%------------------------------------------------------------------------------
% 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/0015_Terms_Positions/prob_00346_012044__13716110_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1706 ( 553 unt; 461 typ;   0 def)
%            Number of atoms       : 3557 (1821 equ;   0 cnn)
%            Maximal formula atoms :   23 (   2 avg)
%            Number of connectives : 13589 ( 576   ~;  90   |; 311   &;11013   @)
%                                         (   0 <=>;1599  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   7 avg)
%            Number of types       :  103 ( 102 usr)
%            Number of type conns  :  988 ( 988   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  360 ( 359 usr;  18 con; 0-3 aty)
%            Number of variables   : 4280 ( 114   ^;3921   !; 245   ?;4280   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:18:17.640
%------------------------------------------------------------------------------
% Could-be-implicit typings (102)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J_J,type,
    set_Pr1636216636623060423rm_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
    produc1601299519461031143rm_a_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
    set_Pr5038301440468608839rm_a_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    set_Pr1542805901266377927at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J_J,type,
    set_Pr4817715314677154882st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    set_Pr2163802558726022747_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J_Mt__List__Olist_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J_J_J,type,
    set_Pr5614382033011752617rm_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    produc7765306094179330407rm_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc6392793444374437607at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_J,type,
    produc4754350896526429996_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    produc4787317212837456354st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J,type,
    produc417292134775302395_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J_Mt__List__Olist_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
    produc5770105429786724425rm_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_Itf__b_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    produc5279506192219892694rm_a_b: $tType ).

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

thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J_J_J,type,
    option7375638066962775414st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    produc2732850333517536310rm_a_b: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J,type,
    list_P4844521053307213409_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J,type,
    set_Pr1811044260758604347_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
    set_Pr8564414093027780873rm_a_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_Mt__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
    set_Pr1190453367779242145st_nat: $tType ).

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

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    produc8196726482729697190st_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
    list_P5570844911817530746rm_a_b: $tType ).

thf(ty_n_t__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,
    produc4708774622424448987_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    produc51424535725745577rm_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_Mt__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
    produc4326814125627636033st_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J,type,
    list_P4418275421265205624_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    list_P1709179899020865016rm_a_b: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    list_P7080749244337468100st_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J,type,
    list_P3322095665850492996_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J,type,
    set_Pr677149416131461714_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    set_Pr7191425930741896914rm_a_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    set_Pr3339623239203724190st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J,type,
    set_Pr8804341697571524894_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    set_Pr7072801126362145067st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    produc5781992300264797426_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    produc3105026996248004978rm_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc6445306749111383102st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    produc9150974310851111358_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    list_P8875379029341186191rm_a_b: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    list_P5023576337803013467st_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J,type,
    list_P4901192995000098612_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
    list_P7973965478605097563rm_a_b: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    list_P2568480811734797659rm_a_b: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    list_P7940050157051400743st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_J,type,
    produc1531442364937356539_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc254973753779126261st_nat: $tType ).

thf(ty_n_t__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,
    set_Pr4386577575007340137rm_a_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    set_Pr534774883469167413st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    set_Pr7303051394255727413rm_a_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    set_Pr3451248702717554689st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Nat__Onat_M_Eo_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    set_Pr1012447495205133252st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    set_Pr5046312416420021961st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
    set_Pr4048851178543822343list_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    produc357393685978478089rm_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc3697673438841856213st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    produc7711739908350443733rm_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc1828647624359046049st_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    list_P2364656488115551307rm_a_b: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    list_P7736225833432154391st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_M_062_It__Nat__Onat_M_Eo_J_J,type,
    produc7845605095797950484_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_Eo_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc4226810134323546766st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc432399132543013523st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    set_Pr1357592988741419896rm_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
    produc9164743771328383783list_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    produc1234881154892807749rm_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc4575160907756185873st_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    list_P6011104703257516679at_nat: $tType ).

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

thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    option5551091909395471437_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    produc6551080308567045442rm_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mtf__b_J,type,
    produc6510227487322075744_a_b_b: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    list_P3592885314253461005_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
    list_P2851791750731487283_nat_a: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    list_list_term_a_b: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
    list_list_list_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    set_Pr4934435412358123699_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
    set_Pr4193341848836149977_nat_a: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    set_list_term_a_b: $tType ).

thf(ty_n_t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
    option_Sum_sum_a_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    set_Product_prod_a_a: $tType ).

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

thf(ty_n_t__Option__Ooption_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    option_term_a_b: $tType ).

thf(ty_n_t__Option__Ooption_It__List__Olist_It__Nat__Onat_J_J,type,
    option_list_nat: $tType ).

thf(ty_n_t__Subterm____and____Context__Octxt_Itf__a_Mtf__b_J,type,
    subterm_and_ctxt_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    product_prod_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
    product_prod_nat_a: $tType ).

thf(ty_n_t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    list_term_a_b: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    list_list_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    set_term_a_b: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    set_list_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    product_prod_a_a: $tType ).

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

thf(ty_n_t__Term__Oterm_Itf__a_Mtf__b_J,type,
    term_a_b: $tType ).

thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
    list_nat: $tType ).

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

thf(ty_n_t__List__Olist_Itf__a_J,type,
    list_a: $tType ).

thf(ty_n_t__Set__Oset_Itf__a_J,type,
    set_a: $tType ).

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

thf(ty_n_tf__b,type,
    b: $tType ).

thf(ty_n_tf__a,type,
    a: $tType ).

% Explicit typings (359)
thf(sy_c_Basic__Utils_Oadd__elem__list__lists_001t__Nat__Onat,type,
    basic_4874698711677410535ts_nat: nat > list_nat > list_list_nat ).

thf(sy_c_Basic__Utils_Oadd__elem__list__lists_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    basic_1593220722155286443rm_a_b: term_a_b > list_term_a_b > list_list_term_a_b ).

thf(sy_c_Basic__Utils_Olist__of__permutation__element__n_001t__Nat__Onat,type,
    basic_7079635023375748421_n_nat: nat > nat > list_nat > list_list_nat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
    plus_plus_nat: nat > nat > nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_HOL_Oundefined_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    undefined_term_a_b: term_a_b ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
    sup_sup_nat: nat > nat > nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    sup_su459911885395995103_a_nat: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat ).

thf(sy_c_List_Oappend_001t__List__Olist_It__Nat__Onat_J,type,
    append_list_nat: list_list_nat > list_list_nat > list_list_nat ).

thf(sy_c_List_Oappend_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    append_list_term_a_b: list_list_term_a_b > list_list_term_a_b > list_list_term_a_b ).

thf(sy_c_List_Oappend_001t__Nat__Onat,type,
    append_nat: list_nat > list_nat > list_nat ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J,type,
    append104611586619867308st_nat: list_P7736225833432154391st_nat > list_P7736225833432154391st_nat > list_P7736225833432154391st_nat ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    append7461743152246115312rm_a_b: list_P7973965478605097563rm_a_b > list_P7973965478605097563rm_a_b > list_P7973965478605097563rm_a_b ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    append985823374593552924at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    append6140445811708759055rm_a_b: list_P5570844911817530746rm_a_b > list_P5570844911817530746rm_a_b > list_P5570844911817530746rm_a_b ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    append7569360072597011785_a_nat: list_P4901192995000098612_a_nat > list_P4901192995000098612_a_nat > list_P4901192995000098612_a_nat ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    append5987703870611264992rm_a_b: list_P2364656488115551307rm_a_b > list_P2364656488115551307rm_a_b > list_P2364656488115551307rm_a_b ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
    append1694031006427026248_nat_a: list_P2851791750731487283_nat_a > list_P2851791750731487283_nat_a > list_P2851791750731487283_nat_a ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    append1152621114427393060rm_a_b: list_P8875379029341186191rm_a_b > list_P8875379029341186191rm_a_b > list_P8875379029341186191rm_a_b ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    append7679239579558125090_a_nat: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat ).

thf(sy_c_List_Oappend_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    append_term_a_b: list_term_a_b > list_term_a_b > list_term_a_b ).

thf(sy_c_List_Oappend_001tf__a,type,
    append_a: list_a > list_a > list_a ).

thf(sy_c_List_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
    bind_nat_nat: list_nat > ( nat > list_nat ) > list_nat ).

thf(sy_c_List_Obind_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    bind_P8747971513157681562st_nat: list_P6011104703257516679at_nat > ( product_prod_nat_nat > list_list_nat ) > list_list_nat ).

thf(sy_c_List_Obind_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    bind_P7742074774332787594at_nat: list_P6011104703257516679at_nat > ( product_prod_nat_nat > list_nat ) > list_nat ).

thf(sy_c_List_Obind_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    bind_P7929151312109640459rm_a_b: list_P6011104703257516679at_nat > ( product_prod_nat_nat > list_P2364656488115551307rm_a_b ) > list_P2364656488115551307rm_a_b ).

thf(sy_c_List_Obind_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    bind_P5100334560823729207_a_nat: list_P6011104703257516679at_nat > ( product_prod_nat_nat > list_P3592885314253461005_a_nat ) > list_P3592885314253461005_a_nat ).

thf(sy_c_List_Obind_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    bind_P1981734077250826446rm_a_b: list_P6011104703257516679at_nat > ( product_prod_nat_nat > list_term_a_b ) > list_term_a_b ).

thf(sy_c_List_Obind_001tf__a_001t__List__Olist_It__Nat__Onat_J,type,
    bind_a_list_nat: list_a > ( a > list_list_nat ) > list_list_nat ).

thf(sy_c_List_Obind_001tf__a_001t__Nat__Onat,type,
    bind_a_nat: list_a > ( a > list_nat ) > list_nat ).

thf(sy_c_List_Oconcat_001t__List__Olist_It__Nat__Onat_J,type,
    concat_list_nat: list_list_list_nat > list_list_nat ).

thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
    concat_nat: list_list_nat > list_nat ).

thf(sy_c_List_Oconcat_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    concat_term_a_b: list_list_term_a_b > list_term_a_b ).

thf(sy_c_List_Ocoset_001t__List__Olist_It__Nat__Onat_J,type,
    coset_list_nat: list_list_nat > set_list_nat ).

thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
    coset_nat: list_nat > set_nat ).

thf(sy_c_List_Ocoset_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    coset_6037984010582723146_a_nat: list_P3592885314253461005_a_nat > set_Pr4934435412358123699_a_nat ).

thf(sy_c_List_Odrop_001t__List__Olist_It__Nat__Onat_J,type,
    drop_list_nat: nat > list_list_nat > list_list_nat ).

thf(sy_c_List_Odrop_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    drop_list_term_a_b: nat > list_list_term_a_b > list_list_term_a_b ).

thf(sy_c_List_Odrop_001t__Nat__Onat,type,
    drop_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Odrop_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    drop_P2883665741211355575_a_nat: nat > list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat ).

thf(sy_c_List_Odrop_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    drop_term_a_b: nat > list_term_a_b > list_term_a_b ).

thf(sy_c_List_Oenumerate_001t__List__Olist_It__Nat__Onat_J,type,
    enumerate_list_nat: nat > list_list_nat > list_P7736225833432154391st_nat ).

thf(sy_c_List_Oenumerate_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    enumer1424976507981524095rm_a_b: nat > list_list_term_a_b > list_P7973965478605097563rm_a_b ).

thf(sy_c_List_Oenumerate_001t__Nat__Onat,type,
    enumerate_nat: nat > list_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Oenumerate_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    enumer1459956249525160746rm_a_b: nat > list_P2364656488115551307rm_a_b > list_P5570844911817530746rm_a_b ).

thf(sy_c_List_Oenumerate_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    enumer3239626523597437528_a_nat: nat > list_P3592885314253461005_a_nat > list_P4901192995000098612_a_nat ).

thf(sy_c_List_Oenumerate_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    enumerate_term_a_b: nat > list_term_a_b > list_P2364656488115551307rm_a_b ).

thf(sy_c_List_Oenumerate_001tf__a,type,
    enumerate_a: nat > list_a > list_P2851791750731487283_nat_a ).

thf(sy_c_List_Oextract_001t__Nat__Onat,type,
    extract_nat: ( nat > $o ) > list_nat > option7375638066962775414st_nat ).

thf(sy_c_List_Ofind_001t__List__Olist_It__Nat__Onat_J,type,
    find_list_nat: ( list_nat > $o ) > list_list_nat > option_list_nat ).

thf(sy_c_List_Ofind_001t__Nat__Onat,type,
    find_nat: ( nat > $o ) > list_nat > option_nat ).

thf(sy_c_List_Ofind_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    find_P6125149089620491617_a_nat: ( product_prod_a_nat > $o ) > list_P3592885314253461005_a_nat > option5551091909395471437_a_nat ).

thf(sy_c_List_Ofind_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    find_term_a_b: ( term_a_b > $o ) > list_term_a_b > option_term_a_b ).

thf(sy_c_List_Ogen__length_001t__List__Olist_It__Nat__Onat_J,type,
    gen_length_list_nat: nat > list_list_nat > nat ).

thf(sy_c_List_Ogen__length_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    gen_le4942244902447077073rm_a_b: nat > list_list_term_a_b > nat ).

thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
    gen_length_nat: nat > list_nat > nat ).

thf(sy_c_List_Ogen__length_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    gen_length_term_a_b: nat > list_term_a_b > nat ).

thf(sy_c_List_Ogen__length_001tf__a,type,
    gen_length_a: nat > list_a > nat ).

thf(sy_c_List_Olenlex_001t__List__Olist_It__Nat__Onat_J,type,
    lenlex_list_nat: set_Pr3451248702717554689st_nat > set_Pr1190453367779242145st_nat ).

thf(sy_c_List_Olenlex_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    lenlex_list_term_a_b: set_Pr8564414093027780873rm_a_b > set_Pr5614382033011752617rm_a_b ).

thf(sy_c_List_Olenlex_001t__Nat__Onat,type,
    lenlex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_List_Olenlex_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    lenlex3166702271860852816rm_a_b: set_Pr5038301440468608839rm_a_b > set_Pr1636216636623060423rm_a_b ).

thf(sy_c_List_Olenlex_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    lenlex6533860390983708594_a_nat: set_Pr1811044260758604347_a_nat > set_Pr2163802558726022747_a_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_Olenlex_001tf__a,type,
    lenlex_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).

thf(sy_c_List_Olex_001t__List__Olist_It__Nat__Onat_J,type,
    lex_list_nat: set_Pr3451248702717554689st_nat > set_Pr1190453367779242145st_nat ).

thf(sy_c_List_Olex_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    lex_list_term_a_b: set_Pr8564414093027780873rm_a_b > set_Pr5614382033011752617rm_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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    lex_Pr8571645452597969515at_nat: set_Pr8693737435421807431at_nat > set_Pr1542805901266377927at_nat ).

thf(sy_c_List_Olex_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    lex_Pr1817077058774561455rm_a_b: set_Pr5038301440468608839rm_a_b > set_Pr1636216636623060423rm_a_b ).

thf(sy_c_List_Olex_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    lex_Pr1126020055417095059_a_nat: set_Pr1811044260758604347_a_nat > set_Pr2163802558726022747_a_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_Olex_001tf__a,type,
    lex_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).

thf(sy_c_List_Olexn_001t__List__Olist_It__Nat__Onat_J,type,
    lexn_list_nat: set_Pr3451248702717554689st_nat > nat > set_Pr1190453367779242145st_nat ).

thf(sy_c_List_Olexn_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    lexn_list_term_a_b: set_Pr8564414093027780873rm_a_b > nat > set_Pr5614382033011752617rm_a_b ).

thf(sy_c_List_Olexn_001t__Nat__Onat,type,
    lexn_nat: set_Pr1261947904930325089at_nat > nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_List_Olexn_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    lexn_term_a_b: set_Pr4386577575007340137rm_a_b > nat > set_Pr8564414093027780873rm_a_b ).

thf(sy_c_List_Olexn_001tf__a,type,
    lexn_a: set_Product_prod_a_a > nat > set_Pr4048851178543822343list_a ).

thf(sy_c_List_Olexord_001t__List__Olist_It__Nat__Onat_J,type,
    lexord_list_nat: set_Pr3451248702717554689st_nat > set_Pr1190453367779242145st_nat ).

thf(sy_c_List_Olexord_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    lexord_list_term_a_b: set_Pr8564414093027780873rm_a_b > set_Pr5614382033011752617rm_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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    lexord2841853652668343668at_nat: set_Pr8693737435421807431at_nat > set_Pr1542805901266377927at_nat ).

thf(sy_c_List_Olexord_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    lexord7943539855291252024rm_a_b: set_Pr5038301440468608839rm_a_b > set_Pr1636216636623060423rm_a_b ).

thf(sy_c_List_Olexord_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    lexord2902578037316800714_a_nat: set_Pr1811044260758604347_a_nat > set_Pr2163802558726022747_a_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_Olexord_001tf__a,type,
    lexord_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).

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__Nat__Onat_Mt__Nat__Onat_J,type,
    cons_P6512896166579812791at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    cons_P2836617062085252091rm_a_b: produc1234881154892807749rm_a_b > list_P2364656488115551307rm_a_b > list_P2364656488115551307rm_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_ONil_001t__List__Olist_It__Nat__Onat_J,type,
    nil_list_nat: list_list_nat ).

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__Nat__Onat_Mt__Nat__Onat_J,type,
    nil_Pr5478986624290739719at_nat: list_P6011104703257516679at_nat ).

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__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_Ohd_001t__List__Olist_It__Nat__Onat_J,type,
    hd_list_nat: list_list_nat > list_nat ).

thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
    hd_nat: list_nat > nat ).

thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    hd_Pro3460610213475200108at_nat: list_P6011104703257516679at_nat > product_prod_nat_nat ).

thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    hd_Pro8935205257713178578_a_nat: list_P3592885314253461005_a_nat > product_prod_a_nat ).

thf(sy_c_List_Olist_Ohd_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    hd_term_a_b: list_term_a_b > term_a_b ).

thf(sy_c_List_Olist_Ohd_001tf__a,type,
    hd_a: list_a > a ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    map_li960784813134754710st_nat: ( list_nat > list_list_nat ) > list_list_nat > list_list_list_nat ).

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_Oset_001t__List__Olist_It__Nat__Onat_J,type,
    set_list_nat2: list_list_nat > set_list_nat ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    set_list_term_a_b2: list_list_term_a_b > set_list_term_a_b ).

thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
    set_nat2: list_nat > set_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    set_Pr3842133991353686454st_nat: list_P7940050157051400743st_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    set_Pr8402832695120333139_a_nat: list_P3322095665850492996_a_nat > set_Pr8804341697571524894_a_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    set_Pr501854238490308330rm_a_b: list_P2568480811734797659rm_a_b > set_Pr7303051394255727413rm_a_b ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    set_Pr5697165133380604883st_nat: list_P7080749244337468100st_nat > set_Pr3339623239203724190st_nat ).

thf(sy_c_List_Olist_Oset_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,
    set_Pr5434493847166161136_a_nat: list_P4844521053307213409_a_nat > set_Pr1811044260758604347_a_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    set_Pr2356885380517226759rm_a_b: list_P1709179899020865016rm_a_b > set_Pr7191425930741896914rm_a_b ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    set_Pr5711159805836496618st_nat: list_P5023576337803013467st_nat > set_Pr534774883469167413st_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    set_Pr5033850684534019207_a_nat: list_P4418275421265205624_a_nat > set_Pr677149416131461714_a_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    set_Pr2370880052973118494rm_a_b: list_P8875379029341186191rm_a_b > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    set_Pr924983374503034536_a_nat: list_P3592885314253461005_a_nat > set_Pr4934435412358123699_a_nat ).

thf(sy_c_List_Olist_Oset_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    set_term_a_b2: list_term_a_b > set_term_a_b ).

thf(sy_c_List_Olist_Oset_001tf__a,type,
    set_a2: list_a > set_a ).

thf(sy_c_List_Olist_Osize__list_001t__List__Olist_It__Nat__Onat_J,type,
    size_list_list_nat: ( list_nat > nat ) > list_list_nat > nat ).

thf(sy_c_List_Olist_Osize__list_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    size_l7588118768477829276rm_a_b: ( list_term_a_b > nat ) > list_list_term_a_b > nat ).

thf(sy_c_List_Olist_Osize__list_001t__Nat__Onat,type,
    size_list_nat: ( nat > nat ) > list_nat > nat ).

thf(sy_c_List_Olist_Osize__list_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    size_l179396747238966901_a_nat: ( product_prod_a_nat > nat ) > list_P3592885314253461005_a_nat > nat ).

thf(sy_c_List_Olist_Osize__list_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    size_list_term_a_b: ( term_a_b > nat ) > list_term_a_b > nat ).

thf(sy_c_List_Olist__update_001t__List__Olist_It__Nat__Onat_J,type,
    list_update_list_nat: list_list_nat > nat > list_nat > list_list_nat ).

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__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    list_u4318310736832020111_a_nat: list_P3592885314253461005_a_nat > nat > product_prod_a_nat > list_P3592885314253461005_a_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_Olist__update_001tf__a,type,
    list_update_a: list_a > nat > a > list_a ).

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_001t__Nat__Onat_001t__Nat__Onat,type,
    listrel_nat_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_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_Omaps_001t__Nat__Onat_001t__Nat__Onat,type,
    maps_nat_nat: ( nat > list_nat ) > list_nat > list_nat ).

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_001t__List__Olist_It__Nat__Onat_J,type,
    nth_list_nat: list_list_nat > nat > list_nat ).

thf(sy_c_List_Onth_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    nth_list_term_a_b: list_list_term_a_b > nat > list_term_a_b ).

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    nth_Pr5300001119514325452rm_a_b: list_P2364656488115551307rm_a_b > nat > produc1234881154892807749rm_a_b ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    nth_Pr1098598711982373392rm_a_b: list_P8875379029341186191rm_a_b > nat > produc357393685978478089rm_a_b ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    nth_Pr8461465654520414006_a_nat: list_P3592885314253461005_a_nat > nat > product_prod_a_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_List_Oshuffles__rel_001t__Nat__Onat,type,
    shuffles_rel_nat: produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o ).

thf(sy_c_List_Osplice_001t__Nat__Onat,type,
    splice_nat: list_nat > list_nat > list_nat ).

thf(sy_c_List_Osplice_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    splice_term_a_b: list_term_a_b > list_term_a_b > list_term_a_b ).

thf(sy_c_List_Osplice__rel_001t__Nat__Onat,type,
    splice_rel_nat: produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o ).

thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
    subseqs_nat: list_nat > list_list_nat ).

thf(sy_c_List_Osubseqs_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    subseqs_term_a_b: list_term_a_b > list_list_term_a_b ).

thf(sy_c_List_Otake_001t__List__Olist_It__Nat__Onat_J,type,
    take_list_nat: nat > list_list_nat > list_list_nat ).

thf(sy_c_List_Otake_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    take_list_term_a_b: nat > list_list_term_a_b > list_list_term_a_b ).

thf(sy_c_List_Otake_001t__Nat__Onat,type,
    take_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Otake_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    take_P7249722199902868751_a_nat: nat > list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat ).

thf(sy_c_List_Otake_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    take_term_a_b: nat > list_term_a_b > list_term_a_b ).

thf(sy_c_List_Ozip_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    zip_li7157463729305086713st_nat: list_list_nat > list_list_nat > list_P7940050157051400743st_nat ).

thf(sy_c_List_Ozip_001t__List__Olist_It__Nat__Onat_J_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    zip_li3266749930535455254_a_nat: list_list_nat > list_P3592885314253461005_a_nat > list_P3322095665850492996_a_nat ).

thf(sy_c_List_Ozip_001t__List__Olist_It__Nat__Onat_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    zip_li391226293398231597rm_a_b: list_list_nat > list_term_a_b > list_P2568480811734797659rm_a_b ).

thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
    zip_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    zip_Pr8052531261761853462st_nat: list_P3592885314253461005_a_nat > list_list_nat > list_P7080749244337468100st_nat ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    zip_Pr1824874466442446003_a_nat: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > list_P4844521053307213409_a_nat ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    zip_Pr1286293825854998346rm_a_b: list_P3592885314253461005_a_nat > list_term_a_b > list_P1709179899020865016rm_a_b ).

thf(sy_c_List_Ozip_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__List__Olist_It__Nat__Onat_J,type,
    zip_te9026489543787896877st_nat: list_term_a_b > list_list_nat > list_P5023576337803013467st_nat ).

thf(sy_c_List_Ozip_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    zip_te9121139956803917130_a_nat: list_term_a_b > list_P3592885314253461005_a_nat > list_P4418275421265205624_a_nat ).

thf(sy_c_List_Ozip_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    zip_te2260252107881041761rm_a_b: list_term_a_b > list_term_a_b > list_P8875379029341186191rm_a_b ).

thf(sy_c_List_Ozip_001tf__a_001t__Nat__Onat,type,
    zip_a_nat: list_a > list_nat > list_P3592885314253461005_a_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_Olist__inter_001t__List__Olist_It__Nat__Onat_J,type,
    missin6532874241183986279st_nat: list_list_nat > list_list_nat > list_list_nat ).

thf(sy_c_Missing__List_Olist__inter_001t__Nat__Onat,type,
    missin6377695591745783511er_nat: list_nat > list_nat > list_nat ).

thf(sy_c_Missing__List_Olist__inter_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    missin2021864273562616452_a_nat: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat ).

thf(sy_c_Missing__List_Olist__inter__rel_001t__List__Olist_It__Nat__Onat_J,type,
    missin517122817877786516st_nat: produc4326814125627636033st_nat > produc4326814125627636033st_nat > $o ).

thf(sy_c_Missing__List_Olist__inter__rel_001t__Nat__Onat,type,
    missin8053613324461657732el_nat: produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o ).

thf(sy_c_Missing__List_Olist__inter__rel_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    missin2597475645481178417_a_nat: produc417292134775302395_a_nat > produc417292134775302395_a_nat > $o ).

thf(sy_c_Missing__List_Omem__idx_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    missin3271338439613487829rm_a_b: term_a_b > list_term_a_b > option_nat ).

thf(sy_c_Missing__List_Omin__list_001t__Nat__Onat,type,
    missing_min_list_nat: list_nat > nat ).

thf(sy_c_Missing__List_Opermut_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    missin5417461770552926780rm_a_b: list_term_a_b > ( nat > nat ) > list_term_a_b ).

thf(sy_c_Missing__List_Opermut__aux_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    missin5661735679256376914rm_a_b: list_term_a_b > ( nat > nat ) > list_term_a_b > list_term_a_b ).

thf(sy_c_Missing__List_Oremove__nth_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    missin7590947408886171483rm_a_b: nat > list_term_a_b > list_term_a_b ).

thf(sy_c_Missing__List_Ounion__list__sorted_001t__Nat__Onat,type,
    missin8019018944680490243ed_nat: list_nat > list_nat > list_nat ).

thf(sy_c_Missing__List_Ounion__list__sorted__rel_001t__Nat__Onat,type,
    missin7611735270441097048el_nat: produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o ).

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    size_s3023201423986296836st_nat: list_list_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    size_s877380706853472072rm_a_b: list_list_term_a_b > 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__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    size_s9035287501014481795st_nat: list_P7736225833432154391st_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
    size_s7698706559401467591rm_a_b: list_P7973965478605097563rm_a_b > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    size_s3663718155697878711rm_a_b: list_P2364656488115551307rm_a_b > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
    size_s243904063682394823_nat_a: list_P2851791750731487283_nat_a > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    size_s984997627204368545_a_nat: list_P3592885314253461005_a_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_Nat_Osize__class_Osize_001t__Nat__Onat,type,
    size_size_nat: nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    size_s1497571686738832145_a_nat: product_prod_a_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    size_size_term_a_b: term_a_b > nat ).

thf(sy_c_Option_Ooption_ONone_001t__List__Olist_It__Nat__Onat_J,type,
    none_list_nat: option_list_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    none_P7817216376769613445st_nat: option7375638066962775414st_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    none_P4102489432868120540_a_nat: option5551091909395471437_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_Option_Ooption_OSome_001t__Nat__Onat,type,
    some_nat: nat > option_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    some_P617598486967069697st_nat: produc8196726482729697190st_nat > option7375638066962775414st_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    some_P6251353102471802712_a_nat: product_prod_a_nat > option5551091909395471437_a_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    some_term_a_b: term_a_b > option_term_a_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__Nat__Onat_J,type,
    ord_less_eq_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_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_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_Eo_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc8587622027977423880st_nat: ( nat > $o ) > list_nat > produc4226810134323546766st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    produc7129799990162260089st_nat: list_list_nat > list_list_nat > produc4326814125627636033st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J_001t__List__Olist_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    produc7557079580777928833rm_a_b: list_list_term_a_b > list_list_term_a_b > produc5770105429786724425rm_a_b ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    produc2887701473640236333_nat_o: list_nat > ( nat > nat > $o ) > produc1531442364937356539_nat_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001_062_It__Nat__Onat_M_Eo_J,type,
    produc8069997065985037382_nat_o: list_nat > ( nat > $o ) > produc7845605095797950484_nat_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc7518127839388293336st_nat: list_nat > produc4575160907756185873st_nat > produc8196726482729697190st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    produc6151120682086626038_a_nat: list_nat > product_prod_a_nat > produc9150974310851111358_a_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    produc5151171985953862413rm_a_b: list_nat > term_a_b > produc7711739908350443733rm_a_b ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc5943733680697469783at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > produc6392793444374437607at_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    produc8811016564285052503rm_a_b: list_P2364656488115551307rm_a_b > list_P2364656488115551307rm_a_b > produc1601299519461031143rm_a_b ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    produc5384655689722402227_a_nat: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > produc417292134775302395_a_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    produc4885699992713594593rm_a_b: list_term_a_b > list_term_a_b > produc51424535725745577rm_a_b ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc4792949784200893581st_nat: list_a > list_nat > produc432399132543013523st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    produc6837034575241423639list_a: list_a > list_a > produc9164743771328383783list_a ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    produc8282810413953273033st_nat: nat > list_nat > produc4575160907756185873st_nat ).

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

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    produc1516572978046417917rm_a_b: nat > term_a_b > produc1234881154892807749rm_a_b ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001tf__a,type,
    product_Pair_nat_a: nat > a > product_prod_nat_a ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    produc3563911201077956380_nat_o: produc1828647624359046049st_nat > ( nat > nat > $o ) > produc4754350896526429996_nat_o ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    produc3440225595649897687rm_a_b: produc1234881154892807749rm_a_b > produc1234881154892807749rm_a_b > produc7765306094179330407rm_a_b ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc1713529976458248438st_nat: product_prod_a_nat > list_nat > produc6445306749111383102st_nat ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    produc2026711137822539155_a_nat: product_prod_a_nat > product_prod_a_nat > produc4708774622424448987_a_nat ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    produc4170664577406169130rm_a_b: product_prod_a_nat > term_a_b > produc3105026996248004978rm_a_b ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_Itf__b_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    produc8030969961714872974rm_a_b: set_Pr4934435412358123699_a_nat > produc6551080308567045442rm_a_b > produc5279506192219892694rm_a_b ).

thf(sy_c_Product__Type_OPair_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc4563063199488751885st_nat: term_a_b > list_nat > produc3697673438841856213st_nat ).

thf(sy_c_Product__Type_OPair_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    produc3812856575676843240rm_a_b: term_a_b > produc7711739908350443733rm_a_b > produc2732850333517536310rm_a_b ).

thf(sy_c_Product__Type_OPair_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    produc2782138671500312106_a_nat: term_a_b > product_prod_a_nat > produc5781992300264797426_a_nat ).

thf(sy_c_Product__Type_OPair_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    produc7020197800436672577rm_a_b: term_a_b > term_a_b > produc357393685978478089rm_a_b ).

thf(sy_c_Product__Type_OPair_001t__Term__Oterm_Itf__a_Mtf__b_J_001tf__b,type,
    produc1666658391573147538_a_b_b: term_a_b > b > produc6510227487322075744_a_b_b ).

thf(sy_c_Product__Type_OPair_001tf__a_001t__Nat__Onat,type,
    product_Pair_a_nat: a > nat > product_prod_a_nat ).

thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
    product_Pair_a_a: a > a > product_prod_a_a ).

thf(sy_c_Product__Type_OPair_001tf__b_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    produc1437816968797971900rm_a_b: b > term_a_b > produc6551080308567045442rm_a_b ).

thf(sy_c_Product__Type_Ocurry_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__List__Olist_It__Nat__Onat_J_001_Eo,type,
    produc1492399064212916507_nat_o: ( produc254973753779126261st_nat > $o ) > ( nat > nat > $o ) > list_nat > $o ).

thf(sy_c_Product__Type_Ocurry_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_001_Eo,type,
    produc2023121225571165330_nat_o: ( produc4787317212837456354st_nat > $o ) > ( nat > nat > $o ) > produc1828647624359046049st_nat > $o ).

thf(sy_c_Product__Type_Ocurry_001_062_It__Nat__Onat_M_Eo_J_001t__List__Olist_It__Nat__Onat_J_001_Eo,type,
    produc3615150704383769922_nat_o: ( produc4226810134323546766st_nat > $o ) > ( nat > $o ) > list_nat > $o ).

thf(sy_c_Product__Type_Ocurry_001t__Nat__Onat_001tf__a_001_Eo,type,
    produc4412725994305482677at_a_o: ( product_prod_nat_a > $o ) > nat > a > $o ).

thf(sy_c_Product__Type_Ocurry_001tf__a_001t__Nat__Onat_001_Eo,type,
    produc5346504556365831315_nat_o: ( product_prod_a_nat > $o ) > a > nat > $o ).

thf(sy_c_Product__Type_Ocurry_001tf__b_001t__Term__Oterm_Itf__a_Mtf__b_J_001_Eo,type,
    produc863636840395689358_a_b_o: ( produc6551080308567045442rm_a_b > $o ) > b > term_a_b > $o ).

thf(sy_c_Product__Type_Oprod_Oswap_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc3374130635869402703st_nat: produc254973753779126261st_nat > produc1531442364937356539_nat_o ).

thf(sy_c_Product__Type_Oprod_Oswap_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,
    produc875157120124018930st_nat: produc4787317212837456354st_nat > produc4754350896526429996_nat_o ).

thf(sy_c_Product__Type_Oprod_Oswap_001_062_It__Nat__Onat_M_Eo_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc5544029460578993768st_nat: produc4226810134323546766st_nat > produc7845605095797950484_nat_o ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__List__Olist_It__Nat__Onat_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    produc1534639687815544717_nat_o: produc1531442364937356539_nat_o > produc254973753779126261st_nat ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__List__Olist_It__Nat__Onat_J_001_062_It__Nat__Onat_M_Eo_J,type,
    produc5026404498586607270_nat_o: produc7845605095797950484_nat_o > produc4226810134323546766st_nat ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__Nat__Onat_001tf__a,type,
    product_swap_nat_a: product_prod_nat_a > product_prod_a_nat ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    produc1311334868336790716_nat_o: produc4754350896526429996_nat_o > produc4787317212837456354st_nat ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__Term__Oterm_Itf__a_Mtf__b_J_001tf__b,type,
    produc1428861184864852466_a_b_b: produc6510227487322075744_a_b_b > produc6551080308567045442rm_a_b ).

thf(sy_c_Product__Type_Oprod_Oswap_001tf__a_001t__Nat__Onat,type,
    product_swap_a_nat: product_prod_a_nat > product_prod_nat_a ).

thf(sy_c_Product__Type_Oprod_Oswap_001tf__b_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    produc1200019762089676828rm_a_b: produc6551080308567045442rm_a_b > produc6510227487322075744_a_b_b ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
    collect_list_nat: ( list_nat > $o ) > set_list_nat ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    collec2649484519605854073rm_a_b: ( list_term_a_b > $o ) > set_list_term_a_b ).

thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: ( nat > $o ) > set_nat ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    collec4464134535221767506_a_nat: ( product_prod_a_nat > $o ) > set_Pr4934435412358123699_a_nat ).

thf(sy_c_Set_OCollect_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    collect_term_a_b: ( term_a_b > $o ) > set_term_a_b ).

thf(sy_c_Set_OCollect_001tf__a,type,
    collect_a: ( a > $o ) > set_a ).

thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
    the_elem_nat: set_nat > nat ).

thf(sy_c_Subterm__and__Context_Osupt_001tf__a_001tf__b,type,
    subterm_and_supt_a_b: set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Subterm__and__Context_Osupteq_001tf__a_001tf__b,type,
    subter523971068842742411eq_a_b: set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Subterm__and__Context_Osupteqp_001tf__a_001tf__b,type,
    subter3057829548224121575qp_a_b: term_a_b > term_a_b > $o ).

thf(sy_c_Subterm__and__Context_Osuptp_001tf__a_001tf__b,type,
    subter8709468037939584051tp_a_b: term_a_b > term_a_b > $o ).

thf(sy_c_Term_Oroot_001tf__a_001tf__b,type,
    root_a_b: term_a_b > option5551091909395471437_a_nat ).

thf(sy_c_Term_Oroot__rel_001tf__a_001tf__b,type,
    root_rel_a_b: term_a_b > term_a_b > $o ).

thf(sy_c_Term_Oterm_OFun_001tf__a_001tf__b,type,
    fun_a_b: a > list_term_a_b > term_a_b ).

thf(sy_c_Term_Oterm_OVar_001tf__b_001tf__a,type,
    var_b_a: b > term_a_b ).

thf(sy_c_Term_Oterm_Oargs_001tf__a_001tf__b,type,
    args_a_b: term_a_b > list_term_a_b ).

thf(sy_c_Term__Context_Octxt__at__pos_001tf__a_001tf__b,type,
    term_ctxt_at_pos_a_b: term_a_b > list_nat > subterm_and_ctxt_a_b ).

thf(sy_c_Term__Context_Ofun__at_001tf__a_001tf__b,type,
    term_fun_at_a_b: term_a_b > list_nat > option_Sum_sum_a_b ).

thf(sy_c_Term__Context_Ofunas__term_001tf__a_001tf__b,type,
    term_funas_term_a_b: term_a_b > set_Pr4934435412358123699_a_nat ).

thf(sy_c_Term__Context_Opos__diff_001t__List__Olist_It__Nat__Onat_J,type,
    term_p7564741194569991203st_nat: list_list_nat > list_list_nat > list_list_nat ).

thf(sy_c_Term__Context_Opos__diff_001t__Nat__Onat,type,
    term_pos_diff_nat: list_nat > list_nat > list_nat ).

thf(sy_c_Term__Context_Opos__diff_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    term_p3376976900432600702at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_Term__Context_Opos__diff_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    term_p4159419441187170626rm_a_b: list_P2364656488115551307rm_a_b > list_P2364656488115551307rm_a_b > list_P2364656488115551307rm_a_b ).

thf(sy_c_Term__Context_Opos__diff_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    term_p8115227756575868480_a_nat: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat ).

thf(sy_c_Term__Context_Opos__diff_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    term_p798503758663136087rm_a_b: list_term_a_b > list_term_a_b > list_term_a_b ).

thf(sy_c_Term__Context_Opos__diff_001tf__a,type,
    term_pos_diff_a: list_a > list_a > list_a ).

thf(sy_c_Term__Context_Oposition__less__eq_001t__List__Olist_It__Nat__Onat_J,type,
    term_p5934426891874639750st_nat: list_list_nat > list_list_nat > $o ).

thf(sy_c_Term__Context_Oposition__less__eq_001t__Nat__Onat,type,
    term_p3503116865373065078eq_nat: list_nat > list_nat > $o ).

thf(sy_c_Term__Context_Oposition__less__eq_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    term_p5452418589371346395at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o ).

thf(sy_c_Term__Context_Oposition__less__eq_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    term_p6496282196689937951rm_a_b: list_P2364656488115551307rm_a_b > list_P2364656488115551307rm_a_b > $o ).

thf(sy_c_Term__Context_Oposition__less__eq_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    term_p4408110555530040355_a_nat: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > $o ).

thf(sy_c_Term__Context_Oposition__less__eq_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    term_p8391561492822560442rm_a_b: list_term_a_b > list_term_a_b > $o ).

thf(sy_c_Term__Context_Oposition__less__eq_001tf__a,type,
    term_p1826803665625515224s_eq_a: list_a > list_a > $o ).

thf(sy_c_Term__Context_Oposition__par_001t__List__Olist_It__Nat__Onat_J,type,
    term_p4950861579910180738st_nat: list_list_nat > list_list_nat > $o ).

thf(sy_c_Term__Context_Oposition__par_001t__Nat__Onat,type,
    term_p5017330785391824242ar_nat: list_nat > list_nat > $o ).

thf(sy_c_Term__Context_Oposition__par_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    term_p8419326880412058847at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o ).

thf(sy_c_Term__Context_Oposition__par_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    term_p5648667246897444383_a_nat: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > $o ).

thf(sy_c_Term__Context_Oposition__par_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    term_p7407996180858101430rm_a_b: list_term_a_b > list_term_a_b > $o ).

thf(sy_c_Term__Context_Oposition__par_001tf__a,type,
    term_position_par_a: list_a > list_a > $o ).

thf(sy_c_Term__Context_Oposs_001tf__a_001tf__b,type,
    term_poss_a_b: term_a_b > set_list_nat ).

thf(sy_c_Term__Context_Oreplace__term__at_001tf__a_001tf__b,type,
    term_r6860082780075436317at_a_b: term_a_b > list_nat > term_a_b > term_a_b ).

thf(sy_c_Term__Context_Oreplace__term__at__rel_001tf__a_001tf__b,type,
    term_r1280879029893354718el_a_b: produc2732850333517536310rm_a_b > produc2732850333517536310rm_a_b > $o ).

thf(sy_c_Term__Context_Osubt__at_001tf__a_001tf__b,type,
    term_subt_at_a_b: term_a_b > list_nat > term_a_b ).

thf(sy_c_Term__Context_Osubt__at__rel_001tf__a_001tf__b,type,
    term_subt_at_rel_a_b: produc3697673438841856213st_nat > produc3697673438841856213st_nat > $o ).

thf(sy_c_Terms__Positions_Oall__ctxt__closed_001tf__a_001tf__b,type,
    terms_5226143800768910156ed_a_b: set_Pr4934435412358123699_a_nat > set_Pr4386577575007340137rm_a_b > $o ).

thf(sy_c_Terms__Positions_Oposs__of__term_001tf__a_001tf__b,type,
    terms_7168686267159881682rm_a_b: term_a_b > term_a_b > set_list_nat ).

thf(sy_c_Utility_Osorted__list__subset_001t__Nat__Onat,type,
    sorted3362290152684031886et_nat: list_nat > list_nat > option_nat ).

thf(sy_c_Utility_Osorted__list__subset__rel_001t__Nat__Onat,type,
    sorted7771632751644887693el_nat: produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_Mt__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
    accp_P5780766562878364362st_nat: ( produc4326814125627636033st_nat > produc4326814125627636033st_nat > $o ) > produc4326814125627636033st_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    accp_P8037286306265792042st_nat: ( produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o ) > produc1828647624359046049st_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J,type,
    accp_P1855726166071684_a_nat: ( produc417292134775302395_a_nat > produc417292134775302395_a_nat > $o ) > produc417292134775302395_a_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    accp_P682940083893826398st_nat: ( produc3697673438841856213st_nat > produc3697673438841856213st_nat > $o ) > produc3697673438841856213st_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    accp_P2729577386226225901rm_a_b: ( produc2732850333517536310rm_a_b > produc2732850333517536310rm_a_b > $o ) > produc2732850333517536310rm_a_b > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    accp_term_a_b: ( term_a_b > term_a_b > $o ) > term_a_b > $o ).

thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
    member_list_nat: list_nat > set_list_nat > $o ).

thf(sy_c_member_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    member_list_term_a_b: list_term_a_b > set_list_term_a_b > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member5350383351084882060st_nat: produc254973753779126261st_nat > set_Pr7072801126362145067st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    member6951660485171671051st_nat: produc4787317212837456354st_nat > set_Pr4817715314677154882st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_Eo_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member2422889256081674661st_nat: produc4226810134323546766st_nat > set_Pr1012447495205133252st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_Mt__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
    member8680655010358287850st_nat: produc4326814125627636033st_nat > set_Pr1190453367779242145st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J_Mt__List__Olist_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
    member8612744848686930930rm_a_b: produc5770105429786724425rm_a_b > set_Pr5614382033011752617rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    member7922532354050488039_a_nat: produc9150974310851111358_a_nat > set_Pr8804341697571524894_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    member4000689696542324350rm_a_b: produc7711739908350443733rm_a_b > set_Pr7303051394255727413rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member6693912407220327184at_nat: produc6392793444374437607at_nat > set_Pr1542805901266377927at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
    member5494928282894704656rm_a_b: produc1601299519461031143rm_a_b > set_Pr1636216636623060423rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J,type,
    member3259931553675508900_a_nat: produc417292134775302395_a_nat > set_Pr2163802558726022747_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    member4405265420456397394rm_a_b: produc51424535725745577rm_a_b > set_Pr8564414093027780873rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member4851138774834033962st_nat: produc432399132543013523st_nat > set_Pr5046312416420021961st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
    member8191768239178080336list_a: produc9164743771328383783list_a > set_Pr4048851178543822343list_a > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__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_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    member8066425057025219984rm_a_b: produc7765306094179330407rm_a_b > set_Pr5038301440468608839rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member5216864792310759783st_nat: produc6445306749111383102st_nat > set_Pr3339623239203724190st_nat > $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__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    member1876585039447381659rm_a_b: produc3105026996248004978rm_a_b > set_Pr7191425930741896914rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member9209995263888512638st_nat: produc3697673438841856213st_nat > set_Pr534774883469167413st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    member4553550343464174107_a_nat: produc5781992300264797426_a_nat > set_Pr677149416131461714_a_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_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_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    member7000133282412810073rm_a_b: produc6551080308567045442rm_a_b > set_Pr1357592988741419896rm_a_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__a,type,
    member_a: a > set_a > $o ).

thf(sy_v_f,type,
    f: a ).

thf(sy_v_n,type,
    n: nat ).

thf(sy_v_t,type,
    t: term_a_b ).

% Relevant facts (1243)
thf(fact_0_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_1_prod_Oinject,axiom,
    ! [X1: nat,X2: a,Y1: nat,Y2: a] :
      ( ( ( product_Pair_nat_a @ X1 @ X2 )
        = ( product_Pair_nat_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_2_prod_Oinject,axiom,
    ! [X1: nat > $o,X2: list_nat,Y1: nat > $o,Y2: list_nat] :
      ( ( ( produc8587622027977423880st_nat @ X1 @ X2 )
        = ( produc8587622027977423880st_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_3_prod_Oinject,axiom,
    ! [X1: nat > nat > $o,X2: produc1828647624359046049st_nat,Y1: nat > nat > $o,Y2: produc1828647624359046049st_nat] :
      ( ( ( produc3127733452865184594st_nat @ X1 @ X2 )
        = ( produc3127733452865184594st_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_4_prod_Oinject,axiom,
    ! [X1: nat > nat > $o,X2: list_nat,Y1: nat > nat > $o,Y2: list_nat] :
      ( ( ( produc4727192421694094319st_nat @ X1 @ X2 )
        = ( produc4727192421694094319st_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_5_prod_Oinject,axiom,
    ! [X1: b,X2: term_a_b,Y1: b,Y2: term_a_b] :
      ( ( ( produc1437816968797971900rm_a_b @ X1 @ X2 )
        = ( produc1437816968797971900rm_a_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_6_prod_Oinject,axiom,
    ! [X1: a,X2: nat,Y1: a,Y2: nat] :
      ( ( ( product_Pair_a_nat @ X1 @ X2 )
        = ( product_Pair_a_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_7_old_Oprod_Oinject,axiom,
    ! [A: nat,B: a,A2: nat,B2: a] :
      ( ( ( product_Pair_nat_a @ A @ B )
        = ( product_Pair_nat_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_8_old_Oprod_Oinject,axiom,
    ! [A: nat > $o,B: list_nat,A2: nat > $o,B2: list_nat] :
      ( ( ( produc8587622027977423880st_nat @ A @ B )
        = ( produc8587622027977423880st_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_9_old_Oprod_Oinject,axiom,
    ! [A: nat > nat > $o,B: produc1828647624359046049st_nat,A2: nat > nat > $o,B2: produc1828647624359046049st_nat] :
      ( ( ( produc3127733452865184594st_nat @ A @ B )
        = ( produc3127733452865184594st_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_10_old_Oprod_Oinject,axiom,
    ! [A: nat > nat > $o,B: list_nat,A2: nat > nat > $o,B2: list_nat] :
      ( ( ( produc4727192421694094319st_nat @ A @ B )
        = ( produc4727192421694094319st_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_11_old_Oprod_Oinject,axiom,
    ! [A: b,B: term_a_b,A2: b,B2: term_a_b] :
      ( ( ( produc1437816968797971900rm_a_b @ A @ B )
        = ( produc1437816968797971900rm_a_b @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_12_old_Oprod_Oinject,axiom,
    ! [A: a,B: nat,A2: a,B2: nat] :
      ( ( ( product_Pair_a_nat @ A @ B )
        = ( product_Pair_a_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_13_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_14_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs: list_list_term_a_b] :
      ( ( size_s877380706853472072rm_a_b @ Xs )
      = N ) ).

% Ex_list_of_length
thf(fact_15_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs: list_list_nat] :
      ( ( size_s3023201423986296836st_nat @ Xs )
      = N ) ).

% Ex_list_of_length
thf(fact_16_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs: list_nat] :
      ( ( size_size_list_nat @ Xs )
      = N ) ).

% Ex_list_of_length
thf(fact_17_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs: list_a] :
      ( ( size_size_list_a @ Xs )
      = N ) ).

% Ex_list_of_length
thf(fact_18_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs: list_term_a_b] :
      ( ( size_s8906293707977694520rm_a_b @ Xs )
      = N ) ).

% Ex_list_of_length
thf(fact_19_neq__if__length__neq,axiom,
    ! [Xs2: list_list_term_a_b,Ys: list_list_term_a_b] :
      ( ( ( size_s877380706853472072rm_a_b @ Xs2 )
       != ( size_s877380706853472072rm_a_b @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_20_neq__if__length__neq,axiom,
    ! [Xs2: list_list_nat,Ys: list_list_nat] :
      ( ( ( size_s3023201423986296836st_nat @ Xs2 )
       != ( size_s3023201423986296836st_nat @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_21_neq__if__length__neq,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_22_neq__if__length__neq,axiom,
    ! [Xs2: list_a,Ys: list_a] :
      ( ( ( size_size_list_a @ Xs2 )
       != ( size_size_list_a @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_23_neq__if__length__neq,axiom,
    ! [Xs2: list_term_a_b,Ys: list_term_a_b] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
       != ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_24_size__neq__size__imp__neq,axiom,
    ! [X: list_list_term_a_b,Y: list_list_term_a_b] :
      ( ( ( size_s877380706853472072rm_a_b @ X )
       != ( size_s877380706853472072rm_a_b @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_25_size__neq__size__imp__neq,axiom,
    ! [X: list_list_nat,Y: list_list_nat] :
      ( ( ( size_s3023201423986296836st_nat @ X )
       != ( size_s3023201423986296836st_nat @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_26_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( ( size_size_list_nat @ X )
       != ( size_size_list_nat @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_27_size__neq__size__imp__neq,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( ( size_size_list_a @ X )
       != ( size_size_list_a @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_28_size__neq__size__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( size_size_nat @ X )
       != ( size_size_nat @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_29_size__neq__size__imp__neq,axiom,
    ! [X: list_term_a_b,Y: list_term_a_b] :
      ( ( ( size_s8906293707977694520rm_a_b @ X )
       != ( size_s8906293707977694520rm_a_b @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_30_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_a] :
      ~ ! [A3: nat,B3: a] :
          ( Y
         != ( product_Pair_nat_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_31_old_Oprod_Oexhaust,axiom,
    ! [Y: produc4226810134323546766st_nat] :
      ~ ! [A3: nat > $o,B3: list_nat] :
          ( Y
         != ( produc8587622027977423880st_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_32_old_Oprod_Oexhaust,axiom,
    ! [Y: produc4787317212837456354st_nat] :
      ~ ! [A3: nat > nat > $o,B3: produc1828647624359046049st_nat] :
          ( Y
         != ( produc3127733452865184594st_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_33_old_Oprod_Oexhaust,axiom,
    ! [Y: produc254973753779126261st_nat] :
      ~ ! [A3: nat > nat > $o,B3: list_nat] :
          ( Y
         != ( produc4727192421694094319st_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_34_old_Oprod_Oexhaust,axiom,
    ! [Y: produc6551080308567045442rm_a_b] :
      ~ ! [A3: b,B3: term_a_b] :
          ( Y
         != ( produc1437816968797971900rm_a_b @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_35_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_a_nat] :
      ~ ! [A3: a,B3: nat] :
          ( Y
         != ( product_Pair_a_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_36_surj__pair,axiom,
    ! [P: product_prod_nat_a] :
    ? [X3: nat,Y3: a] :
      ( P
      = ( product_Pair_nat_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_37_surj__pair,axiom,
    ! [P: produc4226810134323546766st_nat] :
    ? [X3: nat > $o,Y3: list_nat] :
      ( P
      = ( produc8587622027977423880st_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_38_surj__pair,axiom,
    ! [P: produc4787317212837456354st_nat] :
    ? [X3: nat > nat > $o,Y3: produc1828647624359046049st_nat] :
      ( P
      = ( produc3127733452865184594st_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_39_surj__pair,axiom,
    ! [P: produc254973753779126261st_nat] :
    ? [X3: nat > nat > $o,Y3: list_nat] :
      ( P
      = ( produc4727192421694094319st_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_40_surj__pair,axiom,
    ! [P: produc6551080308567045442rm_a_b] :
    ? [X3: b,Y3: term_a_b] :
      ( P
      = ( produc1437816968797971900rm_a_b @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_41_surj__pair,axiom,
    ! [P: product_prod_a_nat] :
    ? [X3: a,Y3: nat] :
      ( P
      = ( product_Pair_a_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_42_prod__cases,axiom,
    ! [P2: product_prod_nat_a > $o,P: product_prod_nat_a] :
      ( ! [A3: nat,B3: a] : ( P2 @ ( product_Pair_nat_a @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_43_prod__cases,axiom,
    ! [P2: produc4226810134323546766st_nat > $o,P: produc4226810134323546766st_nat] :
      ( ! [A3: nat > $o,B3: list_nat] : ( P2 @ ( produc8587622027977423880st_nat @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_44_prod__cases,axiom,
    ! [P2: produc4787317212837456354st_nat > $o,P: produc4787317212837456354st_nat] :
      ( ! [A3: nat > nat > $o,B3: produc1828647624359046049st_nat] : ( P2 @ ( produc3127733452865184594st_nat @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_45_prod__cases,axiom,
    ! [P2: produc254973753779126261st_nat > $o,P: produc254973753779126261st_nat] :
      ( ! [A3: nat > nat > $o,B3: list_nat] : ( P2 @ ( produc4727192421694094319st_nat @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_46_prod__cases,axiom,
    ! [P2: produc6551080308567045442rm_a_b > $o,P: produc6551080308567045442rm_a_b] :
      ( ! [A3: b,B3: term_a_b] : ( P2 @ ( produc1437816968797971900rm_a_b @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_47_prod__cases,axiom,
    ! [P2: product_prod_a_nat > $o,P: product_prod_a_nat] :
      ( ! [A3: a,B3: nat] : ( P2 @ ( product_Pair_a_nat @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_48_Pair__inject,axiom,
    ! [A: nat,B: a,A2: nat,B2: a] :
      ( ( ( product_Pair_nat_a @ A @ B )
        = ( product_Pair_nat_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_49_Pair__inject,axiom,
    ! [A: nat > $o,B: list_nat,A2: nat > $o,B2: list_nat] :
      ( ( ( produc8587622027977423880st_nat @ A @ B )
        = ( produc8587622027977423880st_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_50_Pair__inject,axiom,
    ! [A: nat > nat > $o,B: produc1828647624359046049st_nat,A2: nat > nat > $o,B2: produc1828647624359046049st_nat] :
      ( ( ( produc3127733452865184594st_nat @ A @ B )
        = ( produc3127733452865184594st_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_51_Pair__inject,axiom,
    ! [A: nat > nat > $o,B: list_nat,A2: nat > nat > $o,B2: list_nat] :
      ( ( ( produc4727192421694094319st_nat @ A @ B )
        = ( produc4727192421694094319st_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_52_Pair__inject,axiom,
    ! [A: b,B: term_a_b,A2: b,B2: term_a_b] :
      ( ( ( produc1437816968797971900rm_a_b @ A @ B )
        = ( produc1437816968797971900rm_a_b @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_53_Pair__inject,axiom,
    ! [A: a,B: nat,A2: a,B2: nat] :
      ( ( ( product_Pair_a_nat @ A @ B )
        = ( product_Pair_a_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_54_prod__induct3,axiom,
    ! [P2: produc4787317212837456354st_nat > $o,X: produc4787317212837456354st_nat] :
      ( ! [A3: nat > nat > $o,B3: list_nat,C: list_nat] : ( P2 @ ( produc3127733452865184594st_nat @ A3 @ ( produc2694037385005941721st_nat @ B3 @ C ) ) )
     => ( P2 @ X ) ) ).

% prod_induct3
thf(fact_55_prod__cases3,axiom,
    ! [Y: produc4787317212837456354st_nat] :
      ~ ! [A3: nat > nat > $o,B3: list_nat,C: list_nat] :
          ( Y
         != ( produc3127733452865184594st_nat @ A3 @ ( produc2694037385005941721st_nat @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_56_ssubst__Pair__rhs,axiom,
    ! [R: nat,S: nat,R2: set_Pr1261947904930325089at_nat,S2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_57_ssubst__Pair__rhs,axiom,
    ! [R: nat,S: a,R2: set_Pr4193341848836149977_nat_a,S2: a] :
      ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_58_ssubst__Pair__rhs,axiom,
    ! [R: nat > $o,S: list_nat,R2: set_Pr1012447495205133252st_nat,S2: list_nat] :
      ( ( member2422889256081674661st_nat @ ( produc8587622027977423880st_nat @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member2422889256081674661st_nat @ ( produc8587622027977423880st_nat @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_59_ssubst__Pair__rhs,axiom,
    ! [R: nat > nat > $o,S: produc1828647624359046049st_nat,R2: set_Pr4817715314677154882st_nat,S2: produc1828647624359046049st_nat] :
      ( ( member6951660485171671051st_nat @ ( produc3127733452865184594st_nat @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member6951660485171671051st_nat @ ( produc3127733452865184594st_nat @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_60_ssubst__Pair__rhs,axiom,
    ! [R: nat > nat > $o,S: list_nat,R2: set_Pr7072801126362145067st_nat,S2: list_nat] :
      ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_61_ssubst__Pair__rhs,axiom,
    ! [R: b,S: term_a_b,R2: set_Pr1357592988741419896rm_a_b,S2: term_a_b] :
      ( ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_62_ssubst__Pair__rhs,axiom,
    ! [R: a,S: nat,R2: set_Pr4934435412358123699_a_nat,S2: nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_63_curry__conv,axiom,
    ( produc5346504556365831315_nat_o
    = ( ^ [F: product_prod_a_nat > $o,A4: a,B4: nat] : ( F @ ( product_Pair_a_nat @ A4 @ B4 ) ) ) ) ).

% curry_conv
thf(fact_64_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_65_curryI,axiom,
    ! [F2: product_prod_nat_a > $o,A: nat,B: a] :
      ( ( F2 @ ( product_Pair_nat_a @ A @ B ) )
     => ( produc4412725994305482677at_a_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_66_curryI,axiom,
    ! [F2: produc4226810134323546766st_nat > $o,A: nat > $o,B: list_nat] :
      ( ( F2 @ ( produc8587622027977423880st_nat @ A @ B ) )
     => ( produc3615150704383769922_nat_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_67_curryI,axiom,
    ! [F2: produc4787317212837456354st_nat > $o,A: nat > nat > $o,B: produc1828647624359046049st_nat] :
      ( ( F2 @ ( produc3127733452865184594st_nat @ A @ B ) )
     => ( produc2023121225571165330_nat_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_68_curryI,axiom,
    ! [F2: produc254973753779126261st_nat > $o,A: nat > nat > $o,B: list_nat] :
      ( ( F2 @ ( produc4727192421694094319st_nat @ A @ B ) )
     => ( produc1492399064212916507_nat_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_69_curryI,axiom,
    ! [F2: produc6551080308567045442rm_a_b > $o,A: b,B: term_a_b] :
      ( ( F2 @ ( produc1437816968797971900rm_a_b @ A @ B ) )
     => ( produc863636840395689358_a_b_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_70_curryI,axiom,
    ! [F2: product_prod_a_nat > $o,A: a,B: nat] :
      ( ( F2 @ ( product_Pair_a_nat @ A @ B ) )
     => ( produc5346504556365831315_nat_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_71_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_72_swap__simp,axiom,
    ! [X: list_nat,Y: nat > $o] :
      ( ( produc5026404498586607270_nat_o @ ( produc8069997065985037382_nat_o @ X @ Y ) )
      = ( produc8587622027977423880st_nat @ Y @ X ) ) ).

% swap_simp
thf(fact_73_swap__simp,axiom,
    ! [X: produc1828647624359046049st_nat,Y: nat > nat > $o] :
      ( ( produc1311334868336790716_nat_o @ ( produc3563911201077956380_nat_o @ X @ Y ) )
      = ( produc3127733452865184594st_nat @ Y @ X ) ) ).

% swap_simp
thf(fact_74_swap__simp,axiom,
    ! [X: list_nat,Y: nat > nat > $o] :
      ( ( produc1534639687815544717_nat_o @ ( produc2887701473640236333_nat_o @ X @ Y ) )
      = ( produc4727192421694094319st_nat @ Y @ X ) ) ).

% swap_simp
thf(fact_75_swap__simp,axiom,
    ! [X: term_a_b,Y: b] :
      ( ( produc1428861184864852466_a_b_b @ ( produc1666658391573147538_a_b_b @ X @ Y ) )
      = ( produc1437816968797971900rm_a_b @ Y @ X ) ) ).

% swap_simp
thf(fact_76_swap__simp,axiom,
    ! [X: nat > $o,Y: list_nat] :
      ( ( produc5544029460578993768st_nat @ ( produc8587622027977423880st_nat @ X @ Y ) )
      = ( produc8069997065985037382_nat_o @ Y @ X ) ) ).

% swap_simp
thf(fact_77_swap__simp,axiom,
    ! [X: nat > nat > $o,Y: produc1828647624359046049st_nat] :
      ( ( produc875157120124018930st_nat @ ( produc3127733452865184594st_nat @ X @ Y ) )
      = ( produc3563911201077956380_nat_o @ Y @ X ) ) ).

% swap_simp
thf(fact_78_swap__simp,axiom,
    ! [X: nat > nat > $o,Y: list_nat] :
      ( ( produc3374130635869402703st_nat @ ( produc4727192421694094319st_nat @ X @ Y ) )
      = ( produc2887701473640236333_nat_o @ Y @ X ) ) ).

% swap_simp
thf(fact_79_swap__simp,axiom,
    ! [X: b,Y: term_a_b] :
      ( ( produc1200019762089676828rm_a_b @ ( produc1437816968797971900rm_a_b @ X @ Y ) )
      = ( produc1666658391573147538_a_b_b @ Y @ X ) ) ).

% swap_simp
thf(fact_80_swap__simp,axiom,
    ! [X: nat,Y: a] :
      ( ( product_swap_nat_a @ ( product_Pair_nat_a @ X @ Y ) )
      = ( product_Pair_a_nat @ Y @ X ) ) ).

% swap_simp
thf(fact_81_swap__simp,axiom,
    ! [X: a,Y: nat] :
      ( ( product_swap_a_nat @ ( product_Pair_a_nat @ X @ Y ) )
      = ( product_Pair_nat_a @ Y @ X ) ) ).

% swap_simp
thf(fact_82_lexn__length,axiom,
    ! [Xs2: list_list_term_a_b,Ys: list_list_term_a_b,R: set_Pr8564414093027780873rm_a_b,N: nat] :
      ( ( member8612744848686930930rm_a_b @ ( produc7557079580777928833rm_a_b @ Xs2 @ Ys ) @ ( lexn_list_term_a_b @ R @ N ) )
     => ( ( ( size_s877380706853472072rm_a_b @ Xs2 )
          = N )
        & ( ( size_s877380706853472072rm_a_b @ Ys )
          = N ) ) ) ).

% lexn_length
thf(fact_83_lexn__length,axiom,
    ! [Xs2: list_list_nat,Ys: list_list_nat,R: set_Pr3451248702717554689st_nat,N: nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs2 @ Ys ) @ ( lexn_list_nat @ R @ N ) )
     => ( ( ( size_s3023201423986296836st_nat @ Xs2 )
          = N )
        & ( ( size_s3023201423986296836st_nat @ Ys )
          = N ) ) ) ).

% lexn_length
thf(fact_84_lexn__length,axiom,
    ! [Xs2: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,N: nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( lexn_nat @ R @ N ) )
     => ( ( ( size_size_list_nat @ Xs2 )
          = N )
        & ( ( size_size_list_nat @ Ys )
          = N ) ) ) ).

% lexn_length
thf(fact_85_lexn__length,axiom,
    ! [Xs2: list_a,Ys: list_a,R: set_Product_prod_a_a,N: nat] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys ) @ ( lexn_a @ R @ N ) )
     => ( ( ( size_size_list_a @ Xs2 )
          = N )
        & ( ( size_size_list_a @ Ys )
          = N ) ) ) ).

% lexn_length
thf(fact_86_lexn__length,axiom,
    ! [Xs2: list_term_a_b,Ys: list_term_a_b,R: set_Pr4386577575007340137rm_a_b,N: nat] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs2 @ Ys ) @ ( lexn_term_a_b @ R @ N ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
          = N )
        & ( ( size_s8906293707977694520rm_a_b @ Ys )
          = N ) ) ) ).

% lexn_length
thf(fact_87_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_88_length__enumerate,axiom,
    ! [N: nat,Xs2: list_list_term_a_b] :
      ( ( size_s7698706559401467591rm_a_b @ ( enumer1424976507981524095rm_a_b @ N @ Xs2 ) )
      = ( size_s877380706853472072rm_a_b @ Xs2 ) ) ).

% length_enumerate
thf(fact_89_length__enumerate,axiom,
    ! [N: nat,Xs2: list_list_nat] :
      ( ( size_s9035287501014481795st_nat @ ( enumerate_list_nat @ N @ Xs2 ) )
      = ( size_s3023201423986296836st_nat @ Xs2 ) ) ).

% length_enumerate
thf(fact_90_length__enumerate,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs2 ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_enumerate
thf(fact_91_length__enumerate,axiom,
    ! [N: nat,Xs2: list_a] :
      ( ( size_s243904063682394823_nat_a @ ( enumerate_a @ N @ Xs2 ) )
      = ( size_size_list_a @ Xs2 ) ) ).

% length_enumerate
thf(fact_92_length__enumerate,axiom,
    ! [N: nat,Xs2: list_term_a_b] :
      ( ( size_s3663718155697878711rm_a_b @ ( enumerate_term_a_b @ N @ Xs2 ) )
      = ( size_s8906293707977694520rm_a_b @ Xs2 ) ) ).

% length_enumerate
thf(fact_93_root_Osimps_I2_J,axiom,
    ! [F2: a,Ts: list_term_a_b] :
      ( ( root_a_b @ ( fun_a_b @ F2 @ Ts ) )
      = ( some_P6251353102471802712_a_nat @ ( product_Pair_a_nat @ F2 @ ( size_s8906293707977694520rm_a_b @ Ts ) ) ) ) ).

% root.simps(2)
thf(fact_94_append_Oassoc,axiom,
    ! [A: list_P2364656488115551307rm_a_b,B: list_P2364656488115551307rm_a_b,C2: list_P2364656488115551307rm_a_b] :
      ( ( append5987703870611264992rm_a_b @ ( append5987703870611264992rm_a_b @ A @ B ) @ C2 )
      = ( append5987703870611264992rm_a_b @ A @ ( append5987703870611264992rm_a_b @ B @ C2 ) ) ) ).

% append.assoc
thf(fact_95_append_Oassoc,axiom,
    ! [A: list_P3592885314253461005_a_nat,B: list_P3592885314253461005_a_nat,C2: list_P3592885314253461005_a_nat] :
      ( ( append7679239579558125090_a_nat @ ( append7679239579558125090_a_nat @ A @ B ) @ C2 )
      = ( append7679239579558125090_a_nat @ A @ ( append7679239579558125090_a_nat @ B @ C2 ) ) ) ).

% append.assoc
thf(fact_96_append_Oassoc,axiom,
    ! [A: list_term_a_b,B: list_term_a_b,C2: list_term_a_b] :
      ( ( append_term_a_b @ ( append_term_a_b @ A @ B ) @ C2 )
      = ( append_term_a_b @ A @ ( append_term_a_b @ B @ C2 ) ) ) ).

% append.assoc
thf(fact_97_append_Oassoc,axiom,
    ! [A: list_list_nat,B: list_list_nat,C2: list_list_nat] :
      ( ( append_list_nat @ ( append_list_nat @ A @ B ) @ C2 )
      = ( append_list_nat @ A @ ( append_list_nat @ B @ C2 ) ) ) ).

% append.assoc
thf(fact_98_append_Oassoc,axiom,
    ! [A: list_nat,B: list_nat,C2: list_nat] :
      ( ( append_nat @ ( append_nat @ A @ B ) @ C2 )
      = ( append_nat @ A @ ( append_nat @ B @ C2 ) ) ) ).

% append.assoc
thf(fact_99_append__assoc,axiom,
    ! [Xs2: list_P2364656488115551307rm_a_b,Ys: list_P2364656488115551307rm_a_b,Zs: list_P2364656488115551307rm_a_b] :
      ( ( append5987703870611264992rm_a_b @ ( append5987703870611264992rm_a_b @ Xs2 @ Ys ) @ Zs )
      = ( append5987703870611264992rm_a_b @ Xs2 @ ( append5987703870611264992rm_a_b @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_100_append__assoc,axiom,
    ! [Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat] :
      ( ( append7679239579558125090_a_nat @ ( append7679239579558125090_a_nat @ Xs2 @ Ys ) @ Zs )
      = ( append7679239579558125090_a_nat @ Xs2 @ ( append7679239579558125090_a_nat @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_101_append__assoc,axiom,
    ! [Xs2: list_term_a_b,Ys: list_term_a_b,Zs: list_term_a_b] :
      ( ( append_term_a_b @ ( append_term_a_b @ Xs2 @ Ys ) @ Zs )
      = ( append_term_a_b @ Xs2 @ ( append_term_a_b @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_102_append__assoc,axiom,
    ! [Xs2: list_list_nat,Ys: list_list_nat,Zs: list_list_nat] :
      ( ( append_list_nat @ ( append_list_nat @ Xs2 @ Ys ) @ Zs )
      = ( append_list_nat @ Xs2 @ ( append_list_nat @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_103_append__assoc,axiom,
    ! [Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( append_nat @ ( append_nat @ Xs2 @ Ys ) @ Zs )
      = ( append_nat @ Xs2 @ ( append_nat @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_104_append__same__eq,axiom,
    ! [Ys: list_P2364656488115551307rm_a_b,Xs2: list_P2364656488115551307rm_a_b,Zs: list_P2364656488115551307rm_a_b] :
      ( ( ( append5987703870611264992rm_a_b @ Ys @ Xs2 )
        = ( append5987703870611264992rm_a_b @ Zs @ Xs2 ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_105_append__same__eq,axiom,
    ! [Ys: list_P3592885314253461005_a_nat,Xs2: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat] :
      ( ( ( append7679239579558125090_a_nat @ Ys @ Xs2 )
        = ( append7679239579558125090_a_nat @ Zs @ Xs2 ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_106_append__same__eq,axiom,
    ! [Ys: list_term_a_b,Xs2: list_term_a_b,Zs: list_term_a_b] :
      ( ( ( append_term_a_b @ Ys @ Xs2 )
        = ( append_term_a_b @ Zs @ Xs2 ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_107_append__same__eq,axiom,
    ! [Ys: list_list_nat,Xs2: list_list_nat,Zs: list_list_nat] :
      ( ( ( append_list_nat @ Ys @ Xs2 )
        = ( append_list_nat @ Zs @ Xs2 ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_108_append__same__eq,axiom,
    ! [Ys: list_nat,Xs2: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Ys @ Xs2 )
        = ( append_nat @ Zs @ Xs2 ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_109_same__append__eq,axiom,
    ! [Xs2: list_P2364656488115551307rm_a_b,Ys: list_P2364656488115551307rm_a_b,Zs: list_P2364656488115551307rm_a_b] :
      ( ( ( append5987703870611264992rm_a_b @ Xs2 @ Ys )
        = ( append5987703870611264992rm_a_b @ Xs2 @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_110_same__append__eq,axiom,
    ! [Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat] :
      ( ( ( append7679239579558125090_a_nat @ Xs2 @ Ys )
        = ( append7679239579558125090_a_nat @ Xs2 @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_111_same__append__eq,axiom,
    ! [Xs2: list_term_a_b,Ys: list_term_a_b,Zs: list_term_a_b] :
      ( ( ( append_term_a_b @ Xs2 @ Ys )
        = ( append_term_a_b @ Xs2 @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_112_same__append__eq,axiom,
    ! [Xs2: list_list_nat,Ys: list_list_nat,Zs: list_list_nat] :
      ( ( ( append_list_nat @ Xs2 @ Ys )
        = ( append_list_nat @ Xs2 @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_113_same__append__eq,axiom,
    ! [Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Xs2 @ Ys )
        = ( append_nat @ Xs2 @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_114_swap__swap,axiom,
    ! [P: product_prod_nat_a] :
      ( ( product_swap_a_nat @ ( product_swap_nat_a @ P ) )
      = P ) ).

% swap_swap
thf(fact_115_swap__swap,axiom,
    ! [P: product_prod_a_nat] :
      ( ( product_swap_nat_a @ ( product_swap_a_nat @ P ) )
      = P ) ).

% swap_swap
thf(fact_116_append__eq__append__conv,axiom,
    ! [Xs2: list_P2364656488115551307rm_a_b,Ys: list_P2364656488115551307rm_a_b,Us: list_P2364656488115551307rm_a_b,Vs: list_P2364656488115551307rm_a_b] :
      ( ( ( ( size_s3663718155697878711rm_a_b @ Xs2 )
          = ( size_s3663718155697878711rm_a_b @ Ys ) )
        | ( ( size_s3663718155697878711rm_a_b @ Us )
          = ( size_s3663718155697878711rm_a_b @ Vs ) ) )
     => ( ( ( append5987703870611264992rm_a_b @ Xs2 @ Us )
          = ( append5987703870611264992rm_a_b @ Ys @ Vs ) )
        = ( ( Xs2 = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_117_append__eq__append__conv,axiom,
    ! [Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,Us: list_P3592885314253461005_a_nat,Vs: list_P3592885314253461005_a_nat] :
      ( ( ( ( size_s984997627204368545_a_nat @ Xs2 )
          = ( size_s984997627204368545_a_nat @ Ys ) )
        | ( ( size_s984997627204368545_a_nat @ Us )
          = ( size_s984997627204368545_a_nat @ Vs ) ) )
     => ( ( ( append7679239579558125090_a_nat @ Xs2 @ Us )
          = ( append7679239579558125090_a_nat @ Ys @ Vs ) )
        = ( ( Xs2 = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_118_append__eq__append__conv,axiom,
    ! [Xs2: list_list_term_a_b,Ys: list_list_term_a_b,Us: list_list_term_a_b,Vs: list_list_term_a_b] :
      ( ( ( ( size_s877380706853472072rm_a_b @ Xs2 )
          = ( size_s877380706853472072rm_a_b @ Ys ) )
        | ( ( size_s877380706853472072rm_a_b @ Us )
          = ( size_s877380706853472072rm_a_b @ Vs ) ) )
     => ( ( ( append_list_term_a_b @ Xs2 @ Us )
          = ( append_list_term_a_b @ Ys @ Vs ) )
        = ( ( Xs2 = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_119_append__eq__append__conv,axiom,
    ! [Xs2: list_list_nat,Ys: list_list_nat,Us: list_list_nat,Vs: list_list_nat] :
      ( ( ( ( size_s3023201423986296836st_nat @ Xs2 )
          = ( size_s3023201423986296836st_nat @ Ys ) )
        | ( ( size_s3023201423986296836st_nat @ Us )
          = ( size_s3023201423986296836st_nat @ Vs ) ) )
     => ( ( ( append_list_nat @ Xs2 @ Us )
          = ( append_list_nat @ Ys @ Vs ) )
        = ( ( Xs2 = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_120_append__eq__append__conv,axiom,
    ! [Xs2: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
      ( ( ( ( size_size_list_nat @ Xs2 )
          = ( size_size_list_nat @ Ys ) )
        | ( ( size_size_list_nat @ Us )
          = ( size_size_list_nat @ Vs ) ) )
     => ( ( ( append_nat @ Xs2 @ Us )
          = ( append_nat @ Ys @ Vs ) )
        = ( ( Xs2 = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_121_append__eq__append__conv,axiom,
    ! [Xs2: list_a,Ys: list_a,Us: list_a,Vs: list_a] :
      ( ( ( ( size_size_list_a @ Xs2 )
          = ( size_size_list_a @ Ys ) )
        | ( ( size_size_list_a @ Us )
          = ( size_size_list_a @ Vs ) ) )
     => ( ( ( append_a @ Xs2 @ Us )
          = ( append_a @ Ys @ Vs ) )
        = ( ( Xs2 = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_122_append__eq__append__conv,axiom,
    ! [Xs2: list_term_a_b,Ys: list_term_a_b,Us: list_term_a_b,Vs: list_term_a_b] :
      ( ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
          = ( size_s8906293707977694520rm_a_b @ Ys ) )
        | ( ( size_s8906293707977694520rm_a_b @ Us )
          = ( size_s8906293707977694520rm_a_b @ Vs ) ) )
     => ( ( ( append_term_a_b @ Xs2 @ Us )
          = ( append_term_a_b @ Ys @ Vs ) )
        = ( ( Xs2 = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_123_append__eq__appendI,axiom,
    ! [Xs2: list_P2364656488115551307rm_a_b,Xs1: list_P2364656488115551307rm_a_b,Zs: list_P2364656488115551307rm_a_b,Ys: list_P2364656488115551307rm_a_b,Us: list_P2364656488115551307rm_a_b] :
      ( ( ( append5987703870611264992rm_a_b @ Xs2 @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append5987703870611264992rm_a_b @ Xs1 @ Us ) )
       => ( ( append5987703870611264992rm_a_b @ Xs2 @ Ys )
          = ( append5987703870611264992rm_a_b @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_124_append__eq__appendI,axiom,
    ! [Xs2: list_P3592885314253461005_a_nat,Xs1: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,Us: list_P3592885314253461005_a_nat] :
      ( ( ( append7679239579558125090_a_nat @ Xs2 @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append7679239579558125090_a_nat @ Xs1 @ Us ) )
       => ( ( append7679239579558125090_a_nat @ Xs2 @ Ys )
          = ( append7679239579558125090_a_nat @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_125_append__eq__appendI,axiom,
    ! [Xs2: list_term_a_b,Xs1: list_term_a_b,Zs: list_term_a_b,Ys: list_term_a_b,Us: list_term_a_b] :
      ( ( ( append_term_a_b @ Xs2 @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append_term_a_b @ Xs1 @ Us ) )
       => ( ( append_term_a_b @ Xs2 @ Ys )
          = ( append_term_a_b @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_126_append__eq__appendI,axiom,
    ! [Xs2: list_list_nat,Xs1: list_list_nat,Zs: list_list_nat,Ys: list_list_nat,Us: list_list_nat] :
      ( ( ( append_list_nat @ Xs2 @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append_list_nat @ Xs1 @ Us ) )
       => ( ( append_list_nat @ Xs2 @ Ys )
          = ( append_list_nat @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_127_append__eq__appendI,axiom,
    ! [Xs2: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us: list_nat] :
      ( ( ( append_nat @ Xs2 @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append_nat @ Xs1 @ Us ) )
       => ( ( append_nat @ Xs2 @ Ys )
          = ( append_nat @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_128_append__eq__append__conv2,axiom,
    ! [Xs2: list_P2364656488115551307rm_a_b,Ys: list_P2364656488115551307rm_a_b,Zs: list_P2364656488115551307rm_a_b,Ts: list_P2364656488115551307rm_a_b] :
      ( ( ( append5987703870611264992rm_a_b @ Xs2 @ Ys )
        = ( append5987703870611264992rm_a_b @ Zs @ Ts ) )
      = ( ? [Us2: list_P2364656488115551307rm_a_b] :
            ( ( ( Xs2
                = ( append5987703870611264992rm_a_b @ Zs @ Us2 ) )
              & ( ( append5987703870611264992rm_a_b @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append5987703870611264992rm_a_b @ Xs2 @ Us2 )
                = Zs )
              & ( Ys
                = ( append5987703870611264992rm_a_b @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_129_append__eq__append__conv2,axiom,
    ! [Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat,Ts: list_P3592885314253461005_a_nat] :
      ( ( ( append7679239579558125090_a_nat @ Xs2 @ Ys )
        = ( append7679239579558125090_a_nat @ Zs @ Ts ) )
      = ( ? [Us2: list_P3592885314253461005_a_nat] :
            ( ( ( Xs2
                = ( append7679239579558125090_a_nat @ Zs @ Us2 ) )
              & ( ( append7679239579558125090_a_nat @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append7679239579558125090_a_nat @ Xs2 @ Us2 )
                = Zs )
              & ( Ys
                = ( append7679239579558125090_a_nat @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_130_append__eq__append__conv2,axiom,
    ! [Xs2: list_term_a_b,Ys: list_term_a_b,Zs: list_term_a_b,Ts: list_term_a_b] :
      ( ( ( append_term_a_b @ Xs2 @ Ys )
        = ( append_term_a_b @ Zs @ Ts ) )
      = ( ? [Us2: list_term_a_b] :
            ( ( ( Xs2
                = ( append_term_a_b @ Zs @ Us2 ) )
              & ( ( append_term_a_b @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append_term_a_b @ Xs2 @ Us2 )
                = Zs )
              & ( Ys
                = ( append_term_a_b @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_131_append__eq__append__conv2,axiom,
    ! [Xs2: list_list_nat,Ys: list_list_nat,Zs: list_list_nat,Ts: list_list_nat] :
      ( ( ( append_list_nat @ Xs2 @ Ys )
        = ( append_list_nat @ Zs @ Ts ) )
      = ( ? [Us2: list_list_nat] :
            ( ( ( Xs2
                = ( append_list_nat @ Zs @ Us2 ) )
              & ( ( append_list_nat @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append_list_nat @ Xs2 @ Us2 )
                = Zs )
              & ( Ys
                = ( append_list_nat @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_132_append__eq__append__conv2,axiom,
    ! [Xs2: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
      ( ( ( append_nat @ Xs2 @ Ys )
        = ( append_nat @ Zs @ Ts ) )
      = ( ? [Us2: list_nat] :
            ( ( ( Xs2
                = ( append_nat @ Zs @ Us2 ) )
              & ( ( append_nat @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append_nat @ Xs2 @ Us2 )
                = Zs )
              & ( Ys
                = ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_133_curryD,axiom,
    ! [F2: product_prod_nat_a > $o,A: nat,B: a] :
      ( ( produc4412725994305482677at_a_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_nat_a @ A @ B ) ) ) ).

% curryD
thf(fact_134_curryD,axiom,
    ! [F2: produc4226810134323546766st_nat > $o,A: nat > $o,B: list_nat] :
      ( ( produc3615150704383769922_nat_o @ F2 @ A @ B )
     => ( F2 @ ( produc8587622027977423880st_nat @ A @ B ) ) ) ).

% curryD
thf(fact_135_curryD,axiom,
    ! [F2: produc4787317212837456354st_nat > $o,A: nat > nat > $o,B: produc1828647624359046049st_nat] :
      ( ( produc2023121225571165330_nat_o @ F2 @ A @ B )
     => ( F2 @ ( produc3127733452865184594st_nat @ A @ B ) ) ) ).

% curryD
thf(fact_136_curryD,axiom,
    ! [F2: produc254973753779126261st_nat > $o,A: nat > nat > $o,B: list_nat] :
      ( ( produc1492399064212916507_nat_o @ F2 @ A @ B )
     => ( F2 @ ( produc4727192421694094319st_nat @ A @ B ) ) ) ).

% curryD
thf(fact_137_curryD,axiom,
    ! [F2: produc6551080308567045442rm_a_b > $o,A: b,B: term_a_b] :
      ( ( produc863636840395689358_a_b_o @ F2 @ A @ B )
     => ( F2 @ ( produc1437816968797971900rm_a_b @ A @ B ) ) ) ).

% curryD
thf(fact_138_curryD,axiom,
    ! [F2: product_prod_a_nat > $o,A: a,B: nat] :
      ( ( produc5346504556365831315_nat_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_a_nat @ A @ B ) ) ) ).

% curryD
thf(fact_139_curryE,axiom,
    ! [F2: product_prod_nat_a > $o,A: nat,B: a] :
      ( ( produc4412725994305482677at_a_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_nat_a @ A @ B ) ) ) ).

% curryE
thf(fact_140_curryE,axiom,
    ! [F2: produc4226810134323546766st_nat > $o,A: nat > $o,B: list_nat] :
      ( ( produc3615150704383769922_nat_o @ F2 @ A @ B )
     => ( F2 @ ( produc8587622027977423880st_nat @ A @ B ) ) ) ).

% curryE
thf(fact_141_curryE,axiom,
    ! [F2: produc4787317212837456354st_nat > $o,A: nat > nat > $o,B: produc1828647624359046049st_nat] :
      ( ( produc2023121225571165330_nat_o @ F2 @ A @ B )
     => ( F2 @ ( produc3127733452865184594st_nat @ A @ B ) ) ) ).

% curryE
thf(fact_142_curryE,axiom,
    ! [F2: produc254973753779126261st_nat > $o,A: nat > nat > $o,B: list_nat] :
      ( ( produc1492399064212916507_nat_o @ F2 @ A @ B )
     => ( F2 @ ( produc4727192421694094319st_nat @ A @ B ) ) ) ).

% curryE
thf(fact_143_curryE,axiom,
    ! [F2: produc6551080308567045442rm_a_b > $o,A: b,B: term_a_b] :
      ( ( produc863636840395689358_a_b_o @ F2 @ A @ B )
     => ( F2 @ ( produc1437816968797971900rm_a_b @ A @ B ) ) ) ).

% curryE
thf(fact_144_curryE,axiom,
    ! [F2: product_prod_a_nat > $o,A: a,B: nat] :
      ( ( produc5346504556365831315_nat_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_a_nat @ A @ B ) ) ) ).

% curryE
thf(fact_145_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_146_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_147_mem__Collect__eq,axiom,
    ! [A: term_a_b,P2: term_a_b > $o] :
      ( ( member_term_a_b @ A @ ( collect_term_a_b @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_148_mem__Collect__eq,axiom,
    ! [A: list_term_a_b,P2: list_term_a_b > $o] :
      ( ( member_list_term_a_b @ A @ ( collec2649484519605854073rm_a_b @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_149_mem__Collect__eq,axiom,
    ! [A: nat,P2: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_150_mem__Collect__eq,axiom,
    ! [A: a,P2: a > $o] :
      ( ( member_a @ A @ ( collect_a @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_151_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_152_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_153_Collect__mem__eq,axiom,
    ! [A5: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_154_Collect__mem__eq,axiom,
    ! [A5: set_term_a_b] :
      ( ( collect_term_a_b
        @ ^ [X4: term_a_b] : ( member_term_a_b @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_155_Collect__mem__eq,axiom,
    ! [A5: set_list_term_a_b] :
      ( ( collec2649484519605854073rm_a_b
        @ ^ [X4: list_term_a_b] : ( member_list_term_a_b @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_156_Collect__mem__eq,axiom,
    ! [A5: set_nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : ( member_nat @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_157_Collect__mem__eq,axiom,
    ! [A5: set_a] :
      ( ( collect_a
        @ ^ [X4: a] : ( member_a @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_158_Collect__mem__eq,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat] :
      ( ( collec4464134535221767506_a_nat
        @ ^ [X4: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_159_Collect__mem__eq,axiom,
    ! [A5: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_160_Collect__cong,axiom,
    ! [P2: list_nat > $o,Q2: list_nat > $o] :
      ( ! [X3: list_nat] :
          ( ( P2 @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collect_list_nat @ P2 )
        = ( collect_list_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_161_Collect__cong,axiom,
    ! [P2: product_prod_a_nat > $o,Q2: product_prod_a_nat > $o] :
      ( ! [X3: product_prod_a_nat] :
          ( ( P2 @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collec4464134535221767506_a_nat @ P2 )
        = ( collec4464134535221767506_a_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_162_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_163_option_Oinject,axiom,
    ! [X2: product_prod_a_nat,Y2: product_prod_a_nat] :
      ( ( ( some_P6251353102471802712_a_nat @ X2 )
        = ( some_P6251353102471802712_a_nat @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_164_option_Oinject,axiom,
    ! [X2: produc8196726482729697190st_nat,Y2: produc8196726482729697190st_nat] :
      ( ( ( some_P617598486967069697st_nat @ X2 )
        = ( some_P617598486967069697st_nat @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_165_option_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( some_nat @ X2 )
        = ( some_nat @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_166_option_Oinject,axiom,
    ! [X2: term_a_b,Y2: term_a_b] :
      ( ( ( some_term_a_b @ X2 )
        = ( some_term_a_b @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_167_supteq_Orefl,axiom,
    ! [T: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ T ) @ subter523971068842742411eq_a_b ) ).

% supteq.refl
thf(fact_168_subterm_Oeq__refl,axiom,
    ! [X: term_a_b,Y: term_a_b] :
      ( ( X = Y )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b ) ) ).

% subterm.eq_refl
thf(fact_169_subterm_Oorder__refl,axiom,
    ! [X: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ subter523971068842742411eq_a_b ) ).

% subterm.order_refl
thf(fact_170_subterm_Oorder_Otrans,axiom,
    ! [B: term_a_b,A: term_a_b,C2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ B ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ A ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.order.trans
thf(fact_171_subterm_Oorder__trans,axiom,
    ! [Y: term_a_b,X: term_a_b,Z: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ Y ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ X ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.order_trans
thf(fact_172_subterm_Oantisym__conv,axiom,
    ! [X: term_a_b,Y: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b )
        = ( X = Y ) ) ) ).

% subterm.antisym_conv
thf(fact_173_subterm_Oorder_Oeq__iff,axiom,
    ( ( ^ [Y4: term_a_b,Z2: term_a_b] : ( Y4 = Z2 ) )
    = ( ^ [A4: term_a_b,B4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ A4 ) @ subter523971068842742411eq_a_b )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B4 ) @ subter523971068842742411eq_a_b ) ) ) ) ).

% subterm.order.eq_iff
thf(fact_174_subterm_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: term_a_b,Z2: term_a_b] : ( Y4 = Z2 ) )
    = ( ^ [X4: term_a_b,Y5: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X4 ) @ subter523971068842742411eq_a_b )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ subter523971068842742411eq_a_b ) ) ) ) ).

% subterm.order_eq_iff
thf(fact_175_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_176_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_177_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_178_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_179_subterm_Odual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: term_a_b,Z2: term_a_b] : ( Y4 = Z2 ) )
    = ( ^ [A4: term_a_b,B4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B4 ) @ subter523971068842742411eq_a_b )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ A4 ) @ subter523971068842742411eq_a_b ) ) ) ) ).

% subterm.dual_order.eq_iff
thf(fact_180_subterm_Odual__order_Otrans,axiom,
    ! [A: term_a_b,B: term_a_b,C2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.dual_order.trans
thf(fact_181_subterm_Oord__le__eq__trans,axiom,
    ! [B: term_a_b,A: term_a_b,C2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
     => ( ( B = C2 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ A ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.ord_le_eq_trans
thf(fact_182_subterm_Oord__eq__le__trans,axiom,
    ! [A: term_a_b,B: term_a_b,C2: term_a_b] :
      ( ( A = B )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ B ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ A ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.ord_eq_le_trans
thf(fact_183_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_184_subterm_Oorder__antisym,axiom,
    ! [Y: term_a_b,X: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ subter523971068842742411eq_a_b )
       => ( X = Y ) ) ) ).

% subterm.order_antisym
thf(fact_185_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_186_lenlex__append1,axiom,
    ! [Us: list_P2364656488115551307rm_a_b,Xs2: list_P2364656488115551307rm_a_b,R2: set_Pr5038301440468608839rm_a_b,Vs: list_P2364656488115551307rm_a_b,Ys: list_P2364656488115551307rm_a_b] :
      ( ( member5494928282894704656rm_a_b @ ( produc8811016564285052503rm_a_b @ Us @ Xs2 ) @ ( lenlex3166702271860852816rm_a_b @ R2 ) )
     => ( ( ( size_s3663718155697878711rm_a_b @ Vs )
          = ( size_s3663718155697878711rm_a_b @ Ys ) )
       => ( member5494928282894704656rm_a_b @ ( produc8811016564285052503rm_a_b @ ( append5987703870611264992rm_a_b @ Us @ Vs ) @ ( append5987703870611264992rm_a_b @ Xs2 @ Ys ) ) @ ( lenlex3166702271860852816rm_a_b @ R2 ) ) ) ) ).

% lenlex_append1
thf(fact_187_lenlex__append1,axiom,
    ! [Us: list_P3592885314253461005_a_nat,Xs2: list_P3592885314253461005_a_nat,R2: set_Pr1811044260758604347_a_nat,Vs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
      ( ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ Us @ Xs2 ) @ ( lenlex6533860390983708594_a_nat @ R2 ) )
     => ( ( ( size_s984997627204368545_a_nat @ Vs )
          = ( size_s984997627204368545_a_nat @ Ys ) )
       => ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ ( append7679239579558125090_a_nat @ Us @ Vs ) @ ( append7679239579558125090_a_nat @ Xs2 @ Ys ) ) @ ( lenlex6533860390983708594_a_nat @ R2 ) ) ) ) ).

% lenlex_append1
thf(fact_188_lenlex__append1,axiom,
    ! [Us: list_list_term_a_b,Xs2: list_list_term_a_b,R2: set_Pr8564414093027780873rm_a_b,Vs: list_list_term_a_b,Ys: list_list_term_a_b] :
      ( ( member8612744848686930930rm_a_b @ ( produc7557079580777928833rm_a_b @ Us @ Xs2 ) @ ( lenlex_list_term_a_b @ R2 ) )
     => ( ( ( size_s877380706853472072rm_a_b @ Vs )
          = ( size_s877380706853472072rm_a_b @ Ys ) )
       => ( member8612744848686930930rm_a_b @ ( produc7557079580777928833rm_a_b @ ( append_list_term_a_b @ Us @ Vs ) @ ( append_list_term_a_b @ Xs2 @ Ys ) ) @ ( lenlex_list_term_a_b @ R2 ) ) ) ) ).

% lenlex_append1
thf(fact_189_lenlex__append1,axiom,
    ! [Us: list_list_nat,Xs2: list_list_nat,R2: set_Pr3451248702717554689st_nat,Vs: list_list_nat,Ys: list_list_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Us @ Xs2 ) @ ( lenlex_list_nat @ R2 ) )
     => ( ( ( size_s3023201423986296836st_nat @ Vs )
          = ( size_s3023201423986296836st_nat @ Ys ) )
       => ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( append_list_nat @ Us @ Vs ) @ ( append_list_nat @ Xs2 @ Ys ) ) @ ( lenlex_list_nat @ R2 ) ) ) ) ).

% lenlex_append1
thf(fact_190_lenlex__append1,axiom,
    ! [Us: list_nat,Xs2: list_nat,R2: set_Pr1261947904930325089at_nat,Vs: list_nat,Ys: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us @ Xs2 ) @ ( lenlex_nat @ R2 ) )
     => ( ( ( size_size_list_nat @ Vs )
          = ( size_size_list_nat @ Ys ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Us @ Vs ) @ ( append_nat @ Xs2 @ Ys ) ) @ ( lenlex_nat @ R2 ) ) ) ) ).

% lenlex_append1
thf(fact_191_lenlex__append1,axiom,
    ! [Us: list_a,Xs2: list_a,R2: set_Product_prod_a_a,Vs: list_a,Ys: list_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Us @ Xs2 ) @ ( lenlex_a @ R2 ) )
     => ( ( ( size_size_list_a @ Vs )
          = ( size_size_list_a @ Ys ) )
       => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Us @ Vs ) @ ( append_a @ Xs2 @ Ys ) ) @ ( lenlex_a @ R2 ) ) ) ) ).

% lenlex_append1
thf(fact_192_lenlex__append1,axiom,
    ! [Us: list_term_a_b,Xs2: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b,Vs: list_term_a_b,Ys: list_term_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Us @ Xs2 ) @ ( lenlex_term_a_b @ R2 ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Vs )
          = ( size_s8906293707977694520rm_a_b @ Ys ) )
       => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ Us @ Vs ) @ ( append_term_a_b @ Xs2 @ Ys ) ) @ ( lenlex_term_a_b @ R2 ) ) ) ) ).

% lenlex_append1
thf(fact_193_lex__append__rightI,axiom,
    ! [Xs2: list_P2364656488115551307rm_a_b,Ys: list_P2364656488115551307rm_a_b,R: set_Pr5038301440468608839rm_a_b,Vs: list_P2364656488115551307rm_a_b,Us: list_P2364656488115551307rm_a_b] :
      ( ( member5494928282894704656rm_a_b @ ( produc8811016564285052503rm_a_b @ Xs2 @ Ys ) @ ( lex_Pr1817077058774561455rm_a_b @ R ) )
     => ( ( ( size_s3663718155697878711rm_a_b @ Vs )
          = ( size_s3663718155697878711rm_a_b @ Us ) )
       => ( member5494928282894704656rm_a_b @ ( produc8811016564285052503rm_a_b @ ( append5987703870611264992rm_a_b @ Xs2 @ Us ) @ ( append5987703870611264992rm_a_b @ Ys @ Vs ) ) @ ( lex_Pr1817077058774561455rm_a_b @ R ) ) ) ) ).

% lex_append_rightI
thf(fact_194_lex__append__rightI,axiom,
    ! [Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,R: set_Pr1811044260758604347_a_nat,Vs: list_P3592885314253461005_a_nat,Us: list_P3592885314253461005_a_nat] :
      ( ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ Xs2 @ Ys ) @ ( lex_Pr1126020055417095059_a_nat @ R ) )
     => ( ( ( size_s984997627204368545_a_nat @ Vs )
          = ( size_s984997627204368545_a_nat @ Us ) )
       => ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ ( append7679239579558125090_a_nat @ Xs2 @ Us ) @ ( append7679239579558125090_a_nat @ Ys @ Vs ) ) @ ( lex_Pr1126020055417095059_a_nat @ R ) ) ) ) ).

% lex_append_rightI
thf(fact_195_lex__append__rightI,axiom,
    ! [Xs2: list_list_term_a_b,Ys: list_list_term_a_b,R: set_Pr8564414093027780873rm_a_b,Vs: list_list_term_a_b,Us: list_list_term_a_b] :
      ( ( member8612744848686930930rm_a_b @ ( produc7557079580777928833rm_a_b @ Xs2 @ Ys ) @ ( lex_list_term_a_b @ R ) )
     => ( ( ( size_s877380706853472072rm_a_b @ Vs )
          = ( size_s877380706853472072rm_a_b @ Us ) )
       => ( member8612744848686930930rm_a_b @ ( produc7557079580777928833rm_a_b @ ( append_list_term_a_b @ Xs2 @ Us ) @ ( append_list_term_a_b @ Ys @ Vs ) ) @ ( lex_list_term_a_b @ R ) ) ) ) ).

% lex_append_rightI
thf(fact_196_lex__append__rightI,axiom,
    ! [Xs2: list_list_nat,Ys: list_list_nat,R: set_Pr3451248702717554689st_nat,Vs: list_list_nat,Us: list_list_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs2 @ Ys ) @ ( lex_list_nat @ R ) )
     => ( ( ( size_s3023201423986296836st_nat @ Vs )
          = ( size_s3023201423986296836st_nat @ Us ) )
       => ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( append_list_nat @ Xs2 @ Us ) @ ( append_list_nat @ Ys @ Vs ) ) @ ( lex_list_nat @ R ) ) ) ) ).

% lex_append_rightI
thf(fact_197_lex__append__rightI,axiom,
    ! [Xs2: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,Vs: list_nat,Us: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( lex_nat @ R ) )
     => ( ( ( size_size_list_nat @ Vs )
          = ( size_size_list_nat @ Us ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Us ) @ ( append_nat @ Ys @ Vs ) ) @ ( lex_nat @ R ) ) ) ) ).

% lex_append_rightI
thf(fact_198_lex__append__rightI,axiom,
    ! [Xs2: list_a,Ys: list_a,R: set_Product_prod_a_a,Vs: list_a,Us: list_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys ) @ ( lex_a @ R ) )
     => ( ( ( size_size_list_a @ Vs )
          = ( size_size_list_a @ Us ) )
       => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs2 @ Us ) @ ( append_a @ Ys @ Vs ) ) @ ( lex_a @ R ) ) ) ) ).

% lex_append_rightI
thf(fact_199_lex__append__rightI,axiom,
    ! [Xs2: list_term_a_b,Ys: list_term_a_b,R: set_Pr4386577575007340137rm_a_b,Vs: list_term_a_b,Us: list_term_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs2 @ Ys ) @ ( lex_term_a_b @ R ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Vs )
          = ( size_s8906293707977694520rm_a_b @ Us ) )
       => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ Xs2 @ Us ) @ ( append_term_a_b @ Ys @ Vs ) ) @ ( lex_term_a_b @ R ) ) ) ) ).

% lex_append_rightI
thf(fact_200_lex__append__leftD,axiom,
    ! [R: set_Pr5038301440468608839rm_a_b,Xs2: list_P2364656488115551307rm_a_b,Ys: list_P2364656488115551307rm_a_b,Zs: list_P2364656488115551307rm_a_b] :
      ( ! [X3: produc1234881154892807749rm_a_b] :
          ~ ( member8066425057025219984rm_a_b @ ( produc3440225595649897687rm_a_b @ X3 @ X3 ) @ R )
     => ( ( member5494928282894704656rm_a_b @ ( produc8811016564285052503rm_a_b @ ( append5987703870611264992rm_a_b @ Xs2 @ Ys ) @ ( append5987703870611264992rm_a_b @ Xs2 @ Zs ) ) @ ( lex_Pr1817077058774561455rm_a_b @ R ) )
       => ( member5494928282894704656rm_a_b @ ( produc8811016564285052503rm_a_b @ Ys @ Zs ) @ ( lex_Pr1817077058774561455rm_a_b @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_201_lex__append__leftD,axiom,
    ! [R: set_Pr1811044260758604347_a_nat,Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat] :
      ( ! [X3: product_prod_a_nat] :
          ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ X3 ) @ R )
     => ( ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ ( append7679239579558125090_a_nat @ Xs2 @ Ys ) @ ( append7679239579558125090_a_nat @ Xs2 @ Zs ) ) @ ( lex_Pr1126020055417095059_a_nat @ R ) )
       => ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ Ys @ Zs ) @ ( lex_Pr1126020055417095059_a_nat @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_202_lex__append__leftD,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,Xs2: list_term_a_b,Ys: list_term_a_b,Zs: list_term_a_b] :
      ( ! [X3: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ R )
     => ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ Xs2 @ Ys ) @ ( append_term_a_b @ Xs2 @ Zs ) ) @ ( lex_term_a_b @ R ) )
       => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ys @ Zs ) @ ( lex_term_a_b @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_203_lex__append__leftD,axiom,
    ! [R: set_Pr3451248702717554689st_nat,Xs2: list_list_nat,Ys: list_list_nat,Zs: list_list_nat] :
      ( ! [X3: list_nat] :
          ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ X3 ) @ R )
     => ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( append_list_nat @ Xs2 @ Ys ) @ ( append_list_nat @ Xs2 @ Zs ) ) @ ( lex_list_nat @ R ) )
       => ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Ys @ Zs ) @ ( lex_list_nat @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_204_lex__append__leftD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ! [X3: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Ys ) @ ( append_nat @ Xs2 @ Zs ) ) @ ( lex_nat @ R ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_205_lex__append__left__iff,axiom,
    ! [R: set_Pr5038301440468608839rm_a_b,Xs2: list_P2364656488115551307rm_a_b,Ys: list_P2364656488115551307rm_a_b,Zs: list_P2364656488115551307rm_a_b] :
      ( ! [X3: produc1234881154892807749rm_a_b] :
          ~ ( member8066425057025219984rm_a_b @ ( produc3440225595649897687rm_a_b @ X3 @ X3 ) @ R )
     => ( ( member5494928282894704656rm_a_b @ ( produc8811016564285052503rm_a_b @ ( append5987703870611264992rm_a_b @ Xs2 @ Ys ) @ ( append5987703870611264992rm_a_b @ Xs2 @ Zs ) ) @ ( lex_Pr1817077058774561455rm_a_b @ R ) )
        = ( member5494928282894704656rm_a_b @ ( produc8811016564285052503rm_a_b @ Ys @ Zs ) @ ( lex_Pr1817077058774561455rm_a_b @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_206_lex__append__left__iff,axiom,
    ! [R: set_Pr1811044260758604347_a_nat,Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat] :
      ( ! [X3: product_prod_a_nat] :
          ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ X3 ) @ R )
     => ( ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ ( append7679239579558125090_a_nat @ Xs2 @ Ys ) @ ( append7679239579558125090_a_nat @ Xs2 @ Zs ) ) @ ( lex_Pr1126020055417095059_a_nat @ R ) )
        = ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ Ys @ Zs ) @ ( lex_Pr1126020055417095059_a_nat @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_207_lex__append__left__iff,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,Xs2: list_term_a_b,Ys: list_term_a_b,Zs: list_term_a_b] :
      ( ! [X3: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ R )
     => ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ Xs2 @ Ys ) @ ( append_term_a_b @ Xs2 @ Zs ) ) @ ( lex_term_a_b @ R ) )
        = ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ys @ Zs ) @ ( lex_term_a_b @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_208_lex__append__left__iff,axiom,
    ! [R: set_Pr3451248702717554689st_nat,Xs2: list_list_nat,Ys: list_list_nat,Zs: list_list_nat] :
      ( ! [X3: list_nat] :
          ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ X3 ) @ R )
     => ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( append_list_nat @ Xs2 @ Ys ) @ ( append_list_nat @ Xs2 @ Zs ) ) @ ( lex_list_nat @ R ) )
        = ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Ys @ Zs ) @ ( lex_list_nat @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_209_lex__append__left__iff,axiom,
    ! [R: set_Pr1261947904930325089at_nat,Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ! [X3: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Ys ) @ ( append_nat @ Xs2 @ Zs ) ) @ ( lex_nat @ R ) )
        = ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_210_pos__diff__append__itself,axiom,
    ! [P: list_P2364656488115551307rm_a_b,Q: list_P2364656488115551307rm_a_b] :
      ( ( term_p4159419441187170626rm_a_b @ ( append5987703870611264992rm_a_b @ P @ Q ) @ P )
      = Q ) ).

% pos_diff_append_itself
thf(fact_211_pos__diff__append__itself,axiom,
    ! [P: list_P3592885314253461005_a_nat,Q: list_P3592885314253461005_a_nat] :
      ( ( term_p8115227756575868480_a_nat @ ( append7679239579558125090_a_nat @ P @ Q ) @ P )
      = Q ) ).

% pos_diff_append_itself
thf(fact_212_pos__diff__append__itself,axiom,
    ! [P: list_term_a_b,Q: list_term_a_b] :
      ( ( term_p798503758663136087rm_a_b @ ( append_term_a_b @ P @ Q ) @ P )
      = Q ) ).

% pos_diff_append_itself
thf(fact_213_pos__diff__append__itself,axiom,
    ! [P: list_list_nat,Q: list_list_nat] :
      ( ( term_p7564741194569991203st_nat @ ( append_list_nat @ P @ Q ) @ P )
      = Q ) ).

% pos_diff_append_itself
thf(fact_214_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_215_lexord__sufE,axiom,
    ! [Xs2: list_P2364656488115551307rm_a_b,Zs: list_P2364656488115551307rm_a_b,Ys: list_P2364656488115551307rm_a_b,Qs: list_P2364656488115551307rm_a_b,R: set_Pr5038301440468608839rm_a_b] :
      ( ( member5494928282894704656rm_a_b @ ( produc8811016564285052503rm_a_b @ ( append5987703870611264992rm_a_b @ Xs2 @ Zs ) @ ( append5987703870611264992rm_a_b @ Ys @ Qs ) ) @ ( lexord7943539855291252024rm_a_b @ R ) )
     => ( ( Xs2 != Ys )
       => ( ( ( size_s3663718155697878711rm_a_b @ Xs2 )
            = ( size_s3663718155697878711rm_a_b @ Ys ) )
         => ( ( ( size_s3663718155697878711rm_a_b @ Zs )
              = ( size_s3663718155697878711rm_a_b @ Qs ) )
           => ( member5494928282894704656rm_a_b @ ( produc8811016564285052503rm_a_b @ Xs2 @ Ys ) @ ( lexord7943539855291252024rm_a_b @ R ) ) ) ) ) ) ).

% lexord_sufE
thf(fact_216_lexord__sufE,axiom,
    ! [Xs2: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,Qs: list_P3592885314253461005_a_nat,R: set_Pr1811044260758604347_a_nat] :
      ( ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ ( append7679239579558125090_a_nat @ Xs2 @ Zs ) @ ( append7679239579558125090_a_nat @ Ys @ Qs ) ) @ ( lexord2902578037316800714_a_nat @ R ) )
     => ( ( Xs2 != Ys )
       => ( ( ( size_s984997627204368545_a_nat @ Xs2 )
            = ( size_s984997627204368545_a_nat @ Ys ) )
         => ( ( ( size_s984997627204368545_a_nat @ Zs )
              = ( size_s984997627204368545_a_nat @ Qs ) )
           => ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ Xs2 @ Ys ) @ ( lexord2902578037316800714_a_nat @ R ) ) ) ) ) ) ).

% lexord_sufE
thf(fact_217_lexord__sufE,axiom,
    ! [Xs2: list_list_term_a_b,Zs: list_list_term_a_b,Ys: list_list_term_a_b,Qs: list_list_term_a_b,R: set_Pr8564414093027780873rm_a_b] :
      ( ( member8612744848686930930rm_a_b @ ( produc7557079580777928833rm_a_b @ ( append_list_term_a_b @ Xs2 @ Zs ) @ ( append_list_term_a_b @ Ys @ Qs ) ) @ ( lexord_list_term_a_b @ R ) )
     => ( ( Xs2 != Ys )
       => ( ( ( size_s877380706853472072rm_a_b @ Xs2 )
            = ( size_s877380706853472072rm_a_b @ Ys ) )
         => ( ( ( size_s877380706853472072rm_a_b @ Zs )
              = ( size_s877380706853472072rm_a_b @ Qs ) )
           => ( member8612744848686930930rm_a_b @ ( produc7557079580777928833rm_a_b @ Xs2 @ Ys ) @ ( lexord_list_term_a_b @ R ) ) ) ) ) ) ).

% lexord_sufE
thf(fact_218_lexord__sufE,axiom,
    ! [Xs2: list_list_nat,Zs: list_list_nat,Ys: list_list_nat,Qs: list_list_nat,R: set_Pr3451248702717554689st_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( append_list_nat @ Xs2 @ Zs ) @ ( append_list_nat @ Ys @ Qs ) ) @ ( lexord_list_nat @ R ) )
     => ( ( Xs2 != Ys )
       => ( ( ( size_s3023201423986296836st_nat @ Xs2 )
            = ( size_s3023201423986296836st_nat @ Ys ) )
         => ( ( ( size_s3023201423986296836st_nat @ Zs )
              = ( size_s3023201423986296836st_nat @ Qs ) )
           => ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs2 @ Ys ) @ ( lexord_list_nat @ R ) ) ) ) ) ) ).

% lexord_sufE
thf(fact_219_lexord__sufE,axiom,
    ! [Xs2: list_nat,Zs: list_nat,Ys: list_nat,Qs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Zs ) @ ( append_nat @ Ys @ Qs ) ) @ ( lexord_nat @ R ) )
     => ( ( Xs2 != Ys )
       => ( ( ( size_size_list_nat @ Xs2 )
            = ( size_size_list_nat @ Ys ) )
         => ( ( ( size_size_list_nat @ Zs )
              = ( size_size_list_nat @ Qs ) )
           => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( lexord_nat @ R ) ) ) ) ) ) ).

% lexord_sufE
thf(fact_220_lexord__sufE,axiom,
    ! [Xs2: list_a,Zs: list_a,Ys: list_a,Qs: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs2 @ Zs ) @ ( append_a @ Ys @ Qs ) ) @ ( lexord_a @ R ) )
     => ( ( Xs2 != Ys )
       => ( ( ( size_size_list_a @ Xs2 )
            = ( size_size_list_a @ Ys ) )
         => ( ( ( size_size_list_a @ Zs )
              = ( size_size_list_a @ Qs ) )
           => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys ) @ ( lexord_a @ R ) ) ) ) ) ) ).

% lexord_sufE
thf(fact_221_lexord__sufE,axiom,
    ! [Xs2: list_term_a_b,Zs: list_term_a_b,Ys: list_term_a_b,Qs: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ Xs2 @ Zs ) @ ( append_term_a_b @ Ys @ Qs ) ) @ ( lexord_term_a_b @ R ) )
     => ( ( Xs2 != Ys )
       => ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
            = ( size_s8906293707977694520rm_a_b @ Ys ) )
         => ( ( ( size_s8906293707977694520rm_a_b @ Zs )
              = ( size_s8906293707977694520rm_a_b @ Qs ) )
           => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs2 @ Ys ) @ ( lexord_term_a_b @ R ) ) ) ) ) ) ).

% lexord_sufE
thf(fact_222_lexord__append__leftD,axiom,
    ! [X: list_P2364656488115551307rm_a_b,U: list_P2364656488115551307rm_a_b,V: list_P2364656488115551307rm_a_b,R: set_Pr5038301440468608839rm_a_b] :
      ( ( member5494928282894704656rm_a_b @ ( produc8811016564285052503rm_a_b @ ( append5987703870611264992rm_a_b @ X @ U ) @ ( append5987703870611264992rm_a_b @ X @ V ) ) @ ( lexord7943539855291252024rm_a_b @ R ) )
     => ( ! [A3: produc1234881154892807749rm_a_b] :
            ~ ( member8066425057025219984rm_a_b @ ( produc3440225595649897687rm_a_b @ A3 @ A3 ) @ R )
       => ( member5494928282894704656rm_a_b @ ( produc8811016564285052503rm_a_b @ U @ V ) @ ( lexord7943539855291252024rm_a_b @ R ) ) ) ) ).

% lexord_append_leftD
thf(fact_223_lexord__append__leftD,axiom,
    ! [X: list_term_a_b,U: list_term_a_b,V: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ X @ U ) @ ( append_term_a_b @ X @ V ) ) @ ( lexord_term_a_b @ R ) )
     => ( ! [A3: term_a_b] :
            ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ A3 ) @ R )
       => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ U @ V ) @ ( lexord_term_a_b @ R ) ) ) ) ).

% lexord_append_leftD
thf(fact_224_lexord__append__leftD,axiom,
    ! [X: list_nat,U: list_nat,V: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ X @ U ) @ ( append_nat @ X @ V ) ) @ ( lexord_nat @ R ) )
     => ( ! [A3: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ A3 ) @ R )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ V ) @ ( lexord_nat @ R ) ) ) ) ).

% lexord_append_leftD
thf(fact_225_lexord__append__leftD,axiom,
    ! [X: list_list_nat,U: list_list_nat,V: list_list_nat,R: set_Pr3451248702717554689st_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( append_list_nat @ X @ U ) @ ( append_list_nat @ X @ V ) ) @ ( lexord_list_nat @ R ) )
     => ( ! [A3: list_nat] :
            ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A3 @ A3 ) @ R )
       => ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ U @ V ) @ ( lexord_list_nat @ R ) ) ) ) ).

% lexord_append_leftD
thf(fact_226_lexord__append__leftD,axiom,
    ! [X: list_P3592885314253461005_a_nat,U: list_P3592885314253461005_a_nat,V: list_P3592885314253461005_a_nat,R: set_Pr1811044260758604347_a_nat] :
      ( ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ ( append7679239579558125090_a_nat @ X @ U ) @ ( append7679239579558125090_a_nat @ X @ V ) ) @ ( lexord2902578037316800714_a_nat @ R ) )
     => ( ! [A3: product_prod_a_nat] :
            ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A3 @ A3 ) @ R )
       => ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ U @ V ) @ ( lexord2902578037316800714_a_nat @ R ) ) ) ) ).

% lexord_append_leftD
thf(fact_227_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_228_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_229_enumerate__append__eq,axiom,
    ! [N: nat,Xs2: list_P2364656488115551307rm_a_b,Ys: list_P2364656488115551307rm_a_b] :
      ( ( enumer1459956249525160746rm_a_b @ N @ ( append5987703870611264992rm_a_b @ Xs2 @ Ys ) )
      = ( append6140445811708759055rm_a_b @ ( enumer1459956249525160746rm_a_b @ N @ Xs2 ) @ ( enumer1459956249525160746rm_a_b @ ( plus_plus_nat @ N @ ( size_s3663718155697878711rm_a_b @ Xs2 ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_230_enumerate__append__eq,axiom,
    ! [N: nat,Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
      ( ( enumer3239626523597437528_a_nat @ N @ ( append7679239579558125090_a_nat @ Xs2 @ Ys ) )
      = ( append7569360072597011785_a_nat @ ( enumer3239626523597437528_a_nat @ N @ Xs2 ) @ ( enumer3239626523597437528_a_nat @ ( plus_plus_nat @ N @ ( size_s984997627204368545_a_nat @ Xs2 ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_231_enumerate__append__eq,axiom,
    ! [N: nat,Xs2: list_list_term_a_b,Ys: list_list_term_a_b] :
      ( ( enumer1424976507981524095rm_a_b @ N @ ( append_list_term_a_b @ Xs2 @ Ys ) )
      = ( append7461743152246115312rm_a_b @ ( enumer1424976507981524095rm_a_b @ N @ Xs2 ) @ ( enumer1424976507981524095rm_a_b @ ( plus_plus_nat @ N @ ( size_s877380706853472072rm_a_b @ Xs2 ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_232_enumerate__append__eq,axiom,
    ! [N: nat,Xs2: list_list_nat,Ys: list_list_nat] :
      ( ( enumerate_list_nat @ N @ ( append_list_nat @ Xs2 @ Ys ) )
      = ( append104611586619867308st_nat @ ( enumerate_list_nat @ N @ Xs2 ) @ ( enumerate_list_nat @ ( plus_plus_nat @ N @ ( size_s3023201423986296836st_nat @ Xs2 ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_233_enumerate__append__eq,axiom,
    ! [N: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( enumerate_nat @ N @ ( append_nat @ Xs2 @ Ys ) )
      = ( append985823374593552924at_nat @ ( enumerate_nat @ N @ Xs2 ) @ ( enumerate_nat @ ( plus_plus_nat @ N @ ( size_size_list_nat @ Xs2 ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_234_enumerate__append__eq,axiom,
    ! [N: nat,Xs2: list_a,Ys: list_a] :
      ( ( enumerate_a @ N @ ( append_a @ Xs2 @ Ys ) )
      = ( append1694031006427026248_nat_a @ ( enumerate_a @ N @ Xs2 ) @ ( enumerate_a @ ( plus_plus_nat @ N @ ( size_size_list_a @ Xs2 ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_235_enumerate__append__eq,axiom,
    ! [N: nat,Xs2: list_term_a_b,Ys: list_term_a_b] :
      ( ( enumerate_term_a_b @ N @ ( append_term_a_b @ Xs2 @ Ys ) )
      = ( append5987703870611264992rm_a_b @ ( enumerate_term_a_b @ N @ Xs2 ) @ ( enumerate_term_a_b @ ( plus_plus_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs2 ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_236_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_237_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_238_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_239_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_240_length__append,axiom,
    ! [Xs2: list_P2364656488115551307rm_a_b,Ys: list_P2364656488115551307rm_a_b] :
      ( ( size_s3663718155697878711rm_a_b @ ( append5987703870611264992rm_a_b @ Xs2 @ Ys ) )
      = ( plus_plus_nat @ ( size_s3663718155697878711rm_a_b @ Xs2 ) @ ( size_s3663718155697878711rm_a_b @ Ys ) ) ) ).

% length_append
thf(fact_241_length__append,axiom,
    ! [Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
      ( ( size_s984997627204368545_a_nat @ ( append7679239579558125090_a_nat @ Xs2 @ Ys ) )
      = ( plus_plus_nat @ ( size_s984997627204368545_a_nat @ Xs2 ) @ ( size_s984997627204368545_a_nat @ Ys ) ) ) ).

% length_append
thf(fact_242_length__append,axiom,
    ! [Xs2: list_list_term_a_b,Ys: list_list_term_a_b] :
      ( ( size_s877380706853472072rm_a_b @ ( append_list_term_a_b @ Xs2 @ Ys ) )
      = ( plus_plus_nat @ ( size_s877380706853472072rm_a_b @ Xs2 ) @ ( size_s877380706853472072rm_a_b @ Ys ) ) ) ).

% length_append
thf(fact_243_length__append,axiom,
    ! [Xs2: list_list_nat,Ys: list_list_nat] :
      ( ( size_s3023201423986296836st_nat @ ( append_list_nat @ Xs2 @ Ys ) )
      = ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ ( size_s3023201423986296836st_nat @ Ys ) ) ) ).

% length_append
thf(fact_244_length__append,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat @ ( append_nat @ Xs2 @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_append
thf(fact_245_length__append,axiom,
    ! [Xs2: list_a,Ys: list_a] :
      ( ( size_size_list_a @ ( append_a @ Xs2 @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_a @ Xs2 ) @ ( size_size_list_a @ Ys ) ) ) ).

% length_append
thf(fact_246_length__append,axiom,
    ! [Xs2: list_term_a_b,Ys: list_term_a_b] :
      ( ( size_s8906293707977694520rm_a_b @ ( append_term_a_b @ Xs2 @ Ys ) )
      = ( plus_plus_nat @ ( size_s8906293707977694520rm_a_b @ Xs2 ) @ ( size_s8906293707977694520rm_a_b @ Ys ) ) ) ).

% length_append
thf(fact_247_lexord__lex,axiom,
    ! [X: list_P3592885314253461005_a_nat,Y: list_P3592885314253461005_a_nat,R: set_Pr1811044260758604347_a_nat] :
      ( ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ X @ Y ) @ ( lex_Pr1126020055417095059_a_nat @ R ) )
      = ( ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ X @ Y ) @ ( lexord2902578037316800714_a_nat @ R ) )
        & ( ( size_s984997627204368545_a_nat @ X )
          = ( size_s984997627204368545_a_nat @ Y ) ) ) ) ).

% lexord_lex
thf(fact_248_lexord__lex,axiom,
    ! [X: list_list_term_a_b,Y: list_list_term_a_b,R: set_Pr8564414093027780873rm_a_b] :
      ( ( member8612744848686930930rm_a_b @ ( produc7557079580777928833rm_a_b @ X @ Y ) @ ( lex_list_term_a_b @ R ) )
      = ( ( member8612744848686930930rm_a_b @ ( produc7557079580777928833rm_a_b @ X @ Y ) @ ( lexord_list_term_a_b @ R ) )
        & ( ( size_s877380706853472072rm_a_b @ X )
          = ( size_s877380706853472072rm_a_b @ Y ) ) ) ) ).

% lexord_lex
thf(fact_249_lexord__lex,axiom,
    ! [X: list_list_nat,Y: list_list_nat,R: set_Pr3451248702717554689st_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ X @ Y ) @ ( lex_list_nat @ R ) )
      = ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ X @ Y ) @ ( lexord_list_nat @ R ) )
        & ( ( size_s3023201423986296836st_nat @ X )
          = ( size_s3023201423986296836st_nat @ Y ) ) ) ) ).

% lexord_lex
thf(fact_250_lexord__lex,axiom,
    ! [X: list_nat,Y: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lex_nat @ R ) )
      = ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lexord_nat @ R ) )
        & ( ( size_size_list_nat @ X )
          = ( size_size_list_nat @ Y ) ) ) ) ).

% lexord_lex
thf(fact_251_lexord__lex,axiom,
    ! [X: list_a,Y: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( lex_a @ R ) )
      = ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( lexord_a @ R ) )
        & ( ( size_size_list_a @ X )
          = ( size_size_list_a @ Y ) ) ) ) ).

% lexord_lex
thf(fact_252_lexord__lex,axiom,
    ! [X: list_term_a_b,Y: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X @ Y ) @ ( lex_term_a_b @ R ) )
      = ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X @ Y ) @ ( lexord_term_a_b @ R ) )
        & ( ( size_s8906293707977694520rm_a_b @ X )
          = ( size_s8906293707977694520rm_a_b @ Y ) ) ) ) ).

% lexord_lex
thf(fact_253_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_254_lexord__linear,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,X: list_term_a_b,Y: list_term_a_b] :
      ( ! [A3: term_a_b,B3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ R )
          | ( A3 = B3 )
          | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A3 ) @ R ) )
     => ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X @ Y ) @ ( lexord_term_a_b @ R ) )
        | ( X = Y )
        | ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Y @ X ) @ ( lexord_term_a_b @ R ) ) ) ) ).

% lexord_linear
thf(fact_255_lexord__linear,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X: list_nat,Y: list_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ R )
          | ( A3 = B3 )
          | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A3 ) @ R ) )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lexord_nat @ R ) )
        | ( X = Y )
        | ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y @ X ) @ ( lexord_nat @ R ) ) ) ) ).

% lexord_linear
thf(fact_256_lexord__linear,axiom,
    ! [R: set_Pr3451248702717554689st_nat,X: list_list_nat,Y: list_list_nat] :
      ( ! [A3: list_nat,B3: list_nat] :
          ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A3 @ B3 ) @ R )
          | ( A3 = B3 )
          | ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B3 @ A3 ) @ R ) )
     => ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ X @ Y ) @ ( lexord_list_nat @ R ) )
        | ( X = Y )
        | ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Y @ X ) @ ( lexord_list_nat @ R ) ) ) ) ).

% lexord_linear
thf(fact_257_lexord__linear,axiom,
    ! [R: set_Pr1811044260758604347_a_nat,X: list_P3592885314253461005_a_nat,Y: list_P3592885314253461005_a_nat] :
      ( ! [A3: product_prod_a_nat,B3: product_prod_a_nat] :
          ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A3 @ B3 ) @ R )
          | ( A3 = B3 )
          | ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B3 @ A3 ) @ R ) )
     => ( ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ X @ Y ) @ ( lexord2902578037316800714_a_nat @ R ) )
        | ( X = Y )
        | ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ Y @ X ) @ ( lexord2902578037316800714_a_nat @ R ) ) ) ) ).

% lexord_linear
thf(fact_258_lexord__irreflexive,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,Xs2: list_term_a_b] :
      ( ! [X3: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ R )
     => ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs2 @ Xs2 ) @ ( lexord_term_a_b @ R ) ) ) ).

% lexord_irreflexive
thf(fact_259_lexord__irreflexive,axiom,
    ! [R: set_Pr1261947904930325089at_nat,Xs2: list_nat] :
      ( ! [X3: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R )
     => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Xs2 ) @ ( lexord_nat @ R ) ) ) ).

% lexord_irreflexive
thf(fact_260_lexord__irreflexive,axiom,
    ! [R: set_Pr3451248702717554689st_nat,Xs2: list_list_nat] :
      ( ! [X3: list_nat] :
          ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ X3 ) @ R )
     => ~ ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs2 @ Xs2 ) @ ( lexord_list_nat @ R ) ) ) ).

% lexord_irreflexive
thf(fact_261_lexord__irreflexive,axiom,
    ! [R: set_Pr1811044260758604347_a_nat,Xs2: list_P3592885314253461005_a_nat] :
      ( ! [X3: product_prod_a_nat] :
          ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ X3 ) @ R )
     => ~ ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ Xs2 @ Xs2 ) @ ( lexord2902578037316800714_a_nat @ R ) ) ) ).

% lexord_irreflexive
thf(fact_262_lexord__append__leftI,axiom,
    ! [U: list_P2364656488115551307rm_a_b,V: list_P2364656488115551307rm_a_b,R: set_Pr5038301440468608839rm_a_b,X: list_P2364656488115551307rm_a_b] :
      ( ( member5494928282894704656rm_a_b @ ( produc8811016564285052503rm_a_b @ U @ V ) @ ( lexord7943539855291252024rm_a_b @ R ) )
     => ( member5494928282894704656rm_a_b @ ( produc8811016564285052503rm_a_b @ ( append5987703870611264992rm_a_b @ X @ U ) @ ( append5987703870611264992rm_a_b @ X @ V ) ) @ ( lexord7943539855291252024rm_a_b @ R ) ) ) ).

% lexord_append_leftI
thf(fact_263_lexord__append__leftI,axiom,
    ! [U: list_term_a_b,V: list_term_a_b,R: set_Pr4386577575007340137rm_a_b,X: list_term_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ U @ V ) @ ( lexord_term_a_b @ R ) )
     => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ X @ U ) @ ( append_term_a_b @ X @ V ) ) @ ( lexord_term_a_b @ R ) ) ) ).

% lexord_append_leftI
thf(fact_264_lexord__append__leftI,axiom,
    ! [U: list_nat,V: list_nat,R: set_Pr1261947904930325089at_nat,X: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ V ) @ ( lexord_nat @ R ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ X @ U ) @ ( append_nat @ X @ V ) ) @ ( lexord_nat @ R ) ) ) ).

% lexord_append_leftI
thf(fact_265_lexord__append__leftI,axiom,
    ! [U: list_list_nat,V: list_list_nat,R: set_Pr3451248702717554689st_nat,X: list_list_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ U @ V ) @ ( lexord_list_nat @ R ) )
     => ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( append_list_nat @ X @ U ) @ ( append_list_nat @ X @ V ) ) @ ( lexord_list_nat @ R ) ) ) ).

% lexord_append_leftI
thf(fact_266_lexord__append__leftI,axiom,
    ! [U: list_P3592885314253461005_a_nat,V: list_P3592885314253461005_a_nat,R: set_Pr1811044260758604347_a_nat,X: list_P3592885314253461005_a_nat] :
      ( ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ U @ V ) @ ( lexord2902578037316800714_a_nat @ R ) )
     => ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ ( append7679239579558125090_a_nat @ X @ U ) @ ( append7679239579558125090_a_nat @ X @ V ) ) @ ( lexord2902578037316800714_a_nat @ R ) ) ) ).

% lexord_append_leftI
thf(fact_267_lex__append__leftI,axiom,
    ! [Ys: list_P2364656488115551307rm_a_b,Zs: list_P2364656488115551307rm_a_b,R: set_Pr5038301440468608839rm_a_b,Xs2: list_P2364656488115551307rm_a_b] :
      ( ( member5494928282894704656rm_a_b @ ( produc8811016564285052503rm_a_b @ Ys @ Zs ) @ ( lex_Pr1817077058774561455rm_a_b @ R ) )
     => ( member5494928282894704656rm_a_b @ ( produc8811016564285052503rm_a_b @ ( append5987703870611264992rm_a_b @ Xs2 @ Ys ) @ ( append5987703870611264992rm_a_b @ Xs2 @ Zs ) ) @ ( lex_Pr1817077058774561455rm_a_b @ R ) ) ) ).

% lex_append_leftI
thf(fact_268_lex__append__leftI,axiom,
    ! [Ys: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat,R: set_Pr1811044260758604347_a_nat,Xs2: list_P3592885314253461005_a_nat] :
      ( ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ Ys @ Zs ) @ ( lex_Pr1126020055417095059_a_nat @ R ) )
     => ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ ( append7679239579558125090_a_nat @ Xs2 @ Ys ) @ ( append7679239579558125090_a_nat @ Xs2 @ Zs ) ) @ ( lex_Pr1126020055417095059_a_nat @ R ) ) ) ).

% lex_append_leftI
thf(fact_269_lex__append__leftI,axiom,
    ! [Ys: list_term_a_b,Zs: list_term_a_b,R: set_Pr4386577575007340137rm_a_b,Xs2: list_term_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ys @ Zs ) @ ( lex_term_a_b @ R ) )
     => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ Xs2 @ Ys ) @ ( append_term_a_b @ Xs2 @ Zs ) ) @ ( lex_term_a_b @ R ) ) ) ).

% lex_append_leftI
thf(fact_270_lex__append__leftI,axiom,
    ! [Ys: list_list_nat,Zs: list_list_nat,R: set_Pr3451248702717554689st_nat,Xs2: list_list_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Ys @ Zs ) @ ( lex_list_nat @ R ) )
     => ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( append_list_nat @ Xs2 @ Ys ) @ ( append_list_nat @ Xs2 @ Zs ) ) @ ( lex_list_nat @ R ) ) ) ).

% lex_append_leftI
thf(fact_271_lex__append__leftI,axiom,
    ! [Ys: list_nat,Zs: list_nat,R: set_Pr1261947904930325089at_nat,Xs2: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Ys ) @ ( append_nat @ Xs2 @ Zs ) ) @ ( lex_nat @ R ) ) ) ).

% lex_append_leftI
thf(fact_272_lenlex__irreflexive,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,Xs2: list_term_a_b] :
      ( ! [X3: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ R )
     => ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs2 @ Xs2 ) @ ( lenlex_term_a_b @ R ) ) ) ).

% lenlex_irreflexive
thf(fact_273_lenlex__irreflexive,axiom,
    ! [R: set_Pr1261947904930325089at_nat,Xs2: list_nat] :
      ( ! [X3: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R )
     => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Xs2 ) @ ( lenlex_nat @ R ) ) ) ).

% lenlex_irreflexive
thf(fact_274_add__right__cancel,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add_right_cancel
thf(fact_275_add__left__cancel,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add_left_cancel
thf(fact_276_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_277_pos__les__eq__append__diff,axiom,
    ! [P: list_P2364656488115551307rm_a_b,Q: list_P2364656488115551307rm_a_b] :
      ( ( term_p6496282196689937951rm_a_b @ P @ Q )
     => ( ( append5987703870611264992rm_a_b @ P @ ( term_p4159419441187170626rm_a_b @ Q @ P ) )
        = Q ) ) ).

% pos_les_eq_append_diff
thf(fact_278_pos__les__eq__append__diff,axiom,
    ! [P: list_P3592885314253461005_a_nat,Q: list_P3592885314253461005_a_nat] :
      ( ( term_p4408110555530040355_a_nat @ P @ Q )
     => ( ( append7679239579558125090_a_nat @ P @ ( term_p8115227756575868480_a_nat @ Q @ P ) )
        = Q ) ) ).

% pos_les_eq_append_diff
thf(fact_279_pos__les__eq__append__diff,axiom,
    ! [P: list_term_a_b,Q: list_term_a_b] :
      ( ( term_p8391561492822560442rm_a_b @ P @ Q )
     => ( ( append_term_a_b @ P @ ( term_p798503758663136087rm_a_b @ Q @ P ) )
        = Q ) ) ).

% pos_les_eq_append_diff
thf(fact_280_pos__les__eq__append__diff,axiom,
    ! [P: list_list_nat,Q: list_list_nat] :
      ( ( term_p5934426891874639750st_nat @ P @ Q )
     => ( ( append_list_nat @ P @ ( term_p7564741194569991203st_nat @ Q @ P ) )
        = Q ) ) ).

% pos_les_eq_append_diff
thf(fact_281_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_282_Cons__in__lex,axiom,
    ! [X: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Y: product_prod_a_nat,Ys: list_P3592885314253461005_a_nat,R: set_Pr1811044260758604347_a_nat] :
      ( ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ ( cons_P5205166803686508359_a_nat @ X @ Xs2 ) @ ( cons_P5205166803686508359_a_nat @ Y @ Ys ) ) @ ( lex_Pr1126020055417095059_a_nat @ R ) )
      = ( ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ R )
          & ( ( size_s984997627204368545_a_nat @ Xs2 )
            = ( size_s984997627204368545_a_nat @ Ys ) ) )
        | ( ( X = Y )
          & ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ Xs2 @ Ys ) @ ( lex_Pr1126020055417095059_a_nat @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_283_Cons__in__lex,axiom,
    ! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys ) ) @ ( lex_Pr8571645452597969515at_nat @ R ) )
      = ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R )
          & ( ( size_s5460976970255530739at_nat @ Xs2 )
            = ( size_s5460976970255530739at_nat @ Ys ) ) )
        | ( ( X = Y )
          & ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Ys ) @ ( lex_Pr8571645452597969515at_nat @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_284_Cons__in__lex,axiom,
    ! [X: list_term_a_b,Xs2: list_list_term_a_b,Y: list_term_a_b,Ys: list_list_term_a_b,R: set_Pr8564414093027780873rm_a_b] :
      ( ( member8612744848686930930rm_a_b @ ( produc7557079580777928833rm_a_b @ ( cons_list_term_a_b @ X @ Xs2 ) @ ( cons_list_term_a_b @ Y @ Ys ) ) @ ( lex_list_term_a_b @ R ) )
      = ( ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X @ Y ) @ R )
          & ( ( size_s877380706853472072rm_a_b @ Xs2 )
            = ( size_s877380706853472072rm_a_b @ Ys ) ) )
        | ( ( X = Y )
          & ( member8612744848686930930rm_a_b @ ( produc7557079580777928833rm_a_b @ Xs2 @ Ys ) @ ( lex_list_term_a_b @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_285_Cons__in__lex,axiom,
    ! [X: list_nat,Xs2: list_list_nat,Y: list_nat,Ys: list_list_nat,R: set_Pr3451248702717554689st_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ X @ Xs2 ) @ ( cons_list_nat @ Y @ Ys ) ) @ ( lex_list_nat @ R ) )
      = ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R )
          & ( ( size_s3023201423986296836st_nat @ Xs2 )
            = ( size_s3023201423986296836st_nat @ Ys ) ) )
        | ( ( X = Y )
          & ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs2 @ Ys ) @ ( lex_list_nat @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_286_Cons__in__lex,axiom,
    ! [X: a,Xs2: list_a,Y: a,Ys: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y @ Ys ) ) @ ( lex_a @ R ) )
      = ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
          & ( ( size_size_list_a @ Xs2 )
            = ( size_size_list_a @ Ys ) ) )
        | ( ( X = Y )
          & ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys ) @ ( lex_a @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_287_Cons__in__lex,axiom,
    ! [X: nat,Xs2: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys ) ) @ ( lex_nat @ R ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
          & ( ( size_size_list_nat @ Xs2 )
            = ( size_size_list_nat @ Ys ) ) )
        | ( ( X = Y )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( lex_nat @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_288_Cons__in__lex,axiom,
    ! [X: term_a_b,Xs2: list_term_a_b,Y: term_a_b,Ys: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X @ Xs2 ) @ ( cons_term_a_b @ Y @ Ys ) ) @ ( lex_term_a_b @ R ) )
      = ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
          & ( ( size_s8906293707977694520rm_a_b @ Xs2 )
            = ( size_s8906293707977694520rm_a_b @ Ys ) ) )
        | ( ( X = Y )
          & ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs2 @ Ys ) @ ( lex_term_a_b @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_289_gen__length__def,axiom,
    ( gen_le4942244902447077073rm_a_b
    = ( ^ [N2: nat,Xs3: list_list_term_a_b] : ( plus_plus_nat @ N2 @ ( size_s877380706853472072rm_a_b @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_290_gen__length__def,axiom,
    ( gen_length_list_nat
    = ( ^ [N2: nat,Xs3: list_list_nat] : ( plus_plus_nat @ N2 @ ( size_s3023201423986296836st_nat @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_291_gen__length__def,axiom,
    ( gen_length_nat
    = ( ^ [N2: nat,Xs3: list_nat] : ( plus_plus_nat @ N2 @ ( size_size_list_nat @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_292_gen__length__def,axiom,
    ( gen_length_a
    = ( ^ [N2: nat,Xs3: list_a] : ( plus_plus_nat @ N2 @ ( size_size_list_a @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_293_gen__length__def,axiom,
    ( gen_length_term_a_b
    = ( ^ [N2: nat,Xs3: list_term_a_b] : ( plus_plus_nat @ N2 @ ( size_s8906293707977694520rm_a_b @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_294_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_295_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_296_lexord__sufI,axiom,
    ! [U: list_P2364656488115551307rm_a_b,W: list_P2364656488115551307rm_a_b,R: set_Pr5038301440468608839rm_a_b,V: list_P2364656488115551307rm_a_b,Z: list_P2364656488115551307rm_a_b] :
      ( ( member5494928282894704656rm_a_b @ ( produc8811016564285052503rm_a_b @ U @ W ) @ ( lexord7943539855291252024rm_a_b @ R ) )
     => ( ( ord_less_eq_nat @ ( size_s3663718155697878711rm_a_b @ W ) @ ( size_s3663718155697878711rm_a_b @ U ) )
       => ( member5494928282894704656rm_a_b @ ( produc8811016564285052503rm_a_b @ ( append5987703870611264992rm_a_b @ U @ V ) @ ( append5987703870611264992rm_a_b @ W @ Z ) ) @ ( lexord7943539855291252024rm_a_b @ R ) ) ) ) ).

% lexord_sufI
thf(fact_297_lexord__sufI,axiom,
    ! [U: list_P3592885314253461005_a_nat,W: list_P3592885314253461005_a_nat,R: set_Pr1811044260758604347_a_nat,V: list_P3592885314253461005_a_nat,Z: list_P3592885314253461005_a_nat] :
      ( ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ U @ W ) @ ( lexord2902578037316800714_a_nat @ R ) )
     => ( ( ord_less_eq_nat @ ( size_s984997627204368545_a_nat @ W ) @ ( size_s984997627204368545_a_nat @ U ) )
       => ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ ( append7679239579558125090_a_nat @ U @ V ) @ ( append7679239579558125090_a_nat @ W @ Z ) ) @ ( lexord2902578037316800714_a_nat @ R ) ) ) ) ).

% lexord_sufI
thf(fact_298_lexord__sufI,axiom,
    ! [U: list_list_term_a_b,W: list_list_term_a_b,R: set_Pr8564414093027780873rm_a_b,V: list_list_term_a_b,Z: list_list_term_a_b] :
      ( ( member8612744848686930930rm_a_b @ ( produc7557079580777928833rm_a_b @ U @ W ) @ ( lexord_list_term_a_b @ R ) )
     => ( ( ord_less_eq_nat @ ( size_s877380706853472072rm_a_b @ W ) @ ( size_s877380706853472072rm_a_b @ U ) )
       => ( member8612744848686930930rm_a_b @ ( produc7557079580777928833rm_a_b @ ( append_list_term_a_b @ U @ V ) @ ( append_list_term_a_b @ W @ Z ) ) @ ( lexord_list_term_a_b @ R ) ) ) ) ).

% lexord_sufI
thf(fact_299_lexord__sufI,axiom,
    ! [U: list_list_nat,W: list_list_nat,R: set_Pr3451248702717554689st_nat,V: list_list_nat,Z: list_list_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ U @ W ) @ ( lexord_list_nat @ R ) )
     => ( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ W ) @ ( size_s3023201423986296836st_nat @ U ) )
       => ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( append_list_nat @ U @ V ) @ ( append_list_nat @ W @ Z ) ) @ ( lexord_list_nat @ R ) ) ) ) ).

% lexord_sufI
thf(fact_300_lexord__sufI,axiom,
    ! [U: list_nat,W: list_nat,R: set_Pr1261947904930325089at_nat,V: list_nat,Z: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ W ) @ ( lexord_nat @ R ) )
     => ( ( ord_less_eq_nat @ ( size_size_list_nat @ W ) @ ( size_size_list_nat @ U ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ U @ V ) @ ( append_nat @ W @ Z ) ) @ ( lexord_nat @ R ) ) ) ) ).

% lexord_sufI
thf(fact_301_lexord__sufI,axiom,
    ! [U: list_a,W: list_a,R: set_Product_prod_a_a,V: list_a,Z: list_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ U @ W ) @ ( lexord_a @ R ) )
     => ( ( ord_less_eq_nat @ ( size_size_list_a @ W ) @ ( size_size_list_a @ U ) )
       => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ U @ V ) @ ( append_a @ W @ Z ) ) @ ( lexord_a @ R ) ) ) ) ).

% lexord_sufI
thf(fact_302_lexord__sufI,axiom,
    ! [U: list_term_a_b,W: list_term_a_b,R: set_Pr4386577575007340137rm_a_b,V: list_term_a_b,Z: list_term_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ U @ W ) @ ( lexord_term_a_b @ R ) )
     => ( ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ W ) @ ( size_s8906293707977694520rm_a_b @ U ) )
       => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ U @ V ) @ ( append_term_a_b @ W @ Z ) ) @ ( lexord_term_a_b @ R ) ) ) ) ).

% lexord_sufI
thf(fact_303_lexord__append__left__rightI,axiom,
    ! [A: produc1234881154892807749rm_a_b,B: produc1234881154892807749rm_a_b,R: set_Pr5038301440468608839rm_a_b,U: list_P2364656488115551307rm_a_b,X: list_P2364656488115551307rm_a_b,Y: list_P2364656488115551307rm_a_b] :
      ( ( member8066425057025219984rm_a_b @ ( produc3440225595649897687rm_a_b @ A @ B ) @ R )
     => ( member5494928282894704656rm_a_b @ ( produc8811016564285052503rm_a_b @ ( append5987703870611264992rm_a_b @ U @ ( cons_P2836617062085252091rm_a_b @ A @ X ) ) @ ( append5987703870611264992rm_a_b @ U @ ( cons_P2836617062085252091rm_a_b @ B @ Y ) ) ) @ ( lexord7943539855291252024rm_a_b @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_304_lexord__append__left__rightI,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,U: list_term_a_b,X: list_term_a_b,Y: list_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
     => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ U @ ( cons_term_a_b @ A @ X ) ) @ ( append_term_a_b @ U @ ( cons_term_a_b @ B @ Y ) ) ) @ ( lexord_term_a_b @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_305_lexord__append__left__rightI,axiom,
    ! [A: list_nat,B: list_nat,R: set_Pr3451248702717554689st_nat,U: list_list_nat,X: list_list_nat,Y: list_list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ R )
     => ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( append_list_nat @ U @ ( cons_list_nat @ A @ X ) ) @ ( append_list_nat @ U @ ( cons_list_nat @ B @ Y ) ) ) @ ( lexord_list_nat @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_306_lexord__append__left__rightI,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat,U: list_P3592885314253461005_a_nat,X: list_P3592885314253461005_a_nat,Y: list_P3592885314253461005_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ R )
     => ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ ( append7679239579558125090_a_nat @ U @ ( cons_P5205166803686508359_a_nat @ A @ X ) ) @ ( append7679239579558125090_a_nat @ U @ ( cons_P5205166803686508359_a_nat @ B @ Y ) ) ) @ ( lexord2902578037316800714_a_nat @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_307_lexord__append__left__rightI,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a,U: list_a,X: list_a,Y: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
     => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ U @ ( cons_a @ A @ X ) ) @ ( append_a @ U @ ( cons_a @ B @ Y ) ) ) @ ( lexord_a @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_308_lexord__append__left__rightI,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,U: list_P6011104703257516679at_nat,X: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R )
     => ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( append985823374593552924at_nat @ U @ ( cons_P6512896166579812791at_nat @ A @ X ) ) @ ( append985823374593552924at_nat @ U @ ( cons_P6512896166579812791at_nat @ B @ Y ) ) ) @ ( lexord2841853652668343668at_nat @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_309_lexord__append__left__rightI,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,U: list_nat,X: list_nat,Y: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ U @ ( cons_nat @ A @ X ) ) @ ( append_nat @ U @ ( cons_nat @ B @ Y ) ) ) @ ( lexord_nat @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_310_list_Oinject,axiom,
    ! [X21: term_a_b,X22: list_term_a_b,Y21: term_a_b,Y22: list_term_a_b] :
      ( ( ( cons_term_a_b @ X21 @ X22 )
        = ( cons_term_a_b @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_311_list_Oinject,axiom,
    ! [X21: list_nat,X22: list_list_nat,Y21: list_nat,Y22: list_list_nat] :
      ( ( ( cons_list_nat @ X21 @ X22 )
        = ( cons_list_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_312_list_Oinject,axiom,
    ! [X21: product_prod_a_nat,X22: list_P3592885314253461005_a_nat,Y21: product_prod_a_nat,Y22: list_P3592885314253461005_a_nat] :
      ( ( ( cons_P5205166803686508359_a_nat @ X21 @ X22 )
        = ( cons_P5205166803686508359_a_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_313_list_Oinject,axiom,
    ! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
      ( ( ( cons_a @ X21 @ X22 )
        = ( cons_a @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_314_list_Oinject,axiom,
    ! [X21: product_prod_nat_nat,X22: list_P6011104703257516679at_nat,Y21: product_prod_nat_nat,Y22: list_P6011104703257516679at_nat] :
      ( ( ( cons_P6512896166579812791at_nat @ X21 @ X22 )
        = ( cons_P6512896166579812791at_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_315_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_316_par__not__refl,axiom,
    ! [P: list_nat] :
      ~ ( term_p5017330785391824242ar_nat @ P @ P ) ).

% par_not_refl
thf(fact_317_add__le__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_318_add__le__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_319_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_320_position__less__eq__Cons,axiom,
    ! [I: term_a_b,Ps: list_term_a_b,J: term_a_b,Qs: list_term_a_b] :
      ( ( term_p8391561492822560442rm_a_b @ ( cons_term_a_b @ I @ Ps ) @ ( cons_term_a_b @ J @ Qs ) )
      = ( ( I = J )
        & ( term_p8391561492822560442rm_a_b @ Ps @ Qs ) ) ) ).

% position_less_eq_Cons
thf(fact_321_position__less__eq__Cons,axiom,
    ! [I: list_nat,Ps: list_list_nat,J: list_nat,Qs: list_list_nat] :
      ( ( term_p5934426891874639750st_nat @ ( cons_list_nat @ I @ Ps ) @ ( cons_list_nat @ J @ Qs ) )
      = ( ( I = J )
        & ( term_p5934426891874639750st_nat @ Ps @ Qs ) ) ) ).

% position_less_eq_Cons
thf(fact_322_position__less__eq__Cons,axiom,
    ! [I: product_prod_a_nat,Ps: list_P3592885314253461005_a_nat,J: product_prod_a_nat,Qs: list_P3592885314253461005_a_nat] :
      ( ( term_p4408110555530040355_a_nat @ ( cons_P5205166803686508359_a_nat @ I @ Ps ) @ ( cons_P5205166803686508359_a_nat @ J @ Qs ) )
      = ( ( I = J )
        & ( term_p4408110555530040355_a_nat @ Ps @ Qs ) ) ) ).

% position_less_eq_Cons
thf(fact_323_position__less__eq__Cons,axiom,
    ! [I: a,Ps: list_a,J: a,Qs: list_a] :
      ( ( term_p1826803665625515224s_eq_a @ ( cons_a @ I @ Ps ) @ ( cons_a @ J @ Qs ) )
      = ( ( I = J )
        & ( term_p1826803665625515224s_eq_a @ Ps @ Qs ) ) ) ).

% position_less_eq_Cons
thf(fact_324_position__less__eq__Cons,axiom,
    ! [I: product_prod_nat_nat,Ps: list_P6011104703257516679at_nat,J: product_prod_nat_nat,Qs: list_P6011104703257516679at_nat] :
      ( ( term_p5452418589371346395at_nat @ ( cons_P6512896166579812791at_nat @ I @ Ps ) @ ( cons_P6512896166579812791at_nat @ J @ Qs ) )
      = ( ( I = J )
        & ( term_p5452418589371346395at_nat @ Ps @ Qs ) ) ) ).

% position_less_eq_Cons
thf(fact_325_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_326_position__diff__Cons,axiom,
    ! [I: term_a_b,Ps: list_term_a_b,Qs: list_term_a_b] :
      ( ( term_p798503758663136087rm_a_b @ ( cons_term_a_b @ I @ Ps ) @ ( cons_term_a_b @ I @ Qs ) )
      = ( term_p798503758663136087rm_a_b @ Ps @ Qs ) ) ).

% position_diff_Cons
thf(fact_327_position__diff__Cons,axiom,
    ! [I: list_nat,Ps: list_list_nat,Qs: list_list_nat] :
      ( ( term_p7564741194569991203st_nat @ ( cons_list_nat @ I @ Ps ) @ ( cons_list_nat @ I @ Qs ) )
      = ( term_p7564741194569991203st_nat @ Ps @ Qs ) ) ).

% position_diff_Cons
thf(fact_328_position__diff__Cons,axiom,
    ! [I: product_prod_a_nat,Ps: list_P3592885314253461005_a_nat,Qs: list_P3592885314253461005_a_nat] :
      ( ( term_p8115227756575868480_a_nat @ ( cons_P5205166803686508359_a_nat @ I @ Ps ) @ ( cons_P5205166803686508359_a_nat @ I @ Qs ) )
      = ( term_p8115227756575868480_a_nat @ Ps @ Qs ) ) ).

% position_diff_Cons
thf(fact_329_position__diff__Cons,axiom,
    ! [I: a,Ps: list_a,Qs: list_a] :
      ( ( term_pos_diff_a @ ( cons_a @ I @ Ps ) @ ( cons_a @ I @ Qs ) )
      = ( term_pos_diff_a @ Ps @ Qs ) ) ).

% position_diff_Cons
thf(fact_330_position__diff__Cons,axiom,
    ! [I: product_prod_nat_nat,Ps: list_P6011104703257516679at_nat,Qs: list_P6011104703257516679at_nat] :
      ( ( term_p3376976900432600702at_nat @ ( cons_P6512896166579812791at_nat @ I @ Ps ) @ ( cons_P6512896166579812791at_nat @ I @ Qs ) )
      = ( term_p3376976900432600702at_nat @ Ps @ Qs ) ) ).

% position_diff_Cons
thf(fact_331_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_332_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_333_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_334_lexord__cons__cons,axiom,
    ! [A: term_a_b,X: list_term_a_b,B: term_a_b,Y: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ A @ X ) @ ( cons_term_a_b @ B @ Y ) ) @ ( lexord_term_a_b @ R ) )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
        | ( ( A = B )
          & ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X @ Y ) @ ( lexord_term_a_b @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_335_lexord__cons__cons,axiom,
    ! [A: list_nat,X: list_list_nat,B: list_nat,Y: list_list_nat,R: set_Pr3451248702717554689st_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ A @ X ) @ ( cons_list_nat @ B @ Y ) ) @ ( lexord_list_nat @ R ) )
      = ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ R )
        | ( ( A = B )
          & ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ X @ Y ) @ ( lexord_list_nat @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_336_lexord__cons__cons,axiom,
    ! [A: product_prod_a_nat,X: list_P3592885314253461005_a_nat,B: product_prod_a_nat,Y: list_P3592885314253461005_a_nat,R: set_Pr1811044260758604347_a_nat] :
      ( ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ ( cons_P5205166803686508359_a_nat @ A @ X ) @ ( cons_P5205166803686508359_a_nat @ B @ Y ) ) @ ( lexord2902578037316800714_a_nat @ R ) )
      = ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ R )
        | ( ( A = B )
          & ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ X @ Y ) @ ( lexord2902578037316800714_a_nat @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_337_lexord__cons__cons,axiom,
    ! [A: a,X: list_a,B: a,Y: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ A @ X ) @ ( cons_a @ B @ Y ) ) @ ( lexord_a @ R ) )
      = ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
        | ( ( A = B )
          & ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( lexord_a @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_338_lexord__cons__cons,axiom,
    ! [A: product_prod_nat_nat,X: list_P6011104703257516679at_nat,B: product_prod_nat_nat,Y: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ A @ X ) @ ( cons_P6512896166579812791at_nat @ B @ Y ) ) @ ( lexord2841853652668343668at_nat @ R ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R )
        | ( ( A = B )
          & ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ X @ Y ) @ ( lexord2841853652668343668at_nat @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_339_lexord__cons__cons,axiom,
    ! [A: nat,X: list_nat,B: nat,Y: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ A @ X ) @ ( cons_nat @ B @ Y ) ) @ ( lexord_nat @ R ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
        | ( ( A = B )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lexord_nat @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_340_par__pos__prefix,axiom,
    ! [I: term_a_b,P: list_term_a_b,Q: list_term_a_b] :
      ( ( term_p7407996180858101430rm_a_b @ ( cons_term_a_b @ I @ P ) @ ( cons_term_a_b @ I @ Q ) )
     => ( term_p7407996180858101430rm_a_b @ P @ Q ) ) ).

% par_pos_prefix
thf(fact_341_par__pos__prefix,axiom,
    ! [I: list_nat,P: list_list_nat,Q: list_list_nat] :
      ( ( term_p4950861579910180738st_nat @ ( cons_list_nat @ I @ P ) @ ( cons_list_nat @ I @ Q ) )
     => ( term_p4950861579910180738st_nat @ P @ Q ) ) ).

% par_pos_prefix
thf(fact_342_par__pos__prefix,axiom,
    ! [I: product_prod_a_nat,P: list_P3592885314253461005_a_nat,Q: list_P3592885314253461005_a_nat] :
      ( ( term_p5648667246897444383_a_nat @ ( cons_P5205166803686508359_a_nat @ I @ P ) @ ( cons_P5205166803686508359_a_nat @ I @ Q ) )
     => ( term_p5648667246897444383_a_nat @ P @ Q ) ) ).

% par_pos_prefix
thf(fact_343_par__pos__prefix,axiom,
    ! [I: a,P: list_a,Q: list_a] :
      ( ( term_position_par_a @ ( cons_a @ I @ P ) @ ( cons_a @ I @ Q ) )
     => ( term_position_par_a @ P @ Q ) ) ).

% par_pos_prefix
thf(fact_344_par__pos__prefix,axiom,
    ! [I: product_prod_nat_nat,P: list_P6011104703257516679at_nat,Q: list_P6011104703257516679at_nat] :
      ( ( term_p8419326880412058847at_nat @ ( cons_P6512896166579812791at_nat @ I @ P ) @ ( cons_P6512896166579812791at_nat @ I @ Q ) )
     => ( term_p8419326880412058847at_nat @ P @ Q ) ) ).

% par_pos_prefix
thf(fact_345_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_346_Nat_Oex__has__greatest__nat,axiom,
    ! [P2: nat > $o,K: nat,B: nat] :
      ( ( P2 @ K )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ? [X3: nat] :
            ( ( P2 @ X3 )
            & ! [Y6: nat] :
                ( ( P2 @ Y6 )
               => ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_347_not__Cons__self2,axiom,
    ! [X: term_a_b,Xs2: list_term_a_b] :
      ( ( cons_term_a_b @ X @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_348_not__Cons__self2,axiom,
    ! [X: list_nat,Xs2: list_list_nat] :
      ( ( cons_list_nat @ X @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_349_not__Cons__self2,axiom,
    ! [X: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat] :
      ( ( cons_P5205166803686508359_a_nat @ X @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_350_not__Cons__self2,axiom,
    ! [X: a,Xs2: list_a] :
      ( ( cons_a @ X @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_351_not__Cons__self2,axiom,
    ! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
      ( ( cons_P6512896166579812791at_nat @ X @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_352_not__Cons__self2,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( cons_nat @ X @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_353_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
      | ( ord_less_eq_nat @ N @ M ) ) ).

% nat_le_linear
thf(fact_354_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( M = N ) ) ) ).

% le_antisym
thf(fact_355_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_356_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I @ K ) ) ) ).

% le_trans
thf(fact_357_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_358_position__less__refl,axiom,
    ! [P: list_nat] : ( term_p3503116865373065078eq_nat @ P @ P ) ).

% position_less_refl
thf(fact_359_position__par__def,axiom,
    ( term_p5017330785391824242ar_nat
    = ( ^ [P3: list_nat,Q3: list_nat] :
          ( ~ ( term_p3503116865373065078eq_nat @ P3 @ Q3 )
          & ~ ( term_p3503116865373065078eq_nat @ Q3 @ P3 ) ) ) ) ).

% position_par_def
thf(fact_360_par__Cons__iff,axiom,
    ! [I: term_a_b,Ps: list_term_a_b,J: term_a_b,Qs: list_term_a_b] :
      ( ( term_p7407996180858101430rm_a_b @ ( cons_term_a_b @ I @ Ps ) @ ( cons_term_a_b @ J @ Qs ) )
      = ( ( I != J )
        | ( term_p7407996180858101430rm_a_b @ Ps @ Qs ) ) ) ).

% par_Cons_iff
thf(fact_361_par__Cons__iff,axiom,
    ! [I: list_nat,Ps: list_list_nat,J: list_nat,Qs: list_list_nat] :
      ( ( term_p4950861579910180738st_nat @ ( cons_list_nat @ I @ Ps ) @ ( cons_list_nat @ J @ Qs ) )
      = ( ( I != J )
        | ( term_p4950861579910180738st_nat @ Ps @ Qs ) ) ) ).

% par_Cons_iff
thf(fact_362_par__Cons__iff,axiom,
    ! [I: product_prod_a_nat,Ps: list_P3592885314253461005_a_nat,J: product_prod_a_nat,Qs: list_P3592885314253461005_a_nat] :
      ( ( term_p5648667246897444383_a_nat @ ( cons_P5205166803686508359_a_nat @ I @ Ps ) @ ( cons_P5205166803686508359_a_nat @ J @ Qs ) )
      = ( ( I != J )
        | ( term_p5648667246897444383_a_nat @ Ps @ Qs ) ) ) ).

% par_Cons_iff
thf(fact_363_par__Cons__iff,axiom,
    ! [I: a,Ps: list_a,J: a,Qs: list_a] :
      ( ( term_position_par_a @ ( cons_a @ I @ Ps ) @ ( cons_a @ J @ Qs ) )
      = ( ( I != J )
        | ( term_position_par_a @ Ps @ Qs ) ) ) ).

% par_Cons_iff
thf(fact_364_par__Cons__iff,axiom,
    ! [I: product_prod_nat_nat,Ps: list_P6011104703257516679at_nat,J: product_prod_nat_nat,Qs: list_P6011104703257516679at_nat] :
      ( ( term_p8419326880412058847at_nat @ ( cons_P6512896166579812791at_nat @ I @ Ps ) @ ( cons_P6512896166579812791at_nat @ J @ Qs ) )
      = ( ( I != J )
        | ( term_p8419326880412058847at_nat @ Ps @ Qs ) ) ) ).

% par_Cons_iff
thf(fact_365_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_366_impossible__Cons,axiom,
    ! [Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,X: product_prod_a_nat] :
      ( ( ord_less_eq_nat @ ( size_s984997627204368545_a_nat @ Xs2 ) @ ( size_s984997627204368545_a_nat @ Ys ) )
     => ( Xs2
       != ( cons_P5205166803686508359_a_nat @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_367_impossible__Cons,axiom,
    ! [Xs2: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
      ( ( ord_less_eq_nat @ ( size_s5460976970255530739at_nat @ Xs2 ) @ ( size_s5460976970255530739at_nat @ Ys ) )
     => ( Xs2
       != ( cons_P6512896166579812791at_nat @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_368_impossible__Cons,axiom,
    ! [Xs2: list_list_term_a_b,Ys: list_list_term_a_b,X: list_term_a_b] :
      ( ( ord_less_eq_nat @ ( size_s877380706853472072rm_a_b @ Xs2 ) @ ( size_s877380706853472072rm_a_b @ Ys ) )
     => ( Xs2
       != ( cons_list_term_a_b @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_369_impossible__Cons,axiom,
    ! [Xs2: list_list_nat,Ys: list_list_nat,X: list_nat] :
      ( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ ( size_s3023201423986296836st_nat @ Ys ) )
     => ( Xs2
       != ( cons_list_nat @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_370_impossible__Cons,axiom,
    ! [Xs2: list_a,Ys: list_a,X: a] :
      ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ ( size_size_list_a @ Ys ) )
     => ( Xs2
       != ( cons_a @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_371_impossible__Cons,axiom,
    ! [Xs2: list_nat,Ys: list_nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) )
     => ( Xs2
       != ( cons_nat @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_372_impossible__Cons,axiom,
    ! [Xs2: list_term_a_b,Ys: list_term_a_b,X: term_a_b] :
      ( ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs2 ) @ ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( Xs2
       != ( cons_term_a_b @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_373_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_374_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_375_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_376_add__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_377_add__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).

% add_left_mono
thf(fact_378_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C: nat] :
            ( B
           != ( plus_plus_nat @ A @ C ) ) ) ).

% less_eqE
thf(fact_379_add__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% add_right_mono
thf(fact_380_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
        ? [C3: nat] :
          ( B4
          = ( plus_plus_nat @ A4 @ C3 ) ) ) ) ).

% le_iff_add
thf(fact_381_add__le__imp__le__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_382_add__le__imp__le__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_383_Cons__eq__appendI,axiom,
    ! [X: produc1234881154892807749rm_a_b,Xs1: list_P2364656488115551307rm_a_b,Ys: list_P2364656488115551307rm_a_b,Xs2: list_P2364656488115551307rm_a_b,Zs: list_P2364656488115551307rm_a_b] :
      ( ( ( cons_P2836617062085252091rm_a_b @ X @ Xs1 )
        = Ys )
     => ( ( Xs2
          = ( append5987703870611264992rm_a_b @ Xs1 @ Zs ) )
       => ( ( cons_P2836617062085252091rm_a_b @ X @ Xs2 )
          = ( append5987703870611264992rm_a_b @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_384_Cons__eq__appendI,axiom,
    ! [X: term_a_b,Xs1: list_term_a_b,Ys: list_term_a_b,Xs2: list_term_a_b,Zs: list_term_a_b] :
      ( ( ( cons_term_a_b @ X @ Xs1 )
        = Ys )
     => ( ( Xs2
          = ( append_term_a_b @ Xs1 @ Zs ) )
       => ( ( cons_term_a_b @ X @ Xs2 )
          = ( append_term_a_b @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_385_Cons__eq__appendI,axiom,
    ! [X: list_nat,Xs1: list_list_nat,Ys: list_list_nat,Xs2: list_list_nat,Zs: list_list_nat] :
      ( ( ( cons_list_nat @ X @ Xs1 )
        = Ys )
     => ( ( Xs2
          = ( append_list_nat @ Xs1 @ Zs ) )
       => ( ( cons_list_nat @ X @ Xs2 )
          = ( append_list_nat @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_386_Cons__eq__appendI,axiom,
    ! [X: product_prod_a_nat,Xs1: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,Xs2: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat] :
      ( ( ( cons_P5205166803686508359_a_nat @ X @ Xs1 )
        = Ys )
     => ( ( Xs2
          = ( append7679239579558125090_a_nat @ Xs1 @ Zs ) )
       => ( ( cons_P5205166803686508359_a_nat @ X @ Xs2 )
          = ( append7679239579558125090_a_nat @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_387_Cons__eq__appendI,axiom,
    ! [X: a,Xs1: list_a,Ys: list_a,Xs2: list_a,Zs: list_a] :
      ( ( ( cons_a @ X @ Xs1 )
        = Ys )
     => ( ( Xs2
          = ( append_a @ Xs1 @ Zs ) )
       => ( ( cons_a @ X @ Xs2 )
          = ( append_a @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_388_Cons__eq__appendI,axiom,
    ! [X: product_prod_nat_nat,Xs1: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Xs2: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
      ( ( ( cons_P6512896166579812791at_nat @ X @ Xs1 )
        = Ys )
     => ( ( Xs2
          = ( append985823374593552924at_nat @ Xs1 @ Zs ) )
       => ( ( cons_P6512896166579812791at_nat @ X @ Xs2 )
          = ( append985823374593552924at_nat @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_389_Cons__eq__appendI,axiom,
    ! [X: nat,Xs1: list_nat,Ys: list_nat,Xs2: list_nat,Zs: list_nat] :
      ( ( ( cons_nat @ X @ Xs1 )
        = Ys )
     => ( ( Xs2
          = ( append_nat @ Xs1 @ Zs ) )
       => ( ( cons_nat @ X @ Xs2 )
          = ( append_nat @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_390_append__Cons,axiom,
    ! [X: produc1234881154892807749rm_a_b,Xs2: list_P2364656488115551307rm_a_b,Ys: list_P2364656488115551307rm_a_b] :
      ( ( append5987703870611264992rm_a_b @ ( cons_P2836617062085252091rm_a_b @ X @ Xs2 ) @ Ys )
      = ( cons_P2836617062085252091rm_a_b @ X @ ( append5987703870611264992rm_a_b @ Xs2 @ Ys ) ) ) ).

% append_Cons
thf(fact_391_append__Cons,axiom,
    ! [X: term_a_b,Xs2: list_term_a_b,Ys: list_term_a_b] :
      ( ( append_term_a_b @ ( cons_term_a_b @ X @ Xs2 ) @ Ys )
      = ( cons_term_a_b @ X @ ( append_term_a_b @ Xs2 @ Ys ) ) ) ).

% append_Cons
thf(fact_392_append__Cons,axiom,
    ! [X: list_nat,Xs2: list_list_nat,Ys: list_list_nat] :
      ( ( append_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ Ys )
      = ( cons_list_nat @ X @ ( append_list_nat @ Xs2 @ Ys ) ) ) ).

% append_Cons
thf(fact_393_append__Cons,axiom,
    ! [X: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
      ( ( append7679239579558125090_a_nat @ ( cons_P5205166803686508359_a_nat @ X @ Xs2 ) @ Ys )
      = ( cons_P5205166803686508359_a_nat @ X @ ( append7679239579558125090_a_nat @ Xs2 @ Ys ) ) ) ).

% append_Cons
thf(fact_394_append__Cons,axiom,
    ! [X: a,Xs2: list_a,Ys: list_a] :
      ( ( append_a @ ( cons_a @ X @ Xs2 ) @ Ys )
      = ( cons_a @ X @ ( append_a @ Xs2 @ Ys ) ) ) ).

% append_Cons
thf(fact_395_append__Cons,axiom,
    ! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( append985823374593552924at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ Ys )
      = ( cons_P6512896166579812791at_nat @ X @ ( append985823374593552924at_nat @ Xs2 @ Ys ) ) ) ).

% append_Cons
thf(fact_396_append__Cons,axiom,
    ! [X: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( append_nat @ ( cons_nat @ X @ Xs2 ) @ Ys )
      = ( cons_nat @ X @ ( append_nat @ Xs2 @ Ys ) ) ) ).

% append_Cons
thf(fact_397_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M2: nat,N2: nat] :
        ? [K2: nat] :
          ( N2
          = ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).

% nat_le_iff_add
thf(fact_398_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_le_add2
thf(fact_399_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_le_add1
thf(fact_400_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_le_mono1
thf(fact_401_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_402_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N3: nat] :
          ( L
          = ( plus_plus_nat @ K @ N3 ) ) ) ).

% le_Suc_ex
thf(fact_403_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ K @ N ) ) ).

% add_leD2
thf(fact_404_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% add_leD1
thf(fact_405_le__add2,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).

% le_add2
thf(fact_406_le__add1,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).

% le_add1
thf(fact_407_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ~ ( ( ord_less_eq_nat @ M @ N )
         => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).

% add_leE
thf(fact_408_position__less__eq__def,axiom,
    ( term_p6496282196689937951rm_a_b
    = ( ^ [P3: list_P2364656488115551307rm_a_b,Q3: list_P2364656488115551307rm_a_b] :
        ? [R3: list_P2364656488115551307rm_a_b] :
          ( ( append5987703870611264992rm_a_b @ P3 @ R3 )
          = Q3 ) ) ) ).

% position_less_eq_def
thf(fact_409_position__less__eq__def,axiom,
    ( term_p4408110555530040355_a_nat
    = ( ^ [P3: list_P3592885314253461005_a_nat,Q3: list_P3592885314253461005_a_nat] :
        ? [R3: list_P3592885314253461005_a_nat] :
          ( ( append7679239579558125090_a_nat @ P3 @ R3 )
          = Q3 ) ) ) ).

% position_less_eq_def
thf(fact_410_position__less__eq__def,axiom,
    ( term_p8391561492822560442rm_a_b
    = ( ^ [P3: list_term_a_b,Q3: list_term_a_b] :
        ? [R3: list_term_a_b] :
          ( ( append_term_a_b @ P3 @ R3 )
          = Q3 ) ) ) ).

% position_less_eq_def
thf(fact_411_position__less__eq__def,axiom,
    ( term_p5934426891874639750st_nat
    = ( ^ [P3: list_list_nat,Q3: list_list_nat] :
        ? [R3: list_list_nat] :
          ( ( append_list_nat @ P3 @ R3 )
          = Q3 ) ) ) ).

% position_less_eq_def
thf(fact_412_position__less__eq__def,axiom,
    ( term_p3503116865373065078eq_nat
    = ( ^ [P3: list_nat,Q3: list_nat] :
        ? [R3: list_nat] :
          ( ( append_nat @ P3 @ R3 )
          = Q3 ) ) ) ).

% position_less_eq_def
thf(fact_413_position__less__eq__induct,axiom,
    ! [P: list_P2364656488115551307rm_a_b,Q: list_P2364656488115551307rm_a_b,P2: list_P2364656488115551307rm_a_b > list_P2364656488115551307rm_a_b > $o] :
      ( ( term_p6496282196689937951rm_a_b @ P @ Q )
     => ( ! [P4: list_P2364656488115551307rm_a_b] : ( P2 @ P4 @ P4 )
       => ( ! [P4: list_P2364656488115551307rm_a_b,Q4: list_P2364656488115551307rm_a_b,R4: list_P2364656488115551307rm_a_b] :
              ( ( term_p6496282196689937951rm_a_b @ P4 @ Q4 )
             => ( ( P2 @ P4 @ Q4 )
               => ( P2 @ P4 @ ( append5987703870611264992rm_a_b @ Q4 @ R4 ) ) ) )
         => ( P2 @ P @ Q ) ) ) ) ).

% position_less_eq_induct
thf(fact_414_position__less__eq__induct,axiom,
    ! [P: list_P3592885314253461005_a_nat,Q: list_P3592885314253461005_a_nat,P2: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > $o] :
      ( ( term_p4408110555530040355_a_nat @ P @ Q )
     => ( ! [P4: list_P3592885314253461005_a_nat] : ( P2 @ P4 @ P4 )
       => ( ! [P4: list_P3592885314253461005_a_nat,Q4: list_P3592885314253461005_a_nat,R4: list_P3592885314253461005_a_nat] :
              ( ( term_p4408110555530040355_a_nat @ P4 @ Q4 )
             => ( ( P2 @ P4 @ Q4 )
               => ( P2 @ P4 @ ( append7679239579558125090_a_nat @ Q4 @ R4 ) ) ) )
         => ( P2 @ P @ Q ) ) ) ) ).

% position_less_eq_induct
thf(fact_415_position__less__eq__induct,axiom,
    ! [P: list_term_a_b,Q: list_term_a_b,P2: list_term_a_b > list_term_a_b > $o] :
      ( ( term_p8391561492822560442rm_a_b @ P @ Q )
     => ( ! [P4: list_term_a_b] : ( P2 @ P4 @ P4 )
       => ( ! [P4: list_term_a_b,Q4: list_term_a_b,R4: list_term_a_b] :
              ( ( term_p8391561492822560442rm_a_b @ P4 @ Q4 )
             => ( ( P2 @ P4 @ Q4 )
               => ( P2 @ P4 @ ( append_term_a_b @ Q4 @ R4 ) ) ) )
         => ( P2 @ P @ Q ) ) ) ) ).

% position_less_eq_induct
thf(fact_416_position__less__eq__induct,axiom,
    ! [P: list_list_nat,Q: list_list_nat,P2: list_list_nat > list_list_nat > $o] :
      ( ( term_p5934426891874639750st_nat @ P @ Q )
     => ( ! [P4: list_list_nat] : ( P2 @ P4 @ P4 )
       => ( ! [P4: list_list_nat,Q4: list_list_nat,R4: list_list_nat] :
              ( ( term_p5934426891874639750st_nat @ P4 @ Q4 )
             => ( ( P2 @ P4 @ Q4 )
               => ( P2 @ P4 @ ( append_list_nat @ Q4 @ R4 ) ) ) )
         => ( P2 @ P @ Q ) ) ) ) ).

% position_less_eq_induct
thf(fact_417_position__less__eq__induct,axiom,
    ! [P: list_nat,Q: list_nat,P2: list_nat > list_nat > $o] :
      ( ( term_p3503116865373065078eq_nat @ P @ Q )
     => ( ! [P4: list_nat] : ( P2 @ P4 @ P4 )
       => ( ! [P4: list_nat,Q4: list_nat,R4: list_nat] :
              ( ( term_p3503116865373065078eq_nat @ P4 @ Q4 )
             => ( ( P2 @ P4 @ Q4 )
               => ( P2 @ P4 @ ( append_nat @ Q4 @ R4 ) ) ) )
         => ( P2 @ P @ Q ) ) ) ) ).

% position_less_eq_induct
thf(fact_418_less__eq__poss__append__itself,axiom,
    ! [P: list_P2364656488115551307rm_a_b,Q: list_P2364656488115551307rm_a_b] : ( term_p6496282196689937951rm_a_b @ P @ ( append5987703870611264992rm_a_b @ P @ Q ) ) ).

% less_eq_poss_append_itself
thf(fact_419_less__eq__poss__append__itself,axiom,
    ! [P: list_P3592885314253461005_a_nat,Q: list_P3592885314253461005_a_nat] : ( term_p4408110555530040355_a_nat @ P @ ( append7679239579558125090_a_nat @ P @ Q ) ) ).

% less_eq_poss_append_itself
thf(fact_420_less__eq__poss__append__itself,axiom,
    ! [P: list_term_a_b,Q: list_term_a_b] : ( term_p8391561492822560442rm_a_b @ P @ ( append_term_a_b @ P @ Q ) ) ).

% less_eq_poss_append_itself
thf(fact_421_less__eq__poss__append__itself,axiom,
    ! [P: list_list_nat,Q: list_list_nat] : ( term_p5934426891874639750st_nat @ P @ ( append_list_nat @ P @ Q ) ) ).

% less_eq_poss_append_itself
thf(fact_422_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_423_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_424_supteq__size,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
     => ( ord_less_eq_nat @ ( size_size_term_a_b @ T ) @ ( size_size_term_a_b @ S ) ) ) ).

% supteq_size
thf(fact_425_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_426_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_427_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_nat @ I @ K )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_428_group__cancel_Oadd1,axiom,
    ! [A5: nat,K: nat,A: nat,B: nat] :
      ( ( A5
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A5 @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_429_group__cancel_Oadd2,axiom,
    ! [B5: nat,K: nat,B: nat,A: nat] :
      ( ( B5
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B5 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_430_add_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% add.assoc
thf(fact_431_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).

% add.commute
thf(fact_432_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% add.left_commute
thf(fact_433_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C2 ) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
thf(fact_434_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C2 @ A ) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
thf(fact_435_lexord__append__rightI,axiom,
    ! [Y: list_P2364656488115551307rm_a_b,X: list_P2364656488115551307rm_a_b,R: set_Pr5038301440468608839rm_a_b] :
      ( ? [B6: produc1234881154892807749rm_a_b,Z3: list_P2364656488115551307rm_a_b] :
          ( Y
          = ( cons_P2836617062085252091rm_a_b @ B6 @ Z3 ) )
     => ( member5494928282894704656rm_a_b @ ( produc8811016564285052503rm_a_b @ X @ ( append5987703870611264992rm_a_b @ X @ Y ) ) @ ( lexord7943539855291252024rm_a_b @ R ) ) ) ).

% lexord_append_rightI
thf(fact_436_lexord__append__rightI,axiom,
    ! [Y: list_term_a_b,X: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ? [B6: term_a_b,Z3: list_term_a_b] :
          ( Y
          = ( cons_term_a_b @ B6 @ Z3 ) )
     => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X @ ( append_term_a_b @ X @ Y ) ) @ ( lexord_term_a_b @ R ) ) ) ).

% lexord_append_rightI
thf(fact_437_lexord__append__rightI,axiom,
    ! [Y: list_list_nat,X: list_list_nat,R: set_Pr3451248702717554689st_nat] :
      ( ? [B6: list_nat,Z3: list_list_nat] :
          ( Y
          = ( cons_list_nat @ B6 @ Z3 ) )
     => ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ X @ ( append_list_nat @ X @ Y ) ) @ ( lexord_list_nat @ R ) ) ) ).

% lexord_append_rightI
thf(fact_438_lexord__append__rightI,axiom,
    ! [Y: list_P3592885314253461005_a_nat,X: list_P3592885314253461005_a_nat,R: set_Pr1811044260758604347_a_nat] :
      ( ? [B6: product_prod_a_nat,Z3: list_P3592885314253461005_a_nat] :
          ( Y
          = ( cons_P5205166803686508359_a_nat @ B6 @ Z3 ) )
     => ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ X @ ( append7679239579558125090_a_nat @ X @ Y ) ) @ ( lexord2902578037316800714_a_nat @ R ) ) ) ).

% lexord_append_rightI
thf(fact_439_lexord__append__rightI,axiom,
    ! [Y: list_a,X: list_a,R: set_Product_prod_a_a] :
      ( ? [B6: a,Z3: list_a] :
          ( Y
          = ( cons_a @ B6 @ Z3 ) )
     => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ ( append_a @ X @ Y ) ) @ ( lexord_a @ R ) ) ) ).

% lexord_append_rightI
thf(fact_440_lexord__append__rightI,axiom,
    ! [Y: list_P6011104703257516679at_nat,X: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ? [B6: product_prod_nat_nat,Z3: list_P6011104703257516679at_nat] :
          ( Y
          = ( cons_P6512896166579812791at_nat @ B6 @ Z3 ) )
     => ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ X @ ( append985823374593552924at_nat @ X @ Y ) ) @ ( lexord2841853652668343668at_nat @ R ) ) ) ).

% lexord_append_rightI
thf(fact_441_lexord__append__rightI,axiom,
    ! [Y: list_nat,X: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ? [B6: nat,Z3: list_nat] :
          ( Y
          = ( cons_nat @ B6 @ Z3 ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ ( append_nat @ X @ Y ) ) @ ( lexord_nat @ R ) ) ) ).

% lexord_append_rightI
thf(fact_442_lenlex__length,axiom,
    ! [Ms: list_list_term_a_b,Ns: list_list_term_a_b,R: set_Pr8564414093027780873rm_a_b] :
      ( ( member8612744848686930930rm_a_b @ ( produc7557079580777928833rm_a_b @ Ms @ Ns ) @ ( lenlex_list_term_a_b @ R ) )
     => ( ord_less_eq_nat @ ( size_s877380706853472072rm_a_b @ Ms ) @ ( size_s877380706853472072rm_a_b @ Ns ) ) ) ).

% lenlex_length
thf(fact_443_lenlex__length,axiom,
    ! [Ms: list_list_nat,Ns: list_list_nat,R: set_Pr3451248702717554689st_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Ms @ Ns ) @ ( lenlex_list_nat @ R ) )
     => ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Ms ) @ ( size_s3023201423986296836st_nat @ Ns ) ) ) ).

% lenlex_length
thf(fact_444_lenlex__length,axiom,
    ! [Ms: list_nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R ) )
     => ( ord_less_eq_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) ) ) ).

% lenlex_length
thf(fact_445_lenlex__length,axiom,
    ! [Ms: list_a,Ns: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ms @ Ns ) @ ( lenlex_a @ R ) )
     => ( ord_less_eq_nat @ ( size_size_list_a @ Ms ) @ ( size_size_list_a @ Ns ) ) ) ).

% lenlex_length
thf(fact_446_lenlex__length,axiom,
    ! [Ms: list_term_a_b,Ns: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ms @ Ns ) @ ( lenlex_term_a_b @ R ) )
     => ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Ms ) @ ( size_s8906293707977694520rm_a_b @ Ns ) ) ) ).

% lenlex_length
thf(fact_447_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_448_order__refl,axiom,
    ! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).

% order_refl
thf(fact_449_order__refl,axiom,
    ! [X: set_list_nat] : ( ord_le6045566169113846134st_nat @ X @ X ) ).

% order_refl
thf(fact_450_order__refl,axiom,
    ! [X: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X @ X ) ).

% order_refl
thf(fact_451_order__refl,axiom,
    ! [X: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ X @ X ) ).

% order_refl
thf(fact_452_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_453_dual__order_Orefl,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% dual_order.refl
thf(fact_454_dual__order_Orefl,axiom,
    ! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ A @ A ) ).

% dual_order.refl
thf(fact_455_dual__order_Orefl,axiom,
    ! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ A ) ).

% dual_order.refl
thf(fact_456_dual__order_Orefl,axiom,
    ! [A: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ A @ A ) ).

% dual_order.refl
thf(fact_457_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_458_bind__simps_I2_J,axiom,
    ! [X: a,Xs2: list_a,F2: a > list_list_nat] :
      ( ( bind_a_list_nat @ ( cons_a @ X @ Xs2 ) @ F2 )
      = ( append_list_nat @ ( F2 @ X ) @ ( bind_a_list_nat @ Xs2 @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_459_bind__simps_I2_J,axiom,
    ! [X: a,Xs2: list_a,F2: a > list_nat] :
      ( ( bind_a_nat @ ( cons_a @ X @ Xs2 ) @ F2 )
      = ( append_nat @ ( F2 @ X ) @ ( bind_a_nat @ Xs2 @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_460_bind__simps_I2_J,axiom,
    ! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,F2: product_prod_nat_nat > list_P2364656488115551307rm_a_b] :
      ( ( bind_P7929151312109640459rm_a_b @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ F2 )
      = ( append5987703870611264992rm_a_b @ ( F2 @ X ) @ ( bind_P7929151312109640459rm_a_b @ Xs2 @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_461_bind__simps_I2_J,axiom,
    ! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,F2: product_prod_nat_nat > list_P3592885314253461005_a_nat] :
      ( ( bind_P5100334560823729207_a_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ F2 )
      = ( append7679239579558125090_a_nat @ ( F2 @ X ) @ ( bind_P5100334560823729207_a_nat @ Xs2 @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_462_bind__simps_I2_J,axiom,
    ! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,F2: product_prod_nat_nat > list_term_a_b] :
      ( ( bind_P1981734077250826446rm_a_b @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ F2 )
      = ( append_term_a_b @ ( F2 @ X ) @ ( bind_P1981734077250826446rm_a_b @ Xs2 @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_463_bind__simps_I2_J,axiom,
    ! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,F2: product_prod_nat_nat > list_list_nat] :
      ( ( bind_P8747971513157681562st_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ F2 )
      = ( append_list_nat @ ( F2 @ X ) @ ( bind_P8747971513157681562st_nat @ Xs2 @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_464_bind__simps_I2_J,axiom,
    ! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,F2: product_prod_nat_nat > list_nat] :
      ( ( bind_P7742074774332787594at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ F2 )
      = ( append_nat @ ( F2 @ X ) @ ( bind_P7742074774332787594at_nat @ Xs2 @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_465_Cons__lenlex__iff,axiom,
    ! [M: nat,Ms: list_nat,N: nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R ) )
      = ( ( 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 ) @ R ) )
        | ( ( M = N )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_466_Cons__lenlex__iff,axiom,
    ! [M: term_a_b,Ms: list_term_a_b,N: term_a_b,Ns: list_term_a_b,R: 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 @ R ) )
      = ( ( 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 ) @ R ) )
        | ( ( M = N )
          & ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ms @ Ns ) @ ( lenlex_term_a_b @ R ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_467_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 ) )
       => ( ! [Q4: list_nat] :
              ( ~ ( term_p3503116865373065078eq_nat @ P @ Q4 )
             => ( ( member_list_nat @ Q4 @ ( term_poss_a_b @ S ) )
                = ( member_list_nat @ Q4 @ ( term_poss_a_b @ T ) ) ) )
         => ( ! [Q4: list_nat] :
                ( ( member_list_nat @ Q4 @ ( term_poss_a_b @ S ) )
               => ( ~ ( term_p3503116865373065078eq_nat @ P @ Q4 )
                 => ( ( term_fun_at_a_b @ S @ Q4 )
                    = ( term_fun_at_a_b @ T @ Q4 ) ) ) )
           => ( ( term_ctxt_at_pos_a_b @ S @ P )
              = ( term_ctxt_at_pos_a_b @ T @ P ) ) ) ) ) ) ).

% eq_ctxt_at_pos_by_poss
thf(fact_468_listrel1I,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs2: list_nat,Us: list_nat,Vs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
     => ( ( Xs2
          = ( append_nat @ Us @ ( cons_nat @ X @ Vs ) ) )
       => ( ( Ys
            = ( append_nat @ Us @ ( cons_nat @ Y @ Vs ) ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% listrel1I
thf(fact_469_listrel1E,axiom,
    ! [Xs2: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( listrel1_nat @ R ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
           => ! [Us3: list_nat,Vs2: list_nat] :
                ( ( Xs2
                  = ( append_nat @ Us3 @ ( cons_nat @ X3 @ Vs2 ) ) )
               => ( Ys
                 != ( append_nat @ Us3 @ ( cons_nat @ Y3 @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_470_lexord__Nil__left,axiom,
    ! [Y: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Y ) @ ( lexord_nat @ R ) )
      = ( ? [A4: nat,X4: list_nat] :
            ( Y
            = ( cons_nat @ A4 @ X4 ) ) ) ) ).

% lexord_Nil_left
thf(fact_471_add__less__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_472_add__less__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_473_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_474_append_Oright__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ A @ nil_nat )
      = A ) ).

% append.right_neutral
thf(fact_475_append__Nil2,axiom,
    ! [Xs2: list_nat] :
      ( ( append_nat @ Xs2 @ nil_nat )
      = Xs2 ) ).

% append_Nil2
thf(fact_476_append__self__conv,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs2 @ Ys )
        = Xs2 )
      = ( Ys = nil_nat ) ) ).

% append_self_conv
thf(fact_477_self__append__conv,axiom,
    ! [Y: list_nat,Ys: list_nat] :
      ( ( Y
        = ( append_nat @ Y @ Ys ) )
      = ( Ys = nil_nat ) ) ).

% self_append_conv
thf(fact_478_append__self__conv2,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs2 @ Ys )
        = Ys )
      = ( Xs2 = nil_nat ) ) ).

% append_self_conv2
thf(fact_479_self__append__conv2,axiom,
    ! [Y: list_nat,Xs2: list_nat] :
      ( ( Y
        = ( append_nat @ Xs2 @ Y ) )
      = ( Xs2 = nil_nat ) ) ).

% self_append_conv2
thf(fact_480_Nil__is__append__conv,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( nil_nat
        = ( append_nat @ Xs2 @ Ys ) )
      = ( ( Xs2 = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_481_append__is__Nil__conv,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs2 @ Ys )
        = nil_nat )
      = ( ( Xs2 = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_482_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_483_enumerate__simps_I1_J,axiom,
    ! [N: nat] :
      ( ( enumerate_nat @ N @ nil_nat )
      = nil_Pr5478986624290739719at_nat ) ).

% enumerate_simps(1)
thf(fact_484_position__diff__Nil,axiom,
    ! [Q: list_nat] :
      ( ( term_pos_diff_nat @ Q @ nil_nat )
      = Q ) ).

% position_diff_Nil
thf(fact_485_pos__diff__itself,axiom,
    ! [P: list_nat] :
      ( ( term_pos_diff_nat @ P @ P )
      = nil_nat ) ).

% pos_diff_itself
thf(fact_486_Nil__not__par_I2_J,axiom,
    ! [P: list_nat] :
      ~ ( term_p5017330785391824242ar_nat @ P @ nil_nat ) ).

% Nil_not_par(2)
thf(fact_487_Nil__not__par_I1_J,axiom,
    ! [P: list_nat] :
      ~ ( term_p5017330785391824242ar_nat @ nil_nat @ P ) ).

% Nil_not_par(1)
thf(fact_488_bind__simps_I1_J,axiom,
    ! [F2: nat > list_nat] :
      ( ( bind_nat_nat @ nil_nat @ F2 )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_489_append1__eq__conv,axiom,
    ! [Xs2: list_nat,X: nat,Ys: list_nat,Y: nat] :
      ( ( ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) )
        = ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
      = ( ( Xs2 = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
thf(fact_490_Nil__lenlex__iff1,axiom,
    ! [Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ns ) @ ( lenlex_nat @ R ) )
      = ( Ns != nil_nat ) ) ).

% Nil_lenlex_iff1
thf(fact_491_Cons__listrel1__Cons,axiom,
    ! [X: nat,Xs2: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
          & ( Xs2 = Ys ) )
        | ( ( X = Y )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_492_set__subset__Cons,axiom,
    ! [Xs2: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ ( cons_nat @ X @ Xs2 ) ) ) ).

% set_subset_Cons
thf(fact_493_transpose_Ocases,axiom,
    ! [X: list_list_nat] :
      ( ( X != nil_list_nat )
     => ( ! [Xss: list_list_nat] :
            ( X
           != ( cons_list_nat @ nil_nat @ Xss ) )
       => ~ ! [X3: nat,Xs: list_nat,Xss: list_list_nat] :
              ( X
             != ( cons_list_nat @ ( cons_nat @ X3 @ Xs ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_494_subset__code_I1_J,axiom,
    ! [Xs2: list_P3592885314253461005_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ ( set_Pr924983374503034536_a_nat @ Xs2 ) @ B5 )
      = ( ! [X4: product_prod_a_nat] :
            ( ( member5724188588386418708_a_nat @ X4 @ ( set_Pr924983374503034536_a_nat @ Xs2 ) )
           => ( member5724188588386418708_a_nat @ X4 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_495_subset__code_I1_J,axiom,
    ! [Xs2: list_list_nat,B5: set_list_nat] :
      ( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ B5 )
      = ( ! [X4: list_nat] :
            ( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs2 ) )
           => ( member_list_nat @ X4 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_496_maps__simps_I2_J,axiom,
    ! [F2: nat > list_nat] :
      ( ( maps_nat_nat @ F2 @ nil_nat )
      = nil_nat ) ).

% maps_simps(2)
thf(fact_497_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
      = ( ( ord_less_nat @ M @ N )
        | ( ord_less_nat @ N @ M ) ) ) ).

% nat_neq_iff
thf(fact_498_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_499_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_500_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_501_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_502_nat__less__induct,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N3 )
             => ( P2 @ M3 ) )
         => ( P2 @ N3 ) )
     => ( P2 @ N ) ) ).

% nat_less_induct
thf(fact_503_infinite__descent,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ~ ( P2 @ N3 )
         => ? [M3: nat] :
              ( ( ord_less_nat @ M3 @ N3 )
              & ~ ( P2 @ M3 ) ) )
     => ( P2 @ N ) ) ).

% infinite_descent
thf(fact_504_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE_nat
thf(fact_505_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_506_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_507_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_508_order__less__le__subst1,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_509_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_510_order__le__less__subst1,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_511_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_512_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_513_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_514_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_515_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_516_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_517_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_518_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
          & ( X4 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_519_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_nat @ X4 @ Y5 )
          | ( X4 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_520_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_521_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_522_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_523_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_524_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_525_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_526_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_nat @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_527_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_528_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_529_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_530_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_531_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_nat @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_532_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X )
     => ( ord_less_nat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_533_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
          & ~ ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_534_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_535_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_536_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_537_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_538_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_539_not__Nil__listrel1,axiom,
    ! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs2 ) @ ( listrel1_nat @ R ) ) ).

% not_Nil_listrel1
thf(fact_540_not__listrel1__Nil,axiom,
    ! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) @ ( listrel1_nat @ R ) ) ).

% not_listrel1_Nil
thf(fact_541_list_Oset__intros_I2_J,axiom,
    ! [Y: product_prod_a_nat,X22: list_P3592885314253461005_a_nat,X21: product_prod_a_nat] :
      ( ( member5724188588386418708_a_nat @ Y @ ( set_Pr924983374503034536_a_nat @ X22 ) )
     => ( member5724188588386418708_a_nat @ Y @ ( set_Pr924983374503034536_a_nat @ ( cons_P5205166803686508359_a_nat @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_542_list_Oset__intros_I2_J,axiom,
    ! [Y: list_nat,X22: list_list_nat,X21: list_nat] :
      ( ( member_list_nat @ Y @ ( set_list_nat2 @ X22 ) )
     => ( member_list_nat @ Y @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_543_list_Oset__intros_I2_J,axiom,
    ! [Y: nat,X22: list_nat,X21: nat] :
      ( ( member_nat @ Y @ ( set_nat2 @ X22 ) )
     => ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_544_list_Oset__intros_I1_J,axiom,
    ! [X21: product_prod_a_nat,X22: list_P3592885314253461005_a_nat] : ( member5724188588386418708_a_nat @ X21 @ ( set_Pr924983374503034536_a_nat @ ( cons_P5205166803686508359_a_nat @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_545_list_Oset__intros_I1_J,axiom,
    ! [X21: list_nat,X22: list_list_nat] : ( member_list_nat @ X21 @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_546_list_Oset__intros_I1_J,axiom,
    ! [X21: nat,X22: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_547_list_Oset__cases,axiom,
    ! [E: product_prod_a_nat,A: list_P3592885314253461005_a_nat] :
      ( ( member5724188588386418708_a_nat @ E @ ( set_Pr924983374503034536_a_nat @ A ) )
     => ( ! [Z22: list_P3592885314253461005_a_nat] :
            ( A
           != ( cons_P5205166803686508359_a_nat @ E @ Z22 ) )
       => ~ ! [Z1: product_prod_a_nat,Z22: list_P3592885314253461005_a_nat] :
              ( ( A
                = ( cons_P5205166803686508359_a_nat @ Z1 @ Z22 ) )
             => ~ ( member5724188588386418708_a_nat @ E @ ( set_Pr924983374503034536_a_nat @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_548_list_Oset__cases,axiom,
    ! [E: list_nat,A: list_list_nat] :
      ( ( member_list_nat @ E @ ( set_list_nat2 @ A ) )
     => ( ! [Z22: list_list_nat] :
            ( A
           != ( cons_list_nat @ E @ Z22 ) )
       => ~ ! [Z1: list_nat,Z22: list_list_nat] :
              ( ( A
                = ( cons_list_nat @ Z1 @ Z22 ) )
             => ~ ( member_list_nat @ E @ ( set_list_nat2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_549_list_Oset__cases,axiom,
    ! [E: nat,A: list_nat] :
      ( ( member_nat @ E @ ( set_nat2 @ A ) )
     => ( ! [Z22: list_nat] :
            ( A
           != ( cons_nat @ E @ Z22 ) )
       => ~ ! [Z1: nat,Z22: list_nat] :
              ( ( A
                = ( cons_nat @ Z1 @ Z22 ) )
             => ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_550_set__ConsD,axiom,
    ! [Y: product_prod_a_nat,X: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat] :
      ( ( member5724188588386418708_a_nat @ Y @ ( set_Pr924983374503034536_a_nat @ ( cons_P5205166803686508359_a_nat @ X @ Xs2 ) ) )
     => ( ( Y = X )
        | ( member5724188588386418708_a_nat @ Y @ ( set_Pr924983374503034536_a_nat @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_551_set__ConsD,axiom,
    ! [Y: list_nat,X: list_nat,Xs2: list_list_nat] :
      ( ( member_list_nat @ Y @ ( set_list_nat2 @ ( cons_list_nat @ X @ Xs2 ) ) )
     => ( ( Y = X )
        | ( member_list_nat @ Y @ ( set_list_nat2 @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_552_set__ConsD,axiom,
    ! [Y: nat,X: nat,Xs2: list_nat] :
      ( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs2 ) ) )
     => ( ( Y = X )
        | ( member_nat @ Y @ ( set_nat2 @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_553_add__less__imp__less__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_554_add__less__imp__less__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_555_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_556_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).

% add_strict_left_mono
thf(fact_557_add__strict__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_558_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_559_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_560_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_561_length__induct,axiom,
    ! [P2: list_term_a_b > $o,Xs2: list_term_a_b] :
      ( ! [Xs: list_term_a_b] :
          ( ! [Ys2: list_term_a_b] :
              ( ( ord_less_nat @ ( size_s8906293707977694520rm_a_b @ Ys2 ) @ ( size_s8906293707977694520rm_a_b @ Xs ) )
             => ( P2 @ Ys2 ) )
         => ( P2 @ Xs ) )
     => ( P2 @ Xs2 ) ) ).

% length_induct
thf(fact_562_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M2: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M2 @ N2 )
          & ( M2 != N2 ) ) ) ) ).

% nat_less_le
thf(fact_563_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_imp_le_nat
thf(fact_564_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M2: nat,N2: nat] :
          ( ( ord_less_nat @ M2 @ N2 )
          | ( M2 = N2 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_565_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_nat @ M @ N )
        | ( M = N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_or_eq_imp_le
thf(fact_566_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( M != N )
       => ( ord_less_nat @ M @ N ) ) ) ).

% le_neq_implies_less
thf(fact_567_less__mono__imp__le__mono,axiom,
    ! [F2: nat > nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_nat @ I2 @ J2 )
         => ( ord_less_nat @ ( F2 @ I2 ) @ ( F2 @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_568_list_Odistinct_I1_J,axiom,
    ! [X21: nat,X22: list_nat] :
      ( nil_nat
     != ( cons_nat @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_569_list_OdiscI,axiom,
    ! [List: list_nat,X21: nat,X22: list_nat] :
      ( ( List
        = ( cons_nat @ X21 @ X22 ) )
     => ( List != nil_nat ) ) ).

% list.discI
thf(fact_570_list_Oexhaust,axiom,
    ! [Y: list_nat] :
      ( ( Y != nil_nat )
     => ~ ! [X212: nat,X222: list_nat] :
            ( Y
           != ( cons_nat @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_571_List_Omin__list_Ocases,axiom,
    ! [X: list_nat] :
      ( ! [X3: nat,Xs: list_nat] :
          ( X
         != ( cons_nat @ X3 @ Xs ) )
     => ( X = nil_nat ) ) ).

% List.min_list.cases
thf(fact_572_remdups__adj_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ( ! [X3: nat] :
            ( X
           != ( cons_nat @ X3 @ nil_nat ) )
       => ~ ! [X3: nat,Y3: nat,Xs: list_nat] :
              ( X
             != ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_573_neq__Nil__conv,axiom,
    ! [Xs2: list_nat] :
      ( ( Xs2 != nil_nat )
      = ( ? [Y5: nat,Ys3: list_nat] :
            ( Xs2
            = ( cons_nat @ Y5 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_574_list__induct2_H,axiom,
    ! [P2: list_nat > list_nat > $o,Xs2: list_nat,Ys: list_nat] :
      ( ( P2 @ nil_nat @ nil_nat )
     => ( ! [X3: nat,Xs: list_nat] : ( P2 @ ( cons_nat @ X3 @ Xs ) @ nil_nat )
       => ( ! [Y3: nat,Ys4: list_nat] : ( P2 @ nil_nat @ ( cons_nat @ Y3 @ Ys4 ) )
         => ( ! [X3: nat,Xs: list_nat,Y3: nat,Ys4: list_nat] :
                ( ( P2 @ Xs @ Ys4 )
               => ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
           => ( P2 @ Xs2 @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_575_list__nonempty__induct,axiom,
    ! [Xs2: list_nat,P2: list_nat > $o] :
      ( ( Xs2 != nil_nat )
     => ( ! [X3: nat] : ( P2 @ ( cons_nat @ X3 @ nil_nat ) )
       => ( ! [X3: nat,Xs: list_nat] :
              ( ( Xs != nil_nat )
             => ( ( P2 @ Xs )
               => ( P2 @ ( cons_nat @ X3 @ Xs ) ) ) )
         => ( P2 @ Xs2 ) ) ) ) ).

% list_nonempty_induct
thf(fact_576_successively_Ocases,axiom,
    ! [X: produc254973753779126261st_nat] :
      ( ! [P5: nat > nat > $o] :
          ( X
         != ( produc4727192421694094319st_nat @ P5 @ nil_nat ) )
     => ( ! [P5: nat > nat > $o,X3: nat] :
            ( X
           != ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X3 @ nil_nat ) ) )
       => ~ ! [P5: nat > nat > $o,X3: nat,Y3: nat,Xs: list_nat] :
              ( X
             != ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs ) ) ) ) ) ) ).

% successively.cases
thf(fact_577_term__not__arg,axiom,
    ! [F2: a,Ss: list_term_a_b] :
      ~ ( member_term_a_b @ ( fun_a_b @ F2 @ Ss ) @ ( set_term_a_b2 @ Ss ) ) ).

% term_not_arg
thf(fact_578_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
     => ( ord_less_nat @ I @ K ) ) ).

% add_lessD1
thf(fact_579_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_580_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_581_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_582_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_583_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_less_add1
thf(fact_584_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_less_add2
thf(fact_585_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% less_add_eq_less
thf(fact_586_append__Nil,axiom,
    ! [Ys: list_nat] :
      ( ( append_nat @ nil_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_587_append_Oleft__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ nil_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_588_eq__Nil__appendI,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( Xs2 = Ys )
     => ( Xs2
        = ( append_nat @ nil_nat @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_589_Nil__in__poss,axiom,
    ! [T: term_a_b] : ( member_list_nat @ nil_nat @ ( term_poss_a_b @ T ) ) ).

% Nil_in_poss
thf(fact_590_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_591_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_592_position__less__Nil__is__bot,axiom,
    ! [P: list_nat] : ( term_p3503116865373065078eq_nat @ nil_nat @ P ) ).

% position_less_Nil_is_bot
thf(fact_593_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 ) )
     => ( ! [P4: list_nat] :
            ( ( member_list_nat @ P4 @ ( term_poss_a_b @ S ) )
           => ( ( term_fun_at_a_b @ S @ P4 )
              = ( term_fun_at_a_b @ T @ P4 ) ) )
       => ( S = T ) ) ) ).

% eq_term_by_poss_fun_at
thf(fact_594_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_length_nat @ N @ nil_nat )
      = N ) ).

% gen_length_code(1)
thf(fact_595_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_596_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_597_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_598_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_599_snoc__listrel1__snoc__iff,axiom,
    ! [Xs2: list_nat,X: nat,Ys: list_nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) ) @ ( listrel1_nat @ R ) )
      = ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( listrel1_nat @ R ) )
          & ( X = Y ) )
        | ( ( Xs2 = Ys )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_600_split__list,axiom,
    ! [X: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat] :
      ( ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Xs2 ) )
     => ? [Ys4: list_P3592885314253461005_a_nat,Zs2: list_P3592885314253461005_a_nat] :
          ( Xs2
          = ( append7679239579558125090_a_nat @ Ys4 @ ( cons_P5205166803686508359_a_nat @ X @ Zs2 ) ) ) ) ).

% split_list
thf(fact_601_split__list,axiom,
    ! [X: list_nat,Xs2: list_list_nat] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
     => ? [Ys4: list_list_nat,Zs2: list_list_nat] :
          ( Xs2
          = ( append_list_nat @ Ys4 @ ( cons_list_nat @ X @ Zs2 ) ) ) ) ).

% split_list
thf(fact_602_split__list,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ? [Ys4: list_nat,Zs2: list_nat] :
          ( Xs2
          = ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs2 ) ) ) ) ).

% split_list
thf(fact_603_split__list__last,axiom,
    ! [X: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat] :
      ( ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Xs2 ) )
     => ? [Ys4: list_P3592885314253461005_a_nat,Zs2: list_P3592885314253461005_a_nat] :
          ( ( Xs2
            = ( append7679239579558125090_a_nat @ Ys4 @ ( cons_P5205166803686508359_a_nat @ X @ Zs2 ) ) )
          & ~ ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_604_split__list__last,axiom,
    ! [X: list_nat,Xs2: list_list_nat] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
     => ? [Ys4: list_list_nat,Zs2: list_list_nat] :
          ( ( Xs2
            = ( append_list_nat @ Ys4 @ ( cons_list_nat @ X @ Zs2 ) ) )
          & ~ ( member_list_nat @ X @ ( set_list_nat2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_605_split__list__last,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ? [Ys4: list_nat,Zs2: list_nat] :
          ( ( Xs2
            = ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs2 ) ) )
          & ~ ( member_nat @ X @ ( set_nat2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_606_split__list__prop,axiom,
    ! [Xs2: list_nat,P2: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys4: list_nat,X3: nat] :
          ( ? [Zs2: list_nat] :
              ( Xs2
              = ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs2 ) ) )
          & ( P2 @ X3 ) ) ) ).

% split_list_prop
thf(fact_607_split__list__first,axiom,
    ! [X: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat] :
      ( ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Xs2 ) )
     => ? [Ys4: list_P3592885314253461005_a_nat,Zs2: list_P3592885314253461005_a_nat] :
          ( ( Xs2
            = ( append7679239579558125090_a_nat @ Ys4 @ ( cons_P5205166803686508359_a_nat @ X @ Zs2 ) ) )
          & ~ ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Ys4 ) ) ) ) ).

% split_list_first
thf(fact_608_split__list__first,axiom,
    ! [X: list_nat,Xs2: list_list_nat] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
     => ? [Ys4: list_list_nat,Zs2: list_list_nat] :
          ( ( Xs2
            = ( append_list_nat @ Ys4 @ ( cons_list_nat @ X @ Zs2 ) ) )
          & ~ ( member_list_nat @ X @ ( set_list_nat2 @ Ys4 ) ) ) ) ).

% split_list_first
thf(fact_609_split__list__first,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ? [Ys4: list_nat,Zs2: list_nat] :
          ( ( Xs2
            = ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs2 ) ) )
          & ~ ( member_nat @ X @ ( set_nat2 @ Ys4 ) ) ) ) ).

% split_list_first
thf(fact_610_split__list__propE,axiom,
    ! [Xs2: list_nat,P2: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys4: list_nat,X3: nat] :
            ( ? [Zs2: list_nat] :
                ( Xs2
                = ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs2 ) ) )
           => ~ ( P2 @ X3 ) ) ) ).

% split_list_propE
thf(fact_611_append__Cons__eq__iff,axiom,
    ! [X: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,Xs4: list_P3592885314253461005_a_nat,Ys5: list_P3592885314253461005_a_nat] :
      ( ~ ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Xs2 ) )
     => ( ~ ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Ys ) )
       => ( ( ( append7679239579558125090_a_nat @ Xs2 @ ( cons_P5205166803686508359_a_nat @ X @ Ys ) )
            = ( append7679239579558125090_a_nat @ Xs4 @ ( cons_P5205166803686508359_a_nat @ X @ Ys5 ) ) )
          = ( ( Xs2 = Xs4 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_612_append__Cons__eq__iff,axiom,
    ! [X: list_nat,Xs2: list_list_nat,Ys: list_list_nat,Xs4: list_list_nat,Ys5: list_list_nat] :
      ( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
     => ( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Ys ) )
       => ( ( ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ Ys ) )
            = ( append_list_nat @ Xs4 @ ( cons_list_nat @ X @ Ys5 ) ) )
          = ( ( Xs2 = Xs4 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_613_append__Cons__eq__iff,axiom,
    ! [X: nat,Xs2: list_nat,Ys: list_nat,Xs4: list_nat,Ys5: list_nat] :
      ( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ( ~ ( member_nat @ X @ ( set_nat2 @ Ys ) )
       => ( ( ( append_nat @ Xs2 @ ( cons_nat @ X @ Ys ) )
            = ( append_nat @ Xs4 @ ( cons_nat @ X @ Ys5 ) ) )
          = ( ( Xs2 = Xs4 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_614_in__set__conv__decomp,axiom,
    ! [X: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat] :
      ( ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Xs2 ) )
      = ( ? [Ys3: list_P3592885314253461005_a_nat,Zs3: list_P3592885314253461005_a_nat] :
            ( Xs2
            = ( append7679239579558125090_a_nat @ Ys3 @ ( cons_P5205166803686508359_a_nat @ X @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_615_in__set__conv__decomp,axiom,
    ! [X: list_nat,Xs2: list_list_nat] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
      = ( ? [Ys3: list_list_nat,Zs3: list_list_nat] :
            ( Xs2
            = ( append_list_nat @ Ys3 @ ( cons_list_nat @ X @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_616_in__set__conv__decomp,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
      = ( ? [Ys3: list_nat,Zs3: list_nat] :
            ( Xs2
            = ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_617_split__list__last__prop,axiom,
    ! [Xs2: list_nat,P2: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys4: list_nat,X3: nat,Zs2: list_nat] :
          ( ( Xs2
            = ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs2 ) ) )
          & ( P2 @ X3 )
          & ! [Xa: nat] :
              ( ( member_nat @ Xa @ ( set_nat2 @ Zs2 ) )
             => ~ ( P2 @ Xa ) ) ) ) ).

% split_list_last_prop
thf(fact_618_split__list__first__prop,axiom,
    ! [Xs2: list_nat,P2: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys4: list_nat,X3: nat] :
          ( ? [Zs2: list_nat] :
              ( Xs2
              = ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs2 ) ) )
          & ( P2 @ X3 )
          & ! [Xa: nat] :
              ( ( member_nat @ Xa @ ( set_nat2 @ Ys4 ) )
             => ~ ( P2 @ Xa ) ) ) ) ).

% split_list_first_prop
thf(fact_619_split__list__last__propE,axiom,
    ! [Xs2: list_nat,P2: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys4: list_nat,X3: nat,Zs2: list_nat] :
            ( ( Xs2
              = ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs2 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa: nat] :
                    ( ( member_nat @ Xa @ ( set_nat2 @ Zs2 ) )
                   => ~ ( P2 @ Xa ) ) ) ) ) ).

% split_list_last_propE
thf(fact_620_split__list__first__propE,axiom,
    ! [Xs2: list_nat,P2: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys4: list_nat,X3: nat] :
            ( ? [Zs2: list_nat] :
                ( Xs2
                = ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs2 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa: nat] :
                    ( ( member_nat @ Xa @ ( set_nat2 @ Ys4 ) )
                   => ~ ( P2 @ Xa ) ) ) ) ) ).

% split_list_first_propE
thf(fact_621_in__set__conv__decomp__last,axiom,
    ! [X: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat] :
      ( ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Xs2 ) )
      = ( ? [Ys3: list_P3592885314253461005_a_nat,Zs3: list_P3592885314253461005_a_nat] :
            ( ( Xs2
              = ( append7679239579558125090_a_nat @ Ys3 @ ( cons_P5205166803686508359_a_nat @ X @ Zs3 ) ) )
            & ~ ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_622_in__set__conv__decomp__last,axiom,
    ! [X: list_nat,Xs2: list_list_nat] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
      = ( ? [Ys3: list_list_nat,Zs3: list_list_nat] :
            ( ( Xs2
              = ( append_list_nat @ Ys3 @ ( cons_list_nat @ X @ Zs3 ) ) )
            & ~ ( member_list_nat @ X @ ( set_list_nat2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_623_in__set__conv__decomp__last,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
      = ( ? [Ys3: list_nat,Zs3: list_nat] :
            ( ( Xs2
              = ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) )
            & ~ ( member_nat @ X @ ( set_nat2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_624_in__set__conv__decomp__first,axiom,
    ! [X: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat] :
      ( ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Xs2 ) )
      = ( ? [Ys3: list_P3592885314253461005_a_nat,Zs3: list_P3592885314253461005_a_nat] :
            ( ( Xs2
              = ( append7679239579558125090_a_nat @ Ys3 @ ( cons_P5205166803686508359_a_nat @ X @ Zs3 ) ) )
            & ~ ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_625_in__set__conv__decomp__first,axiom,
    ! [X: list_nat,Xs2: list_list_nat] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
      = ( ? [Ys3: list_list_nat,Zs3: list_list_nat] :
            ( ( Xs2
              = ( append_list_nat @ Ys3 @ ( cons_list_nat @ X @ Zs3 ) ) )
            & ~ ( member_list_nat @ X @ ( set_list_nat2 @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_626_in__set__conv__decomp__first,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
      = ( ? [Ys3: list_nat,Zs3: list_nat] :
            ( ( Xs2
              = ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) )
            & ~ ( member_nat @ X @ ( set_nat2 @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_627_split__list__last__prop__iff,axiom,
    ! [Xs2: list_nat,P2: nat > $o] :
      ( ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys3: list_nat,X4: nat,Zs3: list_nat] :
            ( ( Xs2
              = ( append_nat @ Ys3 @ ( cons_nat @ X4 @ Zs3 ) ) )
            & ( P2 @ X4 )
            & ! [Y5: nat] :
                ( ( member_nat @ Y5 @ ( set_nat2 @ Zs3 ) )
               => ~ ( P2 @ Y5 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_628_split__list__first__prop__iff,axiom,
    ! [Xs2: list_nat,P2: nat > $o] :
      ( ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys3: list_nat,X4: nat] :
            ( ? [Zs3: list_nat] :
                ( Xs2
                = ( append_nat @ Ys3 @ ( cons_nat @ X4 @ Zs3 ) ) )
            & ( P2 @ X4 )
            & ! [Y5: nat] :
                ( ( member_nat @ Y5 @ ( set_nat2 @ Ys3 ) )
               => ~ ( P2 @ Y5 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_629_mono__nat__linear__lb,axiom,
    ! [F2: nat > nat,M: nat,K: nat] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_nat @ M4 @ N3 )
         => ( ord_less_nat @ ( F2 @ M4 ) @ ( F2 @ N3 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F2 @ M ) @ K ) @ ( F2 @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_630_list__induct4,axiom,
    ! [Xs2: 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 @ Xs2 )
        = ( 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 )
           => ( ! [X3: nat,Xs: list_nat,Y3: nat,Ys4: list_nat,Z4: nat,Zs2: list_nat,W2: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P2 @ Xs @ Ys4 @ Zs2 @ Ws2 )
                       => ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs2 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_631_list__induct4,axiom,
    ! [Xs2: 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 @ Xs2 )
        = ( 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 )
           => ( ! [X3: nat,Xs: list_nat,Y3: nat,Ys4: list_nat,Z4: nat,Zs2: list_nat,W2: term_a_b,Ws2: list_term_a_b] :
                  ( ( ( size_size_list_nat @ Xs )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_s8906293707977694520rm_a_b @ Ws2 ) )
                     => ( ( P2 @ Xs @ Ys4 @ Zs2 @ Ws2 )
                       => ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs2 ) @ ( cons_term_a_b @ W2 @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_632_list__induct4,axiom,
    ! [Xs2: 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 @ Xs2 )
        = ( 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 )
           => ( ! [X3: nat,Xs: list_nat,Y3: nat,Ys4: list_nat,Z4: term_a_b,Zs2: list_term_a_b,W2: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_s8906293707977694520rm_a_b @ Zs2 ) )
                   => ( ( ( size_s8906293707977694520rm_a_b @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P2 @ Xs @ Ys4 @ Zs2 @ Ws2 )
                       => ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_term_a_b @ Z4 @ Zs2 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_633_list__induct4,axiom,
    ! [Xs2: 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 @ Xs2 )
        = ( 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 )
           => ( ! [X3: nat,Xs: list_nat,Y3: term_a_b,Ys4: list_term_a_b,Z4: nat,Zs2: list_nat,W2: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs )
                    = ( size_s8906293707977694520rm_a_b @ Ys4 ) )
                 => ( ( ( size_s8906293707977694520rm_a_b @ Ys4 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P2 @ Xs @ Ys4 @ Zs2 @ Ws2 )
                       => ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_term_a_b @ Y3 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs2 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_634_list__induct4,axiom,
    ! [Xs2: 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 @ Xs2 )
        = ( 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 )
           => ( ! [X3: term_a_b,Xs: list_term_a_b,Y3: nat,Ys4: list_nat,Z4: nat,Zs2: list_nat,W2: nat,Ws2: list_nat] :
                  ( ( ( size_s8906293707977694520rm_a_b @ Xs )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P2 @ Xs @ Ys4 @ Zs2 @ Ws2 )
                       => ( P2 @ ( cons_term_a_b @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs2 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_635_list__induct4,axiom,
    ! [Xs2: list_nat,Ys: list_nat,Zs: list_term_a_b,Ws: list_term_a_b,P2: list_nat > list_nat > list_term_a_b > list_term_a_b > $o] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s8906293707977694520rm_a_b @ Zs ) )
       => ( ( ( size_s8906293707977694520rm_a_b @ Zs )
            = ( size_s8906293707977694520rm_a_b @ Ws ) )
         => ( ( P2 @ nil_nat @ nil_nat @ nil_term_a_b @ nil_term_a_b )
           => ( ! [X3: nat,Xs: list_nat,Y3: nat,Ys4: list_nat,Z4: term_a_b,Zs2: list_term_a_b,W2: term_a_b,Ws2: list_term_a_b] :
                  ( ( ( size_size_list_nat @ Xs )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_s8906293707977694520rm_a_b @ Zs2 ) )
                   => ( ( ( size_s8906293707977694520rm_a_b @ Zs2 )
                        = ( size_s8906293707977694520rm_a_b @ Ws2 ) )
                     => ( ( P2 @ Xs @ Ys4 @ Zs2 @ Ws2 )
                       => ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_term_a_b @ Z4 @ Zs2 ) @ ( cons_term_a_b @ W2 @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_636_list__induct4,axiom,
    ! [Xs2: 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 @ Xs2 )
        = ( 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 )
           => ( ! [X3: nat,Xs: list_nat,Y3: term_a_b,Ys4: list_term_a_b,Z4: nat,Zs2: list_nat,W2: term_a_b,Ws2: list_term_a_b] :
                  ( ( ( size_size_list_nat @ Xs )
                    = ( size_s8906293707977694520rm_a_b @ Ys4 ) )
                 => ( ( ( size_s8906293707977694520rm_a_b @ Ys4 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_s8906293707977694520rm_a_b @ Ws2 ) )
                     => ( ( P2 @ Xs @ Ys4 @ Zs2 @ Ws2 )
                       => ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_term_a_b @ Y3 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs2 ) @ ( cons_term_a_b @ W2 @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_637_list__induct4,axiom,
    ! [Xs2: 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 @ Xs2 )
        = ( 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 )
           => ( ! [X3: nat,Xs: list_nat,Y3: term_a_b,Ys4: list_term_a_b,Z4: term_a_b,Zs2: list_term_a_b,W2: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs )
                    = ( size_s8906293707977694520rm_a_b @ Ys4 ) )
                 => ( ( ( size_s8906293707977694520rm_a_b @ Ys4 )
                      = ( size_s8906293707977694520rm_a_b @ Zs2 ) )
                   => ( ( ( size_s8906293707977694520rm_a_b @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P2 @ Xs @ Ys4 @ Zs2 @ Ws2 )
                       => ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_term_a_b @ Y3 @ Ys4 ) @ ( cons_term_a_b @ Z4 @ Zs2 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_638_list__induct4,axiom,
    ! [Xs2: 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 @ Xs2 )
        = ( 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 )
           => ( ! [X3: term_a_b,Xs: list_term_a_b,Y3: nat,Ys4: list_nat,Z4: nat,Zs2: list_nat,W2: term_a_b,Ws2: list_term_a_b] :
                  ( ( ( size_s8906293707977694520rm_a_b @ Xs )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_s8906293707977694520rm_a_b @ Ws2 ) )
                     => ( ( P2 @ Xs @ Ys4 @ Zs2 @ Ws2 )
                       => ( P2 @ ( cons_term_a_b @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs2 ) @ ( cons_term_a_b @ W2 @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_639_list__induct4,axiom,
    ! [Xs2: 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 @ Xs2 )
        = ( 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 )
           => ( ! [X3: term_a_b,Xs: list_term_a_b,Y3: nat,Ys4: list_nat,Z4: term_a_b,Zs2: list_term_a_b,W2: nat,Ws2: list_nat] :
                  ( ( ( size_s8906293707977694520rm_a_b @ Xs )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_s8906293707977694520rm_a_b @ Zs2 ) )
                   => ( ( ( size_s8906293707977694520rm_a_b @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P2 @ Xs @ Ys4 @ Zs2 @ Ws2 )
                       => ( P2 @ ( cons_term_a_b @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_term_a_b @ Z4 @ Zs2 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_640_list__induct3,axiom,
    ! [Xs2: list_nat,Ys: list_nat,Zs: list_nat,P2: list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P2 @ nil_nat @ nil_nat @ nil_nat )
         => ( ! [X3: nat,Xs: list_nat,Y3: nat,Ys4: list_nat,Z4: nat,Zs2: list_nat] :
                ( ( ( size_size_list_nat @ Xs )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ( ( size_size_list_nat @ Ys4 )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P2 @ Xs @ Ys4 @ Zs2 )
                   => ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs2 ) ) ) ) )
           => ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_641_list__induct3,axiom,
    ! [Xs2: list_nat,Ys: list_nat,Zs: list_term_a_b,P2: list_nat > list_nat > list_term_a_b > $o] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s8906293707977694520rm_a_b @ Zs ) )
       => ( ( P2 @ nil_nat @ nil_nat @ nil_term_a_b )
         => ( ! [X3: nat,Xs: list_nat,Y3: nat,Ys4: list_nat,Z4: term_a_b,Zs2: list_term_a_b] :
                ( ( ( size_size_list_nat @ Xs )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ( ( size_size_list_nat @ Ys4 )
                    = ( size_s8906293707977694520rm_a_b @ Zs2 ) )
                 => ( ( P2 @ Xs @ Ys4 @ Zs2 )
                   => ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_term_a_b @ Z4 @ Zs2 ) ) ) ) )
           => ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_642_list__induct3,axiom,
    ! [Xs2: list_nat,Ys: list_term_a_b,Zs: list_nat,P2: list_nat > list_term_a_b > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P2 @ nil_nat @ nil_term_a_b @ nil_nat )
         => ( ! [X3: nat,Xs: list_nat,Y3: term_a_b,Ys4: list_term_a_b,Z4: nat,Zs2: list_nat] :
                ( ( ( size_size_list_nat @ Xs )
                  = ( size_s8906293707977694520rm_a_b @ Ys4 ) )
               => ( ( ( size_s8906293707977694520rm_a_b @ Ys4 )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P2 @ Xs @ Ys4 @ Zs2 )
                   => ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_term_a_b @ Y3 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs2 ) ) ) ) )
           => ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_643_list__induct3,axiom,
    ! [Xs2: 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 @ Xs2 )
        = ( 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 )
         => ( ! [X3: nat,Xs: list_nat,Y3: term_a_b,Ys4: list_term_a_b,Z4: term_a_b,Zs2: list_term_a_b] :
                ( ( ( size_size_list_nat @ Xs )
                  = ( size_s8906293707977694520rm_a_b @ Ys4 ) )
               => ( ( ( size_s8906293707977694520rm_a_b @ Ys4 )
                    = ( size_s8906293707977694520rm_a_b @ Zs2 ) )
                 => ( ( P2 @ Xs @ Ys4 @ Zs2 )
                   => ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_term_a_b @ Y3 @ Ys4 ) @ ( cons_term_a_b @ Z4 @ Zs2 ) ) ) ) )
           => ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_644_list__induct3,axiom,
    ! [Xs2: list_term_a_b,Ys: list_nat,Zs: list_nat,P2: list_term_a_b > list_nat > list_nat > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P2 @ nil_term_a_b @ nil_nat @ nil_nat )
         => ( ! [X3: term_a_b,Xs: list_term_a_b,Y3: nat,Ys4: list_nat,Z4: nat,Zs2: list_nat] :
                ( ( ( size_s8906293707977694520rm_a_b @ Xs )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ( ( size_size_list_nat @ Ys4 )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P2 @ Xs @ Ys4 @ Zs2 )
                   => ( P2 @ ( cons_term_a_b @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs2 ) ) ) ) )
           => ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_645_list__induct3,axiom,
    ! [Xs2: 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 @ Xs2 )
        = ( 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 )
         => ( ! [X3: term_a_b,Xs: list_term_a_b,Y3: nat,Ys4: list_nat,Z4: term_a_b,Zs2: list_term_a_b] :
                ( ( ( size_s8906293707977694520rm_a_b @ Xs )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ( ( size_size_list_nat @ Ys4 )
                    = ( size_s8906293707977694520rm_a_b @ Zs2 ) )
                 => ( ( P2 @ Xs @ Ys4 @ Zs2 )
                   => ( P2 @ ( cons_term_a_b @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_term_a_b @ Z4 @ Zs2 ) ) ) ) )
           => ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_646_list__induct3,axiom,
    ! [Xs2: 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 @ Xs2 )
        = ( 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 )
         => ( ! [X3: term_a_b,Xs: list_term_a_b,Y3: term_a_b,Ys4: list_term_a_b,Z4: nat,Zs2: list_nat] :
                ( ( ( size_s8906293707977694520rm_a_b @ Xs )
                  = ( size_s8906293707977694520rm_a_b @ Ys4 ) )
               => ( ( ( size_s8906293707977694520rm_a_b @ Ys4 )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P2 @ Xs @ Ys4 @ Zs2 )
                   => ( P2 @ ( cons_term_a_b @ X3 @ Xs ) @ ( cons_term_a_b @ Y3 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs2 ) ) ) ) )
           => ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_647_list__induct3,axiom,
    ! [Xs2: 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 @ Xs2 )
        = ( 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 )
         => ( ! [X3: term_a_b,Xs: list_term_a_b,Y3: term_a_b,Ys4: list_term_a_b,Z4: term_a_b,Zs2: list_term_a_b] :
                ( ( ( size_s8906293707977694520rm_a_b @ Xs )
                  = ( size_s8906293707977694520rm_a_b @ Ys4 ) )
               => ( ( ( size_s8906293707977694520rm_a_b @ Ys4 )
                    = ( size_s8906293707977694520rm_a_b @ Zs2 ) )
                 => ( ( P2 @ Xs @ Ys4 @ Zs2 )
                   => ( P2 @ ( cons_term_a_b @ X3 @ Xs ) @ ( cons_term_a_b @ Y3 @ Ys4 ) @ ( cons_term_a_b @ Z4 @ Zs2 ) ) ) ) )
           => ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_648_list__induct2,axiom,
    ! [Xs2: list_nat,Ys: list_nat,P2: list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P2 @ nil_nat @ nil_nat )
       => ( ! [X3: nat,Xs: list_nat,Y3: nat,Ys4: list_nat] :
              ( ( ( size_size_list_nat @ Xs )
                = ( size_size_list_nat @ Ys4 ) )
             => ( ( P2 @ Xs @ Ys4 )
               => ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) )
         => ( P2 @ Xs2 @ Ys ) ) ) ) ).

% list_induct2
thf(fact_649_list__induct2,axiom,
    ! [Xs2: list_nat,Ys: list_term_a_b,P2: list_nat > list_term_a_b > $o] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( P2 @ nil_nat @ nil_term_a_b )
       => ( ! [X3: nat,Xs: list_nat,Y3: term_a_b,Ys4: list_term_a_b] :
              ( ( ( size_size_list_nat @ Xs )
                = ( size_s8906293707977694520rm_a_b @ Ys4 ) )
             => ( ( P2 @ Xs @ Ys4 )
               => ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_term_a_b @ Y3 @ Ys4 ) ) ) )
         => ( P2 @ Xs2 @ Ys ) ) ) ) ).

% list_induct2
thf(fact_650_list__induct2,axiom,
    ! [Xs2: list_term_a_b,Ys: list_nat,P2: list_term_a_b > list_nat > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P2 @ nil_term_a_b @ nil_nat )
       => ( ! [X3: term_a_b,Xs: list_term_a_b,Y3: nat,Ys4: list_nat] :
              ( ( ( size_s8906293707977694520rm_a_b @ Xs )
                = ( size_size_list_nat @ Ys4 ) )
             => ( ( P2 @ Xs @ Ys4 )
               => ( P2 @ ( cons_term_a_b @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) )
         => ( P2 @ Xs2 @ Ys ) ) ) ) ).

% list_induct2
thf(fact_651_list__induct2,axiom,
    ! [Xs2: list_term_a_b,Ys: list_term_a_b,P2: list_term_a_b > list_term_a_b > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( P2 @ nil_term_a_b @ nil_term_a_b )
       => ( ! [X3: term_a_b,Xs: list_term_a_b,Y3: term_a_b,Ys4: list_term_a_b] :
              ( ( ( size_s8906293707977694520rm_a_b @ Xs )
                = ( size_s8906293707977694520rm_a_b @ Ys4 ) )
             => ( ( P2 @ Xs @ Ys4 )
               => ( P2 @ ( cons_term_a_b @ X3 @ Xs ) @ ( cons_term_a_b @ Y3 @ Ys4 ) ) ) )
         => ( P2 @ Xs2 @ Ys ) ) ) ) ).

% list_induct2
thf(fact_652_rev__induct,axiom,
    ! [P2: list_nat > $o,Xs2: list_nat] :
      ( ( P2 @ nil_nat )
     => ( ! [X3: nat,Xs: list_nat] :
            ( ( P2 @ Xs )
           => ( P2 @ ( append_nat @ Xs @ ( cons_nat @ X3 @ nil_nat ) ) ) )
       => ( P2 @ Xs2 ) ) ) ).

% rev_induct
thf(fact_653_rev__exhaust,axiom,
    ! [Xs2: list_nat] :
      ( ( Xs2 != nil_nat )
     => ~ ! [Ys4: list_nat,Y3: nat] :
            ( Xs2
           != ( append_nat @ Ys4 @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) ).

% rev_exhaust
thf(fact_654_Cons__eq__append__conv,axiom,
    ! [X: nat,Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( cons_nat @ X @ Xs2 )
        = ( append_nat @ Ys @ Zs ) )
      = ( ( ( Ys = nil_nat )
          & ( ( cons_nat @ X @ Xs2 )
            = Zs ) )
        | ? [Ys6: list_nat] :
            ( ( ( cons_nat @ X @ Ys6 )
              = Ys )
            & ( Xs2
              = ( append_nat @ Ys6 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_655_append__eq__Cons__conv,axiom,
    ! [Ys: list_nat,Zs: list_nat,X: nat,Xs2: list_nat] :
      ( ( ( append_nat @ Ys @ Zs )
        = ( cons_nat @ X @ Xs2 ) )
      = ( ( ( Ys = nil_nat )
          & ( Zs
            = ( cons_nat @ X @ Xs2 ) ) )
        | ? [Ys6: list_nat] :
            ( ( Ys
              = ( cons_nat @ X @ Ys6 ) )
            & ( ( append_nat @ Ys6 @ Zs )
              = Xs2 ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_656_rev__nonempty__induct,axiom,
    ! [Xs2: list_nat,P2: list_nat > $o] :
      ( ( Xs2 != nil_nat )
     => ( ! [X3: nat] : ( P2 @ ( cons_nat @ X3 @ nil_nat ) )
       => ( ! [X3: nat,Xs: list_nat] :
              ( ( Xs != nil_nat )
             => ( ( P2 @ Xs )
               => ( P2 @ ( append_nat @ Xs @ ( cons_nat @ X3 @ nil_nat ) ) ) ) )
         => ( P2 @ Xs2 ) ) ) ) ).

% rev_nonempty_induct
thf(fact_657_shuffles_Ocases,axiom,
    ! [X: produc1828647624359046049st_nat] :
      ( ! [Ys4: list_nat] :
          ( X
         != ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
     => ( ! [Xs: list_nat] :
            ( X
           != ( produc2694037385005941721st_nat @ Xs @ nil_nat ) )
       => ~ ! [X3: nat,Xs: list_nat,Y3: nat,Ys4: list_nat] :
              ( X
             != ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) ) ) ).

% shuffles.cases
thf(fact_658_remove__prefix_Ocases,axiom,
    ! [X: produc1828647624359046049st_nat] :
      ( ! [X3: nat,Xs: list_nat,Y3: nat,Ys4: list_nat] :
          ( X
         != ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
     => ( ! [Ys4: list_nat] :
            ( X
           != ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
       => ~ ! [V2: nat,Va: list_nat] :
              ( X
             != ( produc2694037385005941721st_nat @ ( cons_nat @ V2 @ Va ) @ nil_nat ) ) ) ) ).

% remove_prefix.cases
thf(fact_659_listrel1I2,axiom,
    ! [Xs2: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,X: nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( listrel1_nat @ R ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ X @ Ys ) ) @ ( listrel1_nat @ R ) ) ) ).

% listrel1I2
thf(fact_660_listrel1__eq__len,axiom,
    ! [Xs2: list_term_a_b,Ys: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs2 @ Ys ) @ ( listrel1_term_a_b @ R ) )
     => ( ( size_s8906293707977694520rm_a_b @ Xs2 )
        = ( size_s8906293707977694520rm_a_b @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_661_lexord__Nil__right,axiom,
    ! [X: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ nil_nat ) @ ( lexord_nat @ R ) ) ).

% lexord_Nil_right
thf(fact_662_Nil__notin__lex,axiom,
    ! [Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys ) @ ( lex_nat @ R ) ) ).

% Nil_notin_lex
thf(fact_663_Nil2__notin__lex,axiom,
    ! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) @ ( lex_nat @ R ) ) ).

% Nil2_notin_lex
thf(fact_664_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_665_Nil__lenlex__iff2,axiom,
    ! [Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ns @ nil_nat ) @ ( lenlex_nat @ R ) ) ).

% Nil_lenlex_iff2
thf(fact_666_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_667_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_668_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_669_ord__eq__le__subst,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C2: nat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_670_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_671_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_672_order__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_673_order__subst1,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_674_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_675_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_676_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_677_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_678_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_679_linorder__wlog,axiom,
    ! [P2: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: nat,B3: nat] :
            ( ( P2 @ B3 @ A3 )
           => ( P2 @ A3 @ B3 ) )
       => ( P2 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_680_order__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_eq_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_681_order_Otrans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% order.trans
thf(fact_682_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_683_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_684_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_685_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
          & ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_686_le__cases3,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z ) )
       => ( ( ( ord_less_eq_nat @ X @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_687_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_688_supteq_Osubt,axiom,
    ! [U: term_a_b,Ss: list_term_a_b,T: term_a_b,F2: a] :
      ( ( member_term_a_b @ U @ ( set_term_a_b2 @ Ss ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ T ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( fun_a_b @ F2 @ Ss ) @ T ) @ subter523971068842742411eq_a_b ) ) ) ).

% supteq.subt
thf(fact_689_supteq_Ocases,axiom,
    ! [A1: term_a_b,A22: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ subter523971068842742411eq_a_b )
     => ( ( A22 != A1 )
       => ~ ! [U2: term_a_b,Ss2: list_term_a_b] :
              ( ? [F3: a] :
                  ( A1
                  = ( fun_a_b @ F3 @ Ss2 ) )
             => ( ( member_term_a_b @ U2 @ ( set_term_a_b2 @ Ss2 ) )
               => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U2 @ A22 ) @ subter523971068842742411eq_a_b ) ) ) ) ) ).

% supteq.cases
thf(fact_690_supteq_Osimps,axiom,
    ! [A1: term_a_b,A22: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ subter523971068842742411eq_a_b )
      = ( ? [T2: term_a_b] :
            ( ( A1 = T2 )
            & ( A22 = T2 ) )
        | ? [U3: term_a_b,Ss3: list_term_a_b,T2: term_a_b,F: a] :
            ( ( A1
              = ( fun_a_b @ F @ Ss3 ) )
            & ( A22 = T2 )
            & ( member_term_a_b @ U3 @ ( set_term_a_b2 @ Ss3 ) )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U3 @ T2 ) @ subter523971068842742411eq_a_b ) ) ) ) ).

% supteq.simps
thf(fact_691_lexord__partial__trans,axiom,
    ! [Xs2: list_P3592885314253461005_a_nat,R: set_Pr1811044260758604347_a_nat,Ys: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat] :
      ( ! [X3: product_prod_a_nat,Y3: product_prod_a_nat,Z4: product_prod_a_nat] :
          ( ( member5724188588386418708_a_nat @ X3 @ ( set_Pr924983374503034536_a_nat @ Xs2 ) )
         => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ R )
           => ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y3 @ Z4 ) @ R )
             => ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Z4 ) @ R ) ) ) )
     => ( ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ Xs2 @ Ys ) @ ( lexord2902578037316800714_a_nat @ R ) )
       => ( ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ Ys @ Zs ) @ ( lexord2902578037316800714_a_nat @ R ) )
         => ( member3259931553675508900_a_nat @ ( produc5384655689722402227_a_nat @ Xs2 @ Zs ) @ ( lexord2902578037316800714_a_nat @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_692_lexord__partial__trans,axiom,
    ! [Xs2: list_list_nat,R: set_Pr3451248702717554689st_nat,Ys: list_list_nat,Zs: list_list_nat] :
      ( ! [X3: list_nat,Y3: list_nat,Z4: list_nat] :
          ( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs2 ) )
         => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ R )
           => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y3 @ Z4 ) @ R )
             => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Z4 ) @ R ) ) ) )
     => ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs2 @ Ys ) @ ( lexord_list_nat @ R ) )
       => ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Ys @ Zs ) @ ( lexord_list_nat @ R ) )
         => ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs2 @ Zs ) @ ( lexord_list_nat @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_693_same__length__different,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( Xs2 != Ys )
     => ( ( ( size_size_list_nat @ Xs2 )
          = ( size_size_list_nat @ Ys ) )
       => ? [Pre: list_nat,X3: nat,Xs5: list_nat,Y3: nat,Ys7: list_nat] :
            ( ( X3 != Y3 )
            & ( Xs2
              = ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X3 @ nil_nat ) @ Xs5 ) ) )
            & ( Ys
              = ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y3 @ nil_nat ) @ Ys7 ) ) ) ) ) ) ).

% same_length_different
thf(fact_694_same__length__different,axiom,
    ! [Xs2: list_term_a_b,Ys: list_term_a_b] :
      ( ( Xs2 != Ys )
     => ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
          = ( size_s8906293707977694520rm_a_b @ Ys ) )
       => ? [Pre: list_term_a_b,X3: term_a_b,Xs5: list_term_a_b,Y3: term_a_b,Ys7: list_term_a_b] :
            ( ( X3 != Y3 )
            & ( Xs2
              = ( append_term_a_b @ Pre @ ( append_term_a_b @ ( cons_term_a_b @ X3 @ nil_term_a_b ) @ Xs5 ) ) )
            & ( Ys
              = ( append_term_a_b @ Pre @ ( append_term_a_b @ ( cons_term_a_b @ Y3 @ nil_term_a_b ) @ Ys7 ) ) ) ) ) ) ).

% same_length_different
thf(fact_695_Cons__listrel1E2,axiom,
    ! [Xs2: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R ) )
     => ( ! [X3: nat] :
            ( ( Xs2
              = ( cons_nat @ X3 @ Ys ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R ) )
       => ~ ! [Zs2: list_nat] :
              ( ( Xs2
                = ( cons_nat @ Y @ Zs2 ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs2 @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_696_Cons__listrel1E1,axiom,
    ! [X: nat,Xs2: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ Ys ) @ ( listrel1_nat @ R ) )
     => ( ! [Y3: nat] :
            ( ( Ys
              = ( cons_nat @ Y3 @ Xs2 ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R ) )
       => ~ ! [Zs2: list_nat] :
              ( ( Ys
                = ( cons_nat @ X @ Zs2 ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Zs2 ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_697_listrel1I1,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs2: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Xs2 ) ) @ ( listrel1_nat @ R ) ) ) ).

% listrel1I1
thf(fact_698_subtract__list__sorted_Ocases,axiom,
    ! [X: produc1828647624359046049st_nat] :
      ( ! [X3: nat,Xs: list_nat,Y3: nat,Ys4: list_nat] :
          ( X
         != ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
     => ( ! [Ys4: list_nat] :
            ( X
           != ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
       => ~ ! [V2: nat,Va: list_nat] :
              ( X
             != ( produc2694037385005941721st_nat @ ( cons_nat @ V2 @ Va ) @ nil_nat ) ) ) ) ).

% subtract_list_sorted.cases
thf(fact_699_union__list__sorted_Ocases,axiom,
    ! [X: produc1828647624359046049st_nat] :
      ( ! [X3: nat,Xs: list_nat,Y3: nat,Ys4: list_nat] :
          ( X
         != ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
     => ( ! [Ys4: list_nat] :
            ( X
           != ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
       => ~ ! [V2: nat,Va: list_nat] :
              ( X
             != ( produc2694037385005941721st_nat @ ( cons_nat @ V2 @ Va ) @ nil_nat ) ) ) ) ).

% union_list_sorted.cases
thf(fact_700_list__inter_Ocases,axiom,
    ! [X: produc1828647624359046049st_nat] :
      ( ! [Bs: list_nat] :
          ( X
         != ( produc2694037385005941721st_nat @ nil_nat @ Bs ) )
     => ~ ! [A3: nat,As: list_nat,Bs: list_nat] :
            ( X
           != ( produc2694037385005941721st_nat @ ( cons_nat @ A3 @ As ) @ Bs ) ) ) ).

% list_inter.cases
thf(fact_701_filter2_Ocases,axiom,
    ! [X: produc4787317212837456354st_nat] :
      ( ! [P5: nat > nat > $o,Uu: list_nat] :
          ( X
         != ( produc3127733452865184594st_nat @ P5 @ ( produc2694037385005941721st_nat @ nil_nat @ Uu ) ) )
     => ( ! [P5: nat > nat > $o,V2: nat,Va: list_nat] :
            ( X
           != ( produc3127733452865184594st_nat @ P5 @ ( produc2694037385005941721st_nat @ ( cons_nat @ V2 @ Va ) @ nil_nat ) ) )
       => ~ ! [P5: nat > nat > $o,A3: nat,As: list_nat,B3: nat,Bs: list_nat] :
              ( X
             != ( produc3127733452865184594st_nat @ P5 @ ( produc2694037385005941721st_nat @ ( cons_nat @ A3 @ As ) @ ( cons_nat @ B3 @ Bs ) ) ) ) ) ) ).

% filter2.cases
thf(fact_702_mem__idx_Ocases,axiom,
    ! [X: produc4575160907756185873st_nat] :
      ( ! [Uu: nat] :
          ( X
         != ( produc8282810413953273033st_nat @ Uu @ nil_nat ) )
     => ~ ! [X3: nat,A3: nat,As: list_nat] :
            ( X
           != ( produc8282810413953273033st_nat @ X3 @ ( cons_nat @ A3 @ As ) ) ) ) ).

% mem_idx.cases
thf(fact_703_max__list_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ~ ! [X3: nat,Xs: list_nat] :
            ( X
           != ( cons_nat @ X3 @ Xs ) ) ) ).

% max_list.cases
thf(fact_704_span_Ocases,axiom,
    ! [X: produc4226810134323546766st_nat] :
      ( ! [P5: nat > $o,X3: nat,Xs: list_nat] :
          ( X
         != ( produc8587622027977423880st_nat @ P5 @ ( cons_nat @ X3 @ Xs ) ) )
     => ~ ! [Uu: nat > $o] :
            ( X
           != ( produc8587622027977423880st_nat @ Uu @ nil_nat ) ) ) ).

% span.cases
thf(fact_705_distinct__eq_Ocases,axiom,
    ! [X: produc254973753779126261st_nat] :
      ( ! [Uu: nat > nat > $o] :
          ( X
         != ( produc4727192421694094319st_nat @ Uu @ nil_nat ) )
     => ~ ! [Eq: nat > nat > $o,X3: nat,Xs: list_nat] :
            ( X
           != ( produc4727192421694094319st_nat @ Eq @ ( cons_nat @ X3 @ Xs ) ) ) ) ).

% distinct_eq.cases
thf(fact_706_Missing__List_Omin__list_Ocases,axiom,
    ! [X: list_nat] :
      ( ! [X3: nat] :
          ( X
         != ( cons_nat @ X3 @ nil_nat ) )
     => ( ! [X3: nat,V2: nat,Va: list_nat] :
            ( X
           != ( cons_nat @ X3 @ ( cons_nat @ V2 @ Va ) ) )
       => ( X = nil_nat ) ) ) ).

% Missing_List.min_list.cases
thf(fact_707_list__3__cases,axiom,
    ! [Xs2: list_nat] :
      ( ( Xs2 != nil_nat )
     => ( ! [X3: nat] :
            ( Xs2
           != ( cons_nat @ X3 @ nil_nat ) )
       => ~ ! [X3: nat,Y3: nat,Ys4: list_nat] :
              ( Xs2
             != ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Ys4 ) ) ) ) ) ).

% list_3_cases
thf(fact_708_list__4__cases,axiom,
    ! [Xs2: list_nat] :
      ( ( Xs2 != nil_nat )
     => ( ! [X3: nat] :
            ( Xs2
           != ( cons_nat @ X3 @ nil_nat ) )
       => ( ! [X3: nat,Y3: nat] :
              ( Xs2
             != ( cons_nat @ X3 @ ( cons_nat @ Y3 @ nil_nat ) ) )
         => ~ ! [X3: nat,Y3: nat,Z4: nat,Zs2: list_nat] :
                ( Xs2
               != ( cons_nat @ X3 @ ( cons_nat @ Y3 @ ( cons_nat @ Z4 @ Zs2 ) ) ) ) ) ) ) ).

% list_4_cases
thf(fact_709_inf__pigeonhole__principle,axiom,
    ! [N: nat,F2: nat > nat > $o] :
      ( ! [K3: nat] :
        ? [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
          & ( F2 @ K3 @ I3 ) )
     => ? [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
          & ! [K4: nat] :
            ? [K5: nat] :
              ( ( ord_less_eq_nat @ K4 @ K5 )
              & ( F2 @ K5 @ I2 ) ) ) ) ).

% inf_pigeonhole_principle
thf(fact_710_subsetI,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
      ( ! [X3: product_prod_a_nat] :
          ( ( member5724188588386418708_a_nat @ X3 @ A5 )
         => ( member5724188588386418708_a_nat @ X3 @ B5 ) )
     => ( ord_le8666007276011122963_a_nat @ A5 @ B5 ) ) ).

% subsetI
thf(fact_711_subsetI,axiom,
    ! [A5: set_list_nat,B5: set_list_nat] :
      ( ! [X3: list_nat] :
          ( ( member_list_nat @ X3 @ A5 )
         => ( member_list_nat @ X3 @ B5 ) )
     => ( ord_le6045566169113846134st_nat @ A5 @ B5 ) ) ).

% subsetI
thf(fact_712_extract__Some__iff,axiom,
    ! [P2: nat > $o,Xs2: list_nat,Ys: list_nat,Y: nat,Zs: list_nat] :
      ( ( ( extract_nat @ P2 @ Xs2 )
        = ( some_P617598486967069697st_nat @ ( produc7518127839388293336st_nat @ Ys @ ( produc8282810413953273033st_nat @ Y @ Zs ) ) ) )
      = ( ( Xs2
          = ( append_nat @ Ys @ ( cons_nat @ Y @ Zs ) ) )
        & ( P2 @ Y )
        & ~ ? [X4: nat] :
              ( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
              & ( P2 @ X4 ) ) ) ) ).

% extract_Some_iff
thf(fact_713_extract__SomeE,axiom,
    ! [P2: nat > $o,Xs2: list_nat,Ys: list_nat,Y: nat,Zs: list_nat] :
      ( ( ( extract_nat @ P2 @ Xs2 )
        = ( some_P617598486967069697st_nat @ ( produc7518127839388293336st_nat @ Ys @ ( produc8282810413953273033st_nat @ Y @ Zs ) ) ) )
     => ( ( Xs2
          = ( append_nat @ Ys @ ( cons_nat @ Y @ Zs ) ) )
        & ( P2 @ Y )
        & ~ ? [X5: nat] :
              ( ( member_nat @ X5 @ ( set_nat2 @ Ys ) )
              & ( P2 @ X5 ) ) ) ) ).

% extract_SomeE
thf(fact_714_subset__iff,axiom,
    ( ord_le8666007276011122963_a_nat
    = ( ^ [A6: set_Pr4934435412358123699_a_nat,B7: set_Pr4934435412358123699_a_nat] :
        ! [T2: product_prod_a_nat] :
          ( ( member5724188588386418708_a_nat @ T2 @ A6 )
         => ( member5724188588386418708_a_nat @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_715_subset__iff,axiom,
    ( ord_le6045566169113846134st_nat
    = ( ^ [A6: set_list_nat,B7: set_list_nat] :
        ! [T2: list_nat] :
          ( ( member_list_nat @ T2 @ A6 )
         => ( member_list_nat @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_716_subset__eq,axiom,
    ( ord_le8666007276011122963_a_nat
    = ( ^ [A6: set_Pr4934435412358123699_a_nat,B7: set_Pr4934435412358123699_a_nat] :
        ! [X4: product_prod_a_nat] :
          ( ( member5724188588386418708_a_nat @ X4 @ A6 )
         => ( member5724188588386418708_a_nat @ X4 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_717_subset__eq,axiom,
    ( ord_le6045566169113846134st_nat
    = ( ^ [A6: set_list_nat,B7: set_list_nat] :
        ! [X4: list_nat] :
          ( ( member_list_nat @ X4 @ A6 )
         => ( member_list_nat @ X4 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_718_subsetD,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,C2: product_prod_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
     => ( ( member5724188588386418708_a_nat @ C2 @ A5 )
       => ( member5724188588386418708_a_nat @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_719_subsetD,axiom,
    ! [A5: set_list_nat,B5: set_list_nat,C2: list_nat] :
      ( ( ord_le6045566169113846134st_nat @ A5 @ B5 )
     => ( ( member_list_nat @ C2 @ A5 )
       => ( member_list_nat @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_720_in__mono,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
     => ( ( member5724188588386418708_a_nat @ X @ A5 )
       => ( member5724188588386418708_a_nat @ X @ B5 ) ) ) ).

% in_mono
thf(fact_721_in__mono,axiom,
    ! [A5: set_list_nat,B5: set_list_nat,X: list_nat] :
      ( ( ord_le6045566169113846134st_nat @ A5 @ B5 )
     => ( ( member_list_nat @ X @ A5 )
       => ( member_list_nat @ X @ B5 ) ) ) ).

% in_mono
thf(fact_722_the__elem__set,axiom,
    ! [X: nat] :
      ( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X @ nil_nat ) ) )
      = X ) ).

% the_elem_set
thf(fact_723_subrelI,axiom,
    ! [R: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
      ( ! [X3: a,Y3: nat] :
          ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y3 ) @ R )
         => ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le8666007276011122963_a_nat @ R @ S ) ) ).

% subrelI
thf(fact_724_nat__descend__induct,axiom,
    ! [N: nat,P2: nat > $o,M: nat] :
      ( ! [K3: nat] :
          ( ( ord_less_nat @ N @ K3 )
         => ( P2 @ K3 ) )
     => ( ! [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N )
           => ( ! [I3: nat] :
                  ( ( ord_less_nat @ K3 @ I3 )
                 => ( P2 @ I3 ) )
             => ( P2 @ K3 ) ) )
       => ( P2 @ M ) ) ) ).

% nat_descend_induct
thf(fact_725_complete__interval,axiom,
    ! [A: nat,B: nat,P2: nat > $o] :
      ( ( ord_less_nat @ A @ B )
     => ( ( P2 @ A )
       => ( ~ ( P2 @ B )
         => ? [C: nat] :
              ( ( ord_less_eq_nat @ A @ C )
              & ( ord_less_eq_nat @ C @ B )
              & ! [X5: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X5 )
                    & ( ord_less_nat @ X5 @ C ) )
                 => ( P2 @ X5 ) )
              & ! [D2: nat] :
                  ( ! [X3: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X3 )
                        & ( ord_less_nat @ X3 @ D2 ) )
                     => ( P2 @ X3 ) )
                 => ( ord_less_eq_nat @ D2 @ C ) ) ) ) ) ) ).

% complete_interval
thf(fact_726_verit__comp__simplify1_I3_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
      = ( ord_less_nat @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_727_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_728_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ ( ord_less_eq_nat @ A @ B )
      | ~ ( ord_less_eq_nat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_729_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z4 )
     => ~ ( ord_less_eq_nat @ T @ X5 ) ) ).

% minf(8)
thf(fact_730_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z4 )
     => ( ord_less_eq_nat @ X5 @ T ) ) ).

% minf(6)
thf(fact_731_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z4 @ X5 )
     => ( ord_less_eq_nat @ T @ X5 ) ) ).

% pinf(8)
thf(fact_732_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z4 @ X5 )
     => ~ ( ord_less_eq_nat @ X5 @ T ) ) ).

% pinf(6)
thf(fact_733_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_734_in__set__product__lists__length,axiom,
    ! [Xs2: list_nat,Xss2: list_list_nat] :
      ( ( member_list_nat @ Xs2 @ ( set_list_nat2 @ ( product_lists_nat @ Xss2 ) ) )
     => ( ( size_size_list_nat @ Xs2 )
        = ( size_s3023201423986296836st_nat @ Xss2 ) ) ) ).

% in_set_product_lists_length
thf(fact_735_in__set__product__lists__length,axiom,
    ! [Xs2: list_term_a_b,Xss2: list_list_term_a_b] :
      ( ( member_list_term_a_b @ Xs2 @ ( set_list_term_a_b2 @ ( produc17669015410068453rm_a_b @ Xss2 ) ) )
     => ( ( size_s8906293707977694520rm_a_b @ Xs2 )
        = ( size_s877380706853472072rm_a_b @ Xss2 ) ) ) ).

% in_set_product_lists_length
thf(fact_736_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_737_suptp_Osimps,axiom,
    ( subter8709468037939584051tp_a_b
    = ( ^ [A12: term_a_b,A23: term_a_b] :
          ( ? [S3: term_a_b,Ss3: list_term_a_b,F: a] :
              ( ( A12
                = ( fun_a_b @ F @ Ss3 ) )
              & ( A23 = S3 )
              & ( member_term_a_b @ S3 @ ( set_term_a_b2 @ Ss3 ) ) )
          | ? [S3: term_a_b,Ss3: list_term_a_b,T2: term_a_b,F: a] :
              ( ( A12
                = ( fun_a_b @ F @ Ss3 ) )
              & ( A23 = T2 )
              & ( member_term_a_b @ S3 @ ( set_term_a_b2 @ Ss3 ) )
              & ( subter8709468037939584051tp_a_b @ S3 @ T2 ) ) ) ) ) ).

% suptp.simps
thf(fact_738_supteqp_Osimps,axiom,
    ( subter3057829548224121575qp_a_b
    = ( ^ [A12: term_a_b,A23: term_a_b] :
          ( ? [T2: term_a_b] :
              ( ( A12 = T2 )
              & ( A23 = T2 ) )
          | ? [U3: term_a_b,Ss3: list_term_a_b,T2: term_a_b,F: a] :
              ( ( A12
                = ( fun_a_b @ F @ Ss3 ) )
              & ( A23 = T2 )
              & ( member_term_a_b @ U3 @ ( set_term_a_b2 @ Ss3 ) )
              & ( subter3057829548224121575qp_a_b @ U3 @ T2 ) ) ) ) ) ).

% supteqp.simps
thf(fact_739_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_740_list__inter_Oelims,axiom,
    ! [X: list_P3592885314253461005_a_nat,Xa2: list_P3592885314253461005_a_nat,Y: list_P3592885314253461005_a_nat] :
      ( ( ( missin2021864273562616452_a_nat @ X @ Xa2 )
        = Y )
     => ( ( ( X = nil_Pr7402525243500994295_a_nat )
         => ( Y != nil_Pr7402525243500994295_a_nat ) )
       => ~ ! [A3: product_prod_a_nat,As: list_P3592885314253461005_a_nat] :
              ( ( X
                = ( cons_P5205166803686508359_a_nat @ A3 @ As ) )
             => ~ ( ( ( member5724188588386418708_a_nat @ A3 @ ( set_Pr924983374503034536_a_nat @ Xa2 ) )
                   => ( Y
                      = ( cons_P5205166803686508359_a_nat @ A3 @ ( missin2021864273562616452_a_nat @ As @ Xa2 ) ) ) )
                  & ( ~ ( member5724188588386418708_a_nat @ A3 @ ( set_Pr924983374503034536_a_nat @ Xa2 ) )
                   => ( Y
                      = ( missin2021864273562616452_a_nat @ As @ Xa2 ) ) ) ) ) ) ) ).

% list_inter.elims
thf(fact_741_list__inter_Oelims,axiom,
    ! [X: list_list_nat,Xa2: list_list_nat,Y: list_list_nat] :
      ( ( ( missin6532874241183986279st_nat @ X @ Xa2 )
        = Y )
     => ( ( ( X = nil_list_nat )
         => ( Y != nil_list_nat ) )
       => ~ ! [A3: list_nat,As: list_list_nat] :
              ( ( X
                = ( cons_list_nat @ A3 @ As ) )
             => ~ ( ( ( member_list_nat @ A3 @ ( set_list_nat2 @ Xa2 ) )
                   => ( Y
                      = ( cons_list_nat @ A3 @ ( missin6532874241183986279st_nat @ As @ Xa2 ) ) ) )
                  & ( ~ ( member_list_nat @ A3 @ ( set_list_nat2 @ Xa2 ) )
                   => ( Y
                      = ( missin6532874241183986279st_nat @ As @ Xa2 ) ) ) ) ) ) ) ).

% list_inter.elims
thf(fact_742_list__inter_Oelims,axiom,
    ! [X: list_nat,Xa2: list_nat,Y: list_nat] :
      ( ( ( missin6377695591745783511er_nat @ X @ Xa2 )
        = Y )
     => ( ( ( X = nil_nat )
         => ( Y != nil_nat ) )
       => ~ ! [A3: nat,As: list_nat] :
              ( ( X
                = ( cons_nat @ A3 @ As ) )
             => ~ ( ( ( member_nat @ A3 @ ( set_nat2 @ Xa2 ) )
                   => ( Y
                      = ( cons_nat @ A3 @ ( missin6377695591745783511er_nat @ As @ Xa2 ) ) ) )
                  & ( ~ ( member_nat @ A3 @ ( set_nat2 @ Xa2 ) )
                   => ( Y
                      = ( missin6377695591745783511er_nat @ As @ Xa2 ) ) ) ) ) ) ) ).

% list_inter.elims
thf(fact_743_sorted__list__subset_Osimps_I3_J,axiom,
    ! [A: nat,Uv: list_nat] :
      ( ( sorted3362290152684031886et_nat @ ( cons_nat @ A @ Uv ) @ nil_nat )
      = ( some_nat @ A ) ) ).

% sorted_list_subset.simps(3)
thf(fact_744_sorted__list__subset_Osimps_I1_J,axiom,
    ! [A: nat,B: nat,As2: list_nat,Bs2: list_nat] :
      ( ( ( A = B )
       => ( ( sorted3362290152684031886et_nat @ ( cons_nat @ A @ As2 ) @ ( cons_nat @ B @ Bs2 ) )
          = ( sorted3362290152684031886et_nat @ As2 @ ( cons_nat @ B @ Bs2 ) ) ) )
      & ( ( A != B )
       => ( ( ( ord_less_nat @ B @ A )
           => ( ( sorted3362290152684031886et_nat @ ( cons_nat @ A @ As2 ) @ ( cons_nat @ B @ Bs2 ) )
              = ( sorted3362290152684031886et_nat @ ( cons_nat @ A @ As2 ) @ Bs2 ) ) )
          & ( ~ ( ord_less_nat @ B @ A )
           => ( ( sorted3362290152684031886et_nat @ ( cons_nat @ A @ As2 ) @ ( cons_nat @ B @ Bs2 ) )
              = ( some_nat @ A ) ) ) ) ) ) ).

% sorted_list_subset.simps(1)
thf(fact_745_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_746_supt_Oarg,axiom,
    ! [S: term_a_b,Ss: list_term_a_b,F2: a] :
      ( ( member_term_a_b @ S @ ( set_term_a_b2 @ Ss ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( fun_a_b @ F2 @ Ss ) @ S ) @ subterm_and_supt_a_b ) ) ).

% supt.arg
thf(fact_747_supt_Osubt,axiom,
    ! [S: term_a_b,Ss: list_term_a_b,T: term_a_b,F2: a] :
      ( ( member_term_a_b @ S @ ( set_term_a_b2 @ Ss ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( fun_a_b @ F2 @ Ss ) @ T ) @ subterm_and_supt_a_b ) ) ) ).

% supt.subt
thf(fact_748_supt_Ocases,axiom,
    ! [A1: term_a_b,A22: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ subterm_and_supt_a_b )
     => ( ! [S4: term_a_b,Ss2: list_term_a_b] :
            ( ? [F3: a] :
                ( A1
                = ( fun_a_b @ F3 @ Ss2 ) )
           => ( ( A22 = S4 )
             => ~ ( member_term_a_b @ S4 @ ( set_term_a_b2 @ Ss2 ) ) ) )
       => ~ ! [S4: term_a_b,Ss2: list_term_a_b] :
              ( ? [F3: a] :
                  ( A1
                  = ( fun_a_b @ F3 @ Ss2 ) )
             => ( ( member_term_a_b @ S4 @ ( set_term_a_b2 @ Ss2 ) )
               => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ A22 ) @ subterm_and_supt_a_b ) ) ) ) ) ).

% supt.cases
thf(fact_749_supt_Osimps,axiom,
    ! [A1: term_a_b,A22: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ subterm_and_supt_a_b )
      = ( ? [S3: term_a_b,Ss3: list_term_a_b,F: a] :
            ( ( A1
              = ( fun_a_b @ F @ Ss3 ) )
            & ( A22 = S3 )
            & ( member_term_a_b @ S3 @ ( set_term_a_b2 @ Ss3 ) ) )
        | ? [S3: term_a_b,Ss3: list_term_a_b,T2: term_a_b,F: a] :
            ( ( A1
              = ( fun_a_b @ F @ Ss3 ) )
            & ( A22 = T2 )
            & ( member_term_a_b @ S3 @ ( set_term_a_b2 @ Ss3 ) )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S3 @ T2 ) @ subterm_and_supt_a_b ) ) ) ) ).

% supt.simps
thf(fact_750_supt__Fun,axiom,
    ! [S: term_a_b,F2: a,Ss: list_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ ( fun_a_b @ F2 @ Ss ) ) @ subterm_and_supt_a_b )
     => ~ ( member_term_a_b @ S @ ( set_term_a_b2 @ Ss ) ) ) ).

% supt_Fun
thf(fact_751_list__inter_Osimps_I2_J,axiom,
    ! [A: product_prod_a_nat,Bs2: list_P3592885314253461005_a_nat,As2: list_P3592885314253461005_a_nat] :
      ( ( ( member5724188588386418708_a_nat @ A @ ( set_Pr924983374503034536_a_nat @ Bs2 ) )
       => ( ( missin2021864273562616452_a_nat @ ( cons_P5205166803686508359_a_nat @ A @ As2 ) @ Bs2 )
          = ( cons_P5205166803686508359_a_nat @ A @ ( missin2021864273562616452_a_nat @ As2 @ Bs2 ) ) ) )
      & ( ~ ( member5724188588386418708_a_nat @ A @ ( set_Pr924983374503034536_a_nat @ Bs2 ) )
       => ( ( missin2021864273562616452_a_nat @ ( cons_P5205166803686508359_a_nat @ A @ As2 ) @ Bs2 )
          = ( missin2021864273562616452_a_nat @ As2 @ Bs2 ) ) ) ) ).

% list_inter.simps(2)
thf(fact_752_list__inter_Osimps_I2_J,axiom,
    ! [A: list_nat,Bs2: list_list_nat,As2: list_list_nat] :
      ( ( ( member_list_nat @ A @ ( set_list_nat2 @ Bs2 ) )
       => ( ( missin6532874241183986279st_nat @ ( cons_list_nat @ A @ As2 ) @ Bs2 )
          = ( cons_list_nat @ A @ ( missin6532874241183986279st_nat @ As2 @ Bs2 ) ) ) )
      & ( ~ ( member_list_nat @ A @ ( set_list_nat2 @ Bs2 ) )
       => ( ( missin6532874241183986279st_nat @ ( cons_list_nat @ A @ As2 ) @ Bs2 )
          = ( missin6532874241183986279st_nat @ As2 @ Bs2 ) ) ) ) ).

% list_inter.simps(2)
thf(fact_753_list__inter_Osimps_I2_J,axiom,
    ! [A: nat,Bs2: list_nat,As2: list_nat] :
      ( ( ( member_nat @ A @ ( set_nat2 @ Bs2 ) )
       => ( ( missin6377695591745783511er_nat @ ( cons_nat @ A @ As2 ) @ Bs2 )
          = ( cons_nat @ A @ ( missin6377695591745783511er_nat @ As2 @ Bs2 ) ) ) )
      & ( ~ ( member_nat @ A @ ( set_nat2 @ Bs2 ) )
       => ( ( missin6377695591745783511er_nat @ ( cons_nat @ A @ As2 ) @ Bs2 )
          = ( missin6377695591745783511er_nat @ As2 @ Bs2 ) ) ) ) ).

% list_inter.simps(2)
thf(fact_754_Fun__supt,axiom,
    ! [F2: a,Ts: list_term_a_b,S: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( fun_a_b @ F2 @ Ts ) @ S ) @ subterm_and_supt_a_b )
     => ~ ! [T3: term_a_b] :
            ( ( member_term_a_b @ T3 @ ( set_term_a_b2 @ Ts ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T3 @ S ) @ subter523971068842742411eq_a_b ) ) ) ).

% Fun_supt
thf(fact_755_set__supteq__into__supt,axiom,
    ! [T: term_a_b,Ts: list_term_a_b,S: term_a_b,F2: a] :
      ( ( member_term_a_b @ T @ ( set_term_a_b2 @ Ts ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( fun_a_b @ F2 @ Ts ) @ S ) @ subterm_and_supt_a_b ) ) ) ).

% set_supteq_into_supt
thf(fact_756_sorted__list__subset_Oelims,axiom,
    ! [X: list_nat,Xa2: list_nat,Y: option_nat] :
      ( ( ( sorted3362290152684031886et_nat @ X @ Xa2 )
        = Y )
     => ( ! [A3: nat,As: list_nat] :
            ( ( X
              = ( cons_nat @ A3 @ As ) )
           => ! [B3: nat,Bs: list_nat] :
                ( ( Xa2
                  = ( cons_nat @ B3 @ Bs ) )
               => ~ ( ( ( A3 = B3 )
                     => ( Y
                        = ( sorted3362290152684031886et_nat @ As @ ( cons_nat @ B3 @ Bs ) ) ) )
                    & ( ( A3 != B3 )
                     => ( ( ( ord_less_nat @ B3 @ A3 )
                         => ( Y
                            = ( sorted3362290152684031886et_nat @ ( cons_nat @ A3 @ As ) @ Bs ) ) )
                        & ( ~ ( ord_less_nat @ B3 @ A3 )
                         => ( Y
                            = ( some_nat @ A3 ) ) ) ) ) ) ) )
       => ( ( ( X = nil_nat )
           => ( Y != none_nat ) )
         => ~ ! [A3: nat] :
                ( ? [Uv2: list_nat] :
                    ( X
                    = ( cons_nat @ A3 @ Uv2 ) )
               => ( ( Xa2 = nil_nat )
                 => ( Y
                   != ( some_nat @ A3 ) ) ) ) ) ) ) ).

% sorted_list_subset.elims
thf(fact_757_subset__code_I3_J,axiom,
    ~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).

% subset_code(3)
thf(fact_758_subseqs_Osimps_I1_J,axiom,
    ( ( subseqs_nat @ nil_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% subseqs.simps(1)
thf(fact_759_subseqs__length__simple,axiom,
    ! [B: list_nat,Xs2: list_nat] :
      ( ( member_list_nat @ B @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) )
     => ( ord_less_eq_nat @ ( size_size_list_nat @ B ) @ ( size_size_list_nat @ Xs2 ) ) ) ).

% subseqs_length_simple
thf(fact_760_subseqs__length__simple,axiom,
    ! [B: list_term_a_b,Xs2: list_term_a_b] :
      ( ( member_list_term_a_b @ B @ ( set_list_term_a_b2 @ ( subseqs_term_a_b @ Xs2 ) ) )
     => ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ B ) @ ( size_s8906293707977694520rm_a_b @ Xs2 ) ) ) ).

% subseqs_length_simple
thf(fact_761_subseqs__refl,axiom,
    ! [Xs2: list_nat] : ( member_list_nat @ Xs2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) ) ).

% subseqs_refl
thf(fact_762_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_763_extract__Nil__code,axiom,
    ! [P2: nat > $o] :
      ( ( extract_nat @ P2 @ nil_nat )
      = none_P7817216376769613445st_nat ) ).

% extract_Nil_code
thf(fact_764_Cons__in__subseqsD,axiom,
    ! [Y: nat,Ys: list_nat,Xs2: list_nat] :
      ( ( member_list_nat @ ( cons_nat @ Y @ Ys ) @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) )
     => ( member_list_nat @ Ys @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_765_subset__code_I2_J,axiom,
    ! [A5: set_Pr4934435412358123699_a_nat,Ys: list_P3592885314253461005_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ A5 @ ( coset_6037984010582723146_a_nat @ Ys ) )
      = ( ! [X4: product_prod_a_nat] :
            ( ( member5724188588386418708_a_nat @ X4 @ ( set_Pr924983374503034536_a_nat @ Ys ) )
           => ~ ( member5724188588386418708_a_nat @ X4 @ A5 ) ) ) ) ).

% subset_code(2)
thf(fact_766_subset__code_I2_J,axiom,
    ! [A5: set_list_nat,Ys: list_list_nat] :
      ( ( ord_le6045566169113846134st_nat @ A5 @ ( coset_list_nat @ Ys ) )
      = ( ! [X4: list_nat] :
            ( ( member_list_nat @ X4 @ ( set_list_nat2 @ Ys ) )
           => ~ ( member_list_nat @ X4 @ A5 ) ) ) ) ).

% subset_code(2)
thf(fact_767_subseqs__length__simple__False,axiom,
    ! [B: list_nat,Xs2: list_nat] :
      ( ( member_list_nat @ B @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) )
     => ~ ( ord_less_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ B ) ) ) ).

% subseqs_length_simple_False
thf(fact_768_subseqs__length__simple__False,axiom,
    ! [B: list_term_a_b,Xs2: list_term_a_b] :
      ( ( member_list_term_a_b @ B @ ( set_list_term_a_b2 @ ( subseqs_term_a_b @ Xs2 ) ) )
     => ~ ( ord_less_nat @ ( size_s8906293707977694520rm_a_b @ Xs2 ) @ ( size_s8906293707977694520rm_a_b @ B ) ) ) ).

% subseqs_length_simple_False
thf(fact_769_root_Oelims,axiom,
    ! [X: term_a_b,Y: option5551091909395471437_a_nat] :
      ( ( ( root_a_b @ X )
        = Y )
     => ( ( ? [X3: b] :
              ( X
              = ( var_b_a @ X3 ) )
         => ( Y != none_P4102489432868120540_a_nat ) )
       => ~ ! [F3: a,Ts2: list_term_a_b] :
              ( ( X
                = ( fun_a_b @ F3 @ Ts2 ) )
             => ( Y
               != ( some_P6251353102471802712_a_nat @ ( product_Pair_a_nat @ F3 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) ) ) ) ) ) ) ).

% root.elims
thf(fact_770_fun__at_Osimps_I3_J,axiom,
    ! [I: nat,Ts: list_term_a_b,F2: a,P: list_nat] :
      ( ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ts ) )
       => ( ( term_fun_at_a_b @ ( fun_a_b @ F2 @ Ts ) @ ( cons_nat @ I @ P ) )
          = ( term_fun_at_a_b @ ( nth_term_a_b @ Ts @ I ) @ P ) ) )
      & ( ~ ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ts ) )
       => ( ( term_fun_at_a_b @ ( fun_a_b @ F2 @ Ts ) @ ( cons_nat @ I @ P ) )
          = none_Sum_sum_a_b ) ) ) ).

% fun_at.simps(3)
thf(fact_771_sorted__list__subset_Opelims,axiom,
    ! [X: list_nat,Xa2: list_nat,Y: option_nat] :
      ( ( ( sorted3362290152684031886et_nat @ X @ Xa2 )
        = Y )
     => ( ( accp_P8037286306265792042st_nat @ sorted7771632751644887693el_nat @ ( produc2694037385005941721st_nat @ X @ Xa2 ) )
       => ( ! [A3: nat,As: list_nat] :
              ( ( X
                = ( cons_nat @ A3 @ As ) )
             => ! [B3: nat,Bs: list_nat] :
                  ( ( Xa2
                    = ( cons_nat @ B3 @ Bs ) )
                 => ( ( ( ( A3 = B3 )
                       => ( Y
                          = ( sorted3362290152684031886et_nat @ As @ ( cons_nat @ B3 @ Bs ) ) ) )
                      & ( ( A3 != B3 )
                       => ( ( ( ord_less_nat @ B3 @ A3 )
                           => ( Y
                              = ( sorted3362290152684031886et_nat @ ( cons_nat @ A3 @ As ) @ Bs ) ) )
                          & ( ~ ( ord_less_nat @ B3 @ A3 )
                           => ( Y
                              = ( some_nat @ A3 ) ) ) ) ) )
                   => ~ ( accp_P8037286306265792042st_nat @ sorted7771632751644887693el_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ A3 @ As ) @ ( cons_nat @ B3 @ Bs ) ) ) ) ) )
         => ( ( ( X = nil_nat )
             => ( ( Y = none_nat )
               => ~ ( accp_P8037286306265792042st_nat @ sorted7771632751644887693el_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xa2 ) ) ) )
           => ~ ! [A3: nat,Uv2: list_nat] :
                  ( ( X
                    = ( cons_nat @ A3 @ Uv2 ) )
                 => ( ( Xa2 = nil_nat )
                   => ( ( Y
                        = ( some_nat @ A3 ) )
                     => ~ ( accp_P8037286306265792042st_nat @ sorted7771632751644887693el_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ A3 @ Uv2 ) @ nil_nat ) ) ) ) ) ) ) ) ) ).

% sorted_list_subset.pelims
thf(fact_772_union__list__sorted_Oelims,axiom,
    ! [X: list_nat,Xa2: list_nat,Y: list_nat] :
      ( ( ( missin8019018944680490243ed_nat @ X @ Xa2 )
        = Y )
     => ( ! [X3: nat,Xs: list_nat] :
            ( ( X
              = ( cons_nat @ X3 @ Xs ) )
           => ! [Y3: nat,Ys4: list_nat] :
                ( ( Xa2
                  = ( cons_nat @ Y3 @ Ys4 ) )
               => ~ ( ( ( X3 = Y3 )
                     => ( Y
                        = ( cons_nat @ X3 @ ( missin8019018944680490243ed_nat @ Xs @ Ys4 ) ) ) )
                    & ( ( X3 != Y3 )
                     => ( ( ( ord_less_nat @ X3 @ Y3 )
                         => ( Y
                            = ( cons_nat @ X3 @ ( missin8019018944680490243ed_nat @ Xs @ ( cons_nat @ Y3 @ Ys4 ) ) ) ) )
                        & ( ~ ( ord_less_nat @ X3 @ Y3 )
                         => ( Y
                            = ( cons_nat @ Y3 @ ( missin8019018944680490243ed_nat @ ( cons_nat @ X3 @ Xs ) @ Ys4 ) ) ) ) ) ) ) ) )
       => ( ( ( X = nil_nat )
           => ( Y != Xa2 ) )
         => ~ ! [V2: nat,Va: list_nat] :
                ( ( X
                  = ( cons_nat @ V2 @ Va ) )
               => ( ( Xa2 = nil_nat )
                 => ( Y
                   != ( cons_nat @ V2 @ Va ) ) ) ) ) ) ) ).

% union_list_sorted.elims
thf(fact_773_nth__append__length,axiom,
    ! [Xs2: list_nat,X: nat,Ys: list_nat] :
      ( ( nth_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs2 ) )
      = X ) ).

% nth_append_length
thf(fact_774_nth__append__length,axiom,
    ! [Xs2: list_term_a_b,X: term_a_b,Ys: list_term_a_b] :
      ( ( nth_term_a_b @ ( append_term_a_b @ Xs2 @ ( cons_term_a_b @ X @ Ys ) ) @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
      = X ) ).

% nth_append_length
thf(fact_775_nth__append__length__plus,axiom,
    ! [Xs2: list_term_a_b,Ys: list_term_a_b,N: nat] :
      ( ( nth_term_a_b @ ( append_term_a_b @ Xs2 @ Ys ) @ ( plus_plus_nat @ ( size_s8906293707977694520rm_a_b @ Xs2 ) @ N ) )
      = ( nth_term_a_b @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_776_funas__term_Ocases,axiom,
    ! [X: term_a_b] :
      ( ! [X3: b] :
          ( X
         != ( var_b_a @ X3 ) )
     => ~ ! [F3: a,Ts2: list_term_a_b] :
            ( X
           != ( fun_a_b @ F3 @ Ts2 ) ) ) ).

% funas_term.cases
thf(fact_777_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_778_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_779_linear__term_Ocases,axiom,
    ! [X: term_a_b] :
      ( ! [Uu: b] :
          ( X
         != ( var_b_a @ Uu ) )
     => ~ ! [Uv2: a,Ts2: list_term_a_b] :
            ( X
           != ( fun_a_b @ Uv2 @ Ts2 ) ) ) ).

% linear_term.cases
thf(fact_780_nth__enumerate__eq,axiom,
    ! [M: nat,Xs2: list_term_a_b,N: nat] :
      ( ( ord_less_nat @ M @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( ( nth_Pr5300001119514325452rm_a_b @ ( enumerate_term_a_b @ N @ Xs2 ) @ M )
        = ( produc1516572978046417917rm_a_b @ ( plus_plus_nat @ N @ M ) @ ( nth_term_a_b @ Xs2 @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_781_nth__equalityI,axiom,
    ! [Xs2: list_term_a_b,Ys: list_term_a_b] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
           => ( ( nth_term_a_b @ Xs2 @ I2 )
              = ( nth_term_a_b @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_782_Skolem__list__nth,axiom,
    ! [K: nat,P2: nat > term_a_b > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X6: term_a_b] : ( P2 @ I4 @ X6 ) ) )
      = ( ? [Xs3: list_term_a_b] :
            ( ( ( size_s8906293707977694520rm_a_b @ Xs3 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P2 @ I4 @ ( nth_term_a_b @ Xs3 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_783_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y4: list_term_a_b,Z2: list_term_a_b] : ( Y4 = Z2 ) )
    = ( ^ [Xs3: list_term_a_b,Ys3: list_term_a_b] :
          ( ( ( size_s8906293707977694520rm_a_b @ Xs3 )
            = ( size_s8906293707977694520rm_a_b @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs3 ) )
             => ( ( nth_term_a_b @ Xs3 @ I4 )
                = ( nth_term_a_b @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_784_Ex__list__of__length__P,axiom,
    ! [N: nat,P2: term_a_b > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ? [X5: term_a_b] : ( P2 @ X5 @ I2 ) )
     => ? [Xs: list_term_a_b] :
          ( ( ( size_s8906293707977694520rm_a_b @ Xs )
            = N )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ N )
             => ( P2 @ ( nth_term_a_b @ Xs @ I3 ) @ I3 ) ) ) ) ).

% Ex_list_of_length_P
thf(fact_785_term__to__sig_Ocases,axiom,
    ! [X: produc5279506192219892694rm_a_b] :
      ( ! [F4: set_Pr4934435412358123699_a_nat,V2: b,X3: b] :
          ( X
         != ( produc8030969961714872974rm_a_b @ F4 @ ( produc1437816968797971900rm_a_b @ V2 @ ( var_b_a @ X3 ) ) ) )
     => ~ ! [F4: set_Pr4934435412358123699_a_nat,V2: b,F3: a,Ts2: list_term_a_b] :
            ( X
           != ( produc8030969961714872974rm_a_b @ F4 @ ( produc1437816968797971900rm_a_b @ V2 @ ( fun_a_b @ F3 @ Ts2 ) ) ) ) ) ).

% term_to_sig.cases
thf(fact_786_depth_Ocases,axiom,
    ! [X: term_a_b] :
      ( ! [X3: b] :
          ( X
         != ( var_b_a @ X3 ) )
     => ( ! [F3: a] :
            ( X
           != ( fun_a_b @ F3 @ nil_term_a_b ) )
       => ~ ! [F3: a,V2: term_a_b,Va: list_term_a_b] :
              ( X
             != ( fun_a_b @ F3 @ ( cons_term_a_b @ V2 @ Va ) ) ) ) ) ).

% depth.cases
thf(fact_787_all__set__conv__all__nth,axiom,
    ! [Xs2: list_term_a_b,P2: term_a_b > $o] :
      ( ( ! [X4: term_a_b] :
            ( ( member_term_a_b @ X4 @ ( set_term_a_b2 @ Xs2 ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
           => ( P2 @ ( nth_term_a_b @ Xs2 @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_788_all__nth__imp__all__set,axiom,
    ! [Xs2: list_P3592885314253461005_a_nat,P2: product_prod_a_nat > $o,X: product_prod_a_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s984997627204368545_a_nat @ Xs2 ) )
         => ( P2 @ ( nth_Pr8461465654520414006_a_nat @ Xs2 @ I2 ) ) )
     => ( ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Xs2 ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_789_all__nth__imp__all__set,axiom,
    ! [Xs2: list_list_nat,P2: list_nat > $o,X: list_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
         => ( P2 @ ( nth_list_nat @ Xs2 @ I2 ) ) )
     => ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_790_all__nth__imp__all__set,axiom,
    ! [Xs2: list_term_a_b,P2: term_a_b > $o,X: term_a_b] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
         => ( P2 @ ( nth_term_a_b @ Xs2 @ I2 ) ) )
     => ( ( member_term_a_b @ X @ ( set_term_a_b2 @ Xs2 ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_791_in__set__conv__nth,axiom,
    ! [X: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat] :
      ( ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Xs2 ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s984997627204368545_a_nat @ Xs2 ) )
            & ( ( nth_Pr8461465654520414006_a_nat @ Xs2 @ I4 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_792_in__set__conv__nth,axiom,
    ! [X: list_nat,Xs2: list_list_nat] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
            & ( ( nth_list_nat @ Xs2 @ I4 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_793_in__set__conv__nth,axiom,
    ! [X: term_a_b,Xs2: list_term_a_b] :
      ( ( member_term_a_b @ X @ ( set_term_a_b2 @ Xs2 ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
            & ( ( nth_term_a_b @ Xs2 @ I4 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_794_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_term_a_b,P2: term_a_b > $o] :
      ( ( ord_less_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( ! [X3: term_a_b] :
            ( ( member_term_a_b @ X3 @ ( set_term_a_b2 @ Xs2 ) )
           => ( P2 @ X3 ) )
       => ( P2 @ ( nth_term_a_b @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_795_nth__mem,axiom,
    ! [N: nat,Xs2: list_P3592885314253461005_a_nat] :
      ( ( ord_less_nat @ N @ ( size_s984997627204368545_a_nat @ Xs2 ) )
     => ( member5724188588386418708_a_nat @ ( nth_Pr8461465654520414006_a_nat @ Xs2 @ N ) @ ( set_Pr924983374503034536_a_nat @ Xs2 ) ) ) ).

% nth_mem
thf(fact_796_nth__mem,axiom,
    ! [N: nat,Xs2: list_list_nat] :
      ( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs2 ) )
     => ( member_list_nat @ ( nth_list_nat @ Xs2 @ N ) @ ( set_list_nat2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_797_nth__mem,axiom,
    ! [N: nat,Xs2: list_term_a_b] :
      ( ( ord_less_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( member_term_a_b @ ( nth_term_a_b @ Xs2 @ N ) @ ( set_term_a_b2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_798_ex__set__conv__ex__nth,axiom,
    ! [Xs2: list_term_a_b,P2: term_a_b > $o] :
      ( ( ? [X4: term_a_b] :
            ( ( member_term_a_b @ X4 @ ( set_term_a_b2 @ Xs2 ) )
            & ( P2 @ X4 ) ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
            & ( P2 @ ( nth_term_a_b @ Xs2 @ I4 ) ) ) ) ) ).

% ex_set_conv_ex_nth
thf(fact_799_append__Cons__nth__middle,axiom,
    ! [I: nat,Xs2: list_nat,Y: nat,Zs: list_nat] :
      ( ( I
        = ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_nat @ ( append_nat @ Xs2 @ ( cons_nat @ Y @ Zs ) ) @ I )
        = Y ) ) ).

% append_Cons_nth_middle
thf(fact_800_append__Cons__nth__middle,axiom,
    ! [I: nat,Xs2: list_term_a_b,Y: term_a_b,Zs: list_term_a_b] :
      ( ( I
        = ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( ( nth_term_a_b @ ( append_term_a_b @ Xs2 @ ( cons_term_a_b @ Y @ Zs ) ) @ I )
        = Y ) ) ).

% append_Cons_nth_middle
thf(fact_801_append__Cons__nth__not__middle,axiom,
    ! [I: nat,Xs2: list_nat,U: nat,Ys: list_nat,Z: nat] :
      ( ( I
       != ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_nat @ ( append_nat @ Xs2 @ ( cons_nat @ U @ Ys ) ) @ I )
        = ( nth_nat @ ( append_nat @ Xs2 @ ( cons_nat @ Z @ Ys ) ) @ I ) ) ) ).

% append_Cons_nth_not_middle
thf(fact_802_append__Cons__nth__not__middle,axiom,
    ! [I: nat,Xs2: list_term_a_b,U: term_a_b,Ys: list_term_a_b,Z: term_a_b] :
      ( ( I
       != ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( ( nth_term_a_b @ ( append_term_a_b @ Xs2 @ ( cons_term_a_b @ U @ Ys ) ) @ I )
        = ( nth_term_a_b @ ( append_term_a_b @ Xs2 @ ( cons_term_a_b @ Z @ Ys ) ) @ I ) ) ) ).

% append_Cons_nth_not_middle
thf(fact_803_P__as__bs__extend,axiom,
    ! [As2: list_term_a_b,Bs2: list_term_a_b,Cs: list_term_a_b,Ds: list_term_a_b,P2: term_a_b > term_a_b > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ As2 )
        = ( size_s8906293707977694520rm_a_b @ Bs2 ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Cs )
          = ( size_s8906293707977694520rm_a_b @ Ds ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Bs2 ) )
             => ( P2 @ ( nth_term_a_b @ As2 @ I2 ) @ ( nth_term_a_b @ Bs2 @ I2 ) ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ds ) )
               => ( P2 @ ( nth_term_a_b @ Cs @ I2 ) @ ( nth_term_a_b @ Ds @ I2 ) ) )
           => ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ ( append_term_a_b @ Bs2 @ Ds ) ) )
               => ( P2 @ ( nth_term_a_b @ ( append_term_a_b @ As2 @ Cs ) @ I3 ) @ ( nth_term_a_b @ ( append_term_a_b @ Bs2 @ Ds ) @ I3 ) ) ) ) ) ) ) ).

% P_as_bs_extend
thf(fact_804_replace__term__at_Ocases,axiom,
    ! [X: produc2732850333517536310rm_a_b] :
      ( ! [S4: term_a_b,T3: term_a_b] :
          ( X
         != ( produc3812856575676843240rm_a_b @ S4 @ ( produc5151171985953862413rm_a_b @ nil_nat @ T3 ) ) )
     => ( ! [X3: b,V2: nat,Va: list_nat,T3: term_a_b] :
            ( X
           != ( produc3812856575676843240rm_a_b @ ( var_b_a @ X3 ) @ ( produc5151171985953862413rm_a_b @ ( cons_nat @ V2 @ Va ) @ T3 ) ) )
       => ~ ! [F3: a,Ts2: list_term_a_b,I2: nat,Ps2: list_nat,T3: term_a_b] :
              ( X
             != ( produc3812856575676843240rm_a_b @ ( fun_a_b @ F3 @ Ts2 ) @ ( produc5151171985953862413rm_a_b @ ( cons_nat @ I2 @ Ps2 ) @ T3 ) ) ) ) ) ).

% replace_term_at.cases
thf(fact_805_ctxt__at__pos_Ocases,axiom,
    ! [X: produc3697673438841856213st_nat] :
      ( ! [S4: term_a_b] :
          ( X
         != ( produc4563063199488751885st_nat @ S4 @ nil_nat ) )
     => ( ! [F3: a,Ss2: list_term_a_b,I2: nat,P4: list_nat] :
            ( X
           != ( produc4563063199488751885st_nat @ ( fun_a_b @ F3 @ Ss2 ) @ ( cons_nat @ I2 @ P4 ) ) )
       => ~ ! [X3: b,V2: nat,Va: list_nat] :
              ( X
             != ( produc4563063199488751885st_nat @ ( var_b_a @ X3 ) @ ( cons_nat @ V2 @ Va ) ) ) ) ) ).

% ctxt_at_pos.cases
thf(fact_806_subt__at_Ocases,axiom,
    ! [X: produc3697673438841856213st_nat] :
      ( ! [S4: term_a_b] :
          ( X
         != ( produc4563063199488751885st_nat @ S4 @ nil_nat ) )
     => ( ! [F3: a,Ss2: list_term_a_b,I2: nat,P4: list_nat] :
            ( X
           != ( produc4563063199488751885st_nat @ ( fun_a_b @ F3 @ Ss2 ) @ ( cons_nat @ I2 @ P4 ) ) )
       => ~ ! [X3: b,V2: nat,Va: list_nat] :
              ( X
             != ( produc4563063199488751885st_nat @ ( var_b_a @ X3 ) @ ( cons_nat @ V2 @ Va ) ) ) ) ) ).

% subt_at.cases
thf(fact_807_fun__at_Ocases,axiom,
    ! [X: produc3697673438841856213st_nat] :
      ( ! [X3: b] :
          ( X
         != ( produc4563063199488751885st_nat @ ( var_b_a @ X3 ) @ nil_nat ) )
     => ( ! [F3: a,Ts2: list_term_a_b] :
            ( X
           != ( produc4563063199488751885st_nat @ ( fun_a_b @ F3 @ Ts2 ) @ nil_nat ) )
       => ( ! [F3: a,Ts2: list_term_a_b,I2: nat,P4: list_nat] :
              ( X
             != ( produc4563063199488751885st_nat @ ( fun_a_b @ F3 @ Ts2 ) @ ( cons_nat @ I2 @ P4 ) ) )
         => ~ ! [Vb: b,V2: nat,Va: list_nat] :
                ( X
               != ( produc4563063199488751885st_nat @ ( var_b_a @ Vb ) @ ( cons_nat @ V2 @ Va ) ) ) ) ) ) ).

% fun_at.cases
thf(fact_808_subt__at_Osimps_I2_J,axiom,
    ! [F2: a,Ss: list_term_a_b,I: nat,P: list_nat] :
      ( ( term_subt_at_a_b @ ( fun_a_b @ F2 @ Ss ) @ ( cons_nat @ I @ P ) )
      = ( term_subt_at_a_b @ ( nth_term_a_b @ Ss @ I ) @ P ) ) ).

% subt_at.simps(2)
thf(fact_809_union__list__sorted_Osimps_I1_J,axiom,
    ! [X: nat,Y: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( ( X = Y )
       => ( ( missin8019018944680490243ed_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys ) )
          = ( cons_nat @ X @ ( missin8019018944680490243ed_nat @ Xs2 @ Ys ) ) ) )
      & ( ( X != Y )
       => ( ( ( ord_less_nat @ X @ Y )
           => ( ( missin8019018944680490243ed_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys ) )
              = ( cons_nat @ X @ ( missin8019018944680490243ed_nat @ Xs2 @ ( cons_nat @ Y @ Ys ) ) ) ) )
          & ( ~ ( ord_less_nat @ X @ Y )
           => ( ( missin8019018944680490243ed_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys ) )
              = ( cons_nat @ Y @ ( missin8019018944680490243ed_nat @ ( cons_nat @ X @ Xs2 ) @ Ys ) ) ) ) ) ) ) ).

% union_list_sorted.simps(1)
thf(fact_810_union__list__sorted_Osimps_I3_J,axiom,
    ! [V: nat,Va2: list_nat] :
      ( ( missin8019018944680490243ed_nat @ ( cons_nat @ V @ Va2 ) @ nil_nat )
      = ( cons_nat @ V @ Va2 ) ) ).

% union_list_sorted.simps(3)
thf(fact_811_poss__of__term__Cons,axiom,
    ! [I: nat,P: list_nat,U: term_a_b,F2: a,Ts: list_term_a_b] :
      ( ( member_list_nat @ ( cons_nat @ I @ P ) @ ( terms_7168686267159881682rm_a_b @ U @ ( fun_a_b @ F2 @ Ts ) ) )
     => ( member_list_nat @ P @ ( terms_7168686267159881682rm_a_b @ U @ ( nth_term_a_b @ Ts @ I ) ) ) ) ).

% poss_of_term_Cons
thf(fact_812_append__Cons__nth__right,axiom,
    ! [Xs2: list_nat,I: nat,U: nat,Ys: list_nat,Z: nat] :
      ( ( ord_less_nat @ ( size_size_list_nat @ Xs2 ) @ I )
     => ( ( nth_nat @ ( append_nat @ Xs2 @ ( cons_nat @ U @ Ys ) ) @ I )
        = ( nth_nat @ ( append_nat @ Xs2 @ ( cons_nat @ Z @ Ys ) ) @ I ) ) ) ).

% append_Cons_nth_right
thf(fact_813_append__Cons__nth__right,axiom,
    ! [Xs2: list_term_a_b,I: nat,U: term_a_b,Ys: list_term_a_b,Z: term_a_b] :
      ( ( ord_less_nat @ ( size_s8906293707977694520rm_a_b @ Xs2 ) @ I )
     => ( ( nth_term_a_b @ ( append_term_a_b @ Xs2 @ ( cons_term_a_b @ U @ Ys ) ) @ I )
        = ( nth_term_a_b @ ( append_term_a_b @ Xs2 @ ( cons_term_a_b @ Z @ Ys ) ) @ I ) ) ) ).

% append_Cons_nth_right
thf(fact_814_append__Cons__nth__left,axiom,
    ! [I: nat,Xs2: list_nat,U: nat,Ys: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_nat @ ( append_nat @ Xs2 @ ( cons_nat @ U @ Ys ) ) @ I )
        = ( nth_nat @ Xs2 @ I ) ) ) ).

% append_Cons_nth_left
thf(fact_815_append__Cons__nth__left,axiom,
    ! [I: nat,Xs2: list_term_a_b,U: term_a_b,Ys: list_term_a_b] :
      ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( ( nth_term_a_b @ ( append_term_a_b @ Xs2 @ ( cons_term_a_b @ U @ Ys ) ) @ I )
        = ( nth_term_a_b @ Xs2 @ I ) ) ) ).

% append_Cons_nth_left
thf(fact_816_all__ctxt__closedD,axiom,
    ! [F5: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b,F2: a,Ss: list_term_a_b,Ts: list_term_a_b] :
      ( ( terms_5226143800768910156ed_a_b @ F5 @ R )
     => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F2 @ ( size_s8906293707977694520rm_a_b @ Ss ) ) @ F5 )
       => ( ( ( size_s8906293707977694520rm_a_b @ Ts )
            = ( size_s8906293707977694520rm_a_b @ Ss ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts ) )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Ts @ I2 ) @ ( nth_term_a_b @ Ss @ I2 ) ) @ R ) )
           => ( ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts ) )
                 => ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( nth_term_a_b @ Ts @ I2 ) ) @ F5 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts ) )
                   => ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( nth_term_a_b @ Ss @ I2 ) ) @ F5 ) )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( fun_a_b @ F2 @ Ts ) @ ( fun_a_b @ F2 @ Ss ) ) @ R ) ) ) ) ) ) ) ).

% all_ctxt_closedD
thf(fact_817_list__inter_Opelims,axiom,
    ! [X: list_P3592885314253461005_a_nat,Xa2: list_P3592885314253461005_a_nat,Y: list_P3592885314253461005_a_nat] :
      ( ( ( missin2021864273562616452_a_nat @ X @ Xa2 )
        = Y )
     => ( ( accp_P1855726166071684_a_nat @ missin2597475645481178417_a_nat @ ( produc5384655689722402227_a_nat @ X @ Xa2 ) )
       => ( ( ( X = nil_Pr7402525243500994295_a_nat )
           => ( ( Y = nil_Pr7402525243500994295_a_nat )
             => ~ ( accp_P1855726166071684_a_nat @ missin2597475645481178417_a_nat @ ( produc5384655689722402227_a_nat @ nil_Pr7402525243500994295_a_nat @ Xa2 ) ) ) )
         => ~ ! [A3: product_prod_a_nat,As: list_P3592885314253461005_a_nat] :
                ( ( X
                  = ( cons_P5205166803686508359_a_nat @ A3 @ As ) )
               => ( ( ( ( member5724188588386418708_a_nat @ A3 @ ( set_Pr924983374503034536_a_nat @ Xa2 ) )
                     => ( Y
                        = ( cons_P5205166803686508359_a_nat @ A3 @ ( missin2021864273562616452_a_nat @ As @ Xa2 ) ) ) )
                    & ( ~ ( member5724188588386418708_a_nat @ A3 @ ( set_Pr924983374503034536_a_nat @ Xa2 ) )
                     => ( Y
                        = ( missin2021864273562616452_a_nat @ As @ Xa2 ) ) ) )
                 => ~ ( accp_P1855726166071684_a_nat @ missin2597475645481178417_a_nat @ ( produc5384655689722402227_a_nat @ ( cons_P5205166803686508359_a_nat @ A3 @ As ) @ Xa2 ) ) ) ) ) ) ) ).

% list_inter.pelims
thf(fact_818_list__inter_Opelims,axiom,
    ! [X: list_list_nat,Xa2: list_list_nat,Y: list_list_nat] :
      ( ( ( missin6532874241183986279st_nat @ X @ Xa2 )
        = Y )
     => ( ( accp_P5780766562878364362st_nat @ missin517122817877786516st_nat @ ( produc7129799990162260089st_nat @ X @ Xa2 ) )
       => ( ( ( X = nil_list_nat )
           => ( ( Y = nil_list_nat )
             => ~ ( accp_P5780766562878364362st_nat @ missin517122817877786516st_nat @ ( produc7129799990162260089st_nat @ nil_list_nat @ Xa2 ) ) ) )
         => ~ ! [A3: list_nat,As: list_list_nat] :
                ( ( X
                  = ( cons_list_nat @ A3 @ As ) )
               => ( ( ( ( member_list_nat @ A3 @ ( set_list_nat2 @ Xa2 ) )
                     => ( Y
                        = ( cons_list_nat @ A3 @ ( missin6532874241183986279st_nat @ As @ Xa2 ) ) ) )
                    & ( ~ ( member_list_nat @ A3 @ ( set_list_nat2 @ Xa2 ) )
                     => ( Y
                        = ( missin6532874241183986279st_nat @ As @ Xa2 ) ) ) )
                 => ~ ( accp_P5780766562878364362st_nat @ missin517122817877786516st_nat @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ A3 @ As ) @ Xa2 ) ) ) ) ) ) ) ).

% list_inter.pelims
thf(fact_819_list__inter_Opelims,axiom,
    ! [X: list_nat,Xa2: list_nat,Y: list_nat] :
      ( ( ( missin6377695591745783511er_nat @ X @ Xa2 )
        = Y )
     => ( ( accp_P8037286306265792042st_nat @ missin8053613324461657732el_nat @ ( produc2694037385005941721st_nat @ X @ Xa2 ) )
       => ( ( ( X = nil_nat )
           => ( ( Y = nil_nat )
             => ~ ( accp_P8037286306265792042st_nat @ missin8053613324461657732el_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xa2 ) ) ) )
         => ~ ! [A3: nat,As: list_nat] :
                ( ( X
                  = ( cons_nat @ A3 @ As ) )
               => ( ( ( ( member_nat @ A3 @ ( set_nat2 @ Xa2 ) )
                     => ( Y
                        = ( cons_nat @ A3 @ ( missin6377695591745783511er_nat @ As @ Xa2 ) ) ) )
                    & ( ~ ( member_nat @ A3 @ ( set_nat2 @ Xa2 ) )
                     => ( Y
                        = ( missin6377695591745783511er_nat @ As @ Xa2 ) ) ) )
                 => ~ ( accp_P8037286306265792042st_nat @ missin8053613324461657732el_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ A3 @ As ) @ Xa2 ) ) ) ) ) ) ) ).

% list_inter.pelims
thf(fact_820_set__list__subset__eq__nth__conv,axiom,
    ! [Xs2: list_P3592885314253461005_a_nat,A5: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ ( set_Pr924983374503034536_a_nat @ Xs2 ) @ A5 )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s984997627204368545_a_nat @ Xs2 ) )
           => ( member5724188588386418708_a_nat @ ( nth_Pr8461465654520414006_a_nat @ Xs2 @ I4 ) @ A5 ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_821_set__list__subset__eq__nth__conv,axiom,
    ! [Xs2: list_list_nat,A5: set_list_nat] :
      ( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ A5 )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
           => ( member_list_nat @ ( nth_list_nat @ Xs2 @ I4 ) @ A5 ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_822_set__list__subset__eq__nth__conv,axiom,
    ! [Xs2: list_term_a_b,A5: set_term_a_b] :
      ( ( ord_le2705286416250468010rm_a_b @ ( set_term_a_b2 @ Xs2 ) @ A5 )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
           => ( member_term_a_b @ ( nth_term_a_b @ Xs2 @ I4 ) @ A5 ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_823_union__list__sorted_Opelims,axiom,
    ! [X: list_nat,Xa2: list_nat,Y: list_nat] :
      ( ( ( missin8019018944680490243ed_nat @ X @ Xa2 )
        = Y )
     => ( ( accp_P8037286306265792042st_nat @ missin7611735270441097048el_nat @ ( produc2694037385005941721st_nat @ X @ Xa2 ) )
       => ( ! [X3: nat,Xs: list_nat] :
              ( ( X
                = ( cons_nat @ X3 @ Xs ) )
             => ! [Y3: nat,Ys4: list_nat] :
                  ( ( Xa2
                    = ( cons_nat @ Y3 @ Ys4 ) )
                 => ( ( ( ( X3 = Y3 )
                       => ( Y
                          = ( cons_nat @ X3 @ ( missin8019018944680490243ed_nat @ Xs @ Ys4 ) ) ) )
                      & ( ( X3 != Y3 )
                       => ( ( ( ord_less_nat @ X3 @ Y3 )
                           => ( Y
                              = ( cons_nat @ X3 @ ( missin8019018944680490243ed_nat @ Xs @ ( cons_nat @ Y3 @ Ys4 ) ) ) ) )
                          & ( ~ ( ord_less_nat @ X3 @ Y3 )
                           => ( Y
                              = ( cons_nat @ Y3 @ ( missin8019018944680490243ed_nat @ ( cons_nat @ X3 @ Xs ) @ Ys4 ) ) ) ) ) ) )
                   => ~ ( accp_P8037286306265792042st_nat @ missin7611735270441097048el_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) ) ) )
         => ( ( ( X = nil_nat )
             => ( ( Y = Xa2 )
               => ~ ( accp_P8037286306265792042st_nat @ missin7611735270441097048el_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xa2 ) ) ) )
           => ~ ! [V2: nat,Va: list_nat] :
                  ( ( X
                    = ( cons_nat @ V2 @ Va ) )
                 => ( ( Xa2 = nil_nat )
                   => ( ( Y
                        = ( cons_nat @ V2 @ Va ) )
                     => ~ ( accp_P8037286306265792042st_nat @ missin7611735270441097048el_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ V2 @ Va ) @ nil_nat ) ) ) ) ) ) ) ) ) ).

% union_list_sorted.pelims
thf(fact_824_in__set__idx,axiom,
    ! [X: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat] :
      ( ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Xs2 ) )
     => ? [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s984997627204368545_a_nat @ Xs2 ) )
          & ( ( nth_Pr8461465654520414006_a_nat @ Xs2 @ I2 )
            = X ) ) ) ).

% in_set_idx
thf(fact_825_in__set__idx,axiom,
    ! [X: list_nat,Xs2: list_list_nat] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
     => ? [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
          & ( ( nth_list_nat @ Xs2 @ I2 )
            = X ) ) ) ).

% in_set_idx
thf(fact_826_in__set__idx,axiom,
    ! [X: term_a_b,Xs2: list_term_a_b] :
      ( ( member_term_a_b @ X @ ( set_term_a_b2 @ Xs2 ) )
     => ? [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
          & ( ( nth_term_a_b @ Xs2 @ I2 )
            = X ) ) ) ).

% in_set_idx
thf(fact_827_all__ctxt__closed__def,axiom,
    ( terms_5226143800768910156ed_a_b
    = ( ^ [F6: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
          ( ! [F: a,Ts3: list_term_a_b,Ss3: list_term_a_b] :
              ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F @ ( size_s8906293707977694520rm_a_b @ Ss3 ) ) @ F6 )
             => ( ( ( size_s8906293707977694520rm_a_b @ Ts3 )
                  = ( size_s8906293707977694520rm_a_b @ Ss3 ) )
               => ( ! [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 @ Ss3 @ I4 ) ) @ R3 ) )
                 => ( ! [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 @ Ss3 @ I4 ) ) ) @ F6 ) )
                   => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( fun_a_b @ F @ Ts3 ) @ ( fun_a_b @ F @ Ss3 ) ) @ R3 ) ) ) ) )
          & ! [X4: b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( var_b_a @ X4 ) @ ( var_b_a @ X4 ) ) @ R3 ) ) ) ) ).

% all_ctxt_closed_def
thf(fact_828_permut__sound,axiom,
    ! [I: nat,As2: list_term_a_b,F2: nat > nat] :
      ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ As2 ) )
     => ( ( nth_term_a_b @ ( missin5417461770552926780rm_a_b @ As2 @ F2 ) @ I )
        = ( nth_term_a_b @ As2 @ ( F2 @ I ) ) ) ) ).

% permut_sound
thf(fact_829_gterm__of__term_Ocases,axiom,
    ! [X: term_a_b] :
      ( ! [F3: a,Ts2: list_term_a_b] :
          ( X
         != ( fun_a_b @ F3 @ Ts2 ) )
     => ~ ! [V2: b] :
            ( X
           != ( var_b_a @ V2 ) ) ) ).

% gterm_of_term.cases
thf(fact_830_le__sup__iff,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
      = ( ( ord_less_eq_nat @ X @ Z )
        & ( ord_less_eq_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_831_sup_Obounded__iff,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% sup.bounded_iff
thf(fact_832_inf__sup__ord_I4_J,axiom,
    ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_833_inf__sup__ord_I3_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_834_le__supE,axiom,
    ! [A: nat,B: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X )
     => ~ ( ( ord_less_eq_nat @ A @ X )
         => ~ ( ord_less_eq_nat @ B @ X ) ) ) ).

% le_supE
thf(fact_835_le__supI,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ X )
     => ( ( ord_less_eq_nat @ B @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_836_sup__ge1,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_837_sup__ge2,axiom,
    ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_838_le__supI1,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X @ A )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_839_le__supI2,axiom,
    ! [X: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ X @ B )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_840_sup_Omono,axiom,
    ! [C2: nat,A: nat,D: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ A )
     => ( ( ord_less_eq_nat @ D @ B )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_841_sup__mono,axiom,
    ! [A: nat,C2: nat,B: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ C2 )
     => ( ( ord_less_eq_nat @ B @ D )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C2 @ D ) ) ) ) ).

% sup_mono
thf(fact_842_sup__least,axiom,
    ! [Y: nat,X: nat,Z: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ Z @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_843_le__iff__sup,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( sup_sup_nat @ X4 @ Y5 )
          = Y5 ) ) ) ).

% le_iff_sup
thf(fact_844_sup_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( sup_sup_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_845_sup_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( sup_sup_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_846_sup__unique,axiom,
    ! [F2: nat > nat > nat,X: nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( F2 @ X3 @ Y3 ) )
     => ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F2 @ X3 @ Y3 ) )
       => ( ! [X3: nat,Y3: nat,Z4: nat] :
              ( ( ord_less_eq_nat @ Y3 @ X3 )
             => ( ( ord_less_eq_nat @ Z4 @ X3 )
               => ( ord_less_eq_nat @ ( F2 @ Y3 @ Z4 ) @ X3 ) ) )
         => ( ( sup_sup_nat @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_847_sup_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( sup_sup_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_848_sup_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( sup_sup_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_849_sup__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( sup_sup_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_850_sup__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( sup_sup_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_851_sup_OboundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% sup.boundedE
thf(fact_852_sup_OboundedI,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ A )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A ) ) ) ).

% sup.boundedI
thf(fact_853_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( A4
          = ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_854_sup_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_855_sup_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_856_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( sup_sup_nat @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_857_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( sup_sup_nat @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_858_sup_OcoboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ A )
     => ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_859_sup_OcoboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C2 @ B )
     => ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_860_replace__term__at_Oelims,axiom,
    ! [X: term_a_b,Xa2: list_nat,Xb: term_a_b,Y: term_a_b] :
      ( ( ( term_r6860082780075436317at_a_b @ X @ Xa2 @ Xb )
        = Y )
     => ( ( ( Xa2 = nil_nat )
         => ( Y != Xb ) )
       => ( ! [X3: b] :
              ( ( X
                = ( var_b_a @ X3 ) )
             => ( ? [V2: nat,Va: list_nat] :
                    ( Xa2
                    = ( cons_nat @ V2 @ Va ) )
               => ( Y
                 != ( var_b_a @ X3 ) ) ) )
         => ~ ! [F3: a,Ts2: list_term_a_b] :
                ( ( X
                  = ( fun_a_b @ F3 @ Ts2 ) )
               => ! [I2: nat,Ps2: list_nat] :
                    ( ( Xa2
                      = ( cons_nat @ I2 @ Ps2 ) )
                   => ~ ( ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
                         => ( Y
                            = ( fun_a_b @ F3 @ ( list_update_term_a_b @ Ts2 @ I2 @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ts2 @ I2 ) @ Ps2 @ Xb ) ) ) ) )
                        & ( ~ ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
                         => ( Y
                            = ( fun_a_b @ F3 @ Ts2 ) ) ) ) ) ) ) ) ) ).

% replace_term_at.elims
thf(fact_861_splice_Opinduct,axiom,
    ! [A0: list_nat,A1: list_nat,P2: list_nat > list_nat > $o] :
      ( ( accp_P8037286306265792042st_nat @ splice_rel_nat @ ( produc2694037385005941721st_nat @ A0 @ A1 ) )
     => ( ! [Ys4: list_nat] :
            ( ( accp_P8037286306265792042st_nat @ splice_rel_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
           => ( P2 @ nil_nat @ Ys4 ) )
       => ( ! [X3: nat,Xs: list_nat,Ys4: list_nat] :
              ( ( accp_P8037286306265792042st_nat @ splice_rel_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ Ys4 ) )
             => ( ( P2 @ Ys4 @ Xs )
               => ( P2 @ ( cons_nat @ X3 @ Xs ) @ Ys4 ) ) )
         => ( P2 @ A0 @ A1 ) ) ) ) ).

% splice.pinduct
thf(fact_862_add__elem__list__lists_Osimps_I1_J,axiom,
    ! [X: nat] :
      ( ( basic_4874698711677410535ts_nat @ X @ nil_nat )
      = ( cons_list_nat @ ( cons_nat @ X @ nil_nat ) @ nil_list_nat ) ) ).

% add_elem_list_lists.simps(1)
thf(fact_863_list__update__nonempty,axiom,
    ! [Xs2: list_nat,K: nat,X: nat] :
      ( ( ( list_update_nat @ Xs2 @ K @ X )
        = nil_nat )
      = ( Xs2 = nil_nat ) ) ).

% list_update_nonempty
thf(fact_864_length__list__update,axiom,
    ! [Xs2: list_term_a_b,I: nat,X: term_a_b] :
      ( ( size_s8906293707977694520rm_a_b @ ( list_update_term_a_b @ Xs2 @ I @ X ) )
      = ( size_s8906293707977694520rm_a_b @ Xs2 ) ) ).

% length_list_update
thf(fact_865_list__update__beyond,axiom,
    ! [Xs2: list_term_a_b,I: nat,X: term_a_b] :
      ( ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs2 ) @ I )
     => ( ( list_update_term_a_b @ Xs2 @ I @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_866_list__update__length,axiom,
    ! [Xs2: list_nat,X: nat,Ys: list_nat,Y: nat] :
      ( ( list_update_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs2 ) @ Y )
      = ( append_nat @ Xs2 @ ( cons_nat @ Y @ Ys ) ) ) ).

% list_update_length
thf(fact_867_list__update__length,axiom,
    ! [Xs2: list_term_a_b,X: term_a_b,Ys: list_term_a_b,Y: term_a_b] :
      ( ( list_update_term_a_b @ ( append_term_a_b @ Xs2 @ ( cons_term_a_b @ X @ Ys ) ) @ ( size_s8906293707977694520rm_a_b @ Xs2 ) @ Y )
      = ( append_term_a_b @ Xs2 @ ( cons_term_a_b @ Y @ Ys ) ) ) ).

% list_update_length
thf(fact_868_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_term_a_b,X: term_a_b] :
      ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( ( nth_term_a_b @ ( list_update_term_a_b @ Xs2 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_869_set__swap,axiom,
    ! [I: nat,Xs2: list_term_a_b,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
       => ( ( set_term_a_b2 @ ( list_update_term_a_b @ ( list_update_term_a_b @ Xs2 @ I @ ( nth_term_a_b @ Xs2 @ J ) ) @ J @ ( nth_term_a_b @ Xs2 @ I ) ) )
          = ( set_term_a_b2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_870_list__update_Osimps_I1_J,axiom,
    ! [I: nat,V: nat] :
      ( ( list_update_nat @ nil_nat @ I @ V )
      = nil_nat ) ).

% list_update.simps(1)
thf(fact_871_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_872_set__update__subsetI,axiom,
    ! [Xs2: list_P3592885314253461005_a_nat,A5: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat,I: nat] :
      ( ( ord_le8666007276011122963_a_nat @ ( set_Pr924983374503034536_a_nat @ Xs2 ) @ A5 )
     => ( ( member5724188588386418708_a_nat @ X @ A5 )
       => ( ord_le8666007276011122963_a_nat @ ( set_Pr924983374503034536_a_nat @ ( list_u4318310736832020111_a_nat @ Xs2 @ I @ X ) ) @ A5 ) ) ) ).

% set_update_subsetI
thf(fact_873_set__update__subsetI,axiom,
    ! [Xs2: list_list_nat,A5: set_list_nat,X: list_nat,I: nat] :
      ( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ A5 )
     => ( ( member_list_nat @ X @ A5 )
       => ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( list_update_list_nat @ Xs2 @ I @ X ) ) @ A5 ) ) ) ).

% set_update_subsetI
thf(fact_874_set__update__memI,axiom,
    ! [N: nat,Xs2: list_P3592885314253461005_a_nat,X: product_prod_a_nat] :
      ( ( ord_less_nat @ N @ ( size_s984997627204368545_a_nat @ Xs2 ) )
     => ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ ( list_u4318310736832020111_a_nat @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_875_set__update__memI,axiom,
    ! [N: nat,Xs2: list_list_nat,X: list_nat] :
      ( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs2 ) )
     => ( member_list_nat @ X @ ( set_list_nat2 @ ( list_update_list_nat @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_876_set__update__memI,axiom,
    ! [N: nat,Xs2: list_term_a_b,X: term_a_b] :
      ( ( ord_less_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( member_term_a_b @ X @ ( set_term_a_b2 @ ( list_update_term_a_b @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_877_list__update__append1,axiom,
    ! [I: nat,Xs2: list_term_a_b,Ys: list_term_a_b,X: term_a_b] :
      ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( ( list_update_term_a_b @ ( append_term_a_b @ Xs2 @ Ys ) @ I @ X )
        = ( append_term_a_b @ ( list_update_term_a_b @ Xs2 @ I @ X ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_878_parallel__list__update,axiom,
    ! [N: nat,R: term_a_b > term_a_b > $o,P: list_term_a_b > $o,Xs2: list_term_a_b,Ys: list_term_a_b] :
      ( ! [Xs: list_term_a_b,I2: nat,Y3: term_a_b] :
          ( ( ( size_s8906293707977694520rm_a_b @ Xs )
            = N )
         => ( ( ord_less_nat @ I2 @ N )
           => ( ( R @ ( nth_term_a_b @ Xs @ I2 ) @ Y3 )
             => ( ( P @ Xs )
               => ( P @ ( list_update_term_a_b @ Xs @ I2 @ Y3 ) ) ) ) ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
          = N )
       => ( ( P @ Xs2 )
         => ( ( ( size_s8906293707977694520rm_a_b @ Ys )
              = N )
           => ( ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ N )
                 => ( R @ ( nth_term_a_b @ Xs2 @ I2 ) @ ( nth_term_a_b @ Ys @ I2 ) ) )
             => ( P @ Ys ) ) ) ) ) ) ).

% parallel_list_update
thf(fact_879_nth__list__update,axiom,
    ! [I: nat,Xs2: list_term_a_b,J: nat,X: term_a_b] :
      ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_term_a_b @ ( list_update_term_a_b @ Xs2 @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_term_a_b @ ( list_update_term_a_b @ Xs2 @ I @ X ) @ J )
            = ( nth_term_a_b @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_880_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_term_a_b,X: term_a_b] :
      ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( ( ( list_update_term_a_b @ Xs2 @ I @ X )
          = Xs2 )
        = ( ( nth_term_a_b @ Xs2 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_881_listrel1__iff__update,axiom,
    ! [Xs2: list_term_a_b,Ys: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs2 @ Ys ) @ ( listrel1_term_a_b @ R ) )
      = ( ? [Y5: term_a_b,N2: nat] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Xs2 @ N2 ) @ Y5 ) @ R )
            & ( ord_less_nat @ N2 @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
            & ( Ys
              = ( list_update_term_a_b @ Xs2 @ N2 @ Y5 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_882_replace__term__at_Osimps_I3_J,axiom,
    ! [I: nat,Ts: list_term_a_b,F2: a,Ps: list_nat,T: term_a_b] :
      ( ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ts ) )
       => ( ( term_r6860082780075436317at_a_b @ ( fun_a_b @ F2 @ Ts ) @ ( cons_nat @ I @ Ps ) @ T )
          = ( fun_a_b @ F2 @ ( list_update_term_a_b @ Ts @ I @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ts @ I ) @ Ps @ T ) ) ) ) )
      & ( ~ ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ts ) )
       => ( ( term_r6860082780075436317at_a_b @ ( fun_a_b @ F2 @ Ts ) @ ( cons_nat @ I @ Ps ) @ T )
          = ( fun_a_b @ F2 @ Ts ) ) ) ) ).

% replace_term_at.simps(3)
thf(fact_883_replace__term__at_Opelims,axiom,
    ! [X: term_a_b,Xa2: list_nat,Xb: term_a_b,Y: term_a_b] :
      ( ( ( term_r6860082780075436317at_a_b @ X @ Xa2 @ Xb )
        = Y )
     => ( ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ X @ ( produc5151171985953862413rm_a_b @ Xa2 @ Xb ) ) )
       => ( ( ( Xa2 = nil_nat )
           => ( ( Y = Xb )
             => ~ ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ X @ ( produc5151171985953862413rm_a_b @ nil_nat @ Xb ) ) ) ) )
         => ( ! [X3: b] :
                ( ( X
                  = ( var_b_a @ X3 ) )
               => ! [V2: nat,Va: list_nat] :
                    ( ( Xa2
                      = ( cons_nat @ V2 @ Va ) )
                   => ( ( Y
                        = ( var_b_a @ X3 ) )
                     => ~ ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ ( var_b_a @ X3 ) @ ( produc5151171985953862413rm_a_b @ ( cons_nat @ V2 @ Va ) @ Xb ) ) ) ) ) )
           => ~ ! [F3: a,Ts2: list_term_a_b] :
                  ( ( X
                    = ( fun_a_b @ F3 @ Ts2 ) )
                 => ! [I2: nat,Ps2: list_nat] :
                      ( ( Xa2
                        = ( cons_nat @ I2 @ Ps2 ) )
                     => ( ( ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
                           => ( Y
                              = ( fun_a_b @ F3 @ ( list_update_term_a_b @ Ts2 @ I2 @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ts2 @ I2 ) @ Ps2 @ Xb ) ) ) ) )
                          & ( ~ ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
                           => ( Y
                              = ( fun_a_b @ F3 @ Ts2 ) ) ) )
                       => ~ ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ ( fun_a_b @ F3 @ Ts2 ) @ ( produc5151171985953862413rm_a_b @ ( cons_nat @ I2 @ Ps2 ) @ Xb ) ) ) ) ) ) ) ) ) ) ).

% replace_term_at.pelims
thf(fact_884_splice_Opelims,axiom,
    ! [X: list_nat,Xa2: list_nat,Y: list_nat] :
      ( ( ( splice_nat @ X @ Xa2 )
        = Y )
     => ( ( accp_P8037286306265792042st_nat @ splice_rel_nat @ ( produc2694037385005941721st_nat @ X @ Xa2 ) )
       => ( ( ( X = nil_nat )
           => ( ( Y = Xa2 )
             => ~ ( accp_P8037286306265792042st_nat @ splice_rel_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xa2 ) ) ) )
         => ~ ! [X3: nat,Xs: list_nat] :
                ( ( X
                  = ( cons_nat @ X3 @ Xs ) )
               => ( ( Y
                    = ( cons_nat @ X3 @ ( splice_nat @ Xa2 @ Xs ) ) )
                 => ~ ( accp_P8037286306265792042st_nat @ splice_rel_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ Xa2 ) ) ) ) ) ) ) ).

% splice.pelims
thf(fact_885_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_886_split__Nil__iff,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( splice_nat @ Xs2 @ Ys )
        = nil_nat )
      = ( ( Xs2 = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% split_Nil_iff
thf(fact_887_splice__Nil2,axiom,
    ! [Xs2: list_nat] :
      ( ( splice_nat @ Xs2 @ nil_nat )
      = Xs2 ) ).

% splice_Nil2
thf(fact_888_length__splice,axiom,
    ! [Xs2: list_term_a_b,Ys: list_term_a_b] :
      ( ( size_s8906293707977694520rm_a_b @ ( splice_term_a_b @ Xs2 @ Ys ) )
      = ( plus_plus_nat @ ( size_s8906293707977694520rm_a_b @ Xs2 ) @ ( size_s8906293707977694520rm_a_b @ Ys ) ) ) ).

% length_splice
thf(fact_889_splice_Osimps_I1_J,axiom,
    ! [Ys: list_nat] :
      ( ( splice_nat @ nil_nat @ Ys )
      = Ys ) ).

% splice.simps(1)
thf(fact_890_splice_Osimps_I2_J,axiom,
    ! [X: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( splice_nat @ ( cons_nat @ X @ Xs2 ) @ Ys )
      = ( cons_nat @ X @ ( splice_nat @ Ys @ Xs2 ) ) ) ).

% splice.simps(2)
thf(fact_891_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_892_splice_Oelims,axiom,
    ! [X: list_nat,Xa2: list_nat,Y: list_nat] :
      ( ( ( splice_nat @ X @ Xa2 )
        = Y )
     => ( ( ( X = nil_nat )
         => ( Y != Xa2 ) )
       => ~ ! [X3: nat,Xs: list_nat] :
              ( ( X
                = ( cons_nat @ X3 @ Xs ) )
             => ( Y
               != ( cons_nat @ X3 @ ( splice_nat @ Xa2 @ Xs ) ) ) ) ) ) ).

% splice.elims
thf(fact_893_splice_Opsimps_I2_J,axiom,
    ! [X: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( accp_P8037286306265792042st_nat @ splice_rel_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ Ys ) )
     => ( ( splice_nat @ ( cons_nat @ X @ Xs2 ) @ Ys )
        = ( cons_nat @ X @ ( splice_nat @ Ys @ Xs2 ) ) ) ) ).

% splice.psimps(2)
thf(fact_894_splice_Opsimps_I1_J,axiom,
    ! [Ys: list_nat] :
      ( ( accp_P8037286306265792042st_nat @ splice_rel_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys ) )
     => ( ( splice_nat @ nil_nat @ Ys )
        = Ys ) ) ).

% splice.psimps(1)
thf(fact_895_shuffles_Opinduct,axiom,
    ! [A0: list_nat,A1: list_nat,P2: list_nat > list_nat > $o] :
      ( ( accp_P8037286306265792042st_nat @ shuffles_rel_nat @ ( produc2694037385005941721st_nat @ A0 @ A1 ) )
     => ( ! [Ys4: list_nat] :
            ( ( accp_P8037286306265792042st_nat @ shuffles_rel_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
           => ( P2 @ nil_nat @ Ys4 ) )
       => ( ! [Xs: list_nat] :
              ( ( accp_P8037286306265792042st_nat @ shuffles_rel_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) )
             => ( P2 @ Xs @ nil_nat ) )
         => ( ! [X3: nat,Xs: list_nat,Y3: nat,Ys4: list_nat] :
                ( ( accp_P8037286306265792042st_nat @ shuffles_rel_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
               => ( ( P2 @ Xs @ ( cons_nat @ Y3 @ Ys4 ) )
                 => ( ( P2 @ ( cons_nat @ X3 @ Xs ) @ Ys4 )
                   => ( P2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) ) )
           => ( P2 @ A0 @ A1 ) ) ) ) ) ).

% shuffles.pinduct
thf(fact_896_subt__at_Oelims,axiom,
    ! [X: term_a_b,Xa2: list_nat,Y: term_a_b] :
      ( ( ( term_subt_at_a_b @ X @ Xa2 )
        = Y )
     => ( ( ( Xa2 = nil_nat )
         => ( Y != X ) )
       => ( ! [F3: a,Ss2: list_term_a_b] :
              ( ( X
                = ( fun_a_b @ F3 @ Ss2 ) )
             => ! [I2: nat,P4: list_nat] :
                  ( ( Xa2
                    = ( cons_nat @ I2 @ P4 ) )
                 => ( Y
                   != ( term_subt_at_a_b @ ( nth_term_a_b @ Ss2 @ I2 ) @ P4 ) ) ) )
         => ~ ( ? [X3: b] :
                  ( X
                  = ( var_b_a @ X3 ) )
             => ( ? [V2: nat,Va: list_nat] :
                    ( Xa2
                    = ( cons_nat @ V2 @ Va ) )
               => ( Y != undefined_term_a_b ) ) ) ) ) ) ).

% subt_at.elims
thf(fact_897_root_Opelims,axiom,
    ! [X: term_a_b,Y: option5551091909395471437_a_nat] :
      ( ( ( root_a_b @ X )
        = Y )
     => ( ( accp_term_a_b @ root_rel_a_b @ X )
       => ( ! [X3: b] :
              ( ( X
                = ( var_b_a @ X3 ) )
             => ( ( Y = none_P4102489432868120540_a_nat )
               => ~ ( accp_term_a_b @ root_rel_a_b @ ( var_b_a @ X3 ) ) ) )
         => ~ ! [F3: a,Ts2: list_term_a_b] :
                ( ( X
                  = ( fun_a_b @ F3 @ Ts2 ) )
               => ( ( Y
                    = ( some_P6251353102471802712_a_nat @ ( product_Pair_a_nat @ F3 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) ) ) )
                 => ~ ( accp_term_a_b @ root_rel_a_b @ ( fun_a_b @ F3 @ Ts2 ) ) ) ) ) ) ) ).

% root.pelims
thf(fact_898_subt__at_Osimps_I3_J,axiom,
    ! [X: b,V: nat,Va2: list_nat] :
      ( ( term_subt_at_a_b @ ( var_b_a @ X ) @ ( cons_nat @ V @ Va2 ) )
      = undefined_term_a_b ) ).

% subt_at.simps(3)
thf(fact_899_subt__at_Opelims,axiom,
    ! [X: term_a_b,Xa2: list_nat,Y: term_a_b] :
      ( ( ( term_subt_at_a_b @ X @ Xa2 )
        = Y )
     => ( ( accp_P682940083893826398st_nat @ term_subt_at_rel_a_b @ ( produc4563063199488751885st_nat @ X @ Xa2 ) )
       => ( ( ( Xa2 = nil_nat )
           => ( ( Y = X )
             => ~ ( accp_P682940083893826398st_nat @ term_subt_at_rel_a_b @ ( produc4563063199488751885st_nat @ X @ nil_nat ) ) ) )
         => ( ! [F3: a,Ss2: list_term_a_b] :
                ( ( X
                  = ( fun_a_b @ F3 @ Ss2 ) )
               => ! [I2: nat,P4: list_nat] :
                    ( ( Xa2
                      = ( cons_nat @ I2 @ P4 ) )
                   => ( ( Y
                        = ( term_subt_at_a_b @ ( nth_term_a_b @ Ss2 @ I2 ) @ P4 ) )
                     => ~ ( accp_P682940083893826398st_nat @ term_subt_at_rel_a_b @ ( produc4563063199488751885st_nat @ ( fun_a_b @ F3 @ Ss2 ) @ ( cons_nat @ I2 @ P4 ) ) ) ) ) )
           => ~ ! [X3: b] :
                  ( ( X
                    = ( var_b_a @ X3 ) )
                 => ! [V2: nat,Va: list_nat] :
                      ( ( Xa2
                        = ( cons_nat @ V2 @ Va ) )
                     => ( ( Y = undefined_term_a_b )
                       => ~ ( accp_P682940083893826398st_nat @ term_subt_at_rel_a_b @ ( produc4563063199488751885st_nat @ ( var_b_a @ X3 ) @ ( cons_nat @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ).

% subt_at.pelims
thf(fact_900_permut__aux__sound,axiom,
    ! [I: nat,As2: list_term_a_b,F2: nat > nat,Bs2: list_term_a_b] :
      ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ As2 ) )
     => ( ( nth_term_a_b @ ( missin5661735679256376914rm_a_b @ As2 @ F2 @ Bs2 ) @ I )
        = ( nth_term_a_b @ Bs2 @ ( F2 @ I ) ) ) ) ).

% permut_aux_sound
thf(fact_901_find__Some__iff,axiom,
    ! [P2: term_a_b > $o,Xs2: list_term_a_b,X: term_a_b] :
      ( ( ( find_term_a_b @ P2 @ Xs2 )
        = ( some_term_a_b @ X ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
            & ( P2 @ ( nth_term_a_b @ Xs2 @ I4 ) )
            & ( X
              = ( nth_term_a_b @ Xs2 @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P2 @ ( nth_term_a_b @ Xs2 @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_902_find__cong,axiom,
    ! [Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,P2: product_prod_a_nat > $o,Q2: product_prod_a_nat > $o] :
      ( ( Xs2 = Ys )
     => ( ! [X3: product_prod_a_nat] :
            ( ( member5724188588386418708_a_nat @ X3 @ ( set_Pr924983374503034536_a_nat @ Ys ) )
           => ( ( P2 @ X3 )
              = ( Q2 @ X3 ) ) )
       => ( ( find_P6125149089620491617_a_nat @ P2 @ Xs2 )
          = ( find_P6125149089620491617_a_nat @ Q2 @ Ys ) ) ) ) ).

% find_cong
thf(fact_903_find__cong,axiom,
    ! [Xs2: list_list_nat,Ys: list_list_nat,P2: list_nat > $o,Q2: list_nat > $o] :
      ( ( Xs2 = Ys )
     => ( ! [X3: list_nat] :
            ( ( member_list_nat @ X3 @ ( set_list_nat2 @ Ys ) )
           => ( ( P2 @ X3 )
              = ( Q2 @ X3 ) ) )
       => ( ( find_list_nat @ P2 @ Xs2 )
          = ( find_list_nat @ Q2 @ Ys ) ) ) ) ).

% find_cong
thf(fact_904_find_Osimps_I2_J,axiom,
    ! [P2: nat > $o,X: nat,Xs2: list_nat] :
      ( ( ( P2 @ X )
       => ( ( find_nat @ P2 @ ( cons_nat @ X @ Xs2 ) )
          = ( some_nat @ X ) ) )
      & ( ~ ( P2 @ X )
       => ( ( find_nat @ P2 @ ( cons_nat @ X @ Xs2 ) )
          = ( find_nat @ P2 @ Xs2 ) ) ) ) ).

% find.simps(2)
thf(fact_905_find_Osimps_I1_J,axiom,
    ! [Uu2: nat > $o] :
      ( ( find_nat @ Uu2 @ nil_nat )
      = none_nat ) ).

% find.simps(1)
thf(fact_906_find__None__iff,axiom,
    ! [P2: product_prod_a_nat > $o,Xs2: list_P3592885314253461005_a_nat] :
      ( ( ( find_P6125149089620491617_a_nat @ P2 @ Xs2 )
        = none_P4102489432868120540_a_nat )
      = ( ~ ? [X4: product_prod_a_nat] :
              ( ( member5724188588386418708_a_nat @ X4 @ ( set_Pr924983374503034536_a_nat @ Xs2 ) )
              & ( P2 @ X4 ) ) ) ) ).

% find_None_iff
thf(fact_907_find__None__iff,axiom,
    ! [P2: list_nat > $o,Xs2: list_list_nat] :
      ( ( ( find_list_nat @ P2 @ Xs2 )
        = none_list_nat )
      = ( ~ ? [X4: list_nat] :
              ( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs2 ) )
              & ( P2 @ X4 ) ) ) ) ).

% find_None_iff
thf(fact_908_find__None__iff2,axiom,
    ! [P2: product_prod_a_nat > $o,Xs2: list_P3592885314253461005_a_nat] :
      ( ( none_P4102489432868120540_a_nat
        = ( find_P6125149089620491617_a_nat @ P2 @ Xs2 ) )
      = ( ~ ? [X4: product_prod_a_nat] :
              ( ( member5724188588386418708_a_nat @ X4 @ ( set_Pr924983374503034536_a_nat @ Xs2 ) )
              & ( P2 @ X4 ) ) ) ) ).

% find_None_iff2
thf(fact_909_find__None__iff2,axiom,
    ! [P2: list_nat > $o,Xs2: list_list_nat] :
      ( ( none_list_nat
        = ( find_list_nat @ P2 @ Xs2 ) )
      = ( ~ ? [X4: list_nat] :
              ( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs2 ) )
              & ( P2 @ X4 ) ) ) ) ).

% find_None_iff2
thf(fact_910_find__Some__iff2,axiom,
    ! [X: term_a_b,P2: term_a_b > $o,Xs2: list_term_a_b] :
      ( ( ( some_term_a_b @ X )
        = ( find_term_a_b @ P2 @ Xs2 ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
            & ( P2 @ ( nth_term_a_b @ Xs2 @ I4 ) )
            & ( X
              = ( nth_term_a_b @ Xs2 @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P2 @ ( nth_term_a_b @ Xs2 @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_911_nth__zip,axiom,
    ! [I: nat,Xs2: list_a,Ys: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr8461465654520414006_a_nat @ ( zip_a_nat @ Xs2 @ Ys ) @ I )
          = ( product_Pair_a_nat @ ( nth_a @ Xs2 @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_912_nth__zip,axiom,
    ! [I: nat,Xs2: list_term_a_b,Ys: list_term_a_b] :
      ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ys ) )
       => ( ( nth_Pr1098598711982373392rm_a_b @ ( zip_te2260252107881041761rm_a_b @ Xs2 @ Ys ) @ I )
          = ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Xs2 @ I ) @ ( nth_term_a_b @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_913_min__list__nth,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
           => ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) )
       => ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs2 ) @ ( missing_min_list_nat @ Ys ) ) ) ) ).

% min_list_nth
thf(fact_914_listrel__iff__nth,axiom,
    ! [Xs2: list_a,Ys: list_nat,R: set_Pr4934435412358123699_a_nat] :
      ( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs2 @ Ys ) @ ( listrel_a_nat @ R ) )
      = ( ( ( size_size_list_a @ Xs2 )
          = ( size_size_list_nat @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs2 ) )
           => ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ ( nth_a @ Xs2 @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_915_listrel__iff__nth,axiom,
    ! [Xs2: list_term_a_b,Ys: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs2 @ Ys ) @ ( listre1194016999521814427rm_a_b @ R ) )
      = ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
          = ( size_s8906293707977694520rm_a_b @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Xs2 @ N2 ) @ ( nth_term_a_b @ Ys @ N2 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_916_zip__Cons__Cons,axiom,
    ! [X: nat,Xs2: list_nat,Y: nat,Ys: list_nat] :
      ( ( zip_nat_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( zip_nat_nat @ Xs2 @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_917_zip__Cons__Cons,axiom,
    ! [X: a,Xs2: list_a,Y: nat,Ys: list_nat] :
      ( ( zip_a_nat @ ( cons_a @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys ) )
      = ( cons_P5205166803686508359_a_nat @ ( product_Pair_a_nat @ X @ Y ) @ ( zip_a_nat @ Xs2 @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_918_Nil__eq__zip__iff,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( nil_Pr5478986624290739719at_nat
        = ( zip_nat_nat @ Xs2 @ Ys ) )
      = ( ( Xs2 = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_919_zip__eq__Nil__iff,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( zip_nat_nat @ Xs2 @ Ys )
        = nil_Pr5478986624290739719at_nat )
      = ( ( Xs2 = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_920_zip__append,axiom,
    ! [Xs2: list_term_a_b,Us: list_term_a_b,Ys: list_term_a_b,Vs: list_term_a_b] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
        = ( size_s8906293707977694520rm_a_b @ Us ) )
     => ( ( zip_te2260252107881041761rm_a_b @ ( append_term_a_b @ Xs2 @ Ys ) @ ( append_term_a_b @ Us @ Vs ) )
        = ( append1152621114427393060rm_a_b @ ( zip_te2260252107881041761rm_a_b @ Xs2 @ Us ) @ ( zip_te2260252107881041761rm_a_b @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_921_zip__update,axiom,
    ! [Xs2: list_a,I: nat,X: a,Ys: list_nat,Y: nat] :
      ( ( zip_a_nat @ ( list_update_a @ Xs2 @ I @ X ) @ ( list_update_nat @ Ys @ I @ Y ) )
      = ( list_u4318310736832020111_a_nat @ ( zip_a_nat @ Xs2 @ Ys ) @ I @ ( product_Pair_a_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_922_zip__same,axiom,
    ! [A: product_prod_a_nat,B: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B ) @ ( set_Pr5434493847166161136_a_nat @ ( zip_Pr1824874466442446003_a_nat @ Xs2 @ Xs2 ) ) )
      = ( ( member5724188588386418708_a_nat @ A @ ( set_Pr924983374503034536_a_nat @ Xs2 ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_923_zip__same,axiom,
    ! [A: list_nat,B: list_nat,Xs2: list_list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( set_Pr3842133991353686454st_nat @ ( zip_li7157463729305086713st_nat @ Xs2 @ Xs2 ) ) )
      = ( ( member_list_nat @ A @ ( set_list_nat2 @ Xs2 ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_924_in__set__zipE,axiom,
    ! [X: product_prod_a_nat,Y: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
      ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( set_Pr5434493847166161136_a_nat @ ( zip_Pr1824874466442446003_a_nat @ Xs2 @ Ys ) ) )
     => ~ ( ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Xs2 ) )
         => ~ ( member5724188588386418708_a_nat @ Y @ ( set_Pr924983374503034536_a_nat @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_925_in__set__zipE,axiom,
    ! [X: product_prod_a_nat,Y: list_nat,Xs2: list_P3592885314253461005_a_nat,Ys: list_list_nat] :
      ( ( member5216864792310759783st_nat @ ( produc1713529976458248438st_nat @ X @ Y ) @ ( set_Pr5697165133380604883st_nat @ ( zip_Pr8052531261761853462st_nat @ Xs2 @ Ys ) ) )
     => ~ ( ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Xs2 ) )
         => ~ ( member_list_nat @ Y @ ( set_list_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_926_in__set__zipE,axiom,
    ! [X: list_nat,Y: product_prod_a_nat,Xs2: list_list_nat,Ys: list_P3592885314253461005_a_nat] :
      ( ( member7922532354050488039_a_nat @ ( produc6151120682086626038_a_nat @ X @ Y ) @ ( set_Pr8402832695120333139_a_nat @ ( zip_li3266749930535455254_a_nat @ Xs2 @ Ys ) ) )
     => ~ ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
         => ~ ( member5724188588386418708_a_nat @ Y @ ( set_Pr924983374503034536_a_nat @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_927_in__set__zipE,axiom,
    ! [X: list_nat,Y: list_nat,Xs2: list_list_nat,Ys: list_list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( set_Pr3842133991353686454st_nat @ ( zip_li7157463729305086713st_nat @ Xs2 @ Ys ) ) )
     => ~ ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
         => ~ ( member_list_nat @ Y @ ( set_list_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_928_in__set__zipE,axiom,
    ! [X: a,Y: nat,Xs2: list_a,Ys: list_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y ) @ ( set_Pr924983374503034536_a_nat @ ( zip_a_nat @ Xs2 @ Ys ) ) )
     => ~ ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
         => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_929_set__zip__leftD,axiom,
    ! [X: a,Y: nat,Xs2: list_a,Ys: list_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y ) @ ( set_Pr924983374503034536_a_nat @ ( zip_a_nat @ Xs2 @ Ys ) ) )
     => ( member_a @ X @ ( set_a2 @ Xs2 ) ) ) ).

% set_zip_leftD
thf(fact_930_set__zip__rightD,axiom,
    ! [X: a,Y: nat,Xs2: list_a,Ys: list_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y ) @ ( set_Pr924983374503034536_a_nat @ ( zip_a_nat @ Xs2 @ Ys ) ) )
     => ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_931_zip__eq__ConsE,axiom,
    ! [Xs2: list_nat,Ys: list_nat,Xy: product_prod_nat_nat,Xys: list_P6011104703257516679at_nat] :
      ( ( ( zip_nat_nat @ Xs2 @ Ys )
        = ( cons_P6512896166579812791at_nat @ Xy @ Xys ) )
     => ~ ! [X3: nat,Xs5: list_nat] :
            ( ( Xs2
              = ( cons_nat @ X3 @ Xs5 ) )
           => ! [Y3: nat,Ys7: list_nat] :
                ( ( Ys
                  = ( cons_nat @ Y3 @ Ys7 ) )
               => ( ( Xy
                    = ( product_Pair_nat_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_nat_nat @ Xs5 @ Ys7 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_932_zip__eq__ConsE,axiom,
    ! [Xs2: list_a,Ys: list_nat,Xy: product_prod_a_nat,Xys: list_P3592885314253461005_a_nat] :
      ( ( ( zip_a_nat @ Xs2 @ Ys )
        = ( cons_P5205166803686508359_a_nat @ Xy @ Xys ) )
     => ~ ! [X3: a,Xs5: list_a] :
            ( ( Xs2
              = ( cons_a @ X3 @ Xs5 ) )
           => ! [Y3: nat,Ys7: list_nat] :
                ( ( Ys
                  = ( cons_nat @ Y3 @ Ys7 ) )
               => ( ( Xy
                    = ( product_Pair_a_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_a_nat @ Xs5 @ Ys7 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_933_listrel_ONil,axiom,
    ! [R: set_Pr1261947904930325089at_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) @ ( listrel_nat_nat @ R ) ) ).

% listrel.Nil
thf(fact_934_listrel__Nil1,axiom,
    ! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs2 ) @ ( listrel_nat_nat @ R ) )
     => ( Xs2 = nil_nat ) ) ).

% listrel_Nil1
thf(fact_935_listrel__Nil2,axiom,
    ! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) @ ( listrel_nat_nat @ R ) )
     => ( Xs2 = nil_nat ) ) ).

% listrel_Nil2
thf(fact_936_listrel__eq__len,axiom,
    ! [Xs2: list_term_a_b,Ys: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs2 @ Ys ) @ ( listre1194016999521814427rm_a_b @ R ) )
     => ( ( size_s8906293707977694520rm_a_b @ Xs2 )
        = ( size_s8906293707977694520rm_a_b @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_937_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_a,Ys: list_nat,X: a] :
      ( ( ( size_size_list_a @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
       => ~ ! [Y3: nat] :
              ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y3 ) @ ( set_Pr924983374503034536_a_nat @ ( zip_a_nat @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_938_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_P3592885314253461005_a_nat,Ys: list_term_a_b,X: product_prod_a_nat] :
      ( ( ( size_s984997627204368545_a_nat @ Xs2 )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Xs2 ) )
       => ~ ! [Y3: term_a_b] :
              ~ ( member1876585039447381659rm_a_b @ ( produc4170664577406169130rm_a_b @ X @ Y3 ) @ ( set_Pr2356885380517226759rm_a_b @ ( zip_Pr1286293825854998346rm_a_b @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_939_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_list_nat,Ys: list_term_a_b,X: list_nat] :
      ( ( ( size_s3023201423986296836st_nat @ Xs2 )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
       => ~ ! [Y3: term_a_b] :
              ~ ( member4000689696542324350rm_a_b @ ( produc5151171985953862413rm_a_b @ X @ Y3 ) @ ( set_Pr501854238490308330rm_a_b @ ( zip_li391226293398231597rm_a_b @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_940_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_term_a_b,Ys: list_term_a_b,X: term_a_b] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( member_term_a_b @ X @ ( set_term_a_b2 @ Xs2 ) )
       => ~ ! [Y3: term_a_b] :
              ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ ( set_Pr2370880052973118494rm_a_b @ ( zip_te2260252107881041761rm_a_b @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_941_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_a,Ys: list_nat,Y: nat] :
      ( ( ( size_size_list_a @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
       => ~ ! [X3: a] :
              ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y ) @ ( set_Pr924983374503034536_a_nat @ ( zip_a_nat @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_942_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_term_a_b,Ys: list_P3592885314253461005_a_nat,Y: product_prod_a_nat] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
        = ( size_s984997627204368545_a_nat @ Ys ) )
     => ( ( member5724188588386418708_a_nat @ Y @ ( set_Pr924983374503034536_a_nat @ Ys ) )
       => ~ ! [X3: term_a_b] :
              ~ ( member4553550343464174107_a_nat @ ( produc2782138671500312106_a_nat @ X3 @ Y ) @ ( set_Pr5033850684534019207_a_nat @ ( zip_te9121139956803917130_a_nat @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_943_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_term_a_b,Ys: list_list_nat,Y: list_nat] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ( member_list_nat @ Y @ ( set_list_nat2 @ Ys ) )
       => ~ ! [X3: term_a_b] :
              ~ ( member9209995263888512638st_nat @ ( produc4563063199488751885st_nat @ X3 @ Y ) @ ( set_Pr5711159805836496618st_nat @ ( zip_te9026489543787896877st_nat @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_944_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_term_a_b,Ys: list_term_a_b,Y: term_a_b] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( member_term_a_b @ Y @ ( set_term_a_b2 @ Ys ) )
       => ~ ! [X3: term_a_b] :
              ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( set_Pr2370880052973118494rm_a_b @ ( zip_te2260252107881041761rm_a_b @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_945_min__list,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs2 ) @ X ) ) ).

% min_list
thf(fact_946_Missing__List_Omin__list_Osimps_I1_J,axiom,
    ! [X: nat] :
      ( ( missing_min_list_nat @ ( cons_nat @ X @ nil_nat ) )
      = X ) ).

% Missing_List.min_list.simps(1)
thf(fact_947_listrel__Cons2,axiom,
    ! [Xs2: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat @ R ) )
     => ~ ! [X3: nat,Xs: list_nat] :
            ( ( Xs2
              = ( cons_nat @ X3 @ Xs ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_948_listrel__Cons2,axiom,
    ! [Xs2: list_a,Y: nat,Ys: list_nat,R: set_Pr4934435412358123699_a_nat] :
      ( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs2 @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_a_nat @ R ) )
     => ~ ! [X3: a,Xs: list_a] :
            ( ( Xs2
              = ( cons_a @ X3 @ Xs ) )
           => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y ) @ R )
             => ~ ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs @ Ys ) @ ( listrel_a_nat @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_949_listrel__Cons1,axiom,
    ! [Y: nat,Ys: list_nat,Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y @ Ys ) @ Xs2 ) @ ( listrel_nat_nat @ R ) )
     => ~ ! [Y3: nat,Ys4: list_nat] :
            ( ( Xs2
              = ( cons_nat @ Y3 @ Ys4 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Y3 ) @ R )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Ys4 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_950_listrel__Cons1,axiom,
    ! [Y: a,Ys: list_a,Xs2: list_nat,R: set_Pr4934435412358123699_a_nat] :
      ( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ ( cons_a @ Y @ Ys ) @ Xs2 ) @ ( listrel_a_nat @ R ) )
     => ~ ! [Y3: nat,Ys4: list_nat] :
            ( ( Xs2
              = ( cons_nat @ Y3 @ Ys4 ) )
           => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ Y @ Y3 ) @ R )
             => ~ ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Ys @ Ys4 ) @ ( listrel_a_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_951_listrel_OCons,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs2: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( listrel_nat_nat @ R ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_952_listrel_OCons,axiom,
    ! [X: a,Y: nat,R: set_Pr4934435412358123699_a_nat,Xs2: list_a,Ys: list_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y ) @ R )
     => ( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs2 @ Ys ) @ ( listrel_a_nat @ R ) )
       => ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ ( cons_a @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_a_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_953_min__list__Cons,axiom,
    ! [X: nat,Y: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ( size_size_list_nat @ Xs2 )
          = ( size_size_list_nat @ Ys ) )
       => ( ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs2 ) @ ( missing_min_list_nat @ Ys ) )
         => ( ord_less_eq_nat @ ( missing_min_list_nat @ ( cons_nat @ X @ Xs2 ) ) @ ( missing_min_list_nat @ ( cons_nat @ Y @ Ys ) ) ) ) ) ) ).

% min_list_Cons
thf(fact_954_listrel_Ocases,axiom,
    ! [A1: list_nat,A22: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R ) )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_nat ) )
       => ~ ! [X3: nat,Y3: nat,Xs: list_nat] :
              ( ( A1
                = ( cons_nat @ X3 @ Xs ) )
             => ! [Ys4: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y3 @ Ys4 ) )
                 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
                   => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys4 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_955_listrel_Ocases,axiom,
    ! [A1: list_a,A22: list_nat,R: set_Pr4934435412358123699_a_nat] :
      ( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ A1 @ A22 ) @ ( listrel_a_nat @ R ) )
     => ( ( ( A1 = nil_a )
         => ( A22 != nil_nat ) )
       => ~ ! [X3: a,Y3: nat,Xs: list_a] :
              ( ( A1
                = ( cons_a @ X3 @ Xs ) )
             => ! [Ys4: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y3 @ Ys4 ) )
                 => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y3 ) @ R )
                   => ~ ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs @ Ys4 ) @ ( listrel_a_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_956_listrel_Osimps,axiom,
    ! [A1: list_nat,A22: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R ) )
      = ( ( ( A1 = nil_nat )
          & ( A22 = nil_nat ) )
        | ? [X4: nat,Y5: nat,Xs3: list_nat,Ys3: list_nat] :
            ( ( A1
              = ( cons_nat @ X4 @ Xs3 ) )
            & ( A22
              = ( cons_nat @ Y5 @ Ys3 ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R )
            & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys3 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_957_listrel_Osimps,axiom,
    ! [A1: list_a,A22: list_nat,R: set_Pr4934435412358123699_a_nat] :
      ( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ A1 @ A22 ) @ ( listrel_a_nat @ R ) )
      = ( ( ( A1 = nil_a )
          & ( A22 = nil_nat ) )
        | ? [X4: a,Y5: nat,Xs3: list_a,Ys3: list_nat] :
            ( ( A1
              = ( cons_a @ X4 @ Xs3 ) )
            & ( A22
              = ( cons_nat @ Y5 @ Ys3 ) )
            & ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X4 @ Y5 ) @ R )
            & ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs3 @ Ys3 ) @ ( listrel_a_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_958_lex__take__index,axiom,
    ! [Xs2: list_term_a_b,Ys: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs2 @ Ys ) @ ( lex_term_a_b @ R ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
           => ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ys ) )
             => ( ( ( take_term_a_b @ I2 @ Xs2 )
                  = ( take_term_a_b @ I2 @ Ys ) )
               => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Xs2 @ I2 ) @ ( nth_term_a_b @ Ys @ I2 ) ) @ R ) ) ) ) ) ).

% lex_take_index
thf(fact_959_nth__equal__first__eq,axiom,
    ! [X: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,N: nat] :
      ( ~ ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Xs2 ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s984997627204368545_a_nat @ Xs2 ) )
       => ( ( ( nth_Pr8461465654520414006_a_nat @ ( cons_P5205166803686508359_a_nat @ X @ Xs2 ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_960_nth__equal__first__eq,axiom,
    ! [X: list_nat,Xs2: list_list_nat,N: nat] :
      ( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s3023201423986296836st_nat @ Xs2 ) )
       => ( ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_961_nth__equal__first__eq,axiom,
    ! [X: nat,Xs2: list_nat,N: nat] :
      ( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs2 ) )
       => ( ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_962_nth__equal__first__eq,axiom,
    ! [X: term_a_b,Xs2: list_term_a_b,N: nat] :
      ( ~ ( member_term_a_b @ X @ ( set_term_a_b2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
       => ( ( ( nth_term_a_b @ ( cons_term_a_b @ X @ Xs2 ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_963_mem__idx__sound__output,axiom,
    ! [X: term_a_b,As2: list_term_a_b,I: nat] :
      ( ( ( missin3271338439613487829rm_a_b @ X @ As2 )
        = ( some_nat @ I ) )
     => ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ As2 ) )
        & ( ( nth_term_a_b @ As2 @ I )
          = X ) ) ) ).

% mem_idx_sound_output
thf(fact_964_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_965_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_966_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_967_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X @ Y ) )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_968_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = zero_zero_nat )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_969_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_970_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_971_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_972_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_973_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_974_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_975_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_976_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_977_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_978_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_979_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_980_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_981_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_982_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_983_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_984_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_985_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_986_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_987_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_988_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_989_length__0__conv,axiom,
    ! [Xs2: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = zero_zero_nat )
      = ( Xs2 = nil_nat ) ) ).

% length_0_conv
thf(fact_990_length__0__conv,axiom,
    ! [Xs2: list_term_a_b] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
        = zero_zero_nat )
      = ( Xs2 = nil_term_a_b ) ) ).

% length_0_conv
thf(fact_991_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_992_nth__Cons__0,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_993_take__eq__Nil2,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( nil_nat
        = ( take_nat @ N @ Xs2 ) )
      = ( ( N = zero_zero_nat )
        | ( Xs2 = nil_nat ) ) ) ).

% take_eq_Nil2
thf(fact_994_take__eq__Nil,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ( take_nat @ N @ Xs2 )
        = nil_nat )
      = ( ( N = zero_zero_nat )
        | ( Xs2 = nil_nat ) ) ) ).

% take_eq_Nil
thf(fact_995_take0,axiom,
    ( ( take_nat @ zero_zero_nat )
    = ( ^ [Xs3: list_nat] : nil_nat ) ) ).

% take0
thf(fact_996_take__all,axiom,
    ! [Xs2: list_term_a_b,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs2 ) @ N )
     => ( ( take_term_a_b @ N @ Xs2 )
        = Xs2 ) ) ).

% take_all
thf(fact_997_take__all__iff,axiom,
    ! [N: nat,Xs2: list_term_a_b] :
      ( ( ( take_term_a_b @ N @ Xs2 )
        = Xs2 )
      = ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs2 ) @ N ) ) ).

% take_all_iff
thf(fact_998_length__greater__0__conv,axiom,
    ! [Xs2: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) )
      = ( Xs2 != nil_nat ) ) ).

% length_greater_0_conv
thf(fact_999_length__greater__0__conv,axiom,
    ! [Xs2: list_term_a_b] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
      = ( Xs2 != nil_term_a_b ) ) ).

% length_greater_0_conv
thf(fact_1000_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_1001_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_1002_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_1003_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_1004_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_1005_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_1006_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_1007_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_1008_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_1009_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_1010_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_1011_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_1012_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_1013_infinite__descent0,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ~ ( P2 @ N3 )
             => ? [M3: nat] :
                  ( ( ord_less_nat @ M3 @ N3 )
                  & ~ ( P2 @ M3 ) ) ) )
       => ( P2 @ N ) ) ) ).

% infinite_descent0
thf(fact_1014_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_1015_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_1016_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_1017_in__set__takeD,axiom,
    ! [X: product_prod_a_nat,N: nat,Xs2: list_P3592885314253461005_a_nat] :
      ( ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ ( take_P7249722199902868751_a_nat @ N @ Xs2 ) ) )
     => ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Xs2 ) ) ) ).

% in_set_takeD
thf(fact_1018_in__set__takeD,axiom,
    ! [X: list_nat,N: nat,Xs2: list_list_nat] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ ( take_list_nat @ N @ Xs2 ) ) )
     => ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) ) ) ).

% in_set_takeD
thf(fact_1019_take__Nil,axiom,
    ! [N: nat] :
      ( ( take_nat @ N @ nil_nat )
      = nil_nat ) ).

% take_Nil
thf(fact_1020_take__0,axiom,
    ! [Xs2: list_nat] :
      ( ( take_nat @ zero_zero_nat @ Xs2 )
      = nil_nat ) ).

% take_0
thf(fact_1021_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_1022_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_1023_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_1024_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_1025_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_1026_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1027_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1028_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_1029_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_1030_add__increasing2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_1031_add__decreasing2,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).

% add_decreasing2
thf(fact_1032_add__increasing,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_increasing
thf(fact_1033_add__decreasing,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).

% add_decreasing
thf(fact_1034_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_1035_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_1036_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C ) )
           => ( C = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_1037_pos__add__strict,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_1038_list_Osize_I3_J,axiom,
    ( ( size_size_list_nat @ nil_nat )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_1039_list_Osize_I3_J,axiom,
    ( ( size_s8906293707977694520rm_a_b @ nil_term_a_b )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_1040_ex__least__nat__le,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ N )
     => ( ~ ( P2 @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K3 )
               => ~ ( P2 @ I3 ) )
            & ( P2 @ K3 ) ) ) ) ).

% ex_least_nat_le
thf(fact_1041_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K3: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K3 )
          & ( ( plus_plus_nat @ I @ K3 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_1042_list__update__code_I2_J,axiom,
    ! [X: nat,Xs2: list_nat,Y: nat] :
      ( ( list_update_nat @ ( cons_nat @ X @ Xs2 ) @ zero_zero_nat @ Y )
      = ( cons_nat @ Y @ Xs2 ) ) ).

% list_update_code(2)
thf(fact_1043_length__code,axiom,
    ( size_s8906293707977694520rm_a_b
    = ( gen_length_term_a_b @ zero_zero_nat ) ) ).

% length_code
thf(fact_1044_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_1045_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_1046_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_1047_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_1048_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_1049_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_1050_length__pos__if__in__set,axiom,
    ! [X: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat] :
      ( ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s984997627204368545_a_nat @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_1051_length__pos__if__in__set,axiom,
    ! [X: list_nat,Xs2: list_list_nat] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_1052_length__pos__if__in__set,axiom,
    ! [X: term_a_b,Xs2: list_term_a_b] :
      ( ( member_term_a_b @ X @ ( set_term_a_b2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s8906293707977694520rm_a_b @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_1053_nth__take__lemma,axiom,
    ! [K: nat,Xs2: list_term_a_b,Ys: list_term_a_b] :
      ( ( ord_less_eq_nat @ K @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s8906293707977694520rm_a_b @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_term_a_b @ Xs2 @ I2 )
                = ( nth_term_a_b @ Ys @ I2 ) ) )
         => ( ( take_term_a_b @ K @ Xs2 )
            = ( take_term_a_b @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_1054_nth__take__prefix,axiom,
    ! [Ys: list_term_a_b,Xs2: list_term_a_b] :
      ( ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Ys ) @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ys ) )
           => ( ( nth_term_a_b @ Xs2 @ I2 )
              = ( nth_term_a_b @ Ys @ I2 ) ) )
       => ( ( take_term_a_b @ ( size_s8906293707977694520rm_a_b @ Ys ) @ Xs2 )
          = Ys ) ) ) ).

% nth_take_prefix
thf(fact_1055_nth__append__take,axiom,
    ! [I: nat,Xs2: list_nat,Y: nat,Ys: list_nat] :
      ( ( ord_less_eq_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_nat @ ( append_nat @ ( take_nat @ I @ Xs2 ) @ ( cons_nat @ Y @ Ys ) ) @ I )
        = Y ) ) ).

% nth_append_take
thf(fact_1056_nth__append__take,axiom,
    ! [I: nat,Xs2: list_term_a_b,Y: term_a_b,Ys: list_term_a_b] :
      ( ( ord_less_eq_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( ( nth_term_a_b @ ( append_term_a_b @ ( take_term_a_b @ I @ Xs2 ) @ ( cons_term_a_b @ Y @ Ys ) ) @ I )
        = Y ) ) ).

% nth_append_take
thf(fact_1057_nth__append__take__is__nth__conv,axiom,
    ! [I: nat,J: nat,Xs2: list_term_a_b,Ys: list_term_a_b] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
       => ( ( nth_term_a_b @ ( append_term_a_b @ ( take_term_a_b @ J @ Xs2 ) @ Ys ) @ I )
          = ( nth_term_a_b @ Xs2 @ I ) ) ) ) ).

% nth_append_take_is_nth_conv
thf(fact_1058_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_1059_length__n__lists__elem,axiom,
    ! [Ys: list_nat,N: nat,Xs2: list_nat] :
      ( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs2 ) ) )
     => ( ( size_size_list_nat @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_1060_length__n__lists__elem,axiom,
    ! [Ys: list_term_a_b,N: nat,Xs2: list_term_a_b] :
      ( ( member_list_term_a_b @ Ys @ ( set_list_term_a_b2 @ ( n_lists_term_a_b @ N @ Xs2 ) ) )
     => ( ( size_s8906293707977694520rm_a_b @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_1061_n__lists_Osimps_I1_J,axiom,
    ! [Xs2: list_nat] :
      ( ( n_lists_nat @ zero_zero_nat @ Xs2 )
      = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% n_lists.simps(1)
thf(fact_1062_list__of__permutation__element__n_Osimps_I1_J,axiom,
    ! [X: nat,L2: list_nat] :
      ( ( basic_7079635023375748421_n_nat @ X @ zero_zero_nat @ L2 )
      = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% list_of_permutation_element_n.simps(1)
thf(fact_1063_take__Suc__conv__app__nth,axiom,
    ! [I: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( take_nat @ ( suc @ I ) @ Xs2 )
        = ( append_nat @ ( take_nat @ I @ Xs2 ) @ ( cons_nat @ ( nth_nat @ Xs2 @ I ) @ nil_nat ) ) ) ) ).

% take_Suc_conv_app_nth
thf(fact_1064_take__Suc__conv__app__nth,axiom,
    ! [I: nat,Xs2: list_term_a_b] :
      ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( ( take_term_a_b @ ( suc @ I ) @ Xs2 )
        = ( append_term_a_b @ ( take_term_a_b @ I @ Xs2 ) @ ( cons_term_a_b @ ( nth_term_a_b @ Xs2 @ I ) @ nil_term_a_b ) ) ) ) ).

% take_Suc_conv_app_nth
thf(fact_1065_add__elem__list__listsI,axiom,
    ! [N: nat,Xs2: list_nat,Ys: list_nat,X: nat] :
      ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( ( Ys
          = ( append_nat @ ( take_nat @ N @ Xs2 ) @ ( cons_nat @ X @ ( drop_nat @ N @ Xs2 ) ) ) )
       => ( member_list_nat @ Ys @ ( set_list_nat2 @ ( basic_4874698711677410535ts_nat @ X @ Xs2 ) ) ) ) ) ).

% add_elem_list_listsI
thf(fact_1066_add__elem__list__listsI,axiom,
    ! [N: nat,Xs2: list_term_a_b,Ys: list_term_a_b,X: term_a_b] :
      ( ( ord_less_eq_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( ( Ys
          = ( append_term_a_b @ ( take_term_a_b @ N @ Xs2 ) @ ( cons_term_a_b @ X @ ( drop_term_a_b @ N @ Xs2 ) ) ) )
       => ( member_list_term_a_b @ Ys @ ( set_list_term_a_b2 @ ( basic_1593220722155286443rm_a_b @ X @ Xs2 ) ) ) ) ) ).

% add_elem_list_listsI
thf(fact_1067_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% nat.inject
thf(fact_1068_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_1069_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_less_eq
thf(fact_1070_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_1071_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_1072_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N @ M ) ) ).

% Suc_le_mono
thf(fact_1073_add__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc_right
thf(fact_1074_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_1075_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_1076_nth__Cons__Suc,axiom,
    ! [X: nat,Xs2: list_nat,N: nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ ( suc @ N ) )
      = ( nth_nat @ Xs2 @ N ) ) ).

% nth_Cons_Suc
thf(fact_1077_take__Suc__Cons,axiom,
    ! [N: nat,X: nat,Xs2: list_nat] :
      ( ( take_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs2 ) )
      = ( cons_nat @ X @ ( take_nat @ N @ Xs2 ) ) ) ).

% take_Suc_Cons
thf(fact_1078_drop__Suc__Cons,axiom,
    ! [N: nat,X: nat,Xs2: list_nat] :
      ( ( drop_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs2 ) )
      = ( drop_nat @ N @ Xs2 ) ) ).

% drop_Suc_Cons
thf(fact_1079_drop__eq__Nil2,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( nil_nat
        = ( drop_nat @ N @ Xs2 ) )
      = ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ).

% drop_eq_Nil2
thf(fact_1080_drop__eq__Nil2,axiom,
    ! [N: nat,Xs2: list_term_a_b] :
      ( ( nil_term_a_b
        = ( drop_term_a_b @ N @ Xs2 ) )
      = ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs2 ) @ N ) ) ).

% drop_eq_Nil2
thf(fact_1081_drop__eq__Nil,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ( drop_nat @ N @ Xs2 )
        = nil_nat )
      = ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ).

% drop_eq_Nil
thf(fact_1082_drop__eq__Nil,axiom,
    ! [N: nat,Xs2: list_term_a_b] :
      ( ( ( drop_term_a_b @ N @ Xs2 )
        = nil_term_a_b )
      = ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs2 ) @ N ) ) ).

% drop_eq_Nil
thf(fact_1083_drop__all,axiom,
    ! [Xs2: list_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N )
     => ( ( drop_nat @ N @ Xs2 )
        = nil_nat ) ) ).

% drop_all
thf(fact_1084_drop__all,axiom,
    ! [Xs2: list_term_a_b,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs2 ) @ N )
     => ( ( drop_term_a_b @ N @ Xs2 )
        = nil_term_a_b ) ) ).

% drop_all
thf(fact_1085_enumerate__simps_I2_J,axiom,
    ! [N: nat,X: nat,Xs2: list_nat] :
      ( ( enumerate_nat @ N @ ( cons_nat @ X @ Xs2 ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ X ) @ ( enumerate_nat @ ( suc @ N ) @ Xs2 ) ) ) ).

% enumerate_simps(2)
thf(fact_1086_nth__drop,axiom,
    ! [N: nat,Xs2: list_term_a_b,I: nat] :
      ( ( ord_less_eq_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( ( nth_term_a_b @ ( drop_term_a_b @ N @ Xs2 ) @ I )
        = ( nth_term_a_b @ Xs2 @ ( plus_plus_nat @ N @ I ) ) ) ) ).

% nth_drop
thf(fact_1087_nth__via__drop,axiom,
    ! [N: nat,Xs2: list_nat,Y: nat,Ys: list_nat] :
      ( ( ( drop_nat @ N @ Xs2 )
        = ( cons_nat @ Y @ Ys ) )
     => ( ( nth_nat @ Xs2 @ N )
        = Y ) ) ).

% nth_via_drop
thf(fact_1088_in__set__dropD,axiom,
    ! [X: product_prod_a_nat,N: nat,Xs2: list_P3592885314253461005_a_nat] :
      ( ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ ( drop_P2883665741211355575_a_nat @ N @ Xs2 ) ) )
     => ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Xs2 ) ) ) ).

% in_set_dropD
thf(fact_1089_in__set__dropD,axiom,
    ! [X: list_nat,N: nat,Xs2: list_list_nat] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ ( drop_list_nat @ N @ Xs2 ) ) )
     => ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) ) ) ).

% in_set_dropD
thf(fact_1090_drop__Nil,axiom,
    ! [N: nat] :
      ( ( drop_nat @ N @ nil_nat )
      = nil_nat ) ).

% drop_Nil
thf(fact_1091_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_1092_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_le_eq
thf(fact_1093_dec__induct,axiom,
    ! [I: nat,J: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P2 @ I )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P2 @ N3 )
                 => ( P2 @ ( suc @ N3 ) ) ) ) )
         => ( P2 @ J ) ) ) ) ).

% dec_induct
thf(fact_1094_inc__induct,axiom,
    ! [I: nat,J: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P2 @ J )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P2 @ ( suc @ N3 ) )
                 => ( P2 @ N3 ) ) ) )
         => ( P2 @ I ) ) ) ) ).

% inc_induct
thf(fact_1095_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_le_lessD
thf(fact_1096_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% le_less_Suc_eq
thf(fact_1097_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% less_Suc_eq_le
thf(fact_1098_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1099_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_1100_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ~ ! [Q4: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M @ Q4 ) ) ) ) ).

% less_natE
thf(fact_1101_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_1102_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_1103_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M2: nat,N2: nat] :
        ? [K2: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_1104_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ? [K3: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_1105_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_1106_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M @ N )
       => ( M
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_1107_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_1108_Suc__le__D,axiom,
    ! [N: nat,M5: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
     => ? [M4: nat] :
          ( M5
          = ( suc @ M4 ) ) ) ).

% Suc_le_D
thf(fact_1109_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M @ N )
        | ( M
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_1110_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_1111_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).

% not_less_eq_eq
thf(fact_1112_full__nat__induct,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
             => ( P2 @ M3 ) )
         => ( P2 @ N3 ) )
     => ( P2 @ N ) ) ).

% full_nat_induct
thf(fact_1113_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P2 @ M )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ M @ N3 )
             => ( ( P2 @ N3 )
               => ( P2 @ ( suc @ N3 ) ) ) )
         => ( P2 @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_1114_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R2: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X3: nat] : ( R2 @ X3 @ X3 )
       => ( ! [X3: nat,Y3: nat,Z4: nat] :
              ( ( R2 @ X3 @ Y3 )
             => ( ( R2 @ Y3 @ Z4 )
               => ( R2 @ X3 @ Z4 ) ) )
         => ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
           => ( R2 @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_1115_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_1116_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_1117_add__Suc__shift,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).

% add_Suc_shift
thf(fact_1118_add__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc
thf(fact_1119_nat__arith_Osuc1,axiom,
    ! [A5: nat,K: nat,A: nat] :
      ( ( A5
        = ( plus_plus_nat @ K @ A ) )
     => ( ( suc @ A5 )
        = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_1120_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_1121_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P2: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] :
            ( ( J
              = ( suc @ I2 ) )
           => ( P2 @ I2 ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ J )
             => ( ( P2 @ ( suc @ I2 ) )
               => ( P2 @ I2 ) ) )
         => ( P2 @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_1122_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P2: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] : ( P2 @ I2 @ ( suc @ I2 ) )
       => ( ! [I2: nat,J2: nat,K3: nat] :
              ( ( ord_less_nat @ I2 @ J2 )
             => ( ( ord_less_nat @ J2 @ K3 )
               => ( ( P2 @ I2 @ J2 )
                 => ( ( P2 @ J2 @ K3 )
                   => ( P2 @ I2 @ K3 ) ) ) ) )
         => ( P2 @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_1123_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_1124_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_less_SucD
thf(fact_1125_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_1126_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M )
      = ( ? [M6: nat] :
            ( ( M
              = ( suc @ M6 ) )
            & ( ord_less_nat @ N @ M6 ) ) ) ) ).

% Suc_less_eq2
thf(fact_1127_All__less__Suc,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
           => ( P2 @ I4 ) ) )
      = ( ( P2 @ N )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
           => ( P2 @ I4 ) ) ) ) ).

% All_less_Suc
thf(fact_1128_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_1129_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( M = N ) ) ) ).

% less_Suc_eq
thf(fact_1130_Ex__less__Suc,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
            & ( P2 @ I4 ) ) )
      = ( ( P2 @ N )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
            & ( P2 @ I4 ) ) ) ) ).

% Ex_less_Suc
thf(fact_1131_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_1132_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_1133_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ( suc @ M )
         != N )
       => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).

% Suc_lessI
thf(fact_1134_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I @ J2 )
           => ( K
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_1135_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_1136_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ I @ K )
     => ( ( K
         != ( suc @ I ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I @ J2 )
             => ( K
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_1137_lift__Suc__antimono__le,axiom,
    ! [F2: nat > nat,N: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_nat @ ( F2 @ N4 ) @ ( F2 @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1138_lift__Suc__mono__le,axiom,
    ! [F2: nat > nat,N: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_nat @ ( F2 @ N ) @ ( F2 @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1139_lift__Suc__mono__less__iff,axiom,
    ! [F2: nat > nat,N: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ ( F2 @ N ) @ ( F2 @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1140_lift__Suc__mono__less,axiom,
    ! [F2: nat > nat,N: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_nat @ ( F2 @ N ) @ ( F2 @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1141_nth__drop__0,axiom,
    ! [Ss: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Ss ) )
     => ( ( cons_nat @ ( nth_nat @ Ss @ zero_zero_nat ) @ ( drop_nat @ ( suc @ zero_zero_nat ) @ Ss ) )
        = Ss ) ) ).

% nth_drop_0
thf(fact_1142_nth__drop__0,axiom,
    ! [Ss: list_term_a_b] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s8906293707977694520rm_a_b @ Ss ) )
     => ( ( cons_term_a_b @ ( nth_term_a_b @ Ss @ zero_zero_nat ) @ ( drop_term_a_b @ ( suc @ zero_zero_nat ) @ Ss ) )
        = Ss ) ) ).

% nth_drop_0
thf(fact_1143_list__update__code_I3_J,axiom,
    ! [X: nat,Xs2: list_nat,I: nat,Y: nat] :
      ( ( list_update_nat @ ( cons_nat @ X @ Xs2 ) @ ( suc @ I ) @ Y )
      = ( cons_nat @ X @ ( list_update_nat @ Xs2 @ I @ Y ) ) ) ).

% list_update_code(3)
thf(fact_1144_Suc__length__conv,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ( suc @ N )
        = ( size_size_list_nat @ Xs2 ) )
      = ( ? [Y5: nat,Ys3: list_nat] :
            ( ( Xs2
              = ( cons_nat @ Y5 @ Ys3 ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_1145_Suc__length__conv,axiom,
    ! [N: nat,Xs2: list_term_a_b] :
      ( ( ( suc @ N )
        = ( size_s8906293707977694520rm_a_b @ Xs2 ) )
      = ( ? [Y5: term_a_b,Ys3: list_term_a_b] :
            ( ( Xs2
              = ( cons_term_a_b @ Y5 @ Ys3 ) )
            & ( ( size_s8906293707977694520rm_a_b @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_1146_length__Suc__conv,axiom,
    ! [Xs2: list_nat,N: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( suc @ N ) )
      = ( ? [Y5: nat,Ys3: list_nat] :
            ( ( Xs2
              = ( cons_nat @ Y5 @ Ys3 ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_1147_length__Suc__conv,axiom,
    ! [Xs2: list_term_a_b,N: nat] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
        = ( suc @ N ) )
      = ( ? [Y5: term_a_b,Ys3: list_term_a_b] :
            ( ( Xs2
              = ( cons_term_a_b @ Y5 @ Ys3 ) )
            & ( ( size_s8906293707977694520rm_a_b @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_1148_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_1149_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_1150_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] :
      ( zero_zero_nat
     != ( suc @ X2 ) ) ).

% nat.distinct(1)
thf(fact_1151_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_1152_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_1153_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat
        = ( suc @ X2 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_1154_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_1155_nat__induct,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( P2 @ N3 )
           => ( P2 @ ( suc @ N3 ) ) )
       => ( P2 @ N ) ) ) ).

% nat_induct
thf(fact_1156_diff__induct,axiom,
    ! [P2: nat > nat > $o,M: nat,N: nat] :
      ( ! [X3: nat] : ( P2 @ X3 @ zero_zero_nat )
     => ( ! [Y3: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y3 ) )
       => ( ! [X3: nat,Y3: nat] :
              ( ( P2 @ X3 @ Y3 )
             => ( P2 @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
         => ( P2 @ M @ N ) ) ) ) ).

% diff_induct
thf(fact_1157_zero__induct,axiom,
    ! [P2: nat > $o,K: nat] :
      ( ( P2 @ K )
     => ( ! [N3: nat] :
            ( ( P2 @ ( suc @ N3 ) )
           => ( P2 @ N3 ) )
       => ( P2 @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_1158_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_1159_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_1160_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_1161_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M4: nat] :
          ( N
          = ( suc @ M4 ) ) ) ).

% not0_implies_Suc
thf(fact_1162_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_1163_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M4: nat] :
          ( N
          = ( suc @ M4 ) ) ) ).

% gr0_implies_Suc
thf(fact_1164_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M2: nat] :
            ( N
            = ( suc @ M2 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_1165_Cons__nth__drop__Suc,axiom,
    ! [I: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( cons_nat @ ( nth_nat @ Xs2 @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs2 ) )
        = ( drop_nat @ I @ Xs2 ) ) ) ).

% Cons_nth_drop_Suc
thf(fact_1166_Cons__nth__drop__Suc,axiom,
    ! [I: nat,Xs2: list_term_a_b] :
      ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( ( cons_term_a_b @ ( nth_term_a_b @ Xs2 @ I ) @ ( drop_term_a_b @ ( suc @ I ) @ Xs2 ) )
        = ( drop_term_a_b @ I @ Xs2 ) ) ) ).

% Cons_nth_drop_Suc
thf(fact_1167_take__drop__imp__nth,axiom,
    ! [I: nat,Ss: list_nat,X: nat] :
      ( ( ( append_nat @ ( take_nat @ I @ Ss ) @ ( cons_nat @ X @ ( drop_nat @ ( suc @ I ) @ Ss ) ) )
        = Ss )
     => ( X
        = ( nth_nat @ Ss @ I ) ) ) ).

% take_drop_imp_nth
thf(fact_1168_gen__length__code_I2_J,axiom,
    ! [N: nat,X: nat,Xs2: list_nat] :
      ( ( gen_length_nat @ N @ ( cons_nat @ X @ Xs2 ) )
      = ( gen_length_nat @ ( suc @ N ) @ Xs2 ) ) ).

% gen_length_code(2)
thf(fact_1169_id__take__nth__drop,axiom,
    ! [I: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( Xs2
        = ( append_nat @ ( take_nat @ I @ Xs2 ) @ ( cons_nat @ ( nth_nat @ Xs2 @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs2 ) ) ) ) ) ).

% id_take_nth_drop
thf(fact_1170_id__take__nth__drop,axiom,
    ! [I: nat,Xs2: list_term_a_b] :
      ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( Xs2
        = ( append_term_a_b @ ( take_term_a_b @ I @ Xs2 ) @ ( cons_term_a_b @ ( nth_term_a_b @ Xs2 @ I ) @ ( drop_term_a_b @ ( suc @ I ) @ Xs2 ) ) ) ) ) ).

% id_take_nth_drop
thf(fact_1171_nth__append__take__drop__is__nth__conv,axiom,
    ! [I: nat,Xs2: list_nat,J: nat,Y: nat] :
      ( ( ord_less_eq_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_eq_nat @ J @ ( size_size_list_nat @ Xs2 ) )
       => ( ( I != J )
         => ( ( nth_nat @ ( append_nat @ ( take_nat @ J @ Xs2 ) @ ( cons_nat @ Y @ ( drop_nat @ ( suc @ J ) @ Xs2 ) ) ) @ I )
            = ( nth_nat @ Xs2 @ I ) ) ) ) ) ).

% nth_append_take_drop_is_nth_conv
thf(fact_1172_nth__append__take__drop__is__nth__conv,axiom,
    ! [I: nat,Xs2: list_term_a_b,J: nat,Y: term_a_b] :
      ( ( ord_less_eq_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( ( ord_less_eq_nat @ J @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
       => ( ( I != J )
         => ( ( nth_term_a_b @ ( append_term_a_b @ ( take_term_a_b @ J @ Xs2 ) @ ( cons_term_a_b @ Y @ ( drop_term_a_b @ ( suc @ J ) @ Xs2 ) ) ) @ I )
            = ( nth_term_a_b @ Xs2 @ I ) ) ) ) ) ).

% nth_append_take_drop_is_nth_conv
thf(fact_1173_upd__conv__take__nth__drop,axiom,
    ! [I: nat,Xs2: list_nat,A: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( list_update_nat @ Xs2 @ I @ A )
        = ( append_nat @ ( take_nat @ I @ Xs2 ) @ ( cons_nat @ A @ ( drop_nat @ ( suc @ I ) @ Xs2 ) ) ) ) ) ).

% upd_conv_take_nth_drop
thf(fact_1174_upd__conv__take__nth__drop,axiom,
    ! [I: nat,Xs2: list_term_a_b,A: term_a_b] :
      ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( ( list_update_term_a_b @ Xs2 @ I @ A )
        = ( append_term_a_b @ ( take_term_a_b @ I @ Xs2 ) @ ( cons_term_a_b @ A @ ( drop_term_a_b @ ( suc @ I ) @ Xs2 ) ) ) ) ) ).

% upd_conv_take_nth_drop
thf(fact_1175_take__drop__update__first,axiom,
    ! [J: nat,Ds: list_term_a_b,Cs: list_term_a_b] :
      ( ( ord_less_nat @ J @ ( size_s8906293707977694520rm_a_b @ Ds ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Cs )
          = ( size_s8906293707977694520rm_a_b @ Ds ) )
       => ( ( list_update_term_a_b @ ( append_term_a_b @ ( take_term_a_b @ J @ Ds ) @ ( drop_term_a_b @ J @ Cs ) ) @ J @ ( nth_term_a_b @ Ds @ J ) )
          = ( append_term_a_b @ ( take_term_a_b @ ( suc @ J ) @ Ds ) @ ( drop_term_a_b @ ( suc @ J ) @ Cs ) ) ) ) ) ).

% take_drop_update_first
thf(fact_1176_append__eq__conv__conj,axiom,
    ! [Xs2: list_term_a_b,Ys: list_term_a_b,Zs: list_term_a_b] :
      ( ( ( append_term_a_b @ Xs2 @ Ys )
        = Zs )
      = ( ( Xs2
          = ( take_term_a_b @ ( size_s8906293707977694520rm_a_b @ Xs2 ) @ Zs ) )
        & ( Ys
          = ( drop_term_a_b @ ( size_s8906293707977694520rm_a_b @ Xs2 ) @ Zs ) ) ) ) ).

% append_eq_conv_conj
thf(fact_1177_Suc__le__length__iff,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs2 ) )
      = ( ? [X4: nat,Ys3: list_nat] :
            ( ( Xs2
              = ( cons_nat @ X4 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_1178_Suc__le__length__iff,axiom,
    ! [N: nat,Xs2: list_term_a_b] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
      = ( ? [X4: term_a_b,Ys3: list_term_a_b] :
            ( ( Xs2
              = ( cons_term_a_b @ X4 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_s8906293707977694520rm_a_b @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_1179_ex__least__nat__less,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ N )
     => ( ~ ( P2 @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ! [I3: nat] :
                ( ( ord_less_eq_nat @ I3 @ K3 )
               => ~ ( P2 @ I3 ) )
            & ( P2 @ ( suc @ K3 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_1180_nth__append__drop__is__nth__conv,axiom,
    ! [J: nat,I: nat,Xs2: list_nat,Y: nat] :
      ( ( ord_less_nat @ J @ I )
     => ( ( ord_less_eq_nat @ J @ ( size_size_list_nat @ Xs2 ) )
       => ( ( ord_less_eq_nat @ I @ ( size_size_list_nat @ Xs2 ) )
         => ( ( nth_nat @ ( append_nat @ ( take_nat @ J @ Xs2 ) @ ( cons_nat @ Y @ ( drop_nat @ ( suc @ J ) @ Xs2 ) ) ) @ I )
            = ( nth_nat @ Xs2 @ I ) ) ) ) ) ).

% nth_append_drop_is_nth_conv
thf(fact_1181_nth__append__drop__is__nth__conv,axiom,
    ! [J: nat,I: nat,Xs2: list_term_a_b,Y: term_a_b] :
      ( ( ord_less_nat @ J @ I )
     => ( ( ord_less_eq_nat @ J @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
       => ( ( ord_less_eq_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
         => ( ( nth_term_a_b @ ( append_term_a_b @ ( take_term_a_b @ J @ Xs2 ) @ ( cons_term_a_b @ Y @ ( drop_term_a_b @ ( suc @ J ) @ Xs2 ) ) ) @ I )
            = ( nth_term_a_b @ Xs2 @ I ) ) ) ) ) ).

% nth_append_drop_is_nth_conv
thf(fact_1182_length__add__elem__list__lists,axiom,
    ! [Ys: list_nat,X: nat,Xs2: list_nat] :
      ( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( basic_4874698711677410535ts_nat @ X @ Xs2 ) ) )
     => ( ( size_size_list_nat @ Ys )
        = ( suc @ ( size_size_list_nat @ Xs2 ) ) ) ) ).

% length_add_elem_list_lists
thf(fact_1183_length__add__elem__list__lists,axiom,
    ! [Ys: list_term_a_b,X: term_a_b,Xs2: list_term_a_b] :
      ( ( member_list_term_a_b @ Ys @ ( set_list_term_a_b2 @ ( basic_1593220722155286443rm_a_b @ X @ Xs2 ) ) )
     => ( ( size_s8906293707977694520rm_a_b @ Ys )
        = ( suc @ ( size_s8906293707977694520rm_a_b @ Xs2 ) ) ) ) ).

% length_add_elem_list_lists
thf(fact_1184_append__eq__append__conv__if,axiom,
    ! [Xs_1: list_term_a_b,Xs_2: list_term_a_b,Ys_1: list_term_a_b,Ys_2: list_term_a_b] :
      ( ( ( append_term_a_b @ Xs_1 @ Xs_2 )
        = ( append_term_a_b @ Ys_1 @ Ys_2 ) )
      = ( ( ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs_1 ) @ ( size_s8906293707977694520rm_a_b @ Ys_1 ) )
         => ( ( Xs_1
              = ( take_term_a_b @ ( size_s8906293707977694520rm_a_b @ Xs_1 ) @ Ys_1 ) )
            & ( Xs_2
              = ( append_term_a_b @ ( drop_term_a_b @ ( size_s8906293707977694520rm_a_b @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs_1 ) @ ( size_s8906293707977694520rm_a_b @ Ys_1 ) )
         => ( ( ( take_term_a_b @ ( size_s8906293707977694520rm_a_b @ Ys_1 ) @ Xs_1 )
              = Ys_1 )
            & ( ( append_term_a_b @ ( drop_term_a_b @ ( size_s8906293707977694520rm_a_b @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
              = Ys_2 ) ) ) ) ) ).

% append_eq_append_conv_if
thf(fact_1185_list_Osize_I4_J,axiom,
    ! [X21: nat,X22: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_1186_list_Osize_I4_J,axiom,
    ! [X21: term_a_b,X22: list_term_a_b] :
      ( ( size_s8906293707977694520rm_a_b @ ( cons_term_a_b @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_s8906293707977694520rm_a_b @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_1187_length__Suc__conv__rev,axiom,
    ! [Xs2: list_nat,N: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( suc @ N ) )
      = ( ? [Y5: nat,Ys3: list_nat] :
            ( ( Xs2
              = ( append_nat @ Ys3 @ ( cons_nat @ Y5 @ nil_nat ) ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_1188_length__Suc__conv__rev,axiom,
    ! [Xs2: list_term_a_b,N: nat] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
        = ( suc @ N ) )
      = ( ? [Y5: term_a_b,Ys3: list_term_a_b] :
            ( ( Xs2
              = ( append_term_a_b @ Ys3 @ ( cons_term_a_b @ Y5 @ nil_term_a_b ) ) )
            & ( ( size_s8906293707977694520rm_a_b @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_1189_take__drop__update__second,axiom,
    ! [J: nat,Ds: list_term_a_b,Cs: list_term_a_b] :
      ( ( ord_less_nat @ J @ ( size_s8906293707977694520rm_a_b @ Ds ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Cs )
          = ( size_s8906293707977694520rm_a_b @ Ds ) )
       => ( ( list_update_term_a_b @ ( append_term_a_b @ ( take_term_a_b @ J @ Ds ) @ ( drop_term_a_b @ J @ Cs ) ) @ J @ ( nth_term_a_b @ Cs @ J ) )
          = ( append_term_a_b @ ( take_term_a_b @ J @ Ds ) @ ( drop_term_a_b @ J @ Cs ) ) ) ) ) ).

% take_drop_update_second
thf(fact_1190_add__elem__list__listsE,axiom,
    ! [Ys: list_nat,X: nat,Xs2: list_nat] :
      ( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( basic_4874698711677410535ts_nat @ X @ Xs2 ) ) )
     => ? [N3: nat] :
          ( ( ord_less_eq_nat @ N3 @ ( size_size_list_nat @ Xs2 ) )
          & ( Ys
            = ( append_nat @ ( take_nat @ N3 @ Xs2 ) @ ( cons_nat @ X @ ( drop_nat @ N3 @ Xs2 ) ) ) ) ) ) ).

% add_elem_list_listsE
thf(fact_1191_add__elem__list__listsE,axiom,
    ! [Ys: list_term_a_b,X: term_a_b,Xs2: list_term_a_b] :
      ( ( member_list_term_a_b @ Ys @ ( set_list_term_a_b2 @ ( basic_1593220722155286443rm_a_b @ X @ Xs2 ) ) )
     => ? [N3: nat] :
          ( ( ord_less_eq_nat @ N3 @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
          & ( Ys
            = ( append_term_a_b @ ( take_term_a_b @ N3 @ Xs2 ) @ ( cons_term_a_b @ X @ ( drop_term_a_b @ N3 @ Xs2 ) ) ) ) ) ) ).

% add_elem_list_listsE
thf(fact_1192_length__append__singleton,axiom,
    ! [Xs2: list_nat,X: nat] :
      ( ( size_size_list_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) )
      = ( suc @ ( size_size_list_nat @ Xs2 ) ) ) ).

% length_append_singleton
thf(fact_1193_length__append__singleton,axiom,
    ! [Xs2: list_term_a_b,X: term_a_b] :
      ( ( size_s8906293707977694520rm_a_b @ ( append_term_a_b @ Xs2 @ ( cons_term_a_b @ X @ nil_term_a_b ) ) )
      = ( suc @ ( size_s8906293707977694520rm_a_b @ Xs2 ) ) ) ).

% length_append_singleton
thf(fact_1194_length__Cons,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X @ Xs2 ) )
      = ( suc @ ( size_size_list_nat @ Xs2 ) ) ) ).

% length_Cons
thf(fact_1195_length__Cons,axiom,
    ! [X: term_a_b,Xs2: list_term_a_b] :
      ( ( size_s8906293707977694520rm_a_b @ ( cons_term_a_b @ X @ Xs2 ) )
      = ( suc @ ( size_s8906293707977694520rm_a_b @ Xs2 ) ) ) ).

% length_Cons
thf(fact_1196_list_Osize__gen_I1_J,axiom,
    ! [X: nat > nat] :
      ( ( size_list_nat @ X @ nil_nat )
      = zero_zero_nat ) ).

% list.size_gen(1)
thf(fact_1197_size__list__pointwise,axiom,
    ! [Xs2: list_P3592885314253461005_a_nat,F2: product_prod_a_nat > nat,G: product_prod_a_nat > nat] :
      ( ! [X3: product_prod_a_nat] :
          ( ( member5724188588386418708_a_nat @ X3 @ ( set_Pr924983374503034536_a_nat @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_nat @ ( size_l179396747238966901_a_nat @ F2 @ Xs2 ) @ ( size_l179396747238966901_a_nat @ G @ Xs2 ) ) ) ).

% size_list_pointwise
thf(fact_1198_size__list__pointwise,axiom,
    ! [Xs2: list_list_nat,F2: list_nat > nat,G: list_nat > nat] :
      ( ! [X3: list_nat] :
          ( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_nat @ ( size_list_list_nat @ F2 @ Xs2 ) @ ( size_list_list_nat @ G @ Xs2 ) ) ) ).

% size_list_pointwise
thf(fact_1199_size__list__estimation_H,axiom,
    ! [X: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Y: nat,F2: product_prod_a_nat > nat] :
      ( ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Xs2 ) )
     => ( ( ord_less_eq_nat @ Y @ ( F2 @ X ) )
       => ( ord_less_eq_nat @ Y @ ( size_l179396747238966901_a_nat @ F2 @ Xs2 ) ) ) ) ).

% size_list_estimation'
thf(fact_1200_size__list__estimation_H,axiom,
    ! [X: list_nat,Xs2: list_list_nat,Y: nat,F2: list_nat > nat] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ Y @ ( F2 @ X ) )
       => ( ord_less_eq_nat @ Y @ ( size_list_list_nat @ F2 @ Xs2 ) ) ) ) ).

% size_list_estimation'
thf(fact_1201_size__list__estimation,axiom,
    ! [X: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Y: nat,F2: product_prod_a_nat > nat] :
      ( ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Xs2 ) )
     => ( ( ord_less_nat @ Y @ ( F2 @ X ) )
       => ( ord_less_nat @ Y @ ( size_l179396747238966901_a_nat @ F2 @ Xs2 ) ) ) ) ).

% size_list_estimation
thf(fact_1202_size__list__estimation,axiom,
    ! [X: list_nat,Xs2: list_list_nat,Y: nat,F2: list_nat > nat] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
     => ( ( ord_less_nat @ Y @ ( F2 @ X ) )
       => ( ord_less_nat @ Y @ ( size_list_list_nat @ F2 @ Xs2 ) ) ) ) ).

% size_list_estimation
thf(fact_1203_size__simp1,axiom,
    ! [T: product_prod_a_nat,Ts: list_P3592885314253461005_a_nat] :
      ( ( member5724188588386418708_a_nat @ T @ ( set_Pr924983374503034536_a_nat @ Ts ) )
     => ( ord_less_nat @ ( size_s1497571686738832145_a_nat @ T ) @ ( suc @ ( size_l179396747238966901_a_nat @ size_s1497571686738832145_a_nat @ Ts ) ) ) ) ).

% size_simp1
thf(fact_1204_size__simp1,axiom,
    ! [T: list_nat,Ts: list_list_nat] :
      ( ( member_list_nat @ T @ ( set_list_nat2 @ Ts ) )
     => ( ord_less_nat @ ( size_size_list_nat @ T ) @ ( suc @ ( size_list_list_nat @ size_size_list_nat @ Ts ) ) ) ) ).

% size_simp1
thf(fact_1205_size__simp1,axiom,
    ! [T: list_term_a_b,Ts: list_list_term_a_b] :
      ( ( member_list_term_a_b @ T @ ( set_list_term_a_b2 @ Ts ) )
     => ( ord_less_nat @ ( size_s8906293707977694520rm_a_b @ T ) @ ( suc @ ( size_l7588118768477829276rm_a_b @ size_s8906293707977694520rm_a_b @ Ts ) ) ) ) ).

% size_simp1
thf(fact_1206_term_Osize_I4_J,axiom,
    ! [X21: a,X22: list_term_a_b] :
      ( ( size_size_term_a_b @ ( fun_a_b @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_list_term_a_b @ size_size_term_a_b @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% term.size(4)
thf(fact_1207_size__simp2,axiom,
    ! [T: product_prod_a_nat,Ts: list_P3592885314253461005_a_nat,S: list_term_a_b] :
      ( ( member5724188588386418708_a_nat @ T @ ( set_Pr924983374503034536_a_nat @ Ts ) )
     => ( ord_less_nat @ ( size_s1497571686738832145_a_nat @ T ) @ ( suc @ ( suc @ ( plus_plus_nat @ ( size_s8906293707977694520rm_a_b @ S ) @ ( size_l179396747238966901_a_nat @ size_s1497571686738832145_a_nat @ Ts ) ) ) ) ) ) ).

% size_simp2
thf(fact_1208_size__simp2,axiom,
    ! [T: list_nat,Ts: list_list_nat,S: list_term_a_b] :
      ( ( member_list_nat @ T @ ( set_list_nat2 @ Ts ) )
     => ( ord_less_nat @ ( size_size_list_nat @ T ) @ ( suc @ ( suc @ ( plus_plus_nat @ ( size_s8906293707977694520rm_a_b @ S ) @ ( size_list_list_nat @ size_size_list_nat @ Ts ) ) ) ) ) ) ).

% size_simp2
thf(fact_1209_size__simp2,axiom,
    ! [T: list_term_a_b,Ts: list_list_term_a_b,S: list_term_a_b] :
      ( ( member_list_term_a_b @ T @ ( set_list_term_a_b2 @ Ts ) )
     => ( ord_less_nat @ ( size_s8906293707977694520rm_a_b @ T ) @ ( suc @ ( suc @ ( plus_plus_nat @ ( size_s8906293707977694520rm_a_b @ S ) @ ( size_l7588118768477829276rm_a_b @ size_s8906293707977694520rm_a_b @ Ts ) ) ) ) ) ) ).

% size_simp2
thf(fact_1210_size__simp4,axiom,
    ! [X: a,Y: nat,Xs2: list_a,Ys: list_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y ) @ ( set_Pr924983374503034536_a_nat @ ( zip_a_nat @ Xs2 @ Ys ) ) )
     => ( ord_less_nat @ ( size_size_nat @ Y ) @ ( suc @ ( size_list_nat @ size_size_nat @ Ys ) ) ) ) ).

% size_simp4
thf(fact_1211_take__hd__drop,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( ( append_nat @ ( take_nat @ N @ Xs2 ) @ ( cons_nat @ ( hd_nat @ ( drop_nat @ N @ Xs2 ) ) @ nil_nat ) )
        = ( take_nat @ ( suc @ N ) @ Xs2 ) ) ) ).

% take_hd_drop
thf(fact_1212_take__hd__drop,axiom,
    ! [N: nat,Xs2: list_term_a_b] :
      ( ( ord_less_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( ( append_term_a_b @ ( take_term_a_b @ N @ Xs2 ) @ ( cons_term_a_b @ ( hd_term_a_b @ ( drop_term_a_b @ N @ Xs2 ) ) @ nil_term_a_b ) )
        = ( take_term_a_b @ ( suc @ N ) @ Xs2 ) ) ) ).

% take_hd_drop
thf(fact_1213_take__nth__drop__concat,axiom,
    ! [I: nat,Xss2: list_list_term_a_b,Ys: list_term_a_b,J: nat,Z: term_a_b] :
      ( ( ord_less_nat @ I @ ( size_s877380706853472072rm_a_b @ Xss2 ) )
     => ( ( ( nth_list_term_a_b @ Xss2 @ I )
          = Ys )
       => ( ( ord_less_nat @ J @ ( size_s8906293707977694520rm_a_b @ Ys ) )
         => ( ( ( nth_term_a_b @ Ys @ J )
              = Z )
           => ? [K3: nat] :
                ( ( ord_less_nat @ K3 @ ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Xss2 ) ) )
                & ( ( take_term_a_b @ K3 @ ( concat_term_a_b @ Xss2 ) )
                  = ( append_term_a_b @ ( concat_term_a_b @ ( take_list_term_a_b @ I @ Xss2 ) ) @ ( take_term_a_b @ J @ Ys ) ) )
                & ( ( nth_term_a_b @ ( concat_term_a_b @ Xss2 ) @ K3 )
                  = ( nth_term_a_b @ ( nth_list_term_a_b @ Xss2 @ I ) @ J ) )
                & ( ( drop_term_a_b @ ( suc @ K3 ) @ ( concat_term_a_b @ Xss2 ) )
                  = ( append_term_a_b @ ( drop_term_a_b @ ( suc @ J ) @ Ys ) @ ( concat_term_a_b @ ( drop_list_term_a_b @ ( suc @ I ) @ Xss2 ) ) ) ) ) ) ) ) ) ).

% take_nth_drop_concat
thf(fact_1214_hd__append2,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( Xs2 != nil_nat )
     => ( ( hd_nat @ ( append_nat @ Xs2 @ Ys ) )
        = ( hd_nat @ Xs2 ) ) ) ).

% hd_append2
thf(fact_1215_Nil__eq__concat__conv,axiom,
    ! [Xss2: list_list_nat] :
      ( ( nil_nat
        = ( concat_nat @ Xss2 ) )
      = ( ! [X4: list_nat] :
            ( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xss2 ) )
           => ( X4 = nil_nat ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_1216_concat__eq__Nil__conv,axiom,
    ! [Xss2: list_list_nat] :
      ( ( ( concat_nat @ Xss2 )
        = nil_nat )
      = ( ! [X4: list_nat] :
            ( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xss2 ) )
           => ( X4 = nil_nat ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_1217_list_Osel_I1_J,axiom,
    ! [X21: nat,X22: list_nat] :
      ( ( hd_nat @ ( cons_nat @ X21 @ X22 ) )
      = X21 ) ).

% list.sel(1)
thf(fact_1218_hd__conv__nth,axiom,
    ! [Xs2: list_nat] :
      ( ( Xs2 != nil_nat )
     => ( ( hd_nat @ Xs2 )
        = ( nth_nat @ Xs2 @ zero_zero_nat ) ) ) ).

% hd_conv_nth
thf(fact_1219_eq__length__concat__nth,axiom,
    ! [Xs2: list_list_term_a_b,Ys: list_list_term_a_b] :
      ( ( ( size_s877380706853472072rm_a_b @ Xs2 )
        = ( size_s877380706853472072rm_a_b @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s877380706853472072rm_a_b @ Xs2 ) )
           => ( ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs2 @ I2 ) )
              = ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Ys @ I2 ) ) ) )
       => ( ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Xs2 ) )
          = ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Ys ) ) ) ) ) ).

% eq_length_concat_nth
thf(fact_1220_hd__zip,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( Xs2 != nil_nat )
     => ( ( Ys != nil_nat )
       => ( ( hd_Pro3460610213475200108at_nat @ ( zip_nat_nat @ Xs2 @ Ys ) )
          = ( product_Pair_nat_nat @ ( hd_nat @ Xs2 ) @ ( hd_nat @ Ys ) ) ) ) ) ).

% hd_zip
thf(fact_1221_hd__zip,axiom,
    ! [Xs2: list_a,Ys: list_nat] :
      ( ( Xs2 != nil_a )
     => ( ( Ys != nil_nat )
       => ( ( hd_Pro8935205257713178578_a_nat @ ( zip_a_nat @ Xs2 @ Ys ) )
          = ( product_Pair_a_nat @ ( hd_a @ Xs2 ) @ ( hd_nat @ Ys ) ) ) ) ) ).

% hd_zip
thf(fact_1222_concat_Osimps_I1_J,axiom,
    ( ( concat_nat @ nil_list_nat )
    = nil_nat ) ).

% concat.simps(1)
thf(fact_1223_list_Oset__sel_I1_J,axiom,
    ! [A: list_P3592885314253461005_a_nat] :
      ( ( A != nil_Pr7402525243500994295_a_nat )
     => ( member5724188588386418708_a_nat @ ( hd_Pro8935205257713178578_a_nat @ A ) @ ( set_Pr924983374503034536_a_nat @ A ) ) ) ).

% list.set_sel(1)
thf(fact_1224_list_Oset__sel_I1_J,axiom,
    ! [A: list_list_nat] :
      ( ( A != nil_list_nat )
     => ( member_list_nat @ ( hd_list_nat @ A ) @ ( set_list_nat2 @ A ) ) ) ).

% list.set_sel(1)
thf(fact_1225_list_Oset__sel_I1_J,axiom,
    ! [A: list_nat] :
      ( ( A != nil_nat )
     => ( member_nat @ ( hd_nat @ A ) @ ( set_nat2 @ A ) ) ) ).

% list.set_sel(1)
thf(fact_1226_hd__in__set,axiom,
    ! [Xs2: list_P3592885314253461005_a_nat] :
      ( ( Xs2 != nil_Pr7402525243500994295_a_nat )
     => ( member5724188588386418708_a_nat @ ( hd_Pro8935205257713178578_a_nat @ Xs2 ) @ ( set_Pr924983374503034536_a_nat @ Xs2 ) ) ) ).

% hd_in_set
thf(fact_1227_hd__in__set,axiom,
    ! [Xs2: list_list_nat] :
      ( ( Xs2 != nil_list_nat )
     => ( member_list_nat @ ( hd_list_nat @ Xs2 ) @ ( set_list_nat2 @ Xs2 ) ) ) ).

% hd_in_set
thf(fact_1228_hd__in__set,axiom,
    ! [Xs2: list_nat] :
      ( ( Xs2 != nil_nat )
     => ( member_nat @ ( hd_nat @ Xs2 ) @ ( set_nat2 @ Xs2 ) ) ) ).

% hd_in_set
thf(fact_1229_longest__common__prefix,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
    ? [Ps2: list_nat,Xs5: list_nat,Ys7: list_nat] :
      ( ( Xs2
        = ( append_nat @ Ps2 @ Xs5 ) )
      & ( Ys
        = ( append_nat @ Ps2 @ Ys7 ) )
      & ( ( Xs5 = nil_nat )
        | ( Ys7 = nil_nat )
        | ( ( hd_nat @ Xs5 )
         != ( hd_nat @ Ys7 ) ) ) ) ).

% longest_common_prefix
thf(fact_1230_hd__append,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( Xs2 = nil_nat )
       => ( ( hd_nat @ ( append_nat @ Xs2 @ Ys ) )
          = ( hd_nat @ Ys ) ) )
      & ( ( Xs2 != nil_nat )
       => ( ( hd_nat @ ( append_nat @ Xs2 @ Ys ) )
          = ( hd_nat @ Xs2 ) ) ) ) ).

% hd_append
thf(fact_1231_hd__concat,axiom,
    ! [Xs2: list_list_nat] :
      ( ( Xs2 != nil_list_nat )
     => ( ( ( hd_list_nat @ Xs2 )
         != nil_nat )
       => ( ( hd_nat @ ( concat_nat @ Xs2 ) )
          = ( hd_nat @ ( hd_list_nat @ Xs2 ) ) ) ) ) ).

% hd_concat
thf(fact_1232_nth__concat__two__lists,axiom,
    ! [I: nat,Xs2: list_list_term_a_b,Ys: list_list_term_a_b] :
      ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Xs2 ) ) )
     => ( ( ( size_s877380706853472072rm_a_b @ Ys )
          = ( size_s877380706853472072rm_a_b @ Xs2 ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s877380706853472072rm_a_b @ Xs2 ) )
             => ( ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Ys @ I2 ) )
                = ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs2 @ I2 ) ) ) )
         => ? [J2: nat,K3: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s877380706853472072rm_a_b @ Xs2 ) )
              & ( ord_less_nat @ K3 @ ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs2 @ J2 ) ) )
              & ( ( nth_term_a_b @ ( concat_term_a_b @ Xs2 ) @ I )
                = ( nth_term_a_b @ ( nth_list_term_a_b @ Xs2 @ J2 ) @ K3 ) )
              & ( ( nth_term_a_b @ ( concat_term_a_b @ Ys ) @ I )
                = ( nth_term_a_b @ ( nth_list_term_a_b @ Ys @ J2 ) @ K3 ) ) ) ) ) ) ).

% nth_concat_two_lists
thf(fact_1233_nth__concat__split,axiom,
    ! [I: nat,Xs2: list_list_term_a_b] :
      ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Xs2 ) ) )
     => ? [J2: nat,K3: nat] :
          ( ( ord_less_nat @ J2 @ ( size_s877380706853472072rm_a_b @ Xs2 ) )
          & ( ord_less_nat @ K3 @ ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs2 @ J2 ) ) )
          & ( ( nth_term_a_b @ ( concat_term_a_b @ Xs2 ) @ I )
            = ( nth_term_a_b @ ( nth_list_term_a_b @ Xs2 @ J2 ) @ K3 ) ) ) ) ).

% nth_concat_split
thf(fact_1234_nth__concat__diff,axiom,
    ! [I1: nat,Xs2: list_list_term_a_b,I22: nat] :
      ( ( ord_less_nat @ I1 @ ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Xs2 ) ) )
     => ( ( ord_less_nat @ I22 @ ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Xs2 ) ) )
       => ( ( I1 != I22 )
         => ? [J1: nat,K1: nat,J22: nat,K22: nat] :
              ( ( ( product_Pair_nat_nat @ J1 @ K1 )
               != ( product_Pair_nat_nat @ J22 @ K22 ) )
              & ( ord_less_nat @ J1 @ ( size_s877380706853472072rm_a_b @ Xs2 ) )
              & ( ord_less_nat @ J22 @ ( size_s877380706853472072rm_a_b @ Xs2 ) )
              & ( ord_less_nat @ K1 @ ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs2 @ J1 ) ) )
              & ( ord_less_nat @ K22 @ ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs2 @ J22 ) ) )
              & ( ( nth_term_a_b @ ( concat_term_a_b @ Xs2 ) @ I1 )
                = ( nth_term_a_b @ ( nth_list_term_a_b @ Xs2 @ J1 ) @ K1 ) )
              & ( ( nth_term_a_b @ ( concat_term_a_b @ Xs2 ) @ I22 )
                = ( nth_term_a_b @ ( nth_list_term_a_b @ Xs2 @ J22 ) @ K22 ) ) ) ) ) ) ).

% nth_concat_diff
thf(fact_1235_concat__all__nth,axiom,
    ! [Xs2: list_list_term_a_b,Ys: list_list_term_a_b,P2: term_a_b > term_a_b > $o] :
      ( ( ( size_s877380706853472072rm_a_b @ Xs2 )
        = ( size_s877380706853472072rm_a_b @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s877380706853472072rm_a_b @ Xs2 ) )
           => ( ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs2 @ I2 ) )
              = ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Ys @ I2 ) ) ) )
       => ( ! [I2: nat,J2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s877380706853472072rm_a_b @ Xs2 ) )
             => ( ( ord_less_nat @ J2 @ ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs2 @ I2 ) ) )
               => ( P2 @ ( nth_term_a_b @ ( nth_list_term_a_b @ Xs2 @ I2 ) @ J2 ) @ ( nth_term_a_b @ ( nth_list_term_a_b @ Ys @ I2 ) @ J2 ) ) ) )
         => ! [K4: nat] :
              ( ( ord_less_nat @ K4 @ ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Xs2 ) ) )
             => ( P2 @ ( nth_term_a_b @ ( concat_term_a_b @ Xs2 ) @ K4 ) @ ( nth_term_a_b @ ( concat_term_a_b @ Ys ) @ K4 ) ) ) ) ) ) ).

% concat_all_nth
thf(fact_1236_hd__drop__conv__nth,axiom,
    ! [N: nat,Xs2: list_term_a_b] :
      ( ( ord_less_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
     => ( ( hd_term_a_b @ ( drop_term_a_b @ N @ Xs2 ) )
        = ( nth_term_a_b @ Xs2 @ N ) ) ) ).

% hd_drop_conv_nth
thf(fact_1237_concat__eq__append__conv,axiom,
    ! [Xss2: list_list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( concat_nat @ Xss2 )
        = ( append_nat @ Ys @ Zs ) )
      = ( ( ( Xss2 = nil_list_nat )
         => ( ( Ys = nil_nat )
            & ( Zs = nil_nat ) ) )
        & ( ( Xss2 != nil_list_nat )
         => ? [Xss1: list_list_nat,Xs3: list_nat,Xs6: list_nat,Xss22: list_list_nat] :
              ( ( Xss2
                = ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs3 @ Xs6 ) @ Xss22 ) ) )
              & ( Ys
                = ( append_nat @ ( concat_nat @ Xss1 ) @ Xs3 ) )
              & ( Zs
                = ( append_nat @ Xs6 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_1238_list__of__permutation__element__n_Oelims,axiom,
    ! [X: nat,Xa2: nat,Xb: list_nat,Y: list_list_nat] :
      ( ( ( basic_7079635023375748421_n_nat @ X @ Xa2 @ Xb )
        = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y
           != ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
       => ~ ! [N3: nat] :
              ( ( Xa2
                = ( suc @ N3 ) )
             => ( Y
               != ( concat_list_nat @ ( map_li960784813134754710st_nat @ ( basic_4874698711677410535ts_nat @ X ) @ ( n_lists_nat @ N3 @ Xb ) ) ) ) ) ) ) ).

% list_of_permutation_element_n.elims
thf(fact_1239_remove__nth__sound__r,axiom,
    ! [N: nat,P: nat,Xs2: list_term_a_b] :
      ( ( ord_less_eq_nat @ N @ P )
     => ( ( ord_less_nat @ P @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
       => ( ( nth_term_a_b @ ( missin7590947408886171483rm_a_b @ N @ Xs2 ) @ P )
          = ( nth_term_a_b @ Xs2 @ ( suc @ P ) ) ) ) ) ).

% remove_nth_sound_r
thf(fact_1240_map__is__Nil__conv,axiom,
    ! [F2: nat > nat,Xs2: list_nat] :
      ( ( ( map_nat_nat @ F2 @ Xs2 )
        = nil_nat )
      = ( Xs2 = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_1241_Nil__is__map__conv,axiom,
    ! [F2: nat > nat,Xs2: list_nat] :
      ( ( nil_nat
        = ( map_nat_nat @ F2 @ Xs2 ) )
      = ( Xs2 = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_1242_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

% Conjectures (2)
thf(conj_0,hypothesis,
    member5724188588386418708_a_nat @ ( product_Pair_a_nat @ f @ n ) @ ( term_funas_term_a_b @ t ) ).

thf(conj_1,conjecture,
    ? [P6: list_nat,Ts4: list_term_a_b] :
      ( ( member_list_nat @ P6 @ ( term_poss_a_b @ t ) )
      & ( ( term_subt_at_a_b @ t @ P6 )
        = ( fun_a_b @ f @ Ts4 ) )
      & ( ( size_s8906293707977694520rm_a_b @ Ts4 )
        = n ) ) ).

%------------------------------------------------------------------------------