ) )
             => ( ( member_b @ A4 @ A2 )
               => ~ ( member_list_b @ L @ ( lists_b @ A2 ) ) ) ) ) ) ).

% lists.cases
thf(fact_868_lists_Ocases,axiom,
    ! [A: list_P8875379029341186191rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
      ( ( member7646937072540562872rm_a_b @ A @ ( lists_3936190290964281341rm_a_b @ A2 ) )
     => ( ( A != nil_Pr6942174756412032271rm_a_b )
       => ~ ! [A4: produc357393685978478089rm_a_b,L: list_P8875379029341186191rm_a_b] :
              ( ( A
                = ( cons_P3536305108106557631rm_a_b @ A4 @ L ) )
             => ( ( member5869715511025134514rm_a_b @ A4 @ A2 )
               => ~ ( member7646937072540562872rm_a_b @ L @ ( lists_3936190290964281341rm_a_b @ A2 ) ) ) ) ) ) ).

% lists.cases
thf(fact_869_lists_Ocases,axiom,
    ! [A: list_P3592885314253461005_a_nat,A2: set_Pr4934435412358123699_a_nat] :
      ( ( member4041938288099225636_a_nat @ A @ ( lists_3983152277583228297_a_nat @ A2 ) )
     => ( ( A != nil_Pr7402525243500994295_a_nat )
       => ~ ! [A4: product_prod_a_nat,L: list_P3592885314253461005_a_nat] :
              ( ( A
                = ( cons_P5205166803686508359_a_nat @ A4 @ L ) )
             => ( ( member5724188588386418708_a_nat @ A4 @ A2 )
               => ~ ( member4041938288099225636_a_nat @ L @ ( lists_3983152277583228297_a_nat @ A2 ) ) ) ) ) ) ).

% lists.cases
thf(fact_870_lists_Ocases,axiom,
    ! [A: list_term_a_b,A2: set_term_a_b] :
      ( ( member_list_term_a_b @ A @ ( lists_term_a_b @ A2 ) )
     => ( ( A != nil_term_a_b )
       => ~ ! [A4: term_a_b,L: list_term_a_b] :
              ( ( A
                = ( cons_term_a_b @ A4 @ L ) )
             => ( ( member_term_a_b @ A4 @ A2 )
               => ~ ( member_list_term_a_b @ L @ ( lists_term_a_b @ A2 ) ) ) ) ) ) ).

% lists.cases
thf(fact_871_lists_Ocases,axiom,
    ! [A: list_nat,A2: set_nat] :
      ( ( member_list_nat @ A @ ( lists_nat @ A2 ) )
     => ( ( A != nil_nat )
       => ~ ! [A4: nat,L: list_nat] :
              ( ( A
                = ( cons_nat @ A4 @ L ) )
             => ( ( member_nat @ A4 @ A2 )
               => ~ ( member_list_nat @ L @ ( lists_nat @ A2 ) ) ) ) ) ) ).

% lists.cases
thf(fact_872_lists_Osimps,axiom,
    ! [A: list_list_nat,A2: set_list_nat] :
      ( ( member_list_list_nat @ A @ ( lists_list_nat @ A2 ) )
      = ( ( A = nil_list_nat )
        | ? [A5: list_nat,L3: list_list_nat] :
            ( ( A
              = ( cons_list_nat @ A5 @ L3 ) )
            & ( member_list_nat @ A5 @ A2 )
            & ( member_list_list_nat @ L3 @ ( lists_list_nat @ A2 ) ) ) ) ) ).

% lists.simps
thf(fact_873_lists_Osimps,axiom,
    ! [A: list_b,A2: set_b] :
      ( ( member_list_b @ A @ ( lists_b @ A2 ) )
      = ( ( A = nil_b )
        | ? [A5: b,L3: list_b] :
            ( ( A
              = ( cons_b @ A5 @ L3 ) )
            & ( member_b @ A5 @ A2 )
            & ( member_list_b @ L3 @ ( lists_b @ A2 ) ) ) ) ) ).

% lists.simps
thf(fact_874_lists_Osimps,axiom,
    ! [A: list_P8875379029341186191rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
      ( ( member7646937072540562872rm_a_b @ A @ ( lists_3936190290964281341rm_a_b @ A2 ) )
      = ( ( A = nil_Pr6942174756412032271rm_a_b )
        | ? [A5: produc357393685978478089rm_a_b,L3: list_P8875379029341186191rm_a_b] :
            ( ( A
              = ( cons_P3536305108106557631rm_a_b @ A5 @ L3 ) )
            & ( member5869715511025134514rm_a_b @ A5 @ A2 )
            & ( member7646937072540562872rm_a_b @ L3 @ ( lists_3936190290964281341rm_a_b @ A2 ) ) ) ) ) ).

% lists.simps
thf(fact_875_lists_Osimps,axiom,
    ! [A: list_P3592885314253461005_a_nat,A2: set_Pr4934435412358123699_a_nat] :
      ( ( member4041938288099225636_a_nat @ A @ ( lists_3983152277583228297_a_nat @ A2 ) )
      = ( ( A = nil_Pr7402525243500994295_a_nat )
        | ? [A5: product_prod_a_nat,L3: list_P3592885314253461005_a_nat] :
            ( ( A
              = ( cons_P5205166803686508359_a_nat @ A5 @ L3 ) )
            & ( member5724188588386418708_a_nat @ A5 @ A2 )
            & ( member4041938288099225636_a_nat @ L3 @ ( lists_3983152277583228297_a_nat @ A2 ) ) ) ) ) ).

% lists.simps
thf(fact_876_lists_Osimps,axiom,
    ! [A: list_term_a_b,A2: set_term_a_b] :
      ( ( member_list_term_a_b @ A @ ( lists_term_a_b @ A2 ) )
      = ( ( A = nil_term_a_b )
        | ? [A5: term_a_b,L3: list_term_a_b] :
            ( ( A
              = ( cons_term_a_b @ A5 @ L3 ) )
            & ( member_term_a_b @ A5 @ A2 )
            & ( member_list_term_a_b @ L3 @ ( lists_term_a_b @ A2 ) ) ) ) ) ).

% lists.simps
thf(fact_877_lists_Osimps,axiom,
    ! [A: list_nat,A2: set_nat] :
      ( ( member_list_nat @ A @ ( lists_nat @ A2 ) )
      = ( ( A = nil_nat )
        | ? [A5: nat,L3: list_nat] :
            ( ( A
              = ( cons_nat @ A5 @ L3 ) )
            & ( member_nat @ A5 @ A2 )
            & ( member_list_nat @ L3 @ ( lists_nat @ A2 ) ) ) ) ) ).

% lists.simps
thf(fact_878_replace__term__at_Osimps_I2_J,axiom,
    ! [X2: b,V4: nat,Va2: list_nat,T: term_a_b] :
      ( ( term_r6860082780075436317at_a_b @ ( var_b_a @ X2 ) @ ( cons_nat @ V4 @ Va2 ) @ T )
      = ( var_b_a @ X2 ) ) ).

% replace_term_at.simps(2)
thf(fact_879_poss__Cons,axiom,
    ! [I: nat,P: list_nat,T: term_a_b] :
      ( ( member_list_nat @ ( cons_nat @ I @ P ) @ ( term_poss_a_b @ T ) )
     => ( member_list_nat @ ( cons_nat @ I @ nil_nat ) @ ( term_poss_a_b @ T ) ) ) ).

% poss_Cons
thf(fact_880_same__length__different,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( Xs != Ys )
     => ( ( ( size_s8906293707977694520rm_a_b @ Xs )
          = ( size_s8906293707977694520rm_a_b @ Ys ) )
       => ? [Pre: list_term_a_b,X: term_a_b,Xs3: list_term_a_b,Y2: term_a_b,Ys5: list_term_a_b] :
            ( ( X != Y2 )
            & ( Xs
              = ( append_term_a_b @ Pre @ ( append_term_a_b @ ( cons_term_a_b @ X @ nil_term_a_b ) @ Xs3 ) ) )
            & ( Ys
              = ( append_term_a_b @ Pre @ ( append_term_a_b @ ( cons_term_a_b @ Y2 @ nil_term_a_b ) @ Ys5 ) ) ) ) ) ) ).

% same_length_different
thf(fact_881_same__length__different,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs != Ys )
     => ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
       => ? [Pre: list_nat,X: nat,Xs3: list_nat,Y2: nat,Ys5: list_nat] :
            ( ( X != Y2 )
            & ( Xs
              = ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X @ nil_nat ) @ Xs3 ) ) )
            & ( Ys
              = ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y2 @ nil_nat ) @ Ys5 ) ) ) ) ) ) ).

% same_length_different
thf(fact_882_replace__term__at_Ocases,axiom,
    ! [X2: produc2732850333517536310rm_a_b] :
      ( ! [S3: term_a_b,T2: term_a_b] :
          ( X2
         != ( produc3812856575676843240rm_a_b @ S3 @ ( produc5151171985953862413rm_a_b @ nil_nat @ T2 ) ) )
     => ( ! [X: b,V: nat,Va: list_nat,T2: term_a_b] :
            ( X2
           != ( produc3812856575676843240rm_a_b @ ( var_b_a @ X ) @ ( produc5151171985953862413rm_a_b @ ( cons_nat @ V @ Va ) @ T2 ) ) )
       => ~ ! [F: a,Ts: list_term_a_b,I2: nat,Ps2: list_nat,T2: term_a_b] :
              ( X2
             != ( produc3812856575676843240rm_a_b @ ( fun_a_b @ F @ Ts ) @ ( produc5151171985953862413rm_a_b @ ( cons_nat @ I2 @ Ps2 ) @ T2 ) ) ) ) ) ).

% replace_term_at.cases
thf(fact_883_fun__at_Ocases,axiom,
    ! [X2: produc3697673438841856213st_nat] :
      ( ! [X: b] :
          ( X2
         != ( produc4563063199488751885st_nat @ ( var_b_a @ X ) @ nil_nat ) )
     => ( ! [F: a,Ts: list_term_a_b] :
            ( X2
           != ( produc4563063199488751885st_nat @ ( fun_a_b @ F @ Ts ) @ nil_nat ) )
       => ( ! [F: a,Ts: list_term_a_b,I2: nat,P3: list_nat] :
              ( X2
             != ( produc4563063199488751885st_nat @ ( fun_a_b @ F @ Ts ) @ ( cons_nat @ I2 @ P3 ) ) )
         => ~ ! [Vb: b,V: nat,Va: list_nat] :
                ( X2
               != ( produc4563063199488751885st_nat @ ( var_b_a @ Vb ) @ ( cons_nat @ V @ Va ) ) ) ) ) ) ).

% fun_at.cases
thf(fact_884_subt__at_Ocases,axiom,
    ! [X2: produc3697673438841856213st_nat] :
      ( ! [S3: term_a_b] :
          ( X2
         != ( produc4563063199488751885st_nat @ S3 @ nil_nat ) )
     => ( ! [F: a,Ss: list_term_a_b,I2: nat,P3: list_nat] :
            ( X2
           != ( produc4563063199488751885st_nat @ ( fun_a_b @ F @ Ss ) @ ( cons_nat @ I2 @ P3 ) ) )
       => ~ ! [X: b,V: nat,Va: list_nat] :
              ( X2
             != ( produc4563063199488751885st_nat @ ( var_b_a @ X ) @ ( cons_nat @ V @ Va ) ) ) ) ) ).

% subt_at.cases
thf(fact_885_ctxt__at__pos_Ocases,axiom,
    ! [X2: produc3697673438841856213st_nat] :
      ( ! [S3: term_a_b] :
          ( X2
         != ( produc4563063199488751885st_nat @ S3 @ nil_nat ) )
     => ( ! [F: a,Ss: list_term_a_b,I2: nat,P3: list_nat] :
            ( X2
           != ( produc4563063199488751885st_nat @ ( fun_a_b @ F @ Ss ) @ ( cons_nat @ I2 @ P3 ) ) )
       => ~ ! [X: b,V: nat,Va: list_nat] :
              ( X2
             != ( produc4563063199488751885st_nat @ ( var_b_a @ X ) @ ( cons_nat @ V @ Va ) ) ) ) ) ).

% ctxt_at_pos.cases
thf(fact_886_lexord__append__rightI,axiom,
    ! [Y: list_nat,X2: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ? [B6: nat,Z5: list_nat] :
          ( Y
          = ( cons_nat @ B6 @ Z5 ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X2 @ ( append_nat @ X2 @ Y ) ) @ ( lexord_nat @ R3 ) ) ) ).

% lexord_append_rightI
thf(fact_887_depth_Ocases,axiom,
    ! [X2: term_a_b] :
      ( ! [X: b] :
          ( X2
         != ( var_b_a @ X ) )
     => ( ! [F: a] :
            ( X2
           != ( fun_a_b @ F @ nil_term_a_b ) )
       => ~ ! [F: a,V: term_a_b,Va: list_term_a_b] :
              ( X2
             != ( fun_a_b @ F @ ( cons_term_a_b @ V @ Va ) ) ) ) ) ).

% depth.cases
thf(fact_888_subt__at__Cons__comp,axiom,
    ! [I: nat,P: list_nat,S: term_a_b] :
      ( ( member_list_nat @ ( cons_nat @ I @ P ) @ ( term_poss_a_b @ S ) )
     => ( ( term_subt_at_a_b @ ( term_subt_at_a_b @ S @ ( cons_nat @ I @ nil_nat ) ) @ P )
        = ( term_subt_at_a_b @ S @ ( cons_nat @ I @ P ) ) ) ) ).

% subt_at_Cons_comp
thf(fact_889_lexord__append__left__rightI,axiom,
    ! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,U: list_nat,X2: list_nat,Y: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ U @ ( cons_nat @ A @ X2 ) ) @ ( append_nat @ U @ ( cons_nat @ B @ Y ) ) ) @ ( lexord_nat @ R3 ) ) ) ).

% lexord_append_left_rightI
thf(fact_890_lexord__append__left__rightI,axiom,
    ! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,U: list_term_a_b,X2: list_term_a_b,Y: list_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
     => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ U @ ( cons_term_a_b @ A @ X2 ) ) @ ( append_term_a_b @ U @ ( cons_term_a_b @ B @ Y ) ) ) @ ( lexord_term_a_b @ R3 ) ) ) ).

% lexord_append_left_rightI
thf(fact_891_filter2_Ocases,axiom,
    ! [X2: produc8215350257458196082rm_a_b] :
      ( ! [P6: term_a_b > term_a_b > $o,Uu: list_term_a_b] :
          ( X2
         != ( produc1278856658843369954rm_a_b @ P6 @ ( produc4885699992713594593rm_a_b @ nil_term_a_b @ Uu ) ) )
     => ( ! [P6: term_a_b > term_a_b > $o,V: term_a_b,Va: list_term_a_b] :
            ( X2
           != ( produc1278856658843369954rm_a_b @ P6 @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ V @ Va ) @ nil_term_a_b ) ) )
       => ~ ! [P6: term_a_b > term_a_b > $o,A4: term_a_b,As: list_term_a_b,B3: term_a_b,Bs: list_term_a_b] :
              ( X2
             != ( produc1278856658843369954rm_a_b @ P6 @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ A4 @ As ) @ ( cons_term_a_b @ B3 @ Bs ) ) ) ) ) ) ).

% filter2.cases
thf(fact_892_filter2_Ocases,axiom,
    ! [X2: produc6817702157447538794st_nat] :
      ( ! [P6: term_a_b > nat > $o,Uu: list_nat] :
          ( X2
         != ( produc7813177934582050522st_nat @ P6 @ ( produc5386604813764302749st_nat @ nil_term_a_b @ Uu ) ) )
     => ( ! [P6: term_a_b > nat > $o,V: term_a_b,Va: list_term_a_b] :
            ( X2
           != ( produc7813177934582050522st_nat @ P6 @ ( produc5386604813764302749st_nat @ ( cons_term_a_b @ V @ Va ) @ nil_nat ) ) )
       => ~ ! [P6: term_a_b > nat > $o,A4: term_a_b,As: list_term_a_b,B3: nat,Bs: list_nat] :
              ( X2
             != ( produc7813177934582050522st_nat @ P6 @ ( produc5386604813764302749st_nat @ ( cons_term_a_b @ A4 @ As ) @ ( cons_nat @ B3 @ Bs ) ) ) ) ) ) ).

% filter2.cases
thf(fact_893_filter2_Ocases,axiom,
    ! [X2: produc8734217775477389290rm_a_b] :
      ( ! [P6: nat > term_a_b > $o,Uu: list_term_a_b] :
          ( X2
         != ( produc2750092164434624090rm_a_b @ P6 @ ( produc4336470666470712605rm_a_b @ nil_nat @ Uu ) ) )
     => ( ! [P6: nat > term_a_b > $o,V: nat,Va: list_nat] :
            ( X2
           != ( produc2750092164434624090rm_a_b @ P6 @ ( produc4336470666470712605rm_a_b @ ( cons_nat @ V @ Va ) @ nil_term_a_b ) ) )
       => ~ ! [P6: nat > term_a_b > $o,A4: nat,As: list_nat,B3: term_a_b,Bs: list_term_a_b] :
              ( X2
             != ( produc2750092164434624090rm_a_b @ P6 @ ( produc4336470666470712605rm_a_b @ ( cons_nat @ A4 @ As ) @ ( cons_term_a_b @ B3 @ Bs ) ) ) ) ) ) ).

% filter2.cases
thf(fact_894_filter2_Ocases,axiom,
    ! [X2: produc4787317212837456354st_nat] :
      ( ! [P6: nat > nat > $o,Uu: list_nat] :
          ( X2
         != ( produc3127733452865184594st_nat @ P6 @ ( produc2694037385005941721st_nat @ nil_nat @ Uu ) ) )
     => ( ! [P6: nat > nat > $o,V: nat,Va: list_nat] :
            ( X2
           != ( produc3127733452865184594st_nat @ P6 @ ( produc2694037385005941721st_nat @ ( cons_nat @ V @ Va ) @ nil_nat ) ) )
       => ~ ! [P6: nat > nat > $o,A4: nat,As: list_nat,B3: nat,Bs: list_nat] :
              ( X2
             != ( produc3127733452865184594st_nat @ P6 @ ( produc2694037385005941721st_nat @ ( cons_nat @ A4 @ As ) @ ( cons_nat @ B3 @ Bs ) ) ) ) ) ) ).

% filter2.cases
thf(fact_895_list__inter_Ocases,axiom,
    ! [X2: produc51424535725745577rm_a_b] :
      ( ! [Bs: list_term_a_b] :
          ( X2
         != ( produc4885699992713594593rm_a_b @ nil_term_a_b @ Bs ) )
     => ~ ! [A4: term_a_b,As: list_term_a_b,Bs: list_term_a_b] :
            ( X2
           != ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ A4 @ As ) @ Bs ) ) ) ).

% list_inter.cases
thf(fact_896_list__inter_Ocases,axiom,
    ! [X2: produc1828647624359046049st_nat] :
      ( ! [Bs: list_nat] :
          ( X2
         != ( produc2694037385005941721st_nat @ nil_nat @ Bs ) )
     => ~ ! [A4: nat,As: list_nat,Bs: list_nat] :
            ( X2
           != ( produc2694037385005941721st_nat @ ( cons_nat @ A4 @ As ) @ Bs ) ) ) ).

% list_inter.cases
thf(fact_897_union__list__sorted_Ocases,axiom,
    ! [X2: produc1828647624359046049st_nat] :
      ( ! [X: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat] :
          ( X2
         != ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
     => ( ! [Ys2: list_nat] :
            ( X2
           != ( produc2694037385005941721st_nat @ nil_nat @ Ys2 ) )
       => ~ ! [V: nat,Va: list_nat] :
              ( X2
             != ( produc2694037385005941721st_nat @ ( cons_nat @ V @ Va ) @ nil_nat ) ) ) ) ).

% union_list_sorted.cases
thf(fact_898_list__4__cases,axiom,
    ! [Xs: list_term_a_b] :
      ( ( Xs != nil_term_a_b )
     => ( ! [X: term_a_b] :
            ( Xs
           != ( cons_term_a_b @ X @ nil_term_a_b ) )
       => ( ! [X: term_a_b,Y2: term_a_b] :
              ( Xs
             != ( cons_term_a_b @ X @ ( cons_term_a_b @ Y2 @ nil_term_a_b ) ) )
         => ~ ! [X: term_a_b,Y2: term_a_b,Z3: term_a_b,Zs3: list_term_a_b] :
                ( Xs
               != ( cons_term_a_b @ X @ ( cons_term_a_b @ Y2 @ ( cons_term_a_b @ Z3 @ Zs3 ) ) ) ) ) ) ) ).

% list_4_cases
thf(fact_899_list__4__cases,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ( ! [X: nat] :
            ( Xs
           != ( cons_nat @ X @ nil_nat ) )
       => ( ! [X: nat,Y2: nat] :
              ( Xs
             != ( cons_nat @ X @ ( cons_nat @ Y2 @ nil_nat ) ) )
         => ~ ! [X: nat,Y2: nat,Z3: nat,Zs3: list_nat] :
                ( Xs
               != ( cons_nat @ X @ ( cons_nat @ Y2 @ ( cons_nat @ Z3 @ Zs3 ) ) ) ) ) ) ) ).

% list_4_cases
thf(fact_900_list__3__cases,axiom,
    ! [Xs: list_term_a_b] :
      ( ( Xs != nil_term_a_b )
     => ( ! [X: term_a_b] :
            ( Xs
           != ( cons_term_a_b @ X @ nil_term_a_b ) )
       => ~ ! [X: term_a_b,Y2: term_a_b,Ys2: list_term_a_b] :
              ( Xs
             != ( cons_term_a_b @ X @ ( cons_term_a_b @ Y2 @ Ys2 ) ) ) ) ) ).

% list_3_cases
thf(fact_901_list__3__cases,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ( ! [X: nat] :
            ( Xs
           != ( cons_nat @ X @ nil_nat ) )
       => ~ ! [X: nat,Y2: nat,Ys2: list_nat] :
              ( Xs
             != ( cons_nat @ X @ ( cons_nat @ Y2 @ Ys2 ) ) ) ) ) ).

% list_3_cases
thf(fact_902_Missing__List_Omin__list_Ocases,axiom,
    ! [X2: list_nat] :
      ( ! [X: nat] :
          ( X2
         != ( cons_nat @ X @ nil_nat ) )
     => ( ! [X: nat,V: nat,Va: list_nat] :
            ( X2
           != ( cons_nat @ X @ ( cons_nat @ V @ Va ) ) )
       => ( X2 = nil_nat ) ) ) ).

% Missing_List.min_list.cases
thf(fact_903_distinct__eq_Ocases,axiom,
    ! [X2: produc8847469151730319409rm_a_b] :
      ( ! [Uu: term_a_b > term_a_b > $o] :
          ( X2
         != ( produc5931190720334929579rm_a_b @ Uu @ nil_term_a_b ) )
     => ~ ! [Eq: term_a_b > term_a_b > $o,X: term_a_b,Xs2: list_term_a_b] :
            ( X2
           != ( produc5931190720334929579rm_a_b @ Eq @ ( cons_term_a_b @ X @ Xs2 ) ) ) ) ).

% distinct_eq.cases
thf(fact_904_distinct__eq_Ocases,axiom,
    ! [X2: produc254973753779126261st_nat] :
      ( ! [Uu: nat > nat > $o] :
          ( X2
         != ( produc4727192421694094319st_nat @ Uu @ nil_nat ) )
     => ~ ! [Eq: nat > nat > $o,X: nat,Xs2: list_nat] :
            ( X2
           != ( produc4727192421694094319st_nat @ Eq @ ( cons_nat @ X @ Xs2 ) ) ) ) ).

% distinct_eq.cases
thf(fact_905_span_Ocases,axiom,
    ! [X2: produc5260046862718186894rm_a_b] :
      ( ! [P6: term_a_b > $o,X: term_a_b,Xs2: list_term_a_b] :
          ( X2
         != ( produc2136336272451886600rm_a_b @ P6 @ ( cons_term_a_b @ X @ Xs2 ) ) )
     => ~ ! [Uu: term_a_b > $o] :
            ( X2
           != ( produc2136336272451886600rm_a_b @ Uu @ nil_term_a_b ) ) ) ).

% span.cases
thf(fact_906_span_Ocases,axiom,
    ! [X2: produc4226810134323546766st_nat] :
      ( ! [P6: nat > $o,X: nat,Xs2: list_nat] :
          ( X2
         != ( produc8587622027977423880st_nat @ P6 @ ( cons_nat @ X @ Xs2 ) ) )
     => ~ ! [Uu: nat > $o] :
            ( X2
           != ( produc8587622027977423880st_nat @ Uu @ nil_nat ) ) ) ).

% span.cases
thf(fact_907_mem__idx_Ocases,axiom,
    ! [X2: produc5674375379913900953rm_a_b] :
      ( ! [Uu: term_a_b] :
          ( X2
         != ( produc967488655884398673rm_a_b @ Uu @ nil_term_a_b ) )
     => ~ ! [X: term_a_b,A4: term_a_b,As: list_term_a_b] :
            ( X2
           != ( produc967488655884398673rm_a_b @ X @ ( cons_term_a_b @ A4 @ As ) ) ) ) ).

% mem_idx.cases
thf(fact_908_mem__idx_Ocases,axiom,
    ! [X2: produc4575160907756185873st_nat] :
      ( ! [Uu: nat] :
          ( X2
         != ( produc8282810413953273033st_nat @ Uu @ nil_nat ) )
     => ~ ! [X: nat,A4: nat,As: list_nat] :
            ( X2
           != ( produc8282810413953273033st_nat @ X @ ( cons_nat @ A4 @ As ) ) ) ) ).

% mem_idx.cases
thf(fact_909_subtract__list__sorted_Ocases,axiom,
    ! [X2: produc1828647624359046049st_nat] :
      ( ! [X: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat] :
          ( X2
         != ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
     => ( ! [Ys2: list_nat] :
            ( X2
           != ( produc2694037385005941721st_nat @ nil_nat @ Ys2 ) )
       => ~ ! [V: nat,Va: list_nat] :
              ( X2
             != ( produc2694037385005941721st_nat @ ( cons_nat @ V @ Va ) @ nil_nat ) ) ) ) ).

% subtract_list_sorted.cases
thf(fact_910_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_lists_term_a_b @ N @ nil_term_a_b )
          = ( cons_list_term_a_b @ nil_term_a_b @ nil_list_term_a_b ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_lists_term_a_b @ N @ nil_term_a_b )
          = nil_list_term_a_b ) ) ) ).

% n_lists_Nil
thf(fact_911_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_lists_nat @ N @ nil_nat )
          = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_lists_nat @ N @ nil_nat )
          = nil_list_nat ) ) ) ).

% n_lists_Nil
thf(fact_912_list__of__permutation__element__n_Osimps_I1_J,axiom,
    ! [X2: term_a_b,L4: list_term_a_b] :
      ( ( basic_7447718476515137801rm_a_b @ X2 @ zero_zero_nat @ L4 )
      = ( cons_list_term_a_b @ nil_term_a_b @ nil_list_term_a_b ) ) ).

% list_of_permutation_element_n.simps(1)
thf(fact_913_list__of__permutation__element__n_Osimps_I1_J,axiom,
    ! [X2: nat,L4: list_nat] :
      ( ( basic_7079635023375748421_n_nat @ X2 @ zero_zero_nat @ L4 )
      = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% list_of_permutation_element_n.simps(1)
thf(fact_914_add__elem__list__lists_Osimps_I1_J,axiom,
    ! [X2: term_a_b] :
      ( ( basic_1593220722155286443rm_a_b @ X2 @ nil_term_a_b )
      = ( cons_list_term_a_b @ ( cons_term_a_b @ X2 @ nil_term_a_b ) @ nil_list_term_a_b ) ) ).

% add_elem_list_lists.simps(1)
thf(fact_915_add__elem__list__lists_Osimps_I1_J,axiom,
    ! [X2: nat] :
      ( ( basic_4874698711677410535ts_nat @ X2 @ nil_nat )
      = ( cons_list_nat @ ( cons_nat @ X2 @ nil_nat ) @ nil_list_nat ) ) ).

% add_elem_list_lists.simps(1)
thf(fact_916_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_term_a_b] :
      ( ( n_lists_term_a_b @ zero_zero_nat @ Xs )
      = ( cons_list_term_a_b @ nil_term_a_b @ nil_list_term_a_b ) ) ).

% n_lists.simps(1)
thf(fact_917_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_nat] :
      ( ( n_lists_nat @ zero_zero_nat @ Xs )
      = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% n_lists.simps(1)
thf(fact_918_max__list_Ocases,axiom,
    ! [X2: list_nat] :
      ( ( X2 != nil_nat )
     => ~ ! [X: nat,Xs2: list_nat] :
            ( X2
           != ( cons_nat @ X @ Xs2 ) ) ) ).

% max_list.cases
thf(fact_919_SuccD,axiom,
    ! [K: list_nat,Kl: set_list_list_nat,Kl2: list_list_nat] :
      ( ( member_list_nat @ K @ ( bNF_Gr3053708287304744325st_nat @ Kl @ Kl2 ) )
     => ( member_list_list_nat @ ( append_list_nat @ Kl2 @ ( cons_list_nat @ K @ nil_list_nat ) ) @ Kl ) ) ).

% SuccD
thf(fact_920_SuccD,axiom,
    ! [K: b,Kl: set_list_b,Kl2: list_b] :
      ( ( member_b @ K @ ( bNF_Greatest_Succ_b @ Kl @ Kl2 ) )
     => ( member_list_b @ ( append_b @ Kl2 @ ( cons_b @ K @ nil_b ) ) @ Kl ) ) ).

% SuccD
thf(fact_921_SuccD,axiom,
    ! [K: produc357393685978478089rm_a_b,Kl: set_li7311725489111463791rm_a_b,Kl2: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ K @ ( bNF_Gr7233806482624125860rm_a_b @ Kl @ Kl2 ) )
     => ( member7646937072540562872rm_a_b @ ( append1152621114427393060rm_a_b @ Kl2 @ ( cons_P3536305108106557631rm_a_b @ K @ nil_Pr6942174756412032271rm_a_b ) ) @ Kl ) ) ).

% SuccD
thf(fact_922_SuccD,axiom,
    ! [K: product_prod_a_nat,Kl: set_li7590439355475177155_a_nat,Kl2: list_P3592885314253461005_a_nat] :
      ( ( member5724188588386418708_a_nat @ K @ ( bNF_Gr7525783304380262050_a_nat @ Kl @ Kl2 ) )
     => ( member4041938288099225636_a_nat @ ( append7679239579558125090_a_nat @ Kl2 @ ( cons_P5205166803686508359_a_nat @ K @ nil_Pr7402525243500994295_a_nat ) ) @ Kl ) ) ).

% SuccD
thf(fact_923_SuccD,axiom,
    ! [K: term_a_b,Kl: set_list_term_a_b,Kl2: list_term_a_b] :
      ( ( member_term_a_b @ K @ ( bNF_Gr5510842888252665017rm_a_b @ Kl @ Kl2 ) )
     => ( member_list_term_a_b @ ( append_term_a_b @ Kl2 @ ( cons_term_a_b @ K @ nil_term_a_b ) ) @ Kl ) ) ).

% SuccD
thf(fact_924_SuccD,axiom,
    ! [K: nat,Kl: set_list_nat,Kl2: list_nat] :
      ( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ Kl2 ) )
     => ( member_list_nat @ ( append_nat @ Kl2 @ ( cons_nat @ K @ nil_nat ) ) @ Kl ) ) ).

% SuccD
thf(fact_925_SuccI,axiom,
    ! [Kl2: list_list_nat,K: list_nat,Kl: set_list_list_nat] :
      ( ( member_list_list_nat @ ( append_list_nat @ Kl2 @ ( cons_list_nat @ K @ nil_list_nat ) ) @ Kl )
     => ( member_list_nat @ K @ ( bNF_Gr3053708287304744325st_nat @ Kl @ Kl2 ) ) ) ).

% SuccI
thf(fact_926_SuccI,axiom,
    ! [Kl2: list_b,K: b,Kl: set_list_b] :
      ( ( member_list_b @ ( append_b @ Kl2 @ ( cons_b @ K @ nil_b ) ) @ Kl )
     => ( member_b @ K @ ( bNF_Greatest_Succ_b @ Kl @ Kl2 ) ) ) ).

% SuccI
thf(fact_927_SuccI,axiom,
    ! [Kl2: list_P8875379029341186191rm_a_b,K: produc357393685978478089rm_a_b,Kl: set_li7311725489111463791rm_a_b] :
      ( ( member7646937072540562872rm_a_b @ ( append1152621114427393060rm_a_b @ Kl2 @ ( cons_P3536305108106557631rm_a_b @ K @ nil_Pr6942174756412032271rm_a_b ) ) @ Kl )
     => ( member5869715511025134514rm_a_b @ K @ ( bNF_Gr7233806482624125860rm_a_b @ Kl @ Kl2 ) ) ) ).

% SuccI
thf(fact_928_SuccI,axiom,
    ! [Kl2: list_P3592885314253461005_a_nat,K: product_prod_a_nat,Kl: set_li7590439355475177155_a_nat] :
      ( ( member4041938288099225636_a_nat @ ( append7679239579558125090_a_nat @ Kl2 @ ( cons_P5205166803686508359_a_nat @ K @ nil_Pr7402525243500994295_a_nat ) ) @ Kl )
     => ( member5724188588386418708_a_nat @ K @ ( bNF_Gr7525783304380262050_a_nat @ Kl @ Kl2 ) ) ) ).

% SuccI
thf(fact_929_SuccI,axiom,
    ! [Kl2: list_term_a_b,K: term_a_b,Kl: set_list_term_a_b] :
      ( ( member_list_term_a_b @ ( append_term_a_b @ Kl2 @ ( cons_term_a_b @ K @ nil_term_a_b ) ) @ Kl )
     => ( member_term_a_b @ K @ ( bNF_Gr5510842888252665017rm_a_b @ Kl @ Kl2 ) ) ) ).

% SuccI
thf(fact_930_SuccI,axiom,
    ! [Kl2: list_nat,K: nat,Kl: set_list_nat] :
      ( ( member_list_nat @ ( append_nat @ Kl2 @ ( cons_nat @ K @ nil_nat ) ) @ Kl )
     => ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ Kl2 ) ) ) ).

% SuccI
thf(fact_931_empty__Shift,axiom,
    ! [Kl: set_list_list_nat,K: list_nat] :
      ( ( member_list_list_nat @ nil_list_nat @ Kl )
     => ( ( member_list_nat @ K @ ( bNF_Gr3053708287304744325st_nat @ Kl @ nil_list_nat ) )
       => ( member_list_list_nat @ nil_list_nat @ ( bNF_Gr9051742241863529473st_nat @ Kl @ K ) ) ) ) ).

% empty_Shift
thf(fact_932_empty__Shift,axiom,
    ! [Kl: set_list_b,K: b] :
      ( ( member_list_b @ nil_b @ Kl )
     => ( ( member_b @ K @ ( bNF_Greatest_Succ_b @ Kl @ nil_b ) )
       => ( member_list_b @ nil_b @ ( bNF_Greatest_Shift_b @ Kl @ K ) ) ) ) ).

% empty_Shift
thf(fact_933_empty__Shift,axiom,
    ! [Kl: set_li7311725489111463791rm_a_b,K: produc357393685978478089rm_a_b] :
      ( ( member7646937072540562872rm_a_b @ nil_Pr6942174756412032271rm_a_b @ Kl )
     => ( ( member5869715511025134514rm_a_b @ K @ ( bNF_Gr7233806482624125860rm_a_b @ Kl @ nil_Pr6942174756412032271rm_a_b ) )
       => ( member7646937072540562872rm_a_b @ nil_Pr6942174756412032271rm_a_b @ ( bNF_Gr369565347726608424rm_a_b @ Kl @ K ) ) ) ) ).

% empty_Shift
thf(fact_934_empty__Shift,axiom,
    ! [Kl: set_li7590439355475177155_a_nat,K: product_prod_a_nat] :
      ( ( member4041938288099225636_a_nat @ nil_Pr7402525243500994295_a_nat @ Kl )
     => ( ( member5724188588386418708_a_nat @ K @ ( bNF_Gr7525783304380262050_a_nat @ Kl @ nil_Pr7402525243500994295_a_nat ) )
       => ( member4041938288099225636_a_nat @ nil_Pr7402525243500994295_a_nat @ ( bNF_Gr4939858184705699614_a_nat @ Kl @ K ) ) ) ) ).

% empty_Shift
thf(fact_935_empty__Shift,axiom,
    ! [Kl: set_list_term_a_b,K: term_a_b] :
      ( ( member_list_term_a_b @ nil_term_a_b @ Kl )
     => ( ( member_term_a_b @ K @ ( bNF_Gr5510842888252665017rm_a_b @ Kl @ nil_term_a_b ) )
       => ( member_list_term_a_b @ nil_term_a_b @ ( bNF_Gr2285504805956674357rm_a_b @ Kl @ K ) ) ) ) ).

% empty_Shift
thf(fact_936_empty__Shift,axiom,
    ! [Kl: set_list_nat,K: nat] :
      ( ( member_list_nat @ nil_nat @ Kl )
     => ( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ nil_nat ) )
       => ( member_list_nat @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl @ K ) ) ) ) ).

% empty_Shift
thf(fact_937_bind__simps_I2_J,axiom,
    ! [X2: nat,Xs: list_nat,F2: nat > list_nat] :
      ( ( bind_nat_nat @ ( cons_nat @ X2 @ Xs ) @ F2 )
      = ( append_nat @ ( F2 @ X2 ) @ ( bind_nat_nat @ Xs @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_938_product__lists_Osimps_I1_J,axiom,
    ( ( produc17669015410068453rm_a_b @ nil_list_term_a_b )
    = ( cons_list_term_a_b @ nil_term_a_b @ nil_list_term_a_b ) ) ).

% product_lists.simps(1)
thf(fact_939_product__lists_Osimps_I1_J,axiom,
    ( ( product_lists_nat @ nil_list_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% product_lists.simps(1)
thf(fact_940_bind__simps_I1_J,axiom,
    ! [F2: term_a_b > list_term_a_b] :
      ( ( bind_t2899167355749536305rm_a_b @ nil_term_a_b @ F2 )
      = nil_term_a_b ) ).

% bind_simps(1)
thf(fact_941_bind__simps_I1_J,axiom,
    ! [F2: term_a_b > list_nat] :
      ( ( bind_term_a_b_nat @ nil_term_a_b @ F2 )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_942_bind__simps_I1_J,axiom,
    ! [F2: nat > list_term_a_b] :
      ( ( bind_nat_term_a_b @ nil_nat @ F2 )
      = nil_term_a_b ) ).

% bind_simps(1)
thf(fact_943_bind__simps_I1_J,axiom,
    ! [F2: nat > list_nat] :
      ( ( bind_nat_nat @ nil_nat @ F2 )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_944_concat__lists_Osimps_I1_J,axiom,
    ( ( missin8060632096978918206rm_a_b @ nil_list_term_a_b )
    = ( cons_list_term_a_b @ nil_term_a_b @ nil_list_term_a_b ) ) ).

% concat_lists.simps(1)
thf(fact_945_concat__lists_Osimps_I1_J,axiom,
    ( ( missin4567272213201432058ts_nat @ nil_list_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% concat_lists.simps(1)
thf(fact_946_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_nat,X2: nat,Ys: list_nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) ) @ ( listrel1_nat @ R3 ) )
      = ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R3 ) )
          & ( X2 = Y ) )
        | ( ( Xs = Ys )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R3 ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_947_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_term_a_b,X2: term_a_b,Ys: list_term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ X2 @ nil_term_a_b ) ) @ ( append_term_a_b @ Ys @ ( cons_term_a_b @ Y @ nil_term_a_b ) ) ) @ ( listrel1_term_a_b @ R3 ) )
      = ( ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listrel1_term_a_b @ R3 ) )
          & ( X2 = Y ) )
        | ( ( Xs = Ys )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ R3 ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_948_Cons__lenlex__iff,axiom,
    ! [M: nat,Ms: list_nat,N: nat,Ns: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R3 ) )
      = ( ( ord_less_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) )
        | ( ( ( size_size_list_nat @ Ms )
            = ( size_size_list_nat @ Ns ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ R3 ) )
        | ( ( M = N )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R3 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_949_Cons__lenlex__iff,axiom,
    ! [M: term_a_b,Ms: list_term_a_b,N: term_a_b,Ns: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ M @ Ms ) @ ( cons_term_a_b @ N @ Ns ) ) @ ( lenlex_term_a_b @ R3 ) )
      = ( ( ord_less_nat @ ( size_s8906293707977694520rm_a_b @ Ms ) @ ( size_s8906293707977694520rm_a_b @ Ns ) )
        | ( ( ( size_s8906293707977694520rm_a_b @ Ms )
            = ( size_s8906293707977694520rm_a_b @ Ns ) )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ M @ N ) @ R3 ) )
        | ( ( M = N )
          & ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ms @ Ns ) @ ( lenlex_term_a_b @ R3 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_950_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_951_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_952_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_953_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_954_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
      = ( ord_less_nat @ M @ N ) ) ).

% zero_less_diff
thf(fact_955_length__greater__0__conv,axiom,
    ! [Xs: list_term_a_b] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) )
      = ( Xs != nil_term_a_b ) ) ).

% length_greater_0_conv
thf(fact_956_length__greater__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
      = ( Xs != nil_nat ) ) ).

% length_greater_0_conv
thf(fact_957_Cons__listrel1__Cons,axiom,
    ! [X2: nat,Xs: list_nat,Y: nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R3 ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R3 )
          & ( Xs = Ys ) )
        | ( ( X2 = Y )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R3 ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_958_Cons__listrel1__Cons,axiom,
    ! [X2: term_a_b,Xs: list_term_a_b,Y: term_a_b,Ys: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X2 @ Xs ) @ ( cons_term_a_b @ Y @ Ys ) ) @ ( listrel1_term_a_b @ R3 ) )
      = ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ R3 )
          & ( Xs = Ys ) )
        | ( ( X2 = Y )
          & ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listrel1_term_a_b @ R3 ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_959_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_960_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_961_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_962_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_963_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_964_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_965_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_966_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_967_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_968_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_969_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_970_infinite__descent0,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ~ ( P2 @ N2 )
             => ? [M3: nat] :
                  ( ( ord_less_nat @ M3 @ N2 )
                  & ~ ( P2 @ M3 ) ) ) )
       => ( P2 @ N ) ) ) ).

% infinite_descent0
thf(fact_971_ex__least__nat__le,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ N )
     => ( ~ ( P2 @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K2 )
               => ~ ( P2 @ I3 ) )
            & ( P2 @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_972_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).

% diff_less
thf(fact_973_acyclicI__order,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,F2: term_a_b > nat] :
      ( ! [A4: term_a_b,B3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ R3 )
         => ( ord_less_nat @ ( F2 @ B3 ) @ ( F2 @ A4 ) ) )
     => ( transi5314701259734769157rm_a_b @ R3 ) ) ).

% acyclicI_order
thf(fact_974_listrel1I2,axiom,
    ! [Xs: list_nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat,X2: nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R3 ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ X2 @ Ys ) ) @ ( listrel1_nat @ R3 ) ) ) ).

% listrel1I2
thf(fact_975_rtrancl__listrel1__ConsI1,axiom,
    ! [Xs: list_nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat,X2: nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( transi5285580207609517981st_nat @ ( listrel1_nat @ R3 ) ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ X2 @ Ys ) ) @ ( transi5285580207609517981st_nat @ ( listrel1_nat @ R3 ) ) ) ) ).

% rtrancl_listrel1_ConsI1
thf(fact_976_not__listrel1__Nil,axiom,
    ! [Xs: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ nil_term_a_b ) @ ( listrel1_term_a_b @ R3 ) ) ).

% not_listrel1_Nil
thf(fact_977_not__listrel1__Nil,axiom,
    ! [Xs: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel1_nat @ R3 ) ) ).

% not_listrel1_Nil
thf(fact_978_not__Nil__listrel1,axiom,
    ! [Xs: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ nil_term_a_b @ Xs ) @ ( listrel1_term_a_b @ R3 ) ) ).

% not_Nil_listrel1
thf(fact_979_not__Nil__listrel1,axiom,
    ! [Xs: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel1_nat @ R3 ) ) ).

% not_Nil_listrel1
thf(fact_980_append__listrel1I,axiom,
    ! [Xs: list_nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat,Us2: list_nat,Vs: list_nat] :
      ( ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R3 ) )
          & ( Us2 = Vs ) )
        | ( ( Xs = Ys )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us2 @ Vs ) @ ( listrel1_nat @ R3 ) ) ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Us2 ) @ ( append_nat @ Ys @ Vs ) ) @ ( listrel1_nat @ R3 ) ) ) ).

% append_listrel1I
thf(fact_981_wf__bounded__measure,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,Ub: term_a_b > nat,F2: term_a_b > nat] :
      ( ! [A4: term_a_b,B3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A4 ) @ R3 )
         => ( ( ord_less_eq_nat @ ( Ub @ B3 ) @ ( Ub @ A4 ) )
            & ( ord_less_eq_nat @ ( F2 @ B3 ) @ ( Ub @ A4 ) )
            & ( ord_less_nat @ ( F2 @ A4 ) @ ( F2 @ B3 ) ) ) )
     => ( wf_term_a_b @ R3 ) ) ).

% wf_bounded_measure
thf(fact_982_Cons__listrel1E2,axiom,
    ! [Xs: list_nat,Y: nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R3 ) )
     => ( ! [X: nat] :
            ( ( Xs
              = ( cons_nat @ X @ Ys ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 ) )
       => ~ ! [Zs3: list_nat] :
              ( ( Xs
                = ( cons_nat @ Y @ Zs3 ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs3 @ Ys ) @ ( listrel1_nat @ R3 ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_983_Cons__listrel1E2,axiom,
    ! [Xs: list_term_a_b,Y: term_a_b,Ys: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ ( cons_term_a_b @ Y @ Ys ) ) @ ( listrel1_term_a_b @ R3 ) )
     => ( ! [X: term_a_b] :
            ( ( Xs
              = ( cons_term_a_b @ X @ Ys ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R3 ) )
       => ~ ! [Zs3: list_term_a_b] :
              ( ( Xs
                = ( cons_term_a_b @ Y @ Zs3 ) )
             => ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Zs3 @ Ys ) @ ( listrel1_term_a_b @ R3 ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_984_Cons__listrel1E1,axiom,
    ! [X2: nat,Xs: list_nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ Ys ) @ ( listrel1_nat @ R3 ) )
     => ( ! [Y2: nat] :
            ( ( Ys
              = ( cons_nat @ Y2 @ Xs ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R3 ) )
       => ~ ! [Zs3: list_nat] :
              ( ( Ys
                = ( cons_nat @ X2 @ Zs3 ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs3 ) @ ( listrel1_nat @ R3 ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_985_Cons__listrel1E1,axiom,
    ! [X2: term_a_b,Xs: list_term_a_b,Ys: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X2 @ Xs ) @ Ys ) @ ( listrel1_term_a_b @ R3 ) )
     => ( ! [Y2: term_a_b] :
            ( ( Ys
              = ( cons_term_a_b @ Y2 @ Xs ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y2 ) @ R3 ) )
       => ~ ! [Zs3: list_term_a_b] :
              ( ( Ys
                = ( cons_term_a_b @ X2 @ Zs3 ) )
             => ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Zs3 ) @ ( listrel1_term_a_b @ R3 ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_986_listrel1I1,axiom,
    ! [X2: nat,Y: nat,R3: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R3 )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Xs ) ) @ ( listrel1_nat @ R3 ) ) ) ).

% listrel1I1
thf(fact_987_listrel1I1,axiom,
    ! [X2: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ R3 )
     => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X2 @ Xs ) @ ( cons_term_a_b @ Y @ Xs ) ) @ ( listrel1_term_a_b @ R3 ) ) ) ).

% listrel1I1
thf(fact_988_trancl__power,axiom,
    ! [P: produc357393685978478089rm_a_b,R5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ P @ ( transi7922773638565587891rm_a_b @ R5 ) )
      = ( ? [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
            & ( member5869715511025134514rm_a_b @ P @ ( compow4057154403645558940rm_a_b @ N3 @ R5 ) ) ) ) ) ).

% trancl_power
thf(fact_989_listrel1E,axiom,
    ! [Xs: list_nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R3 ) )
     => ~ ! [X: nat,Y2: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ R3 )
           => ! [Us3: list_nat,Vs3: list_nat] :
                ( ( Xs
                  = ( append_nat @ Us3 @ ( cons_nat @ X @ Vs3 ) ) )
               => ( Ys
                 != ( append_nat @ Us3 @ ( cons_nat @ Y2 @ Vs3 ) ) ) ) ) ) ).

% listrel1E
thf(fact_990_listrel1E,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listrel1_term_a_b @ R3 ) )
     => ~ ! [X: term_a_b,Y2: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ R3 )
           => ! [Us3: list_term_a_b,Vs3: list_term_a_b] :
                ( ( Xs
                  = ( append_term_a_b @ Us3 @ ( cons_term_a_b @ X @ Vs3 ) ) )
               => ( Ys
                 != ( append_term_a_b @ Us3 @ ( cons_term_a_b @ Y2 @ Vs3 ) ) ) ) ) ) ).

% listrel1E
thf(fact_991_listrel1I,axiom,
    ! [X2: nat,Y: nat,R3: set_Pr1261947904930325089at_nat,Xs: list_nat,Us2: list_nat,Vs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R3 )
     => ( ( Xs
          = ( append_nat @ Us2 @ ( cons_nat @ X2 @ Vs ) ) )
       => ( ( Ys
            = ( append_nat @ Us2 @ ( cons_nat @ Y @ Vs ) ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R3 ) ) ) ) ) ).

% listrel1I
thf(fact_992_listrel1I,axiom,
    ! [X2: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b,Us2: list_term_a_b,Vs: list_term_a_b,Ys: list_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ R3 )
     => ( ( Xs
          = ( append_term_a_b @ Us2 @ ( cons_term_a_b @ X2 @ Vs ) ) )
       => ( ( Ys
            = ( append_term_a_b @ Us2 @ ( cons_term_a_b @ Y @ Vs ) ) )
         => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listrel1_term_a_b @ R3 ) ) ) ) ) ).

% listrel1I
thf(fact_993_rtrancl__listrel1__ConsI2,axiom,
    ! [X2: nat,Y: nat,R3: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ ( transi2905341329935302413cl_nat @ R3 ) )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( transi5285580207609517981st_nat @ ( listrel1_nat @ R3 ) ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( transi5285580207609517981st_nat @ ( listrel1_nat @ R3 ) ) ) ) ) ).

% rtrancl_listrel1_ConsI2
thf(fact_994_rtrancl__listrel1__ConsI2,axiom,
    ! [X2: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
     => ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( transi911495901722478305rm_a_b @ ( listrel1_term_a_b @ R3 ) ) )
       => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X2 @ Xs ) @ ( cons_term_a_b @ Y @ Ys ) ) @ ( transi911495901722478305rm_a_b @ ( listrel1_term_a_b @ R3 ) ) ) ) ) ).

% rtrancl_listrel1_ConsI2
thf(fact_995_in__measure,axiom,
    ! [X2: term_a_b,Y: term_a_b,F2: term_a_b > nat] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( measure_term_a_b @ F2 ) )
      = ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y ) ) ) ).

% in_measure
thf(fact_996_nsrsteps__with__root__step__step__on__args,axiom,
    ! [S: term_a_b,T: term_a_b,F4: set_Pr4934435412358123699_a_nat,R4: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) )
     => ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F4 @ R4 ) )
       => ? [F: a,Ss: list_term_a_b,Ts: list_term_a_b] :
            ( ( S
              = ( fun_a_b @ F @ Ss ) )
            & ( T
              = ( fun_a_b @ F @ Ts ) )
            & ( ( size_s8906293707977694520rm_a_b @ Ss )
              = ( size_s8906293707977694520rm_a_b @ Ts ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Ts ) )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Ss @ I3 ) @ ( nth_term_a_b @ Ts @ I3 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R4 ) ) ) ) ) ) ) ) ).

% nsrsteps_with_root_step_step_on_args
thf(fact_997_all__ctxt__closed__def,axiom,
    ( terms_5226143800768910156ed_a_b
    = ( ^ [F6: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
          ( ! [F7: a,Ts3: list_term_a_b,Ss2: list_term_a_b] :
              ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F7 @ ( size_s8906293707977694520rm_a_b @ Ss2 ) ) @ F6 )
             => ( ( ( size_s8906293707977694520rm_a_b @ Ts3 )
                  = ( size_s8906293707977694520rm_a_b @ Ss2 ) )
               => ( ! [I4: nat] :
                      ( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Ts3 ) )
                     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Ts3 @ I4 ) @ ( nth_term_a_b @ Ss2 @ I4 ) ) @ R2 ) )
                 => ( ! [I4: nat] :
                        ( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Ts3 ) )
                       => ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ ( term_funas_term_a_b @ ( nth_term_a_b @ Ts3 @ I4 ) ) @ ( term_funas_term_a_b @ ( nth_term_a_b @ Ss2 @ I4 ) ) ) @ F6 ) )
                   => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( fun_a_b @ F7 @ Ts3 ) @ ( fun_a_b @ F7 @ Ss2 ) ) @ R2 ) ) ) ) )
          & ! [X3: b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( var_b_a @ X3 ) @ ( var_b_a @ X3 ) ) @ R2 ) ) ) ) ).

% all_ctxt_closed_def
thf(fact_998_nth__Cons__0,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat )
      = X2 ) ).

% nth_Cons_0
thf(fact_999_nth__append__length,axiom,
    ! [Xs: list_nat,X2: nat,Ys: list_nat] :
      ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) ) @ ( size_size_list_nat @ Xs ) )
      = X2 ) ).

% nth_append_length
thf(fact_1000_append__Cons__nth__middle,axiom,
    ! [I: nat,Xs: list_nat,Y: nat,Zs: list_nat] :
      ( ( I
        = ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ Y @ Zs ) ) @ I )
        = Y ) ) ).

% append_Cons_nth_middle
thf(fact_1001_append__Cons__nth__not__middle,axiom,
    ! [I: nat,Xs: list_nat,U: nat,Ys: list_nat,Z: nat] :
      ( ( I
       != ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ U @ Ys ) ) @ I )
        = ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ Z @ Ys ) ) @ I ) ) ) ).

% append_Cons_nth_not_middle
thf(fact_1002_P__as__bs__extend,axiom,
    ! [As2: list_nat,Bs2: list_nat,Cs: list_nat,Ds: list_nat,P2: nat > nat > $o] :
      ( ( ( size_size_list_nat @ As2 )
        = ( size_size_list_nat @ Bs2 ) )
     => ( ( ( size_size_list_nat @ Cs )
          = ( size_size_list_nat @ Ds ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Bs2 ) )
             => ( P2 @ ( nth_nat @ As2 @ I2 ) @ ( nth_nat @ Bs2 @ I2 ) ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ds ) )
               => ( P2 @ ( nth_nat @ Cs @ I2 ) @ ( nth_nat @ Ds @ I2 ) ) )
           => ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ ( append_nat @ Bs2 @ Ds ) ) )
               => ( P2 @ ( nth_nat @ ( append_nat @ As2 @ Cs ) @ I3 ) @ ( nth_nat @ ( append_nat @ Bs2 @ Ds ) @ I3 ) ) ) ) ) ) ) ).

% P_as_bs_extend
thf(fact_1003_subt__at_Osimps_I2_J,axiom,
    ! [F2: a,Ss3: list_term_a_b,I: nat,P: list_nat] :
      ( ( term_subt_at_a_b @ ( fun_a_b @ F2 @ Ss3 ) @ ( cons_nat @ I @ P ) )
      = ( term_subt_at_a_b @ ( nth_term_a_b @ Ss3 @ I ) @ P ) ) ).

% subt_at.simps(2)
thf(fact_1004_poss__of__term__Cons,axiom,
    ! [I: nat,P: list_nat,U: term_a_b,F2: a,Ts2: list_term_a_b] :
      ( ( member_list_nat @ ( cons_nat @ I @ P ) @ ( terms_7168686267159881682rm_a_b @ U @ ( fun_a_b @ F2 @ Ts2 ) ) )
     => ( member_list_nat @ P @ ( terms_7168686267159881682rm_a_b @ U @ ( nth_term_a_b @ Ts2 @ I ) ) ) ) ).

% poss_of_term_Cons
thf(fact_1005_append__Cons__nth__left,axiom,
    ! [I: nat,Xs: list_nat,U: nat,Ys: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ U @ Ys ) ) @ I )
        = ( nth_nat @ Xs @ I ) ) ) ).

% append_Cons_nth_left
thf(fact_1006_append__Cons__nth__right,axiom,
    ! [Xs: list_nat,I: nat,U: nat,Ys: list_nat,Z: nat] :
      ( ( ord_less_nat @ ( size_size_list_nat @ Xs ) @ I )
     => ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ U @ Ys ) ) @ I )
        = ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ Z @ Ys ) ) @ I ) ) ) ).

% append_Cons_nth_right
thf(fact_1007_nth__append,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
          = ( nth_nat @ Xs @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
          = ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).

% nth_append
thf(fact_1008_all__ctxt__closedD,axiom,
    ! [F8: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,F2: a,Ss3: list_term_a_b,Ts2: list_term_a_b] :
      ( ( terms_5226143800768910156ed_a_b @ F8 @ R3 )
     => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F2 @ ( size_s8906293707977694520rm_a_b @ Ss3 ) ) @ F8 )
       => ( ( ( size_s8906293707977694520rm_a_b @ Ts2 )
            = ( size_s8906293707977694520rm_a_b @ Ss3 ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Ts2 @ I2 ) @ ( nth_term_a_b @ Ss3 @ I2 ) ) @ R3 ) )
           => ( ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
                 => ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( nth_term_a_b @ Ts2 @ I2 ) ) @ F8 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
                   => ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( nth_term_a_b @ Ss3 @ I2 ) ) @ F8 ) )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( fun_a_b @ F2 @ Ts2 ) @ ( fun_a_b @ F2 @ Ss3 ) ) @ R3 ) ) ) ) ) ) ) ).

% all_ctxt_closedD
thf(fact_1009_replace__term__at_Oelims,axiom,
    ! [X2: term_a_b,Xa: list_nat,Xb: term_a_b,Y: term_a_b] :
      ( ( ( term_r6860082780075436317at_a_b @ X2 @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_nat )
         => ( Y != Xb ) )
       => ( ! [X: b] :
              ( ( X2
                = ( var_b_a @ X ) )
             => ( ? [V: nat,Va: list_nat] :
                    ( Xa
                    = ( cons_nat @ V @ Va ) )
               => ( Y
                 != ( var_b_a @ X ) ) ) )
         => ~ ! [F: a,Ts: list_term_a_b] :
                ( ( X2
                  = ( fun_a_b @ F @ Ts ) )
               => ! [I2: nat,Ps2: list_nat] :
                    ( ( Xa
                      = ( cons_nat @ I2 @ Ps2 ) )
                   => ~ ( ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts ) )
                         => ( Y
                            = ( fun_a_b @ F @ ( list_update_term_a_b @ Ts @ I2 @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ts @ I2 ) @ Ps2 @ Xb ) ) ) ) )
                        & ( ~ ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts ) )
                         => ( Y
                            = ( fun_a_b @ F @ Ts ) ) ) ) ) ) ) ) ) ).

% replace_term_at.elims
thf(fact_1010_poss__Cons__poss,axiom,
    ! [I: nat,Q: list_nat,T: term_a_b] :
      ( ( member_list_nat @ ( cons_nat @ I @ Q ) @ ( term_poss_a_b @ T ) )
      = ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ ( args_a_b @ T ) ) )
        & ( member_list_nat @ Q @ ( term_poss_a_b @ ( nth_term_a_b @ ( args_a_b @ T ) @ I ) ) ) ) ) ).

% poss_Cons_poss
thf(fact_1011_list__update__nonempty,axiom,
    ! [Xs: list_term_a_b,K: nat,X2: term_a_b] :
      ( ( ( list_update_term_a_b @ Xs @ K @ X2 )
        = nil_term_a_b )
      = ( Xs = nil_term_a_b ) ) ).

% list_update_nonempty
thf(fact_1012_list__update__nonempty,axiom,
    ! [Xs: list_nat,K: nat,X2: nat] :
      ( ( ( list_update_nat @ Xs @ K @ X2 )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% list_update_nonempty
thf(fact_1013_list__update__length,axiom,
    ! [Xs: list_nat,X2: nat,Ys: list_nat,Y: nat] :
      ( ( list_update_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) ) @ ( size_size_list_nat @ Xs ) @ Y )
      = ( append_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) ) ).

% list_update_length
thf(fact_1014_list__update__code_I2_J,axiom,
    ! [X2: nat,Xs: list_nat,Y: nat] :
      ( ( list_update_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat @ Y )
      = ( cons_nat @ Y @ Xs ) ) ).

% list_update_code(2)
thf(fact_1015_term_Osel_I4_J,axiom,
    ! [X21: a,X22: list_term_a_b] :
      ( ( args_a_b @ ( fun_a_b @ X21 @ X22 ) )
      = X22 ) ).

% term.sel(4)
thf(fact_1016_list__update_Osimps_I1_J,axiom,
    ! [I: nat,V4: term_a_b] :
      ( ( list_update_term_a_b @ nil_term_a_b @ I @ V4 )
      = nil_term_a_b ) ).

% list_update.simps(1)
thf(fact_1017_list__update_Osimps_I1_J,axiom,
    ! [I: nat,V4: nat] :
      ( ( list_update_nat @ nil_nat @ I @ V4 )
      = nil_nat ) ).

% list_update.simps(1)
thf(fact_1018_list__update__code_I1_J,axiom,
    ! [I: nat,Y: term_a_b] :
      ( ( list_update_term_a_b @ nil_term_a_b @ I @ Y )
      = nil_term_a_b ) ).

% list_update_code(1)
thf(fact_1019_list__update__code_I1_J,axiom,
    ! [I: nat,Y: nat] :
      ( ( list_update_nat @ nil_nat @ I @ Y )
      = nil_nat ) ).

% list_update_code(1)
thf(fact_1020_list__update__append1,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_nat,X2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ I @ X2 )
        = ( append_nat @ ( list_update_nat @ Xs @ I @ X2 ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_1021_term_Osel_I3_J,axiom,
    ! [X1: b] :
      ( ( args_a_b @ ( var_b_a @ X1 ) )
      = nil_term_a_b ) ).

% term.sel(3)
thf(fact_1022_list__update__append,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat,X2: nat] :
      ( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X2 )
          = ( append_nat @ ( list_update_nat @ Xs @ N @ X2 ) @ Ys ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X2 )
          = ( append_nat @ Xs @ ( list_update_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ X2 ) ) ) ) ) ).

% list_update_append
thf(fact_1023_listrel1__iff__update,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listrel1_term_a_b @ R3 ) )
      = ( ? [Y4: term_a_b,N3: nat] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Xs @ N3 ) @ Y4 ) @ R3 )
            & ( ord_less_nat @ N3 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
            & ( Ys
              = ( list_update_term_a_b @ Xs @ N3 @ Y4 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1024_replace__term__at_Osimps_I3_J,axiom,
    ! [I: nat,Ts2: list_term_a_b,F2: a,Ps: list_nat,T: term_a_b] :
      ( ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
       => ( ( term_r6860082780075436317at_a_b @ ( fun_a_b @ F2 @ Ts2 ) @ ( cons_nat @ I @ Ps ) @ T )
          = ( fun_a_b @ F2 @ ( list_update_term_a_b @ Ts2 @ I @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ts2 @ I ) @ Ps @ T ) ) ) ) )
      & ( ~ ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
       => ( ( term_r6860082780075436317at_a_b @ ( fun_a_b @ F2 @ Ts2 ) @ ( cons_nat @ I @ Ps ) @ T )
          = ( fun_a_b @ F2 @ Ts2 ) ) ) ) ).

% replace_term_at.simps(3)
thf(fact_1025_replace__term__at_Opelims,axiom,
    ! [X2: term_a_b,Xa: list_nat,Xb: term_a_b,Y: term_a_b] :
      ( ( ( term_r6860082780075436317at_a_b @ X2 @ Xa @ Xb )
        = Y )
     => ( ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ X2 @ ( produc5151171985953862413rm_a_b @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_nat )
           => ( ( Y = Xb )
             => ~ ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ X2 @ ( produc5151171985953862413rm_a_b @ nil_nat @ Xb ) ) ) ) )
         => ( ! [X: b] :
                ( ( X2
                  = ( var_b_a @ X ) )
               => ! [V: nat,Va: list_nat] :
                    ( ( Xa
                      = ( cons_nat @ V @ Va ) )
                   => ( ( Y
                        = ( var_b_a @ X ) )
                     => ~ ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ ( var_b_a @ X ) @ ( produc5151171985953862413rm_a_b @ ( cons_nat @ V @ Va ) @ Xb ) ) ) ) ) )
           => ~ ! [F: a,Ts: list_term_a_b] :
                  ( ( X2
                    = ( fun_a_b @ F @ Ts ) )
                 => ! [I2: nat,Ps2: list_nat] :
                      ( ( Xa
                        = ( cons_nat @ I2 @ Ps2 ) )
                     => ( ( ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts ) )
                           => ( Y
                              = ( fun_a_b @ F @ ( list_update_term_a_b @ Ts @ I2 @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ts @ I2 ) @ Ps2 @ Xb ) ) ) ) )
                          & ( ~ ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts ) )
                           => ( Y
                              = ( fun_a_b @ F @ Ts ) ) ) )
                       => ~ ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ ( fun_a_b @ F @ Ts ) @ ( produc5151171985953862413rm_a_b @ ( cons_nat @ I2 @ Ps2 ) @ Xb ) ) ) ) ) ) ) ) ) ) ).

% replace_term_at.pelims
thf(fact_1026_trans__ctxt__sig__imp__all__ctxt__closed,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,F8: set_Pr4934435412358123699_a_nat] :
      ( ( trans_on_term_a_b @ top_top_set_term_a_b @ R3 )
     => ( ! [T2: term_a_b] :
            ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ F8 )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ T2 ) @ R3 ) )
       => ( ! [C2: subterm_and_ctxt_a_b,S3: term_a_b,T2: term_a_b] :
              ( ( ord_le8666007276011122963_a_nat @ ( term_funas_ctxt_a_b @ C2 ) @ F8 )
             => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S3 ) @ F8 )
               => ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ F8 )
                 => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S3 @ T2 ) @ R3 )
                   => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C2 @ S3 ) @ ( subter2376574525758040790rm_a_b @ C2 @ T2 ) ) @ R3 ) ) ) ) )
         => ( terms_5226143800768910156ed_a_b @ F8 @ R3 ) ) ) ) ).

% trans_ctxt_sig_imp_all_ctxt_closed
thf(fact_1027_transD,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,X2: term_a_b,Y: term_a_b,Z: term_a_b] :
      ( ( trans_on_term_a_b @ top_top_set_term_a_b @ R3 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ R3 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ R3 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z ) @ R3 ) ) ) ) ).

% transD
thf(fact_1028_transE,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,X2: term_a_b,Y: term_a_b,Z: term_a_b] :
      ( ( trans_on_term_a_b @ top_top_set_term_a_b @ R3 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ R3 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ R3 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z ) @ R3 ) ) ) ) ).

% transE
thf(fact_1029_transI,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b] :
      ( ! [X: term_a_b,Y2: term_a_b,Z3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ R3 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z3 ) @ R3 )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z3 ) @ R3 ) ) )
     => ( trans_on_term_a_b @ top_top_set_term_a_b @ R3 ) ) ).

% transI
thf(fact_1030_trans__on__def,axiom,
    ( trans_on_term_a_b
    = ( ^ [A6: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
        ! [X3: term_a_b] :
          ( ( member_term_a_b @ X3 @ A6 )
         => ! [Y4: term_a_b] :
              ( ( member_term_a_b @ Y4 @ A6 )
             => ! [Z4: term_a_b] :
                  ( ( member_term_a_b @ Z4 @ A6 )
                 => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R2 )
                   => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Z4 ) @ R2 )
                     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z4 ) @ R2 ) ) ) ) ) ) ) ) ).

% trans_on_def
thf(fact_1031_trans__onI,axiom,
    ! [A2: set_list_nat,R3: set_Pr3451248702717554689st_nat] :
      ( ! [X: list_nat,Y2: list_nat,Z3: list_nat] :
          ( ( member_list_nat @ X @ A2 )
         => ( ( member_list_nat @ Y2 @ A2 )
           => ( ( member_list_nat @ Z3 @ A2 )
             => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y2 ) @ R3 )
               => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y2 @ Z3 ) @ R3 )
                 => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Z3 ) @ R3 ) ) ) ) ) )
     => ( trans_on_list_nat @ A2 @ R3 ) ) ).

% trans_onI
thf(fact_1032_trans__onI,axiom,
    ! [A2: set_b,R3: set_Product_prod_b_b] :
      ( ! [X: b,Y2: b,Z3: b] :
          ( ( member_b @ X @ A2 )
         => ( ( member_b @ Y2 @ A2 )
           => ( ( member_b @ Z3 @ A2 )
             => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y2 ) @ R3 )
               => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y2 @ Z3 ) @ R3 )
                 => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Z3 ) @ R3 ) ) ) ) ) )
     => ( trans_on_b @ A2 @ R3 ) ) ).

% trans_onI
thf(fact_1033_trans__onI,axiom,
    ! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
      ( ! [X: produc357393685978478089rm_a_b,Y2: produc357393685978478089rm_a_b,Z3: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X @ A2 )
         => ( ( member5869715511025134514rm_a_b @ Y2 @ A2 )
           => ( ( member5869715511025134514rm_a_b @ Z3 @ A2 )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y2 ) @ R3 )
               => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ Z3 ) @ R3 )
                 => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z3 ) @ R3 ) ) ) ) ) )
     => ( trans_5404704112166290345rm_a_b @ A2 @ R3 ) ) ).

% trans_onI
thf(fact_1034_trans__onI,axiom,
    ! [A2: set_Pr4934435412358123699_a_nat,R3: set_Pr1811044260758604347_a_nat] :
      ( ! [X: product_prod_a_nat,Y2: product_prod_a_nat,Z3: product_prod_a_nat] :
          ( ( member5724188588386418708_a_nat @ X @ A2 )
         => ( ( member5724188588386418708_a_nat @ Y2 @ A2 )
           => ( ( member5724188588386418708_a_nat @ Z3 @ A2 )
             => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y2 ) @ R3 )
               => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y2 @ Z3 ) @ R3 )
                 => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Z3 ) @ R3 ) ) ) ) ) )
     => ( trans_5746013079982352605_a_nat @ A2 @ R3 ) ) ).

% trans_onI
thf(fact_1035_trans__onI,axiom,
    ! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ! [X: term_a_b,Y2: term_a_b,Z3: term_a_b] :
          ( ( member_term_a_b @ X @ A2 )
         => ( ( member_term_a_b @ Y2 @ A2 )
           => ( ( member_term_a_b @ Z3 @ A2 )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ R3 )
               => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z3 ) @ R3 )
                 => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z3 ) @ R3 ) ) ) ) ) )
     => ( trans_on_term_a_b @ A2 @ R3 ) ) ).

% trans_onI
thf(fact_1036_trans__onD,axiom,
    ! [A2: set_list_nat,R3: set_Pr3451248702717554689st_nat,X2: list_nat,Y: list_nat,Z: list_nat] :
      ( ( trans_on_list_nat @ A2 @ R3 )
     => ( ( member_list_nat @ X2 @ A2 )
       => ( ( member_list_nat @ Y @ A2 )
         => ( ( member_list_nat @ Z @ A2 )
           => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X2 @ Y ) @ R3 )
             => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y @ Z ) @ R3 )
               => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X2 @ Z ) @ R3 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_1037_trans__onD,axiom,
    ! [A2: set_b,R3: set_Product_prod_b_b,X2: b,Y: b,Z: b] :
      ( ( trans_on_b @ A2 @ R3 )
     => ( ( member_b @ X2 @ A2 )
       => ( ( member_b @ Y @ A2 )
         => ( ( member_b @ Z @ A2 )
           => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X2 @ Y ) @ R3 )
             => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y @ Z ) @ R3 )
               => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X2 @ Z ) @ R3 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_1038_trans__onD,axiom,
    ! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b,X2: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
      ( ( trans_5404704112166290345rm_a_b @ A2 @ R3 )
     => ( ( member5869715511025134514rm_a_b @ X2 @ A2 )
       => ( ( member5869715511025134514rm_a_b @ Y @ A2 )
         => ( ( member5869715511025134514rm_a_b @ Z @ A2 )
           => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ Y ) @ R3 )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z ) @ R3 )
               => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ Z ) @ R3 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_1039_trans__onD,axiom,
    ! [A2: set_Pr4934435412358123699_a_nat,R3: set_Pr1811044260758604347_a_nat,X2: product_prod_a_nat,Y: product_prod_a_nat,Z: product_prod_a_nat] :
      ( ( trans_5746013079982352605_a_nat @ A2 @ R3 )
     => ( ( member5724188588386418708_a_nat @ X2 @ A2 )
       => ( ( member5724188588386418708_a_nat @ Y @ A2 )
         => ( ( member5724188588386418708_a_nat @ Z @ A2 )
           => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X2 @ Y ) @ R3 )
             => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y @ Z ) @ R3 )
               => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X2 @ Z ) @ R3 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_1040_trans__onD,axiom,
    ! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,X2: term_a_b,Y: term_a_b,Z: term_a_b] :
      ( ( trans_on_term_a_b @ A2 @ R3 )
     => ( ( member_term_a_b @ X2 @ A2 )
       => ( ( member_term_a_b @ Y @ A2 )
         => ( ( member_term_a_b @ Z @ A2 )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ R3 )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ R3 )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z ) @ R3 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_1041_relcomp3__transI,axiom,
    ! [B5: set_Pr4386577575007340137rm_a_b,T: term_a_b,U: term_a_b,A2: set_Pr4386577575007340137rm_a_b,S: term_a_b,V4: term_a_b] :
      ( ( trans_on_term_a_b @ top_top_set_term_a_b @ B5 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ B5 @ ( relcom370159955682700863rm_a_b @ A2 @ B5 ) ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ B5 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ V4 ) @ B5 )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ V4 ) @ ( relcom370159955682700863rm_a_b @ B5 @ ( relcom370159955682700863rm_a_b @ A2 @ B5 ) ) ) ) ) ) ) ).

% relcomp3_transI
thf(fact_1042_trans__singleton,axiom,
    ! [A: term_a_b] : ( trans_on_term_a_b @ top_top_set_term_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ bot_bo197521221353338581rm_a_b ) ) ).

% trans_singleton
thf(fact_1043_subt__at_Oelims,axiom,
    ! [X2: term_a_b,Xa: list_nat,Y: term_a_b] :
      ( ( ( term_subt_at_a_b @ X2 @ Xa )
        = Y )
     => ( ( ( Xa = nil_nat )
         => ( Y != X2 ) )
       => ( ! [F: a,Ss: list_term_a_b] :
              ( ( X2
                = ( fun_a_b @ F @ Ss ) )
             => ! [I2: nat,P3: list_nat] :
                  ( ( Xa
                    = ( cons_nat @ I2 @ P3 ) )
                 => ( Y
                   != ( term_subt_at_a_b @ ( nth_term_a_b @ Ss @ I2 ) @ P3 ) ) ) )
         => ~ ( ? [X: b] :
                  ( X2
                  = ( var_b_a @ X ) )
             => ( ? [V: nat,Va: list_nat] :
                    ( Xa
                    = ( cons_nat @ V @ Va ) )
               => ( Y != undefined_term_a_b ) ) ) ) ) ) ).

% subt_at.elims
thf(fact_1044_listrel_ONil,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b] : ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ nil_term_a_b @ nil_term_a_b ) @ ( listre1194016999521814427rm_a_b @ R3 ) ) ).

% listrel.Nil
thf(fact_1045_listrel_ONil,axiom,
    ! [R3: set_Pr6708357783813671845_b_nat] : ( member8889939629616814094st_nat @ ( produc5386604813764302749st_nat @ nil_term_a_b @ nil_nat ) @ ( listrel_term_a_b_nat @ R3 ) ) ).

% listrel.Nil
thf(fact_1046_listrel_ONil,axiom,
    ! [R3: set_Pr4549835640365387557rm_a_b] : ( member7814915433699591566rm_a_b @ ( produc4336470666470712605rm_a_b @ nil_nat @ nil_term_a_b ) @ ( listrel_nat_term_a_b @ R3 ) ) ).

% listrel.Nil
thf(fact_1047_listrel_ONil,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) @ ( listrel_nat_nat @ R3 ) ) ).

% listrel.Nil
thf(fact_1048_listrel__Nil1,axiom,
    ! [Xs: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ nil_term_a_b @ Xs ) @ ( listre1194016999521814427rm_a_b @ R3 ) )
     => ( Xs = nil_term_a_b ) ) ).

% listrel_Nil1
thf(fact_1049_listrel__Nil1,axiom,
    ! [Xs: list_nat,R3: set_Pr6708357783813671845_b_nat] :
      ( ( member8889939629616814094st_nat @ ( produc5386604813764302749st_nat @ nil_term_a_b @ Xs ) @ ( listrel_term_a_b_nat @ R3 ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil1
thf(fact_1050_listrel__Nil1,axiom,
    ! [Xs: list_term_a_b,R3: set_Pr4549835640365387557rm_a_b] :
      ( ( member7814915433699591566rm_a_b @ ( produc4336470666470712605rm_a_b @ nil_nat @ Xs ) @ ( listrel_nat_term_a_b @ R3 ) )
     => ( Xs = nil_term_a_b ) ) ).

% listrel_Nil1
thf(fact_1051_listrel__Nil1,axiom,
    ! [Xs: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel_nat_nat @ R3 ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil1
thf(fact_1052_listrel__Nil2,axiom,
    ! [Xs: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ nil_term_a_b ) @ ( listre1194016999521814427rm_a_b @ R3 ) )
     => ( Xs = nil_term_a_b ) ) ).

% listrel_Nil2
thf(fact_1053_listrel__Nil2,axiom,
    ! [Xs: list_nat,R3: set_Pr4549835640365387557rm_a_b] :
      ( ( member7814915433699591566rm_a_b @ ( produc4336470666470712605rm_a_b @ Xs @ nil_term_a_b ) @ ( listrel_nat_term_a_b @ R3 ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil2
thf(fact_1054_listrel__Nil2,axiom,
    ! [Xs: list_term_a_b,R3: set_Pr6708357783813671845_b_nat] :
      ( ( member8889939629616814094st_nat @ ( produc5386604813764302749st_nat @ Xs @ nil_nat ) @ ( listrel_term_a_b_nat @ R3 ) )
     => ( Xs = nil_term_a_b ) ) ).

% listrel_Nil2
thf(fact_1055_listrel__Nil2,axiom,
    ! [Xs: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel_nat_nat @ R3 ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil2
thf(fact_1056_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y: nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat @ R3 ) )
     => ~ ! [X: nat,Xs2: list_nat] :
            ( ( Xs
              = ( cons_nat @ X @ Xs2 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( listrel_nat_nat @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_1057_listrel__Cons2,axiom,
    ! [Xs: list_term_a_b,Y: term_a_b,Ys: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ ( cons_term_a_b @ Y @ Ys ) ) @ ( listre1194016999521814427rm_a_b @ R3 ) )
     => ~ ! [X: term_a_b,Xs2: list_term_a_b] :
            ( ( Xs
              = ( cons_term_a_b @ X @ Xs2 ) )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R3 )
             => ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs2 @ Ys ) @ ( listre1194016999521814427rm_a_b @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_1058_listrel__Cons2,axiom,
    ! [Xs: list_a,Y: nat,Ys: list_nat,R3: set_Pr4934435412358123699_a_nat] :
      ( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_a_nat @ R3 ) )
     => ~ ! [X: a,Xs2: list_a] :
            ( ( Xs
              = ( cons_a @ X @ Xs2 ) )
           => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y ) @ R3 )
             => ~ ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs2 @ Ys ) @ ( listrel_a_nat @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_1059_listrel__Cons2,axiom,
    ! [Xs: list_nat_nat,Y: nat,Ys: list_nat,R3: set_Pr9093778441882193744at_nat] :
      ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat_nat @ R3 ) )
     => ~ ! [X: nat > nat,Xs2: list_nat_nat] :
            ( ( Xs
              = ( cons_nat_nat @ X @ Xs2 ) )
           => ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y ) @ R3 )
             => ~ ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs2 @ Ys ) @ ( listrel_nat_nat_nat @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_1060_listrel__Cons1,axiom,
    ! [Y: nat,Ys: list_nat,Xs: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y @ Ys ) @ Xs ) @ ( listrel_nat_nat @ R3 ) )
     => ~ ! [Y2: nat,Ys2: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y2 @ Ys2 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Y2 ) @ R3 )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Ys2 ) @ ( listrel_nat_nat @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_1061_listrel__Cons1,axiom,
    ! [Y: term_a_b,Ys: list_term_a_b,Xs: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ Y @ Ys ) @ Xs ) @ ( listre1194016999521814427rm_a_b @ R3 ) )
     => ~ ! [Y2: term_a_b,Ys2: list_term_a_b] :
            ( ( Xs
              = ( cons_term_a_b @ Y2 @ Ys2 ) )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Y2 ) @ R3 )
             => ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ys @ Ys2 ) @ ( listre1194016999521814427rm_a_b @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_1062_listrel__Cons1,axiom,
    ! [Y: a,Ys: list_a,Xs: list_nat,R3: set_Pr4934435412358123699_a_nat] :
      ( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ ( cons_a @ Y @ Ys ) @ Xs ) @ ( listrel_a_nat @ R3 ) )
     => ~ ! [Y2: nat,Ys2: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y2 @ Ys2 ) )
           => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ Y @ Y2 ) @ R3 )
             => ~ ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Ys @ Ys2 ) @ ( listrel_a_nat @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_1063_listrel__Cons1,axiom,
    ! [Y: nat > nat,Ys: list_nat_nat,Xs: list_nat,R3: set_Pr9093778441882193744at_nat] :
      ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ ( cons_nat_nat @ Y @ Ys ) @ Xs ) @ ( listrel_nat_nat_nat @ R3 ) )
     => ~ ! [Y2: nat,Ys2: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y2 @ Ys2 ) )
           => ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ Y @ Y2 ) @ R3 )
             => ~ ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Ys @ Ys2 ) @ ( listrel_nat_nat_nat @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_1064_listrel_OCons,axiom,
    ! [X2: nat,Y: nat,R3: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R3 )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R3 ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_1065_listrel_OCons,axiom,
    ! [X2: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ R3 )
     => ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listre1194016999521814427rm_a_b @ R3 ) )
       => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X2 @ Xs ) @ ( cons_term_a_b @ Y @ Ys ) ) @ ( listre1194016999521814427rm_a_b @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_1066_listrel_OCons,axiom,
    ! [X2: a,Y: nat,R3: set_Pr4934435412358123699_a_nat,Xs: list_a,Ys: list_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y ) @ R3 )
     => ( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs @ Ys ) @ ( listrel_a_nat @ R3 ) )
       => ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ ( cons_a @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_a_nat @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_1067_listrel_OCons,axiom,
    ! [X2: nat > nat,Y: nat,R3: set_Pr9093778441882193744at_nat,Xs: list_nat_nat,Ys: list_nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X2 @ Y ) @ R3 )
     => ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs @ Ys ) @ ( listrel_nat_nat_nat @ R3 ) )
       => ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ ( cons_nat_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat_nat @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_1068_subt__at_Osimps_I3_J,axiom,
    ! [X2: b,V4: nat,Va2: list_nat] :
      ( ( term_subt_at_a_b @ ( var_b_a @ X2 ) @ ( cons_nat @ V4 @ Va2 ) )
      = undefined_term_a_b ) ).

% subt_at.simps(3)
thf(fact_1069_listrel_Ocases,axiom,
    ! [A1: list_term_a_b,A22: list_nat,R3: set_Pr6708357783813671845_b_nat] :
      ( ( member8889939629616814094st_nat @ ( produc5386604813764302749st_nat @ A1 @ A22 ) @ ( listrel_term_a_b_nat @ R3 ) )
     => ( ( ( A1 = nil_term_a_b )
         => ( A22 != nil_nat ) )
       => ~ ! [X: term_a_b,Y2: nat,Xs2: list_term_a_b] :
              ( ( A1
                = ( cons_term_a_b @ X @ Xs2 ) )
             => ! [Ys2: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y2 @ Ys2 ) )
                 => ( ( member9000184225777080558_b_nat @ ( produc2478027105211390077_b_nat @ X @ Y2 ) @ R3 )
                   => ~ ( member8889939629616814094st_nat @ ( produc5386604813764302749st_nat @ Xs2 @ Ys2 ) @ ( listrel_term_a_b_nat @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_1070_listrel_Ocases,axiom,
    ! [A1: list_nat,A22: list_term_a_b,R3: set_Pr4549835640365387557rm_a_b] :
      ( ( member7814915433699591566rm_a_b @ ( produc4336470666470712605rm_a_b @ A1 @ A22 ) @ ( listrel_nat_term_a_b @ R3 ) )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_term_a_b ) )
       => ~ ! [X: nat,Y2: term_a_b,Xs2: list_nat] :
              ( ( A1
                = ( cons_nat @ X @ Xs2 ) )
             => ! [Ys2: list_term_a_b] :
                  ( ( A22
                    = ( cons_term_a_b @ Y2 @ Ys2 ) )
                 => ( ( member7514656309480873070rm_a_b @ ( produc1516572978046417917rm_a_b @ X @ Y2 ) @ R3 )
                   => ~ ( member7814915433699591566rm_a_b @ ( produc4336470666470712605rm_a_b @ Xs2 @ Ys2 ) @ ( listrel_nat_term_a_b @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_1071_listrel_Ocases,axiom,
    ! [A1: list_nat,A22: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R3 ) )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_nat ) )
       => ~ ! [X: nat,Y2: nat,Xs2: list_nat] :
              ( ( A1
                = ( cons_nat @ X @ Xs2 ) )
             => ! [Ys2: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y2 @ Ys2 ) )
                 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ R3 )
                   => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_1072_listrel_Ocases,axiom,
    ! [A1: list_term_a_b,A22: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A1 @ A22 ) @ ( listre1194016999521814427rm_a_b @ R3 ) )
     => ( ( ( A1 = nil_term_a_b )
         => ( A22 != nil_term_a_b ) )
       => ~ ! [X: term_a_b,Y2: term_a_b,Xs2: list_term_a_b] :
              ( ( A1
                = ( cons_term_a_b @ X @ Xs2 ) )
             => ! [Ys2: list_term_a_b] :
                  ( ( A22
                    = ( cons_term_a_b @ Y2 @ Ys2 ) )
                 => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ R3 )
                   => ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs2 @ Ys2 ) @ ( listre1194016999521814427rm_a_b @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_1073_listrel_Ocases,axiom,
    ! [A1: list_a,A22: list_nat,R3: set_Pr4934435412358123699_a_nat] :
      ( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ A1 @ A22 ) @ ( listrel_a_nat @ R3 ) )
     => ( ( ( A1 = nil_a )
         => ( A22 != nil_nat ) )
       => ~ ! [X: a,Y2: nat,Xs2: list_a] :
              ( ( A1
                = ( cons_a @ X @ Xs2 ) )
             => ! [Ys2: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y2 @ Ys2 ) )
                 => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y2 ) @ R3 )
                   => ~ ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs2 @ Ys2 ) @ ( listrel_a_nat @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_1074_listrel_Ocases,axiom,
    ! [A1: list_nat_nat,A22: list_nat,R3: set_Pr9093778441882193744at_nat] :
      ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ A1 @ A22 ) @ ( listrel_nat_nat_nat @ R3 ) )
     => ( ( ( A1 = nil_nat_nat )
         => ( A22 != nil_nat ) )
       => ~ ! [X: nat > nat,Y2: nat,Xs2: list_nat_nat] :
              ( ( A1
                = ( cons_nat_nat @ X @ Xs2 ) )
             => ! [Ys2: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y2 @ Ys2 ) )
                 => ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y2 ) @ R3 )
                   => ~ ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat_nat @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_1075_listrel_Osimps,axiom,
    ! [A1: list_term_a_b,A22: list_nat,R3: set_Pr6708357783813671845_b_nat] :
      ( ( member8889939629616814094st_nat @ ( produc5386604813764302749st_nat @ A1 @ A22 ) @ ( listrel_term_a_b_nat @ R3 ) )
      = ( ( ( A1 = nil_term_a_b )
          & ( A22 = nil_nat ) )
        | ? [X3: term_a_b,Y4: nat,Xs4: list_term_a_b,Ys3: list_nat] :
            ( ( A1
              = ( cons_term_a_b @ X3 @ Xs4 ) )
            & ( A22
              = ( cons_nat @ Y4 @ Ys3 ) )
            & ( member9000184225777080558_b_nat @ ( produc2478027105211390077_b_nat @ X3 @ Y4 ) @ R3 )
            & ( member8889939629616814094st_nat @ ( produc5386604813764302749st_nat @ Xs4 @ Ys3 ) @ ( listrel_term_a_b_nat @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_1076_listrel_Osimps,axiom,
    ! [A1: list_nat,A22: list_term_a_b,R3: set_Pr4549835640365387557rm_a_b] :
      ( ( member7814915433699591566rm_a_b @ ( produc4336470666470712605rm_a_b @ A1 @ A22 ) @ ( listrel_nat_term_a_b @ R3 ) )
      = ( ( ( A1 = nil_nat )
          & ( A22 = nil_term_a_b ) )
        | ? [X3: nat,Y4: term_a_b,Xs4: list_nat,Ys3: list_term_a_b] :
            ( ( A1
              = ( cons_nat @ X3 @ Xs4 ) )
            & ( A22
              = ( cons_term_a_b @ Y4 @ Ys3 ) )
            & ( member7514656309480873070rm_a_b @ ( produc1516572978046417917rm_a_b @ X3 @ Y4 ) @ R3 )
            & ( member7814915433699591566rm_a_b @ ( produc4336470666470712605rm_a_b @ Xs4 @ Ys3 ) @ ( listrel_nat_term_a_b @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_1077_listrel_Osimps,axiom,
    ! [A1: list_nat,A22: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R3 ) )
      = ( ( ( A1 = nil_nat )
          & ( A22 = nil_nat ) )
        | ? [X3: nat,Y4: nat,Xs4: list_nat,Ys3: list_nat] :
            ( ( A1
              = ( cons_nat @ X3 @ Xs4 ) )
            & ( A22
              = ( cons_nat @ Y4 @ Ys3 ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R3 )
            & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs4 @ Ys3 ) @ ( listrel_nat_nat @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_1078_listrel_Osimps,axiom,
    ! [A1: list_term_a_b,A22: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A1 @ A22 ) @ ( listre1194016999521814427rm_a_b @ R3 ) )
      = ( ( ( A1 = nil_term_a_b )
          & ( A22 = nil_term_a_b ) )
        | ? [X3: term_a_b,Y4: term_a_b,Xs4: list_term_a_b,Ys3: list_term_a_b] :
            ( ( A1
              = ( cons_term_a_b @ X3 @ Xs4 ) )
            & ( A22
              = ( cons_term_a_b @ Y4 @ Ys3 ) )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R3 )
            & ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs4 @ Ys3 ) @ ( listre1194016999521814427rm_a_b @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_1079_listrel_Osimps,axiom,
    ! [A1: list_a,A22: list_nat,R3: set_Pr4934435412358123699_a_nat] :
      ( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ A1 @ A22 ) @ ( listrel_a_nat @ R3 ) )
      = ( ( ( A1 = nil_a )
          & ( A22 = nil_nat ) )
        | ? [X3: a,Y4: nat,Xs4: list_a,Ys3: list_nat] :
            ( ( A1
              = ( cons_a @ X3 @ Xs4 ) )
            & ( A22
              = ( cons_nat @ Y4 @ Ys3 ) )
            & ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y4 ) @ R3 )
            & ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs4 @ Ys3 ) @ ( listrel_a_nat @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_1080_listrel_Osimps,axiom,
    ! [A1: list_nat_nat,A22: list_nat,R3: set_Pr9093778441882193744at_nat] :
      ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ A1 @ A22 ) @ ( listrel_nat_nat_nat @ R3 ) )
      = ( ( ( A1 = nil_nat_nat )
          & ( A22 = nil_nat ) )
        | ? [X3: nat > nat,Y4: nat,Xs4: list_nat_nat,Ys3: list_nat] :
            ( ( A1
              = ( cons_nat_nat @ X3 @ Xs4 ) )
            & ( A22
              = ( cons_nat @ Y4 @ Ys3 ) )
            & ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X3 @ Y4 ) @ R3 )
            & ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs4 @ Ys3 ) @ ( listrel_nat_nat_nat @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_1081_listrel__iff__nth,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listre1194016999521814427rm_a_b @ R3 ) )
      = ( ( ( size_s8906293707977694520rm_a_b @ Xs )
          = ( size_s8906293707977694520rm_a_b @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Xs @ N3 ) @ ( nth_term_a_b @ Ys @ N3 ) ) @ R3 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1082_listrel__iff__nth,axiom,
    ! [Xs: list_a,Ys: list_nat,R3: set_Pr4934435412358123699_a_nat] :
      ( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs @ Ys ) @ ( listrel_a_nat @ R3 ) )
      = ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs ) )
           => ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ ( nth_a @ Xs @ N3 ) @ ( nth_nat @ Ys @ N3 ) ) @ R3 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1083_listrel__iff__nth,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat,R3: set_Pr9093778441882193744at_nat] :
      ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs @ Ys ) @ ( listrel_nat_nat_nat @ R3 ) )
      = ( ( ( size_s8208510060688613859at_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s8208510060688613859at_nat @ Xs ) )
           => ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ ( nth_nat_nat @ Xs @ N3 ) @ ( nth_nat @ Ys @ N3 ) ) @ R3 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1084_subt__at_Opelims,axiom,
    ! [X2: term_a_b,Xa: list_nat,Y: term_a_b] :
      ( ( ( term_subt_at_a_b @ X2 @ Xa )
        = Y )
     => ( ( accp_P682940083893826398st_nat @ term_subt_at_rel_a_b @ ( produc4563063199488751885st_nat @ X2 @ Xa ) )
       => ( ( ( Xa = nil_nat )
           => ( ( Y = X2 )
             => ~ ( accp_P682940083893826398st_nat @ term_subt_at_rel_a_b @ ( produc4563063199488751885st_nat @ X2 @ nil_nat ) ) ) )
         => ( ! [F: a,Ss: list_term_a_b] :
                ( ( X2
                  = ( fun_a_b @ F @ Ss ) )
               => ! [I2: nat,P3: list_nat] :
                    ( ( Xa
                      = ( cons_nat @ I2 @ P3 ) )
                   => ( ( Y
                        = ( term_subt_at_a_b @ ( nth_term_a_b @ Ss @ I2 ) @ P3 ) )
                     => ~ ( accp_P682940083893826398st_nat @ term_subt_at_rel_a_b @ ( produc4563063199488751885st_nat @ ( fun_a_b @ F @ Ss ) @ ( cons_nat @ I2 @ P3 ) ) ) ) ) )
           => ~ ! [X: b] :
                  ( ( X2
                    = ( var_b_a @ X ) )
                 => ! [V: nat,Va: list_nat] :
                      ( ( Xa
                        = ( cons_nat @ V @ Va ) )
                     => ( ( Y = undefined_term_a_b )
                       => ~ ( accp_P682940083893826398st_nat @ term_subt_at_rel_a_b @ ( produc4563063199488751885st_nat @ ( var_b_a @ X ) @ ( cons_nat @ V @ Va ) ) ) ) ) ) ) ) ) ) ).

% subt_at.pelims
thf(fact_1085_fun__at_Osimps_I3_J,axiom,
    ! [I: nat,Ts2: list_term_a_b,F2: a,P: list_nat] :
      ( ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
       => ( ( term_fun_at_a_b @ ( fun_a_b @ F2 @ Ts2 ) @ ( cons_nat @ I @ P ) )
          = ( term_fun_at_a_b @ ( nth_term_a_b @ Ts2 @ I ) @ P ) ) )
      & ( ~ ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
       => ( ( term_fun_at_a_b @ ( fun_a_b @ F2 @ Ts2 ) @ ( cons_nat @ I @ P ) )
          = none_Sum_sum_a_b ) ) ) ).

% fun_at.simps(3)
thf(fact_1086_fun__at__None__nposs__iff,axiom,
    ! [T: term_a_b,P: list_nat] :
      ( ( ( term_fun_at_a_b @ T @ P )
        = none_Sum_sum_a_b )
      = ( ~ ( member_list_nat @ P @ ( term_poss_a_b @ T ) ) ) ) ).

% fun_at_None_nposs_iff
thf(fact_1087_fun__at_Osimps_I4_J,axiom,
    ! [Vb2: b,V4: nat,Va2: list_nat] :
      ( ( term_fun_at_a_b @ ( var_b_a @ Vb2 ) @ ( cons_nat @ V4 @ Va2 ) )
      = none_Sum_sum_a_b ) ).

% fun_at.simps(4)
thf(fact_1088_nth__Cons__pos,axiom,
    ! [N: nat,X2: nat,Xs: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
        = ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_1089_nth__append__Cons,axiom,
    ! [I: nat,Xs: list_nat,Y: nat,Zs: list_nat] :
      ( ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ Y @ Zs ) ) @ I )
          = ( nth_nat @ Xs @ I ) ) )
      & ( ~ ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( ( I
              = ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ Y @ Zs ) ) @ I )
              = Y ) )
          & ( ( I
             != ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ Y @ Zs ) ) @ I )
              = ( nth_nat @ Zs @ ( minus_minus_nat @ I @ ( suc @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ) ) ) ).

% nth_append_Cons
thf(fact_1090_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_1091_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_1092_nth__Cons__Suc,axiom,
    ! [X2: nat,Xs: list_nat,N: nat] :
      ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ ( suc @ N ) )
      = ( nth_nat @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_1093_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_1094_Suc__pred,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% Suc_pred
thf(fact_1095_Suc__diff__1,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
        = N ) ) ).

% Suc_diff_1
thf(fact_1096_list__update__code_I3_J,axiom,
    ! [X2: nat,Xs: list_nat,I: nat,Y: nat] :
      ( ( list_update_nat @ ( cons_nat @ X2 @ Xs ) @ ( suc @ I ) @ Y )
      = ( cons_nat @ X2 @ ( list_update_nat @ Xs @ I @ Y ) ) ) ).

% list_update_code(3)
thf(fact_1097_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_1098_Suc__pred_H,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( N
        = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_1099_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M4: nat] :
            ( N
            = ( suc @ M4 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_1100_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M2: nat] :
          ( N
          = ( suc @ M2 ) ) ) ).

% gr0_implies_Suc
thf(fact_1101_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( M = zero_zero_nat )
        | ? [J2: nat] :
            ( ( M
              = ( suc @ J2 ) )
            & ( ord_less_nat @ J2 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_1102_nat__induct__non__zero,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P2 @ one_one_nat )
       => ( ! [N2: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( ( P2 @ N2 )
               => ( P2 @ ( suc @ N2 ) ) ) )
         => ( P2 @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_1103_ex__Suc__conv,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
            & ( P2 @ I4 ) ) )
      = ( ( P2 @ zero_zero_nat )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
            & ( P2 @ ( suc @ I4 ) ) ) ) ) ).

% ex_Suc_conv
thf(fact_1104_all__Suc__conv,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
           => ( P2 @ I4 ) ) )
      = ( ( P2 @ zero_zero_nat )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
           => ( P2 @ ( suc @ I4 ) ) ) ) ) ).

% all_Suc_conv
thf(fact_1105_all__less__two,axiom,
    ! [P2: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ ( suc @ zero_zero_nat ) ) )
           => ( P2 @ I4 ) ) )
      = ( ( P2 @ zero_zero_nat )
        & ( P2 @ ( suc @ zero_zero_nat ) ) ) ) ).

% all_less_two
thf(fact_1106_Abstract__Rewriting_Ochain__mono,axiom,
    ! [R9: set_Pr4386577575007340137rm_a_b,R5: set_Pr4386577575007340137rm_a_b,Seq: nat > term_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R9 @ R5 )
     => ( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( Seq @ I2 ) @ ( Seq @ ( suc @ I2 ) ) ) @ R9 )
       => ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( Seq @ I3 ) @ ( Seq @ ( suc @ I3 ) ) ) @ R5 ) ) ) ).

% Abstract_Rewriting.chain_mono
thf(fact_1107_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_1108_relpow__Suc__E,axiom,
    ! [X2: term_a_b,Z: term_a_b,N: nat,R5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R5 ) )
     => ~ ! [Y2: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y2 ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z ) @ R5 ) ) ) ).

% relpow_Suc_E
thf(fact_1109_relpow__Suc__I,axiom,
    ! [X2: term_a_b,Y: term_a_b,N: nat,R5: set_Pr4386577575007340137rm_a_b,Z: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ R5 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R5 ) ) ) ) ).

% relpow_Suc_I
thf(fact_1110_relpow__Suc__D2,axiom,
    ! [X2: term_a_b,Z: term_a_b,N: nat,R5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R5 ) )
     => ? [Y2: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y2 ) @ R5 )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) ) ) ) ).

% relpow_Suc_D2
thf(fact_1111_relpow__Suc__E2,axiom,
    ! [X2: term_a_b,Z: term_a_b,N: nat,R5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R5 ) )
     => ~ ! [Y2: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y2 ) @ R5 )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) ) ) ) ).

% relpow_Suc_E2
thf(fact_1112_relpow__Suc__I2,axiom,
    ! [X2: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b,Z: term_a_b,N: nat] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ R5 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R5 ) ) ) ) ).

% relpow_Suc_I2
thf(fact_1113_length__Suc__conv,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( suc @ N ) )
      = ( ? [Y4: nat,Ys3: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y4 @ Ys3 ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_1114_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ( suc @ N )
        = ( size_size_list_nat @ Xs ) )
      = ( ? [Y4: nat,Ys3: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y4 @ Ys3 ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_1115_nat_Odistinct_I1_J,axiom,
    ! [X23: nat] :
      ( zero_zero_nat
     != ( suc @ X23 ) ) ).

% nat.distinct(1)
thf(fact_1116_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat: nat] :
      ( ( suc @ Nat )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_1117_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat: nat] :
      ( zero_zero_nat
     != ( suc @ Nat ) ) ).

% old.nat.distinct(1)
thf(fact_1118_nat_OdiscI,axiom,
    ! [Nat2: nat,X23: nat] :
      ( ( Nat2
        = ( suc @ X23 ) )
     => ( Nat2 != zero_zero_nat ) ) ).

% nat.discI
thf(fact_1119_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_1120_nat__induct,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P2 @ N2 )
           => ( P2 @ ( suc @ N2 ) ) )
       => ( P2 @ N ) ) ) ).

% nat_induct
thf(fact_1121_diff__induct,axiom,
    ! [P2: nat > nat > $o,M: nat,N: nat] :
      ( ! [X: nat] : ( P2 @ X @ zero_zero_nat )
     => ( ! [Y2: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y2 ) )
       => ( ! [X: nat,Y2: nat] :
              ( ( P2 @ X @ Y2 )
             => ( P2 @ ( suc @ X ) @ ( suc @ Y2 ) ) )
         => ( P2 @ M @ N ) ) ) ) ).

% diff_induct
thf(fact_1122_zero__induct,axiom,
    ! [P2: nat > $o,K: nat] :
      ( ( P2 @ K )
     => ( ! [N2: nat] :
            ( ( P2 @ ( suc @ N2 ) )
           => ( P2 @ N2 ) )
       => ( P2 @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_1123_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_1124_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_1125_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_1126_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M2: nat] :
          ( N
          = ( suc @ M2 ) ) ) ).

% not0_implies_Suc
thf(fact_1127_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_1128_one__reorient,axiom,
    ! [X2: nat] :
      ( ( one_one_nat = X2 )
      = ( X2 = one_one_nat ) ) ).

% one_reorient
thf(fact_1129_gen__length__code_I2_J,axiom,
    ! [N: nat,X2: nat,Xs: list_nat] :
      ( ( gen_length_nat @ N @ ( cons_nat @ X2 @ Xs ) )
      = ( gen_length_nat @ ( suc @ N ) @ Xs ) ) ).

% gen_length_code(2)
thf(fact_1130_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
      = ( ? [X3: nat,Ys3: list_nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_1131_ex__least__nat__less,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ N )
     => ( ~ ( P2 @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N )
            & ! [I3: nat] :
                ( ( ord_less_eq_nat @ I3 @ K2 )
               => ~ ( P2 @ I3 ) )
            & ( P2 @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_1132_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).

% diff_Suc_less
thf(fact_1133_subterm_Olift__Suc__mono__less__iff,axiom,
    ! [F2: nat > term_a_b,M: nat,N: nat] :
      ( ! [N2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ M ) @ ( F2 @ N ) ) @ subterm_and_supt_a_b )
        = ( ord_less_nat @ N @ M ) ) ) ).

% subterm.lift_Suc_mono_less_iff
thf(fact_1134_subterm_Olift__Suc__mono__less,axiom,
    ! [F2: nat > term_a_b,N: nat,N4: nat] :
      ( ! [N2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) ) @ subterm_and_supt_a_b )
     => ( ( ord_less_nat @ N @ N4 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ N4 ) @ ( F2 @ N ) ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.lift_Suc_mono_less
thf(fact_1135_subterm_Olift__Suc__mono__le,axiom,
    ! [F2: nat > term_a_b,N: nat,N4: nat] :
      ( ! [N2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) ) @ subter523971068842742411eq_a_b )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ N4 ) @ ( F2 @ N ) ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.lift_Suc_mono_le
thf(fact_1136_subterm_Olift__Suc__antimono__le,axiom,
    ! [F2: nat > term_a_b,N: nat,N4: nat] :
      ( ! [N2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) ) @ subter523971068842742411eq_a_b )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ N ) @ ( F2 @ N4 ) ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.lift_Suc_antimono_le
thf(fact_1137_relpow__E,axiom,
    ! [X2: term_a_b,Z: term_a_b,N: nat,R5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
     => ( ( ( N = zero_zero_nat )
         => ( X2 != Z ) )
       => ~ ! [Y2: term_a_b,M2: nat] :
              ( ( N
                = ( suc @ M2 ) )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y2 ) @ ( compow4057154403645558940rm_a_b @ M2 @ R5 ) )
               => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z ) @ R5 ) ) ) ) ) ).

% relpow_E
thf(fact_1138_relpow__E2,axiom,
    ! [X2: term_a_b,Z: term_a_b,N: nat,R5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
     => ( ( ( N = zero_zero_nat )
         => ( X2 != Z ) )
       => ~ ! [Y2: term_a_b,M2: nat] :
              ( ( N
                = ( suc @ M2 ) )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y2 ) @ R5 )
               => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z ) @ ( compow4057154403645558940rm_a_b @ M2 @ R5 ) ) ) ) ) ) ).

% relpow_E2
thf(fact_1139_relpow__Suc__E2_H,axiom,
    ! [X2: term_a_b,Z: term_a_b,N: nat,A2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ A2 ) )
     => ~ ! [Y2: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y2 ) @ A2 )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z ) @ ( transi7742714808557438673rm_a_b @ A2 ) ) ) ) ).

% relpow_Suc_E2'
thf(fact_1140_rtrancl__fun__conv,axiom,
    ! [S: term_a_b,T: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
      = ( ? [F7: nat > term_a_b,N3: nat] :
            ( ( ( F7 @ zero_zero_nat )
              = S )
            & ( ( F7 @ N3 )
              = T )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ N3 )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F7 @ I4 ) @ ( F7 @ ( suc @ I4 ) ) ) @ R5 ) ) ) ) ) ).

% rtrancl_fun_conv
thf(fact_1141_rtrancl__imp__seq,axiom,
    ! [X2: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
     => ? [F: nat > term_a_b,N2: nat] :
          ( ( ( F @ zero_zero_nat )
            = X2 )
          & ( ( F @ N2 )
            = Y )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ N2 )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) ) @ R3 ) ) ) ) ).

% rtrancl_imp_seq
thf(fact_1142_length__append__singleton,axiom,
    ! [Xs: list_term_a_b,X2: term_a_b] :
      ( ( size_s8906293707977694520rm_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ X2 @ nil_term_a_b ) ) )
      = ( suc @ ( size_s8906293707977694520rm_a_b @ Xs ) ) ) ).

% length_append_singleton
thf(fact_1143_length__append__singleton,axiom,
    ! [Xs: list_nat,X2: nat] :
      ( ( size_size_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
      = ( suc @ ( size_size_list_nat @ Xs ) ) ) ).

% length_append_singleton
thf(fact_1144_length__Suc__conv__rev,axiom,
    ! [Xs: list_term_a_b,N: nat] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( suc @ N ) )
      = ( ? [Y4: term_a_b,Ys3: list_term_a_b] :
            ( ( Xs
              = ( append_term_a_b @ Ys3 @ ( cons_term_a_b @ Y4 @ nil_term_a_b ) ) )
            & ( ( size_s8906293707977694520rm_a_b @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_1145_length__Suc__conv__rev,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( suc @ N ) )
      = ( ? [Y4: nat,Ys3: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys3 @ ( cons_nat @ Y4 @ nil_nat ) ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_1146_relpow__fun__conv,axiom,
    ! [A: term_a_b,B: term_a_b,N: nat,R5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
      = ( ? [F7: nat > term_a_b] :
            ( ( ( F7 @ zero_zero_nat )
              = A )
            & ( ( F7 @ N )
              = B )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ N )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F7 @ I4 ) @ ( F7 @ ( suc @ I4 ) ) ) @ R5 ) ) ) ) ) ).

% relpow_fun_conv
thf(fact_1147_relpow__seq,axiom,
    ! [X2: term_a_b,Y: term_a_b,N: nat,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) )
     => ? [F: nat > term_a_b] :
          ( ( ( F @ zero_zero_nat )
            = X2 )
          & ( ( F @ N )
            = Y )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ N )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) ) @ R3 ) ) ) ) ).

% relpow_seq
thf(fact_1148_nth__Cons_H,axiom,
    ! [N: nat,X2: nat,Xs: list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
          = X2 ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
          = ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_1149_nth__non__equal__first__eq,axiom,
    ! [X2: nat,Y: nat,Xs: list_nat,N: nat] :
      ( ( X2 != Y )
     => ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
          = Y )
        = ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_1150_chain__imp__trancl,axiom,
    ! [S6: nat > term_a_b,R3: set_Pr4386577575007340137rm_a_b,I: nat,J: nat] :
      ( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S6 @ I2 ) @ ( S6 @ ( suc @ I2 ) ) ) @ R3 )
     => ( ( ord_less_nat @ I @ J )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S6 @ I ) @ ( S6 @ J ) ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ).

% chain_imp_trancl
thf(fact_1151_chain__imp__rtrancl,axiom,
    ! [S6: nat > term_a_b,R3: set_Pr4386577575007340137rm_a_b,I: nat,J: nat] :
      ( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S6 @ I2 ) @ ( S6 @ ( suc @ I2 ) ) ) @ R3 )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S6 @ I ) @ ( S6 @ J ) ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ) ).

% chain_imp_rtrancl
thf(fact_1152_inf__concat__simple_Ocases,axiom,
    ! [X2: produc8199716216217303280at_nat] :
      ( ! [F: nat > nat] :
          ( X2
         != ( produc72220940542539688at_nat @ F @ zero_zero_nat ) )
     => ~ ! [F: nat > nat,N2: nat] :
            ( X2
           != ( produc72220940542539688at_nat @ F @ ( suc @ N2 ) ) ) ) ).

% inf_concat_simple.cases
thf(fact_1153_wf__iff__no__infinite__down__chain,axiom,
    ( wf_term_a_b
    = ( ^ [R2: set_Pr4386577575007340137rm_a_b] :
          ~ ? [F7: nat > term_a_b] :
            ! [I4: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F7 @ ( suc @ I4 ) ) @ ( F7 @ I4 ) ) @ R2 ) ) ) ).

% wf_iff_no_infinite_down_chain
thf(fact_1154_wf__no__infinite__down__chainE,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,F2: nat > term_a_b] :
      ( ( wf_term_a_b @ R3 )
     => ~ ! [K2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ ( suc @ K2 ) ) @ ( F2 @ K2 ) ) @ R3 ) ) ).

% wf_no_infinite_down_chainE
thf(fact_1155_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_1156_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_1157_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_1158_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one_class.zero_le_one
thf(fact_1159_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1160_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_1161_adjust__idx__rev__def,axiom,
    ( missin3815256168798769645dx_rev
    = ( ^ [I4: nat,J2: nat] : ( if_nat @ ( ord_less_nat @ J2 @ I4 ) @ J2 @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) ) ).

% adjust_idx_rev_def
thf(fact_1162_replace__term__context__at_Osimps_I2_J,axiom,
    ! [I: nat,Ss3: list_term_a_b,F2: a,C3: subterm_and_ctxt_a_b,Ts2: list_term_a_b,Ps: list_nat,U: term_a_b] :
      ( ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ss3 ) )
       => ( ( terms_4774307173741787698at_a_b @ ( subterm_and_More_a_b @ F2 @ Ss3 @ C3 @ Ts2 ) @ ( cons_nat @ I @ Ps ) @ U )
          = ( subterm_and_More_a_b @ F2 @ ( list_update_term_a_b @ Ss3 @ I @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ss3 @ I ) @ Ps @ U ) ) @ C3 @ Ts2 ) ) )
      & ( ~ ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ss3 ) )
       => ( ( ( I
              = ( size_s8906293707977694520rm_a_b @ Ss3 ) )
           => ( ( terms_4774307173741787698at_a_b @ ( subterm_and_More_a_b @ F2 @ Ss3 @ C3 @ Ts2 ) @ ( cons_nat @ I @ Ps ) @ U )
              = ( subterm_and_More_a_b @ F2 @ Ss3 @ ( terms_4774307173741787698at_a_b @ C3 @ Ps @ U ) @ Ts2 ) ) )
          & ( ( I
             != ( size_s8906293707977694520rm_a_b @ Ss3 ) )
           => ( ( terms_4774307173741787698at_a_b @ ( subterm_and_More_a_b @ F2 @ Ss3 @ C3 @ Ts2 ) @ ( cons_nat @ I @ Ps ) @ U )
              = ( subterm_and_More_a_b @ F2 @ Ss3 @ C3 @ ( list_update_term_a_b @ Ts2 @ ( minus_minus_nat @ I @ ( suc @ ( size_s8906293707977694520rm_a_b @ Ss3 ) ) ) @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ts2 @ ( minus_minus_nat @ I @ ( suc @ ( size_s8906293707977694520rm_a_b @ Ss3 ) ) ) ) @ Ps @ U ) ) ) ) ) ) ) ) ).

% replace_term_context_at.simps(2)
thf(fact_1163_ctxt__apply__term_Osimps_I2_J,axiom,
    ! [F2: a,Ss1: list_term_a_b,C3: subterm_and_ctxt_a_b,Ss22: list_term_a_b,S: term_a_b] :
      ( ( subter2376574525758040790rm_a_b @ ( subterm_and_More_a_b @ F2 @ Ss1 @ C3 @ Ss22 ) @ S )
      = ( fun_a_b @ F2 @ ( append_term_a_b @ Ss1 @ ( cons_term_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ S ) @ Ss22 ) ) ) ) ).

% ctxt_apply_term.simps(2)
thf(fact_1164_hole__pos_Osimps_I2_J,axiom,
    ! [F2: a,Ss3: list_term_a_b,D: subterm_and_ctxt_a_b,Ts2: list_term_a_b] :
      ( ( term_hole_pos_a_b @ ( subterm_and_More_a_b @ F2 @ Ss3 @ D @ Ts2 ) )
      = ( cons_nat @ ( size_s8906293707977694520rm_a_b @ Ss3 ) @ ( term_hole_pos_a_b @ D ) ) ) ).

% hole_pos.simps(2)
thf(fact_1165_inv__const__ctxt_H_Osimps_I2_J,axiom,
    ! [F2: a,Ss3: list_term_a_b,Ts2: list_term_a_b,F4: set_Pr4934435412358123699_a_nat,V4: b,C3: subterm_and_ctxt_a_b] :
      ( ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F2 @ ( suc @ ( plus_plus_nat @ ( size_s8906293707977694520rm_a_b @ Ss3 ) @ ( size_s8906293707977694520rm_a_b @ Ts2 ) ) ) ) @ F4 )
       => ( ( terms_130083692264552600xt_a_b @ F4 @ V4 @ ( subterm_and_More_a_b @ F2 @ Ss3 @ C3 @ Ts2 ) )
          = ( fun_a_b @ F2 @ ( append_term_a_b @ ( map_te2328734355998148206rm_a_b @ ( terms_8519481630511763164ig_a_b @ F4 @ V4 ) @ Ss3 ) @ ( cons_term_a_b @ ( terms_130083692264552600xt_a_b @ F4 @ V4 @ C3 ) @ ( map_te2328734355998148206rm_a_b @ ( terms_8519481630511763164ig_a_b @ F4 @ V4 ) @ Ts2 ) ) ) ) ) )
      & ( ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F2 @ ( suc @ ( plus_plus_nat @ ( size_s8906293707977694520rm_a_b @ Ss3 ) @ ( size_s8906293707977694520rm_a_b @ Ts2 ) ) ) ) @ F4 )
       => ( ( terms_130083692264552600xt_a_b @ F4 @ V4 @ ( subterm_and_More_a_b @ F2 @ Ss3 @ C3 @ Ts2 ) )
          = ( var_b_a @ V4 ) ) ) ) ).

% inv_const_ctxt'.simps(2)
thf(fact_1166_non__strict__ending,axiom,
    ! [T: nat > term_a_b,R5: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
      ( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ I2 ) @ ( T @ ( suc @ I2 ) ) ) @ ( sup_su6776935440552674877rm_a_b @ R5 @ S6 ) )
     => ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R5 @ S6 ) @ S6 )
       => ( ( abstra4720023341729745482rm_a_b @ S6 @ ( insert_term_a_b @ ( T @ zero_zero_nat ) @ bot_bot_set_term_a_b ) )
         => ? [J3: nat] :
            ! [I3: nat] :
              ( ( ord_less_eq_nat @ J3 @ I3 )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ I3 ) @ ( T @ ( suc @ I3 ) ) ) @ ( minus_5192120951422937424rm_a_b @ R5 @ S6 ) ) ) ) ) ) ).

% non_strict_ending
thf(fact_1167_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_1168_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_1169_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_1170_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_1171_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_1172_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_1173_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_1174_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ A ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_1175_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ A @ B ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_1176_add__eq__0__iff__both__eq__0,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X2 @ Y )
        = zero_zero_nat )
      = ( ( X2 = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_1177_zero__eq__add__iff__both__eq__0,axiom,
    ! [X2: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X2 @ Y ) )
      = ( ( X2 = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_1178_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_1179_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_1180_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_1181_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_1182_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_1183_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_1184_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_1185_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_1186_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_1187_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel2
thf(fact_1188_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel1
thf(fact_1189_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_1190_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_1191_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel2
thf(fact_1192_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel1
thf(fact_1193_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_1194_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_1195_diff__add__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_1196_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_1197_length__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat @ ( append_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_append
thf(fact_1198_not__SN__onI,axiom,
    ! [F2: nat > list_nat,X6: set_list_nat,R5: set_Pr3451248702717554689st_nat] :
      ( ( member_list_nat @ ( F2 @ zero_zero_nat ) @ X6 )
     => ( ! [I2: nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( F2 @ I2 ) @ ( F2 @ ( suc @ I2 ) ) ) @ R5 )
       => ~ ( abstra2262888740781824790st_nat @ R5 @ X6 ) ) ) ).

% not_SN_onI
thf(fact_1199_not__SN__onI,axiom,
    ! [F2: nat > b,X6: set_b,R5: set_Product_prod_b_b] :
      ( ( member_b @ ( F2 @ zero_zero_nat ) @ X6 )
     => ( ! [I2: nat] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( F2 @ I2 ) @ ( F2 @ ( suc @ I2 ) ) ) @ R5 )
       => ~ ( abstract_SN_on_b @ R5 @ X6 ) ) ) ).

% not_SN_onI
thf(fact_1200_not__SN__onI,axiom,
    ! [F2: nat > produc357393685978478089rm_a_b,X6: set_Pr4386577575007340137rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( F2 @ zero_zero_nat ) @ X6 )
     => ( ! [I2: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F2 @ I2 ) @ ( F2 @ ( suc @ I2 ) ) ) @ R5 )
       => ~ ( abstra2398554102055911763rm_a_b @ R5 @ X6 ) ) ) ).

% not_SN_onI
thf(fact_1201_not__SN__onI,axiom,
    ! [F2: nat > product_prod_a_nat,X6: set_Pr4934435412358123699_a_nat,R5: set_Pr1811044260758604347_a_nat] :
      ( ( member5724188588386418708_a_nat @ ( F2 @ zero_zero_nat ) @ X6 )
     => ( ! [I2: nat] : ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( F2 @ I2 ) @ ( F2 @ ( suc @ I2 ) ) ) @ R5 )
       => ~ ( abstra8653715922312955827_a_nat @ R5 @ X6 ) ) ) ).

% not_SN_onI
thf(fact_1202_not__SN__onI,axiom,
    ! [F2: nat > term_a_b,X6: set_term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
      ( ( member_term_a_b @ ( F2 @ zero_zero_nat ) @ X6 )
     => ( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ I2 ) @ ( F2 @ ( suc @ I2 ) ) ) @ R5 )
       => ~ ( abstra4720023341729745482rm_a_b @ R5 @ X6 ) ) ) ).

% not_SN_onI
thf(fact_1203_SN__onI,axiom,
    ! [A2: set_list_nat,R3: set_Pr3451248702717554689st_nat] :
      ( ! [F: nat > list_nat] :
          ( ( member_list_nat @ ( F @ zero_zero_nat ) @ A2 )
         => ~ ! [I3: nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) ) @ R3 ) )
     => ( abstra2262888740781824790st_nat @ R3 @ A2 ) ) ).

% SN_onI
thf(fact_1204_SN__onI,axiom,
    ! [A2: set_b,R3: set_Product_prod_b_b] :
      ( ! [F: nat > b] :
          ( ( member_b @ ( F @ zero_zero_nat ) @ A2 )
         => ~ ! [I3: nat] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) ) @ R3 ) )
     => ( abstract_SN_on_b @ R3 @ A2 ) ) ).

% SN_onI
thf(fact_1205_SN__onI,axiom,
    ! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
      ( ! [F: nat > produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( F @ zero_zero_nat ) @ A2 )
         => ~ ! [I3: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) ) @ R3 ) )
     => ( abstra2398554102055911763rm_a_b @ R3 @ A2 ) ) ).

% SN_onI
thf(fact_1206_SN__onI,axiom,
    ! [A2: set_Pr4934435412358123699_a_nat,R3: set_Pr1811044260758604347_a_nat] :
      ( ! [F: nat > product_prod_a_nat] :
          ( ( member5724188588386418708_a_nat @ ( F @ zero_zero_nat ) @ A2 )
         => ~ ! [I3: nat] : ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) ) @ R3 ) )
     => ( abstra8653715922312955827_a_nat @ R3 @ A2 ) ) ).

% SN_onI
thf(fact_1207_SN__onI,axiom,
    ! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ! [F: nat > term_a_b] :
          ( ( member_term_a_b @ ( F @ zero_zero_nat ) @ A2 )
         => ~ ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) ) @ R3 ) )
     => ( abstra4720023341729745482rm_a_b @ R3 @ A2 ) ) ).

% SN_onI
thf(fact_1208_nth__append__length__plus,axiom,
    ! [Xs: list_nat,Ys: list_nat,N: nat] :
      ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N ) )
      = ( nth_nat @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_1209_SN__on__induct,axiom,
    ! [R5: set_Pr3451248702717554689st_nat,A2: set_list_nat,S: list_nat,P2: list_nat > $o] :
      ( ( abstra2262888740781824790st_nat @ R5 @ A2 )
     => ( ( member_list_nat @ S @ A2 )
       => ( ! [T2: list_nat] :
              ( ! [U4: list_nat] :
                  ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ T2 @ U4 ) @ R5 )
                 => ( P2 @ U4 ) )
             => ( P2 @ T2 ) )
         => ( P2 @ S ) ) ) ) ).

% SN_on_induct
thf(fact_1210_SN__on__induct,axiom,
    ! [R5: set_Product_prod_b_b,A2: set_b,S: b,P2: b > $o] :
      ( ( abstract_SN_on_b @ R5 @ A2 )
     => ( ( member_b @ S @ A2 )
       => ( ! [T2: b] :
              ( ! [U4: b] :
                  ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ T2 @ U4 ) @ R5 )
                 => ( P2 @ U4 ) )
             => ( P2 @ T2 ) )
         => ( P2 @ S ) ) ) ) ).

% SN_on_induct
thf(fact_1211_SN__on__induct,axiom,
    ! [R5: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b,S: produc357393685978478089rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
      ( ( abstra2398554102055911763rm_a_b @ R5 @ A2 )
     => ( ( member5869715511025134514rm_a_b @ S @ A2 )
       => ( ! [T2: produc357393685978478089rm_a_b] :
              ( ! [U4: produc357393685978478089rm_a_b] :
                  ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T2 @ U4 ) @ R5 )
                 => ( P2 @ U4 ) )
             => ( P2 @ T2 ) )
         => ( P2 @ S ) ) ) ) ).

% SN_on_induct
thf(fact_1212_SN__on__induct,axiom,
    ! [R5: set_Pr1811044260758604347_a_nat,A2: set_Pr4934435412358123699_a_nat,S: product_prod_a_nat,P2: product_prod_a_nat > $o] :
      ( ( abstra8653715922312955827_a_nat @ R5 @ A2 )
     => ( ( member5724188588386418708_a_nat @ S @ A2 )
       => ( ! [T2: product_prod_a_nat] :
              ( ! [U4: product_prod_a_nat] :
                  ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ T2 @ U4 ) @ R5 )
                 => ( P2 @ U4 ) )
             => ( P2 @ T2 ) )
         => ( P2 @ S ) ) ) ) ).

% SN_on_induct
thf(fact_1213_SN__on__induct,axiom,
    ! [R5: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b,S: term_a_b,P2: term_a_b > $o] :
      ( ( abstra4720023341729745482rm_a_b @ R5 @ A2 )
     => ( ( member_term_a_b @ S @ A2 )
       => ( ! [T2: term_a_b] :
              ( ! [U4: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ U4 ) @ R5 )
                 => ( P2 @ U4 ) )
             => ( P2 @ T2 ) )
         => ( P2 @ S ) ) ) ) ).

% SN_on_induct
thf(fact_1214_SN__on__irrefl,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R3 @ A2 )
     => ! [X4: term_a_b] :
          ( ( member_term_a_b @ X4 @ A2 )
         => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ X4 ) @ R3 ) ) ) ).

% SN_on_irrefl
thf(fact_1215_SN__induct__rule,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o,A: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R3 @ top_top_set_term_a_b )
     => ( ! [A4: term_a_b] :
            ( ! [B6: term_a_b] :
                ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B6 ) @ R3 )
               => ( P2 @ B6 ) )
           => ( P2 @ A4 ) )
       => ( P2 @ A ) ) ) ).

% SN_induct_rule
thf(fact_1216_SN__imp__minimal,axiom,
    ! [A2: set_Pr3451248702717554689st_nat] :
      ( ( abstra2262888740781824790st_nat @ A2 @ top_top_set_list_nat )
     => ! [Q8: set_list_nat] :
          ( ? [X4: list_nat] : ( member_list_nat @ X4 @ Q8 )
         => ? [X: list_nat] :
              ( ( member_list_nat @ X @ Q8 )
              & ! [Y5: list_nat] :
                  ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y5 ) @ A2 )
                 => ~ ( member_list_nat @ Y5 @ Q8 ) ) ) ) ) ).

% SN_imp_minimal
thf(fact_1217_SN__imp__minimal,axiom,
    ! [A2: set_Product_prod_b_b] :
      ( ( abstract_SN_on_b @ A2 @ top_top_set_b )
     => ! [Q8: set_b] :
          ( ? [X4: b] : ( member_b @ X4 @ Q8 )
         => ? [X: b] :
              ( ( member_b @ X @ Q8 )
              & ! [Y5: b] :
                  ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y5 ) @ A2 )
                 => ~ ( member_b @ Y5 @ Q8 ) ) ) ) ) ).

% SN_imp_minimal
thf(fact_1218_SN__imp__minimal,axiom,
    ! [A2: set_Pr2972776593051762503rm_a_b] :
      ( ( abstra2398554102055911763rm_a_b @ A2 @ top_to1314267278846557113rm_a_b )
     => ! [Q8: set_Pr4386577575007340137rm_a_b] :
          ( ? [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ Q8 )
         => ? [X: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ X @ Q8 )
              & ! [Y5: produc357393685978478089rm_a_b] :
                  ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y5 ) @ A2 )
                 => ~ ( member5869715511025134514rm_a_b @ Y5 @ Q8 ) ) ) ) ) ).

% SN_imp_minimal
thf(fact_1219_SN__imp__minimal,axiom,
    ! [A2: set_Pr1811044260758604347_a_nat] :
      ( ( abstra8653715922312955827_a_nat @ A2 @ top_to3353692345378799459_a_nat )
     => ! [Q8: set_Pr4934435412358123699_a_nat] :
          ( ? [X4: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X4 @ Q8 )
         => ? [X: product_prod_a_nat] :
              ( ( member5724188588386418708_a_nat @ X @ Q8 )
              & ! [Y5: product_prod_a_nat] :
                  ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y5 ) @ A2 )
                 => ~ ( member5724188588386418708_a_nat @ Y5 @ Q8 ) ) ) ) ) ).

% SN_imp_minimal
thf(fact_1220_SN__imp__minimal,axiom,
    ! [A2: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ A2 @ top_top_set_term_a_b )
     => ! [Q8: set_term_a_b] :
          ( ? [X4: term_a_b] : ( member_term_a_b @ X4 @ Q8 )
         => ? [X: term_a_b] :
              ( ( member_term_a_b @ X @ Q8 )
              & ! [Y5: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y5 ) @ A2 )
                 => ~ ( member_term_a_b @ Y5 @ Q8 ) ) ) ) ) ).

% SN_imp_minimal
thf(fact_1221_refl__not__SN,axiom,
    ! [X2: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ X2 ) @ R5 )
     => ~ ( abstra4720023341729745482rm_a_b @ R5 @ top_top_set_term_a_b ) ) ).

% refl_not_SN
thf(fact_1222_SN__onE,axiom,
    ! [R3: set_Pr3451248702717554689st_nat,A2: set_list_nat] :
      ( ( abstra2262888740781824790st_nat @ R3 @ A2 )
     => ~ ? [F9: nat > list_nat] :
            ( ( member_list_nat @ ( F9 @ zero_zero_nat ) @ A2 )
            & ! [I2: nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( F9 @ I2 ) @ ( F9 @ ( suc @ I2 ) ) ) @ R3 ) ) ) ).

% SN_onE
thf(fact_1223_SN__onE,axiom,
    ! [R3: set_Product_prod_b_b,A2: set_b] :
      ( ( abstract_SN_on_b @ R3 @ A2 )
     => ~ ? [F9: nat > b] :
            ( ( member_b @ ( F9 @ zero_zero_nat ) @ A2 )
            & ! [I2: nat] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( F9 @ I2 ) @ ( F9 @ ( suc @ I2 ) ) ) @ R3 ) ) ) ).

% SN_onE
thf(fact_1224_SN__onE,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
      ( ( abstra2398554102055911763rm_a_b @ R3 @ A2 )
     => ~ ? [F9: nat > produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( F9 @ zero_zero_nat ) @ A2 )
            & ! [I2: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F9 @ I2 ) @ ( F9 @ ( suc @ I2 ) ) ) @ R3 ) ) ) ).

% SN_onE
thf(fact_1225_SN__onE,axiom,
    ! [R3: set_Pr1811044260758604347_a_nat,A2: set_Pr4934435412358123699_a_nat] :
      ( ( abstra8653715922312955827_a_nat @ R3 @ A2 )
     => ~ ? [F9: nat > product_prod_a_nat] :
            ( ( member5724188588386418708_a_nat @ ( F9 @ zero_zero_nat ) @ A2 )
            & ! [I2: nat] : ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( F9 @ I2 ) @ ( F9 @ ( suc @ I2 ) ) ) @ R3 ) ) ) ).

% SN_onE
thf(fact_1226_SN__onE,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R3 @ A2 )
     => ~ ? [F9: nat > term_a_b] :
            ( ( member_term_a_b @ ( F9 @ zero_zero_nat ) @ A2 )
            & ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F9 @ I2 ) @ ( F9 @ ( suc @ I2 ) ) ) @ R3 ) ) ) ).

% SN_onE
thf(fact_1227_SN__on__def,axiom,
    ( abstra2262888740781824790st_nat
    = ( ^ [R2: set_Pr3451248702717554689st_nat,A6: set_list_nat] :
          ~ ? [F7: nat > list_nat] :
              ( ( member_list_nat @ ( F7 @ zero_zero_nat ) @ A6 )
              & ! [I4: nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( F7 @ I4 ) @ ( F7 @ ( suc @ I4 ) ) ) @ R2 ) ) ) ) ).

% SN_on_def
thf(fact_1228_SN__on__def,axiom,
    ( abstract_SN_on_b
    = ( ^ [R2: set_Product_prod_b_b,A6: set_b] :
          ~ ? [F7: nat > b] :
              ( ( member_b @ ( F7 @ zero_zero_nat ) @ A6 )
              & ! [I4: nat] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( F7 @ I4 ) @ ( F7 @ ( suc @ I4 ) ) ) @ R2 ) ) ) ) ).

% SN_on_def
thf(fact_1229_SN__on__def,axiom,
    ( abstra2398554102055911763rm_a_b
    = ( ^ [R2: set_Pr2972776593051762503rm_a_b,A6: set_Pr4386577575007340137rm_a_b] :
          ~ ? [F7: nat > produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( F7 @ zero_zero_nat ) @ A6 )
              & ! [I4: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F7 @ I4 ) @ ( F7 @ ( suc @ I4 ) ) ) @ R2 ) ) ) ) ).

% SN_on_def
thf(fact_1230_SN__on__def,axiom,
    ( abstra8653715922312955827_a_nat
    = ( ^ [R2: set_Pr1811044260758604347_a_nat,A6: set_Pr4934435412358123699_a_nat] :
          ~ ? [F7: nat > product_prod_a_nat] :
              ( ( member5724188588386418708_a_nat @ ( F7 @ zero_zero_nat ) @ A6 )
              & ! [I4: nat] : ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( F7 @ I4 ) @ ( F7 @ ( suc @ I4 ) ) ) @ R2 ) ) ) ) ).

% SN_on_def
thf(fact_1231_SN__on__def,axiom,
    ( abstra4720023341729745482rm_a_b
    = ( ^ [R2: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
          ~ ? [F7: nat > term_a_b] :
              ( ( member_term_a_b @ ( F7 @ zero_zero_nat ) @ A6 )
              & ! [I4: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F7 @ I4 ) @ ( F7 @ ( suc @ I4 ) ) ) @ R2 ) ) ) ) ).

% SN_on_def
thf(fact_1232_not__SN__onE,axiom,
    ! [R3: set_Pr3451248702717554689st_nat,A2: set_list_nat] :
      ( ~ ( abstra2262888740781824790st_nat @ R3 @ A2 )
     => ~ ! [F: nat > list_nat] :
            ( ( member_list_nat @ ( F @ zero_zero_nat ) @ A2 )
           => ~ ! [I3: nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) ) @ R3 ) ) ) ).

% not_SN_onE
thf(fact_1233_not__SN__onE,axiom,
    ! [R3: set_Product_prod_b_b,A2: set_b] :
      ( ~ ( abstract_SN_on_b @ R3 @ A2 )
     => ~ ! [F: nat > b] :
            ( ( member_b @ ( F @ zero_zero_nat ) @ A2 )
           => ~ ! [I3: nat] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) ) @ R3 ) ) ) ).

% not_SN_onE
thf(fact_1234_not__SN__onE,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
      ( ~ ( abstra2398554102055911763rm_a_b @ R3 @ A2 )
     => ~ ! [F: nat > produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( F @ zero_zero_nat ) @ A2 )
           => ~ ! [I3: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) ) @ R3 ) ) ) ).

% not_SN_onE
thf(fact_1235_not__SN__onE,axiom,
    ! [R3: set_Pr1811044260758604347_a_nat,A2: set_Pr4934435412358123699_a_nat] :
      ( ~ ( abstra8653715922312955827_a_nat @ R3 @ A2 )
     => ~ ! [F: nat > product_prod_a_nat] :
            ( ( member5724188588386418708_a_nat @ ( F @ zero_zero_nat ) @ A2 )
           => ~ ! [I3: nat] : ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) ) @ R3 ) ) ) ).

% not_SN_onE
thf(fact_1236_not__SN__onE,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b] :
      ( ~ ( abstra4720023341729745482rm_a_b @ R3 @ A2 )
     => ~ ! [F: nat > term_a_b] :
            ( ( member_term_a_b @ ( F @ zero_zero_nat ) @ A2 )
           => ~ ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) ) @ R3 ) ) ) ).

% not_SN_onE
thf(fact_1237_SN__on__induct_H,axiom,
    ! [R5: set_Pr4386577575007340137rm_a_b,S: term_a_b,P2: term_a_b > $o] :
      ( ( abstra4720023341729745482rm_a_b @ R5 @ ( insert_term_a_b @ S @ bot_bot_set_term_a_b ) )
     => ( ! [T2: term_a_b] :
            ( ! [U4: term_a_b] :
                ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ U4 ) @ R5 )
               => ( P2 @ U4 ) )
           => ( P2 @ T2 ) )
       => ( P2 @ S ) ) ) ).

% SN_on_induct'
thf(fact_1238_step__reflects__SN__on,axiom,
    ! [A: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ! [B3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ R3 )
         => ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ B3 @ bot_bot_set_term_a_b ) ) )
     => ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) ) ).

% step_reflects_SN_on
thf(fact_1239_SN__on__imp__on__minimal,axiom,
    ! [R3: set_Pr3451248702717554689st_nat,X2: list_nat] :
      ( ( abstra2262888740781824790st_nat @ R3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) )
     => ! [Q8: set_list_nat] :
          ( ( member_list_nat @ X2 @ Q8 )
         => ? [X: list_nat] :
              ( ( member_list_nat @ X @ Q8 )
              & ! [Y5: list_nat] :
                  ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y5 ) @ R3 )
                 => ~ ( member_list_nat @ Y5 @ Q8 ) ) ) ) ) ).

% SN_on_imp_on_minimal
thf(fact_1240_SN__on__imp__on__minimal,axiom,
    ! [R3: set_Product_prod_b_b,X2: b] :
      ( ( abstract_SN_on_b @ R3 @ ( insert_b @ X2 @ bot_bot_set_b ) )
     => ! [Q8: set_b] :
          ( ( member_b @ X2 @ Q8 )
         => ? [X: b] :
              ( ( member_b @ X @ Q8 )
              & ! [Y5: b] :
                  ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y5 ) @ R3 )
                 => ~ ( member_b @ Y5 @ Q8 ) ) ) ) ) ).

% SN_on_imp_on_minimal
thf(fact_1241_SN__on__imp__on__minimal,axiom,
    ! [R3: set_Pr2972776593051762503rm_a_b,X2: produc357393685978478089rm_a_b] :
      ( ( abstra2398554102055911763rm_a_b @ R3 @ ( insert7009541432154983385rm_a_b @ X2 @ bot_bo197521221353338581rm_a_b ) )
     => ! [Q8: set_Pr4386577575007340137rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X2 @ Q8 )
         => ? [X: produc357393685978478089rm_a_b] :
              ( ( member5869715511025134514rm_a_b @ X @ Q8 )
              & ! [Y5: produc357393685978478089rm_a_b] :
                  ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y5 ) @ R3 )
                 => ~ ( member5869715511025134514rm_a_b @ Y5 @ Q8 ) ) ) ) ) ).

% SN_on_imp_on_minimal
thf(fact_1242_SN__on__imp__on__minimal,axiom,
    ! [R3: set_Pr1811044260758604347_a_nat,X2: product_prod_a_nat] :
      ( ( abstra8653715922312955827_a_nat @ R3 @ ( insert8054603423593749677_a_nat @ X2 @ bot_bo9049108969261143879_a_nat ) )
     => ! [Q8: set_Pr4934435412358123699_a_nat] :
          ( ( member5724188588386418708_a_nat @ X2 @ Q8 )
         => ? [X: product_prod_a_nat] :
              ( ( member5724188588386418708_a_nat @ X @ Q8 )
              & ! [Y5: product_prod_a_nat] :
                  ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y5 ) @ R3 )
                 => ~ ( member5724188588386418708_a_nat @ Y5 @ Q8 ) ) ) ) ) ).

% SN_on_imp_on_minimal
thf(fact_1243_SN__on__imp__on__minimal,axiom,
    ! [R3: set_Pr4386577575007340137rm_a_b,X2: term_a_b] :
      ( ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ X2 @ bot_bot_set_term_a_b ) )
     => ! [Q8: set_term_a_b] :
          ( ( member_term_a_b @ X2 @ Q8 )
         => ? [X: term_a_b] :
              ( ( member_term_a_b @ X @ Q8 )
              & ! [Y5: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y5 ) @ R3 )
                 => ~ ( member_term_a_b @ Y5 @ Q8 ) ) ) ) ) ).

% SN_on_imp_on_minimal
thf(fact_1244_step__preserves__SN__on,axiom,
    ! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
     => ( ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
       => ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) ) ) ).

% step_preserves_SN_on
thf(fact_1245_SN__on__induct__acc__style,axiom,
    ! [R5: set_Pr4386577575007340137rm_a_b,A: term_a_b,P2: term_a_b > $o] :
      ( ( abstra4720023341729745482rm_a_b @ R5 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
     => ( ! [X: term_a_b] :
            ( ( abstra4720023341729745482rm_a_b @ R5 @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
           => ( ! [Y5: term_a_b] :
                  ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y5 ) @ R5 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X ) ) )
       => ( P2 @ A ) ) ) ).

% SN_on_induct_acc_style
thf(fact_1246_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_1247_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_1248_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_1249_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_1250_diff__add__0,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
      = zero_zero_nat ) ).

% diff_add_0

% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y: nat] :
      ( ( if_nat @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y: nat] :
      ( ( if_nat @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_3_1_If_001t__Term__Oterm_Itf__a_Mtf__b_J_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

thf(help_If_2_1_If_001t__Term__Oterm_Itf__a_Mtf__b_J_T,axiom,
    ! [X2: term_a_b,Y: term_a_b] :
      ( ( if_term_a_b @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Term__Oterm_Itf__a_Mtf__b_J_T,axiom,
    ! [X2: term_a_b,Y: term_a_b] :
      ( ( if_term_a_b @ $true @ X2 @ Y )
      = X2 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    member_list_nat @ ( append_nat @ ( term_hole_pos_a_b @ c ) @ ( append_nat @ q @ ( term_pos_diff_nat @ q2 @ v ) ) ) @ ( terms_7168686267159881682rm_a_b @ ( fun_a_b @ c2 @ nil_term_a_b ) @ t ) ).

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