%------------------------------------------------------------------------------
% 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    : FO_Theory_Rewriting/0074_Type_Instances_Impl/prob_00040_001171__18804706_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1950 ( 563 unt; 683 typ;   0 def)
%            Number of atoms       : 3538 (1695 equ;   0 cnn)
%            Maximal formula atoms :   22 (   2 avg)
%            Number of connectives : 13314 ( 502   ~;  51   |; 251   &;10824   @)
%                                         (   0 <=>;1686  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   7 avg)
%            Number of types       :  150 ( 149 usr)
%            Number of type conns  : 1713 (1713   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  537 ( 534 usr;  40 con; 0-4 aty)
%            Number of variables   : 4070 ( 157   ^;3783   !; 130   ?;4070   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 12:02:43.086
%------------------------------------------------------------------------------
% Could-be-implicit typings (149)
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J_J,type,
    list_P4088901747568347245st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_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_Pr915550674247830855st_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_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    produc1342688880896445543st_nat: $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__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__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Bot____Terms__Obot____term_Itf__f_J_J_Mt__List__Olist_It__Bot____Terms__Obot____term_Itf__f_J_J_J_J,type,
    set_Pr488239067965912317term_f: $tType ).

thf(ty_n_t__List__Olist_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,
    list_P8469869581646625389at_nat: $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__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
    list_P4072332127704817200st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
    set_Pr4879555682012792790st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    produc6029997274002168412et_nat: $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__List__Olist_It__Bot____Terms__Obot____term_Itf__f_J_J_Mt__List__Olist_It__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    produc7885403226499153821term_f: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J_J,type,
    list_P710685493942699252term_f: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    list_P184876202426519672at_nat: $tType ).

thf(ty_n_t__List__Olist_It__Bot____Terms__Obot____term_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J_J,type,
    list_B7941409156082897212st_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__Set__Oset_It__Bot____Terms__Obot____term_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J_J,type,
    set_Bo2954278103884806626st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_Mt__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    set_Pr2113329861893509485et_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_It__Nat__Onat_J_Mt__Bot____Terms__Obot____term_It__Nat__Onat_J_J_J,type,
    set_Pr1483163517390835655rm_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
    produc3922691147685405600st_nat: $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__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    list_P1780566427699531651term_f: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    produc3532251127348314084term_f: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc5523380839273745768at_nat: $tType ).

thf(ty_n_t__Bot____Terms__Obot____term_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    bot_bo5929273105904447532st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    set_Pr6841673932529763677term_f: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_Mt__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
    produc8518942320679187981st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_Mt__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    produc1024980458911918861et_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Bot____Terms__Obot____term_It__Nat__Onat_J_Mt__Bot____Terms__Obot____term_It__Nat__Onat_J_J,type,
    produc4207152732626004967rm_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    list_P9018165577420421589term_f: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    list_P3080432348914762709st_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    list_P5953897101696565392at_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    list_P1909269847677398966at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    set_Pr4986122518544458671term_f: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    set_Pr8271761326893575599st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    set_Pr2539167527615954998at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    set_Pr7717912310451564380at_nat: $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__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__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    produc250624644672406269term_f: $tType ).

thf(ty_n_t__List__Olist_It__Bot____Terms__Obot____term_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    list_B7173093809959871388at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
    produc4252680794130103549st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    produc922974302667211389et_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    list_P6254988961118846195et_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Bot____Terms__Obot____term_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    set_Bo8614249060733510210at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    set_Pr5488025237498180813et_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__String__Ochar_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    list_P2124888226753994182term_f: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__String__Ochar_J_J,type,
    list_P5158871451256932422f_char: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_M_062_It__Nat__Onat_M_Eo_J_J,type,
    produc4696479697851519326_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_Eo_J_Mt__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
    produc7905234009183233476st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__String__Ochar_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    set_Pr7448874415078007456term_f: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__String__Ochar_J_J,type,
    set_Pr1259485602726169888f_char: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    produc4771067281011818575term_f: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc5499396499697224271st_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    list_P3405996666353486021term_f: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Nat__Onat_J_J,type,
    list_P8521082412280730693_f_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    produc8373899037510109440at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc7248412053542808358at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    set_Pr460048518328002463term_f: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Nat__Onat_J_J,type,
    set_Pr5575134264255247135_f_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__String__Ochar_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    list_P8844850915902724120st_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__String__Ochar_J_J,type,
    list_P1202297530648401304t_char: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__String__Ochar_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    set_Pr3349125756617969906st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__String__Ochar_J_J,type,
    set_Pr4929944408218422898t_char: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__f_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    list_P2506729754372859472term_f: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mtf__f_J_J,type,
    list_P3672777798497168634rm_f_f: $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__List__Olist_It__Nat__Onat_J_J_J,type,
    list_P7736225833432154391st_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    list_P5364314822750548887at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__f_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    set_Pr7649113431310191094term_f: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mtf__f_J_J,type,
    set_Pr8815161475434500256rm_f_f: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    set_Pr698032948827214833st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    set_Pr7549493975000385137at_nat: $tType ).

thf(ty_n_t__Bot____Terms__Obot____term_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bot_bo655077203622184332at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    produc7819656566062154093et_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__String__Ochar_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    produc4569640511456206400term_f: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__String__Ochar_J,type,
    produc8828238612114966720f_char: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_M_062_It__Nat__Onat_M_Eo_J_J,type,
    produc7205429901713235566_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_Eo_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    produc3959935651712317876et_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__f_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    list_P4278292882394854562st_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mtf__f_J_J,type,
    list_P3776949671930543656_nat_f: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__f_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    set_Pr173388199362880584st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mtf__f_J_J,type,
    set_Pr8895417025753345486_nat_f: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    produc5418599768131837375term_f: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Nat__Onat_J,type,
    produc8417799323881384255_f_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_Eo_J_M_062_It__Nat__Onat_M_Eo_J_J,type,
    produc7360885484439884007_nat_o: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__String__Ochar_Mt__String__Ochar_J_J,type,
    list_P3261013228118292233r_char: $tType ).

thf(ty_n_t__List__Olist_It__Bot____Terms__Obot____term_It__List__Olist_It__Nat__Onat_J_J_J,type,
    list_B5078311858792894385st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__String__Ochar_Mt__String__Ochar_J_J,type,
    set_Pr3309569728657787619r_char: $tType ).

thf(ty_n_t__Set__Oset_It__Bot____Terms__Obot____term_It__List__Olist_It__Nat__Onat_J_J_J,type,
    set_Bo5126868359332389771st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__String__Ochar_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc5995604340556737426st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__String__Ochar_J,type,
    produc3186512776854530322t_char: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__String__Ochar_Mt__Nat__Onat_J_J,type,
    list_P3884813413850506760ar_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__String__Ochar_J_J,type,
    list_P5398368247131344008t_char: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__String__Ochar_Mt__Nat__Onat_J_J,type,
    set_Pr1413074515586503650ar_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__String__Ochar_J_J,type,
    set_Pr2926629348867340898t_char: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__f_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    produc112510866124124352term_f: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mtf__f_J,type,
    produc8536842985679875178rm_f_f: $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__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    produc1540777390238407569at_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__List__Olist_It__Product____Type__Oprod_Itf__f_Mt__String__Ochar_J_J,type,
    list_P494527469494163091f_char: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__String__Ochar_Mtf__f_J_J,type,
    list_P3726693972706002423char_f: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__f_Mt__String__Ochar_J_J,type,
    set_Pr3444705082972973881f_char: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__String__Ochar_Mtf__f_J_J,type,
    set_Pr6676871586184813213char_f: $tType ).

thf(ty_n_t__List__Olist_It__Bot____Terms__Obot____term_It__String__Ochar_J_J,type,
    list_B1576150950144988706m_char: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__f_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc5771559414059236882st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mtf__f_J,type,
    produc6459537420886108824_nat_f: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J,type,
    list_P3903862279629787026_f_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mtf__f_J_J,type,
    list_P3206961950963217208_nat_f: $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__Bot____Terms__Obot____term_It__String__Ochar_J_J,type,
    set_Bo5728432886497068284m_char: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J,type,
    set_Pr5245412377734449720_f_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mtf__f_J_J,type,
    set_Pr4548512049067879902_nat_f: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
    set_list_list_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__String__Ochar_Mt__String__Ochar_J,type,
    produc143505574896075907r_char: $tType ).

thf(ty_n_t__List__Olist_It__Bot____Terms__Obot____term_It__Nat__Onat_J_J,type,
    list_B2641752247266983457rm_nat: $tType ).

thf(ty_n_t__Bot____Terms__Obot____term_It__List__Olist_It__Nat__Onat_J_J,type,
    bot_bo5408180104339501355st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Bot____Terms__Obot____term_It__Nat__Onat_J_J,type,
    set_Bot_bot_term_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__String__Ochar_Mt__Nat__Onat_J,type,
    produc3922234789054302210ar_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__String__Ochar_J,type,
    produc4130232531789655682t_char: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__f_Mtf__f_J_J,type,
    list_P7797237717227858669od_f_f: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__f_Mtf__f_J_J,type,
    set_Product_prod_f_f: $tType ).

thf(ty_n_t__List__Olist_It__Bot____Terms__Obot____term_Itf__f_J_J,type,
    list_Bot_bot_term_f: $tType ).

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

thf(ty_n_t__Set__Oset_It__Bot____Terms__Obot____term_Itf__f_J_J,type,
    set_Bot_bot_term_f: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__f_Mt__String__Ochar_J,type,
    product_prod_f_char: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__String__Ochar_Mtf__f_J,type,
    product_prod_char_f: $tType ).

thf(ty_n_t__Bot____Terms__Obot____term_It__String__Ochar_J,type,
    bot_bot_term_char: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J,type,
    product_prod_f_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mtf__f_J,type,
    product_prod_nat_f: $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__List__Olist_It__Nat__Onat_J_J,type,
    set_list_nat: $tType ).

thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
    list_set_nat: $tType ).

thf(ty_n_t__Bot____Terms__Obot____term_It__Nat__Onat_J,type,
    bot_bot_term_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    set_set_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__f_Mtf__f_J,type,
    product_prod_f_f: $tType ).

thf(ty_n_t__Bot____Terms__Obot____term_Itf__f_J,type,
    bot_bot_term_f: $tType ).

thf(ty_n_t__List__Olist_It__String__Ochar_J,type,
    list_char: $tType ).

thf(ty_n_t__Set__Oset_It__String__Ochar_J,type,
    set_char: $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__f_J,type,
    list_f: $tType ).

thf(ty_n_t__Set__Oset_Itf__f_J,type,
    set_f: $tType ).

thf(ty_n_t__String__Ochar,type,
    char: $tType ).

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

thf(ty_n_tf__f,type,
    f: $tType ).

% Explicit typings (534)
thf(sy_c_AssocList_Ozip__with__index__from_001t__Nat__Onat,type,
    zip_wi3407404960461264653om_nat: nat > list_nat > list_P6011104703257516679at_nat ).

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_Bot__Terms_Obless__eq_001t__Nat__Onat,type,
    bot_bless_eq_nat: set_Pr1483163517390835655rm_nat ).

thf(sy_c_Bot__Terms_Obless__eq_001tf__f,type,
    bot_bless_eq_f: set_Pr6841673932529763677term_f ).

thf(sy_c_Bot__Terms_Obless__eqp_001t__Nat__Onat,type,
    bot_bless_eqp_nat: bot_bot_term_nat > bot_bot_term_nat > $o ).

thf(sy_c_Bot__Terms_Obless__eqp_001tf__f,type,
    bot_bless_eqp_f: bot_bot_term_f > bot_bot_term_f > $o ).

thf(sy_c_Bot__Terms_Obot__term_OBFun_001t__List__Olist_It__Nat__Onat_J,type,
    bot_bo885581241535429109st_nat: list_nat > list_B5078311858792894385st_nat > bot_bo5408180104339501355st_nat ).

thf(sy_c_Bot__Terms_Obot__term_OBFun_001t__Nat__Onat,type,
    bot_bot_BFun_nat: nat > list_B2641752247266983457rm_nat > bot_bot_term_nat ).

thf(sy_c_Bot__Terms_Obot__term_OBFun_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    bot_bo202339342019657932st_nat: produc1828647624359046049st_nat > list_B7941409156082897212st_nat > bot_bo5929273105904447532st_nat ).

thf(sy_c_Bot__Terms_Obot__term_OBFun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bot_bo1451043769174509868at_nat: product_prod_nat_nat > list_B7173093809959871388at_nat > bot_bo655077203622184332at_nat ).

thf(sy_c_Bot__Terms_Obot__term_OBFun_001t__String__Ochar,type,
    bot_bot_BFun_char: char > list_B1576150950144988706m_char > bot_bot_term_char ).

thf(sy_c_Bot__Terms_Obot__term_OBFun_001tf__f,type,
    bot_bot_BFun_f: f > list_Bot_bot_term_f > bot_bot_term_f ).

thf(sy_c_Bot__Terms_Obot__term_OBot_001t__List__Olist_It__Nat__Onat_J,type,
    bot_bot_Bot_list_nat: bot_bo5408180104339501355st_nat ).

thf(sy_c_Bot__Terms_Obot__term_OBot_001t__Nat__Onat,type,
    bot_bot_Bot_nat: bot_bot_term_nat ).

thf(sy_c_Bot__Terms_Obot__term_OBot_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    bot_bo6406203247828191920st_nat: bot_bo5929273105904447532st_nat ).

thf(sy_c_Bot__Terms_Obot__term_OBot_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bot_bo5215575770921047824at_nat: bot_bo655077203622184332at_nat ).

thf(sy_c_Bot__Terms_Obot__term_OBot_001t__String__Ochar,type,
    bot_bot_Bot_char: bot_bot_term_char ).

thf(sy_c_Bot__Terms_Obot__term_OBot_001tf__f,type,
    bot_bot_Bot_f: bot_bot_term_f ).

thf(sy_c_Bot__Terms_Obot__term_Oargs_001t__List__Olist_It__Nat__Onat_J,type,
    bot_bo2362305990503692245st_nat: bot_bo5408180104339501355st_nat > list_B5078311858792894385st_nat ).

thf(sy_c_Bot__Terms_Obot__term_Oargs_001t__Nat__Onat,type,
    bot_bot_args_nat: bot_bot_term_nat > list_B2641752247266983457rm_nat ).

thf(sy_c_Bot__Terms_Obot__term_Oargs_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    bot_bo2187226529533958892st_nat: bot_bo5929273105904447532st_nat > list_B7941409156082897212st_nat ).

thf(sy_c_Bot__Terms_Obot__term_Oargs_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bot_bo1125064044073681740at_nat: bot_bo655077203622184332at_nat > list_B7173093809959871388at_nat ).

thf(sy_c_Bot__Terms_Obot__term_Oargs_001t__String__Ochar,type,
    bot_bot_args_char: bot_bot_term_char > list_B1576150950144988706m_char ).

thf(sy_c_Bot__Terms_Obot__term_Oargs_001tf__f,type,
    bot_bot_args_f: bot_bot_term_f > list_Bot_bot_term_f ).

thf(sy_c_Bot__Terms_Obot__term_Orel__bot__term_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    bot_bo492821383238646227st_nat: ( list_nat > list_nat > $o ) > bot_bo5408180104339501355st_nat > bot_bo5408180104339501355st_nat > $o ).

thf(sy_c_Bot__Terms_Obot__term_Orel__bot__term_001t__Nat__Onat_001t__Nat__Onat,type,
    bot_bo1849039309831764531at_nat: ( nat > nat > $o ) > bot_bot_term_nat > bot_bot_term_nat > $o ).

thf(sy_c_Bot__Terms_Obot__term_Orel__bot__term_001t__Nat__Onat_001t__String__Ochar,type,
    bot_bo7254656895009639220t_char: ( nat > char > $o ) > bot_bot_term_nat > bot_bot_term_char > $o ).

thf(sy_c_Bot__Terms_Obot__term_Orel__bot__term_001t__Nat__Onat_001tf__f,type,
    bot_bo4401546630823235424_nat_f: ( nat > f > $o ) > bot_bot_term_nat > bot_bot_term_f > $o ).

thf(sy_c_Bot__Terms_Obot__term_Orel__bot__term_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    bot_bo6591283685554716113st_nat: ( produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o ) > bot_bo5929273105904447532st_nat > bot_bo5929273105904447532st_nat > $o ).

thf(sy_c_Bot__Terms_Obot__term_Orel__bot__term_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bot_bo1244605953094563793at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > bot_bo655077203622184332at_nat > bot_bo655077203622184332at_nat > $o ).

thf(sy_c_Bot__Terms_Obot__term_Orel__bot__term_001t__String__Ochar_001t__List__Olist_It__Nat__Onat_J,type,
    bot_bo3035317541177340228st_nat: ( char > list_nat > $o ) > bot_bot_term_char > bot_bo5408180104339501355st_nat > $o ).

thf(sy_c_Bot__Terms_Obot__term_Orel__bot__term_001t__String__Ochar_001t__Nat__Onat,type,
    bot_bo4339835670485063732ar_nat: ( char > nat > $o ) > bot_bot_term_char > bot_bot_term_nat > $o ).

thf(sy_c_Bot__Terms_Obot__term_Orel__bot__term_001t__String__Ochar_001t__String__Ochar,type,
    bot_bo2694460441194989621r_char: ( char > char > $o ) > bot_bot_term_char > bot_bot_term_char > $o ).

thf(sy_c_Bot__Terms_Obot__term_Orel__bot__term_001t__String__Ochar_001tf__f,type,
    bot_bo3338816352050261919char_f: ( char > f > $o ) > bot_bot_term_char > bot_bot_term_f > $o ).

thf(sy_c_Bot__Terms_Obot__term_Orel__bot__term_001tf__f_001t__Nat__Onat,type,
    bot_bo6546836774426921474_f_nat: ( f > nat > $o ) > bot_bot_term_f > bot_bot_term_nat > $o ).

thf(sy_c_Bot__Terms_Obot__term_Orel__bot__term_001tf__f_001t__String__Ochar,type,
    bot_bo6232401481379736579f_char: ( f > char > $o ) > bot_bot_term_f > bot_bot_term_char > $o ).

thf(sy_c_Bot__Terms_Obot__term_Orel__bot__term_001tf__f_001tf__f,type,
    bot_bo6868847914503738705rm_f_f: ( f > f > $o ) > bot_bot_term_f > bot_bot_term_f > $o ).

thf(sy_c_Bot__Terms_Obot__term_Oset__bot__term_001t__List__Olist_It__Nat__Onat_J,type,
    bot_bo7758324655462117769st_nat: bot_bo5408180104339501355st_nat > set_list_nat ).

thf(sy_c_Bot__Terms_Obot__term_Oset__bot__term_001t__Nat__Onat,type,
    bot_bo2410032770233908985rm_nat: bot_bot_term_nat > set_nat ).

thf(sy_c_Bot__Terms_Obot__term_Oset__bot__term_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    bot_bo4281033497674357304st_nat: bot_bo5929273105904447532st_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_Bot__Terms_Obot__term_Oset__bot__term_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bot_bo6066252797605031576at_nat: bot_bo655077203622184332at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Bot__Terms_Obot__term_Oset__bot__term_001t__String__Ochar,type,
    bot_bo3082694273163266554m_char: bot_bot_term_char > set_char ).

thf(sy_c_Bot__Terms_Obot__term_Oset__bot__term_001tf__f,type,
    bot_bo658131349715431450term_f: bot_bot_term_f > set_f ).

thf(sy_c_Bot__Terms_Obot__term_Osize__bot__term_001t__Nat__Onat,type,
    bot_bo2025320268590814892rm_nat: ( nat > nat ) > bot_bot_term_nat > nat ).

thf(sy_c_Bot__Terms_Obot__term_Osize__bot__term_001tf__f,type,
    bot_bo4099906974609907495term_f: ( f > nat ) > bot_bot_term_f > nat ).

thf(sy_c_Bot__Terms_Obot__term_Oun__BFun1_001t__List__Olist_It__Nat__Onat_J,type,
    bot_bo1689184820244159237st_nat: bot_bo5408180104339501355st_nat > list_nat ).

thf(sy_c_Bot__Terms_Obot__term_Oun__BFun1_001t__Nat__Onat,type,
    bot_bot_un_BFun1_nat: bot_bot_term_nat > nat ).

thf(sy_c_Bot__Terms_Obot__term_Oun__BFun1_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    bot_bo8841542196867011516st_nat: bot_bo5929273105904447532st_nat > produc1828647624359046049st_nat ).

thf(sy_c_Bot__Terms_Obot__term_Oun__BFun1_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bot_bo673623650730560540at_nat: bot_bo655077203622184332at_nat > product_prod_nat_nat ).

thf(sy_c_Bot__Terms_Obot__term_Oun__BFun1_001t__String__Ochar,type,
    bot_bo12013370452354758301_char: bot_bot_term_char > char ).

thf(sy_c_Bot__Terms_Obot__term_Oun__BFun1_001tf__f,type,
    bot_bot_un_BFun1_f: bot_bot_term_f > f ).

thf(sy_c_Bot__Terms_OmergeP_001t__Nat__Onat,type,
    bot_mergeP_nat: set_Pr1483163517390835655rm_nat ).

thf(sy_c_Bot__Terms_OmergeP_001tf__f,type,
    bot_mergeP_f: set_Pr6841673932529763677term_f ).

thf(sy_c_Bot__Terms_OmergePp_001tf__f,type,
    bot_mergePp_f: bot_bot_term_f > bot_bot_term_f > $o ).

thf(sy_c_Bot__Terms_Omerge__terms_001t__Nat__Onat,type,
    bot_merge_terms_nat: bot_bot_term_nat > bot_bot_term_nat > bot_bot_term_nat ).

thf(sy_c_Bot__Terms_Omerge__terms_001tf__f,type,
    bot_merge_terms_f: bot_bot_term_f > bot_bot_term_f > bot_bot_term_f ).

thf(sy_c_Bot__Terms_Omerge__terms__rel_001tf__f,type,
    bot_me5685897738211366532_rel_f: produc250624644672406269term_f > produc250624644672406269term_f > $o ).

thf(sy_c_Cardinality_Ois__list__UNIV_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    is_lis1896903852932488767term_f: list_Bot_bot_term_f > $o ).

thf(sy_c_Cardinality_Ois__list__UNIV_001t__List__Olist_It__Nat__Onat_J,type,
    is_lis4115413270075476113st_nat: list_list_nat > $o ).

thf(sy_c_Cardinality_Ois__list__UNIV_001t__Nat__Onat,type,
    is_list_UNIV_nat: list_nat > $o ).

thf(sy_c_Cardinality_Ois__list__UNIV_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    is_lis8713152955824212368at_nat: list_P6011104703257516679at_nat > $o ).

thf(sy_c_Cardinality_Ois__list__UNIV_001t__String__Ochar,type,
    is_list_UNIV_char: list_char > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: nat > nat > nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
    minus_minus_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: 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_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
    groups4561878855575611511st_nat: list_nat > nat ).

thf(sy_c_HOL_Oundefined_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    undefi3706045505595579942term_f: bot_bot_term_f ).

thf(sy_c_HOL_Oundefined_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    undefi3946296454836805481at_nat: product_prod_nat_nat ).

thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
    if_list_nat: $o > list_nat > list_nat > list_nat ).

thf(sy_c_If_001t__Nat__Onat,type,
    if_nat: $o > nat > nat > nat ).

thf(sy_c_List_Oappend_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    append3968237287470817971term_f: list_Bot_bot_term_f > list_Bot_bot_term_f > list_Bot_bot_term_f ).

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__Nat__Onat,type,
    append_nat: list_nat > list_nat > list_nat ).

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_Obutlast_001t__Nat__Onat,type,
    butlast_nat: list_nat > list_nat ).

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

thf(sy_c_List_Ocount__list_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    count_166185096570534930term_f: list_Bot_bot_term_f > bot_bot_term_f > nat ).

thf(sy_c_List_Ocount__list_001t__List__Olist_It__Nat__Onat_J,type,
    count_list_list_nat: list_list_nat > list_nat > nat ).

thf(sy_c_List_Ocount__list_001t__Nat__Onat,type,
    count_list_nat: list_nat > nat > nat ).

thf(sy_c_List_Ocount__list_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    count_8697817428404038429st_nat: list_P7940050157051400743st_nat > produc1828647624359046049st_nat > nat ).

thf(sy_c_List_Ocount__list_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    count_4203492906077236349at_nat: list_P6011104703257516679at_nat > product_prod_nat_nat > nat ).

thf(sy_c_List_Ocount__list_001t__String__Ochar,type,
    count_list_char: list_char > char > nat ).

thf(sy_c_List_Ocount__list_001tf__f,type,
    count_list_f: list_f > f > nat ).

thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
    distinct_nat: list_nat > $o ).

thf(sy_c_List_Odrop_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    drop_Bot_bot_term_f: nat > list_Bot_bot_term_f > list_Bot_bot_term_f ).

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__Nat__Onat,type,
    drop_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Odrop_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    drop_P8868858903918902087at_nat: nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

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

thf(sy_c_List_Ogen__length_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    gen_le6202096371021315387term_f: nat > list_Bot_bot_term_f > nat ).

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__Nat__Onat,type,
    gen_length_nat: nat > list_nat > nat ).

thf(sy_c_List_Ogen__length_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    gen_le2383899666085517716at_nat: nat > list_P6011104703257516679at_nat > nat ).

thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
    insert_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Olast_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    last_Bot_bot_term_f: list_Bot_bot_term_f > bot_bot_term_f ).

thf(sy_c_List_Olast_001t__Nat__Onat,type,
    last_nat: list_nat > nat ).

thf(sy_c_List_Olast_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    last_P6484183829340986144at_nat: list_P6011104703257516679at_nat > product_prod_nat_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__Nat__Onat,type,
    lenlex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_List_Olenlex_001t__Set__Oset_It__Nat__Onat_J,type,
    lenlex_set_nat: set_Pr5488025237498180813et_nat > set_Pr2113329861893509485et_nat ).

thf(sy_c_List_Olex_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    lex_Bot_bot_term_f: set_Pr6841673932529763677term_f > set_Pr488239067965912317term_f ).

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__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__Set__Oset_It__Nat__Onat_J,type,
    lex_set_nat: set_Pr5488025237498180813et_nat > set_Pr2113329861893509485et_nat ).

thf(sy_c_List_Olexn_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    lexn_Bot_bot_term_f: set_Pr6841673932529763677term_f > nat > set_Pr488239067965912317term_f ).

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__Nat__Onat,type,
    lexn_nat: set_Pr1261947904930325089at_nat > nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_List_Olexn_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    lexn_P849066182667564199at_nat: set_Pr8693737435421807431at_nat > nat > set_Pr1542805901266377927at_nat ).

thf(sy_c_List_Olexord_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    lexord943918551247894875term_f: set_Pr6841673932529763677term_f > set_Pr488239067965912317term_f ).

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__Nat__Onat,type,
    lexord_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_List_Olexord_001t__Set__Oset_It__Nat__Onat_J,type,
    lexord_set_nat: set_Pr5488025237498180813et_nat > set_Pr2113329861893509485et_nat ).

thf(sy_c_List_Olist_OCons_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    cons_Bot_bot_term_f: bot_bot_term_f > list_Bot_bot_term_f > list_Bot_bot_term_f ).

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__Nat__Onat,type,
    cons_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    cons_P3711653291972799923term_f: produc250624644672406269term_f > list_P1780566427699531651term_f > list_P1780566427699531651term_f ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    cons_P8785853885591068933st_nat: produc5499396499697224271st_nat > list_P3080432348914762709st_nat > list_P3080432348914762709st_nat ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Nat__Onat_J,type,
    cons_P8275894281584890613_f_nat: produc8417799323881384255_f_nat > list_P8521082412280730693_f_nat > list_P8521082412280730693_f_nat ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    cons_P8057524666905663237term_f: produc4771067281011818575term_f > list_P9018165577420421589term_f > list_P9018165577420421589term_f ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    cons_P5007559046487125591st_nat: produc1828647624359046049st_nat > list_P7940050157051400743st_nat > list_P7940050157051400743st_nat ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    cons_P3142513297430851911at_nat: produc1540777390238407569at_nat > list_P5364314822750548887at_nat > list_P5364314822750548887at_nat ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    cons_P5276694725835343733term_f: produc5418599768131837375term_f > list_P3405996666353486021term_f > list_P3405996666353486021term_f ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J,type,
    cons_P6176896814948630215st_nat: produc4575160907756185873st_nat > list_P7736225833432154391st_nat > list_P7736225833432154391st_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__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    cons_P7941000567243684387et_nat: produc7819656566062154093et_nat > list_P6254988961118846195et_nat > list_P6254988961118846195et_nat ).

thf(sy_c_List_Olist_OCons_001t__Set__Oset_It__Nat__Onat_J,type,
    cons_set_nat: set_nat > list_set_nat > list_set_nat ).

thf(sy_c_List_Olist_OCons_001t__String__Ochar,type,
    cons_char: char > list_char > list_char ).

thf(sy_c_List_Olist_OCons_001tf__f,type,
    cons_f: f > list_f > list_f ).

thf(sy_c_List_Olist_ONil_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    nil_Bot_bot_term_f: list_Bot_bot_term_f ).

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_Ohd_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    hd_Bot_bot_term_f: list_Bot_bot_term_f > bot_bot_term_f ).

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

thf(sy_c_List_Olist_Olist__all2_001t__Bot____Terms__Obot____term_It__Nat__Onat_J_001t__Bot____Terms__Obot____term_It__Nat__Onat_J,type,
    list_a3175995857480936519rm_nat: ( bot_bot_term_nat > bot_bot_term_nat > $o ) > list_B2641752247266983457rm_nat > list_B2641752247266983457rm_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Bot____Terms__Obot____term_It__Nat__Onat_J_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    list_a2360236856457976674term_f: ( bot_bot_term_nat > bot_bot_term_f > $o ) > list_B2641752247266983457rm_nat > list_Bot_bot_term_f > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Bot____Terms__Obot____term_It__Nat__Onat_J,type,
    list_a7573497065701696196rm_nat: ( bot_bot_term_f > bot_bot_term_nat > $o ) > list_Bot_bot_term_f > list_B2641752247266983457rm_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    list_a8926285605142776357term_f: ( bot_bot_term_f > bot_bot_term_f > $o ) > list_Bot_bot_term_f > list_Bot_bot_term_f > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Bot____Terms__Obot____term_Itf__f_J_001t__List__Olist_It__Nat__Onat_J,type,
    list_a470444602825883767st_nat: ( bot_bot_term_f > list_nat > $o ) > list_Bot_bot_term_f > list_list_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Nat__Onat,type,
    list_a1093197441716247783_f_nat: ( bot_bot_term_f > nat > $o ) > list_Bot_bot_term_f > list_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Bot____Terms__Obot____term_Itf__f_J_001tf__f,type,
    list_a6572579317876404652rm_f_f: ( bot_bot_term_f > f > $o ) > list_Bot_bot_term_f > list_f > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__List__Olist_It__Nat__Onat_J_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    list_a6660390693657896311term_f: ( list_nat > bot_bot_term_f > $o ) > list_list_nat > list_Bot_bot_term_f > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    list_a4684442626689269705st_nat: ( list_nat > list_nat > $o ) > list_list_nat > list_list_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
    list_a7394835724046789945at_nat: ( list_nat > nat > $o ) > list_list_nat > list_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Nat__Onat_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    list_a8962304667658831463term_f: ( nat > bot_bot_term_f > $o ) > list_nat > list_Bot_bot_term_f > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    list_a5155878676884504761st_nat: ( nat > list_nat > $o ) > list_nat > list_list_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Nat__Onat_001t__Nat__Onat,type,
    list_all2_nat_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    list_a8436948092790885224at_nat: ( nat > product_prod_nat_nat > $o ) > list_nat > list_P6011104703257516679at_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Nat__Onat_001t__String__Ochar,type,
    list_all2_nat_char: ( nat > char > $o ) > list_nat > list_char > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Nat__Onat_001tf__f,type,
    list_all2_nat_f: ( nat > f > $o ) > list_nat > list_f > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    list_a3335923554785236167st_nat: ( produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o ) > list_P7940050157051400743st_nat > list_P7940050157051400743st_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    list_a5076900699748170122at_nat: ( product_prod_nat_nat > nat > $o ) > list_P6011104703257516679at_nat > list_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    list_a2393213337517202119at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__String__Ochar_001t__Nat__Onat,type,
    list_all2_char_nat: ( char > nat > $o ) > list_char > list_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__String__Ochar_001t__String__Ochar,type,
    list_all2_char_char: ( char > char > $o ) > list_char > list_char > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__String__Ochar_001tf__f,type,
    list_all2_char_f: ( char > f > $o ) > list_char > list_f > $o ).

thf(sy_c_List_Olist_Olist__all2_001tf__f_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    list_a8222785964074318922term_f: ( f > bot_bot_term_f > $o ) > list_f > list_Bot_bot_term_f > $o ).

thf(sy_c_List_Olist_Olist__all2_001tf__f_001t__Nat__Onat,type,
    list_all2_f_nat: ( f > nat > $o ) > list_f > list_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001tf__f_001t__String__Ochar,type,
    list_all2_f_char: ( f > char > $o ) > list_f > list_char > $o ).

thf(sy_c_List_Olist_Olist__all2_001tf__f_001tf__f,type,
    list_all2_f_f: ( f > f > $o ) > list_f > list_f > $o ).

thf(sy_c_List_Olist_Omap_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    map_Bo3245817867514345570term_f: ( bot_bot_term_f > bot_bot_term_f ) > list_Bot_bot_term_f > list_Bot_bot_term_f ).

thf(sy_c_List_Olist_Omap_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Nat__Onat,type,
    map_Bo1399257005622609828_f_nat: ( bot_bot_term_f > nat ) > list_Bot_bot_term_f > list_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
    map_list_nat_nat: ( list_nat > nat ) > list_list_nat > list_nat ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    map_na44992194710417700term_f: ( nat > bot_bot_term_f ) > list_nat > list_Bot_bot_term_f ).

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

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Nat__Onat_J,type,
    map_na489546787450227497_f_nat: ( nat > produc8417799323881384255_f_nat ) > list_nat > list_P8521082412280730693_f_nat ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    map_na7298421622053143531at_nat: ( nat > product_prod_nat_nat ) > list_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    map_Pr7995175553647191367term_f: ( produc250624644672406269term_f > bot_bot_term_f ) > list_P1780566427699531651term_f > list_Bot_bot_term_f ).

thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    map_Pr3938374229010428429at_nat: ( product_prod_nat_nat > nat ) > list_P6011104703257516679at_nat > list_nat ).

thf(sy_c_List_Olist_Oset_001t__Bot____Terms__Obot____term_It__List__Olist_It__Nat__Onat_J_J,type,
    set_Bo3558769202554720832st_nat: list_B5078311858792894385st_nat > set_Bo5126868359332389771st_nat ).

thf(sy_c_List_Olist_Oset_001t__Bot____Terms__Obot____term_It__Nat__Onat_J,type,
    set_Bot_bot_term_nat2: list_B2641752247266983457rm_nat > set_Bot_bot_term_nat ).

thf(sy_c_List_Olist_Oset_001t__Bot____Terms__Obot____term_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    set_Bo5869970306151722839st_nat: list_B7941409156082897212st_nat > set_Bo2954278103884806626st_nat ).

thf(sy_c_List_Olist_Oset_001t__Bot____Terms__Obot____term_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    set_Bo1863287301388052407at_nat: list_B7173093809959871388at_nat > set_Bo8614249060733510210at_nat ).

thf(sy_c_List_Olist_Oset_001t__Bot____Terms__Obot____term_It__String__Ochar_J,type,
    set_Bo5011433680641223729m_char: list_B1576150950144988706m_char > set_Bo5728432886497068284m_char ).

thf(sy_c_List_Olist_Oset_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    set_Bot_bot_term_f2: list_Bot_bot_term_f > set_Bot_bot_term_f ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    set_list_list_nat2: list_list_list_nat > set_list_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__Nat__Onat,type,
    set_nat2: list_nat > set_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    set_Pr1368550810757897490term_f: list_P1780566427699531651term_f > set_Pr6841673932529763677term_f ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    set_Pr835449257019969124st_nat: list_P3080432348914762709st_nat > set_Pr8271761326893575599st_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Nat__Onat_J,type,
    set_Pr6427426422077602772_f_nat: list_P8521082412280730693_f_nat > set_Pr5575134264255247135_f_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__String__Ochar_J,type,
    set_Pr6267679966350894677f_char: list_P5158871451256932422f_char > set_Pr1259485602726169888f_char ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mtf__f_J,type,
    set_Pr4453812288428152085rm_f_f: list_P3672777798497168634rm_f_f > set_Pr8815161475434500256rm_f_f ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    set_Pr107120038334563428term_f: list_P9018165577420421589term_f > set_Pr4986122518544458671term_f ).

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__Nat__Onat_J,type,
    set_Pr7715406932454250534at_nat: list_P5364314822750548887at_nat > set_Pr7549493975000385137at_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__String__Ochar_J,type,
    set_Pr6151334496301494439t_char: list_P1202297530648401304t_char > set_Pr4929944408218422898t_char ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mtf__f_J,type,
    set_Pr7185350880610756931_nat_f: list_P3776949671930543656_nat_f > set_Pr8895417025753345486_nat_f ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    set_Pr3428226866328055892term_f: list_P3405996666353486021term_f > set_Pr460048518328002463term_f ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J,type,
    set_Pr1526418413117253030st_nat: list_P7736225833432154391st_nat > set_Pr698032948827214833st_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    set_Pr6710958862608470481at_nat: list_P1909269847677398966at_nat > set_Pr7717912310451564380at_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__String__Ochar_J,type,
    set_Pr7707541591655318679t_char: list_P5398368247131344008t_char > set_Pr2926629348867340898t_char ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__f_J,type,
    set_Pr4163146859742855507_nat_f: list_P3206961950963217208_nat_f > set_Pr4548512049067879902_nat_f ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    set_Pr7183939682275281404st_nat: list_P4088901747568347245st_nat > set_Pr915550674247830855st_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
    set_Pr87060279690929995st_nat: list_P4072332127704817200st_nat > set_Pr4879555682012792790st_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    set_Pr7836445846575771563at_nat: list_P5953897101696565392at_nat > set_Pr2539167527615954998at_nat ).

thf(sy_c_List_Olist_Oset_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,
    set_Pr5518436109238095868at_nat: list_P8469869581646625389at_nat > set_Pr8693737435421807431at_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    set_Pr9040384385603167362et_nat: list_P6254988961118846195et_nat > set_Pr5488025237498180813et_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__String__Ochar_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    set_Pr2009081865692134357term_f: list_P2124888226753994182term_f > set_Pr7448874415078007456term_f ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__String__Ochar_Mt__List__Olist_It__Nat__Onat_J_J,type,
    set_Pr8960426060003701543st_nat: list_P8844850915902724120st_nat > set_Pr3349125756617969906st_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__String__Ochar_Mt__Nat__Onat_J,type,
    set_Pr7499543848919965207ar_nat: list_P3884813413850506760ar_nat > set_Pr1413074515586503650ar_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__String__Ochar_Mt__String__Ochar_J,type,
    set_Pr7517466709966071192r_char: list_P3261013228118292233r_char > set_Pr3309569728657787619r_char ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__String__Ochar_Mtf__f_J,type,
    set_Pr5571943315962559890char_f: list_P3726693972706002423char_f > set_Pr6676871586184813213char_f ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_Itf__f_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    set_Pr5252852205727177067term_f: list_P2506729754372859472term_f > set_Pr7649113431310191094term_f ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_Itf__f_Mt__List__Olist_It__Nat__Onat_J_J,type,
    set_Pr6497372873783884989st_nat: list_P4278292882394854562st_nat > set_Pr173388199362880584st_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J,type,
    set_Pr7102205019285007021_f_nat: list_P3903862279629787026_f_nat > set_Pr5245412377734449720_f_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_Itf__f_Mt__String__Ochar_J,type,
    set_Pr6552472979441277998f_char: list_P494527469494163091f_char > set_Pr3444705082972973881f_char ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_Itf__f_Mtf__f_J,type,
    set_Product_prod_f_f2: list_P7797237717227858669od_f_f > set_Product_prod_f_f ).

thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
    set_set_nat2: list_set_nat > set_set_nat ).

thf(sy_c_List_Olist_Oset_001t__String__Ochar,type,
    set_char2: list_char > set_char ).

thf(sy_c_List_Olist_Oset_001tf__f,type,
    set_f2: list_f > set_f ).

thf(sy_c_List_Olist_Osize__list_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    size_l1747024920156826246term_f: ( bot_bot_term_f > nat ) > list_Bot_bot_term_f > nat ).

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__Nat__Onat,type,
    size_list_nat: ( nat > nat ) > list_nat > nat ).

thf(sy_c_List_Olist_Osize__list_001t__String__Ochar,type,
    size_list_char: ( char > nat ) > list_char > nat ).

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

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

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

thf(sy_c_List_Onth_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    nth_Bot_bot_term_f: list_Bot_bot_term_f > nat > bot_bot_term_f ).

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__Nat__Onat,type,
    nth_nat: list_nat > nat > nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).

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

thf(sy_c_List_Oremdups__adj_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    remdup6045283763748034386term_f: list_Bot_bot_term_f > list_Bot_bot_term_f ).

thf(sy_c_List_Oremdups__adj_001t__Nat__Onat,type,
    remdups_adj_nat: list_nat > list_nat ).

thf(sy_c_List_Oreplicate_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    replic6958083601664011982term_f: nat > bot_bot_term_f > list_Bot_bot_term_f ).

thf(sy_c_List_Oreplicate_001t__List__Olist_It__Nat__Onat_J,type,
    replicate_list_nat: nat > list_nat > list_list_nat ).

thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
    replicate_nat: nat > nat > list_nat ).

thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    replic5523370671267941053term_f: nat > produc250624644672406269term_f > list_P1780566427699531651term_f ).

thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Nat__Onat_J,type,
    replic5744860322987233535_f_nat: nat > produc8417799323881384255_f_nat > list_P8521082412280730693_f_nat ).

thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    replic6518106175821116840at_nat: nat > produc5523380839273745768at_nat > list_P184876202426519672at_nat ).

thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    replic3948718270753518433st_nat: nat > produc1828647624359046049st_nat > list_P7940050157051400743st_nat ).

thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    replic2745660767237686655term_f: nat > produc5418599768131837375term_f > list_P3405996666353486021term_f ).

thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    replic4235873036481779905at_nat: nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    replic4526976463895685156term_f: nat > produc3532251127348314084term_f > list_P710685493942699252term_f ).

thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    replic4101501219377410918at_nat: nat > produc7248412053542808358at_nat > list_P1909269847677398966at_nat ).

thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    replic5226988203344712000at_nat: nat > produc8373899037510109440at_nat > list_P5953897101696565392at_nat ).

thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    replic8313448869583292717et_nat: nat > produc7819656566062154093et_nat > list_P6254988961118846195et_nat ).

thf(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
    replicate_set_nat: nat > set_nat > list_set_nat ).

thf(sy_c_List_Oreplicate_001t__String__Ochar,type,
    replicate_char: nat > char > list_char ).

thf(sy_c_List_Oreplicate_001tf__f,type,
    replicate_f: nat > f > list_f ).

thf(sy_c_List_Otake_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    take_Bot_bot_term_f: nat > list_Bot_bot_term_f > list_Bot_bot_term_f ).

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__Nat__Onat,type,
    take_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Otake_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    take_P2173866234530122223at_nat: nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Ozip_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    zip_Bo7813815397675234645term_f: list_Bot_bot_term_f > list_Bot_bot_term_f > list_P1780566427699531651term_f ).

thf(sy_c_List_Ozip_001t__Bot____Terms__Obot____term_Itf__f_J_001t__List__Olist_It__Nat__Onat_J,type,
    zip_Bo1182424480293800871st_nat: list_Bot_bot_term_f > list_list_nat > list_P3080432348914762709st_nat ).

thf(sy_c_List_Ozip_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Nat__Onat,type,
    zip_Bo4866563780717559831_f_nat: list_Bot_bot_term_f > list_nat > list_P8521082412280730693_f_nat ).

thf(sy_c_List_Ozip_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    zip_Bo4081489131828187130at_nat: list_Bot_bot_term_f > list_P6011104703257516679at_nat > list_P184876202426519672at_nat ).

thf(sy_c_List_Ozip_001t__Bot____Terms__Obot____term_Itf__f_J_001t__String__Ochar,type,
    zip_Bo7523927990824414488f_char: list_Bot_bot_term_f > list_char > list_P5158871451256932422f_char ).

thf(sy_c_List_Ozip_001t__Bot____Terms__Obot____term_Itf__f_J_001tf__f,type,
    zip_Bot_bot_term_f_f: list_Bot_bot_term_f > list_f > list_P3672777798497168634rm_f_f ).

thf(sy_c_List_Ozip_001t__List__Olist_It__Nat__Onat_J_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    zip_li7372370571125813415term_f: list_list_nat > list_Bot_bot_term_f > list_P9018165577420421589term_f ).

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__Nat__Onat,type,
    zip_list_nat_nat: list_list_nat > list_nat > list_P5364314822750548887at_nat ).

thf(sy_c_List_Ozip_001t__List__Olist_It__Nat__Onat_J_001t__String__Ochar,type,
    zip_list_nat_char: list_list_nat > list_char > list_P1202297530648401304t_char ).

thf(sy_c_List_Ozip_001t__List__Olist_It__Nat__Onat_J_001tf__f,type,
    zip_list_nat_f: list_list_nat > list_f > list_P3776949671930543656_nat_f ).

thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    zip_na3512298969805367703term_f: list_nat > list_Bot_bot_term_f > list_P3405996666353486021term_f ).

thf(sy_c_List_Ozip_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    zip_nat_list_nat: list_nat > list_list_nat > list_P7736225833432154391st_nat ).

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__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    zip_na5084685201233939318term_f: list_nat > list_P3405996666353486021term_f > list_P710685493942699252term_f ).

thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    zip_na1006125974040638520at_nat: list_nat > list_P6011104703257516679at_nat > list_P1909269847677398966at_nat ).

thf(sy_c_List_Ozip_001t__Nat__Onat_001t__String__Ochar,type,
    zip_nat_char: list_nat > list_char > list_P5398368247131344008t_char ).

thf(sy_c_List_Ozip_001t__Nat__Onat_001tf__f,type,
    zip_nat_f: list_nat > list_f > list_P3206961950963217208_nat_f ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    zip_Pr4899438369145407479st_nat: list_P7940050157051400743st_nat > list_P7940050157051400743st_nat > list_P4088901747568347245st_nat ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    zip_Pr8472585369470713850st_nat: list_P6011104703257516679at_nat > list_list_list_nat > list_P4072332127704817200st_nat ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    zip_Pr6869450617852699226at_nat: list_P6011104703257516679at_nat > list_nat > list_P5953897101696565392at_nat ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    zip_Pr4664179122662387191at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P8469869581646625389at_nat ).

thf(sy_c_List_Ozip_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    zip_set_nat_set_nat: list_set_nat > list_set_nat > list_P6254988961118846195et_nat ).

thf(sy_c_List_Ozip_001t__String__Ochar_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    zip_ch2203441153213786904term_f: list_char > list_Bot_bot_term_f > list_P2124888226753994182term_f ).

thf(sy_c_List_Ozip_001t__String__Ochar_001t__List__Olist_It__Nat__Onat_J,type,
    zip_char_list_nat: list_char > list_list_nat > list_P8844850915902724120st_nat ).

thf(sy_c_List_Ozip_001t__String__Ochar_001t__Nat__Onat,type,
    zip_char_nat: list_char > list_nat > list_P3884813413850506760ar_nat ).

thf(sy_c_List_Ozip_001t__String__Ochar_001t__String__Ochar,type,
    zip_char_char: list_char > list_char > list_P3261013228118292233r_char ).

thf(sy_c_List_Ozip_001t__String__Ochar_001tf__f,type,
    zip_char_f: list_char > list_f > list_P3726693972706002423char_f ).

thf(sy_c_List_Ozip_001tf__f_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    zip_f_Bot_bot_term_f: list_f > list_Bot_bot_term_f > list_P2506729754372859472term_f ).

thf(sy_c_List_Ozip_001tf__f_001t__List__Olist_It__Nat__Onat_J,type,
    zip_f_list_nat: list_f > list_list_nat > list_P4278292882394854562st_nat ).

thf(sy_c_List_Ozip_001tf__f_001t__Nat__Onat,type,
    zip_f_nat: list_f > list_nat > list_P3903862279629787026_f_nat ).

thf(sy_c_List_Ozip_001tf__f_001t__String__Ochar,type,
    zip_f_char: list_f > list_char > list_P494527469494163091f_char ).

thf(sy_c_List_Ozip_001tf__f_001tf__f,type,
    zip_f_f: list_f > list_f > list_P7797237717227858669od_f_f ).

thf(sy_c_Missing__List_Oadjust__idx,type,
    missing_adjust_idx: nat > nat > nat ).

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

thf(sy_c_Missing__List_Ofilter2_001t__Nat__Onat_001t__Nat__Onat,type,
    missin7656342814955748836at_nat: ( nat > nat > $o ) > list_nat > list_nat > produc1828647624359046049st_nat ).

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

thf(sy_c_Missing__List_Opartition2_001t__Nat__Onat_001t__Nat__Onat,type,
    missin5516618432537284226at_nat: ( nat > nat > $o ) > list_nat > list_nat > produc1342688880896445543st_nat ).

thf(sy_c_Missing__List_Opermut_001t__Nat__Onat,type,
    missing_permut_nat: list_nat > ( nat > nat ) > list_nat ).

thf(sy_c_Missing__List_Opermut__aux_001t__Nat__Onat,type,
    missin1888654203714970382ux_nat: list_nat > ( nat > nat ) > list_nat > list_nat ).

thf(sy_c_Missing__List_Oremdups__sort_001t__List__Olist_It__Nat__Onat_J,type,
    missin2640887136080315381st_nat: list_list_nat > list_list_nat ).

thf(sy_c_Missing__List_Oremdups__sort_001t__Nat__Onat,type,
    missin6101193410121742181rt_nat: list_nat > list_nat ).

thf(sy_c_Missing__List_Oremdups__sort_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    missin1163383351516593964at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_Missing__List_Oremdups__sort_001t__String__Ochar,type,
    missin7692318653990180454t_char: list_char > list_char ).

thf(sy_c_Missing__List_Oremove__nth_001t__Nat__Onat,type,
    missin7175274867594579095th_nat: nat > list_nat > list_nat ).

thf(sy_c_Multihole__Context_Opartition__by_001t__Nat__Onat,type,
    multih8544690682651757648by_nat: list_nat > list_nat > list_list_nat ).

thf(sy_c_Multihole__Context_Opartition__by__idx,type,
    multih7279076602905066526by_idx: nat > list_nat > nat > nat > nat ).

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

thf(sy_c_Nat_Osize__class_Osize_001t__Bot____Terms__Obot____term_It__Nat__Onat_J,type,
    size_s9110728481667089031rm_nat: bot_bot_term_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    size_s7982331069984003362term_f: bot_bot_term_f > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Bot____Terms__Obot____term_It__Nat__Onat_J_J,type,
    size_s7068041265931086989rm_nat: list_B2641752247266983457rm_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__f_J_J,type,
    size_s1458857753309042354term_f: list_Bot_bot_term_f > 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__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__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    size_s9037951226665808879term_f: list_P1780566427699531651term_f > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    size_s5711386298565001281st_nat: list_P3080432348914762709st_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Nat__Onat_J_J,type,
    size_s6959986635380143793_f_nat: list_P8521082412280730693_f_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    size_s2425747490215884353term_f: list_P9018165577420421589term_f > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    size_s7664791237847770771st_nat: list_P7940050157051400743st_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    size_s6663376490332876291at_nat: list_P5364314822750548887at_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    size_s1844900889452899121term_f: list_P3405996666353486021term_f > 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__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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    size_s3596537876095621194at_nat: list_P1909269847677398966at_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    size_s7641165130114787620at_nat: list_P5953897101696565392at_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
    size_s3254054031482475050et_nat: list_set_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__String__Ochar_J,type,
    size_size_list_char: list_char > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__f_J,type,
    size_size_list_f: list_f > 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__String__Ochar,type,
    size_size_char: char > nat ).

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    ord_le1203424502768444845at_nat: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_set_nat: set_nat > set_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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    ord_le8460144461188290721at_nat: product_prod_nat_nat > product_prod_nat_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_Omin_001_062_It__Nat__Onat_M_Eo_J,type,
    ord_min_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: nat > nat > nat ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    ord_mi2418711518901664340et_nat: produc7819656566062154093et_nat > produc7819656566062154093et_nat > produc7819656566062154093et_nat ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    ord_min_set_list_nat: set_list_nat > set_list_nat > set_list_nat ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_min_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    top_top_list_nat_o: list_nat > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
    top_top_nat_o: nat > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_M_Eo_J,type,
    top_to3312871037213017292_nat_o: produc1828647624359046049st_nat > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    top_to3137496036531963500_nat_o: product_prod_nat_nat > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__String__Ochar_M_Eo_J,type,
    top_top_char_o: char > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__f_M_Eo_J,type,
    top_top_f_o: f > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_Eo,type,
    top_top_o: $o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_Eo_J_M_062_It__Nat__Onat_M_Eo_J_J,type,
    top_to3543860447766875703_nat_o: produc7360885484439884007_nat_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_Eo_J_Mt__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
    top_to8731938375264029812st_nat: produc7905234009183233476st_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_Eo_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    top_to8170200190876219492et_nat: produc3959935651712317876et_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_M_062_It__Nat__Onat_M_Eo_J_J,type,
    top_to5523184063932315662_nat_o: produc4696479697851519326_nat_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_Mt__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
    top_to6809973963829177181st_nat: produc8518942320679187981st_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    top_to8453353327228641229et_nat: produc922974302667211389et_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_M_062_It__Nat__Onat_M_Eo_J_J,type,
    top_to2192322404022361374_nat_o: produc7205429901713235566_nat_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    top_to973740076627931404et_nat: produc6029997274002168412et_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
    top_to2559687781836757581st_nat: produc4252680794130103549st_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    top_to7593079806418470589et_nat: produc7819656566062154093et_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Bot____Terms__Obot____term_Itf__f_J_J,type,
    top_to4570605823607218804term_f: set_Bot_bot_term_f ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
    top_to8617644770344506070st_nat: set_list_list_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    top_top_set_list_nat: set_list_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    top_to378938406556771665st_nat: set_Pr3451248702717554689st_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    top_top_set_set_nat: set_set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__f_J,type,
    top_top_set_f: set_f ).

thf(sy_c_Parity_Ounique__euclidean__semiring__with__nat__division__class_Odivides__aux_001t__Nat__Onat,type,
    unique5332122412489317741ux_nat: product_prod_nat_nat > $o ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
    produc5701653931590357335_nat_o: ( nat > $o ) > ( nat > $o ) > produc7360885484439884007_nat_o ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_Eo_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    produc6101318194171510334st_nat: ( nat > $o ) > set_list_nat > produc7905234009183233476st_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc3803981486355964078et_nat: ( nat > $o ) > set_nat > produc3959935651712317876et_nat ).

thf(sy_c_Product__Type_OPair_001t__Bot____Terms__Obot____term_It__Nat__Onat_J_001t__Bot____Terms__Obot____term_It__Nat__Onat_J,type,
    produc1196330319720724567rm_nat: bot_bot_term_nat > bot_bot_term_nat > produc4207152732626004967rm_nat ).

thf(sy_c_Product__Type_OPair_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    produc2865564700782054965term_f: bot_bot_term_f > bot_bot_term_f > produc250624644672406269term_f ).

thf(sy_c_Product__Type_OPair_001t__Bot____Terms__Obot____term_Itf__f_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc6629678290114667655st_nat: bot_bot_term_f > list_nat > produc5499396499697224271st_nat ).

thf(sy_c_Product__Type_OPair_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Nat__Onat,type,
    produc6944956222323935479_f_nat: bot_bot_term_f > nat > produc8417799323881384255_f_nat ).

thf(sy_c_Product__Type_OPair_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc3618334562033856282at_nat: bot_bot_term_f > product_prod_nat_nat > produc5523380839273745768at_nat ).

thf(sy_c_Product__Type_OPair_001t__Bot____Terms__Obot____term_Itf__f_J_001t__String__Ochar,type,
    produc900452663052621304f_char: bot_bot_term_f > char > produc8828238612114966720f_char ).

thf(sy_c_Product__Type_OPair_001t__Bot____Terms__Obot____term_Itf__f_J_001tf__f,type,
    produc2373499641792309660rm_f_f: bot_bot_term_f > f > produc8536842985679875178rm_f_f ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__f_J_J_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__f_J_J,type,
    produc1421358604722026965term_f: list_Bot_bot_term_f > list_Bot_bot_term_f > produc7885403226499153821term_f ).

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__Nat__Onat_J_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    produc3596252344091904391term_f: list_nat > bot_bot_term_f > produc4771067281011818575term_f ).

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__Nat__Onat,type,
    produc1298395424260782409at_nat: list_nat > nat > produc1540777390238407569at_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__String__Ochar,type,
    produc6989213019823764554t_char: list_nat > char > produc3186512776854530322t_char ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001tf__f,type,
    produc1615968307924265418_nat_f: list_nat > f > produc6459537420886108824_nat_f ).

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__Set__Oset_It__Nat__Onat_J_J_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
    produc2728781817031571013et_nat: list_set_nat > list_set_nat > produc1024980458911918861et_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    produc5590691411411743351term_f: nat > bot_bot_term_f > produc5418599768131837375term_f ).

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__Product____Type__Oprod_It__Nat__Onat_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    produc4621530631439608470term_f: nat > produc5418599768131837375term_f > produc3532251127348314084term_f ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__String__Ochar,type,
    produc8245124647523730746t_char: nat > char > produc4130232531789655682t_char ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001tf__f,type,
    product_Pair_nat_f: nat > f > product_prod_nat_f ).

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_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc988773521843941591st_nat: produc1828647624359046049st_nat > produc1828647624359046049st_nat > produc1342688880896445543st_nat ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    produc8649917717998259994st_nat: product_prod_nat_nat > list_list_nat > produc3922691147685405600st_nat ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    produc6350711070570205562at_nat: product_prod_nat_nat > nat > produc8373899037510109440at_nat ).

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__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001_062_It__Nat__Onat_M_Eo_J,type,
    produc3625103166019475856_nat_o: set_list_nat > ( nat > $o ) > produc4696479697851519326_nat_o ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    produc6003640849254547781st_nat: set_list_nat > set_list_nat > produc8518942320679187981st_nat ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc1215867243663872821et_nat: set_list_nat > set_nat > produc922974302667211389et_nat ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Nat__Onat_M_Eo_J,type,
    produc7429560629904325408_nat_o: set_nat > ( nat > $o ) > produc7205429901713235566_nat_o ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    produc8907222138397203598et_nat: set_nat > produc7819656566062154093et_nat > produc6029997274002168412et_nat ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    produc4193119345494528949st_nat: set_nat > set_list_nat > produc4252680794130103549st_nat ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat ).

thf(sy_c_Product__Type_OPair_001t__String__Ochar_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    produc4803337862296769528term_f: char > bot_bot_term_f > produc4569640511456206400term_f ).

thf(sy_c_Product__Type_OPair_001t__String__Ochar_001t__List__Olist_It__Nat__Onat_J,type,
    produc1818140537056686154st_nat: char > list_nat > produc5995604340556737426st_nat ).

thf(sy_c_Product__Type_OPair_001t__String__Ochar_001t__Nat__Onat,type,
    produc5330303422999155258ar_nat: char > nat > produc3922234789054302210ar_nat ).

thf(sy_c_Product__Type_OPair_001t__String__Ochar_001t__String__Ochar,type,
    produc2420258056075623483r_char: char > char > produc143505574896075907r_char ).

thf(sy_c_Product__Type_OPair_001t__String__Ochar_001tf__f,type,
    product_Pair_char_f: char > f > product_prod_char_f ).

thf(sy_c_Product__Type_OPair_001tf__f_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    produc4023706287990223930term_f: f > bot_bot_term_f > produc112510866124124352term_f ).

thf(sy_c_Product__Type_OPair_001tf__f_001t__List__Olist_It__Nat__Onat_J,type,
    produc4084253757333640332st_nat: f > list_nat > produc5771559414059236882st_nat ).

thf(sy_c_Product__Type_OPair_001tf__f_001t__Nat__Onat,type,
    product_Pair_f_nat: f > nat > product_prod_f_nat ).

thf(sy_c_Product__Type_OPair_001tf__f_001t__String__Ochar,type,
    product_Pair_f_char: f > char > product_prod_f_char ).

thf(sy_c_Product__Type_OPair_001tf__f_001tf__f,type,
    product_Pair_f_f: f > f > product_prod_f_f ).

thf(sy_c_Product__Type_Ocurry_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    produc2177247631251368695term_f: ( produc250624644672406269term_f > bot_bot_term_f ) > bot_bot_term_f > bot_bot_term_f > bot_bot_term_f ).

thf(sy_c_Product__Type_Ocurry_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J_001_Eo,type,
    produc6139401228252118475_nat_o: ( produc1828647624359046049st_nat > $o ) > list_nat > list_nat > $o ).

thf(sy_c_Product__Type_Ocurry_001t__Nat__Onat_001t__Bot____Terms__Obot____term_Itf__f_J_001_Eo,type,
    produc7017639609257223213rm_f_o: ( produc5418599768131837375term_f > $o ) > nat > bot_bot_term_f > $o ).

thf(sy_c_Product__Type_Ocurry_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
    produc1310100445399344235_nat_o: ( product_prod_nat_nat > $o ) > nat > nat > $o ).

thf(sy_c_Product__Type_Ocurry_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001_Eo,type,
    produc6878766516837139888_nat_o: ( produc3922691147685405600st_nat > $o ) > product_prod_nat_nat > list_list_nat > $o ).

thf(sy_c_Product__Type_Ocurry_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
    produc8481458855017059839_nat_o: ( produc7819656566062154093et_nat > $o ) > set_nat > set_nat > $o ).

thf(sy_c_Product__Type_Ointernal__case__prod_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    produc30303847659289774term_f: ( bot_bot_term_f > bot_bot_term_f > bot_bot_term_f ) > produc250624644672406269term_f > bot_bot_term_f ).

thf(sy_c_Product__Type_Ointernal__case__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
    produc9139807813499881076_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Bot____Terms__Obot____term_Itf__f_J_001_Eo,type,
    produc8393149739075698894rm_f_o: ( bot_bot_term_f > bot_bot_term_f > $o ) > produc250624644672406269term_f > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    produc1677810408741468632term_f: ( bot_bot_term_f > bot_bot_term_f > bot_bot_term_f ) > produc250624644672406269term_f > bot_bot_term_f ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Bot____Terms__Obot____term_Itf__f_J_001t__List__Olist_It__Nat__Onat_J_001_Eo,type,
    produc6481302613249257980_nat_o: ( bot_bot_term_f > list_nat > $o ) > produc5499396499697224271st_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Nat__Onat_001_Eo,type,
    produc5892416418908211084_nat_o: ( bot_bot_term_f > nat > $o ) > produc8417799323881384255_f_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__List__Olist_It__Nat__Onat_J_001t__Bot____Terms__Obot____term_Itf__f_J_001_Eo,type,
    produc5575963490392018172rm_f_o: ( list_nat > bot_bot_term_f > $o ) > produc4771067281011818575term_f > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J_001_Eo,type,
    produc6382697403170970154_nat_o: ( list_nat > list_nat > $o ) > produc1828647624359046049st_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc7822140737630736756et_nat: ( list_nat > list_nat > set_nat ) > produc1828647624359046049st_nat > set_nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__String__Ochar_J,type,
    produc5516737219024788469t_char: ( list_nat > list_nat > set_char ) > produc1828647624359046049st_nat > set_char ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_Itf__f_J,type,
    produc9005509448160470901_set_f: ( list_nat > list_nat > set_f ) > produc1828647624359046049st_nat > set_f ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat_001_Eo,type,
    produc7308574044827705018_nat_o: ( list_nat > nat > $o ) > produc1540777390238407569at_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Bot____Terms__Obot____term_Itf__f_J_001_Eo,type,
    produc450604121858260492rm_f_o: ( nat > bot_bot_term_f > $o ) > produc5418599768131837375term_f > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc6142541587184120338et_nat: ( nat > bot_bot_term_f > set_nat ) > produc5418599768131837375term_f > set_nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Set__Oset_It__String__Ochar_J,type,
    produc8097753862531554963t_char: ( nat > bot_bot_term_f > set_char ) > produc5418599768131837375term_f > set_char ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Set__Oset_Itf__f_J,type,
    produc2816659430985364055_set_f: ( nat > bot_bot_term_f > set_f ) > produc5418599768131837375term_f > set_f ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J_001_Eo,type,
    produc5265810065896090426_nat_o: ( nat > list_nat > $o ) > produc4575160907756185873st_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
    produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    produc5054468372610444900st_nat: ( nat > nat > set_list_nat ) > product_prod_nat_nat > set_list_nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    produc6189476227299908564et_nat: ( nat > nat > set_nat ) > product_prod_nat_nat > set_nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__String__Ochar_J,type,
    produc1561271593892038229t_char: ( nat > nat > set_char ) > product_prod_nat_nat > set_char ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_Itf__f_J,type,
    produc8052394805345727765_set_f: ( nat > nat > set_f ) > product_prod_nat_nat > set_f ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo,type,
    produc5864757623865647827_nat_o: ( nat > product_prod_nat_nat > $o ) > produc7248412053542808358at_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_001_Eo,type,
    produc4929175875791559596_nat_o: ( produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o ) > produc1342688880896445543st_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001_Eo,type,
    produc8758269395934548017_nat_o: ( product_prod_nat_nat > nat > $o ) > produc8373899037510109440at_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    produc5789359262401207841term_f: produc250624644672406269term_f > bot_bot_term_f ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Nat__Onat,type,
    produc8216198084029437667_f_nat: produc8417799323881384255_f_nat > bot_bot_term_f ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc1382935764643595205st_nat: produc1828647624359046049st_nat > list_nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    produc6861933273117245539term_f: produc5418599768131837375term_f > nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
    product_fst_nat_nat: product_prod_nat_nat > nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc7458156121412276931st_nat: produc1342688880896445543st_nat > produc1828647624359046049st_nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    produc8387770181635782702st_nat: produc3922691147685405600st_nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc2731966376809613713et_nat: produc7819656566062154093et_nat > set_nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    produc656023363694804067term_f: produc250624644672406269term_f > bot_bot_term_f ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Bot____Terms__Obot____term_Itf__f_J_001t__Nat__Onat,type,
    produc8038346297805799973_f_nat: produc8417799323881384255_f_nat > nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc5865812112468994567st_nat: produc1828647624359046049st_nat > list_nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    produc6684081486893607845term_f: produc5418599768131837375term_f > bot_bot_term_f ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
    product_snd_nat_nat: product_prod_nat_nat > nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc7615485059283090437st_nat: produc1342688880896445543st_nat > produc1828647624359046049st_nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    produc8555213167096887148st_nat: produc3922691147685405600st_nat > list_list_nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc397104031625245139et_nat: produc7819656566062154093et_nat > set_nat ).

thf(sy_c_Relation_Oasym__on_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    asym_o1052983387450567102st_nat: set_list_list_nat > set_Pr1190453367779242145st_nat > $o ).

thf(sy_c_Relation_Oasym__on_001t__List__Olist_It__Nat__Onat_J,type,
    asym_on_list_nat: set_list_nat > set_Pr3451248702717554689st_nat > $o ).

thf(sy_c_Relation_Oasym__on_001t__Nat__Onat,type,
    asym_on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Relation_Oasym__on_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    asym_o2914609666330817491st_nat: set_Pr3451248702717554689st_nat > set_Pr915550674247830855st_nat > $o ).

thf(sy_c_Relation_Oasym__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    asym_o8546584678760569651at_nat: set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_Relation_Oasym__on_001t__Set__Oset_It__Nat__Onat_J,type,
    asym_on_set_nat: set_set_nat > set_Pr5488025237498180813et_nat > $o ).

thf(sy_c_Relation_Oasym__on_001t__String__Ochar,type,
    asym_on_char: set_char > set_Pr3309569728657787619r_char > $o ).

thf(sy_c_Relation_Oasym__on_001tf__f,type,
    asym_on_f: set_f > set_Product_prod_f_f > $o ).

thf(sy_c_Relation_Oinv__image_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    inv_im7901436520293687572st_nat: set_Pr3451248702717554689st_nat > ( list_nat > list_nat ) > set_Pr3451248702717554689st_nat ).

thf(sy_c_Relation_Oinv__image_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
    inv_im1429632248634617860at_nat: set_Pr3451248702717554689st_nat > ( nat > list_nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Relation_Oinv__image_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    inv_im1139933451183655866et_nat: set_Pr3451248702717554689st_nat > ( set_nat > list_nat ) > set_Pr5488025237498180813et_nat ).

thf(sy_c_Relation_Oinv__image_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    inv_im8414047238327108484st_nat: set_Pr1261947904930325089at_nat > ( list_nat > nat ) > set_Pr3451248702717554689st_nat ).

thf(sy_c_Relation_Oinv__image_001t__Nat__Onat_001t__Nat__Onat,type,
    inv_image_nat_nat: set_Pr1261947904930325089at_nat > ( nat > nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Relation_Oinv__image_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    inv_im5077416110856988202et_nat: set_Pr1261947904930325089at_nat > ( set_nat > nat ) > set_Pr5488025237498180813et_nat ).

thf(sy_c_Relation_Oinv__image_001t__Set__Oset_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    inv_im4790160404546336826st_nat: set_Pr5488025237498180813et_nat > ( list_nat > set_nat ) > set_Pr3451248702717554689st_nat ).

thf(sy_c_Relation_Oinv__image_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
    inv_im1511781206201618986at_nat: set_Pr5488025237498180813et_nat > ( nat > set_nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Relation_Oinv__image_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    inv_im3688519544425059808et_nat: set_Pr5488025237498180813et_nat > ( set_nat > set_nat ) > set_Pr5488025237498180813et_nat ).

thf(sy_c_Relation_Oirrefl__on_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    irrefl1702768165254659578st_nat: set_list_list_nat > set_Pr1190453367779242145st_nat > $o ).

thf(sy_c_Relation_Oirrefl__on_001t__List__Olist_It__Nat__Onat_J,type,
    irrefl_on_list_nat: set_list_nat > set_Pr3451248702717554689st_nat > $o ).

thf(sy_c_Relation_Oirrefl__on_001t__Nat__Onat,type,
    irrefl_on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).

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

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    collec1570431334306492044st_nat: ( produc1828647624359046049st_nat > $o ) > set_Pr3451248702717554689st_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_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    collec7387078234793787346st_nat: ( produc1342688880896445543st_nat > $o ) > set_Pr915550674247830855st_nat ).

thf(sy_c_Set_OCollect_001t__String__Ochar,type,
    collect_char: ( char > $o ) > set_char ).

thf(sy_c_Set_OCollect_001tf__f,type,
    collect_f: ( f > $o ) > set_f ).

thf(sy_c_Transitive__Closure_Ortrancl_001t__List__Olist_It__Nat__Onat_J,type,
    transi5285580207609517981st_nat: set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
    transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Utils_Oconcat__index__split_001t__Nat__Onat,type,
    concat5568775588498683858it_nat: product_prod_nat_nat > list_list_nat > product_prod_nat_nat ).

thf(sy_c_Utils_Oconcat__index__split__rel_001t__Nat__Onat,type,
    concat1575402509940520777el_nat: produc3922691147685405600st_nat > produc3922691147685405600st_nat > $o ).

thf(sy_c_Utils_Otrancl__list_001t__Nat__Onat,type,
    trancl_list_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_Utils_Otrancl__listp_001t__Nat__Onat,type,
    trancl_listp_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    accp_P5091932899567907206term_f: ( produc250624644672406269term_f > produc250624644672406269term_f > $o ) > produc250624644672406269term_f > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
    accp_P6748712283191960023st_nat: ( produc3922691147685405600st_nat > produc3922691147685405600st_nat > $o ) > produc3922691147685405600st_nat > $o ).

thf(sy_c_member_001t__Bot____Terms__Obot____term_It__List__Olist_It__Nat__Onat_J_J,type,
    member8381404166258140628st_nat: bot_bo5408180104339501355st_nat > set_Bo5126868359332389771st_nat > $o ).

thf(sy_c_member_001t__Bot____Terms__Obot____term_It__Nat__Onat_J,type,
    member2055285333220477764rm_nat: bot_bot_term_nat > set_Bot_bot_term_nat > $o ).

thf(sy_c_member_001t__Bot____Terms__Obot____term_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    member274174779753688259st_nat: bot_bo5929273105904447532st_nat > set_Bo2954278103884806626st_nat > $o ).

thf(sy_c_member_001t__Bot____Terms__Obot____term_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member2038346494265176355at_nat: bot_bo655077203622184332at_nat > set_Bo8614249060733510210at_nat > $o ).

thf(sy_c_member_001t__Bot____Terms__Obot____term_It__String__Ochar_J,type,
    member6716525252696868293m_char: bot_bot_term_char > set_Bo5728432886497068284m_char > $o ).

thf(sy_c_member_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    member2723211850833494437term_f: bot_bot_term_f > set_Bot_bot_term_f > $o ).

thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    member_list_list_nat: list_list_nat > set_list_list_nat > $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__Nat__Onat,type,
    member_nat: nat > set_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_It__Nat__Onat_J_Mt__Bot____Terms__Obot____term_It__Nat__Onat_J_J,type,
    member8370420911751880208rm_nat: produc4207152732626004967rm_nat > set_Pr1483163517390835655rm_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    member4687689947140789414term_f: produc250624644672406269term_f > set_Pr6841673932529763677term_f > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member4663489044743269880st_nat: produc5499396499697224271st_nat > set_Pr8271761326893575599st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Nat__Onat_J,type,
    member7779002035170981224_f_nat: produc8417799323881384255_f_nat > set_Pr5575134264255247135_f_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__String__Ochar_J,type,
    member1593658988870650857f_char: produc8828238612114966720f_char > set_Pr1259485602726169888f_char > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mtf__f_J,type,
    member2818554726470178945rm_f_f: produc8536842985679875178rm_f_f > set_Pr8815161475434500256rm_f_f > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Bot____Terms__Obot____term_Itf__f_J_J_Mt__List__Olist_It__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    member878789856639434822term_f: produc7885403226499153821term_f > set_Pr488239067965912317term_f > $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__Nat__Onat_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    member3935159826057864184term_f: produc4771067281011818575term_f > set_Pr4986122518544458671term_f > $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__Nat__Onat_J,type,
    member7820552544826472890at_nat: produc1540777390238407569at_nat > set_Pr7549493975000385137at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__String__Ochar_J,type,
    member2213537244704226875t_char: produc3186512776854530322t_char > set_Pr4929944408218422898t_char > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mtf__f_J,type,
    member6908590394731873455_nat_f: produc6459537420886108824_nat_f > set_Pr8895417025753345486_nat_f > $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__Set__Oset_It__Nat__Onat_J_J_Mt__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    member911990583396735158et_nat: produc1024980458911918861et_nat > set_Pr2113329861893509485et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    member4779802479421434344term_f: produc5418599768131837375term_f > set_Pr460048518328002463term_f > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member1631564025489475386st_nat: produc4575160907756185873st_nat > set_Pr698032948827214833st_nat > $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_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member2223272150424702269at_nat: produc7248412053542808358at_nat > set_Pr7717912310451564380at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__String__Ochar_J,type,
    member333921081531690027t_char: produc4130232531789655682t_char > set_Pr2926629348867340898t_char > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__f_J,type,
    member8962352073626239679_nat_f: product_prod_nat_f > set_Pr4548512049067879902_nat_f > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    member5236317644330119056st_nat: produc1342688880896445543st_nat > set_Pr915550674247830855st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
    member5740615348714444471st_nat: produc3922691147685405600st_nat > set_Pr4879555682012792790st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    member3348759134392003351at_nat: produc8373899037510109440at_nat > set_Pr2539167527615954998at_nat > $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__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__String__Ochar_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    member6558432925066666345term_f: produc4569640511456206400term_f > set_Pr7448874415078007456term_f > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__String__Ochar_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member5022628808406433979st_nat: produc5995604340556737426st_nat > set_Pr3349125756617969906st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__String__Ochar_Mt__Nat__Onat_J,type,
    member125923338796336555ar_nat: produc3922234789054302210ar_nat > set_Pr1413074515586503650ar_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__String__Ochar_Mt__String__Ochar_J,type,
    member3116729636814715180r_char: produc143505574896075907r_char > set_Pr3309569728657787619r_char > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__String__Ochar_Mtf__f_J,type,
    member6477383651500708606char_f: product_prod_char_f > set_Pr6676871586184813213char_f > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__f_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    member3617594643769203927term_f: produc112510866124124352term_f > set_Pr7649113431310191094term_f > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__f_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member6220612387905001513st_nat: produc5771559414059236882st_nat > set_Pr173388199362880584st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J,type,
    member2678038196313615385_f_nat: product_prod_f_nat > set_Pr5245412377734449720_f_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__f_Mt__String__Ochar_J,type,
    member7457913314979426714f_char: product_prod_f_char > set_Pr3444705082972973881f_char > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__f_Mtf__f_J,type,
    member5935997662887750672od_f_f: product_prod_f_f > set_Product_prod_f_f > $o ).

thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: set_nat > set_set_nat > $o ).

thf(sy_c_member_001t__String__Ochar,type,
    member_char: char > set_char > $o ).

thf(sy_c_member_001tf__f,type,
    member_f: f > set_f > $o ).

thf(sy_v_f____,type,
    f2: f ).

% Relevant facts (1261)
thf(fact_0_bot__term_Oinject,axiom,
    ! [X21: nat,X22: list_B2641752247266983457rm_nat,Y21: nat,Y22: list_B2641752247266983457rm_nat] :
      ( ( ( bot_bot_BFun_nat @ X21 @ X22 )
        = ( bot_bot_BFun_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% bot_term.inject
thf(fact_1_bot__term_Oinject,axiom,
    ! [X21: f,X22: list_Bot_bot_term_f,Y21: f,Y22: list_Bot_bot_term_f] :
      ( ( ( bot_bot_BFun_f @ X21 @ X22 )
        = ( bot_bot_BFun_f @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% bot_term.inject
thf(fact_2_UNIV__I,axiom,
    ! [X: produc1828647624359046049st_nat] : ( member7340969449405702474st_nat @ X @ top_to378938406556771665st_nat ) ).

% UNIV_I
thf(fact_3_UNIV__I,axiom,
    ! [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ top_to4669805908274784177at_nat ) ).

% UNIV_I
thf(fact_4_UNIV__I,axiom,
    ! [X: char] : ( member_char @ X @ top_top_set_char ) ).

% UNIV_I
thf(fact_5_UNIV__I,axiom,
    ! [X: f] : ( member_f @ X @ top_top_set_f ) ).

% UNIV_I
thf(fact_6_UNIV__I,axiom,
    ! [X: list_nat] : ( member_list_nat @ X @ top_top_set_list_nat ) ).

% UNIV_I
thf(fact_7_UNIV__I,axiom,
    ! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).

% UNIV_I
thf(fact_8_iso__tuple__UNIV__I,axiom,
    ! [X: produc1828647624359046049st_nat] : ( member7340969449405702474st_nat @ X @ top_to378938406556771665st_nat ) ).

% iso_tuple_UNIV_I
thf(fact_9_iso__tuple__UNIV__I,axiom,
    ! [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ top_to4669805908274784177at_nat ) ).

% iso_tuple_UNIV_I
thf(fact_10_iso__tuple__UNIV__I,axiom,
    ! [X: char] : ( member_char @ X @ top_top_set_char ) ).

% iso_tuple_UNIV_I
thf(fact_11_iso__tuple__UNIV__I,axiom,
    ! [X: f] : ( member_f @ X @ top_top_set_f ) ).

% iso_tuple_UNIV_I
thf(fact_12_iso__tuple__UNIV__I,axiom,
    ! [X: list_nat] : ( member_list_nat @ X @ top_top_set_list_nat ) ).

% iso_tuple_UNIV_I
thf(fact_13_iso__tuple__UNIV__I,axiom,
    ! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).

% iso_tuple_UNIV_I
thf(fact_14_top__apply,axiom,
    ( top_top_nat_o
    = ( ^ [X2: nat] : top_top_o ) ) ).

% top_apply
thf(fact_15_bot__term_Odistinct_I1_J,axiom,
    ! [X21: nat,X22: list_B2641752247266983457rm_nat] :
      ( bot_bot_Bot_nat
     != ( bot_bot_BFun_nat @ X21 @ X22 ) ) ).

% bot_term.distinct(1)
thf(fact_16_bot__term_Odistinct_I1_J,axiom,
    ! [X21: f,X22: list_Bot_bot_term_f] :
      ( bot_bot_Bot_f
     != ( bot_bot_BFun_f @ X21 @ X22 ) ) ).

% bot_term.distinct(1)
thf(fact_17_bot__term_OdiscI,axiom,
    ! [Bot_term: bot_bot_term_nat,X21: nat,X22: list_B2641752247266983457rm_nat] :
      ( ( Bot_term
        = ( bot_bot_BFun_nat @ X21 @ X22 ) )
     => ( Bot_term != bot_bot_Bot_nat ) ) ).

% bot_term.discI
thf(fact_18_bot__term_OdiscI,axiom,
    ! [Bot_term: bot_bot_term_f,X21: f,X22: list_Bot_bot_term_f] :
      ( ( Bot_term
        = ( bot_bot_BFun_f @ X21 @ X22 ) )
     => ( Bot_term != bot_bot_Bot_f ) ) ).

% bot_term.discI
thf(fact_19_bot__term_Oexhaust,axiom,
    ! [Y: bot_bot_term_nat] :
      ( ( Y != bot_bot_Bot_nat )
     => ~ ! [X212: nat,X222: list_B2641752247266983457rm_nat] :
            ( Y
           != ( bot_bot_BFun_nat @ X212 @ X222 ) ) ) ).

% bot_term.exhaust
thf(fact_20_bot__term_Oexhaust,axiom,
    ! [Y: bot_bot_term_f] :
      ( ( Y != bot_bot_Bot_f )
     => ~ ! [X212: f,X222: list_Bot_bot_term_f] :
            ( Y
           != ( bot_bot_BFun_f @ X212 @ X222 ) ) ) ).

% bot_term.exhaust
thf(fact_21_funas__bot__term_Ocases,axiom,
    ! [X: bot_bot_term_nat] :
      ( ( X != bot_bot_Bot_nat )
     => ~ ! [F: nat,Ss: list_B2641752247266983457rm_nat] :
            ( X
           != ( bot_bot_BFun_nat @ F @ Ss ) ) ) ).

% funas_bot_term.cases
thf(fact_22_funas__bot__term_Ocases,axiom,
    ! [X: bot_bot_term_f] :
      ( ( X != bot_bot_Bot_f )
     => ~ ! [F: f,Ss: list_Bot_bot_term_f] :
            ( X
           != ( bot_bot_BFun_f @ F @ Ss ) ) ) ).

% funas_bot_term.cases
thf(fact_23_UNIV__eq__I,axiom,
    ! [A: set_Pr3451248702717554689st_nat] :
      ( ! [X3: produc1828647624359046049st_nat] : ( member7340969449405702474st_nat @ X3 @ A )
     => ( top_to378938406556771665st_nat = A ) ) ).

% UNIV_eq_I
thf(fact_24_UNIV__eq__I,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A )
     => ( top_to4669805908274784177at_nat = A ) ) ).

% UNIV_eq_I
thf(fact_25_UNIV__eq__I,axiom,
    ! [A: set_char] :
      ( ! [X3: char] : ( member_char @ X3 @ A )
     => ( top_top_set_char = A ) ) ).

% UNIV_eq_I
thf(fact_26_UNIV__eq__I,axiom,
    ! [A: set_f] :
      ( ! [X3: f] : ( member_f @ X3 @ A )
     => ( top_top_set_f = A ) ) ).

% UNIV_eq_I
thf(fact_27_UNIV__eq__I,axiom,
    ! [A: set_list_nat] :
      ( ! [X3: list_nat] : ( member_list_nat @ X3 @ A )
     => ( top_top_set_list_nat = A ) ) ).

% UNIV_eq_I
thf(fact_28_UNIV__eq__I,axiom,
    ! [A: set_nat] :
      ( ! [X3: nat] : ( member_nat @ X3 @ A )
     => ( top_top_set_nat = A ) ) ).

% UNIV_eq_I
thf(fact_29_UNIV__witness,axiom,
    ? [X3: produc1828647624359046049st_nat] : ( member7340969449405702474st_nat @ X3 @ top_to378938406556771665st_nat ) ).

% UNIV_witness
thf(fact_30_UNIV__witness,axiom,
    ? [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ top_to4669805908274784177at_nat ) ).

% UNIV_witness
thf(fact_31_UNIV__witness,axiom,
    ? [X3: char] : ( member_char @ X3 @ top_top_set_char ) ).

% UNIV_witness
thf(fact_32_UNIV__witness,axiom,
    ? [X3: f] : ( member_f @ X3 @ top_top_set_f ) ).

% UNIV_witness
thf(fact_33_UNIV__witness,axiom,
    ? [X3: list_nat] : ( member_list_nat @ X3 @ top_top_set_list_nat ) ).

% UNIV_witness
thf(fact_34_UNIV__witness,axiom,
    ? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).

% UNIV_witness
thf(fact_35_top__set__def,axiom,
    ( top_top_set_list_nat
    = ( collect_list_nat @ top_top_list_nat_o ) ) ).

% top_set_def
thf(fact_36_top__set__def,axiom,
    ( top_top_set_nat
    = ( collect_nat @ top_top_nat_o ) ) ).

% top_set_def
thf(fact_37_top__empty__eq,axiom,
    ( top_to3312871037213017292_nat_o
    = ( ^ [X2: produc1828647624359046049st_nat] : ( member7340969449405702474st_nat @ X2 @ top_to378938406556771665st_nat ) ) ) ).

% top_empty_eq
thf(fact_38_top__empty__eq,axiom,
    ( top_to3137496036531963500_nat_o
    = ( ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ top_to4669805908274784177at_nat ) ) ) ).

% top_empty_eq
thf(fact_39_top__empty__eq,axiom,
    ( top_top_char_o
    = ( ^ [X2: char] : ( member_char @ X2 @ top_top_set_char ) ) ) ).

% top_empty_eq
thf(fact_40_top__empty__eq,axiom,
    ( top_top_f_o
    = ( ^ [X2: f] : ( member_f @ X2 @ top_top_set_f ) ) ) ).

% top_empty_eq
thf(fact_41_top__empty__eq,axiom,
    ( top_top_list_nat_o
    = ( ^ [X2: list_nat] : ( member_list_nat @ X2 @ top_top_set_list_nat ) ) ) ).

% top_empty_eq
thf(fact_42_top__empty__eq,axiom,
    ( top_top_nat_o
    = ( ^ [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ) ) ).

% top_empty_eq
thf(fact_43_merge__terms_Osimps_I2_J,axiom,
    ! [V: nat,Va: list_B2641752247266983457rm_nat] :
      ( ( bot_merge_terms_nat @ ( bot_bot_BFun_nat @ V @ Va ) @ bot_bot_Bot_nat )
      = ( bot_bot_BFun_nat @ V @ Va ) ) ).

% merge_terms.simps(2)
thf(fact_44_merge__terms_Osimps_I2_J,axiom,
    ! [V: f,Va: list_Bot_bot_term_f] :
      ( ( bot_merge_terms_f @ ( bot_bot_BFun_f @ V @ Va ) @ bot_bot_Bot_f )
      = ( bot_bot_BFun_f @ V @ Va ) ) ).

% merge_terms.simps(2)
thf(fact_45_merge__terms_Ocases,axiom,
    ! [X: produc4207152732626004967rm_nat] :
      ( ! [S: bot_bot_term_nat] :
          ( X
         != ( produc1196330319720724567rm_nat @ bot_bot_Bot_nat @ S ) )
     => ( ! [V2: nat,Va2: list_B2641752247266983457rm_nat] :
            ( X
           != ( produc1196330319720724567rm_nat @ ( bot_bot_BFun_nat @ V2 @ Va2 ) @ bot_bot_Bot_nat ) )
       => ~ ! [F: nat,Ss: list_B2641752247266983457rm_nat,G: nat,Ts: list_B2641752247266983457rm_nat] :
              ( X
             != ( produc1196330319720724567rm_nat @ ( bot_bot_BFun_nat @ F @ Ss ) @ ( bot_bot_BFun_nat @ G @ Ts ) ) ) ) ) ).

% merge_terms.cases
thf(fact_46_merge__terms_Ocases,axiom,
    ! [X: produc250624644672406269term_f] :
      ( ! [S: bot_bot_term_f] :
          ( X
         != ( produc2865564700782054965term_f @ bot_bot_Bot_f @ S ) )
     => ( ! [V2: f,Va2: list_Bot_bot_term_f] :
            ( X
           != ( produc2865564700782054965term_f @ ( bot_bot_BFun_f @ V2 @ Va2 ) @ bot_bot_Bot_f ) )
       => ~ ! [F: f,Ss: list_Bot_bot_term_f,G: f,Ts: list_Bot_bot_term_f] :
              ( X
             != ( produc2865564700782054965term_f @ ( bot_bot_BFun_f @ F @ Ss ) @ ( bot_bot_BFun_f @ G @ Ts ) ) ) ) ) ).

% merge_terms.cases
thf(fact_47_bless__eqp_Obase__l,axiom,
    ! [T: bot_bot_term_nat] : ( bot_bless_eqp_nat @ bot_bot_Bot_nat @ T ) ).

% bless_eqp.base_l
thf(fact_48_bless__eqp_Obase__l,axiom,
    ! [T: bot_bot_term_f] : ( bot_bless_eqp_f @ bot_bot_Bot_f @ T ) ).

% bless_eqp.base_l
thf(fact_49_merge__terms__bot__rhs,axiom,
    ! [S2: bot_bot_term_nat] :
      ( ( bot_merge_terms_nat @ S2 @ bot_bot_Bot_nat )
      = S2 ) ).

% merge_terms_bot_rhs
thf(fact_50_merge__terms__bot__rhs,axiom,
    ! [S2: bot_bot_term_f] :
      ( ( bot_merge_terms_f @ S2 @ bot_bot_Bot_f )
      = S2 ) ).

% merge_terms_bot_rhs
thf(fact_51_bot__term_Orel__distinct_I1_J,axiom,
    ! [R: nat > f > $o,Y21: f,Y22: list_Bot_bot_term_f] :
      ~ ( bot_bo4401546630823235424_nat_f @ R @ bot_bot_Bot_nat @ ( bot_bot_BFun_f @ Y21 @ Y22 ) ) ).

% bot_term.rel_distinct(1)
thf(fact_52_bot__term_Orel__distinct_I1_J,axiom,
    ! [R: f > nat > $o,Y21: nat,Y22: list_B2641752247266983457rm_nat] :
      ~ ( bot_bo6546836774426921474_f_nat @ R @ bot_bot_Bot_f @ ( bot_bot_BFun_nat @ Y21 @ Y22 ) ) ).

% bot_term.rel_distinct(1)
thf(fact_53_bot__term_Orel__distinct_I1_J,axiom,
    ! [R: nat > nat > $o,Y21: nat,Y22: list_B2641752247266983457rm_nat] :
      ~ ( bot_bo1849039309831764531at_nat @ R @ bot_bot_Bot_nat @ ( bot_bot_BFun_nat @ Y21 @ Y22 ) ) ).

% bot_term.rel_distinct(1)
thf(fact_54_bot__term_Orel__distinct_I1_J,axiom,
    ! [R: f > f > $o,Y21: f,Y22: list_Bot_bot_term_f] :
      ~ ( bot_bo6868847914503738705rm_f_f @ R @ bot_bot_Bot_f @ ( bot_bot_BFun_f @ Y21 @ Y22 ) ) ).

% bot_term.rel_distinct(1)
thf(fact_55_bot__term_Orel__distinct_I2_J,axiom,
    ! [R: f > nat > $o,Y21: f,Y22: list_Bot_bot_term_f] :
      ~ ( bot_bo6546836774426921474_f_nat @ R @ ( bot_bot_BFun_f @ Y21 @ Y22 ) @ bot_bot_Bot_nat ) ).

% bot_term.rel_distinct(2)
thf(fact_56_bot__term_Orel__distinct_I2_J,axiom,
    ! [R: nat > f > $o,Y21: nat,Y22: list_B2641752247266983457rm_nat] :
      ~ ( bot_bo4401546630823235424_nat_f @ R @ ( bot_bot_BFun_nat @ Y21 @ Y22 ) @ bot_bot_Bot_f ) ).

% bot_term.rel_distinct(2)
thf(fact_57_bot__term_Orel__distinct_I2_J,axiom,
    ! [R: nat > nat > $o,Y21: nat,Y22: list_B2641752247266983457rm_nat] :
      ~ ( bot_bo1849039309831764531at_nat @ R @ ( bot_bot_BFun_nat @ Y21 @ Y22 ) @ bot_bot_Bot_nat ) ).

% bot_term.rel_distinct(2)
thf(fact_58_bot__term_Orel__distinct_I2_J,axiom,
    ! [R: f > f > $o,Y21: f,Y22: list_Bot_bot_term_f] :
      ~ ( bot_bo6868847914503738705rm_f_f @ R @ ( bot_bot_BFun_f @ Y21 @ Y22 ) @ bot_bot_Bot_f ) ).

% bot_term.rel_distinct(2)
thf(fact_59_top1I,axiom,
    ! [X: nat] : ( top_top_nat_o @ X ) ).

% top1I
thf(fact_60_bot__term_Orel__eq,axiom,
    ( ( bot_bo6868847914503738705rm_f_f
      @ ^ [Y2: f,Z: f] : ( Y2 = Z ) )
    = ( ^ [Y2: bot_bot_term_f,Z: bot_bot_term_f] : ( Y2 = Z ) ) ) ).

% bot_term.rel_eq
thf(fact_61_bot__term_Orel__eq,axiom,
    ( ( bot_bo1849039309831764531at_nat
      @ ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
    = ( ^ [Y2: bot_bot_term_nat,Z: bot_bot_term_nat] : ( Y2 = Z ) ) ) ).

% bot_term.rel_eq
thf(fact_62_bot__term_Orel__refl,axiom,
    ! [Ra: f > f > $o,X: bot_bot_term_f] :
      ( ! [X3: f] : ( Ra @ X3 @ X3 )
     => ( bot_bo6868847914503738705rm_f_f @ Ra @ X @ X ) ) ).

% bot_term.rel_refl
thf(fact_63_bot__term_Orel__refl,axiom,
    ! [Ra: nat > nat > $o,X: bot_bot_term_nat] :
      ( ! [X3: nat] : ( Ra @ X3 @ X3 )
     => ( bot_bo1849039309831764531at_nat @ Ra @ X @ X ) ) ).

% bot_term.rel_refl
thf(fact_64_merge__terms__idem,axiom,
    ! [S2: bot_bot_term_f] :
      ( ( bot_merge_terms_f @ S2 @ S2 )
      = S2 ) ).

% merge_terms_idem
thf(fact_65_merge__terms__commutative,axiom,
    ( bot_merge_terms_f
    = ( ^ [S3: bot_bot_term_f,T2: bot_bot_term_f] : ( bot_merge_terms_f @ T2 @ S3 ) ) ) ).

% merge_terms_commutative
thf(fact_66_bot__term_Octr__transfer_I1_J,axiom,
    ! [R: f > nat > $o] : ( bot_bo6546836774426921474_f_nat @ R @ bot_bot_Bot_f @ bot_bot_Bot_nat ) ).

% bot_term.ctr_transfer(1)
thf(fact_67_bot__term_Octr__transfer_I1_J,axiom,
    ! [R: nat > f > $o] : ( bot_bo4401546630823235424_nat_f @ R @ bot_bot_Bot_nat @ bot_bot_Bot_f ) ).

% bot_term.ctr_transfer(1)
thf(fact_68_bot__term_Octr__transfer_I1_J,axiom,
    ! [R: nat > nat > $o] : ( bot_bo1849039309831764531at_nat @ R @ bot_bot_Bot_nat @ bot_bot_Bot_nat ) ).

% bot_term.ctr_transfer(1)
thf(fact_69_bot__term_Octr__transfer_I1_J,axiom,
    ! [R: f > f > $o] : ( bot_bo6868847914503738705rm_f_f @ R @ bot_bot_Bot_f @ bot_bot_Bot_f ) ).

% bot_term.ctr_transfer(1)
thf(fact_70_merge__terms_Osimps_I1_J,axiom,
    ! [S2: bot_bot_term_nat] :
      ( ( bot_merge_terms_nat @ bot_bot_Bot_nat @ S2 )
      = S2 ) ).

% merge_terms.simps(1)
thf(fact_71_merge__terms_Osimps_I1_J,axiom,
    ! [S2: bot_bot_term_f] :
      ( ( bot_merge_terms_f @ bot_bot_Bot_f @ S2 )
      = S2 ) ).

% merge_terms.simps(1)
thf(fact_72_old_Oprod_Oinject,axiom,
    ! [A2: product_prod_nat_nat,B: list_list_nat,A3: product_prod_nat_nat,B2: list_list_nat] :
      ( ( ( produc8649917717998259994st_nat @ A2 @ B )
        = ( produc8649917717998259994st_nat @ A3 @ B2 ) )
      = ( ( A2 = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_73_old_Oprod_Oinject,axiom,
    ! [A2: list_nat,B: list_nat,A3: list_nat,B2: list_nat] :
      ( ( ( produc2694037385005941721st_nat @ A2 @ B )
        = ( produc2694037385005941721st_nat @ A3 @ B2 ) )
      = ( ( A2 = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_74_old_Oprod_Oinject,axiom,
    ! [A2: set_nat,B: set_nat,A3: set_nat,B2: set_nat] :
      ( ( ( produc4532415448927165861et_nat @ A2 @ B )
        = ( produc4532415448927165861et_nat @ A3 @ B2 ) )
      = ( ( A2 = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_75_old_Oprod_Oinject,axiom,
    ! [A2: nat,B: bot_bot_term_f,A3: nat,B2: bot_bot_term_f] :
      ( ( ( produc5590691411411743351term_f @ A2 @ B )
        = ( produc5590691411411743351term_f @ A3 @ B2 ) )
      = ( ( A2 = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_76_old_Oprod_Oinject,axiom,
    ! [A2: nat,B: nat,A3: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A2 @ B )
        = ( product_Pair_nat_nat @ A3 @ B2 ) )
      = ( ( A2 = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_77_prod_Oinject,axiom,
    ! [X1: product_prod_nat_nat,X23: list_list_nat,Y1: product_prod_nat_nat,Y23: list_list_nat] :
      ( ( ( produc8649917717998259994st_nat @ X1 @ X23 )
        = ( produc8649917717998259994st_nat @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_78_prod_Oinject,axiom,
    ! [X1: list_nat,X23: list_nat,Y1: list_nat,Y23: list_nat] :
      ( ( ( produc2694037385005941721st_nat @ X1 @ X23 )
        = ( produc2694037385005941721st_nat @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_79_prod_Oinject,axiom,
    ! [X1: set_nat,X23: set_nat,Y1: set_nat,Y23: set_nat] :
      ( ( ( produc4532415448927165861et_nat @ X1 @ X23 )
        = ( produc4532415448927165861et_nat @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_80_prod_Oinject,axiom,
    ! [X1: nat,X23: bot_bot_term_f,Y1: nat,Y23: bot_bot_term_f] :
      ( ( ( produc5590691411411743351term_f @ X1 @ X23 )
        = ( produc5590691411411743351term_f @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_81_prod_Oinject,axiom,
    ! [X1: nat,X23: nat,Y1: nat,Y23: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X23 )
        = ( product_Pair_nat_nat @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_82_top__prod__def,axiom,
    ( top_to7593079806418470589et_nat
    = ( produc4532415448927165861et_nat @ top_top_set_nat @ top_top_set_nat ) ) ).

% top_prod_def
thf(fact_83_top__prod__def,axiom,
    ( top_to2192322404022361374_nat_o
    = ( produc7429560629904325408_nat_o @ top_top_set_nat @ top_top_nat_o ) ) ).

% top_prod_def
thf(fact_84_top__prod__def,axiom,
    ( top_to2559687781836757581st_nat
    = ( produc4193119345494528949st_nat @ top_top_set_nat @ top_top_set_list_nat ) ) ).

% top_prod_def
thf(fact_85_top__prod__def,axiom,
    ( top_to8170200190876219492et_nat
    = ( produc3803981486355964078et_nat @ top_top_nat_o @ top_top_set_nat ) ) ).

% top_prod_def
thf(fact_86_top__prod__def,axiom,
    ( top_to8453353327228641229et_nat
    = ( produc1215867243663872821et_nat @ top_top_set_list_nat @ top_top_set_nat ) ) ).

% top_prod_def
thf(fact_87_top__prod__def,axiom,
    ( top_to3543860447766875703_nat_o
    = ( produc5701653931590357335_nat_o @ top_top_nat_o @ top_top_nat_o ) ) ).

% top_prod_def
thf(fact_88_top__prod__def,axiom,
    ( top_to8731938375264029812st_nat
    = ( produc6101318194171510334st_nat @ top_top_nat_o @ top_top_set_list_nat ) ) ).

% top_prod_def
thf(fact_89_top__prod__def,axiom,
    ( top_to5523184063932315662_nat_o
    = ( produc3625103166019475856_nat_o @ top_top_set_list_nat @ top_top_nat_o ) ) ).

% top_prod_def
thf(fact_90_top__prod__def,axiom,
    ( top_to6809973963829177181st_nat
    = ( produc6003640849254547781st_nat @ top_top_set_list_nat @ top_top_set_list_nat ) ) ).

% top_prod_def
thf(fact_91_top__prod__def,axiom,
    ( top_to973740076627931404et_nat
    = ( produc8907222138397203598et_nat @ top_top_set_nat @ top_to7593079806418470589et_nat ) ) ).

% top_prod_def
thf(fact_92_BFun__leq__Bot__False,axiom,
    ! [F2: nat,Ts2: list_B2641752247266983457rm_nat] :
      ~ ( member8370420911751880208rm_nat @ ( produc1196330319720724567rm_nat @ ( bot_bot_BFun_nat @ F2 @ Ts2 ) @ bot_bot_Bot_nat ) @ bot_bless_eq_nat ) ).

% BFun_leq_Bot_False
thf(fact_93_BFun__leq__Bot__False,axiom,
    ! [F2: f,Ts2: list_Bot_bot_term_f] :
      ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_bot_BFun_f @ F2 @ Ts2 ) @ bot_bot_Bot_f ) @ bot_bless_eq_f ) ).

% BFun_leq_Bot_False
thf(fact_94_top__conj_I2_J,axiom,
    ! [P: $o,X: nat] :
      ( ( P
        & ( top_top_nat_o @ X ) )
      = P ) ).

% top_conj(2)
thf(fact_95_top__conj_I1_J,axiom,
    ! [X: nat,P: $o] :
      ( ( ( top_top_nat_o @ X )
        & P )
      = P ) ).

% top_conj(1)
thf(fact_96_asymI,axiom,
    ! [R2: set_Pr5488025237498180813et_nat] :
      ( ! [X3: set_nat,Y3: set_nat] :
          ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X3 @ Y3 ) @ R2 )
         => ~ ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Y3 @ X3 ) @ R2 ) )
     => ( asym_on_set_nat @ top_top_set_set_nat @ R2 ) ) ).

% asymI
thf(fact_97_asymI,axiom,
    ! [R2: set_Pr3451248702717554689st_nat] :
      ( ! [X3: list_nat,Y3: list_nat] :
          ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ R2 )
         => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y3 @ X3 ) @ R2 ) )
     => ( asym_on_list_nat @ top_top_set_list_nat @ R2 ) ) ).

% asymI
thf(fact_98_asymI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
         => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X3 ) @ R2 ) )
     => ( asym_on_nat @ top_top_set_nat @ R2 ) ) ).

% asymI
thf(fact_99_megeP__ass,axiom,
    ! [S2: bot_bot_term_f,T: bot_bot_term_f,U: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S2 @ ( bot_merge_terms_f @ T @ U ) ) @ bot_mergeP_f )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ T @ U ) @ bot_mergeP_f )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_merge_terms_f @ S2 @ T ) @ U ) @ bot_mergeP_f ) ) ) ).

% megeP_ass
thf(fact_100_merge__dist,axiom,
    ! [S2: bot_bot_term_f,T: bot_bot_term_f,U: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S2 @ ( bot_merge_terms_f @ T @ U ) ) @ bot_mergeP_f )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ T @ U ) @ bot_mergeP_f )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S2 @ T ) @ bot_mergeP_f ) ) ) ).

% merge_dist
thf(fact_101_merge__terms__assoc,axiom,
    ! [S2: bot_bot_term_f,T: bot_bot_term_f,U: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S2 @ T ) @ bot_mergeP_f )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ T @ U ) @ bot_mergeP_f )
       => ( ( bot_merge_terms_f @ ( bot_merge_terms_f @ S2 @ T ) @ U )
          = ( bot_merge_terms_f @ S2 @ ( bot_merge_terms_f @ T @ U ) ) ) ) ) ).

% merge_terms_assoc
thf(fact_102_is__list__UNIV__iff,axiom,
    ( is_lis1896903852932488767term_f
    = ( ^ [Xs: list_Bot_bot_term_f] :
          ( ( set_Bot_bot_term_f2 @ Xs )
          = top_to4570605823607218804term_f ) ) ) ).

% is_list_UNIV_iff
thf(fact_103_is__list__UNIV__iff,axiom,
    ( is_lis8713152955824212368at_nat
    = ( ^ [Xs: list_P6011104703257516679at_nat] :
          ( ( set_Pr5648618587558075414at_nat @ Xs )
          = top_to4669805908274784177at_nat ) ) ) ).

% is_list_UNIV_iff
thf(fact_104_is__list__UNIV__iff,axiom,
    ( is_list_UNIV_char
    = ( ^ [Xs: list_char] :
          ( ( set_char2 @ Xs )
          = top_top_set_char ) ) ) ).

% is_list_UNIV_iff
thf(fact_105_is__list__UNIV__iff,axiom,
    ( is_lis4115413270075476113st_nat
    = ( ^ [Xs: list_list_nat] :
          ( ( set_list_nat2 @ Xs )
          = top_top_set_list_nat ) ) ) ).

% is_list_UNIV_iff
thf(fact_106_is__list__UNIV__iff,axiom,
    ( is_list_UNIV_nat
    = ( ^ [Xs: list_nat] :
          ( ( set_nat2 @ Xs )
          = top_top_set_nat ) ) ) ).

% is_list_UNIV_iff
thf(fact_107_asym__onI,axiom,
    ! [A: set_Pr3451248702717554689st_nat,R2: set_Pr915550674247830855st_nat] :
      ( ! [X3: produc1828647624359046049st_nat,Y3: produc1828647624359046049st_nat] :
          ( ( member7340969449405702474st_nat @ X3 @ A )
         => ( ( member7340969449405702474st_nat @ Y3 @ A )
           => ( ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ X3 @ Y3 ) @ R2 )
             => ~ ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ Y3 @ X3 ) @ R2 ) ) ) )
     => ( asym_o2914609666330817491st_nat @ A @ R2 ) ) ).

% asym_onI
thf(fact_108_asym__onI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A )
         => ( ( member8440522571783428010at_nat @ Y3 @ A )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R2 )
             => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ X3 ) @ R2 ) ) ) )
     => ( asym_o8546584678760569651at_nat @ A @ R2 ) ) ).

% asym_onI
thf(fact_109_asym__onI,axiom,
    ! [A: set_char,R2: set_Pr3309569728657787619r_char] :
      ( ! [X3: char,Y3: char] :
          ( ( member_char @ X3 @ A )
         => ( ( member_char @ Y3 @ A )
           => ( ( member3116729636814715180r_char @ ( produc2420258056075623483r_char @ X3 @ Y3 ) @ R2 )
             => ~ ( member3116729636814715180r_char @ ( produc2420258056075623483r_char @ Y3 @ X3 ) @ R2 ) ) ) )
     => ( asym_on_char @ A @ R2 ) ) ).

% asym_onI
thf(fact_110_asym__onI,axiom,
    ! [A: set_f,R2: set_Product_prod_f_f] :
      ( ! [X3: f,Y3: f] :
          ( ( member_f @ X3 @ A )
         => ( ( member_f @ Y3 @ A )
           => ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Y3 ) @ R2 )
             => ~ ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ Y3 @ X3 ) @ R2 ) ) ) )
     => ( asym_on_f @ A @ R2 ) ) ).

% asym_onI
thf(fact_111_asym__onI,axiom,
    ! [A: set_list_nat,R2: set_Pr3451248702717554689st_nat] :
      ( ! [X3: list_nat,Y3: list_nat] :
          ( ( member_list_nat @ X3 @ A )
         => ( ( member_list_nat @ Y3 @ A )
           => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ R2 )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y3 @ X3 ) @ R2 ) ) ) )
     => ( asym_on_list_nat @ A @ R2 ) ) ).

% asym_onI
thf(fact_112_asym__onI,axiom,
    ! [A: set_set_nat,R2: set_Pr5488025237498180813et_nat] :
      ( ! [X3: set_nat,Y3: set_nat] :
          ( ( member_set_nat @ X3 @ A )
         => ( ( member_set_nat @ Y3 @ A )
           => ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X3 @ Y3 ) @ R2 )
             => ~ ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Y3 @ X3 ) @ R2 ) ) ) )
     => ( asym_on_set_nat @ A @ R2 ) ) ).

% asym_onI
thf(fact_113_asym__onI,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( ( member_nat @ Y3 @ A )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X3 ) @ R2 ) ) ) )
     => ( asym_on_nat @ A @ R2 ) ) ).

% asym_onI
thf(fact_114_bless__eq__mergeP,axiom,
    ! [S2: bot_bot_term_f,T: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S2 @ T ) @ bot_bless_eq_f )
     => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S2 @ T ) @ bot_mergeP_f ) ) ).

% bless_eq_mergeP
thf(fact_115_mem__Collect__eq,axiom,
    ! [A2: produc1828647624359046049st_nat,P: produc1828647624359046049st_nat > $o] :
      ( ( member7340969449405702474st_nat @ A2 @ ( collec1570431334306492044st_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_116_mem__Collect__eq,axiom,
    ! [A2: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A2 @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_117_mem__Collect__eq,axiom,
    ! [A2: char,P: char > $o] :
      ( ( member_char @ A2 @ ( collect_char @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_118_mem__Collect__eq,axiom,
    ! [A2: list_nat,P: list_nat > $o] :
      ( ( member_list_nat @ A2 @ ( collect_list_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_119_mem__Collect__eq,axiom,
    ! [A2: f,P: f > $o] :
      ( ( member_f @ A2 @ ( collect_f @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_120_mem__Collect__eq,axiom,
    ! [A2: nat,P: nat > $o] :
      ( ( member_nat @ A2 @ ( collect_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_121_Collect__mem__eq,axiom,
    ! [A: set_Pr3451248702717554689st_nat] :
      ( ( collec1570431334306492044st_nat
        @ ^ [X2: produc1828647624359046049st_nat] : ( member7340969449405702474st_nat @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_122_Collect__mem__eq,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_123_Collect__mem__eq,axiom,
    ! [A: set_char] :
      ( ( collect_char
        @ ^ [X2: char] : ( member_char @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_124_Collect__mem__eq,axiom,
    ! [A: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_125_Collect__mem__eq,axiom,
    ! [A: set_f] :
      ( ( collect_f
        @ ^ [X2: f] : ( member_f @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_126_Collect__mem__eq,axiom,
    ! [A: set_nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_127_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_128_asym__onD,axiom,
    ! [A: set_Pr3451248702717554689st_nat,R2: set_Pr915550674247830855st_nat,X: produc1828647624359046049st_nat,Y: produc1828647624359046049st_nat] :
      ( ( asym_o2914609666330817491st_nat @ A @ R2 )
     => ( ( member7340969449405702474st_nat @ X @ A )
       => ( ( member7340969449405702474st_nat @ Y @ A )
         => ( ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ X @ Y ) @ R2 )
           => ~ ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ Y @ X ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_129_asym__onD,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( asym_o8546584678760569651at_nat @ A @ R2 )
     => ( ( member8440522571783428010at_nat @ X @ A )
       => ( ( member8440522571783428010at_nat @ Y @ A )
         => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
           => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ X ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_130_asym__onD,axiom,
    ! [A: set_char,R2: set_Pr3309569728657787619r_char,X: char,Y: char] :
      ( ( asym_on_char @ A @ R2 )
     => ( ( member_char @ X @ A )
       => ( ( member_char @ Y @ A )
         => ( ( member3116729636814715180r_char @ ( produc2420258056075623483r_char @ X @ Y ) @ R2 )
           => ~ ( member3116729636814715180r_char @ ( produc2420258056075623483r_char @ Y @ X ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_131_asym__onD,axiom,
    ! [A: set_f,R2: set_Product_prod_f_f,X: f,Y: f] :
      ( ( asym_on_f @ A @ R2 )
     => ( ( member_f @ X @ A )
       => ( ( member_f @ Y @ A )
         => ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X @ Y ) @ R2 )
           => ~ ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ Y @ X ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_132_asym__onD,axiom,
    ! [A: set_list_nat,R2: set_Pr3451248702717554689st_nat,X: list_nat,Y: list_nat] :
      ( ( asym_on_list_nat @ A @ R2 )
     => ( ( member_list_nat @ X @ A )
       => ( ( member_list_nat @ Y @ A )
         => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R2 )
           => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y @ X ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_133_asym__onD,axiom,
    ! [A: set_set_nat,R2: set_Pr5488025237498180813et_nat,X: set_nat,Y: set_nat] :
      ( ( asym_on_set_nat @ A @ R2 )
     => ( ( member_set_nat @ X @ A )
       => ( ( member_set_nat @ Y @ A )
         => ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X @ Y ) @ R2 )
           => ~ ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Y @ X ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_134_asym__onD,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( asym_on_nat @ A @ R2 )
     => ( ( member_nat @ X @ A )
       => ( ( member_nat @ Y @ A )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_135_asym__on__def,axiom,
    ( asym_on_list_nat
    = ( ^ [A4: set_list_nat,R3: set_Pr3451248702717554689st_nat] :
        ! [X2: list_nat] :
          ( ( member_list_nat @ X2 @ A4 )
         => ! [Y4: list_nat] :
              ( ( member_list_nat @ Y4 @ A4 )
             => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X2 @ Y4 ) @ R3 )
               => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y4 @ X2 ) @ R3 ) ) ) ) ) ) ).

% asym_on_def
thf(fact_136_asym__on__def,axiom,
    ( asym_on_set_nat
    = ( ^ [A4: set_set_nat,R3: set_Pr5488025237498180813et_nat] :
        ! [X2: set_nat] :
          ( ( member_set_nat @ X2 @ A4 )
         => ! [Y4: set_nat] :
              ( ( member_set_nat @ Y4 @ A4 )
             => ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X2 @ Y4 ) @ R3 )
               => ~ ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Y4 @ X2 ) @ R3 ) ) ) ) ) ) ).

% asym_on_def
thf(fact_137_asym__on__def,axiom,
    ( asym_on_nat
    = ( ^ [A4: set_nat,R3: set_Pr1261947904930325089at_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A4 )
         => ! [Y4: nat] :
              ( ( member_nat @ Y4 @ A4 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ R3 )
               => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X2 ) @ R3 ) ) ) ) ) ) ).

% asym_on_def
thf(fact_138_merge__bot__args__bless__eq__merge,axiom,
    ! [S2: bot_bot_term_f,T: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S2 @ T ) @ bot_mergeP_f )
     => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S2 @ ( bot_merge_terms_f @ S2 @ T ) ) @ bot_bless_eq_f ) ) ).

% merge_bot_args_bless_eq_merge
thf(fact_139_bless__eq__closued__under__merge,axiom,
    ! [S2: bot_bot_term_f,T: bot_bot_term_f,U: bot_bot_term_f,V: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S2 @ T ) @ bot_mergeP_f )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ U @ V ) @ bot_mergeP_f )
       => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S2 @ U ) @ bot_bless_eq_f )
         => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ T @ V ) @ bot_bless_eq_f )
           => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_merge_terms_f @ S2 @ T ) @ ( bot_merge_terms_f @ U @ V ) ) @ bot_bless_eq_f ) ) ) ) ) ).

% bless_eq_closued_under_merge
thf(fact_140_bless__eq__anti__sym,axiom,
    ! [S2: bot_bot_term_f,T: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S2 @ T ) @ bot_bless_eq_f )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ T @ S2 ) @ bot_bless_eq_f )
       => ( S2 = T ) ) ) ).

% bless_eq_anti_sym
thf(fact_141_bless__eq__trans,axiom,
    ! [S2: bot_bot_term_f,T: bot_bot_term_f,U: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S2 @ T ) @ bot_bless_eq_f )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ T @ U ) @ bot_bless_eq_f )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S2 @ U ) @ bot_bless_eq_f ) ) ) ).

% bless_eq_trans
thf(fact_142_bless__eq__refl,axiom,
    ! [S2: bot_bot_term_f] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S2 @ S2 ) @ bot_bless_eq_f ) ).

% bless_eq_refl
thf(fact_143_merge__symmetric,axiom,
    ! [S2: bot_bot_term_f,T: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S2 @ T ) @ bot_mergeP_f )
     => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ T @ S2 ) @ bot_mergeP_f ) ) ).

% merge_symmetric
thf(fact_144_merge__refl,axiom,
    ! [S2: bot_bot_term_f] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S2 @ S2 ) @ bot_mergeP_f ) ).

% merge_refl
thf(fact_145_asym__iff,axiom,
    ! [R2: set_Pr5488025237498180813et_nat] :
      ( ( asym_on_set_nat @ top_top_set_set_nat @ R2 )
      = ( ! [X2: set_nat,Y4: set_nat] :
            ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X2 @ Y4 ) @ R2 )
           => ~ ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Y4 @ X2 ) @ R2 ) ) ) ) ).

% asym_iff
thf(fact_146_asym__iff,axiom,
    ! [R2: set_Pr3451248702717554689st_nat] :
      ( ( asym_on_list_nat @ top_top_set_list_nat @ R2 )
      = ( ! [X2: list_nat,Y4: list_nat] :
            ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X2 @ Y4 ) @ R2 )
           => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y4 @ X2 ) @ R2 ) ) ) ) ).

% asym_iff
thf(fact_147_asym__iff,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R2 )
      = ( ! [X2: nat,Y4: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ R2 )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X2 ) @ R2 ) ) ) ) ).

% asym_iff
thf(fact_148_asymD,axiom,
    ! [R2: set_Pr5488025237498180813et_nat,X: set_nat,Y: set_nat] :
      ( ( asym_on_set_nat @ top_top_set_set_nat @ R2 )
     => ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X @ Y ) @ R2 )
       => ~ ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Y @ X ) @ R2 ) ) ) ).

% asymD
thf(fact_149_asymD,axiom,
    ! [R2: set_Pr3451248702717554689st_nat,X: list_nat,Y: list_nat] :
      ( ( asym_on_list_nat @ top_top_set_list_nat @ R2 )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R2 )
       => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y @ X ) @ R2 ) ) ) ).

% asymD
thf(fact_150_asymD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
       => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R2 ) ) ) ).

% asymD
thf(fact_151_bless__eq_Obase__l,axiom,
    ! [T: bot_bot_term_nat] : ( member8370420911751880208rm_nat @ ( produc1196330319720724567rm_nat @ bot_bot_Bot_nat @ T ) @ bot_bless_eq_nat ) ).

% bless_eq.base_l
thf(fact_152_bless__eq_Obase__l,axiom,
    ! [T: bot_bot_term_f] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ bot_bot_Bot_f @ T ) @ bot_bless_eq_f ) ).

% bless_eq.base_l
thf(fact_153_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3922691147685405600st_nat] :
      ~ ! [A5: product_prod_nat_nat,B3: list_list_nat] :
          ( Y
         != ( produc8649917717998259994st_nat @ A5 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_154_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1828647624359046049st_nat] :
      ~ ! [A5: list_nat,B3: list_nat] :
          ( Y
         != ( produc2694037385005941721st_nat @ A5 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_155_old_Oprod_Oexhaust,axiom,
    ! [Y: produc7819656566062154093et_nat] :
      ~ ! [A5: set_nat,B3: set_nat] :
          ( Y
         != ( produc4532415448927165861et_nat @ A5 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_156_old_Oprod_Oexhaust,axiom,
    ! [Y: produc5418599768131837375term_f] :
      ~ ! [A5: nat,B3: bot_bot_term_f] :
          ( Y
         != ( produc5590691411411743351term_f @ A5 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_157_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A5: nat,B3: nat] :
          ( Y
         != ( product_Pair_nat_nat @ A5 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_158_surj__pair,axiom,
    ! [P2: produc3922691147685405600st_nat] :
    ? [X3: product_prod_nat_nat,Y3: list_list_nat] :
      ( P2
      = ( produc8649917717998259994st_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_159_surj__pair,axiom,
    ! [P2: produc1828647624359046049st_nat] :
    ? [X3: list_nat,Y3: list_nat] :
      ( P2
      = ( produc2694037385005941721st_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_160_surj__pair,axiom,
    ! [P2: produc7819656566062154093et_nat] :
    ? [X3: set_nat,Y3: set_nat] :
      ( P2
      = ( produc4532415448927165861et_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_161_surj__pair,axiom,
    ! [P2: produc5418599768131837375term_f] :
    ? [X3: nat,Y3: bot_bot_term_f] :
      ( P2
      = ( produc5590691411411743351term_f @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_162_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X3: nat,Y3: nat] :
      ( P2
      = ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_163_prod__cases,axiom,
    ! [P: produc3922691147685405600st_nat > $o,P2: produc3922691147685405600st_nat] :
      ( ! [A5: product_prod_nat_nat,B3: list_list_nat] : ( P @ ( produc8649917717998259994st_nat @ A5 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_164_prod__cases,axiom,
    ! [P: produc1828647624359046049st_nat > $o,P2: produc1828647624359046049st_nat] :
      ( ! [A5: list_nat,B3: list_nat] : ( P @ ( produc2694037385005941721st_nat @ A5 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_165_prod__cases,axiom,
    ! [P: produc7819656566062154093et_nat > $o,P2: produc7819656566062154093et_nat] :
      ( ! [A5: set_nat,B3: set_nat] : ( P @ ( produc4532415448927165861et_nat @ A5 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_166_prod__cases,axiom,
    ! [P: produc5418599768131837375term_f > $o,P2: produc5418599768131837375term_f] :
      ( ! [A5: nat,B3: bot_bot_term_f] : ( P @ ( produc5590691411411743351term_f @ A5 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_167_prod__cases,axiom,
    ! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
      ( ! [A5: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A5 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_168_Pair__inject,axiom,
    ! [A2: product_prod_nat_nat,B: list_list_nat,A3: product_prod_nat_nat,B2: list_list_nat] :
      ( ( ( produc8649917717998259994st_nat @ A2 @ B )
        = ( produc8649917717998259994st_nat @ A3 @ B2 ) )
     => ~ ( ( A2 = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_169_Pair__inject,axiom,
    ! [A2: list_nat,B: list_nat,A3: list_nat,B2: list_nat] :
      ( ( ( produc2694037385005941721st_nat @ A2 @ B )
        = ( produc2694037385005941721st_nat @ A3 @ B2 ) )
     => ~ ( ( A2 = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_170_Pair__inject,axiom,
    ! [A2: set_nat,B: set_nat,A3: set_nat,B2: set_nat] :
      ( ( ( produc4532415448927165861et_nat @ A2 @ B )
        = ( produc4532415448927165861et_nat @ A3 @ B2 ) )
     => ~ ( ( A2 = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_171_Pair__inject,axiom,
    ! [A2: nat,B: bot_bot_term_f,A3: nat,B2: bot_bot_term_f] :
      ( ( ( produc5590691411411743351term_f @ A2 @ B )
        = ( produc5590691411411743351term_f @ A3 @ B2 ) )
     => ~ ( ( A2 = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_172_Pair__inject,axiom,
    ! [A2: nat,B: nat,A3: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A2 @ B )
        = ( product_Pair_nat_nat @ A3 @ B2 ) )
     => ~ ( ( A2 = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_173_mergeP_Obase__l,axiom,
    ! [T: bot_bot_term_nat] : ( member8370420911751880208rm_nat @ ( produc1196330319720724567rm_nat @ bot_bot_Bot_nat @ T ) @ bot_mergeP_nat ) ).

% mergeP.base_l
thf(fact_174_mergeP_Obase__l,axiom,
    ! [T: bot_bot_term_f] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ bot_bot_Bot_f @ T ) @ bot_mergeP_f ) ).

% mergeP.base_l
thf(fact_175_mergeP_Obase__r,axiom,
    ! [T: bot_bot_term_nat] : ( member8370420911751880208rm_nat @ ( produc1196330319720724567rm_nat @ T @ bot_bot_Bot_nat ) @ bot_mergeP_nat ) ).

% mergeP.base_r
thf(fact_176_mergeP_Obase__r,axiom,
    ! [T: bot_bot_term_f] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ T @ bot_bot_Bot_f ) @ bot_mergeP_f ) ).

% mergeP.base_r
thf(fact_177_bless__eq__closued__under__supremum,axiom,
    ! [S2: bot_bot_term_f,U: bot_bot_term_f,T: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S2 @ U ) @ bot_bless_eq_f )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ T @ U ) @ bot_bless_eq_f )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_merge_terms_f @ S2 @ T ) @ U ) @ bot_bless_eq_f ) ) ) ).

% bless_eq_closued_under_supremum
thf(fact_178_BFun__lesseqE,axiom,
    ! [F2: nat,Ts2: list_B2641752247266983457rm_nat,T: bot_bot_term_nat] :
      ( ( member8370420911751880208rm_nat @ ( produc1196330319720724567rm_nat @ ( bot_bot_BFun_nat @ F2 @ Ts2 ) @ T ) @ bot_bless_eq_nat )
     => ~ ! [Us: list_B2641752247266983457rm_nat] :
            ( ( ( size_s7068041265931086989rm_nat @ Ts2 )
              = ( size_s7068041265931086989rm_nat @ Us ) )
           => ( T
             != ( bot_bot_BFun_nat @ F2 @ Us ) ) ) ) ).

% BFun_lesseqE
thf(fact_179_BFun__lesseqE,axiom,
    ! [F2: f,Ts2: list_Bot_bot_term_f,T: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_bot_BFun_f @ F2 @ Ts2 ) @ T ) @ bot_bless_eq_f )
     => ~ ! [Us: list_Bot_bot_term_f] :
            ( ( ( size_s1458857753309042354term_f @ Ts2 )
              = ( size_s1458857753309042354term_f @ Us ) )
           => ( T
             != ( bot_bot_BFun_f @ F2 @ Us ) ) ) ) ).

% BFun_lesseqE
thf(fact_180_asym__inv__image,axiom,
    ! [R: set_Pr1261947904930325089at_nat,F2: list_nat > nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R )
     => ( asym_on_list_nat @ top_top_set_list_nat @ ( inv_im8414047238327108484st_nat @ R @ F2 ) ) ) ).

% asym_inv_image
thf(fact_181_asym__inv__image,axiom,
    ! [R: set_Pr3451248702717554689st_nat,F2: nat > list_nat] :
      ( ( asym_on_list_nat @ top_top_set_list_nat @ R )
     => ( asym_on_nat @ top_top_set_nat @ ( inv_im1429632248634617860at_nat @ R @ F2 ) ) ) ).

% asym_inv_image
thf(fact_182_asym__inv__image,axiom,
    ! [R: set_Pr3451248702717554689st_nat,F2: list_nat > list_nat] :
      ( ( asym_on_list_nat @ top_top_set_list_nat @ R )
     => ( asym_on_list_nat @ top_top_set_list_nat @ ( inv_im7901436520293687572st_nat @ R @ F2 ) ) ) ).

% asym_inv_image
thf(fact_183_asym__inv__image,axiom,
    ! [R: set_Pr1261947904930325089at_nat,F2: nat > nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R )
     => ( asym_on_nat @ top_top_set_nat @ ( inv_image_nat_nat @ R @ F2 ) ) ) ).

% asym_inv_image
thf(fact_184_bot__term_Oset__intros_I2_J,axiom,
    ! [Y: bot_bo5929273105904447532st_nat,X22: list_B7941409156082897212st_nat,Ya: produc1828647624359046049st_nat,X21: produc1828647624359046049st_nat] :
      ( ( member274174779753688259st_nat @ Y @ ( set_Bo5869970306151722839st_nat @ X22 ) )
     => ( ( member7340969449405702474st_nat @ Ya @ ( bot_bo4281033497674357304st_nat @ Y ) )
       => ( member7340969449405702474st_nat @ Ya @ ( bot_bo4281033497674357304st_nat @ ( bot_bo202339342019657932st_nat @ X21 @ X22 ) ) ) ) ) ).

% bot_term.set_intros(2)
thf(fact_185_bot__term_Oset__intros_I2_J,axiom,
    ! [Y: bot_bo655077203622184332at_nat,X22: list_B7173093809959871388at_nat,Ya: product_prod_nat_nat,X21: product_prod_nat_nat] :
      ( ( member2038346494265176355at_nat @ Y @ ( set_Bo1863287301388052407at_nat @ X22 ) )
     => ( ( member8440522571783428010at_nat @ Ya @ ( bot_bo6066252797605031576at_nat @ Y ) )
       => ( member8440522571783428010at_nat @ Ya @ ( bot_bo6066252797605031576at_nat @ ( bot_bo1451043769174509868at_nat @ X21 @ X22 ) ) ) ) ) ).

% bot_term.set_intros(2)
thf(fact_186_bot__term_Oset__intros_I2_J,axiom,
    ! [Y: bot_bot_term_char,X22: list_B1576150950144988706m_char,Ya: char,X21: char] :
      ( ( member6716525252696868293m_char @ Y @ ( set_Bo5011433680641223729m_char @ X22 ) )
     => ( ( member_char @ Ya @ ( bot_bo3082694273163266554m_char @ Y ) )
       => ( member_char @ Ya @ ( bot_bo3082694273163266554m_char @ ( bot_bot_BFun_char @ X21 @ X22 ) ) ) ) ) ).

% bot_term.set_intros(2)
thf(fact_187_bot__term_Oset__intros_I2_J,axiom,
    ! [Y: bot_bo5408180104339501355st_nat,X22: list_B5078311858792894385st_nat,Ya: list_nat,X21: list_nat] :
      ( ( member8381404166258140628st_nat @ Y @ ( set_Bo3558769202554720832st_nat @ X22 ) )
     => ( ( member_list_nat @ Ya @ ( bot_bo7758324655462117769st_nat @ Y ) )
       => ( member_list_nat @ Ya @ ( bot_bo7758324655462117769st_nat @ ( bot_bo885581241535429109st_nat @ X21 @ X22 ) ) ) ) ) ).

% bot_term.set_intros(2)
thf(fact_188_bot__term_Oset__intros_I2_J,axiom,
    ! [Y: bot_bot_term_nat,X22: list_B2641752247266983457rm_nat,Ya: nat,X21: nat] :
      ( ( member2055285333220477764rm_nat @ Y @ ( set_Bot_bot_term_nat2 @ X22 ) )
     => ( ( member_nat @ Ya @ ( bot_bo2410032770233908985rm_nat @ Y ) )
       => ( member_nat @ Ya @ ( bot_bo2410032770233908985rm_nat @ ( bot_bot_BFun_nat @ X21 @ X22 ) ) ) ) ) ).

% bot_term.set_intros(2)
thf(fact_189_bot__term_Oset__intros_I2_J,axiom,
    ! [Y: bot_bot_term_f,X22: list_Bot_bot_term_f,Ya: f,X21: f] :
      ( ( member2723211850833494437term_f @ Y @ ( set_Bot_bot_term_f2 @ X22 ) )
     => ( ( member_f @ Ya @ ( bot_bo658131349715431450term_f @ Y ) )
       => ( member_f @ Ya @ ( bot_bo658131349715431450term_f @ ( bot_bot_BFun_f @ X21 @ X22 ) ) ) ) ) ).

% bot_term.set_intros(2)
thf(fact_190_bot__term_Oset__cases,axiom,
    ! [E: produc1828647624359046049st_nat,A2: bot_bo5929273105904447532st_nat] :
      ( ( member7340969449405702474st_nat @ E @ ( bot_bo4281033497674357304st_nat @ A2 ) )
     => ( ! [Z2: list_B7941409156082897212st_nat] :
            ( A2
           != ( bot_bo202339342019657932st_nat @ E @ Z2 ) )
       => ~ ! [Z1: produc1828647624359046049st_nat,Z2: list_B7941409156082897212st_nat] :
              ( ( A2
                = ( bot_bo202339342019657932st_nat @ Z1 @ Z2 ) )
             => ! [X3: bot_bo5929273105904447532st_nat] :
                  ( ( member274174779753688259st_nat @ X3 @ ( set_Bo5869970306151722839st_nat @ Z2 ) )
                 => ~ ( member7340969449405702474st_nat @ E @ ( bot_bo4281033497674357304st_nat @ X3 ) ) ) ) ) ) ).

% bot_term.set_cases
thf(fact_191_bot__term_Oset__cases,axiom,
    ! [E: product_prod_nat_nat,A2: bot_bo655077203622184332at_nat] :
      ( ( member8440522571783428010at_nat @ E @ ( bot_bo6066252797605031576at_nat @ A2 ) )
     => ( ! [Z2: list_B7173093809959871388at_nat] :
            ( A2
           != ( bot_bo1451043769174509868at_nat @ E @ Z2 ) )
       => ~ ! [Z1: product_prod_nat_nat,Z2: list_B7173093809959871388at_nat] :
              ( ( A2
                = ( bot_bo1451043769174509868at_nat @ Z1 @ Z2 ) )
             => ! [X3: bot_bo655077203622184332at_nat] :
                  ( ( member2038346494265176355at_nat @ X3 @ ( set_Bo1863287301388052407at_nat @ Z2 ) )
                 => ~ ( member8440522571783428010at_nat @ E @ ( bot_bo6066252797605031576at_nat @ X3 ) ) ) ) ) ) ).

% bot_term.set_cases
thf(fact_192_bot__term_Oset__cases,axiom,
    ! [E: char,A2: bot_bot_term_char] :
      ( ( member_char @ E @ ( bot_bo3082694273163266554m_char @ A2 ) )
     => ( ! [Z2: list_B1576150950144988706m_char] :
            ( A2
           != ( bot_bot_BFun_char @ E @ Z2 ) )
       => ~ ! [Z1: char,Z2: list_B1576150950144988706m_char] :
              ( ( A2
                = ( bot_bot_BFun_char @ Z1 @ Z2 ) )
             => ! [X3: bot_bot_term_char] :
                  ( ( member6716525252696868293m_char @ X3 @ ( set_Bo5011433680641223729m_char @ Z2 ) )
                 => ~ ( member_char @ E @ ( bot_bo3082694273163266554m_char @ X3 ) ) ) ) ) ) ).

% bot_term.set_cases
thf(fact_193_bot__term_Oset__cases,axiom,
    ! [E: list_nat,A2: bot_bo5408180104339501355st_nat] :
      ( ( member_list_nat @ E @ ( bot_bo7758324655462117769st_nat @ A2 ) )
     => ( ! [Z2: list_B5078311858792894385st_nat] :
            ( A2
           != ( bot_bo885581241535429109st_nat @ E @ Z2 ) )
       => ~ ! [Z1: list_nat,Z2: list_B5078311858792894385st_nat] :
              ( ( A2
                = ( bot_bo885581241535429109st_nat @ Z1 @ Z2 ) )
             => ! [X3: bot_bo5408180104339501355st_nat] :
                  ( ( member8381404166258140628st_nat @ X3 @ ( set_Bo3558769202554720832st_nat @ Z2 ) )
                 => ~ ( member_list_nat @ E @ ( bot_bo7758324655462117769st_nat @ X3 ) ) ) ) ) ) ).

% bot_term.set_cases
thf(fact_194_bot__term_Oset__cases,axiom,
    ! [E: nat,A2: bot_bot_term_nat] :
      ( ( member_nat @ E @ ( bot_bo2410032770233908985rm_nat @ A2 ) )
     => ( ! [Z2: list_B2641752247266983457rm_nat] :
            ( A2
           != ( bot_bot_BFun_nat @ E @ Z2 ) )
       => ~ ! [Z1: nat,Z2: list_B2641752247266983457rm_nat] :
              ( ( A2
                = ( bot_bot_BFun_nat @ Z1 @ Z2 ) )
             => ! [X3: bot_bot_term_nat] :
                  ( ( member2055285333220477764rm_nat @ X3 @ ( set_Bot_bot_term_nat2 @ Z2 ) )
                 => ~ ( member_nat @ E @ ( bot_bo2410032770233908985rm_nat @ X3 ) ) ) ) ) ) ).

% bot_term.set_cases
thf(fact_195_bot__term_Oset__cases,axiom,
    ! [E: f,A2: bot_bot_term_f] :
      ( ( member_f @ E @ ( bot_bo658131349715431450term_f @ A2 ) )
     => ( ! [Z2: list_Bot_bot_term_f] :
            ( A2
           != ( bot_bot_BFun_f @ E @ Z2 ) )
       => ~ ! [Z1: f,Z2: list_Bot_bot_term_f] :
              ( ( A2
                = ( bot_bot_BFun_f @ Z1 @ Z2 ) )
             => ! [X3: bot_bot_term_f] :
                  ( ( member2723211850833494437term_f @ X3 @ ( set_Bot_bot_term_f2 @ Z2 ) )
                 => ~ ( member_f @ E @ ( bot_bo658131349715431450term_f @ X3 ) ) ) ) ) ) ).

% bot_term.set_cases
thf(fact_196_ssubst__Pair__rhs,axiom,
    ! [R2: product_prod_nat_nat,S2: list_list_nat,R: set_Pr4879555682012792790st_nat,S4: list_list_nat] :
      ( ( member5740615348714444471st_nat @ ( produc8649917717998259994st_nat @ R2 @ S2 ) @ R )
     => ( ( S4 = S2 )
       => ( member5740615348714444471st_nat @ ( produc8649917717998259994st_nat @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_197_ssubst__Pair__rhs,axiom,
    ! [R2: list_nat,S2: list_nat,R: set_Pr3451248702717554689st_nat,S4: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ R2 @ S2 ) @ R )
     => ( ( S4 = S2 )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_198_ssubst__Pair__rhs,axiom,
    ! [R2: set_nat,S2: set_nat,R: set_Pr5488025237498180813et_nat,S4: set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ R2 @ S2 ) @ R )
     => ( ( S4 = S2 )
       => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_199_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S2: bot_bot_term_f,R: set_Pr460048518328002463term_f,S4: bot_bot_term_f] :
      ( ( member4779802479421434344term_f @ ( produc5590691411411743351term_f @ R2 @ S2 ) @ R )
     => ( ( S4 = S2 )
       => ( member4779802479421434344term_f @ ( produc5590691411411743351term_f @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_200_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S2: nat,R: set_Pr1261947904930325089at_nat,S4: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S2 ) @ R )
     => ( ( S4 = S2 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_201_bot__term_Orel__induct,axiom,
    ! [R: f > nat > $o,X: bot_bot_term_f,Y: bot_bot_term_nat,Q: bot_bot_term_f > bot_bot_term_nat > $o] :
      ( ( bot_bo6546836774426921474_f_nat @ R @ X @ Y )
     => ( ( Q @ bot_bot_Bot_f @ bot_bot_Bot_nat )
       => ( ! [A21: f,A22: list_Bot_bot_term_f,B21: nat,B22: list_B2641752247266983457rm_nat] :
              ( ( R @ A21 @ B21 )
             => ( ( list_a7573497065701696196rm_nat @ Q @ A22 @ B22 )
               => ( Q @ ( bot_bot_BFun_f @ A21 @ A22 ) @ ( bot_bot_BFun_nat @ B21 @ B22 ) ) ) )
         => ( Q @ X @ Y ) ) ) ) ).

% bot_term.rel_induct
thf(fact_202_bot__term_Orel__induct,axiom,
    ! [R: nat > f > $o,X: bot_bot_term_nat,Y: bot_bot_term_f,Q: bot_bot_term_nat > bot_bot_term_f > $o] :
      ( ( bot_bo4401546630823235424_nat_f @ R @ X @ Y )
     => ( ( Q @ bot_bot_Bot_nat @ bot_bot_Bot_f )
       => ( ! [A21: nat,A22: list_B2641752247266983457rm_nat,B21: f,B22: list_Bot_bot_term_f] :
              ( ( R @ A21 @ B21 )
             => ( ( list_a2360236856457976674term_f @ Q @ A22 @ B22 )
               => ( Q @ ( bot_bot_BFun_nat @ A21 @ A22 ) @ ( bot_bot_BFun_f @ B21 @ B22 ) ) ) )
         => ( Q @ X @ Y ) ) ) ) ).

% bot_term.rel_induct
thf(fact_203_bot__term_Orel__induct,axiom,
    ! [R: nat > nat > $o,X: bot_bot_term_nat,Y: bot_bot_term_nat,Q: bot_bot_term_nat > bot_bot_term_nat > $o] :
      ( ( bot_bo1849039309831764531at_nat @ R @ X @ Y )
     => ( ( Q @ bot_bot_Bot_nat @ bot_bot_Bot_nat )
       => ( ! [A21: nat,A22: list_B2641752247266983457rm_nat,B21: nat,B22: list_B2641752247266983457rm_nat] :
              ( ( R @ A21 @ B21 )
             => ( ( list_a3175995857480936519rm_nat @ Q @ A22 @ B22 )
               => ( Q @ ( bot_bot_BFun_nat @ A21 @ A22 ) @ ( bot_bot_BFun_nat @ B21 @ B22 ) ) ) )
         => ( Q @ X @ Y ) ) ) ) ).

% bot_term.rel_induct
thf(fact_204_bot__term_Orel__induct,axiom,
    ! [R: f > f > $o,X: bot_bot_term_f,Y: bot_bot_term_f,Q: bot_bot_term_f > bot_bot_term_f > $o] :
      ( ( bot_bo6868847914503738705rm_f_f @ R @ X @ Y )
     => ( ( Q @ bot_bot_Bot_f @ bot_bot_Bot_f )
       => ( ! [A21: f,A22: list_Bot_bot_term_f,B21: f,B22: list_Bot_bot_term_f] :
              ( ( R @ A21 @ B21 )
             => ( ( list_a8926285605142776357term_f @ Q @ A22 @ B22 )
               => ( Q @ ( bot_bot_BFun_f @ A21 @ A22 ) @ ( bot_bot_BFun_f @ B21 @ B22 ) ) ) )
         => ( Q @ X @ Y ) ) ) ) ).

% bot_term.rel_induct
thf(fact_205_bot__term_Orel__cases,axiom,
    ! [R: f > nat > $o,A2: bot_bot_term_f,B: bot_bot_term_nat] :
      ( ( bot_bo6546836774426921474_f_nat @ R @ A2 @ B )
     => ( ( ( A2 = bot_bot_Bot_f )
         => ( B != bot_bot_Bot_nat ) )
       => ~ ! [X12: f,X24: list_Bot_bot_term_f] :
              ( ( A2
                = ( bot_bot_BFun_f @ X12 @ X24 ) )
             => ! [Y12: nat,Y24: list_B2641752247266983457rm_nat] :
                  ( ( B
                    = ( bot_bot_BFun_nat @ Y12 @ Y24 ) )
                 => ( ( R @ X12 @ Y12 )
                   => ~ ( list_a7573497065701696196rm_nat @ ( bot_bo6546836774426921474_f_nat @ R ) @ X24 @ Y24 ) ) ) ) ) ) ).

% bot_term.rel_cases
thf(fact_206_bot__term_Orel__cases,axiom,
    ! [R: nat > f > $o,A2: bot_bot_term_nat,B: bot_bot_term_f] :
      ( ( bot_bo4401546630823235424_nat_f @ R @ A2 @ B )
     => ( ( ( A2 = bot_bot_Bot_nat )
         => ( B != bot_bot_Bot_f ) )
       => ~ ! [X12: nat,X24: list_B2641752247266983457rm_nat] :
              ( ( A2
                = ( bot_bot_BFun_nat @ X12 @ X24 ) )
             => ! [Y12: f,Y24: list_Bot_bot_term_f] :
                  ( ( B
                    = ( bot_bot_BFun_f @ Y12 @ Y24 ) )
                 => ( ( R @ X12 @ Y12 )
                   => ~ ( list_a2360236856457976674term_f @ ( bot_bo4401546630823235424_nat_f @ R ) @ X24 @ Y24 ) ) ) ) ) ) ).

% bot_term.rel_cases
thf(fact_207_bot__term_Orel__cases,axiom,
    ! [R: nat > nat > $o,A2: bot_bot_term_nat,B: bot_bot_term_nat] :
      ( ( bot_bo1849039309831764531at_nat @ R @ A2 @ B )
     => ( ( ( A2 = bot_bot_Bot_nat )
         => ( B != bot_bot_Bot_nat ) )
       => ~ ! [X12: nat,X24: list_B2641752247266983457rm_nat] :
              ( ( A2
                = ( bot_bot_BFun_nat @ X12 @ X24 ) )
             => ! [Y12: nat,Y24: list_B2641752247266983457rm_nat] :
                  ( ( B
                    = ( bot_bot_BFun_nat @ Y12 @ Y24 ) )
                 => ( ( R @ X12 @ Y12 )
                   => ~ ( list_a3175995857480936519rm_nat @ ( bot_bo1849039309831764531at_nat @ R ) @ X24 @ Y24 ) ) ) ) ) ) ).

% bot_term.rel_cases
thf(fact_208_bot__term_Orel__cases,axiom,
    ! [R: f > f > $o,A2: bot_bot_term_f,B: bot_bot_term_f] :
      ( ( bot_bo6868847914503738705rm_f_f @ R @ A2 @ B )
     => ( ( ( A2 = bot_bot_Bot_f )
         => ( B != bot_bot_Bot_f ) )
       => ~ ! [X12: f,X24: list_Bot_bot_term_f] :
              ( ( A2
                = ( bot_bot_BFun_f @ X12 @ X24 ) )
             => ! [Y12: f,Y24: list_Bot_bot_term_f] :
                  ( ( B
                    = ( bot_bot_BFun_f @ Y12 @ Y24 ) )
                 => ( ( R @ X12 @ Y12 )
                   => ~ ( list_a8926285605142776357term_f @ ( bot_bo6868847914503738705rm_f_f @ R ) @ X24 @ Y24 ) ) ) ) ) ) ).

% bot_term.rel_cases
thf(fact_209_curryI,axiom,
    ! [F2: produc3922691147685405600st_nat > $o,A2: product_prod_nat_nat,B: list_list_nat] :
      ( ( F2 @ ( produc8649917717998259994st_nat @ A2 @ B ) )
     => ( produc6878766516837139888_nat_o @ F2 @ A2 @ B ) ) ).

% curryI
thf(fact_210_curryI,axiom,
    ! [F2: produc1828647624359046049st_nat > $o,A2: list_nat,B: list_nat] :
      ( ( F2 @ ( produc2694037385005941721st_nat @ A2 @ B ) )
     => ( produc6139401228252118475_nat_o @ F2 @ A2 @ B ) ) ).

% curryI
thf(fact_211_curryI,axiom,
    ! [F2: produc7819656566062154093et_nat > $o,A2: set_nat,B: set_nat] :
      ( ( F2 @ ( produc4532415448927165861et_nat @ A2 @ B ) )
     => ( produc8481458855017059839_nat_o @ F2 @ A2 @ B ) ) ).

% curryI
thf(fact_212_curryI,axiom,
    ! [F2: produc5418599768131837375term_f > $o,A2: nat,B: bot_bot_term_f] :
      ( ( F2 @ ( produc5590691411411743351term_f @ A2 @ B ) )
     => ( produc7017639609257223213rm_f_o @ F2 @ A2 @ B ) ) ).

% curryI
thf(fact_213_curryI,axiom,
    ! [F2: product_prod_nat_nat > $o,A2: nat,B: nat] :
      ( ( F2 @ ( product_Pair_nat_nat @ A2 @ B ) )
     => ( produc1310100445399344235_nat_o @ F2 @ A2 @ B ) ) ).

% curryI
thf(fact_214_in__inv__image,axiom,
    ! [X: list_nat,Y: list_nat,R2: set_Pr3451248702717554689st_nat,F2: list_nat > list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( inv_im7901436520293687572st_nat @ R2 @ F2 ) )
      = ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_215_in__inv__image,axiom,
    ! [X: list_nat,Y: list_nat,R2: set_Pr5488025237498180813et_nat,F2: list_nat > set_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( inv_im4790160404546336826st_nat @ R2 @ F2 ) )
      = ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_216_in__inv__image,axiom,
    ! [X: list_nat,Y: list_nat,R2: set_Pr1261947904930325089at_nat,F2: list_nat > nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( inv_im8414047238327108484st_nat @ R2 @ F2 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_217_in__inv__image,axiom,
    ! [X: set_nat,Y: set_nat,R2: set_Pr3451248702717554689st_nat,F2: set_nat > list_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X @ Y ) @ ( inv_im1139933451183655866et_nat @ R2 @ F2 ) )
      = ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_218_in__inv__image,axiom,
    ! [X: set_nat,Y: set_nat,R2: set_Pr5488025237498180813et_nat,F2: set_nat > set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X @ Y ) @ ( inv_im3688519544425059808et_nat @ R2 @ F2 ) )
      = ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_219_in__inv__image,axiom,
    ! [X: set_nat,Y: set_nat,R2: set_Pr1261947904930325089at_nat,F2: set_nat > nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X @ Y ) @ ( inv_im5077416110856988202et_nat @ R2 @ F2 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_220_in__inv__image,axiom,
    ! [X: nat,Y: nat,R2: set_Pr3451248702717554689st_nat,F2: nat > list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_im1429632248634617860at_nat @ R2 @ F2 ) )
      = ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_221_in__inv__image,axiom,
    ! [X: nat,Y: nat,R2: set_Pr5488025237498180813et_nat,F2: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_im1511781206201618986at_nat @ R2 @ F2 ) )
      = ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_222_in__inv__image,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,F2: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_image_nat_nat @ R2 @ F2 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_223_bot__term_Oset__intros_I1_J,axiom,
    ! [X21: produc1828647624359046049st_nat,X22: list_B7941409156082897212st_nat] : ( member7340969449405702474st_nat @ X21 @ ( bot_bo4281033497674357304st_nat @ ( bot_bo202339342019657932st_nat @ X21 @ X22 ) ) ) ).

% bot_term.set_intros(1)
thf(fact_224_bot__term_Oset__intros_I1_J,axiom,
    ! [X21: product_prod_nat_nat,X22: list_B7173093809959871388at_nat] : ( member8440522571783428010at_nat @ X21 @ ( bot_bo6066252797605031576at_nat @ ( bot_bo1451043769174509868at_nat @ X21 @ X22 ) ) ) ).

% bot_term.set_intros(1)
thf(fact_225_bot__term_Oset__intros_I1_J,axiom,
    ! [X21: char,X22: list_B1576150950144988706m_char] : ( member_char @ X21 @ ( bot_bo3082694273163266554m_char @ ( bot_bot_BFun_char @ X21 @ X22 ) ) ) ).

% bot_term.set_intros(1)
thf(fact_226_bot__term_Oset__intros_I1_J,axiom,
    ! [X21: list_nat,X22: list_B5078311858792894385st_nat] : ( member_list_nat @ X21 @ ( bot_bo7758324655462117769st_nat @ ( bot_bo885581241535429109st_nat @ X21 @ X22 ) ) ) ).

% bot_term.set_intros(1)
thf(fact_227_bot__term_Oset__intros_I1_J,axiom,
    ! [X21: nat,X22: list_B2641752247266983457rm_nat] : ( member_nat @ X21 @ ( bot_bo2410032770233908985rm_nat @ ( bot_bot_BFun_nat @ X21 @ X22 ) ) ) ).

% bot_term.set_intros(1)
thf(fact_228_bot__term_Oset__intros_I1_J,axiom,
    ! [X21: f,X22: list_Bot_bot_term_f] : ( member_f @ X21 @ ( bot_bo658131349715431450term_f @ ( bot_bot_BFun_f @ X21 @ X22 ) ) ) ).

% bot_term.set_intros(1)
thf(fact_229_bot__term_Orel__cong,axiom,
    ! [X: bot_bot_term_nat,Ya: bot_bot_term_nat,Y: bot_bot_term_nat,Xa: bot_bot_term_nat,R: nat > nat > $o,Ra: nat > nat > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z3: nat,Yb: nat] :
              ( ( member_nat @ Z3 @ ( bot_bo2410032770233908985rm_nat @ Ya ) )
             => ( ( member_nat @ Yb @ ( bot_bo2410032770233908985rm_nat @ Xa ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( bot_bo1849039309831764531at_nat @ R @ X @ Y )
            = ( bot_bo1849039309831764531at_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).

% bot_term.rel_cong
thf(fact_230_bot__term_Orel__cong,axiom,
    ! [X: bot_bot_term_char,Ya: bot_bot_term_char,Y: bot_bot_term_char,Xa: bot_bot_term_char,R: char > char > $o,Ra: char > char > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z3: char,Yb: char] :
              ( ( member_char @ Z3 @ ( bot_bo3082694273163266554m_char @ Ya ) )
             => ( ( member_char @ Yb @ ( bot_bo3082694273163266554m_char @ Xa ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( bot_bo2694460441194989621r_char @ R @ X @ Y )
            = ( bot_bo2694460441194989621r_char @ Ra @ Ya @ Xa ) ) ) ) ) ).

% bot_term.rel_cong
thf(fact_231_bot__term_Orel__cong,axiom,
    ! [X: bot_bot_term_char,Ya: bot_bot_term_char,Y: bot_bot_term_f,Xa: bot_bot_term_f,R: char > f > $o,Ra: char > f > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z3: char,Yb: f] :
              ( ( member_char @ Z3 @ ( bot_bo3082694273163266554m_char @ Ya ) )
             => ( ( member_f @ Yb @ ( bot_bo658131349715431450term_f @ Xa ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( bot_bo3338816352050261919char_f @ R @ X @ Y )
            = ( bot_bo3338816352050261919char_f @ Ra @ Ya @ Xa ) ) ) ) ) ).

% bot_term.rel_cong
thf(fact_232_bot__term_Orel__cong,axiom,
    ! [X: bot_bot_term_char,Ya: bot_bot_term_char,Y: bot_bot_term_nat,Xa: bot_bot_term_nat,R: char > nat > $o,Ra: char > nat > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z3: char,Yb: nat] :
              ( ( member_char @ Z3 @ ( bot_bo3082694273163266554m_char @ Ya ) )
             => ( ( member_nat @ Yb @ ( bot_bo2410032770233908985rm_nat @ Xa ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( bot_bo4339835670485063732ar_nat @ R @ X @ Y )
            = ( bot_bo4339835670485063732ar_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).

% bot_term.rel_cong
thf(fact_233_bot__term_Orel__cong,axiom,
    ! [X: bot_bot_term_f,Ya: bot_bot_term_f,Y: bot_bot_term_char,Xa: bot_bot_term_char,R: f > char > $o,Ra: f > char > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z3: f,Yb: char] :
              ( ( member_f @ Z3 @ ( bot_bo658131349715431450term_f @ Ya ) )
             => ( ( member_char @ Yb @ ( bot_bo3082694273163266554m_char @ Xa ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( bot_bo6232401481379736579f_char @ R @ X @ Y )
            = ( bot_bo6232401481379736579f_char @ Ra @ Ya @ Xa ) ) ) ) ) ).

% bot_term.rel_cong
thf(fact_234_bot__term_Orel__cong,axiom,
    ! [X: bot_bot_term_f,Ya: bot_bot_term_f,Y: bot_bot_term_f,Xa: bot_bot_term_f,R: f > f > $o,Ra: f > f > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z3: f,Yb: f] :
              ( ( member_f @ Z3 @ ( bot_bo658131349715431450term_f @ Ya ) )
             => ( ( member_f @ Yb @ ( bot_bo658131349715431450term_f @ Xa ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( bot_bo6868847914503738705rm_f_f @ R @ X @ Y )
            = ( bot_bo6868847914503738705rm_f_f @ Ra @ Ya @ Xa ) ) ) ) ) ).

% bot_term.rel_cong
thf(fact_235_bot__term_Orel__cong,axiom,
    ! [X: bot_bot_term_f,Ya: bot_bot_term_f,Y: bot_bot_term_nat,Xa: bot_bot_term_nat,R: f > nat > $o,Ra: f > nat > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z3: f,Yb: nat] :
              ( ( member_f @ Z3 @ ( bot_bo658131349715431450term_f @ Ya ) )
             => ( ( member_nat @ Yb @ ( bot_bo2410032770233908985rm_nat @ Xa ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( bot_bo6546836774426921474_f_nat @ R @ X @ Y )
            = ( bot_bo6546836774426921474_f_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).

% bot_term.rel_cong
thf(fact_236_bot__term_Orel__cong,axiom,
    ! [X: bot_bot_term_nat,Ya: bot_bot_term_nat,Y: bot_bot_term_char,Xa: bot_bot_term_char,R: nat > char > $o,Ra: nat > char > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z3: nat,Yb: char] :
              ( ( member_nat @ Z3 @ ( bot_bo2410032770233908985rm_nat @ Ya ) )
             => ( ( member_char @ Yb @ ( bot_bo3082694273163266554m_char @ Xa ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( bot_bo7254656895009639220t_char @ R @ X @ Y )
            = ( bot_bo7254656895009639220t_char @ Ra @ Ya @ Xa ) ) ) ) ) ).

% bot_term.rel_cong
thf(fact_237_bot__term_Orel__cong,axiom,
    ! [X: bot_bot_term_nat,Ya: bot_bot_term_nat,Y: bot_bot_term_f,Xa: bot_bot_term_f,R: nat > f > $o,Ra: nat > f > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z3: nat,Yb: f] :
              ( ( member_nat @ Z3 @ ( bot_bo2410032770233908985rm_nat @ Ya ) )
             => ( ( member_f @ Yb @ ( bot_bo658131349715431450term_f @ Xa ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( bot_bo4401546630823235424_nat_f @ R @ X @ Y )
            = ( bot_bo4401546630823235424_nat_f @ Ra @ Ya @ Xa ) ) ) ) ) ).

% bot_term.rel_cong
thf(fact_238_bot__term_Orel__cong,axiom,
    ! [X: bot_bot_term_char,Ya: bot_bot_term_char,Y: bot_bo5408180104339501355st_nat,Xa: bot_bo5408180104339501355st_nat,R: char > list_nat > $o,Ra: char > list_nat > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z3: char,Yb: list_nat] :
              ( ( member_char @ Z3 @ ( bot_bo3082694273163266554m_char @ Ya ) )
             => ( ( member_list_nat @ Yb @ ( bot_bo7758324655462117769st_nat @ Xa ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( bot_bo3035317541177340228st_nat @ R @ X @ Y )
            = ( bot_bo3035317541177340228st_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).

% bot_term.rel_cong
thf(fact_239_bot__term_Orel__mono__strong,axiom,
    ! [R: nat > nat > $o,X: bot_bot_term_nat,Y: bot_bot_term_nat,Ra: nat > nat > $o] :
      ( ( bot_bo1849039309831764531at_nat @ R @ X @ Y )
     => ( ! [Z3: nat,Yb: nat] :
            ( ( member_nat @ Z3 @ ( bot_bo2410032770233908985rm_nat @ X ) )
           => ( ( member_nat @ Yb @ ( bot_bo2410032770233908985rm_nat @ Y ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( bot_bo1849039309831764531at_nat @ Ra @ X @ Y ) ) ) ).

% bot_term.rel_mono_strong
thf(fact_240_bot__term_Orel__mono__strong,axiom,
    ! [R: char > char > $o,X: bot_bot_term_char,Y: bot_bot_term_char,Ra: char > char > $o] :
      ( ( bot_bo2694460441194989621r_char @ R @ X @ Y )
     => ( ! [Z3: char,Yb: char] :
            ( ( member_char @ Z3 @ ( bot_bo3082694273163266554m_char @ X ) )
           => ( ( member_char @ Yb @ ( bot_bo3082694273163266554m_char @ Y ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( bot_bo2694460441194989621r_char @ Ra @ X @ Y ) ) ) ).

% bot_term.rel_mono_strong
thf(fact_241_bot__term_Orel__mono__strong,axiom,
    ! [R: char > f > $o,X: bot_bot_term_char,Y: bot_bot_term_f,Ra: char > f > $o] :
      ( ( bot_bo3338816352050261919char_f @ R @ X @ Y )
     => ( ! [Z3: char,Yb: f] :
            ( ( member_char @ Z3 @ ( bot_bo3082694273163266554m_char @ X ) )
           => ( ( member_f @ Yb @ ( bot_bo658131349715431450term_f @ Y ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( bot_bo3338816352050261919char_f @ Ra @ X @ Y ) ) ) ).

% bot_term.rel_mono_strong
thf(fact_242_bot__term_Orel__mono__strong,axiom,
    ! [R: char > nat > $o,X: bot_bot_term_char,Y: bot_bot_term_nat,Ra: char > nat > $o] :
      ( ( bot_bo4339835670485063732ar_nat @ R @ X @ Y )
     => ( ! [Z3: char,Yb: nat] :
            ( ( member_char @ Z3 @ ( bot_bo3082694273163266554m_char @ X ) )
           => ( ( member_nat @ Yb @ ( bot_bo2410032770233908985rm_nat @ Y ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( bot_bo4339835670485063732ar_nat @ Ra @ X @ Y ) ) ) ).

% bot_term.rel_mono_strong
thf(fact_243_bot__term_Orel__mono__strong,axiom,
    ! [R: f > char > $o,X: bot_bot_term_f,Y: bot_bot_term_char,Ra: f > char > $o] :
      ( ( bot_bo6232401481379736579f_char @ R @ X @ Y )
     => ( ! [Z3: f,Yb: char] :
            ( ( member_f @ Z3 @ ( bot_bo658131349715431450term_f @ X ) )
           => ( ( member_char @ Yb @ ( bot_bo3082694273163266554m_char @ Y ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( bot_bo6232401481379736579f_char @ Ra @ X @ Y ) ) ) ).

% bot_term.rel_mono_strong
thf(fact_244_bot__term_Orel__mono__strong,axiom,
    ! [R: f > f > $o,X: bot_bot_term_f,Y: bot_bot_term_f,Ra: f > f > $o] :
      ( ( bot_bo6868847914503738705rm_f_f @ R @ X @ Y )
     => ( ! [Z3: f,Yb: f] :
            ( ( member_f @ Z3 @ ( bot_bo658131349715431450term_f @ X ) )
           => ( ( member_f @ Yb @ ( bot_bo658131349715431450term_f @ Y ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( bot_bo6868847914503738705rm_f_f @ Ra @ X @ Y ) ) ) ).

% bot_term.rel_mono_strong
thf(fact_245_bot__term_Orel__mono__strong,axiom,
    ! [R: f > nat > $o,X: bot_bot_term_f,Y: bot_bot_term_nat,Ra: f > nat > $o] :
      ( ( bot_bo6546836774426921474_f_nat @ R @ X @ Y )
     => ( ! [Z3: f,Yb: nat] :
            ( ( member_f @ Z3 @ ( bot_bo658131349715431450term_f @ X ) )
           => ( ( member_nat @ Yb @ ( bot_bo2410032770233908985rm_nat @ Y ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( bot_bo6546836774426921474_f_nat @ Ra @ X @ Y ) ) ) ).

% bot_term.rel_mono_strong
thf(fact_246_bot__term_Orel__mono__strong,axiom,
    ! [R: nat > char > $o,X: bot_bot_term_nat,Y: bot_bot_term_char,Ra: nat > char > $o] :
      ( ( bot_bo7254656895009639220t_char @ R @ X @ Y )
     => ( ! [Z3: nat,Yb: char] :
            ( ( member_nat @ Z3 @ ( bot_bo2410032770233908985rm_nat @ X ) )
           => ( ( member_char @ Yb @ ( bot_bo3082694273163266554m_char @ Y ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( bot_bo7254656895009639220t_char @ Ra @ X @ Y ) ) ) ).

% bot_term.rel_mono_strong
thf(fact_247_bot__term_Orel__mono__strong,axiom,
    ! [R: nat > f > $o,X: bot_bot_term_nat,Y: bot_bot_term_f,Ra: nat > f > $o] :
      ( ( bot_bo4401546630823235424_nat_f @ R @ X @ Y )
     => ( ! [Z3: nat,Yb: f] :
            ( ( member_nat @ Z3 @ ( bot_bo2410032770233908985rm_nat @ X ) )
           => ( ( member_f @ Yb @ ( bot_bo658131349715431450term_f @ Y ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( bot_bo4401546630823235424_nat_f @ Ra @ X @ Y ) ) ) ).

% bot_term.rel_mono_strong
thf(fact_248_bot__term_Orel__mono__strong,axiom,
    ! [R: char > list_nat > $o,X: bot_bot_term_char,Y: bot_bo5408180104339501355st_nat,Ra: char > list_nat > $o] :
      ( ( bot_bo3035317541177340228st_nat @ R @ X @ Y )
     => ( ! [Z3: char,Yb: list_nat] :
            ( ( member_char @ Z3 @ ( bot_bo3082694273163266554m_char @ X ) )
           => ( ( member_list_nat @ Yb @ ( bot_bo7758324655462117769st_nat @ Y ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( bot_bo3035317541177340228st_nat @ Ra @ X @ Y ) ) ) ).

% bot_term.rel_mono_strong
thf(fact_249_bot__term_Orel__refl__strong,axiom,
    ! [X: bot_bo5929273105904447532st_nat,Ra: produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o] :
      ( ! [Z3: produc1828647624359046049st_nat] :
          ( ( member7340969449405702474st_nat @ Z3 @ ( bot_bo4281033497674357304st_nat @ X ) )
         => ( Ra @ Z3 @ Z3 ) )
     => ( bot_bo6591283685554716113st_nat @ Ra @ X @ X ) ) ).

% bot_term.rel_refl_strong
thf(fact_250_bot__term_Orel__refl__strong,axiom,
    ! [X: bot_bo655077203622184332at_nat,Ra: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [Z3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ Z3 @ ( bot_bo6066252797605031576at_nat @ X ) )
         => ( Ra @ Z3 @ Z3 ) )
     => ( bot_bo1244605953094563793at_nat @ Ra @ X @ X ) ) ).

% bot_term.rel_refl_strong
thf(fact_251_bot__term_Orel__refl__strong,axiom,
    ! [X: bot_bot_term_char,Ra: char > char > $o] :
      ( ! [Z3: char] :
          ( ( member_char @ Z3 @ ( bot_bo3082694273163266554m_char @ X ) )
         => ( Ra @ Z3 @ Z3 ) )
     => ( bot_bo2694460441194989621r_char @ Ra @ X @ X ) ) ).

% bot_term.rel_refl_strong
thf(fact_252_bot__term_Orel__refl__strong,axiom,
    ! [X: bot_bo5408180104339501355st_nat,Ra: list_nat > list_nat > $o] :
      ( ! [Z3: list_nat] :
          ( ( member_list_nat @ Z3 @ ( bot_bo7758324655462117769st_nat @ X ) )
         => ( Ra @ Z3 @ Z3 ) )
     => ( bot_bo492821383238646227st_nat @ Ra @ X @ X ) ) ).

% bot_term.rel_refl_strong
thf(fact_253_bot__term_Orel__refl__strong,axiom,
    ! [X: bot_bot_term_f,Ra: f > f > $o] :
      ( ! [Z3: f] :
          ( ( member_f @ Z3 @ ( bot_bo658131349715431450term_f @ X ) )
         => ( Ra @ Z3 @ Z3 ) )
     => ( bot_bo6868847914503738705rm_f_f @ Ra @ X @ X ) ) ).

% bot_term.rel_refl_strong
thf(fact_254_bot__term_Orel__refl__strong,axiom,
    ! [X: bot_bot_term_nat,Ra: nat > nat > $o] :
      ( ! [Z3: nat] :
          ( ( member_nat @ Z3 @ ( bot_bo2410032770233908985rm_nat @ X ) )
         => ( Ra @ Z3 @ Z3 ) )
     => ( bot_bo1849039309831764531at_nat @ Ra @ X @ X ) ) ).

% bot_term.rel_refl_strong
thf(fact_255_curryE,axiom,
    ! [F2: produc3922691147685405600st_nat > $o,A2: product_prod_nat_nat,B: list_list_nat] :
      ( ( produc6878766516837139888_nat_o @ F2 @ A2 @ B )
     => ( F2 @ ( produc8649917717998259994st_nat @ A2 @ B ) ) ) ).

% curryE
thf(fact_256_curryE,axiom,
    ! [F2: produc1828647624359046049st_nat > $o,A2: list_nat,B: list_nat] :
      ( ( produc6139401228252118475_nat_o @ F2 @ A2 @ B )
     => ( F2 @ ( produc2694037385005941721st_nat @ A2 @ B ) ) ) ).

% curryE
thf(fact_257_curryE,axiom,
    ! [F2: produc7819656566062154093et_nat > $o,A2: set_nat,B: set_nat] :
      ( ( produc8481458855017059839_nat_o @ F2 @ A2 @ B )
     => ( F2 @ ( produc4532415448927165861et_nat @ A2 @ B ) ) ) ).

% curryE
thf(fact_258_curryE,axiom,
    ! [F2: produc5418599768131837375term_f > $o,A2: nat,B: bot_bot_term_f] :
      ( ( produc7017639609257223213rm_f_o @ F2 @ A2 @ B )
     => ( F2 @ ( produc5590691411411743351term_f @ A2 @ B ) ) ) ).

% curryE
thf(fact_259_curryE,axiom,
    ! [F2: product_prod_nat_nat > $o,A2: nat,B: nat] :
      ( ( produc1310100445399344235_nat_o @ F2 @ A2 @ B )
     => ( F2 @ ( product_Pair_nat_nat @ A2 @ B ) ) ) ).

% curryE
thf(fact_260_curryD,axiom,
    ! [F2: produc3922691147685405600st_nat > $o,A2: product_prod_nat_nat,B: list_list_nat] :
      ( ( produc6878766516837139888_nat_o @ F2 @ A2 @ B )
     => ( F2 @ ( produc8649917717998259994st_nat @ A2 @ B ) ) ) ).

% curryD
thf(fact_261_curryD,axiom,
    ! [F2: produc1828647624359046049st_nat > $o,A2: list_nat,B: list_nat] :
      ( ( produc6139401228252118475_nat_o @ F2 @ A2 @ B )
     => ( F2 @ ( produc2694037385005941721st_nat @ A2 @ B ) ) ) ).

% curryD
thf(fact_262_curryD,axiom,
    ! [F2: produc7819656566062154093et_nat > $o,A2: set_nat,B: set_nat] :
      ( ( produc8481458855017059839_nat_o @ F2 @ A2 @ B )
     => ( F2 @ ( produc4532415448927165861et_nat @ A2 @ B ) ) ) ).

% curryD
thf(fact_263_curryD,axiom,
    ! [F2: produc5418599768131837375term_f > $o,A2: nat,B: bot_bot_term_f] :
      ( ( produc7017639609257223213rm_f_o @ F2 @ A2 @ B )
     => ( F2 @ ( produc5590691411411743351term_f @ A2 @ B ) ) ) ).

% curryD
thf(fact_264_curryD,axiom,
    ! [F2: product_prod_nat_nat > $o,A2: nat,B: nat] :
      ( ( produc1310100445399344235_nat_o @ F2 @ A2 @ B )
     => ( F2 @ ( product_Pair_nat_nat @ A2 @ B ) ) ) ).

% curryD
thf(fact_265_bot__term_Orel__intros_I2_J,axiom,
    ! [R: f > nat > $o,X21: f,Y21: nat,X22: list_Bot_bot_term_f,Y22: list_B2641752247266983457rm_nat] :
      ( ( R @ X21 @ Y21 )
     => ( ( list_a7573497065701696196rm_nat @ ( bot_bo6546836774426921474_f_nat @ R ) @ X22 @ Y22 )
       => ( bot_bo6546836774426921474_f_nat @ R @ ( bot_bot_BFun_f @ X21 @ X22 ) @ ( bot_bot_BFun_nat @ Y21 @ Y22 ) ) ) ) ).

% bot_term.rel_intros(2)
thf(fact_266_bot__term_Orel__intros_I2_J,axiom,
    ! [R: nat > f > $o,X21: nat,Y21: f,X22: list_B2641752247266983457rm_nat,Y22: list_Bot_bot_term_f] :
      ( ( R @ X21 @ Y21 )
     => ( ( list_a2360236856457976674term_f @ ( bot_bo4401546630823235424_nat_f @ R ) @ X22 @ Y22 )
       => ( bot_bo4401546630823235424_nat_f @ R @ ( bot_bot_BFun_nat @ X21 @ X22 ) @ ( bot_bot_BFun_f @ Y21 @ Y22 ) ) ) ) ).

% bot_term.rel_intros(2)
thf(fact_267_bot__term_Orel__intros_I2_J,axiom,
    ! [R: nat > nat > $o,X21: nat,Y21: nat,X22: list_B2641752247266983457rm_nat,Y22: list_B2641752247266983457rm_nat] :
      ( ( R @ X21 @ Y21 )
     => ( ( list_a3175995857480936519rm_nat @ ( bot_bo1849039309831764531at_nat @ R ) @ X22 @ Y22 )
       => ( bot_bo1849039309831764531at_nat @ R @ ( bot_bot_BFun_nat @ X21 @ X22 ) @ ( bot_bot_BFun_nat @ Y21 @ Y22 ) ) ) ) ).

% bot_term.rel_intros(2)
thf(fact_268_bot__term_Orel__intros_I2_J,axiom,
    ! [R: f > f > $o,X21: f,Y21: f,X22: list_Bot_bot_term_f,Y22: list_Bot_bot_term_f] :
      ( ( R @ X21 @ Y21 )
     => ( ( list_a8926285605142776357term_f @ ( bot_bo6868847914503738705rm_f_f @ R ) @ X22 @ Y22 )
       => ( bot_bo6868847914503738705rm_f_f @ R @ ( bot_bot_BFun_f @ X21 @ X22 ) @ ( bot_bot_BFun_f @ Y21 @ Y22 ) ) ) ) ).

% bot_term.rel_intros(2)
thf(fact_269_bot__term_Orel__inject_I2_J,axiom,
    ! [R: f > nat > $o,X21: f,X22: list_Bot_bot_term_f,Y21: nat,Y22: list_B2641752247266983457rm_nat] :
      ( ( bot_bo6546836774426921474_f_nat @ R @ ( bot_bot_BFun_f @ X21 @ X22 ) @ ( bot_bot_BFun_nat @ Y21 @ Y22 ) )
      = ( ( R @ X21 @ Y21 )
        & ( list_a7573497065701696196rm_nat @ ( bot_bo6546836774426921474_f_nat @ R ) @ X22 @ Y22 ) ) ) ).

% bot_term.rel_inject(2)
thf(fact_270_bot__term_Orel__inject_I2_J,axiom,
    ! [R: nat > f > $o,X21: nat,X22: list_B2641752247266983457rm_nat,Y21: f,Y22: list_Bot_bot_term_f] :
      ( ( bot_bo4401546630823235424_nat_f @ R @ ( bot_bot_BFun_nat @ X21 @ X22 ) @ ( bot_bot_BFun_f @ Y21 @ Y22 ) )
      = ( ( R @ X21 @ Y21 )
        & ( list_a2360236856457976674term_f @ ( bot_bo4401546630823235424_nat_f @ R ) @ X22 @ Y22 ) ) ) ).

% bot_term.rel_inject(2)
thf(fact_271_bot__term_Orel__inject_I2_J,axiom,
    ! [R: nat > nat > $o,X21: nat,X22: list_B2641752247266983457rm_nat,Y21: nat,Y22: list_B2641752247266983457rm_nat] :
      ( ( bot_bo1849039309831764531at_nat @ R @ ( bot_bot_BFun_nat @ X21 @ X22 ) @ ( bot_bot_BFun_nat @ Y21 @ Y22 ) )
      = ( ( R @ X21 @ Y21 )
        & ( list_a3175995857480936519rm_nat @ ( bot_bo1849039309831764531at_nat @ R ) @ X22 @ Y22 ) ) ) ).

% bot_term.rel_inject(2)
thf(fact_272_bot__term_Orel__inject_I2_J,axiom,
    ! [R: f > f > $o,X21: f,X22: list_Bot_bot_term_f,Y21: f,Y22: list_Bot_bot_term_f] :
      ( ( bot_bo6868847914503738705rm_f_f @ R @ ( bot_bot_BFun_f @ X21 @ X22 ) @ ( bot_bot_BFun_f @ Y21 @ Y22 ) )
      = ( ( R @ X21 @ Y21 )
        & ( list_a8926285605142776357term_f @ ( bot_bo6868847914503738705rm_f_f @ R ) @ X22 @ Y22 ) ) ) ).

% bot_term.rel_inject(2)
thf(fact_273_length__replicate,axiom,
    ! [N: nat,X: produc5418599768131837375term_f] :
      ( ( size_s1844900889452899121term_f @ ( replic2745660767237686655term_f @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_274_length__replicate,axiom,
    ! [N: nat,X: produc8417799323881384255_f_nat] :
      ( ( size_s6959986635380143793_f_nat @ ( replic5744860322987233535_f_nat @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_275_length__replicate,axiom,
    ! [N: nat,X: produc250624644672406269term_f] :
      ( ( size_s9037951226665808879term_f @ ( replic5523370671267941053term_f @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_276_length__replicate,axiom,
    ! [N: nat,X: product_prod_nat_nat] :
      ( ( size_s5460976970255530739at_nat @ ( replic4235873036481779905at_nat @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_277_length__replicate,axiom,
    ! [N: nat,X: list_nat] :
      ( ( size_s3023201423986296836st_nat @ ( replicate_list_nat @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_278_length__replicate,axiom,
    ! [N: nat,X: bot_bot_term_f] :
      ( ( size_s1458857753309042354term_f @ ( replic6958083601664011982term_f @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_279_length__replicate,axiom,
    ! [N: nat,X: nat] :
      ( ( size_size_list_nat @ ( replicate_nat @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_280_replicate__length__same,axiom,
    ! [Xs2: list_char,X: char] :
      ( ! [X3: char] :
          ( ( member_char @ X3 @ ( set_char2 @ Xs2 ) )
         => ( X3 = X ) )
     => ( ( replicate_char @ ( size_size_list_char @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_281_replicate__length__same,axiom,
    ! [Xs2: list_P3405996666353486021term_f,X: produc5418599768131837375term_f] :
      ( ! [X3: produc5418599768131837375term_f] :
          ( ( member4779802479421434344term_f @ X3 @ ( set_Pr3428226866328055892term_f @ Xs2 ) )
         => ( X3 = X ) )
     => ( ( replic2745660767237686655term_f @ ( size_s1844900889452899121term_f @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_282_replicate__length__same,axiom,
    ! [Xs2: list_P8521082412280730693_f_nat,X: produc8417799323881384255_f_nat] :
      ( ! [X3: produc8417799323881384255_f_nat] :
          ( ( member7779002035170981224_f_nat @ X3 @ ( set_Pr6427426422077602772_f_nat @ Xs2 ) )
         => ( X3 = X ) )
     => ( ( replic5744860322987233535_f_nat @ ( size_s6959986635380143793_f_nat @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_283_replicate__length__same,axiom,
    ! [Xs2: list_P1780566427699531651term_f,X: produc250624644672406269term_f] :
      ( ! [X3: produc250624644672406269term_f] :
          ( ( member4687689947140789414term_f @ X3 @ ( set_Pr1368550810757897490term_f @ Xs2 ) )
         => ( X3 = X ) )
     => ( ( replic5523370671267941053term_f @ ( size_s9037951226665808879term_f @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_284_replicate__length__same,axiom,
    ! [Xs2: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
         => ( X3 = X ) )
     => ( ( replic4235873036481779905at_nat @ ( size_s5460976970255530739at_nat @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_285_replicate__length__same,axiom,
    ! [Xs2: list_list_nat,X: list_nat] :
      ( ! [X3: list_nat] :
          ( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs2 ) )
         => ( X3 = X ) )
     => ( ( replicate_list_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_286_replicate__length__same,axiom,
    ! [Xs2: list_Bot_bot_term_f,X: bot_bot_term_f] :
      ( ! [X3: bot_bot_term_f] :
          ( ( member2723211850833494437term_f @ X3 @ ( set_Bot_bot_term_f2 @ Xs2 ) )
         => ( X3 = X ) )
     => ( ( replic6958083601664011982term_f @ ( size_s1458857753309042354term_f @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_287_replicate__length__same,axiom,
    ! [Xs2: list_nat,X: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( X3 = X ) )
     => ( ( replicate_nat @ ( size_size_list_nat @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_288_replicate__eqI,axiom,
    ! [Xs2: list_P7940050157051400743st_nat,N: nat,X: produc1828647624359046049st_nat] :
      ( ( ( size_s7664791237847770771st_nat @ Xs2 )
        = N )
     => ( ! [Y3: produc1828647624359046049st_nat] :
            ( ( member7340969449405702474st_nat @ Y3 @ ( set_Pr3842133991353686454st_nat @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replic3948718270753518433st_nat @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_289_replicate__eqI,axiom,
    ! [Xs2: list_f,N: nat,X: f] :
      ( ( ( size_size_list_f @ Xs2 )
        = N )
     => ( ! [Y3: f] :
            ( ( member_f @ Y3 @ ( set_f2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_f @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_290_replicate__eqI,axiom,
    ! [Xs2: list_char,N: nat,X: char] :
      ( ( ( size_size_list_char @ Xs2 )
        = N )
     => ( ! [Y3: char] :
            ( ( member_char @ Y3 @ ( set_char2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_char @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_291_replicate__eqI,axiom,
    ! [Xs2: list_P3405996666353486021term_f,N: nat,X: produc5418599768131837375term_f] :
      ( ( ( size_s1844900889452899121term_f @ Xs2 )
        = N )
     => ( ! [Y3: produc5418599768131837375term_f] :
            ( ( member4779802479421434344term_f @ Y3 @ ( set_Pr3428226866328055892term_f @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replic2745660767237686655term_f @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_292_replicate__eqI,axiom,
    ! [Xs2: list_P8521082412280730693_f_nat,N: nat,X: produc8417799323881384255_f_nat] :
      ( ( ( size_s6959986635380143793_f_nat @ Xs2 )
        = N )
     => ( ! [Y3: produc8417799323881384255_f_nat] :
            ( ( member7779002035170981224_f_nat @ Y3 @ ( set_Pr6427426422077602772_f_nat @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replic5744860322987233535_f_nat @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_293_replicate__eqI,axiom,
    ! [Xs2: list_P1780566427699531651term_f,N: nat,X: produc250624644672406269term_f] :
      ( ( ( size_s9037951226665808879term_f @ Xs2 )
        = N )
     => ( ! [Y3: produc250624644672406269term_f] :
            ( ( member4687689947140789414term_f @ Y3 @ ( set_Pr1368550810757897490term_f @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replic5523370671267941053term_f @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_294_replicate__eqI,axiom,
    ! [Xs2: list_P6011104703257516679at_nat,N: nat,X: product_prod_nat_nat] :
      ( ( ( size_s5460976970255530739at_nat @ Xs2 )
        = N )
     => ( ! [Y3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ Y3 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replic4235873036481779905at_nat @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_295_replicate__eqI,axiom,
    ! [Xs2: list_list_nat,N: nat,X: list_nat] :
      ( ( ( size_s3023201423986296836st_nat @ Xs2 )
        = N )
     => ( ! [Y3: list_nat] :
            ( ( member_list_nat @ Y3 @ ( set_list_nat2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_list_nat @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_296_replicate__eqI,axiom,
    ! [Xs2: list_Bot_bot_term_f,N: nat,X: bot_bot_term_f] :
      ( ( ( size_s1458857753309042354term_f @ Xs2 )
        = N )
     => ( ! [Y3: bot_bot_term_f] :
            ( ( member2723211850833494437term_f @ Y3 @ ( set_Bot_bot_term_f2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replic6958083601664011982term_f @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_297_replicate__eqI,axiom,
    ! [Xs2: list_nat,N: nat,X: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = N )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ ( set_nat2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_nat @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_298_list__all2__in__set2,axiom,
    ! [P: nat > nat > $o,Xs2: list_nat,Ys: list_nat,Y: nat] :
      ( ( list_all2_nat_nat @ P @ Xs2 @ Ys )
     => ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
       => ~ ! [X3: nat] :
              ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
             => ~ ( P @ X3 @ Y ) ) ) ) ).

% list_all2_in_set2
thf(fact_299_list__all2__in__set2,axiom,
    ! [P: f > f > $o,Xs2: list_f,Ys: list_f,Y: f] :
      ( ( list_all2_f_f @ P @ Xs2 @ Ys )
     => ( ( member_f @ Y @ ( set_f2 @ Ys ) )
       => ~ ! [X3: f] :
              ( ( member_f @ X3 @ ( set_f2 @ Xs2 ) )
             => ~ ( P @ X3 @ Y ) ) ) ) ).

% list_all2_in_set2
thf(fact_300_list__all2__in__set2,axiom,
    ! [P: char > f > $o,Xs2: list_char,Ys: list_f,Y: f] :
      ( ( list_all2_char_f @ P @ Xs2 @ Ys )
     => ( ( member_f @ Y @ ( set_f2 @ Ys ) )
       => ~ ! [X3: char] :
              ( ( member_char @ X3 @ ( set_char2 @ Xs2 ) )
             => ~ ( P @ X3 @ Y ) ) ) ) ).

% list_all2_in_set2
thf(fact_301_list__all2__in__set2,axiom,
    ! [P: nat > f > $o,Xs2: list_nat,Ys: list_f,Y: f] :
      ( ( list_all2_nat_f @ P @ Xs2 @ Ys )
     => ( ( member_f @ Y @ ( set_f2 @ Ys ) )
       => ~ ! [X3: nat] :
              ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
             => ~ ( P @ X3 @ Y ) ) ) ) ).

% list_all2_in_set2
thf(fact_302_list__all2__in__set2,axiom,
    ! [P: f > char > $o,Xs2: list_f,Ys: list_char,Y: char] :
      ( ( list_all2_f_char @ P @ Xs2 @ Ys )
     => ( ( member_char @ Y @ ( set_char2 @ Ys ) )
       => ~ ! [X3: f] :
              ( ( member_f @ X3 @ ( set_f2 @ Xs2 ) )
             => ~ ( P @ X3 @ Y ) ) ) ) ).

% list_all2_in_set2
thf(fact_303_list__all2__in__set2,axiom,
    ! [P: char > char > $o,Xs2: list_char,Ys: list_char,Y: char] :
      ( ( list_all2_char_char @ P @ Xs2 @ Ys )
     => ( ( member_char @ Y @ ( set_char2 @ Ys ) )
       => ~ ! [X3: char] :
              ( ( member_char @ X3 @ ( set_char2 @ Xs2 ) )
             => ~ ( P @ X3 @ Y ) ) ) ) ).

% list_all2_in_set2
thf(fact_304_list__all2__in__set2,axiom,
    ! [P: nat > char > $o,Xs2: list_nat,Ys: list_char,Y: char] :
      ( ( list_all2_nat_char @ P @ Xs2 @ Ys )
     => ( ( member_char @ Y @ ( set_char2 @ Ys ) )
       => ~ ! [X3: nat] :
              ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
             => ~ ( P @ X3 @ Y ) ) ) ) ).

% list_all2_in_set2
thf(fact_305_list__all2__in__set2,axiom,
    ! [P: f > nat > $o,Xs2: list_f,Ys: list_nat,Y: nat] :
      ( ( list_all2_f_nat @ P @ Xs2 @ Ys )
     => ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
       => ~ ! [X3: f] :
              ( ( member_f @ X3 @ ( set_f2 @ Xs2 ) )
             => ~ ( P @ X3 @ Y ) ) ) ) ).

% list_all2_in_set2
thf(fact_306_list__all2__in__set2,axiom,
    ! [P: char > nat > $o,Xs2: list_char,Ys: list_nat,Y: nat] :
      ( ( list_all2_char_nat @ P @ Xs2 @ Ys )
     => ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
       => ~ ! [X3: char] :
              ( ( member_char @ X3 @ ( set_char2 @ Xs2 ) )
             => ~ ( P @ X3 @ Y ) ) ) ) ).

% list_all2_in_set2
thf(fact_307_list__all2__in__set2,axiom,
    ! [P: bot_bot_term_f > f > $o,Xs2: list_Bot_bot_term_f,Ys: list_f,Y: f] :
      ( ( list_a6572579317876404652rm_f_f @ P @ Xs2 @ Ys )
     => ( ( member_f @ Y @ ( set_f2 @ Ys ) )
       => ~ ! [X3: bot_bot_term_f] :
              ( ( member2723211850833494437term_f @ X3 @ ( set_Bot_bot_term_f2 @ Xs2 ) )
             => ~ ( P @ X3 @ Y ) ) ) ) ).

% list_all2_in_set2
thf(fact_308_list__all2__same,axiom,
    ! [P: product_prod_nat_nat > product_prod_nat_nat > $o,Xs2: list_P6011104703257516679at_nat] :
      ( ( list_a2393213337517202119at_nat @ P @ Xs2 @ Xs2 )
      = ( ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
           => ( P @ X2 @ X2 ) ) ) ) ).

% list_all2_same
thf(fact_309_list__all2__same,axiom,
    ! [P: char > char > $o,Xs2: list_char] :
      ( ( list_all2_char_char @ P @ Xs2 @ Xs2 )
      = ( ! [X2: char] :
            ( ( member_char @ X2 @ ( set_char2 @ Xs2 ) )
           => ( P @ X2 @ X2 ) ) ) ) ).

% list_all2_same
thf(fact_310_list__all2__same,axiom,
    ! [P: list_nat > list_nat > $o,Xs2: list_list_nat] :
      ( ( list_a4684442626689269705st_nat @ P @ Xs2 @ Xs2 )
      = ( ! [X2: list_nat] :
            ( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs2 ) )
           => ( P @ X2 @ X2 ) ) ) ) ).

% list_all2_same
thf(fact_311_list__all2__same,axiom,
    ! [P: bot_bot_term_f > bot_bot_term_f > $o,Xs2: list_Bot_bot_term_f] :
      ( ( list_a8926285605142776357term_f @ P @ Xs2 @ Xs2 )
      = ( ! [X2: bot_bot_term_f] :
            ( ( member2723211850833494437term_f @ X2 @ ( set_Bot_bot_term_f2 @ Xs2 ) )
           => ( P @ X2 @ X2 ) ) ) ) ).

% list_all2_same
thf(fact_312_list__all2__same,axiom,
    ! [P: nat > nat > $o,Xs2: list_nat] :
      ( ( list_all2_nat_nat @ P @ Xs2 @ Xs2 )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
           => ( P @ X2 @ X2 ) ) ) ) ).

% list_all2_same
thf(fact_313_list_Orel__refl__strong,axiom,
    ! [X: list_P7940050157051400743st_nat,Ra: produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o] :
      ( ! [Z3: produc1828647624359046049st_nat] :
          ( ( member7340969449405702474st_nat @ Z3 @ ( set_Pr3842133991353686454st_nat @ X ) )
         => ( Ra @ Z3 @ Z3 ) )
     => ( list_a3335923554785236167st_nat @ Ra @ X @ X ) ) ).

% list.rel_refl_strong
thf(fact_314_list_Orel__refl__strong,axiom,
    ! [X: list_f,Ra: f > f > $o] :
      ( ! [Z3: f] :
          ( ( member_f @ Z3 @ ( set_f2 @ X ) )
         => ( Ra @ Z3 @ Z3 ) )
     => ( list_all2_f_f @ Ra @ X @ X ) ) ).

% list.rel_refl_strong
thf(fact_315_list_Orel__refl__strong,axiom,
    ! [X: list_P6011104703257516679at_nat,Ra: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [Z3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ Z3 @ ( set_Pr5648618587558075414at_nat @ X ) )
         => ( Ra @ Z3 @ Z3 ) )
     => ( list_a2393213337517202119at_nat @ Ra @ X @ X ) ) ).

% list.rel_refl_strong
thf(fact_316_list_Orel__refl__strong,axiom,
    ! [X: list_char,Ra: char > char > $o] :
      ( ! [Z3: char] :
          ( ( member_char @ Z3 @ ( set_char2 @ X ) )
         => ( Ra @ Z3 @ Z3 ) )
     => ( list_all2_char_char @ Ra @ X @ X ) ) ).

% list.rel_refl_strong
thf(fact_317_list_Orel__refl__strong,axiom,
    ! [X: list_list_nat,Ra: list_nat > list_nat > $o] :
      ( ! [Z3: list_nat] :
          ( ( member_list_nat @ Z3 @ ( set_list_nat2 @ X ) )
         => ( Ra @ Z3 @ Z3 ) )
     => ( list_a4684442626689269705st_nat @ Ra @ X @ X ) ) ).

% list.rel_refl_strong
thf(fact_318_list_Orel__refl__strong,axiom,
    ! [X: list_Bot_bot_term_f,Ra: bot_bot_term_f > bot_bot_term_f > $o] :
      ( ! [Z3: bot_bot_term_f] :
          ( ( member2723211850833494437term_f @ Z3 @ ( set_Bot_bot_term_f2 @ X ) )
         => ( Ra @ Z3 @ Z3 ) )
     => ( list_a8926285605142776357term_f @ Ra @ X @ X ) ) ).

% list.rel_refl_strong
thf(fact_319_list_Orel__refl__strong,axiom,
    ! [X: list_nat,Ra: nat > nat > $o] :
      ( ! [Z3: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ X ) )
         => ( Ra @ Z3 @ Z3 ) )
     => ( list_all2_nat_nat @ Ra @ X @ X ) ) ).

% list.rel_refl_strong
thf(fact_320_list_Orel__mono__strong,axiom,
    ! [R: nat > nat > $o,X: list_nat,Y: list_nat,Ra: nat > nat > $o] :
      ( ( list_all2_nat_nat @ R @ X @ Y )
     => ( ! [Z3: nat,Yb: nat] :
            ( ( member_nat @ Z3 @ ( set_nat2 @ X ) )
           => ( ( member_nat @ Yb @ ( set_nat2 @ Y ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( list_all2_nat_nat @ Ra @ X @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_321_list_Orel__mono__strong,axiom,
    ! [R: f > f > $o,X: list_f,Y: list_f,Ra: f > f > $o] :
      ( ( list_all2_f_f @ R @ X @ Y )
     => ( ! [Z3: f,Yb: f] :
            ( ( member_f @ Z3 @ ( set_f2 @ X ) )
           => ( ( member_f @ Yb @ ( set_f2 @ Y ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( list_all2_f_f @ Ra @ X @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_322_list_Orel__mono__strong,axiom,
    ! [R: f > char > $o,X: list_f,Y: list_char,Ra: f > char > $o] :
      ( ( list_all2_f_char @ R @ X @ Y )
     => ( ! [Z3: f,Yb: char] :
            ( ( member_f @ Z3 @ ( set_f2 @ X ) )
           => ( ( member_char @ Yb @ ( set_char2 @ Y ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( list_all2_f_char @ Ra @ X @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_323_list_Orel__mono__strong,axiom,
    ! [R: f > nat > $o,X: list_f,Y: list_nat,Ra: f > nat > $o] :
      ( ( list_all2_f_nat @ R @ X @ Y )
     => ( ! [Z3: f,Yb: nat] :
            ( ( member_f @ Z3 @ ( set_f2 @ X ) )
           => ( ( member_nat @ Yb @ ( set_nat2 @ Y ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( list_all2_f_nat @ Ra @ X @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_324_list_Orel__mono__strong,axiom,
    ! [R: char > f > $o,X: list_char,Y: list_f,Ra: char > f > $o] :
      ( ( list_all2_char_f @ R @ X @ Y )
     => ( ! [Z3: char,Yb: f] :
            ( ( member_char @ Z3 @ ( set_char2 @ X ) )
           => ( ( member_f @ Yb @ ( set_f2 @ Y ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( list_all2_char_f @ Ra @ X @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_325_list_Orel__mono__strong,axiom,
    ! [R: char > char > $o,X: list_char,Y: list_char,Ra: char > char > $o] :
      ( ( list_all2_char_char @ R @ X @ Y )
     => ( ! [Z3: char,Yb: char] :
            ( ( member_char @ Z3 @ ( set_char2 @ X ) )
           => ( ( member_char @ Yb @ ( set_char2 @ Y ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( list_all2_char_char @ Ra @ X @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_326_list_Orel__mono__strong,axiom,
    ! [R: char > nat > $o,X: list_char,Y: list_nat,Ra: char > nat > $o] :
      ( ( list_all2_char_nat @ R @ X @ Y )
     => ( ! [Z3: char,Yb: nat] :
            ( ( member_char @ Z3 @ ( set_char2 @ X ) )
           => ( ( member_nat @ Yb @ ( set_nat2 @ Y ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( list_all2_char_nat @ Ra @ X @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_327_list_Orel__mono__strong,axiom,
    ! [R: nat > f > $o,X: list_nat,Y: list_f,Ra: nat > f > $o] :
      ( ( list_all2_nat_f @ R @ X @ Y )
     => ( ! [Z3: nat,Yb: f] :
            ( ( member_nat @ Z3 @ ( set_nat2 @ X ) )
           => ( ( member_f @ Yb @ ( set_f2 @ Y ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( list_all2_nat_f @ Ra @ X @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_328_list_Orel__mono__strong,axiom,
    ! [R: nat > char > $o,X: list_nat,Y: list_char,Ra: nat > char > $o] :
      ( ( list_all2_nat_char @ R @ X @ Y )
     => ( ! [Z3: nat,Yb: char] :
            ( ( member_nat @ Z3 @ ( set_nat2 @ X ) )
           => ( ( member_char @ Yb @ ( set_char2 @ Y ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( list_all2_nat_char @ Ra @ X @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_329_list_Orel__mono__strong,axiom,
    ! [R: f > bot_bot_term_f > $o,X: list_f,Y: list_Bot_bot_term_f,Ra: f > bot_bot_term_f > $o] :
      ( ( list_a8222785964074318922term_f @ R @ X @ Y )
     => ( ! [Z3: f,Yb: bot_bot_term_f] :
            ( ( member_f @ Z3 @ ( set_f2 @ X ) )
           => ( ( member2723211850833494437term_f @ Yb @ ( set_Bot_bot_term_f2 @ Y ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( list_a8222785964074318922term_f @ Ra @ X @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_330_list_Orel__cong,axiom,
    ! [X: list_nat,Ya: list_nat,Y: list_nat,Xa: list_nat,R: nat > nat > $o,Ra: nat > nat > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z3: nat,Yb: nat] :
              ( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
             => ( ( member_nat @ Yb @ ( set_nat2 @ Xa ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( list_all2_nat_nat @ R @ X @ Y )
            = ( list_all2_nat_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_331_list_Orel__cong,axiom,
    ! [X: list_f,Ya: list_f,Y: list_f,Xa: list_f,R: f > f > $o,Ra: f > f > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z3: f,Yb: f] :
              ( ( member_f @ Z3 @ ( set_f2 @ Ya ) )
             => ( ( member_f @ Yb @ ( set_f2 @ Xa ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( list_all2_f_f @ R @ X @ Y )
            = ( list_all2_f_f @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_332_list_Orel__cong,axiom,
    ! [X: list_f,Ya: list_f,Y: list_char,Xa: list_char,R: f > char > $o,Ra: f > char > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z3: f,Yb: char] :
              ( ( member_f @ Z3 @ ( set_f2 @ Ya ) )
             => ( ( member_char @ Yb @ ( set_char2 @ Xa ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( list_all2_f_char @ R @ X @ Y )
            = ( list_all2_f_char @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_333_list_Orel__cong,axiom,
    ! [X: list_f,Ya: list_f,Y: list_nat,Xa: list_nat,R: f > nat > $o,Ra: f > nat > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z3: f,Yb: nat] :
              ( ( member_f @ Z3 @ ( set_f2 @ Ya ) )
             => ( ( member_nat @ Yb @ ( set_nat2 @ Xa ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( list_all2_f_nat @ R @ X @ Y )
            = ( list_all2_f_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_334_list_Orel__cong,axiom,
    ! [X: list_char,Ya: list_char,Y: list_f,Xa: list_f,R: char > f > $o,Ra: char > f > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z3: char,Yb: f] :
              ( ( member_char @ Z3 @ ( set_char2 @ Ya ) )
             => ( ( member_f @ Yb @ ( set_f2 @ Xa ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( list_all2_char_f @ R @ X @ Y )
            = ( list_all2_char_f @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_335_list_Orel__cong,axiom,
    ! [X: list_char,Ya: list_char,Y: list_char,Xa: list_char,R: char > char > $o,Ra: char > char > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z3: char,Yb: char] :
              ( ( member_char @ Z3 @ ( set_char2 @ Ya ) )
             => ( ( member_char @ Yb @ ( set_char2 @ Xa ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( list_all2_char_char @ R @ X @ Y )
            = ( list_all2_char_char @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_336_list_Orel__cong,axiom,
    ! [X: list_char,Ya: list_char,Y: list_nat,Xa: list_nat,R: char > nat > $o,Ra: char > nat > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z3: char,Yb: nat] :
              ( ( member_char @ Z3 @ ( set_char2 @ Ya ) )
             => ( ( member_nat @ Yb @ ( set_nat2 @ Xa ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( list_all2_char_nat @ R @ X @ Y )
            = ( list_all2_char_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_337_list_Orel__cong,axiom,
    ! [X: list_nat,Ya: list_nat,Y: list_f,Xa: list_f,R: nat > f > $o,Ra: nat > f > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z3: nat,Yb: f] :
              ( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
             => ( ( member_f @ Yb @ ( set_f2 @ Xa ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( list_all2_nat_f @ R @ X @ Y )
            = ( list_all2_nat_f @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_338_list_Orel__cong,axiom,
    ! [X: list_nat,Ya: list_nat,Y: list_char,Xa: list_char,R: nat > char > $o,Ra: nat > char > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z3: nat,Yb: char] :
              ( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
             => ( ( member_char @ Yb @ ( set_char2 @ Xa ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( list_all2_nat_char @ R @ X @ Y )
            = ( list_all2_nat_char @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_339_list_Orel__cong,axiom,
    ! [X: list_f,Ya: list_f,Y: list_Bot_bot_term_f,Xa: list_Bot_bot_term_f,R: f > bot_bot_term_f > $o,Ra: f > bot_bot_term_f > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z3: f,Yb: bot_bot_term_f] :
              ( ( member_f @ Z3 @ ( set_f2 @ Ya ) )
             => ( ( member2723211850833494437term_f @ Yb @ ( set_Bot_bot_term_f2 @ Xa ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( list_a8222785964074318922term_f @ R @ X @ Y )
            = ( list_a8222785964074318922term_f @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_340_list__all2__lengthD,axiom,
    ! [P: nat > nat > $o,Xs2: list_nat,Ys: list_nat] :
      ( ( list_all2_nat_nat @ P @ Xs2 @ Ys )
     => ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) ) ) ).

% list_all2_lengthD
thf(fact_341_list__all2__lengthD,axiom,
    ! [P: nat > bot_bot_term_f > $o,Xs2: list_nat,Ys: list_Bot_bot_term_f] :
      ( ( list_a8962304667658831463term_f @ P @ Xs2 @ Ys )
     => ( ( size_size_list_nat @ Xs2 )
        = ( size_s1458857753309042354term_f @ Ys ) ) ) ).

% list_all2_lengthD
thf(fact_342_list__all2__lengthD,axiom,
    ! [P: nat > list_nat > $o,Xs2: list_nat,Ys: list_list_nat] :
      ( ( list_a5155878676884504761st_nat @ P @ Xs2 @ Ys )
     => ( ( size_size_list_nat @ Xs2 )
        = ( size_s3023201423986296836st_nat @ Ys ) ) ) ).

% list_all2_lengthD
thf(fact_343_list__all2__lengthD,axiom,
    ! [P: bot_bot_term_f > nat > $o,Xs2: list_Bot_bot_term_f,Ys: list_nat] :
      ( ( list_a1093197441716247783_f_nat @ P @ Xs2 @ Ys )
     => ( ( size_s1458857753309042354term_f @ Xs2 )
        = ( size_size_list_nat @ Ys ) ) ) ).

% list_all2_lengthD
thf(fact_344_list__all2__lengthD,axiom,
    ! [P: list_nat > nat > $o,Xs2: list_list_nat,Ys: list_nat] :
      ( ( list_a7394835724046789945at_nat @ P @ Xs2 @ Ys )
     => ( ( size_s3023201423986296836st_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) ) ) ).

% list_all2_lengthD
thf(fact_345_list__all2__lengthD,axiom,
    ! [P: nat > product_prod_nat_nat > $o,Xs2: list_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( list_a8436948092790885224at_nat @ P @ Xs2 @ Ys )
     => ( ( size_size_list_nat @ Xs2 )
        = ( size_s5460976970255530739at_nat @ Ys ) ) ) ).

% list_all2_lengthD
thf(fact_346_list__all2__lengthD,axiom,
    ! [P: bot_bot_term_f > bot_bot_term_f > $o,Xs2: list_Bot_bot_term_f,Ys: list_Bot_bot_term_f] :
      ( ( list_a8926285605142776357term_f @ P @ Xs2 @ Ys )
     => ( ( size_s1458857753309042354term_f @ Xs2 )
        = ( size_s1458857753309042354term_f @ Ys ) ) ) ).

% list_all2_lengthD
thf(fact_347_list__all2__lengthD,axiom,
    ! [P: bot_bot_term_f > list_nat > $o,Xs2: list_Bot_bot_term_f,Ys: list_list_nat] :
      ( ( list_a470444602825883767st_nat @ P @ Xs2 @ Ys )
     => ( ( size_s1458857753309042354term_f @ Xs2 )
        = ( size_s3023201423986296836st_nat @ Ys ) ) ) ).

% list_all2_lengthD
thf(fact_348_list__all2__lengthD,axiom,
    ! [P: product_prod_nat_nat > nat > $o,Xs2: list_P6011104703257516679at_nat,Ys: list_nat] :
      ( ( list_a5076900699748170122at_nat @ P @ Xs2 @ Ys )
     => ( ( size_s5460976970255530739at_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) ) ) ).

% list_all2_lengthD
thf(fact_349_list__all2__lengthD,axiom,
    ! [P: list_nat > bot_bot_term_f > $o,Xs2: list_list_nat,Ys: list_Bot_bot_term_f] :
      ( ( list_a6660390693657896311term_f @ P @ Xs2 @ Ys )
     => ( ( size_s3023201423986296836st_nat @ Xs2 )
        = ( size_s1458857753309042354term_f @ Ys ) ) ) ).

% list_all2_lengthD
thf(fact_350_bot__term_Oset__sel_I2_J,axiom,
    ! [A2: bot_bo5929273105904447532st_nat,X: bot_bo5929273105904447532st_nat,Xa: produc1828647624359046049st_nat] :
      ( ( A2 != bot_bo6406203247828191920st_nat )
     => ( ( member274174779753688259st_nat @ X @ ( set_Bo5869970306151722839st_nat @ ( bot_bo2187226529533958892st_nat @ A2 ) ) )
       => ( ( member7340969449405702474st_nat @ Xa @ ( bot_bo4281033497674357304st_nat @ X ) )
         => ( member7340969449405702474st_nat @ Xa @ ( bot_bo4281033497674357304st_nat @ A2 ) ) ) ) ) ).

% bot_term.set_sel(2)
thf(fact_351_bot__term_Oset__sel_I2_J,axiom,
    ! [A2: bot_bo655077203622184332at_nat,X: bot_bo655077203622184332at_nat,Xa: product_prod_nat_nat] :
      ( ( A2 != bot_bo5215575770921047824at_nat )
     => ( ( member2038346494265176355at_nat @ X @ ( set_Bo1863287301388052407at_nat @ ( bot_bo1125064044073681740at_nat @ A2 ) ) )
       => ( ( member8440522571783428010at_nat @ Xa @ ( bot_bo6066252797605031576at_nat @ X ) )
         => ( member8440522571783428010at_nat @ Xa @ ( bot_bo6066252797605031576at_nat @ A2 ) ) ) ) ) ).

% bot_term.set_sel(2)
thf(fact_352_bot__term_Oset__sel_I2_J,axiom,
    ! [A2: bot_bot_term_char,X: bot_bot_term_char,Xa: char] :
      ( ( A2 != bot_bot_Bot_char )
     => ( ( member6716525252696868293m_char @ X @ ( set_Bo5011433680641223729m_char @ ( bot_bot_args_char @ A2 ) ) )
       => ( ( member_char @ Xa @ ( bot_bo3082694273163266554m_char @ X ) )
         => ( member_char @ Xa @ ( bot_bo3082694273163266554m_char @ A2 ) ) ) ) ) ).

% bot_term.set_sel(2)
thf(fact_353_bot__term_Oset__sel_I2_J,axiom,
    ! [A2: bot_bo5408180104339501355st_nat,X: bot_bo5408180104339501355st_nat,Xa: list_nat] :
      ( ( A2 != bot_bot_Bot_list_nat )
     => ( ( member8381404166258140628st_nat @ X @ ( set_Bo3558769202554720832st_nat @ ( bot_bo2362305990503692245st_nat @ A2 ) ) )
       => ( ( member_list_nat @ Xa @ ( bot_bo7758324655462117769st_nat @ X ) )
         => ( member_list_nat @ Xa @ ( bot_bo7758324655462117769st_nat @ A2 ) ) ) ) ) ).

% bot_term.set_sel(2)
thf(fact_354_bot__term_Oset__sel_I2_J,axiom,
    ! [A2: bot_bot_term_nat,X: bot_bot_term_nat,Xa: nat] :
      ( ( A2 != bot_bot_Bot_nat )
     => ( ( member2055285333220477764rm_nat @ X @ ( set_Bot_bot_term_nat2 @ ( bot_bot_args_nat @ A2 ) ) )
       => ( ( member_nat @ Xa @ ( bot_bo2410032770233908985rm_nat @ X ) )
         => ( member_nat @ Xa @ ( bot_bo2410032770233908985rm_nat @ A2 ) ) ) ) ) ).

% bot_term.set_sel(2)
thf(fact_355_bot__term_Oset__sel_I2_J,axiom,
    ! [A2: bot_bot_term_f,X: bot_bot_term_f,Xa: f] :
      ( ( A2 != bot_bot_Bot_f )
     => ( ( member2723211850833494437term_f @ X @ ( set_Bot_bot_term_f2 @ ( bot_bot_args_f @ A2 ) ) )
       => ( ( member_f @ Xa @ ( bot_bo658131349715431450term_f @ X ) )
         => ( member_f @ Xa @ ( bot_bo658131349715431450term_f @ A2 ) ) ) ) ) ).

% bot_term.set_sel(2)
thf(fact_356_bot__term_Osel_I2_J,axiom,
    ! [X21: nat,X22: list_B2641752247266983457rm_nat] :
      ( ( bot_bot_args_nat @ ( bot_bot_BFun_nat @ X21 @ X22 ) )
      = X22 ) ).

% bot_term.sel(2)
thf(fact_357_bot__term_Osel_I2_J,axiom,
    ! [X21: f,X22: list_Bot_bot_term_f] :
      ( ( bot_bot_args_f @ ( bot_bot_BFun_f @ X21 @ X22 ) )
      = X22 ) ).

% bot_term.sel(2)
thf(fact_358_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_Bot_bot_term_f] :
      ( ( size_s1458857753309042354term_f @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_359_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_P6011104703257516679at_nat] :
      ( ( size_s5460976970255530739at_nat @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_360_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_list_nat] :
      ( ( size_s3023201423986296836st_nat @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_361_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_362_neq__if__length__neq,axiom,
    ! [Xs2: list_Bot_bot_term_f,Ys: list_Bot_bot_term_f] :
      ( ( ( size_s1458857753309042354term_f @ Xs2 )
       != ( size_s1458857753309042354term_f @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_363_neq__if__length__neq,axiom,
    ! [Xs2: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( ( size_s5460976970255530739at_nat @ Xs2 )
       != ( size_s5460976970255530739at_nat @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_364_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_365_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_366_list_Orel__eq,axiom,
    ( ( list_a8926285605142776357term_f
      @ ^ [Y2: bot_bot_term_f,Z: bot_bot_term_f] : ( Y2 = Z ) )
    = ( ^ [Y2: list_Bot_bot_term_f,Z: list_Bot_bot_term_f] : ( Y2 = Z ) ) ) ).

% list.rel_eq
thf(fact_367_list_Orel__eq,axiom,
    ( ( list_all2_nat_nat
      @ ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
    = ( ^ [Y2: list_nat,Z: list_nat] : ( Y2 = Z ) ) ) ).

% list.rel_eq
thf(fact_368_list_Orel__refl,axiom,
    ! [Ra: bot_bot_term_f > bot_bot_term_f > $o,X: list_Bot_bot_term_f] :
      ( ! [X3: bot_bot_term_f] : ( Ra @ X3 @ X3 )
     => ( list_a8926285605142776357term_f @ Ra @ X @ X ) ) ).

% list.rel_refl
thf(fact_369_list_Orel__refl,axiom,
    ! [Ra: nat > nat > $o,X: list_nat] :
      ( ! [X3: nat] : ( Ra @ X3 @ X3 )
     => ( list_all2_nat_nat @ Ra @ X @ X ) ) ).

% list.rel_refl
thf(fact_370_list__all2__eq,axiom,
    ( ( ^ [Y2: list_Bot_bot_term_f,Z: list_Bot_bot_term_f] : ( Y2 = Z ) )
    = ( list_a8926285605142776357term_f
      @ ^ [Y2: bot_bot_term_f,Z: bot_bot_term_f] : ( Y2 = Z ) ) ) ).

% list_all2_eq
thf(fact_371_list__all2__eq,axiom,
    ( ( ^ [Y2: list_nat,Z: list_nat] : ( Y2 = Z ) )
    = ( list_all2_nat_nat
      @ ^ [Y2: nat,Z: nat] : ( Y2 = Z ) ) ) ).

% list_all2_eq
thf(fact_372_list__all2__mono,axiom,
    ! [P: bot_bot_term_f > bot_bot_term_f > $o,Xs2: list_Bot_bot_term_f,Ys: list_Bot_bot_term_f,Q: bot_bot_term_f > bot_bot_term_f > $o] :
      ( ( list_a8926285605142776357term_f @ P @ Xs2 @ Ys )
     => ( ! [Xs3: bot_bot_term_f,Ys2: bot_bot_term_f] :
            ( ( P @ Xs3 @ Ys2 )
           => ( Q @ Xs3 @ Ys2 ) )
       => ( list_a8926285605142776357term_f @ Q @ Xs2 @ Ys ) ) ) ).

% list_all2_mono
thf(fact_373_list__all2__mono,axiom,
    ! [P: nat > nat > $o,Xs2: list_nat,Ys: list_nat,Q: nat > nat > $o] :
      ( ( list_all2_nat_nat @ P @ Xs2 @ Ys )
     => ( ! [Xs3: nat,Ys2: nat] :
            ( ( P @ Xs3 @ Ys2 )
           => ( Q @ Xs3 @ Ys2 ) )
       => ( list_all2_nat_nat @ Q @ Xs2 @ Ys ) ) ) ).

% list_all2_mono
thf(fact_374_list__all2__refl,axiom,
    ! [P: bot_bot_term_f > bot_bot_term_f > $o,Xs2: list_Bot_bot_term_f] :
      ( ! [X3: bot_bot_term_f] : ( P @ X3 @ X3 )
     => ( list_a8926285605142776357term_f @ P @ Xs2 @ Xs2 ) ) ).

% list_all2_refl
thf(fact_375_list__all2__refl,axiom,
    ! [P: nat > nat > $o,Xs2: list_nat] :
      ( ! [X3: nat] : ( P @ X3 @ X3 )
     => ( list_all2_nat_nat @ P @ Xs2 @ Xs2 ) ) ).

% list_all2_refl
thf(fact_376_list__all2__trans,axiom,
    ! [P1: bot_bot_term_f > bot_bot_term_f > $o,P22: bot_bot_term_f > bot_bot_term_f > $o,P3: bot_bot_term_f > bot_bot_term_f > $o,As: list_Bot_bot_term_f,Bs: list_Bot_bot_term_f,Cs: list_Bot_bot_term_f] :
      ( ! [A5: bot_bot_term_f,B3: bot_bot_term_f,C: bot_bot_term_f] :
          ( ( P1 @ A5 @ B3 )
         => ( ( P22 @ B3 @ C )
           => ( P3 @ A5 @ C ) ) )
     => ( ( list_a8926285605142776357term_f @ P1 @ As @ Bs )
       => ( ( list_a8926285605142776357term_f @ P22 @ Bs @ Cs )
         => ( list_a8926285605142776357term_f @ P3 @ As @ Cs ) ) ) ) ).

% list_all2_trans
thf(fact_377_list__all2__trans,axiom,
    ! [P1: nat > nat > $o,P22: nat > nat > $o,P3: nat > nat > $o,As: list_nat,Bs: list_nat,Cs: list_nat] :
      ( ! [A5: nat,B3: nat,C: nat] :
          ( ( P1 @ A5 @ B3 )
         => ( ( P22 @ B3 @ C )
           => ( P3 @ A5 @ C ) ) )
     => ( ( list_all2_nat_nat @ P1 @ As @ Bs )
       => ( ( list_all2_nat_nat @ P22 @ Bs @ Cs )
         => ( list_all2_nat_nat @ P3 @ As @ Cs ) ) ) ) ).

% list_all2_trans
thf(fact_378_list__all2__antisym,axiom,
    ! [P: bot_bot_term_f > bot_bot_term_f > $o,Q: bot_bot_term_f > bot_bot_term_f > $o,Xs2: list_Bot_bot_term_f,Ys: list_Bot_bot_term_f] :
      ( ! [X3: bot_bot_term_f,Y3: bot_bot_term_f] :
          ( ( P @ X3 @ Y3 )
         => ( ( Q @ Y3 @ X3 )
           => ( X3 = Y3 ) ) )
     => ( ( list_a8926285605142776357term_f @ P @ Xs2 @ Ys )
       => ( ( list_a8926285605142776357term_f @ Q @ Ys @ Xs2 )
         => ( Xs2 = Ys ) ) ) ) ).

% list_all2_antisym
thf(fact_379_list__all2__antisym,axiom,
    ! [P: nat > nat > $o,Q: nat > nat > $o,Xs2: list_nat,Ys: list_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( P @ X3 @ Y3 )
         => ( ( Q @ Y3 @ X3 )
           => ( X3 = Y3 ) ) )
     => ( ( list_all2_nat_nat @ P @ Xs2 @ Ys )
       => ( ( list_all2_nat_nat @ Q @ Ys @ Xs2 )
         => ( Xs2 = Ys ) ) ) ) ).

% list_all2_antisym
thf(fact_380_bot__term_Orel__sel,axiom,
    ( bot_bo6546836774426921474_f_nat
    = ( ^ [R4: f > nat > $o,A6: bot_bot_term_f,B4: bot_bot_term_nat] :
          ( ( ( A6 = bot_bot_Bot_f )
            = ( B4 = bot_bot_Bot_nat ) )
          & ( ( A6 != bot_bot_Bot_f )
           => ( ( B4 != bot_bot_Bot_nat )
             => ( ( R4 @ ( bot_bot_un_BFun1_f @ A6 ) @ ( bot_bot_un_BFun1_nat @ B4 ) )
                & ( list_a7573497065701696196rm_nat @ ( bot_bo6546836774426921474_f_nat @ R4 ) @ ( bot_bot_args_f @ A6 ) @ ( bot_bot_args_nat @ B4 ) ) ) ) ) ) ) ) ).

% bot_term.rel_sel
thf(fact_381_bot__term_Orel__sel,axiom,
    ( bot_bo4401546630823235424_nat_f
    = ( ^ [R4: nat > f > $o,A6: bot_bot_term_nat,B4: bot_bot_term_f] :
          ( ( ( A6 = bot_bot_Bot_nat )
            = ( B4 = bot_bot_Bot_f ) )
          & ( ( A6 != bot_bot_Bot_nat )
           => ( ( B4 != bot_bot_Bot_f )
             => ( ( R4 @ ( bot_bot_un_BFun1_nat @ A6 ) @ ( bot_bot_un_BFun1_f @ B4 ) )
                & ( list_a2360236856457976674term_f @ ( bot_bo4401546630823235424_nat_f @ R4 ) @ ( bot_bot_args_nat @ A6 ) @ ( bot_bot_args_f @ B4 ) ) ) ) ) ) ) ) ).

% bot_term.rel_sel
thf(fact_382_bot__term_Orel__sel,axiom,
    ( bot_bo1849039309831764531at_nat
    = ( ^ [R4: nat > nat > $o,A6: bot_bot_term_nat,B4: bot_bot_term_nat] :
          ( ( ( A6 = bot_bot_Bot_nat )
            = ( B4 = bot_bot_Bot_nat ) )
          & ( ( A6 != bot_bot_Bot_nat )
           => ( ( B4 != bot_bot_Bot_nat )
             => ( ( R4 @ ( bot_bot_un_BFun1_nat @ A6 ) @ ( bot_bot_un_BFun1_nat @ B4 ) )
                & ( list_a3175995857480936519rm_nat @ ( bot_bo1849039309831764531at_nat @ R4 ) @ ( bot_bot_args_nat @ A6 ) @ ( bot_bot_args_nat @ B4 ) ) ) ) ) ) ) ) ).

% bot_term.rel_sel
thf(fact_383_bot__term_Orel__sel,axiom,
    ( bot_bo6868847914503738705rm_f_f
    = ( ^ [R4: f > f > $o,A6: bot_bot_term_f,B4: bot_bot_term_f] :
          ( ( ( A6 = bot_bot_Bot_f )
            = ( B4 = bot_bot_Bot_f ) )
          & ( ( A6 != bot_bot_Bot_f )
           => ( ( B4 != bot_bot_Bot_f )
             => ( ( R4 @ ( bot_bot_un_BFun1_f @ A6 ) @ ( bot_bot_un_BFun1_f @ B4 ) )
                & ( list_a8926285605142776357term_f @ ( bot_bo6868847914503738705rm_f_f @ R4 ) @ ( bot_bot_args_f @ A6 ) @ ( bot_bot_args_f @ B4 ) ) ) ) ) ) ) ) ).

% bot_term.rel_sel
thf(fact_384_bot__term_Ocollapse,axiom,
    ! [Bot_term: bot_bot_term_nat] :
      ( ( Bot_term != bot_bot_Bot_nat )
     => ( ( bot_bot_BFun_nat @ ( bot_bot_un_BFun1_nat @ Bot_term ) @ ( bot_bot_args_nat @ Bot_term ) )
        = Bot_term ) ) ).

% bot_term.collapse
thf(fact_385_bot__term_Ocollapse,axiom,
    ! [Bot_term: bot_bot_term_f] :
      ( ( Bot_term != bot_bot_Bot_f )
     => ( ( bot_bot_BFun_f @ ( bot_bot_un_BFun1_f @ Bot_term ) @ ( bot_bot_args_f @ Bot_term ) )
        = Bot_term ) ) ).

% bot_term.collapse
thf(fact_386_remdups__sort_I2_J,axiom,
    ! [Xs2: list_P6011104703257516679at_nat] :
      ( ( set_Pr5648618587558075414at_nat @ ( missin1163383351516593964at_nat @ Xs2 ) )
      = ( set_Pr5648618587558075414at_nat @ Xs2 ) ) ).

% remdups_sort(2)
thf(fact_387_remdups__sort_I2_J,axiom,
    ! [Xs2: list_char] :
      ( ( set_char2 @ ( missin7692318653990180454t_char @ Xs2 ) )
      = ( set_char2 @ Xs2 ) ) ).

% remdups_sort(2)
thf(fact_388_remdups__sort_I2_J,axiom,
    ! [Xs2: list_list_nat] :
      ( ( set_list_nat2 @ ( missin2640887136080315381st_nat @ Xs2 ) )
      = ( set_list_nat2 @ Xs2 ) ) ).

% remdups_sort(2)
thf(fact_389_remdups__sort_I2_J,axiom,
    ! [Xs2: list_nat] :
      ( ( set_nat2 @ ( missin6101193410121742181rt_nat @ Xs2 ) )
      = ( set_nat2 @ Xs2 ) ) ).

% remdups_sort(2)
thf(fact_390_bot__term_Oexhaust__sel,axiom,
    ! [Bot_term: bot_bot_term_nat] :
      ( ( Bot_term != bot_bot_Bot_nat )
     => ( Bot_term
        = ( bot_bot_BFun_nat @ ( bot_bot_un_BFun1_nat @ Bot_term ) @ ( bot_bot_args_nat @ Bot_term ) ) ) ) ).

% bot_term.exhaust_sel
thf(fact_391_bot__term_Oexhaust__sel,axiom,
    ! [Bot_term: bot_bot_term_f] :
      ( ( Bot_term != bot_bot_Bot_f )
     => ( Bot_term
        = ( bot_bot_BFun_f @ ( bot_bot_un_BFun1_f @ Bot_term ) @ ( bot_bot_args_f @ Bot_term ) ) ) ) ).

% bot_term.exhaust_sel
thf(fact_392_bot__term_Oexpand,axiom,
    ! [Bot_term: bot_bot_term_nat,Bot_term2: bot_bot_term_nat] :
      ( ( ( Bot_term = bot_bot_Bot_nat )
        = ( Bot_term2 = bot_bot_Bot_nat ) )
     => ( ( ( Bot_term != bot_bot_Bot_nat )
         => ( ( Bot_term2 != bot_bot_Bot_nat )
           => ( ( ( bot_bot_un_BFun1_nat @ Bot_term )
                = ( bot_bot_un_BFun1_nat @ Bot_term2 ) )
              & ( ( bot_bot_args_nat @ Bot_term )
                = ( bot_bot_args_nat @ Bot_term2 ) ) ) ) )
       => ( Bot_term = Bot_term2 ) ) ) ).

% bot_term.expand
thf(fact_393_bot__term_Oexpand,axiom,
    ! [Bot_term: bot_bot_term_f,Bot_term2: bot_bot_term_f] :
      ( ( ( Bot_term = bot_bot_Bot_f )
        = ( Bot_term2 = bot_bot_Bot_f ) )
     => ( ( ( Bot_term != bot_bot_Bot_f )
         => ( ( Bot_term2 != bot_bot_Bot_f )
           => ( ( ( bot_bot_un_BFun1_f @ Bot_term )
                = ( bot_bot_un_BFun1_f @ Bot_term2 ) )
              & ( ( bot_bot_args_f @ Bot_term )
                = ( bot_bot_args_f @ Bot_term2 ) ) ) ) )
       => ( Bot_term = Bot_term2 ) ) ) ).

% bot_term.expand
thf(fact_394_set__zip__rightD,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,Xs2: list_Bot_bot_term_f,Ys: list_Bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( set_Pr1368550810757897490term_f @ ( zip_Bo7813815397675234645term_f @ Xs2 @ Ys ) ) )
     => ( member2723211850833494437term_f @ Y @ ( set_Bot_bot_term_f2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_395_set__zip__rightD,axiom,
    ! [X: bot_bot_term_f,Y: nat,Xs2: list_Bot_bot_term_f,Ys: list_nat] :
      ( ( member7779002035170981224_f_nat @ ( produc6944956222323935479_f_nat @ X @ Y ) @ ( set_Pr6427426422077602772_f_nat @ ( zip_Bo4866563780717559831_f_nat @ Xs2 @ Ys ) ) )
     => ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_396_set__zip__rightD,axiom,
    ! [X: product_prod_nat_nat,Y: list_list_nat,Xs2: list_P6011104703257516679at_nat,Ys: list_list_list_nat] :
      ( ( member5740615348714444471st_nat @ ( produc8649917717998259994st_nat @ X @ Y ) @ ( set_Pr87060279690929995st_nat @ ( zip_Pr8472585369470713850st_nat @ Xs2 @ Ys ) ) )
     => ( member_list_list_nat @ Y @ ( set_list_list_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_397_set__zip__rightD,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 @ Y @ ( set_list_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_398_set__zip__rightD,axiom,
    ! [X: set_nat,Y: set_nat,Xs2: list_set_nat,Ys: list_set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X @ Y ) @ ( set_Pr9040384385603167362et_nat @ ( zip_set_nat_set_nat @ Xs2 @ Ys ) ) )
     => ( member_set_nat @ Y @ ( set_set_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_399_set__zip__rightD,axiom,
    ! [X: nat,Y: bot_bot_term_f,Xs2: list_nat,Ys: list_Bot_bot_term_f] :
      ( ( member4779802479421434344term_f @ ( produc5590691411411743351term_f @ X @ Y ) @ ( set_Pr3428226866328055892term_f @ ( zip_na3512298969805367703term_f @ Xs2 @ Ys ) ) )
     => ( member2723211850833494437term_f @ Y @ ( set_Bot_bot_term_f2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_400_set__zip__rightD,axiom,
    ! [X: nat,Y: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs2 @ Ys ) ) )
     => ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_401_set__zip__leftD,axiom,
    ! [X: bot_bot_term_f,Y: nat,Xs2: list_Bot_bot_term_f,Ys: list_nat] :
      ( ( member7779002035170981224_f_nat @ ( produc6944956222323935479_f_nat @ X @ Y ) @ ( set_Pr6427426422077602772_f_nat @ ( zip_Bo4866563780717559831_f_nat @ Xs2 @ Ys ) ) )
     => ( member2723211850833494437term_f @ X @ ( set_Bot_bot_term_f2 @ Xs2 ) ) ) ).

% set_zip_leftD
thf(fact_402_set__zip__leftD,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,Xs2: list_Bot_bot_term_f,Ys: list_Bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( set_Pr1368550810757897490term_f @ ( zip_Bo7813815397675234645term_f @ Xs2 @ Ys ) ) )
     => ( member2723211850833494437term_f @ X @ ( set_Bot_bot_term_f2 @ Xs2 ) ) ) ).

% set_zip_leftD
thf(fact_403_set__zip__leftD,axiom,
    ! [X: product_prod_nat_nat,Y: list_list_nat,Xs2: list_P6011104703257516679at_nat,Ys: list_list_list_nat] :
      ( ( member5740615348714444471st_nat @ ( produc8649917717998259994st_nat @ X @ Y ) @ ( set_Pr87060279690929995st_nat @ ( zip_Pr8472585369470713850st_nat @ Xs2 @ Ys ) ) )
     => ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) ) ) ).

% set_zip_leftD
thf(fact_404_set__zip__leftD,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 ) ) ) ).

% set_zip_leftD
thf(fact_405_set__zip__leftD,axiom,
    ! [X: set_nat,Y: set_nat,Xs2: list_set_nat,Ys: list_set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X @ Y ) @ ( set_Pr9040384385603167362et_nat @ ( zip_set_nat_set_nat @ Xs2 @ Ys ) ) )
     => ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) ) ) ).

% set_zip_leftD
thf(fact_406_set__zip__leftD,axiom,
    ! [X: nat,Y: bot_bot_term_f,Xs2: list_nat,Ys: list_Bot_bot_term_f] :
      ( ( member4779802479421434344term_f @ ( produc5590691411411743351term_f @ X @ Y ) @ ( set_Pr3428226866328055892term_f @ ( zip_na3512298969805367703term_f @ Xs2 @ Ys ) ) )
     => ( member_nat @ X @ ( set_nat2 @ Xs2 ) ) ) ).

% set_zip_leftD
thf(fact_407_set__zip__leftD,axiom,
    ! [X: nat,Y: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs2 @ Ys ) ) )
     => ( member_nat @ X @ ( set_nat2 @ Xs2 ) ) ) ).

% set_zip_leftD
thf(fact_408_in__set__zipE,axiom,
    ! [X: nat,Y: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs2 @ Ys ) ) )
     => ~ ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
         => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_409_in__set__zipE,axiom,
    ! [X: f,Y: f,Xs2: list_f,Ys: list_f] :
      ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X @ Y ) @ ( set_Product_prod_f_f2 @ ( zip_f_f @ Xs2 @ Ys ) ) )
     => ~ ( ( member_f @ X @ ( set_f2 @ Xs2 ) )
         => ~ ( member_f @ Y @ ( set_f2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_410_in__set__zipE,axiom,
    ! [X: f,Y: char,Xs2: list_f,Ys: list_char] :
      ( ( member7457913314979426714f_char @ ( product_Pair_f_char @ X @ Y ) @ ( set_Pr6552472979441277998f_char @ ( zip_f_char @ Xs2 @ Ys ) ) )
     => ~ ( ( member_f @ X @ ( set_f2 @ Xs2 ) )
         => ~ ( member_char @ Y @ ( set_char2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_411_in__set__zipE,axiom,
    ! [X: f,Y: nat,Xs2: list_f,Ys: list_nat] :
      ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X @ Y ) @ ( set_Pr7102205019285007021_f_nat @ ( zip_f_nat @ Xs2 @ Ys ) ) )
     => ~ ( ( member_f @ X @ ( set_f2 @ Xs2 ) )
         => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_412_in__set__zipE,axiom,
    ! [X: char,Y: f,Xs2: list_char,Ys: list_f] :
      ( ( member6477383651500708606char_f @ ( product_Pair_char_f @ X @ Y ) @ ( set_Pr5571943315962559890char_f @ ( zip_char_f @ Xs2 @ Ys ) ) )
     => ~ ( ( member_char @ X @ ( set_char2 @ Xs2 ) )
         => ~ ( member_f @ Y @ ( set_f2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_413_in__set__zipE,axiom,
    ! [X: char,Y: char,Xs2: list_char,Ys: list_char] :
      ( ( member3116729636814715180r_char @ ( produc2420258056075623483r_char @ X @ Y ) @ ( set_Pr7517466709966071192r_char @ ( zip_char_char @ Xs2 @ Ys ) ) )
     => ~ ( ( member_char @ X @ ( set_char2 @ Xs2 ) )
         => ~ ( member_char @ Y @ ( set_char2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_414_in__set__zipE,axiom,
    ! [X: char,Y: nat,Xs2: list_char,Ys: list_nat] :
      ( ( member125923338796336555ar_nat @ ( produc5330303422999155258ar_nat @ X @ Y ) @ ( set_Pr7499543848919965207ar_nat @ ( zip_char_nat @ Xs2 @ Ys ) ) )
     => ~ ( ( member_char @ X @ ( set_char2 @ Xs2 ) )
         => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_415_in__set__zipE,axiom,
    ! [X: nat,Y: f,Xs2: list_nat,Ys: list_f] :
      ( ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ X @ Y ) @ ( set_Pr4163146859742855507_nat_f @ ( zip_nat_f @ Xs2 @ Ys ) ) )
     => ~ ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
         => ~ ( member_f @ Y @ ( set_f2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_416_in__set__zipE,axiom,
    ! [X: nat,Y: char,Xs2: list_nat,Ys: list_char] :
      ( ( member333921081531690027t_char @ ( produc8245124647523730746t_char @ X @ Y ) @ ( set_Pr7707541591655318679t_char @ ( zip_nat_char @ Xs2 @ Ys ) ) )
     => ~ ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
         => ~ ( member_char @ Y @ ( set_char2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_417_in__set__zipE,axiom,
    ! [X: f,Y: bot_bot_term_f,Xs2: list_f,Ys: list_Bot_bot_term_f] :
      ( ( member3617594643769203927term_f @ ( produc4023706287990223930term_f @ X @ Y ) @ ( set_Pr5252852205727177067term_f @ ( zip_f_Bot_bot_term_f @ Xs2 @ Ys ) ) )
     => ~ ( ( member_f @ X @ ( set_f2 @ Xs2 ) )
         => ~ ( member2723211850833494437term_f @ Y @ ( set_Bot_bot_term_f2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_418_zip__same,axiom,
    ! [A2: produc1828647624359046049st_nat,B: produc1828647624359046049st_nat,Xs2: list_P7940050157051400743st_nat] :
      ( ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ A2 @ B ) @ ( set_Pr7183939682275281404st_nat @ ( zip_Pr4899438369145407479st_nat @ Xs2 @ Xs2 ) ) )
      = ( ( member7340969449405702474st_nat @ A2 @ ( set_Pr3842133991353686454st_nat @ Xs2 ) )
        & ( A2 = B ) ) ) ).

% zip_same
thf(fact_419_zip__same,axiom,
    ! [A2: f,B: f,Xs2: list_f] :
      ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A2 @ B ) @ ( set_Product_prod_f_f2 @ ( zip_f_f @ Xs2 @ Xs2 ) ) )
      = ( ( member_f @ A2 @ ( set_f2 @ Xs2 ) )
        & ( A2 = B ) ) ) ).

% zip_same
thf(fact_420_zip__same,axiom,
    ! [A2: bot_bot_term_f,B: bot_bot_term_f,Xs2: list_Bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A2 @ B ) @ ( set_Pr1368550810757897490term_f @ ( zip_Bo7813815397675234645term_f @ Xs2 @ Xs2 ) ) )
      = ( ( member2723211850833494437term_f @ A2 @ ( set_Bot_bot_term_f2 @ Xs2 ) )
        & ( A2 = B ) ) ) ).

% zip_same
thf(fact_421_zip__same,axiom,
    ! [A2: product_prod_nat_nat,B: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A2 @ B ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs2 @ Xs2 ) ) )
      = ( ( member8440522571783428010at_nat @ A2 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
        & ( A2 = B ) ) ) ).

% zip_same
thf(fact_422_zip__same,axiom,
    ! [A2: char,B: char,Xs2: list_char] :
      ( ( member3116729636814715180r_char @ ( produc2420258056075623483r_char @ A2 @ B ) @ ( set_Pr7517466709966071192r_char @ ( zip_char_char @ Xs2 @ Xs2 ) ) )
      = ( ( member_char @ A2 @ ( set_char2 @ Xs2 ) )
        & ( A2 = B ) ) ) ).

% zip_same
thf(fact_423_zip__same,axiom,
    ! [A2: list_nat,B: list_nat,Xs2: list_list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A2 @ B ) @ ( set_Pr3842133991353686454st_nat @ ( zip_li7157463729305086713st_nat @ Xs2 @ Xs2 ) ) )
      = ( ( member_list_nat @ A2 @ ( set_list_nat2 @ Xs2 ) )
        & ( A2 = B ) ) ) ).

% zip_same
thf(fact_424_zip__same,axiom,
    ! [A2: set_nat,B: set_nat,Xs2: list_set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A2 @ B ) @ ( set_Pr9040384385603167362et_nat @ ( zip_set_nat_set_nat @ Xs2 @ Xs2 ) ) )
      = ( ( member_set_nat @ A2 @ ( set_set_nat2 @ Xs2 ) )
        & ( A2 = B ) ) ) ).

% zip_same
thf(fact_425_zip__same,axiom,
    ! [A2: nat,B: nat,Xs2: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs2 @ Xs2 ) ) )
      = ( ( member_nat @ A2 @ ( set_nat2 @ Xs2 ) )
        & ( A2 = B ) ) ) ).

% zip_same
thf(fact_426_bot__term_Osel_I1_J,axiom,
    ! [X21: nat,X22: list_B2641752247266983457rm_nat] :
      ( ( bot_bot_un_BFun1_nat @ ( bot_bot_BFun_nat @ X21 @ X22 ) )
      = X21 ) ).

% bot_term.sel(1)
thf(fact_427_bot__term_Osel_I1_J,axiom,
    ! [X21: f,X22: list_Bot_bot_term_f] :
      ( ( bot_bot_un_BFun1_f @ ( bot_bot_BFun_f @ X21 @ X22 ) )
      = X21 ) ).

% bot_term.sel(1)
thf(fact_428_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_nat,Ys: list_nat,Y: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
       => ~ ! [X3: nat] :
              ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_429_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_nat,Ys: list_f,Y: f] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_f @ Ys ) )
     => ( ( member_f @ Y @ ( set_f2 @ Ys ) )
       => ~ ! [X3: nat] :
              ~ ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ X3 @ Y ) @ ( set_Pr4163146859742855507_nat_f @ ( zip_nat_f @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_430_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_nat,Ys: list_char,Y: char] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_char @ Ys ) )
     => ( ( member_char @ Y @ ( set_char2 @ Ys ) )
       => ~ ! [X3: nat] :
              ~ ( member333921081531690027t_char @ ( produc8245124647523730746t_char @ X3 @ Y ) @ ( set_Pr7707541591655318679t_char @ ( zip_nat_char @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_431_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_nat,Ys: list_Bot_bot_term_f,Y: bot_bot_term_f] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_s1458857753309042354term_f @ Ys ) )
     => ( ( member2723211850833494437term_f @ Y @ ( set_Bot_bot_term_f2 @ Ys ) )
       => ~ ! [X3: nat] :
              ~ ( member4779802479421434344term_f @ ( produc5590691411411743351term_f @ X3 @ Y ) @ ( set_Pr3428226866328055892term_f @ ( zip_na3512298969805367703term_f @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_432_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_nat,Ys: list_list_nat,Y: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ( member_list_nat @ Y @ ( set_list_nat2 @ Ys ) )
       => ~ ! [X3: nat] :
              ~ ( member1631564025489475386st_nat @ ( produc8282810413953273033st_nat @ X3 @ Y ) @ ( set_Pr1526418413117253030st_nat @ ( zip_nat_list_nat @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_433_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_Bot_bot_term_f,Ys: list_f,Y: f] :
      ( ( ( size_s1458857753309042354term_f @ Xs2 )
        = ( size_size_list_f @ Ys ) )
     => ( ( member_f @ Y @ ( set_f2 @ Ys ) )
       => ~ ! [X3: bot_bot_term_f] :
              ~ ( member2818554726470178945rm_f_f @ ( produc2373499641792309660rm_f_f @ X3 @ Y ) @ ( set_Pr4453812288428152085rm_f_f @ ( zip_Bot_bot_term_f_f @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_434_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_Bot_bot_term_f,Ys: list_char,Y: char] :
      ( ( ( size_s1458857753309042354term_f @ Xs2 )
        = ( size_size_list_char @ Ys ) )
     => ( ( member_char @ Y @ ( set_char2 @ Ys ) )
       => ~ ! [X3: bot_bot_term_f] :
              ~ ( member1593658988870650857f_char @ ( produc900452663052621304f_char @ X3 @ Y ) @ ( set_Pr6267679966350894677f_char @ ( zip_Bo7523927990824414488f_char @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_435_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_Bot_bot_term_f,Ys: list_nat,Y: nat] :
      ( ( ( size_s1458857753309042354term_f @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
       => ~ ! [X3: bot_bot_term_f] :
              ~ ( member7779002035170981224_f_nat @ ( produc6944956222323935479_f_nat @ X3 @ Y ) @ ( set_Pr6427426422077602772_f_nat @ ( zip_Bo4866563780717559831_f_nat @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_436_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_list_nat,Ys: list_f,Y: f] :
      ( ( ( size_s3023201423986296836st_nat @ Xs2 )
        = ( size_size_list_f @ Ys ) )
     => ( ( member_f @ Y @ ( set_f2 @ Ys ) )
       => ~ ! [X3: list_nat] :
              ~ ( member6908590394731873455_nat_f @ ( produc1615968307924265418_nat_f @ X3 @ Y ) @ ( set_Pr7185350880610756931_nat_f @ ( zip_list_nat_f @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_437_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_list_nat,Ys: list_char,Y: char] :
      ( ( ( size_s3023201423986296836st_nat @ Xs2 )
        = ( size_size_list_char @ Ys ) )
     => ( ( member_char @ Y @ ( set_char2 @ Ys ) )
       => ~ ! [X3: list_nat] :
              ~ ( member2213537244704226875t_char @ ( produc6989213019823764554t_char @ X3 @ Y ) @ ( set_Pr6151334496301494439t_char @ ( zip_list_nat_char @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_438_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_nat,Ys: list_nat,X: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
       => ~ ! [Y3: nat] :
              ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_439_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_f,Ys: list_nat,X: f] :
      ( ( ( size_size_list_f @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_f @ X @ ( set_f2 @ Xs2 ) )
       => ~ ! [Y3: nat] :
              ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X @ Y3 ) @ ( set_Pr7102205019285007021_f_nat @ ( zip_f_nat @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_440_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_char,Ys: list_nat,X: char] :
      ( ( ( size_size_list_char @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_char @ X @ ( set_char2 @ Xs2 ) )
       => ~ ! [Y3: nat] :
              ~ ( member125923338796336555ar_nat @ ( produc5330303422999155258ar_nat @ X @ Y3 ) @ ( set_Pr7499543848919965207ar_nat @ ( zip_char_nat @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_441_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_f,Ys: list_Bot_bot_term_f,X: f] :
      ( ( ( size_size_list_f @ Xs2 )
        = ( size_s1458857753309042354term_f @ Ys ) )
     => ( ( member_f @ X @ ( set_f2 @ Xs2 ) )
       => ~ ! [Y3: bot_bot_term_f] :
              ~ ( member3617594643769203927term_f @ ( produc4023706287990223930term_f @ X @ Y3 ) @ ( set_Pr5252852205727177067term_f @ ( zip_f_Bot_bot_term_f @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_442_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_char,Ys: list_Bot_bot_term_f,X: char] :
      ( ( ( size_size_list_char @ Xs2 )
        = ( size_s1458857753309042354term_f @ Ys ) )
     => ( ( member_char @ X @ ( set_char2 @ Xs2 ) )
       => ~ ! [Y3: bot_bot_term_f] :
              ~ ( member6558432925066666345term_f @ ( produc4803337862296769528term_f @ X @ Y3 ) @ ( set_Pr2009081865692134357term_f @ ( zip_ch2203441153213786904term_f @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_443_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_f,Ys: list_list_nat,X: f] :
      ( ( ( size_size_list_f @ Xs2 )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ( member_f @ X @ ( set_f2 @ Xs2 ) )
       => ~ ! [Y3: list_nat] :
              ~ ( member6220612387905001513st_nat @ ( produc4084253757333640332st_nat @ X @ Y3 ) @ ( set_Pr6497372873783884989st_nat @ ( zip_f_list_nat @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_444_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_char,Ys: list_list_nat,X: char] :
      ( ( ( size_size_list_char @ Xs2 )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ( member_char @ X @ ( set_char2 @ Xs2 ) )
       => ~ ! [Y3: list_nat] :
              ~ ( member5022628808406433979st_nat @ ( produc1818140537056686154st_nat @ X @ Y3 ) @ ( set_Pr8960426060003701543st_nat @ ( zip_char_list_nat @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_445_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_nat,Ys: list_Bot_bot_term_f,X: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_s1458857753309042354term_f @ Ys ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
       => ~ ! [Y3: bot_bot_term_f] :
              ~ ( member4779802479421434344term_f @ ( produc5590691411411743351term_f @ X @ Y3 ) @ ( set_Pr3428226866328055892term_f @ ( zip_na3512298969805367703term_f @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_446_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_nat,Ys: list_list_nat,X: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
       => ~ ! [Y3: list_nat] :
              ~ ( member1631564025489475386st_nat @ ( produc8282810413953273033st_nat @ X @ Y3 ) @ ( set_Pr1526418413117253030st_nat @ ( zip_nat_list_nat @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_447_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_Bot_bot_term_f,Ys: list_nat,X: bot_bot_term_f] :
      ( ( ( size_s1458857753309042354term_f @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member2723211850833494437term_f @ X @ ( set_Bot_bot_term_f2 @ Xs2 ) )
       => ~ ! [Y3: nat] :
              ~ ( member7779002035170981224_f_nat @ ( produc6944956222323935479_f_nat @ X @ Y3 ) @ ( set_Pr6427426422077602772_f_nat @ ( zip_Bo4866563780717559831_f_nat @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_448_bot__term_Oset__sel_I1_J,axiom,
    ! [A2: bot_bo5929273105904447532st_nat] :
      ( ( A2 != bot_bo6406203247828191920st_nat )
     => ( member7340969449405702474st_nat @ ( bot_bo8841542196867011516st_nat @ A2 ) @ ( bot_bo4281033497674357304st_nat @ A2 ) ) ) ).

% bot_term.set_sel(1)
thf(fact_449_bot__term_Oset__sel_I1_J,axiom,
    ! [A2: bot_bo655077203622184332at_nat] :
      ( ( A2 != bot_bo5215575770921047824at_nat )
     => ( member8440522571783428010at_nat @ ( bot_bo673623650730560540at_nat @ A2 ) @ ( bot_bo6066252797605031576at_nat @ A2 ) ) ) ).

% bot_term.set_sel(1)
thf(fact_450_bot__term_Oset__sel_I1_J,axiom,
    ! [A2: bot_bot_term_char] :
      ( ( A2 != bot_bot_Bot_char )
     => ( member_char @ ( bot_bo12013370452354758301_char @ A2 ) @ ( bot_bo3082694273163266554m_char @ A2 ) ) ) ).

% bot_term.set_sel(1)
thf(fact_451_bot__term_Oset__sel_I1_J,axiom,
    ! [A2: bot_bo5408180104339501355st_nat] :
      ( ( A2 != bot_bot_Bot_list_nat )
     => ( member_list_nat @ ( bot_bo1689184820244159237st_nat @ A2 ) @ ( bot_bo7758324655462117769st_nat @ A2 ) ) ) ).

% bot_term.set_sel(1)
thf(fact_452_bot__term_Oset__sel_I1_J,axiom,
    ! [A2: bot_bot_term_nat] :
      ( ( A2 != bot_bot_Bot_nat )
     => ( member_nat @ ( bot_bot_un_BFun1_nat @ A2 ) @ ( bot_bo2410032770233908985rm_nat @ A2 ) ) ) ).

% bot_term.set_sel(1)
thf(fact_453_bot__term_Oset__sel_I1_J,axiom,
    ! [A2: bot_bot_term_f] :
      ( ( A2 != bot_bot_Bot_f )
     => ( member_f @ ( bot_bot_un_BFun1_f @ A2 ) @ ( bot_bo658131349715431450term_f @ A2 ) ) ) ).

% bot_term.set_sel(1)
thf(fact_454_list__all2I,axiom,
    ! [A2: list_nat,B: list_nat,P: nat > nat > $o] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ A2 @ B ) ) )
         => ( produc6081775807080527818_nat_o @ P @ X3 ) )
     => ( ( ( size_size_list_nat @ A2 )
          = ( size_size_list_nat @ B ) )
       => ( list_all2_nat_nat @ P @ A2 @ B ) ) ) ).

% list_all2I
thf(fact_455_list__all2I,axiom,
    ! [A2: list_nat,B: list_Bot_bot_term_f,P: nat > bot_bot_term_f > $o] :
      ( ! [X3: produc5418599768131837375term_f] :
          ( ( member4779802479421434344term_f @ X3 @ ( set_Pr3428226866328055892term_f @ ( zip_na3512298969805367703term_f @ A2 @ B ) ) )
         => ( produc450604121858260492rm_f_o @ P @ X3 ) )
     => ( ( ( size_size_list_nat @ A2 )
          = ( size_s1458857753309042354term_f @ B ) )
       => ( list_a8962304667658831463term_f @ P @ A2 @ B ) ) ) ).

% list_all2I
thf(fact_456_list__all2I,axiom,
    ! [A2: list_nat,B: list_list_nat,P: nat > list_nat > $o] :
      ( ! [X3: produc4575160907756185873st_nat] :
          ( ( member1631564025489475386st_nat @ X3 @ ( set_Pr1526418413117253030st_nat @ ( zip_nat_list_nat @ A2 @ B ) ) )
         => ( produc5265810065896090426_nat_o @ P @ X3 ) )
     => ( ( ( size_size_list_nat @ A2 )
          = ( size_s3023201423986296836st_nat @ B ) )
       => ( list_a5155878676884504761st_nat @ P @ A2 @ B ) ) ) ).

% list_all2I
thf(fact_457_list__all2I,axiom,
    ! [A2: list_Bot_bot_term_f,B: list_nat,P: bot_bot_term_f > nat > $o] :
      ( ! [X3: produc8417799323881384255_f_nat] :
          ( ( member7779002035170981224_f_nat @ X3 @ ( set_Pr6427426422077602772_f_nat @ ( zip_Bo4866563780717559831_f_nat @ A2 @ B ) ) )
         => ( produc5892416418908211084_nat_o @ P @ X3 ) )
     => ( ( ( size_s1458857753309042354term_f @ A2 )
          = ( size_size_list_nat @ B ) )
       => ( list_a1093197441716247783_f_nat @ P @ A2 @ B ) ) ) ).

% list_all2I
thf(fact_458_list__all2I,axiom,
    ! [A2: list_list_nat,B: list_nat,P: list_nat > nat > $o] :
      ( ! [X3: produc1540777390238407569at_nat] :
          ( ( member7820552544826472890at_nat @ X3 @ ( set_Pr7715406932454250534at_nat @ ( zip_list_nat_nat @ A2 @ B ) ) )
         => ( produc7308574044827705018_nat_o @ P @ X3 ) )
     => ( ( ( size_s3023201423986296836st_nat @ A2 )
          = ( size_size_list_nat @ B ) )
       => ( list_a7394835724046789945at_nat @ P @ A2 @ B ) ) ) ).

% list_all2I
thf(fact_459_list__all2I,axiom,
    ! [A2: list_nat,B: list_P6011104703257516679at_nat,P: nat > product_prod_nat_nat > $o] :
      ( ! [X3: produc7248412053542808358at_nat] :
          ( ( member2223272150424702269at_nat @ X3 @ ( set_Pr6710958862608470481at_nat @ ( zip_na1006125974040638520at_nat @ A2 @ B ) ) )
         => ( produc5864757623865647827_nat_o @ P @ X3 ) )
     => ( ( ( size_size_list_nat @ A2 )
          = ( size_s5460976970255530739at_nat @ B ) )
       => ( list_a8436948092790885224at_nat @ P @ A2 @ B ) ) ) ).

% list_all2I
thf(fact_460_list__all2I,axiom,
    ! [A2: list_Bot_bot_term_f,B: list_Bot_bot_term_f,P: bot_bot_term_f > bot_bot_term_f > $o] :
      ( ! [X3: produc250624644672406269term_f] :
          ( ( member4687689947140789414term_f @ X3 @ ( set_Pr1368550810757897490term_f @ ( zip_Bo7813815397675234645term_f @ A2 @ B ) ) )
         => ( produc8393149739075698894rm_f_o @ P @ X3 ) )
     => ( ( ( size_s1458857753309042354term_f @ A2 )
          = ( size_s1458857753309042354term_f @ B ) )
       => ( list_a8926285605142776357term_f @ P @ A2 @ B ) ) ) ).

% list_all2I
thf(fact_461_list__all2I,axiom,
    ! [A2: list_Bot_bot_term_f,B: list_list_nat,P: bot_bot_term_f > list_nat > $o] :
      ( ! [X3: produc5499396499697224271st_nat] :
          ( ( member4663489044743269880st_nat @ X3 @ ( set_Pr835449257019969124st_nat @ ( zip_Bo1182424480293800871st_nat @ A2 @ B ) ) )
         => ( produc6481302613249257980_nat_o @ P @ X3 ) )
     => ( ( ( size_s1458857753309042354term_f @ A2 )
          = ( size_s3023201423986296836st_nat @ B ) )
       => ( list_a470444602825883767st_nat @ P @ A2 @ B ) ) ) ).

% list_all2I
thf(fact_462_list__all2I,axiom,
    ! [A2: list_P6011104703257516679at_nat,B: list_nat,P: product_prod_nat_nat > nat > $o] :
      ( ! [X3: produc8373899037510109440at_nat] :
          ( ( member3348759134392003351at_nat @ X3 @ ( set_Pr7836445846575771563at_nat @ ( zip_Pr6869450617852699226at_nat @ A2 @ B ) ) )
         => ( produc8758269395934548017_nat_o @ P @ X3 ) )
     => ( ( ( size_s5460976970255530739at_nat @ A2 )
          = ( size_size_list_nat @ B ) )
       => ( list_a5076900699748170122at_nat @ P @ A2 @ B ) ) ) ).

% list_all2I
thf(fact_463_list__all2I,axiom,
    ! [A2: list_list_nat,B: list_Bot_bot_term_f,P: list_nat > bot_bot_term_f > $o] :
      ( ! [X3: produc4771067281011818575term_f] :
          ( ( member3935159826057864184term_f @ X3 @ ( set_Pr107120038334563428term_f @ ( zip_li7372370571125813415term_f @ A2 @ B ) ) )
         => ( produc5575963490392018172rm_f_o @ P @ X3 ) )
     => ( ( ( size_s3023201423986296836st_nat @ A2 )
          = ( size_s1458857753309042354term_f @ B ) )
       => ( list_a6660390693657896311term_f @ P @ A2 @ B ) ) ) ).

% list_all2I
thf(fact_464_zip__replicate,axiom,
    ! [I: nat,X: nat,J: nat,Y: nat] :
      ( ( zip_nat_nat @ ( replicate_nat @ I @ X ) @ ( replicate_nat @ J @ Y ) )
      = ( replic4235873036481779905at_nat @ ( ord_min_nat @ I @ J ) @ ( product_Pair_nat_nat @ X @ Y ) ) ) ).

% zip_replicate
thf(fact_465_zip__replicate,axiom,
    ! [I: nat,X: bot_bot_term_f,J: nat,Y: nat] :
      ( ( zip_Bo4866563780717559831_f_nat @ ( replic6958083601664011982term_f @ I @ X ) @ ( replicate_nat @ J @ Y ) )
      = ( replic5744860322987233535_f_nat @ ( ord_min_nat @ I @ J ) @ ( produc6944956222323935479_f_nat @ X @ Y ) ) ) ).

% zip_replicate
thf(fact_466_zip__replicate,axiom,
    ! [I: nat,X: nat,J: nat,Y: bot_bot_term_f] :
      ( ( zip_na3512298969805367703term_f @ ( replicate_nat @ I @ X ) @ ( replic6958083601664011982term_f @ J @ Y ) )
      = ( replic2745660767237686655term_f @ ( ord_min_nat @ I @ J ) @ ( produc5590691411411743351term_f @ X @ Y ) ) ) ).

% zip_replicate
thf(fact_467_zip__replicate,axiom,
    ! [I: nat,X: bot_bot_term_f,J: nat,Y: bot_bot_term_f] :
      ( ( zip_Bo7813815397675234645term_f @ ( replic6958083601664011982term_f @ I @ X ) @ ( replic6958083601664011982term_f @ J @ Y ) )
      = ( replic5523370671267941053term_f @ ( ord_min_nat @ I @ J ) @ ( produc2865564700782054965term_f @ X @ Y ) ) ) ).

% zip_replicate
thf(fact_468_zip__replicate,axiom,
    ! [I: nat,X: nat,J: nat,Y: product_prod_nat_nat] :
      ( ( zip_na1006125974040638520at_nat @ ( replicate_nat @ I @ X ) @ ( replic4235873036481779905at_nat @ J @ Y ) )
      = ( replic4101501219377410918at_nat @ ( ord_min_nat @ I @ J ) @ ( produc487386426758144856at_nat @ X @ Y ) ) ) ).

% zip_replicate
thf(fact_469_zip__replicate,axiom,
    ! [I: nat,X: product_prod_nat_nat,J: nat,Y: nat] :
      ( ( zip_Pr6869450617852699226at_nat @ ( replic4235873036481779905at_nat @ I @ X ) @ ( replicate_nat @ J @ Y ) )
      = ( replic5226988203344712000at_nat @ ( ord_min_nat @ I @ J ) @ ( produc6350711070570205562at_nat @ X @ Y ) ) ) ).

% zip_replicate
thf(fact_470_zip__replicate,axiom,
    ! [I: nat,X: list_nat,J: nat,Y: list_nat] :
      ( ( zip_li7157463729305086713st_nat @ ( replicate_list_nat @ I @ X ) @ ( replicate_list_nat @ J @ Y ) )
      = ( replic3948718270753518433st_nat @ ( ord_min_nat @ I @ J ) @ ( produc2694037385005941721st_nat @ X @ Y ) ) ) ).

% zip_replicate
thf(fact_471_zip__replicate,axiom,
    ! [I: nat,X: set_nat,J: nat,Y: set_nat] :
      ( ( zip_set_nat_set_nat @ ( replicate_set_nat @ I @ X ) @ ( replicate_set_nat @ J @ Y ) )
      = ( replic8313448869583292717et_nat @ ( ord_min_nat @ I @ J ) @ ( produc4532415448927165861et_nat @ X @ Y ) ) ) ).

% zip_replicate
thf(fact_472_zip__replicate,axiom,
    ! [I: nat,X: bot_bot_term_f,J: nat,Y: product_prod_nat_nat] :
      ( ( zip_Bo4081489131828187130at_nat @ ( replic6958083601664011982term_f @ I @ X ) @ ( replic4235873036481779905at_nat @ J @ Y ) )
      = ( replic6518106175821116840at_nat @ ( ord_min_nat @ I @ J ) @ ( produc3618334562033856282at_nat @ X @ Y ) ) ) ).

% zip_replicate
thf(fact_473_zip__replicate,axiom,
    ! [I: nat,X: nat,J: nat,Y: produc5418599768131837375term_f] :
      ( ( zip_na5084685201233939318term_f @ ( replicate_nat @ I @ X ) @ ( replic2745660767237686655term_f @ J @ Y ) )
      = ( replic4526976463895685156term_f @ ( ord_min_nat @ I @ J ) @ ( produc4621530631439608470term_f @ X @ Y ) ) ) ).

% zip_replicate
thf(fact_474_asym__lenlex,axiom,
    ! [R: set_Pr3451248702717554689st_nat] :
      ( ( asym_on_list_nat @ top_top_set_list_nat @ R )
     => ( asym_o1052983387450567102st_nat @ top_to8617644770344506070st_nat @ ( lenlex_list_nat @ R ) ) ) ).

% asym_lenlex
thf(fact_475_asym__lenlex,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R )
     => ( asym_on_list_nat @ top_top_set_list_nat @ ( lenlex_nat @ R ) ) ) ).

% asym_lenlex
thf(fact_476_size__neq__size__imp__neq,axiom,
    ! [X: list_Bot_bot_term_f,Y: list_Bot_bot_term_f] :
      ( ( ( size_s1458857753309042354term_f @ X )
       != ( size_s1458857753309042354term_f @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_477_size__neq__size__imp__neq,axiom,
    ! [X: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( size_s5460976970255530739at_nat @ X )
       != ( size_s5460976970255530739at_nat @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_478_size__neq__size__imp__neq,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ( ( size_s7982331069984003362term_f @ X )
       != ( size_s7982331069984003362term_f @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_479_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_480_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_481_size__neq__size__imp__neq,axiom,
    ! [X: char,Y: char] :
      ( ( ( size_size_char @ X )
       != ( size_size_char @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_482_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_483_zip__fst,axiom,
    ! [P2: produc8417799323881384255_f_nat,As: list_Bot_bot_term_f,Bs: list_nat] :
      ( ( member7779002035170981224_f_nat @ P2 @ ( set_Pr6427426422077602772_f_nat @ ( zip_Bo4866563780717559831_f_nat @ As @ Bs ) ) )
     => ( member2723211850833494437term_f @ ( produc8216198084029437667_f_nat @ P2 ) @ ( set_Bot_bot_term_f2 @ As ) ) ) ).

% zip_fst
thf(fact_484_zip__fst,axiom,
    ! [P2: produc250624644672406269term_f,As: list_Bot_bot_term_f,Bs: list_Bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ P2 @ ( set_Pr1368550810757897490term_f @ ( zip_Bo7813815397675234645term_f @ As @ Bs ) ) )
     => ( member2723211850833494437term_f @ ( produc5789359262401207841term_f @ P2 ) @ ( set_Bot_bot_term_f2 @ As ) ) ) ).

% zip_fst
thf(fact_485_zip__fst,axiom,
    ! [P2: produc5418599768131837375term_f,As: list_nat,Bs: list_Bot_bot_term_f] :
      ( ( member4779802479421434344term_f @ P2 @ ( set_Pr3428226866328055892term_f @ ( zip_na3512298969805367703term_f @ As @ Bs ) ) )
     => ( member_nat @ ( produc6861933273117245539term_f @ P2 ) @ ( set_nat2 @ As ) ) ) ).

% zip_fst
thf(fact_486_zip__fst,axiom,
    ! [P2: product_prod_nat_nat,As: list_nat,Bs: list_nat] :
      ( ( member8440522571783428010at_nat @ P2 @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ As @ Bs ) ) )
     => ( member_nat @ ( product_fst_nat_nat @ P2 ) @ ( set_nat2 @ As ) ) ) ).

% zip_fst
thf(fact_487_zip__fst,axiom,
    ! [P2: produc1828647624359046049st_nat,As: list_list_nat,Bs: list_list_nat] :
      ( ( member7340969449405702474st_nat @ P2 @ ( set_Pr3842133991353686454st_nat @ ( zip_li7157463729305086713st_nat @ As @ Bs ) ) )
     => ( member_list_nat @ ( produc1382935764643595205st_nat @ P2 ) @ ( set_list_nat2 @ As ) ) ) ).

% zip_fst
thf(fact_488_zip__fst,axiom,
    ! [P2: produc1342688880896445543st_nat,As: list_P7940050157051400743st_nat,Bs: list_P7940050157051400743st_nat] :
      ( ( member5236317644330119056st_nat @ P2 @ ( set_Pr7183939682275281404st_nat @ ( zip_Pr4899438369145407479st_nat @ As @ Bs ) ) )
     => ( member7340969449405702474st_nat @ ( produc7458156121412276931st_nat @ P2 ) @ ( set_Pr3842133991353686454st_nat @ As ) ) ) ).

% zip_fst
thf(fact_489_zip__snd,axiom,
    ! [P2: produc5418599768131837375term_f,As: list_nat,Bs: list_Bot_bot_term_f] :
      ( ( member4779802479421434344term_f @ P2 @ ( set_Pr3428226866328055892term_f @ ( zip_na3512298969805367703term_f @ As @ Bs ) ) )
     => ( member2723211850833494437term_f @ ( produc6684081486893607845term_f @ P2 ) @ ( set_Bot_bot_term_f2 @ Bs ) ) ) ).

% zip_snd
thf(fact_490_zip__snd,axiom,
    ! [P2: produc250624644672406269term_f,As: list_Bot_bot_term_f,Bs: list_Bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ P2 @ ( set_Pr1368550810757897490term_f @ ( zip_Bo7813815397675234645term_f @ As @ Bs ) ) )
     => ( member2723211850833494437term_f @ ( produc656023363694804067term_f @ P2 ) @ ( set_Bot_bot_term_f2 @ Bs ) ) ) ).

% zip_snd
thf(fact_491_zip__snd,axiom,
    ! [P2: produc8417799323881384255_f_nat,As: list_Bot_bot_term_f,Bs: list_nat] :
      ( ( member7779002035170981224_f_nat @ P2 @ ( set_Pr6427426422077602772_f_nat @ ( zip_Bo4866563780717559831_f_nat @ As @ Bs ) ) )
     => ( member_nat @ ( produc8038346297805799973_f_nat @ P2 ) @ ( set_nat2 @ Bs ) ) ) ).

% zip_snd
thf(fact_492_zip__snd,axiom,
    ! [P2: product_prod_nat_nat,As: list_nat,Bs: list_nat] :
      ( ( member8440522571783428010at_nat @ P2 @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ As @ Bs ) ) )
     => ( member_nat @ ( product_snd_nat_nat @ P2 ) @ ( set_nat2 @ Bs ) ) ) ).

% zip_snd
thf(fact_493_zip__snd,axiom,
    ! [P2: produc1828647624359046049st_nat,As: list_list_nat,Bs: list_list_nat] :
      ( ( member7340969449405702474st_nat @ P2 @ ( set_Pr3842133991353686454st_nat @ ( zip_li7157463729305086713st_nat @ As @ Bs ) ) )
     => ( member_list_nat @ ( produc5865812112468994567st_nat @ P2 ) @ ( set_list_nat2 @ Bs ) ) ) ).

% zip_snd
thf(fact_494_zip__snd,axiom,
    ! [P2: produc1342688880896445543st_nat,As: list_P7940050157051400743st_nat,Bs: list_P7940050157051400743st_nat] :
      ( ( member5236317644330119056st_nat @ P2 @ ( set_Pr7183939682275281404st_nat @ ( zip_Pr4899438369145407479st_nat @ As @ Bs ) ) )
     => ( member7340969449405702474st_nat @ ( produc7615485059283090437st_nat @ P2 ) @ ( set_Pr3842133991353686454st_nat @ Bs ) ) ) ).

% zip_snd
thf(fact_495_asym__lex,axiom,
    ! [R: set_Pr3451248702717554689st_nat] :
      ( ( asym_on_list_nat @ top_top_set_list_nat @ R )
     => ( asym_o1052983387450567102st_nat @ top_to8617644770344506070st_nat @ ( lex_list_nat @ R ) ) ) ).

% asym_lex
thf(fact_496_asym__lex,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R )
     => ( asym_on_list_nat @ top_top_set_list_nat @ ( lex_nat @ R ) ) ) ).

% asym_lex
thf(fact_497_Ball__set__replicate,axiom,
    ! [N: nat,A2: char,P: char > $o] :
      ( ( ! [X2: char] :
            ( ( member_char @ X2 @ ( set_char2 @ ( replicate_char @ N @ A2 ) ) )
           => ( P @ X2 ) ) )
      = ( ( P @ A2 )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_498_Ball__set__replicate,axiom,
    ! [N: nat,A2: list_nat,P: list_nat > $o] :
      ( ( ! [X2: list_nat] :
            ( ( member_list_nat @ X2 @ ( set_list_nat2 @ ( replicate_list_nat @ N @ A2 ) ) )
           => ( P @ X2 ) ) )
      = ( ( P @ A2 )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_499_Ball__set__replicate,axiom,
    ! [N: nat,A2: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ A2 ) ) )
           => ( P @ X2 ) ) )
      = ( ( P @ A2 )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_500_Ball__set__replicate,axiom,
    ! [N: nat,A2: produc5418599768131837375term_f,P: produc5418599768131837375term_f > $o] :
      ( ( ! [X2: produc5418599768131837375term_f] :
            ( ( member4779802479421434344term_f @ X2 @ ( set_Pr3428226866328055892term_f @ ( replic2745660767237686655term_f @ N @ A2 ) ) )
           => ( P @ X2 ) ) )
      = ( ( P @ A2 )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_501_Ball__set__replicate,axiom,
    ! [N: nat,A2: produc8417799323881384255_f_nat,P: produc8417799323881384255_f_nat > $o] :
      ( ( ! [X2: produc8417799323881384255_f_nat] :
            ( ( member7779002035170981224_f_nat @ X2 @ ( set_Pr6427426422077602772_f_nat @ ( replic5744860322987233535_f_nat @ N @ A2 ) ) )
           => ( P @ X2 ) ) )
      = ( ( P @ A2 )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_502_Ball__set__replicate,axiom,
    ! [N: nat,A2: produc250624644672406269term_f,P: produc250624644672406269term_f > $o] :
      ( ( ! [X2: produc250624644672406269term_f] :
            ( ( member4687689947140789414term_f @ X2 @ ( set_Pr1368550810757897490term_f @ ( replic5523370671267941053term_f @ N @ A2 ) ) )
           => ( P @ X2 ) ) )
      = ( ( P @ A2 )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_503_Ball__set__replicate,axiom,
    ! [N: nat,A2: bot_bot_term_f,P: bot_bot_term_f > $o] :
      ( ( ! [X2: bot_bot_term_f] :
            ( ( member2723211850833494437term_f @ X2 @ ( set_Bot_bot_term_f2 @ ( replic6958083601664011982term_f @ N @ A2 ) ) )
           => ( P @ X2 ) ) )
      = ( ( P @ A2 )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_504_Ball__set__replicate,axiom,
    ! [N: nat,A2: nat,P: nat > $o] :
      ( ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ ( replicate_nat @ N @ A2 ) ) )
           => ( P @ X2 ) ) )
      = ( ( P @ A2 )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_505_Bex__set__replicate,axiom,
    ! [N: nat,A2: char,P: char > $o] :
      ( ( ? [X2: char] :
            ( ( member_char @ X2 @ ( set_char2 @ ( replicate_char @ N @ A2 ) ) )
            & ( P @ X2 ) ) )
      = ( ( P @ A2 )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_506_Bex__set__replicate,axiom,
    ! [N: nat,A2: list_nat,P: list_nat > $o] :
      ( ( ? [X2: list_nat] :
            ( ( member_list_nat @ X2 @ ( set_list_nat2 @ ( replicate_list_nat @ N @ A2 ) ) )
            & ( P @ X2 ) ) )
      = ( ( P @ A2 )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_507_Bex__set__replicate,axiom,
    ! [N: nat,A2: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( ? [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ A2 ) ) )
            & ( P @ X2 ) ) )
      = ( ( P @ A2 )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_508_Bex__set__replicate,axiom,
    ! [N: nat,A2: produc5418599768131837375term_f,P: produc5418599768131837375term_f > $o] :
      ( ( ? [X2: produc5418599768131837375term_f] :
            ( ( member4779802479421434344term_f @ X2 @ ( set_Pr3428226866328055892term_f @ ( replic2745660767237686655term_f @ N @ A2 ) ) )
            & ( P @ X2 ) ) )
      = ( ( P @ A2 )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_509_Bex__set__replicate,axiom,
    ! [N: nat,A2: produc8417799323881384255_f_nat,P: produc8417799323881384255_f_nat > $o] :
      ( ( ? [X2: produc8417799323881384255_f_nat] :
            ( ( member7779002035170981224_f_nat @ X2 @ ( set_Pr6427426422077602772_f_nat @ ( replic5744860322987233535_f_nat @ N @ A2 ) ) )
            & ( P @ X2 ) ) )
      = ( ( P @ A2 )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_510_Bex__set__replicate,axiom,
    ! [N: nat,A2: produc250624644672406269term_f,P: produc250624644672406269term_f > $o] :
      ( ( ? [X2: produc250624644672406269term_f] :
            ( ( member4687689947140789414term_f @ X2 @ ( set_Pr1368550810757897490term_f @ ( replic5523370671267941053term_f @ N @ A2 ) ) )
            & ( P @ X2 ) ) )
      = ( ( P @ A2 )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_511_Bex__set__replicate,axiom,
    ! [N: nat,A2: bot_bot_term_f,P: bot_bot_term_f > $o] :
      ( ( ? [X2: bot_bot_term_f] :
            ( ( member2723211850833494437term_f @ X2 @ ( set_Bot_bot_term_f2 @ ( replic6958083601664011982term_f @ N @ A2 ) ) )
            & ( P @ X2 ) ) )
      = ( ( P @ A2 )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_512_Bex__set__replicate,axiom,
    ! [N: nat,A2: nat,P: nat > $o] :
      ( ( ? [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ ( replicate_nat @ N @ A2 ) ) )
            & ( P @ X2 ) ) )
      = ( ( P @ A2 )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_513_min__top2,axiom,
    ! [X: nat > $o] :
      ( ( ord_min_nat_o @ X @ top_top_nat_o )
      = X ) ).

% min_top2
thf(fact_514_min__top2,axiom,
    ! [X: produc7819656566062154093et_nat] :
      ( ( ord_mi2418711518901664340et_nat @ X @ top_to7593079806418470589et_nat )
      = X ) ).

% min_top2
thf(fact_515_min__top2,axiom,
    ! [X: set_list_nat] :
      ( ( ord_min_set_list_nat @ X @ top_top_set_list_nat )
      = X ) ).

% min_top2
thf(fact_516_min__top2,axiom,
    ! [X: set_nat] :
      ( ( ord_min_set_nat @ X @ top_top_set_nat )
      = X ) ).

% min_top2
thf(fact_517_min__top,axiom,
    ! [X: nat > $o] :
      ( ( ord_min_nat_o @ top_top_nat_o @ X )
      = X ) ).

% min_top
thf(fact_518_min__top,axiom,
    ! [X: produc7819656566062154093et_nat] :
      ( ( ord_mi2418711518901664340et_nat @ top_to7593079806418470589et_nat @ X )
      = X ) ).

% min_top
thf(fact_519_min__top,axiom,
    ! [X: set_list_nat] :
      ( ( ord_min_set_list_nat @ top_top_set_list_nat @ X )
      = X ) ).

% min_top
thf(fact_520_min__top,axiom,
    ! [X: set_nat] :
      ( ( ord_min_set_nat @ top_top_set_nat @ X )
      = X ) ).

% min_top
thf(fact_521_replicate__eq__replicate,axiom,
    ! [M: nat,X: product_prod_nat_nat,N: nat,Y: product_prod_nat_nat] :
      ( ( ( replic4235873036481779905at_nat @ M @ X )
        = ( replic4235873036481779905at_nat @ N @ Y ) )
      = ( ( M = N )
        & ( ( M != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
thf(fact_522_replicate__eq__replicate,axiom,
    ! [M: nat,X: produc5418599768131837375term_f,N: nat,Y: produc5418599768131837375term_f] :
      ( ( ( replic2745660767237686655term_f @ M @ X )
        = ( replic2745660767237686655term_f @ N @ Y ) )
      = ( ( M = N )
        & ( ( M != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
thf(fact_523_replicate__eq__replicate,axiom,
    ! [M: nat,X: produc8417799323881384255_f_nat,N: nat,Y: produc8417799323881384255_f_nat] :
      ( ( ( replic5744860322987233535_f_nat @ M @ X )
        = ( replic5744860322987233535_f_nat @ N @ Y ) )
      = ( ( M = N )
        & ( ( M != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
thf(fact_524_replicate__eq__replicate,axiom,
    ! [M: nat,X: produc250624644672406269term_f,N: nat,Y: produc250624644672406269term_f] :
      ( ( ( replic5523370671267941053term_f @ M @ X )
        = ( replic5523370671267941053term_f @ N @ Y ) )
      = ( ( M = N )
        & ( ( M != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
thf(fact_525_replicate__eq__replicate,axiom,
    ! [M: nat,X: bot_bot_term_f,N: nat,Y: bot_bot_term_f] :
      ( ( ( replic6958083601664011982term_f @ M @ X )
        = ( replic6958083601664011982term_f @ N @ Y ) )
      = ( ( M = N )
        & ( ( M != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
thf(fact_526_replicate__eq__replicate,axiom,
    ! [M: nat,X: nat,N: nat,Y: nat] :
      ( ( ( replicate_nat @ M @ X )
        = ( replicate_nat @ N @ Y ) )
      = ( ( M = N )
        & ( ( M != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
thf(fact_527_curry__case__prod,axiom,
    ! [F2: nat > nat > $o] :
      ( ( produc1310100445399344235_nat_o @ ( produc6081775807080527818_nat_o @ F2 ) )
      = F2 ) ).

% curry_case_prod
thf(fact_528_curry__case__prod,axiom,
    ! [F2: bot_bot_term_f > bot_bot_term_f > bot_bot_term_f] :
      ( ( produc2177247631251368695term_f @ ( produc1677810408741468632term_f @ F2 ) )
      = F2 ) ).

% curry_case_prod
thf(fact_529_case__prod__curry,axiom,
    ! [F2: product_prod_nat_nat > $o] :
      ( ( produc6081775807080527818_nat_o @ ( produc1310100445399344235_nat_o @ F2 ) )
      = F2 ) ).

% case_prod_curry
thf(fact_530_case__prod__curry,axiom,
    ! [F2: produc250624644672406269term_f > bot_bot_term_f] :
      ( ( produc1677810408741468632term_f @ ( produc2177247631251368695term_f @ F2 ) )
      = F2 ) ).

% case_prod_curry
thf(fact_531_prod_Ocollapse,axiom,
    ! [Prod: produc3922691147685405600st_nat] :
      ( ( produc8649917717998259994st_nat @ ( produc8387770181635782702st_nat @ Prod ) @ ( produc8555213167096887148st_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_532_prod_Ocollapse,axiom,
    ! [Prod: produc7819656566062154093et_nat] :
      ( ( produc4532415448927165861et_nat @ ( produc2731966376809613713et_nat @ Prod ) @ ( produc397104031625245139et_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_533_prod_Ocollapse,axiom,
    ! [Prod: produc5418599768131837375term_f] :
      ( ( produc5590691411411743351term_f @ ( produc6861933273117245539term_f @ Prod ) @ ( produc6684081486893607845term_f @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_534_prod_Ocollapse,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_535_prod_Ocollapse,axiom,
    ! [Prod: produc1828647624359046049st_nat] :
      ( ( produc2694037385005941721st_nat @ ( produc1382935764643595205st_nat @ Prod ) @ ( produc5865812112468994567st_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_536_prod_Ocollapse,axiom,
    ! [Prod: produc1342688880896445543st_nat] :
      ( ( produc988773521843941591st_nat @ ( produc7458156121412276931st_nat @ Prod ) @ ( produc7615485059283090437st_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_537_in__set__replicate,axiom,
    ! [X: produc1828647624359046049st_nat,N: nat,Y: produc1828647624359046049st_nat] :
      ( ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ ( replic3948718270753518433st_nat @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_538_in__set__replicate,axiom,
    ! [X: f,N: nat,Y: f] :
      ( ( member_f @ X @ ( set_f2 @ ( replicate_f @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_539_in__set__replicate,axiom,
    ! [X: char,N: nat,Y: char] :
      ( ( member_char @ X @ ( set_char2 @ ( replicate_char @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_540_in__set__replicate,axiom,
    ! [X: list_nat,N: nat,Y: list_nat] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ ( replicate_list_nat @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_541_in__set__replicate,axiom,
    ! [X: product_prod_nat_nat,N: nat,Y: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_542_in__set__replicate,axiom,
    ! [X: produc5418599768131837375term_f,N: nat,Y: produc5418599768131837375term_f] :
      ( ( member4779802479421434344term_f @ X @ ( set_Pr3428226866328055892term_f @ ( replic2745660767237686655term_f @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_543_in__set__replicate,axiom,
    ! [X: produc8417799323881384255_f_nat,N: nat,Y: produc8417799323881384255_f_nat] :
      ( ( member7779002035170981224_f_nat @ X @ ( set_Pr6427426422077602772_f_nat @ ( replic5744860322987233535_f_nat @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_544_in__set__replicate,axiom,
    ! [X: produc250624644672406269term_f,N: nat,Y: produc250624644672406269term_f] :
      ( ( member4687689947140789414term_f @ X @ ( set_Pr1368550810757897490term_f @ ( replic5523370671267941053term_f @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_545_in__set__replicate,axiom,
    ! [X: bot_bot_term_f,N: nat,Y: bot_bot_term_f] :
      ( ( member2723211850833494437term_f @ X @ ( set_Bot_bot_term_f2 @ ( replic6958083601664011982term_f @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_546_in__set__replicate,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_547_length__zip,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( zip_nat_nat @ Xs2 @ Ys ) )
      = ( ord_min_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_zip
thf(fact_548_length__zip,axiom,
    ! [Xs2: list_nat,Ys: list_Bot_bot_term_f] :
      ( ( size_s1844900889452899121term_f @ ( zip_na3512298969805367703term_f @ Xs2 @ Ys ) )
      = ( ord_min_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s1458857753309042354term_f @ Ys ) ) ) ).

% length_zip
thf(fact_549_length__zip,axiom,
    ! [Xs2: list_nat,Ys: list_list_nat] :
      ( ( size_s9035287501014481795st_nat @ ( zip_nat_list_nat @ Xs2 @ Ys ) )
      = ( ord_min_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s3023201423986296836st_nat @ Ys ) ) ) ).

% length_zip
thf(fact_550_length__zip,axiom,
    ! [Xs2: list_Bot_bot_term_f,Ys: list_nat] :
      ( ( size_s6959986635380143793_f_nat @ ( zip_Bo4866563780717559831_f_nat @ Xs2 @ Ys ) )
      = ( ord_min_nat @ ( size_s1458857753309042354term_f @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_zip
thf(fact_551_length__zip,axiom,
    ! [Xs2: list_list_nat,Ys: list_nat] :
      ( ( size_s6663376490332876291at_nat @ ( zip_list_nat_nat @ Xs2 @ Ys ) )
      = ( ord_min_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_zip
thf(fact_552_length__zip,axiom,
    ! [Xs2: list_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( size_s3596537876095621194at_nat @ ( zip_na1006125974040638520at_nat @ Xs2 @ Ys ) )
      = ( ord_min_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s5460976970255530739at_nat @ Ys ) ) ) ).

% length_zip
thf(fact_553_length__zip,axiom,
    ! [Xs2: list_Bot_bot_term_f,Ys: list_Bot_bot_term_f] :
      ( ( size_s9037951226665808879term_f @ ( zip_Bo7813815397675234645term_f @ Xs2 @ Ys ) )
      = ( ord_min_nat @ ( size_s1458857753309042354term_f @ Xs2 ) @ ( size_s1458857753309042354term_f @ Ys ) ) ) ).

% length_zip
thf(fact_554_length__zip,axiom,
    ! [Xs2: list_Bot_bot_term_f,Ys: list_list_nat] :
      ( ( size_s5711386298565001281st_nat @ ( zip_Bo1182424480293800871st_nat @ Xs2 @ Ys ) )
      = ( ord_min_nat @ ( size_s1458857753309042354term_f @ Xs2 ) @ ( size_s3023201423986296836st_nat @ Ys ) ) ) ).

% length_zip
thf(fact_555_length__zip,axiom,
    ! [Xs2: list_P6011104703257516679at_nat,Ys: list_nat] :
      ( ( size_s7641165130114787620at_nat @ ( zip_Pr6869450617852699226at_nat @ Xs2 @ Ys ) )
      = ( ord_min_nat @ ( size_s5460976970255530739at_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_zip
thf(fact_556_length__zip,axiom,
    ! [Xs2: list_list_nat,Ys: list_Bot_bot_term_f] :
      ( ( size_s2425747490215884353term_f @ ( zip_li7372370571125813415term_f @ Xs2 @ Ys ) )
      = ( ord_min_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ ( size_s1458857753309042354term_f @ Ys ) ) ) ).

% length_zip
thf(fact_557_prod_Osplit__sel__asm,axiom,
    ! [P: $o > $o,F2: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( P @ ( produc6081775807080527818_nat_o @ F2 @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
            & ~ ( P @ ( F2 @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_558_prod_Osplit__sel__asm,axiom,
    ! [P: bot_bot_term_f > $o,F2: bot_bot_term_f > bot_bot_term_f > bot_bot_term_f,Prod: produc250624644672406269term_f] :
      ( ( P @ ( produc1677810408741468632term_f @ F2 @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc2865564700782054965term_f @ ( produc5789359262401207841term_f @ Prod ) @ ( produc656023363694804067term_f @ Prod ) ) )
            & ~ ( P @ ( F2 @ ( produc5789359262401207841term_f @ Prod ) @ ( produc656023363694804067term_f @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_559_prod_Osplit__sel,axiom,
    ! [P: $o > $o,F2: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( P @ ( produc6081775807080527818_nat_o @ F2 @ Prod ) )
      = ( ( Prod
          = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
       => ( P @ ( F2 @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_560_prod_Osplit__sel,axiom,
    ! [P: bot_bot_term_f > $o,F2: bot_bot_term_f > bot_bot_term_f > bot_bot_term_f,Prod: produc250624644672406269term_f] :
      ( ( P @ ( produc1677810408741468632term_f @ F2 @ Prod ) )
      = ( ( Prod
          = ( produc2865564700782054965term_f @ ( produc5789359262401207841term_f @ Prod ) @ ( produc656023363694804067term_f @ Prod ) ) )
       => ( P @ ( F2 @ ( produc5789359262401207841term_f @ Prod ) @ ( produc656023363694804067term_f @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_561_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3922691147685405600st_nat] :
      ( Prod
      = ( produc8649917717998259994st_nat @ ( produc8387770181635782702st_nat @ Prod ) @ ( produc8555213167096887148st_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_562_prod_Oexhaust__sel,axiom,
    ! [Prod: produc7819656566062154093et_nat] :
      ( Prod
      = ( produc4532415448927165861et_nat @ ( produc2731966376809613713et_nat @ Prod ) @ ( produc397104031625245139et_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_563_prod_Oexhaust__sel,axiom,
    ! [Prod: produc5418599768131837375term_f] :
      ( Prod
      = ( produc5590691411411743351term_f @ ( produc6861933273117245539term_f @ Prod ) @ ( produc6684081486893607845term_f @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_564_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( Prod
      = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_565_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1828647624359046049st_nat] :
      ( Prod
      = ( produc2694037385005941721st_nat @ ( produc1382935764643595205st_nat @ Prod ) @ ( produc5865812112468994567st_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_566_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1342688880896445543st_nat] :
      ( Prod
      = ( produc988773521843941591st_nat @ ( produc7458156121412276931st_nat @ Prod ) @ ( produc7615485059283090437st_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_567_surjective__pairing,axiom,
    ! [T: produc3922691147685405600st_nat] :
      ( T
      = ( produc8649917717998259994st_nat @ ( produc8387770181635782702st_nat @ T ) @ ( produc8555213167096887148st_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_568_surjective__pairing,axiom,
    ! [T: produc7819656566062154093et_nat] :
      ( T
      = ( produc4532415448927165861et_nat @ ( produc2731966376809613713et_nat @ T ) @ ( produc397104031625245139et_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_569_surjective__pairing,axiom,
    ! [T: produc5418599768131837375term_f] :
      ( T
      = ( produc5590691411411743351term_f @ ( produc6861933273117245539term_f @ T ) @ ( produc6684081486893607845term_f @ T ) ) ) ).

% surjective_pairing
thf(fact_570_surjective__pairing,axiom,
    ! [T: product_prod_nat_nat] :
      ( T
      = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ T ) @ ( product_snd_nat_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_571_surjective__pairing,axiom,
    ! [T: produc1828647624359046049st_nat] :
      ( T
      = ( produc2694037385005941721st_nat @ ( produc1382935764643595205st_nat @ T ) @ ( produc5865812112468994567st_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_572_surjective__pairing,axiom,
    ! [T: produc1342688880896445543st_nat] :
      ( T
      = ( produc988773521843941591st_nat @ ( produc7458156121412276931st_nat @ T ) @ ( produc7615485059283090437st_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_573_Product__Type_OCollect__case__prodD,axiom,
    ! [X: produc1828647624359046049st_nat,A: list_nat > list_nat > $o] :
      ( ( member7340969449405702474st_nat @ X @ ( collec1570431334306492044st_nat @ ( produc6382697403170970154_nat_o @ A ) ) )
     => ( A @ ( produc1382935764643595205st_nat @ X ) @ ( produc5865812112468994567st_nat @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_574_Product__Type_OCollect__case__prodD,axiom,
    ! [X: produc1342688880896445543st_nat,A: produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o] :
      ( ( member5236317644330119056st_nat @ X @ ( collec7387078234793787346st_nat @ ( produc4929175875791559596_nat_o @ A ) ) )
     => ( A @ ( produc7458156121412276931st_nat @ X ) @ ( produc7615485059283090437st_nat @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_575_Product__Type_OCollect__case__prodD,axiom,
    ! [X: product_prod_nat_nat,A: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ X @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ A ) ) )
     => ( A @ ( product_fst_nat_nat @ X ) @ ( product_snd_nat_nat @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_576_case__prod__beta,axiom,
    ( produc6081775807080527818_nat_o
    = ( ^ [F3: nat > nat > $o,P4: product_prod_nat_nat] : ( F3 @ ( product_fst_nat_nat @ P4 ) @ ( product_snd_nat_nat @ P4 ) ) ) ) ).

% case_prod_beta
thf(fact_577_case__prod__beta,axiom,
    ( produc1677810408741468632term_f
    = ( ^ [F3: bot_bot_term_f > bot_bot_term_f > bot_bot_term_f,P4: produc250624644672406269term_f] : ( F3 @ ( produc5789359262401207841term_f @ P4 ) @ ( produc656023363694804067term_f @ P4 ) ) ) ) ).

% case_prod_beta
thf(fact_578_prod__eq__iff,axiom,
    ( ( ^ [Y2: product_prod_nat_nat,Z: product_prod_nat_nat] : ( Y2 = Z ) )
    = ( ^ [S3: product_prod_nat_nat,T2: product_prod_nat_nat] :
          ( ( ( product_fst_nat_nat @ S3 )
            = ( product_fst_nat_nat @ T2 ) )
          & ( ( product_snd_nat_nat @ S3 )
            = ( product_snd_nat_nat @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_579_prod__eq__iff,axiom,
    ( ( ^ [Y2: produc1828647624359046049st_nat,Z: produc1828647624359046049st_nat] : ( Y2 = Z ) )
    = ( ^ [S3: produc1828647624359046049st_nat,T2: produc1828647624359046049st_nat] :
          ( ( ( produc1382935764643595205st_nat @ S3 )
            = ( produc1382935764643595205st_nat @ T2 ) )
          & ( ( produc5865812112468994567st_nat @ S3 )
            = ( produc5865812112468994567st_nat @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_580_prod__eq__iff,axiom,
    ( ( ^ [Y2: produc1342688880896445543st_nat,Z: produc1342688880896445543st_nat] : ( Y2 = Z ) )
    = ( ^ [S3: produc1342688880896445543st_nat,T2: produc1342688880896445543st_nat] :
          ( ( ( produc7458156121412276931st_nat @ S3 )
            = ( produc7458156121412276931st_nat @ T2 ) )
          & ( ( produc7615485059283090437st_nat @ S3 )
            = ( produc7615485059283090437st_nat @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_581_split__beta,axiom,
    ( produc6081775807080527818_nat_o
    = ( ^ [F3: nat > nat > $o,Prod2: product_prod_nat_nat] : ( F3 @ ( product_fst_nat_nat @ Prod2 ) @ ( product_snd_nat_nat @ Prod2 ) ) ) ) ).

% split_beta
thf(fact_582_split__beta,axiom,
    ( produc1677810408741468632term_f
    = ( ^ [F3: bot_bot_term_f > bot_bot_term_f > bot_bot_term_f,Prod2: produc250624644672406269term_f] : ( F3 @ ( produc5789359262401207841term_f @ Prod2 ) @ ( produc656023363694804067term_f @ Prod2 ) ) ) ) ).

% split_beta
thf(fact_583_prod__eqI,axiom,
    ! [P2: product_prod_nat_nat,Q2: product_prod_nat_nat] :
      ( ( ( product_fst_nat_nat @ P2 )
        = ( product_fst_nat_nat @ Q2 ) )
     => ( ( ( product_snd_nat_nat @ P2 )
          = ( product_snd_nat_nat @ Q2 ) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
thf(fact_584_prod__eqI,axiom,
    ! [P2: produc1828647624359046049st_nat,Q2: produc1828647624359046049st_nat] :
      ( ( ( produc1382935764643595205st_nat @ P2 )
        = ( produc1382935764643595205st_nat @ Q2 ) )
     => ( ( ( produc5865812112468994567st_nat @ P2 )
          = ( produc5865812112468994567st_nat @ Q2 ) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
thf(fact_585_prod__eqI,axiom,
    ! [P2: produc1342688880896445543st_nat,Q2: produc1342688880896445543st_nat] :
      ( ( ( produc7458156121412276931st_nat @ P2 )
        = ( produc7458156121412276931st_nat @ Q2 ) )
     => ( ( ( produc7615485059283090437st_nat @ P2 )
          = ( produc7615485059283090437st_nat @ Q2 ) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
thf(fact_586_prod_Oexpand,axiom,
    ! [Prod: product_prod_nat_nat,Prod3: product_prod_nat_nat] :
      ( ( ( ( product_fst_nat_nat @ Prod )
          = ( product_fst_nat_nat @ Prod3 ) )
        & ( ( product_snd_nat_nat @ Prod )
          = ( product_snd_nat_nat @ Prod3 ) ) )
     => ( Prod = Prod3 ) ) ).

% prod.expand
thf(fact_587_prod_Oexpand,axiom,
    ! [Prod: produc1828647624359046049st_nat,Prod3: produc1828647624359046049st_nat] :
      ( ( ( ( produc1382935764643595205st_nat @ Prod )
          = ( produc1382935764643595205st_nat @ Prod3 ) )
        & ( ( produc5865812112468994567st_nat @ Prod )
          = ( produc5865812112468994567st_nat @ Prod3 ) ) )
     => ( Prod = Prod3 ) ) ).

% prod.expand
thf(fact_588_prod_Oexpand,axiom,
    ! [Prod: produc1342688880896445543st_nat,Prod3: produc1342688880896445543st_nat] :
      ( ( ( ( produc7458156121412276931st_nat @ Prod )
          = ( produc7458156121412276931st_nat @ Prod3 ) )
        & ( ( produc7615485059283090437st_nat @ Prod )
          = ( produc7615485059283090437st_nat @ Prod3 ) ) )
     => ( Prod = Prod3 ) ) ).

% prod.expand
thf(fact_589_bot__term_Osize_I3_J,axiom,
    ( ( size_s9110728481667089031rm_nat @ bot_bot_Bot_nat )
    = zero_zero_nat ) ).

% bot_term.size(3)
thf(fact_590_bot__term_Osize_I3_J,axiom,
    ( ( size_s7982331069984003362term_f @ bot_bot_Bot_f )
    = zero_zero_nat ) ).

% bot_term.size(3)
thf(fact_591_snd__eqD,axiom,
    ! [X: product_prod_nat_nat,Y: list_list_nat,A2: list_list_nat] :
      ( ( ( produc8555213167096887148st_nat @ ( produc8649917717998259994st_nat @ X @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_592_snd__eqD,axiom,
    ! [X: set_nat,Y: set_nat,A2: set_nat] :
      ( ( ( produc397104031625245139et_nat @ ( produc4532415448927165861et_nat @ X @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_593_snd__eqD,axiom,
    ! [X: nat,Y: bot_bot_term_f,A2: bot_bot_term_f] :
      ( ( ( produc6684081486893607845term_f @ ( produc5590691411411743351term_f @ X @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_594_snd__eqD,axiom,
    ! [X: nat,Y: nat,A2: nat] :
      ( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_595_snd__eqD,axiom,
    ! [X: list_nat,Y: list_nat,A2: list_nat] :
      ( ( ( produc5865812112468994567st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_596_snd__eqD,axiom,
    ! [X: produc1828647624359046049st_nat,Y: produc1828647624359046049st_nat,A2: produc1828647624359046049st_nat] :
      ( ( ( produc7615485059283090437st_nat @ ( produc988773521843941591st_nat @ X @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_597_snd__conv,axiom,
    ! [X1: product_prod_nat_nat,X23: list_list_nat] :
      ( ( produc8555213167096887148st_nat @ ( produc8649917717998259994st_nat @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_598_snd__conv,axiom,
    ! [X1: set_nat,X23: set_nat] :
      ( ( produc397104031625245139et_nat @ ( produc4532415448927165861et_nat @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_599_snd__conv,axiom,
    ! [X1: nat,X23: bot_bot_term_f] :
      ( ( produc6684081486893607845term_f @ ( produc5590691411411743351term_f @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_600_snd__conv,axiom,
    ! [X1: nat,X23: nat] :
      ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_601_snd__conv,axiom,
    ! [X1: list_nat,X23: list_nat] :
      ( ( produc5865812112468994567st_nat @ ( produc2694037385005941721st_nat @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_602_snd__conv,axiom,
    ! [X1: produc1828647624359046049st_nat,X23: produc1828647624359046049st_nat] :
      ( ( produc7615485059283090437st_nat @ ( produc988773521843941591st_nat @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_603_fst__eqD,axiom,
    ! [X: product_prod_nat_nat,Y: list_list_nat,A2: product_prod_nat_nat] :
      ( ( ( produc8387770181635782702st_nat @ ( produc8649917717998259994st_nat @ X @ Y ) )
        = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
thf(fact_604_fst__eqD,axiom,
    ! [X: set_nat,Y: set_nat,A2: set_nat] :
      ( ( ( produc2731966376809613713et_nat @ ( produc4532415448927165861et_nat @ X @ Y ) )
        = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
thf(fact_605_fst__eqD,axiom,
    ! [X: nat,Y: bot_bot_term_f,A2: nat] :
      ( ( ( produc6861933273117245539term_f @ ( produc5590691411411743351term_f @ X @ Y ) )
        = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
thf(fact_606_fst__eqD,axiom,
    ! [X: nat,Y: nat,A2: nat] :
      ( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
        = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
thf(fact_607_fst__eqD,axiom,
    ! [X: list_nat,Y: list_nat,A2: list_nat] :
      ( ( ( produc1382935764643595205st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) )
        = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
thf(fact_608_fst__eqD,axiom,
    ! [X: produc1828647624359046049st_nat,Y: produc1828647624359046049st_nat,A2: produc1828647624359046049st_nat] :
      ( ( ( produc7458156121412276931st_nat @ ( produc988773521843941591st_nat @ X @ Y ) )
        = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
thf(fact_609_fst__conv,axiom,
    ! [X1: product_prod_nat_nat,X23: list_list_nat] :
      ( ( produc8387770181635782702st_nat @ ( produc8649917717998259994st_nat @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_610_fst__conv,axiom,
    ! [X1: set_nat,X23: set_nat] :
      ( ( produc2731966376809613713et_nat @ ( produc4532415448927165861et_nat @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_611_fst__conv,axiom,
    ! [X1: nat,X23: bot_bot_term_f] :
      ( ( produc6861933273117245539term_f @ ( produc5590691411411743351term_f @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_612_fst__conv,axiom,
    ! [X1: nat,X23: nat] :
      ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_613_fst__conv,axiom,
    ! [X1: list_nat,X23: list_nat] :
      ( ( produc1382935764643595205st_nat @ ( produc2694037385005941721st_nat @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_614_fst__conv,axiom,
    ! [X1: produc1828647624359046049st_nat,X23: produc1828647624359046049st_nat] :
      ( ( produc7458156121412276931st_nat @ ( produc988773521843941591st_nat @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_615_old_Oprod_Ocase,axiom,
    ! [F2: nat > nat > $o,X1: nat,X23: nat] :
      ( ( produc6081775807080527818_nat_o @ F2 @ ( product_Pair_nat_nat @ X1 @ X23 ) )
      = ( F2 @ X1 @ X23 ) ) ).

% old.prod.case
thf(fact_616_old_Oprod_Ocase,axiom,
    ! [F2: bot_bot_term_f > bot_bot_term_f > bot_bot_term_f,X1: bot_bot_term_f,X23: bot_bot_term_f] :
      ( ( produc1677810408741468632term_f @ F2 @ ( produc2865564700782054965term_f @ X1 @ X23 ) )
      = ( F2 @ X1 @ X23 ) ) ).

% old.prod.case
thf(fact_617_lenlex__irreflexive,axiom,
    ! [R2: set_Pr3451248702717554689st_nat,Xs2: list_list_nat] :
      ( ! [X3: list_nat] :
          ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ X3 ) @ R2 )
     => ~ ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs2 @ Xs2 ) @ ( lenlex_list_nat @ R2 ) ) ) ).

% lenlex_irreflexive
thf(fact_618_lenlex__irreflexive,axiom,
    ! [R2: set_Pr5488025237498180813et_nat,Xs2: list_set_nat] :
      ( ! [X3: set_nat] :
          ~ ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X3 @ X3 ) @ R2 )
     => ~ ( member911990583396735158et_nat @ ( produc2728781817031571013et_nat @ Xs2 @ Xs2 ) @ ( lenlex_set_nat @ R2 ) ) ) ).

% lenlex_irreflexive
thf(fact_619_lenlex__irreflexive,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,Xs2: list_nat] :
      ( ! [X3: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R2 )
     => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Xs2 ) @ ( lenlex_nat @ R2 ) ) ) ).

% lenlex_irreflexive
thf(fact_620_internal__case__prod__def,axiom,
    produc9139807813499881076_nat_o = produc6081775807080527818_nat_o ).

% internal_case_prod_def
thf(fact_621_internal__case__prod__def,axiom,
    produc30303847659289774term_f = produc1677810408741468632term_f ).

% internal_case_prod_def
thf(fact_622_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: product_prod_nat_nat > list_list_nat > $o,X: product_prod_nat_nat,Y: list_list_nat,A2: produc3922691147685405600st_nat] :
      ( ( P @ X @ Y )
     => ( ( A2
          = ( produc8649917717998259994st_nat @ X @ Y ) )
       => ( P @ ( produc8387770181635782702st_nat @ A2 ) @ ( produc8555213167096887148st_nat @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_623_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: set_nat > set_nat > $o,X: set_nat,Y: set_nat,A2: produc7819656566062154093et_nat] :
      ( ( P @ X @ Y )
     => ( ( A2
          = ( produc4532415448927165861et_nat @ X @ Y ) )
       => ( P @ ( produc2731966376809613713et_nat @ A2 ) @ ( produc397104031625245139et_nat @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_624_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: nat > bot_bot_term_f > $o,X: nat,Y: bot_bot_term_f,A2: produc5418599768131837375term_f] :
      ( ( P @ X @ Y )
     => ( ( A2
          = ( produc5590691411411743351term_f @ X @ Y ) )
       => ( P @ ( produc6861933273117245539term_f @ A2 ) @ ( produc6684081486893607845term_f @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_625_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: nat > nat > $o,X: nat,Y: nat,A2: product_prod_nat_nat] :
      ( ( P @ X @ Y )
     => ( ( A2
          = ( product_Pair_nat_nat @ X @ Y ) )
       => ( P @ ( product_fst_nat_nat @ A2 ) @ ( product_snd_nat_nat @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_626_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: list_nat > list_nat > $o,X: list_nat,Y: list_nat,A2: produc1828647624359046049st_nat] :
      ( ( P @ X @ Y )
     => ( ( A2
          = ( produc2694037385005941721st_nat @ X @ Y ) )
       => ( P @ ( produc1382935764643595205st_nat @ A2 ) @ ( produc5865812112468994567st_nat @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_627_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o,X: produc1828647624359046049st_nat,Y: produc1828647624359046049st_nat,A2: produc1342688880896445543st_nat] :
      ( ( P @ X @ Y )
     => ( ( A2
          = ( produc988773521843941591st_nat @ X @ Y ) )
       => ( P @ ( produc7458156121412276931st_nat @ A2 ) @ ( produc7615485059283090437st_nat @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_628_conjI__realizer,axiom,
    ! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat,Q: list_list_nat > $o,Q2: list_list_nat] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc8387770181635782702st_nat @ ( produc8649917717998259994st_nat @ P2 @ Q2 ) ) )
          & ( Q @ ( produc8555213167096887148st_nat @ ( produc8649917717998259994st_nat @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_629_conjI__realizer,axiom,
    ! [P: set_nat > $o,P2: set_nat,Q: set_nat > $o,Q2: set_nat] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc2731966376809613713et_nat @ ( produc4532415448927165861et_nat @ P2 @ Q2 ) ) )
          & ( Q @ ( produc397104031625245139et_nat @ ( produc4532415448927165861et_nat @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_630_conjI__realizer,axiom,
    ! [P: nat > $o,P2: nat,Q: bot_bot_term_f > $o,Q2: bot_bot_term_f] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc6861933273117245539term_f @ ( produc5590691411411743351term_f @ P2 @ Q2 ) ) )
          & ( Q @ ( produc6684081486893607845term_f @ ( produc5590691411411743351term_f @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_631_conjI__realizer,axiom,
    ! [P: nat > $o,P2: nat,Q: nat > $o,Q2: nat] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P2 @ Q2 ) ) )
          & ( Q @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_632_conjI__realizer,axiom,
    ! [P: list_nat > $o,P2: list_nat,Q: list_nat > $o,Q2: list_nat] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc1382935764643595205st_nat @ ( produc2694037385005941721st_nat @ P2 @ Q2 ) ) )
          & ( Q @ ( produc5865812112468994567st_nat @ ( produc2694037385005941721st_nat @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_633_conjI__realizer,axiom,
    ! [P: produc1828647624359046049st_nat > $o,P2: produc1828647624359046049st_nat,Q: produc1828647624359046049st_nat > $o,Q2: produc1828647624359046049st_nat] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc7458156121412276931st_nat @ ( produc988773521843941591st_nat @ P2 @ Q2 ) ) )
          & ( Q @ ( produc7615485059283090437st_nat @ ( produc988773521843941591st_nat @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_634_exI__realizer,axiom,
    ! [P: list_list_nat > product_prod_nat_nat > $o,Y: list_list_nat,X: product_prod_nat_nat] :
      ( ( P @ Y @ X )
     => ( P @ ( produc8555213167096887148st_nat @ ( produc8649917717998259994st_nat @ X @ Y ) ) @ ( produc8387770181635782702st_nat @ ( produc8649917717998259994st_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_635_exI__realizer,axiom,
    ! [P: set_nat > set_nat > $o,Y: set_nat,X: set_nat] :
      ( ( P @ Y @ X )
     => ( P @ ( produc397104031625245139et_nat @ ( produc4532415448927165861et_nat @ X @ Y ) ) @ ( produc2731966376809613713et_nat @ ( produc4532415448927165861et_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_636_exI__realizer,axiom,
    ! [P: bot_bot_term_f > nat > $o,Y: bot_bot_term_f,X: nat] :
      ( ( P @ Y @ X )
     => ( P @ ( produc6684081486893607845term_f @ ( produc5590691411411743351term_f @ X @ Y ) ) @ ( produc6861933273117245539term_f @ ( produc5590691411411743351term_f @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_637_exI__realizer,axiom,
    ! [P: nat > nat > $o,Y: nat,X: nat] :
      ( ( P @ Y @ X )
     => ( P @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_638_exI__realizer,axiom,
    ! [P: list_nat > list_nat > $o,Y: list_nat,X: list_nat] :
      ( ( P @ Y @ X )
     => ( P @ ( produc5865812112468994567st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) ) @ ( produc1382935764643595205st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_639_exI__realizer,axiom,
    ! [P: produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o,Y: produc1828647624359046049st_nat,X: produc1828647624359046049st_nat] :
      ( ( P @ Y @ X )
     => ( P @ ( produc7615485059283090437st_nat @ ( produc988773521843941591st_nat @ X @ Y ) ) @ ( produc7458156121412276931st_nat @ ( produc988773521843941591st_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_640_size__char__eq__0,axiom,
    ( size_size_char
    = ( ^ [C2: char] : zero_zero_nat ) ) ).

% size_char_eq_0
thf(fact_641_eq__snd__iff,axiom,
    ! [B: list_list_nat,P2: produc3922691147685405600st_nat] :
      ( ( B
        = ( produc8555213167096887148st_nat @ P2 ) )
      = ( ? [A6: product_prod_nat_nat] :
            ( P2
            = ( produc8649917717998259994st_nat @ A6 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_642_eq__snd__iff,axiom,
    ! [B: set_nat,P2: produc7819656566062154093et_nat] :
      ( ( B
        = ( produc397104031625245139et_nat @ P2 ) )
      = ( ? [A6: set_nat] :
            ( P2
            = ( produc4532415448927165861et_nat @ A6 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_643_eq__snd__iff,axiom,
    ! [B: bot_bot_term_f,P2: produc5418599768131837375term_f] :
      ( ( B
        = ( produc6684081486893607845term_f @ P2 ) )
      = ( ? [A6: nat] :
            ( P2
            = ( produc5590691411411743351term_f @ A6 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_644_eq__snd__iff,axiom,
    ! [B: nat,P2: product_prod_nat_nat] :
      ( ( B
        = ( product_snd_nat_nat @ P2 ) )
      = ( ? [A6: nat] :
            ( P2
            = ( product_Pair_nat_nat @ A6 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_645_eq__snd__iff,axiom,
    ! [B: list_nat,P2: produc1828647624359046049st_nat] :
      ( ( B
        = ( produc5865812112468994567st_nat @ P2 ) )
      = ( ? [A6: list_nat] :
            ( P2
            = ( produc2694037385005941721st_nat @ A6 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_646_eq__snd__iff,axiom,
    ! [B: produc1828647624359046049st_nat,P2: produc1342688880896445543st_nat] :
      ( ( B
        = ( produc7615485059283090437st_nat @ P2 ) )
      = ( ? [A6: produc1828647624359046049st_nat] :
            ( P2
            = ( produc988773521843941591st_nat @ A6 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_647_sndI,axiom,
    ! [X: produc3922691147685405600st_nat,Y: product_prod_nat_nat,Z4: list_list_nat] :
      ( ( X
        = ( produc8649917717998259994st_nat @ Y @ Z4 ) )
     => ( ( produc8555213167096887148st_nat @ X )
        = Z4 ) ) ).

% sndI
thf(fact_648_sndI,axiom,
    ! [X: produc7819656566062154093et_nat,Y: set_nat,Z4: set_nat] :
      ( ( X
        = ( produc4532415448927165861et_nat @ Y @ Z4 ) )
     => ( ( produc397104031625245139et_nat @ X )
        = Z4 ) ) ).

% sndI
thf(fact_649_sndI,axiom,
    ! [X: produc5418599768131837375term_f,Y: nat,Z4: bot_bot_term_f] :
      ( ( X
        = ( produc5590691411411743351term_f @ Y @ Z4 ) )
     => ( ( produc6684081486893607845term_f @ X )
        = Z4 ) ) ).

% sndI
thf(fact_650_sndI,axiom,
    ! [X: product_prod_nat_nat,Y: nat,Z4: nat] :
      ( ( X
        = ( product_Pair_nat_nat @ Y @ Z4 ) )
     => ( ( product_snd_nat_nat @ X )
        = Z4 ) ) ).

% sndI
thf(fact_651_sndI,axiom,
    ! [X: produc1828647624359046049st_nat,Y: list_nat,Z4: list_nat] :
      ( ( X
        = ( produc2694037385005941721st_nat @ Y @ Z4 ) )
     => ( ( produc5865812112468994567st_nat @ X )
        = Z4 ) ) ).

% sndI
thf(fact_652_sndI,axiom,
    ! [X: produc1342688880896445543st_nat,Y: produc1828647624359046049st_nat,Z4: produc1828647624359046049st_nat] :
      ( ( X
        = ( produc988773521843941591st_nat @ Y @ Z4 ) )
     => ( ( produc7615485059283090437st_nat @ X )
        = Z4 ) ) ).

% sndI
thf(fact_653_eq__fst__iff,axiom,
    ! [A2: product_prod_nat_nat,P2: produc3922691147685405600st_nat] :
      ( ( A2
        = ( produc8387770181635782702st_nat @ P2 ) )
      = ( ? [B4: list_list_nat] :
            ( P2
            = ( produc8649917717998259994st_nat @ A2 @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_654_eq__fst__iff,axiom,
    ! [A2: set_nat,P2: produc7819656566062154093et_nat] :
      ( ( A2
        = ( produc2731966376809613713et_nat @ P2 ) )
      = ( ? [B4: set_nat] :
            ( P2
            = ( produc4532415448927165861et_nat @ A2 @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_655_eq__fst__iff,axiom,
    ! [A2: nat,P2: produc5418599768131837375term_f] :
      ( ( A2
        = ( produc6861933273117245539term_f @ P2 ) )
      = ( ? [B4: bot_bot_term_f] :
            ( P2
            = ( produc5590691411411743351term_f @ A2 @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_656_eq__fst__iff,axiom,
    ! [A2: nat,P2: product_prod_nat_nat] :
      ( ( A2
        = ( product_fst_nat_nat @ P2 ) )
      = ( ? [B4: nat] :
            ( P2
            = ( product_Pair_nat_nat @ A2 @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_657_eq__fst__iff,axiom,
    ! [A2: list_nat,P2: produc1828647624359046049st_nat] :
      ( ( A2
        = ( produc1382935764643595205st_nat @ P2 ) )
      = ( ? [B4: list_nat] :
            ( P2
            = ( produc2694037385005941721st_nat @ A2 @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_658_eq__fst__iff,axiom,
    ! [A2: produc1828647624359046049st_nat,P2: produc1342688880896445543st_nat] :
      ( ( A2
        = ( produc7458156121412276931st_nat @ P2 ) )
      = ( ? [B4: produc1828647624359046049st_nat] :
            ( P2
            = ( produc988773521843941591st_nat @ A2 @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_659_fstI,axiom,
    ! [X: produc3922691147685405600st_nat,Y: product_prod_nat_nat,Z4: list_list_nat] :
      ( ( X
        = ( produc8649917717998259994st_nat @ Y @ Z4 ) )
     => ( ( produc8387770181635782702st_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_660_fstI,axiom,
    ! [X: produc7819656566062154093et_nat,Y: set_nat,Z4: set_nat] :
      ( ( X
        = ( produc4532415448927165861et_nat @ Y @ Z4 ) )
     => ( ( produc2731966376809613713et_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_661_fstI,axiom,
    ! [X: produc5418599768131837375term_f,Y: nat,Z4: bot_bot_term_f] :
      ( ( X
        = ( produc5590691411411743351term_f @ Y @ Z4 ) )
     => ( ( produc6861933273117245539term_f @ X )
        = Y ) ) ).

% fstI
thf(fact_662_fstI,axiom,
    ! [X: product_prod_nat_nat,Y: nat,Z4: nat] :
      ( ( X
        = ( product_Pair_nat_nat @ Y @ Z4 ) )
     => ( ( product_fst_nat_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_663_fstI,axiom,
    ! [X: produc1828647624359046049st_nat,Y: list_nat,Z4: list_nat] :
      ( ( X
        = ( produc2694037385005941721st_nat @ Y @ Z4 ) )
     => ( ( produc1382935764643595205st_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_664_fstI,axiom,
    ! [X: produc1342688880896445543st_nat,Y: produc1828647624359046049st_nat,Z4: produc1828647624359046049st_nat] :
      ( ( X
        = ( produc988773521843941591st_nat @ Y @ Z4 ) )
     => ( ( produc7458156121412276931st_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_665_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F2: nat > nat > $o,G2: nat > nat > $o,P2: product_prod_nat_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( ( product_Pair_nat_nat @ X3 @ Y3 )
            = Q2 )
         => ( ( F2 @ X3 @ Y3 )
            = ( G2 @ X3 @ Y3 ) ) )
     => ( ( P2 = Q2 )
       => ( ( produc6081775807080527818_nat_o @ F2 @ P2 )
          = ( produc6081775807080527818_nat_o @ G2 @ Q2 ) ) ) ) ).

% split_cong
thf(fact_666_split__cong,axiom,
    ! [Q2: produc250624644672406269term_f,F2: bot_bot_term_f > bot_bot_term_f > bot_bot_term_f,G2: bot_bot_term_f > bot_bot_term_f > bot_bot_term_f,P2: produc250624644672406269term_f] :
      ( ! [X3: bot_bot_term_f,Y3: bot_bot_term_f] :
          ( ( ( produc2865564700782054965term_f @ X3 @ Y3 )
            = Q2 )
         => ( ( F2 @ X3 @ Y3 )
            = ( G2 @ X3 @ Y3 ) ) )
     => ( ( P2 = Q2 )
       => ( ( produc1677810408741468632term_f @ F2 @ P2 )
          = ( produc1677810408741468632term_f @ G2 @ Q2 ) ) ) ) ).

% split_cong
thf(fact_667_mem__case__prodE,axiom,
    ! [Z4: nat,C3: nat > nat > set_nat,P2: product_prod_nat_nat] :
      ( ( member_nat @ Z4 @ ( produc6189476227299908564et_nat @ C3 @ P2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z4 @ ( C3 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_668_mem__case__prodE,axiom,
    ! [Z4: char,C3: nat > nat > set_char,P2: product_prod_nat_nat] :
      ( ( member_char @ Z4 @ ( produc1561271593892038229t_char @ C3 @ P2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member_char @ Z4 @ ( C3 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_669_mem__case__prodE,axiom,
    ! [Z4: f,C3: nat > nat > set_f,P2: product_prod_nat_nat] :
      ( ( member_f @ Z4 @ ( produc8052394805345727765_set_f @ C3 @ P2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member_f @ Z4 @ ( C3 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_670_mem__case__prodE,axiom,
    ! [Z4: nat,C3: nat > bot_bot_term_f > set_nat,P2: produc5418599768131837375term_f] :
      ( ( member_nat @ Z4 @ ( produc6142541587184120338et_nat @ C3 @ P2 ) )
     => ~ ! [X3: nat,Y3: bot_bot_term_f] :
            ( ( P2
              = ( produc5590691411411743351term_f @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z4 @ ( C3 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_671_mem__case__prodE,axiom,
    ! [Z4: char,C3: nat > bot_bot_term_f > set_char,P2: produc5418599768131837375term_f] :
      ( ( member_char @ Z4 @ ( produc8097753862531554963t_char @ C3 @ P2 ) )
     => ~ ! [X3: nat,Y3: bot_bot_term_f] :
            ( ( P2
              = ( produc5590691411411743351term_f @ X3 @ Y3 ) )
           => ~ ( member_char @ Z4 @ ( C3 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_672_mem__case__prodE,axiom,
    ! [Z4: f,C3: nat > bot_bot_term_f > set_f,P2: produc5418599768131837375term_f] :
      ( ( member_f @ Z4 @ ( produc2816659430985364055_set_f @ C3 @ P2 ) )
     => ~ ! [X3: nat,Y3: bot_bot_term_f] :
            ( ( P2
              = ( produc5590691411411743351term_f @ X3 @ Y3 ) )
           => ~ ( member_f @ Z4 @ ( C3 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_673_mem__case__prodE,axiom,
    ! [Z4: list_nat,C3: nat > nat > set_list_nat,P2: product_prod_nat_nat] :
      ( ( member_list_nat @ Z4 @ ( produc5054468372610444900st_nat @ C3 @ P2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member_list_nat @ Z4 @ ( C3 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_674_mem__case__prodE,axiom,
    ! [Z4: nat,C3: list_nat > list_nat > set_nat,P2: produc1828647624359046049st_nat] :
      ( ( member_nat @ Z4 @ ( produc7822140737630736756et_nat @ C3 @ P2 ) )
     => ~ ! [X3: list_nat,Y3: list_nat] :
            ( ( P2
              = ( produc2694037385005941721st_nat @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z4 @ ( C3 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_675_mem__case__prodE,axiom,
    ! [Z4: char,C3: list_nat > list_nat > set_char,P2: produc1828647624359046049st_nat] :
      ( ( member_char @ Z4 @ ( produc5516737219024788469t_char @ C3 @ P2 ) )
     => ~ ! [X3: list_nat,Y3: list_nat] :
            ( ( P2
              = ( produc2694037385005941721st_nat @ X3 @ Y3 ) )
           => ~ ( member_char @ Z4 @ ( C3 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_676_mem__case__prodE,axiom,
    ! [Z4: f,C3: list_nat > list_nat > set_f,P2: produc1828647624359046049st_nat] :
      ( ( member_f @ Z4 @ ( produc9005509448160470901_set_f @ C3 @ P2 ) )
     => ~ ! [X3: list_nat,Y3: list_nat] :
            ( ( P2
              = ( produc2694037385005941721st_nat @ X3 @ Y3 ) )
           => ~ ( member_f @ Z4 @ ( C3 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_677_lexord__asym,axiom,
    ! [R: set_Pr3451248702717554689st_nat] :
      ( ( asym_on_list_nat @ top_top_set_list_nat @ R )
     => ( asym_o1052983387450567102st_nat @ top_to8617644770344506070st_nat @ ( lexord_list_nat @ R ) ) ) ).

% lexord_asym
thf(fact_678_lexord__asym,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R )
     => ( asym_on_list_nat @ top_top_set_list_nat @ ( lexord_nat @ R ) ) ) ).

% lexord_asym
thf(fact_679_Cons__in__lex,axiom,
    ! [X: set_nat,Xs2: list_set_nat,Y: set_nat,Ys: list_set_nat,R2: set_Pr5488025237498180813et_nat] :
      ( ( member911990583396735158et_nat @ ( produc2728781817031571013et_nat @ ( cons_set_nat @ X @ Xs2 ) @ ( cons_set_nat @ Y @ Ys ) ) @ ( lex_set_nat @ R2 ) )
      = ( ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X @ Y ) @ R2 )
          & ( ( size_s3254054031482475050et_nat @ Xs2 )
            = ( size_s3254054031482475050et_nat @ Ys ) ) )
        | ( ( X = Y )
          & ( member911990583396735158et_nat @ ( produc2728781817031571013et_nat @ Xs2 @ Ys ) @ ( lex_set_nat @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_680_Cons__in__lex,axiom,
    ! [X: bot_bot_term_f,Xs2: list_Bot_bot_term_f,Y: bot_bot_term_f,Ys: list_Bot_bot_term_f,R2: set_Pr6841673932529763677term_f] :
      ( ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ ( cons_Bot_bot_term_f @ X @ Xs2 ) @ ( cons_Bot_bot_term_f @ Y @ Ys ) ) @ ( lex_Bot_bot_term_f @ R2 ) )
      = ( ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ R2 )
          & ( ( size_s1458857753309042354term_f @ Xs2 )
            = ( size_s1458857753309042354term_f @ Ys ) ) )
        | ( ( X = Y )
          & ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Xs2 @ Ys ) @ ( lex_Bot_bot_term_f @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_681_Cons__in__lex,axiom,
    ! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys ) ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) )
      = ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
          & ( ( size_s5460976970255530739at_nat @ Xs2 )
            = ( size_s5460976970255530739at_nat @ Ys ) ) )
        | ( ( X = Y )
          & ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Ys ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_682_Cons__in__lex,axiom,
    ! [X: list_nat,Xs2: list_list_nat,Y: list_nat,Ys: list_list_nat,R2: set_Pr3451248702717554689st_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ X @ Xs2 ) @ ( cons_list_nat @ Y @ Ys ) ) @ ( lex_list_nat @ R2 ) )
      = ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R2 )
          & ( ( size_s3023201423986296836st_nat @ Xs2 )
            = ( size_s3023201423986296836st_nat @ Ys ) ) )
        | ( ( X = Y )
          & ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs2 @ Ys ) @ ( lex_list_nat @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_683_Cons__in__lex,axiom,
    ! [X: nat,Xs2: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys ) ) @ ( lex_nat @ R2 ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
          & ( ( size_size_list_nat @ Xs2 )
            = ( size_size_list_nat @ Ys ) ) )
        | ( ( X = Y )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( lex_nat @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_684_bot__term_Osize__gen_I1_J,axiom,
    ! [X: nat > nat] :
      ( ( bot_bo2025320268590814892rm_nat @ X @ bot_bot_Bot_nat )
      = zero_zero_nat ) ).

% bot_term.size_gen(1)
thf(fact_685_bot__term_Osize__gen_I1_J,axiom,
    ! [X: f > nat] :
      ( ( bot_bo4099906974609907495term_f @ X @ bot_bot_Bot_f )
      = zero_zero_nat ) ).

% bot_term.size_gen(1)
thf(fact_686_length__code,axiom,
    ( size_s1458857753309042354term_f
    = ( gen_le6202096371021315387term_f @ zero_zero_nat ) ) ).

% length_code
thf(fact_687_length__code,axiom,
    ( size_s5460976970255530739at_nat
    = ( gen_le2383899666085517716at_nat @ zero_zero_nat ) ) ).

% length_code
thf(fact_688_length__code,axiom,
    ( size_s3023201423986296836st_nat
    = ( gen_length_list_nat @ zero_zero_nat ) ) ).

% length_code
thf(fact_689_length__code,axiom,
    ( size_size_list_nat
    = ( gen_length_nat @ zero_zero_nat ) ) ).

% length_code
thf(fact_690_lexn__length,axiom,
    ! [Xs2: list_Bot_bot_term_f,Ys: list_Bot_bot_term_f,R2: set_Pr6841673932529763677term_f,N: nat] :
      ( ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Xs2 @ Ys ) @ ( lexn_Bot_bot_term_f @ R2 @ N ) )
     => ( ( ( size_s1458857753309042354term_f @ Xs2 )
          = N )
        & ( ( size_s1458857753309042354term_f @ Ys )
          = N ) ) ) ).

% lexn_length
thf(fact_691_lexn__length,axiom,
    ! [Xs2: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat,N: nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Ys ) @ ( lexn_P849066182667564199at_nat @ R2 @ N ) )
     => ( ( ( size_s5460976970255530739at_nat @ Xs2 )
          = N )
        & ( ( size_s5460976970255530739at_nat @ Ys )
          = N ) ) ) ).

% lexn_length
thf(fact_692_lexn__length,axiom,
    ! [Xs2: list_list_nat,Ys: list_list_nat,R2: set_Pr3451248702717554689st_nat,N: nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs2 @ Ys ) @ ( lexn_list_nat @ R2 @ N ) )
     => ( ( ( size_s3023201423986296836st_nat @ Xs2 )
          = N )
        & ( ( size_s3023201423986296836st_nat @ Ys )
          = N ) ) ) ).

% lexn_length
thf(fact_693_lexn__length,axiom,
    ! [Xs2: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat,N: nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( lexn_nat @ R2 @ N ) )
     => ( ( ( size_size_list_nat @ Xs2 )
          = N )
        & ( ( size_size_list_nat @ Ys )
          = N ) ) ) ).

% lexn_length
thf(fact_694_lexl__not__refl,axiom,
    ! [R2: set_Pr3451248702717554689st_nat,X: list_list_nat] :
      ( ( irrefl_on_list_nat @ top_top_set_list_nat @ R2 )
     => ~ ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ X @ X ) @ ( lex_list_nat @ R2 ) ) ) ).

% lexl_not_refl
thf(fact_695_lexl__not__refl,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X: list_nat] :
      ( ( irrefl_on_nat @ top_top_set_nat @ R2 )
     => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ X ) @ ( lex_nat @ R2 ) ) ) ).

% lexl_not_refl
thf(fact_696_divides__aux__eq,axiom,
    ! [Q2: nat,R2: nat] :
      ( ( unique5332122412489317741ux_nat @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
      = ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_697_count__notin,axiom,
    ! [X: produc1828647624359046049st_nat,Xs2: list_P7940050157051400743st_nat] :
      ( ~ ( member7340969449405702474st_nat @ X @ ( set_Pr3842133991353686454st_nat @ Xs2 ) )
     => ( ( count_8697817428404038429st_nat @ Xs2 @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_698_count__notin,axiom,
    ! [X: f,Xs2: list_f] :
      ( ~ ( member_f @ X @ ( set_f2 @ Xs2 ) )
     => ( ( count_list_f @ Xs2 @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_699_count__notin,axiom,
    ! [X: bot_bot_term_f,Xs2: list_Bot_bot_term_f] :
      ( ~ ( member2723211850833494437term_f @ X @ ( set_Bot_bot_term_f2 @ Xs2 ) )
     => ( ( count_166185096570534930term_f @ Xs2 @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_700_count__notin,axiom,
    ! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
     => ( ( count_4203492906077236349at_nat @ Xs2 @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_701_count__notin,axiom,
    ! [X: char,Xs2: list_char] :
      ( ~ ( member_char @ X @ ( set_char2 @ Xs2 ) )
     => ( ( count_list_char @ Xs2 @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_702_count__notin,axiom,
    ! [X: list_nat,Xs2: list_list_nat] :
      ( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
     => ( ( count_list_list_nat @ Xs2 @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_703_count__notin,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ( ( count_list_nat @ Xs2 @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_704_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_705_list_Oinject,axiom,
    ! [X21: bot_bot_term_f,X22: list_Bot_bot_term_f,Y21: bot_bot_term_f,Y22: list_Bot_bot_term_f] :
      ( ( ( cons_Bot_bot_term_f @ X21 @ X22 )
        = ( cons_Bot_bot_term_f @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_706_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_707_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_708_zip__Cons__Cons,axiom,
    ! [X: nat,Xs2: list_nat,Y: list_nat,Ys: list_list_nat] :
      ( ( zip_nat_list_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_list_nat @ Y @ Ys ) )
      = ( cons_P6176896814948630215st_nat @ ( produc8282810413953273033st_nat @ X @ Y ) @ ( zip_nat_list_nat @ Xs2 @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_709_zip__Cons__Cons,axiom,
    ! [X: bot_bot_term_f,Xs2: list_Bot_bot_term_f,Y: nat,Ys: list_nat] :
      ( ( zip_Bo4866563780717559831_f_nat @ ( cons_Bot_bot_term_f @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys ) )
      = ( cons_P8275894281584890613_f_nat @ ( produc6944956222323935479_f_nat @ X @ Y ) @ ( zip_Bo4866563780717559831_f_nat @ Xs2 @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_710_zip__Cons__Cons,axiom,
    ! [X: list_nat,Xs2: list_list_nat,Y: nat,Ys: list_nat] :
      ( ( zip_list_nat_nat @ ( cons_list_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys ) )
      = ( cons_P3142513297430851911at_nat @ ( produc1298395424260782409at_nat @ X @ Y ) @ ( zip_list_nat_nat @ Xs2 @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_711_zip__Cons__Cons,axiom,
    ! [X: nat,Xs2: list_nat,Y: bot_bot_term_f,Ys: list_Bot_bot_term_f] :
      ( ( zip_na3512298969805367703term_f @ ( cons_nat @ X @ Xs2 ) @ ( cons_Bot_bot_term_f @ Y @ Ys ) )
      = ( cons_P5276694725835343733term_f @ ( produc5590691411411743351term_f @ X @ Y ) @ ( zip_na3512298969805367703term_f @ Xs2 @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_712_zip__Cons__Cons,axiom,
    ! [X: bot_bot_term_f,Xs2: list_Bot_bot_term_f,Y: bot_bot_term_f,Ys: list_Bot_bot_term_f] :
      ( ( zip_Bo7813815397675234645term_f @ ( cons_Bot_bot_term_f @ X @ Xs2 ) @ ( cons_Bot_bot_term_f @ Y @ Ys ) )
      = ( cons_P3711653291972799923term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( zip_Bo7813815397675234645term_f @ Xs2 @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_713_zip__Cons__Cons,axiom,
    ! [X: bot_bot_term_f,Xs2: list_Bot_bot_term_f,Y: list_nat,Ys: list_list_nat] :
      ( ( zip_Bo1182424480293800871st_nat @ ( cons_Bot_bot_term_f @ X @ Xs2 ) @ ( cons_list_nat @ Y @ Ys ) )
      = ( cons_P8785853885591068933st_nat @ ( produc6629678290114667655st_nat @ X @ Y ) @ ( zip_Bo1182424480293800871st_nat @ Xs2 @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_714_zip__Cons__Cons,axiom,
    ! [X: list_nat,Xs2: list_list_nat,Y: bot_bot_term_f,Ys: list_Bot_bot_term_f] :
      ( ( zip_li7372370571125813415term_f @ ( cons_list_nat @ X @ Xs2 ) @ ( cons_Bot_bot_term_f @ Y @ Ys ) )
      = ( cons_P8057524666905663237term_f @ ( produc3596252344091904391term_f @ X @ Y ) @ ( zip_li7372370571125813415term_f @ Xs2 @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_715_zip__Cons__Cons,axiom,
    ! [X: list_nat,Xs2: list_list_nat,Y: list_nat,Ys: list_list_nat] :
      ( ( zip_li7157463729305086713st_nat @ ( cons_list_nat @ X @ Xs2 ) @ ( cons_list_nat @ Y @ Ys ) )
      = ( cons_P5007559046487125591st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( zip_li7157463729305086713st_nat @ Xs2 @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_716_zip__Cons__Cons,axiom,
    ! [X: set_nat,Xs2: list_set_nat,Y: set_nat,Ys: list_set_nat] :
      ( ( zip_set_nat_set_nat @ ( cons_set_nat @ X @ Xs2 ) @ ( cons_set_nat @ Y @ Ys ) )
      = ( cons_P7941000567243684387et_nat @ ( produc4532415448927165861et_nat @ X @ Y ) @ ( zip_set_nat_set_nat @ Xs2 @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_717_lexord__cons__cons,axiom,
    ! [A2: bot_bot_term_f,X: list_Bot_bot_term_f,B: bot_bot_term_f,Y: list_Bot_bot_term_f,R2: set_Pr6841673932529763677term_f] :
      ( ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ ( cons_Bot_bot_term_f @ A2 @ X ) @ ( cons_Bot_bot_term_f @ B @ Y ) ) @ ( lexord943918551247894875term_f @ R2 ) )
      = ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A2 @ B ) @ R2 )
        | ( ( A2 = B )
          & ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ X @ Y ) @ ( lexord943918551247894875term_f @ R2 ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_718_lexord__cons__cons,axiom,
    ! [A2: list_nat,X: list_list_nat,B: list_nat,Y: list_list_nat,R2: set_Pr3451248702717554689st_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ A2 @ X ) @ ( cons_list_nat @ B @ Y ) ) @ ( lexord_list_nat @ R2 ) )
      = ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A2 @ B ) @ R2 )
        | ( ( A2 = B )
          & ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ X @ Y ) @ ( lexord_list_nat @ R2 ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_719_lexord__cons__cons,axiom,
    ! [A2: set_nat,X: list_set_nat,B: set_nat,Y: list_set_nat,R2: set_Pr5488025237498180813et_nat] :
      ( ( member911990583396735158et_nat @ ( produc2728781817031571013et_nat @ ( cons_set_nat @ A2 @ X ) @ ( cons_set_nat @ B @ Y ) ) @ ( lexord_set_nat @ R2 ) )
      = ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A2 @ B ) @ R2 )
        | ( ( A2 = B )
          & ( member911990583396735158et_nat @ ( produc2728781817031571013et_nat @ X @ Y ) @ ( lexord_set_nat @ R2 ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_720_lexord__cons__cons,axiom,
    ! [A2: nat,X: list_nat,B: nat,Y: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ A2 @ X ) @ ( cons_nat @ B @ Y ) ) @ ( lexord_nat @ R2 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
        | ( ( A2 = B )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lexord_nat @ R2 ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_721_not__Cons__self2,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( cons_nat @ X @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_722_not__Cons__self2,axiom,
    ! [X: bot_bot_term_f,Xs2: list_Bot_bot_term_f] :
      ( ( cons_Bot_bot_term_f @ X @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_723_not__Cons__self2,axiom,
    ! [X: list_nat,Xs2: list_list_nat] :
      ( ( cons_list_nat @ X @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_724_lexord__irrefl,axiom,
    ! [R: set_Pr3451248702717554689st_nat] :
      ( ( irrefl_on_list_nat @ top_top_set_list_nat @ R )
     => ( irrefl1702768165254659578st_nat @ top_to8617644770344506070st_nat @ ( lexord_list_nat @ R ) ) ) ).

% lexord_irrefl
thf(fact_725_lexord__irrefl,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( irrefl_on_nat @ top_top_set_nat @ R )
     => ( irrefl_on_list_nat @ top_top_set_list_nat @ ( lexord_nat @ R ) ) ) ).

% lexord_irrefl
thf(fact_726_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,Xs4: list_nat] :
            ( ( Xs2
              = ( cons_nat @ X3 @ Xs4 ) )
           => ! [Y3: nat,Ys3: list_nat] :
                ( ( Ys
                  = ( cons_nat @ Y3 @ Ys3 ) )
               => ( ( Xy
                    = ( product_Pair_nat_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_nat_nat @ Xs4 @ Ys3 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_727_zip__eq__ConsE,axiom,
    ! [Xs2: list_nat,Ys: list_list_nat,Xy: produc4575160907756185873st_nat,Xys: list_P7736225833432154391st_nat] :
      ( ( ( zip_nat_list_nat @ Xs2 @ Ys )
        = ( cons_P6176896814948630215st_nat @ Xy @ Xys ) )
     => ~ ! [X3: nat,Xs4: list_nat] :
            ( ( Xs2
              = ( cons_nat @ X3 @ Xs4 ) )
           => ! [Y3: list_nat,Ys3: list_list_nat] :
                ( ( Ys
                  = ( cons_list_nat @ Y3 @ Ys3 ) )
               => ( ( Xy
                    = ( produc8282810413953273033st_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_nat_list_nat @ Xs4 @ Ys3 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_728_zip__eq__ConsE,axiom,
    ! [Xs2: list_Bot_bot_term_f,Ys: list_nat,Xy: produc8417799323881384255_f_nat,Xys: list_P8521082412280730693_f_nat] :
      ( ( ( zip_Bo4866563780717559831_f_nat @ Xs2 @ Ys )
        = ( cons_P8275894281584890613_f_nat @ Xy @ Xys ) )
     => ~ ! [X3: bot_bot_term_f,Xs4: list_Bot_bot_term_f] :
            ( ( Xs2
              = ( cons_Bot_bot_term_f @ X3 @ Xs4 ) )
           => ! [Y3: nat,Ys3: list_nat] :
                ( ( Ys
                  = ( cons_nat @ Y3 @ Ys3 ) )
               => ( ( Xy
                    = ( produc6944956222323935479_f_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_Bo4866563780717559831_f_nat @ Xs4 @ Ys3 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_729_zip__eq__ConsE,axiom,
    ! [Xs2: list_list_nat,Ys: list_nat,Xy: produc1540777390238407569at_nat,Xys: list_P5364314822750548887at_nat] :
      ( ( ( zip_list_nat_nat @ Xs2 @ Ys )
        = ( cons_P3142513297430851911at_nat @ Xy @ Xys ) )
     => ~ ! [X3: list_nat,Xs4: list_list_nat] :
            ( ( Xs2
              = ( cons_list_nat @ X3 @ Xs4 ) )
           => ! [Y3: nat,Ys3: list_nat] :
                ( ( Ys
                  = ( cons_nat @ Y3 @ Ys3 ) )
               => ( ( Xy
                    = ( produc1298395424260782409at_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_list_nat_nat @ Xs4 @ Ys3 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_730_zip__eq__ConsE,axiom,
    ! [Xs2: list_nat,Ys: list_Bot_bot_term_f,Xy: produc5418599768131837375term_f,Xys: list_P3405996666353486021term_f] :
      ( ( ( zip_na3512298969805367703term_f @ Xs2 @ Ys )
        = ( cons_P5276694725835343733term_f @ Xy @ Xys ) )
     => ~ ! [X3: nat,Xs4: list_nat] :
            ( ( Xs2
              = ( cons_nat @ X3 @ Xs4 ) )
           => ! [Y3: bot_bot_term_f,Ys3: list_Bot_bot_term_f] :
                ( ( Ys
                  = ( cons_Bot_bot_term_f @ Y3 @ Ys3 ) )
               => ( ( Xy
                    = ( produc5590691411411743351term_f @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_na3512298969805367703term_f @ Xs4 @ Ys3 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_731_zip__eq__ConsE,axiom,
    ! [Xs2: list_Bot_bot_term_f,Ys: list_Bot_bot_term_f,Xy: produc250624644672406269term_f,Xys: list_P1780566427699531651term_f] :
      ( ( ( zip_Bo7813815397675234645term_f @ Xs2 @ Ys )
        = ( cons_P3711653291972799923term_f @ Xy @ Xys ) )
     => ~ ! [X3: bot_bot_term_f,Xs4: list_Bot_bot_term_f] :
            ( ( Xs2
              = ( cons_Bot_bot_term_f @ X3 @ Xs4 ) )
           => ! [Y3: bot_bot_term_f,Ys3: list_Bot_bot_term_f] :
                ( ( Ys
                  = ( cons_Bot_bot_term_f @ Y3 @ Ys3 ) )
               => ( ( Xy
                    = ( produc2865564700782054965term_f @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_Bo7813815397675234645term_f @ Xs4 @ Ys3 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_732_zip__eq__ConsE,axiom,
    ! [Xs2: list_Bot_bot_term_f,Ys: list_list_nat,Xy: produc5499396499697224271st_nat,Xys: list_P3080432348914762709st_nat] :
      ( ( ( zip_Bo1182424480293800871st_nat @ Xs2 @ Ys )
        = ( cons_P8785853885591068933st_nat @ Xy @ Xys ) )
     => ~ ! [X3: bot_bot_term_f,Xs4: list_Bot_bot_term_f] :
            ( ( Xs2
              = ( cons_Bot_bot_term_f @ X3 @ Xs4 ) )
           => ! [Y3: list_nat,Ys3: list_list_nat] :
                ( ( Ys
                  = ( cons_list_nat @ Y3 @ Ys3 ) )
               => ( ( Xy
                    = ( produc6629678290114667655st_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_Bo1182424480293800871st_nat @ Xs4 @ Ys3 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_733_zip__eq__ConsE,axiom,
    ! [Xs2: list_list_nat,Ys: list_Bot_bot_term_f,Xy: produc4771067281011818575term_f,Xys: list_P9018165577420421589term_f] :
      ( ( ( zip_li7372370571125813415term_f @ Xs2 @ Ys )
        = ( cons_P8057524666905663237term_f @ Xy @ Xys ) )
     => ~ ! [X3: list_nat,Xs4: list_list_nat] :
            ( ( Xs2
              = ( cons_list_nat @ X3 @ Xs4 ) )
           => ! [Y3: bot_bot_term_f,Ys3: list_Bot_bot_term_f] :
                ( ( Ys
                  = ( cons_Bot_bot_term_f @ Y3 @ Ys3 ) )
               => ( ( Xy
                    = ( produc3596252344091904391term_f @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_li7372370571125813415term_f @ Xs4 @ Ys3 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_734_zip__eq__ConsE,axiom,
    ! [Xs2: list_list_nat,Ys: list_list_nat,Xy: produc1828647624359046049st_nat,Xys: list_P7940050157051400743st_nat] :
      ( ( ( zip_li7157463729305086713st_nat @ Xs2 @ Ys )
        = ( cons_P5007559046487125591st_nat @ Xy @ Xys ) )
     => ~ ! [X3: list_nat,Xs4: list_list_nat] :
            ( ( Xs2
              = ( cons_list_nat @ X3 @ Xs4 ) )
           => ! [Y3: list_nat,Ys3: list_list_nat] :
                ( ( Ys
                  = ( cons_list_nat @ Y3 @ Ys3 ) )
               => ( ( Xy
                    = ( produc2694037385005941721st_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_li7157463729305086713st_nat @ Xs4 @ Ys3 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_735_zip__eq__ConsE,axiom,
    ! [Xs2: list_set_nat,Ys: list_set_nat,Xy: produc7819656566062154093et_nat,Xys: list_P6254988961118846195et_nat] :
      ( ( ( zip_set_nat_set_nat @ Xs2 @ Ys )
        = ( cons_P7941000567243684387et_nat @ Xy @ Xys ) )
     => ~ ! [X3: set_nat,Xs4: list_set_nat] :
            ( ( Xs2
              = ( cons_set_nat @ X3 @ Xs4 ) )
           => ! [Y3: set_nat,Ys3: list_set_nat] :
                ( ( Ys
                  = ( cons_set_nat @ Y3 @ Ys3 ) )
               => ( ( Xy
                    = ( produc4532415448927165861et_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_set_nat_set_nat @ Xs4 @ Ys3 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_736_list_Oset__intros_I2_J,axiom,
    ! [Y: produc1828647624359046049st_nat,X22: list_P7940050157051400743st_nat,X21: produc1828647624359046049st_nat] :
      ( ( member7340969449405702474st_nat @ Y @ ( set_Pr3842133991353686454st_nat @ X22 ) )
     => ( member7340969449405702474st_nat @ Y @ ( set_Pr3842133991353686454st_nat @ ( cons_P5007559046487125591st_nat @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_737_list_Oset__intros_I2_J,axiom,
    ! [Y: f,X22: list_f,X21: f] :
      ( ( member_f @ Y @ ( set_f2 @ X22 ) )
     => ( member_f @ Y @ ( set_f2 @ ( cons_f @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_738_list_Oset__intros_I2_J,axiom,
    ! [Y: product_prod_nat_nat,X22: list_P6011104703257516679at_nat,X21: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ X22 ) )
     => ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_739_list_Oset__intros_I2_J,axiom,
    ! [Y: char,X22: list_char,X21: char] :
      ( ( member_char @ Y @ ( set_char2 @ X22 ) )
     => ( member_char @ Y @ ( set_char2 @ ( cons_char @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_740_list_Oset__intros_I2_J,axiom,
    ! [Y: bot_bot_term_f,X22: list_Bot_bot_term_f,X21: bot_bot_term_f] :
      ( ( member2723211850833494437term_f @ Y @ ( set_Bot_bot_term_f2 @ X22 ) )
     => ( member2723211850833494437term_f @ Y @ ( set_Bot_bot_term_f2 @ ( cons_Bot_bot_term_f @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_741_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_742_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_743_list_Oset__intros_I1_J,axiom,
    ! [X21: produc1828647624359046049st_nat,X22: list_P7940050157051400743st_nat] : ( member7340969449405702474st_nat @ X21 @ ( set_Pr3842133991353686454st_nat @ ( cons_P5007559046487125591st_nat @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_744_list_Oset__intros_I1_J,axiom,
    ! [X21: f,X22: list_f] : ( member_f @ X21 @ ( set_f2 @ ( cons_f @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_745_list_Oset__intros_I1_J,axiom,
    ! [X21: product_prod_nat_nat,X22: list_P6011104703257516679at_nat] : ( member8440522571783428010at_nat @ X21 @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_746_list_Oset__intros_I1_J,axiom,
    ! [X21: char,X22: list_char] : ( member_char @ X21 @ ( set_char2 @ ( cons_char @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_747_list_Oset__intros_I1_J,axiom,
    ! [X21: bot_bot_term_f,X22: list_Bot_bot_term_f] : ( member2723211850833494437term_f @ X21 @ ( set_Bot_bot_term_f2 @ ( cons_Bot_bot_term_f @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_748_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_749_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_750_list_Oset__cases,axiom,
    ! [E: nat,A2: list_nat] :
      ( ( member_nat @ E @ ( set_nat2 @ A2 ) )
     => ( ! [Z2: list_nat] :
            ( A2
           != ( cons_nat @ E @ Z2 ) )
       => ~ ! [Z1: nat,Z2: list_nat] :
              ( ( A2
                = ( cons_nat @ Z1 @ Z2 ) )
             => ~ ( member_nat @ E @ ( set_nat2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_751_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_752_irrefl__onD,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat] :
      ( ( irrefl_on_nat @ A @ R2 )
     => ( ( member_nat @ A2 @ A )
       => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ R2 ) ) ) ).

% irrefl_onD
thf(fact_753_irrefl__onI,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ! [A5: nat] :
          ( ( member_nat @ A5 @ A )
         => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ A5 ) @ R2 ) )
     => ( irrefl_on_nat @ A @ R2 ) ) ).

% irrefl_onI
thf(fact_754_irrefl__lex,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( irrefl_on_nat @ top_top_set_nat @ R2 )
     => ( irrefl_on_list_nat @ top_top_set_list_nat @ ( lex_nat @ R2 ) ) ) ).

% irrefl_lex
thf(fact_755_irreflI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [A5: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ A5 ) @ R2 )
     => ( irrefl_on_nat @ top_top_set_nat @ R2 ) ) ).

% irreflI
thf(fact_756_irreflD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X: nat] :
      ( ( irrefl_on_nat @ top_top_set_nat @ R2 )
     => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R2 ) ) ).

% irreflD
thf(fact_757_count__list__0__iff,axiom,
    ! [Xs2: list_nat,X: nat] :
      ( ( ( count_list_nat @ Xs2 @ X )
        = zero_zero_nat )
      = ( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) ) ) ) ).

% count_list_0_iff
thf(fact_758_lexord__partial__trans,axiom,
    ! [Xs2: list_nat,R2: set_Pr1261947904930325089at_nat,Ys: list_nat,Zs: list_nat] :
      ( ! [X3: nat,Y3: nat,Z3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z3 ) @ R2 )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z3 ) @ R2 ) ) ) )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( lexord_nat @ R2 ) )
       => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lexord_nat @ R2 ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Zs ) @ ( lexord_nat @ R2 ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_759_lexord__asymmetric,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A2: list_nat,B: list_nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A2 @ B ) @ ( lexord_nat @ R ) )
       => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ A2 ) @ ( lexord_nat @ R ) ) ) ) ).

% lexord_asymmetric
thf(fact_760_lexord__lex,axiom,
    ! [X: list_nat,Y: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lex_nat @ R2 ) )
      = ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lexord_nat @ R2 ) )
        & ( ( size_size_list_nat @ X )
          = ( size_size_list_nat @ Y ) ) ) ) ).

% lexord_lex
thf(fact_761_in__set__simps_I1_J,axiom,
    ! [X: nat,Y: nat,Z4: nat,Ys: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( cons_nat @ Y @ ( cons_nat @ Z4 @ Ys ) ) ) )
      = ( ( X = Y )
        | ( member_nat @ X @ ( set_nat2 @ ( cons_nat @ Z4 @ Ys ) ) ) ) ) ).

% in_set_simps(1)
thf(fact_762_Cons__lenlex__iff,axiom,
    ! [M: nat,Ms: list_nat,N: nat,Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R2 ) )
      = ( ( 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 ) @ R2 ) )
        | ( ( M = N )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R2 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_763_lenlex__append2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,Us2: list_nat,Xs2: list_nat,Ys: list_nat] :
      ( ( irrefl_on_nat @ top_top_set_nat @ R )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Us2 @ Xs2 ) @ ( append_nat @ Us2 @ Ys ) ) @ ( lenlex_nat @ R ) )
        = ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( lenlex_nat @ R ) ) ) ) ).

% lenlex_append2
thf(fact_764_lexord__same__pref__if__irrefl,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( irrefl_on_nat @ top_top_set_nat @ R2 )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Ys ) @ ( append_nat @ Xs2 @ Zs ) ) @ ( lexord_nat @ R2 ) )
        = ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lexord_nat @ R2 ) ) ) ) ).

% lexord_same_pref_if_irrefl
thf(fact_765_not__in__set__insert,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ( ( insert_nat @ X @ Xs2 )
        = ( cons_nat @ X @ Xs2 ) ) ) ).

% not_in_set_insert
thf(fact_766_append__eq__append__conv,axiom,
    ! [Xs2: list_nat,Ys: list_nat,Us2: list_nat,Vs: list_nat] :
      ( ( ( ( size_size_list_nat @ Xs2 )
          = ( size_size_list_nat @ Ys ) )
        | ( ( size_size_list_nat @ Us2 )
          = ( size_size_list_nat @ Vs ) ) )
     => ( ( ( append_nat @ Xs2 @ Us2 )
          = ( append_nat @ Ys @ Vs ) )
        = ( ( Xs2 = Ys )
          & ( Us2 = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_767_in__set__insert,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ( ( insert_nat @ X @ Xs2 )
        = Xs2 ) ) ).

% in_set_insert
thf(fact_768_length__0__conv,axiom,
    ! [Xs2: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = zero_zero_nat )
      = ( Xs2 = nil_nat ) ) ).

% length_0_conv
thf(fact_769_replicate__empty,axiom,
    ! [N: nat,X: bot_bot_term_f] :
      ( ( ( replic6958083601664011982term_f @ N @ X )
        = nil_Bot_bot_term_f )
      = ( N = zero_zero_nat ) ) ).

% replicate_empty
thf(fact_770_replicate__empty,axiom,
    ! [N: nat,X: nat] :
      ( ( ( replicate_nat @ N @ X )
        = nil_nat )
      = ( N = zero_zero_nat ) ) ).

% replicate_empty
thf(fact_771_empty__replicate,axiom,
    ! [N: nat,X: bot_bot_term_f] :
      ( ( nil_Bot_bot_term_f
        = ( replic6958083601664011982term_f @ N @ X ) )
      = ( N = zero_zero_nat ) ) ).

% empty_replicate
thf(fact_772_empty__replicate,axiom,
    ! [N: nat,X: nat] :
      ( ( nil_nat
        = ( replicate_nat @ N @ X ) )
      = ( N = zero_zero_nat ) ) ).

% empty_replicate
thf(fact_773_zip__append,axiom,
    ! [Xs2: list_nat,Us2: list_nat,Ys: list_nat,Vs: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Us2 ) )
     => ( ( zip_nat_nat @ ( append_nat @ Xs2 @ Ys ) @ ( append_nat @ Us2 @ Vs ) )
        = ( append985823374593552924at_nat @ ( zip_nat_nat @ Xs2 @ Us2 ) @ ( zip_nat_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_774_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_775_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).

% gt_ex
thf(fact_776_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_777_order_Oasym,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ~ ( ord_less_nat @ B @ A2 ) ) ).

% order.asym
thf(fact_778_ord__eq__less__trans,axiom,
    ! [A2: nat,B: nat,C3: nat] :
      ( ( A2 = B )
     => ( ( ord_less_nat @ B @ C3 )
       => ( ord_less_nat @ A2 @ C3 ) ) ) ).

% ord_eq_less_trans
thf(fact_779_ord__less__eq__trans,axiom,
    ! [A2: nat,B: nat,C3: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ( B = C3 )
       => ( ord_less_nat @ A2 @ C3 ) ) ) ).

% ord_less_eq_trans
thf(fact_780_less__induct,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ! [X3: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X3 )
             => ( P @ Y5 ) )
         => ( P @ X3 ) )
     => ( P @ A2 ) ) ).

% less_induct
thf(fact_781_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_782_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_783_dual__order_Oasym,axiom,
    ! [B: nat,A2: nat] :
      ( ( ord_less_nat @ B @ A2 )
     => ~ ( ord_less_nat @ A2 @ B ) ) ).

% dual_order.asym
thf(fact_784_dual__order_Oirrefl,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_785_exists__least__iff,axiom,
    ( ( ^ [P5: nat > $o] :
        ? [X4: nat] : ( P5 @ X4 ) )
    = ( ^ [P6: nat > $o] :
        ? [N2: nat] :
          ( ( P6 @ N2 )
          & ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N2 )
             => ~ ( P6 @ M2 ) ) ) ) ) ).

% exists_least_iff
thf(fact_786_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B: nat] :
      ( ! [A5: nat,B3: nat] :
          ( ( ord_less_nat @ A5 @ B3 )
         => ( P @ A5 @ B3 ) )
     => ( ! [A5: nat] : ( P @ A5 @ A5 )
       => ( ! [A5: nat,B3: nat] :
              ( ( P @ B3 @ A5 )
             => ( P @ A5 @ B3 ) )
         => ( P @ A2 @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_787_order_Ostrict__trans,axiom,
    ! [A2: nat,B: nat,C3: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ( ord_less_nat @ B @ C3 )
       => ( ord_less_nat @ A2 @ C3 ) ) ) ).

% order.strict_trans
thf(fact_788_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_789_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A2: nat,C3: nat] :
      ( ( ord_less_nat @ B @ A2 )
     => ( ( ord_less_nat @ C3 @ B )
       => ( ord_less_nat @ C3 @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_790_order_Ostrict__implies__not__eq,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( A2 != B ) ) ).

% order.strict_implies_not_eq
thf(fact_791_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A2: nat] :
      ( ( ord_less_nat @ B @ A2 )
     => ( A2 != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_792_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_793_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_794_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      = ( ( ord_less_nat @ X @ Y )
        | ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_795_order__less__asym_H,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ~ ( ord_less_nat @ B @ A2 ) ) ).

% order_less_asym'
thf(fact_796_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z4: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z4 )
       => ( ord_less_nat @ X @ Z4 ) ) ) ).

% order_less_trans
thf(fact_797_ord__eq__less__subst,axiom,
    ! [A2: nat,F2: nat > nat,B: nat,C3: nat] :
      ( ( A2
        = ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A2 @ ( F2 @ C3 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_798_ord__less__eq__subst,axiom,
    ! [A2: nat,B: nat,F2: nat > nat,C3: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ( ( F2 @ B )
          = C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A2 ) @ C3 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_799_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_800_order__less__subst1,axiom,
    ! [A2: nat,F2: nat > nat,B: nat,C3: nat] :
      ( ( ord_less_nat @ A2 @ ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A2 @ ( F2 @ C3 ) ) ) ) ) ).

% order_less_subst1
thf(fact_801_order__less__subst2,axiom,
    ! [A2: nat,B: nat,F2: nat > nat,C3: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ( ord_less_nat @ ( F2 @ B ) @ C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A2 ) @ C3 ) ) ) ) ).

% order_less_subst2
thf(fact_802_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_803_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_804_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
      | ( X = Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_805_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_806_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_807_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_808_in__set__simps_I3_J,axiom,
    ! [X: nat] :
      ~ ( member_nat @ X @ ( set_nat2 @ nil_nat ) ) ).

% in_set_simps(3)
thf(fact_809_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,Xs4: list_nat,Y3: nat,Ys3: list_nat] :
            ( ( X3 != Y3 )
            & ( Xs2
              = ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X3 @ nil_nat ) @ Xs4 ) ) )
            & ( Ys
              = ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y3 @ nil_nat ) @ Ys3 ) ) ) ) ) ) ).

% same_length_different
thf(fact_810_less__prod__def_H,axiom,
    ( ord_le1203424502768444845at_nat
    = ( ^ [X2: product_prod_nat_nat,Y4: product_prod_nat_nat] :
          ( ( ord_less_nat @ ( product_fst_nat_nat @ X2 ) @ ( product_fst_nat_nat @ Y4 ) )
          | ( ( ( product_fst_nat_nat @ X2 )
              = ( product_fst_nat_nat @ Y4 ) )
            & ( ord_less_nat @ ( product_snd_nat_nat @ X2 ) @ ( product_snd_nat_nat @ Y4 ) ) ) ) ) ) ).

% less_prod_def'
thf(fact_811_replicate__append__same,axiom,
    ! [I: nat,X: bot_bot_term_f] :
      ( ( append3968237287470817971term_f @ ( replic6958083601664011982term_f @ I @ X ) @ ( cons_Bot_bot_term_f @ X @ nil_Bot_bot_term_f ) )
      = ( cons_Bot_bot_term_f @ X @ ( replic6958083601664011982term_f @ I @ X ) ) ) ).

% replicate_append_same
thf(fact_812_replicate__append__same,axiom,
    ! [I: nat,X: nat] :
      ( ( append_nat @ ( replicate_nat @ I @ X ) @ ( cons_nat @ X @ nil_nat ) )
      = ( cons_nat @ X @ ( replicate_nat @ I @ X ) ) ) ).

% replicate_append_same
thf(fact_813_top_Oextremum__strict,axiom,
    ! [A2: set_nat] :
      ~ ( ord_less_set_nat @ top_top_set_nat @ A2 ) ).

% top.extremum_strict
thf(fact_814_top_Onot__eq__extremum,axiom,
    ! [A2: set_nat] :
      ( ( A2 != top_top_set_nat )
      = ( ord_less_set_nat @ A2 @ top_top_set_nat ) ) ).

% top.not_eq_extremum
thf(fact_815_length__induct,axiom,
    ! [P: list_nat > $o,Xs2: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys4: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys4 ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P @ Ys4 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_816_append__replicate__commute,axiom,
    ! [N: nat,X: bot_bot_term_f,K: nat] :
      ( ( append3968237287470817971term_f @ ( replic6958083601664011982term_f @ N @ X ) @ ( replic6958083601664011982term_f @ K @ X ) )
      = ( append3968237287470817971term_f @ ( replic6958083601664011982term_f @ K @ X ) @ ( replic6958083601664011982term_f @ N @ X ) ) ) ).

% append_replicate_commute
thf(fact_817_append__replicate__commute,axiom,
    ! [N: nat,X: nat,K: nat] :
      ( ( append_nat @ ( replicate_nat @ N @ X ) @ ( replicate_nat @ K @ X ) )
      = ( append_nat @ ( replicate_nat @ K @ X ) @ ( replicate_nat @ N @ X ) ) ) ).

% append_replicate_commute
thf(fact_818_in__set__simps_I2_J,axiom,
    ! [X: nat,Y: nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( cons_nat @ Y @ nil_nat ) ) )
      = ( X = Y ) ) ).

% in_set_simps(2)
thf(fact_819_split__list,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ? [Ys2: list_nat,Zs2: list_nat] :
          ( Xs2
          = ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs2 ) ) ) ) ).

% split_list
thf(fact_820_split__list__last,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ? [Ys2: list_nat,Zs2: list_nat] :
          ( ( Xs2
            = ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs2 ) ) )
          & ~ ( member_nat @ X @ ( set_nat2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_821_split__list__first,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ? [Ys2: list_nat,Zs2: list_nat] :
          ( ( Xs2
            = ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs2 ) ) )
          & ~ ( member_nat @ X @ ( set_nat2 @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_822_append__Cons__eq__iff,axiom,
    ! [X: nat,Xs2: list_nat,Ys: list_nat,Xs5: 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 @ Xs5 @ ( cons_nat @ X @ Ys5 ) ) )
          = ( ( Xs2 = Xs5 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_823_in__set__conv__decomp,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
      = ( ? [Ys6: list_nat,Zs3: list_nat] :
            ( Xs2
            = ( append_nat @ Ys6 @ ( cons_nat @ X @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_824_in__set__conv__decomp__last,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
      = ( ? [Ys6: list_nat,Zs3: list_nat] :
            ( ( Xs2
              = ( append_nat @ Ys6 @ ( cons_nat @ X @ Zs3 ) ) )
            & ~ ( member_nat @ X @ ( set_nat2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_825_in__set__conv__decomp__first,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
      = ( ? [Ys6: list_nat,Zs3: list_nat] :
            ( ( Xs2
              = ( append_nat @ Ys6 @ ( cons_nat @ X @ Zs3 ) ) )
            & ~ ( member_nat @ X @ ( set_nat2 @ Ys6 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_826_list__all2__append2,axiom,
    ! [P: nat > nat > $o,Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( list_all2_nat_nat @ P @ Xs2 @ ( append_nat @ Ys @ Zs ) )
      = ( ? [Us3: list_nat,Vs2: list_nat] :
            ( ( Xs2
              = ( append_nat @ Us3 @ Vs2 ) )
            & ( ( size_size_list_nat @ Us3 )
              = ( size_size_list_nat @ Ys ) )
            & ( ( size_size_list_nat @ Vs2 )
              = ( size_size_list_nat @ Zs ) )
            & ( list_all2_nat_nat @ P @ Us3 @ Ys )
            & ( list_all2_nat_nat @ P @ Vs2 @ Zs ) ) ) ) ).

% list_all2_append2
thf(fact_827_list__all2__append1,axiom,
    ! [P: nat > nat > $o,Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( list_all2_nat_nat @ P @ ( append_nat @ Xs2 @ Ys ) @ Zs )
      = ( ? [Us3: list_nat,Vs2: list_nat] :
            ( ( Zs
              = ( append_nat @ Us3 @ Vs2 ) )
            & ( ( size_size_list_nat @ Us3 )
              = ( size_size_list_nat @ Xs2 ) )
            & ( ( size_size_list_nat @ Vs2 )
              = ( size_size_list_nat @ Ys ) )
            & ( list_all2_nat_nat @ P @ Xs2 @ Us3 )
            & ( list_all2_nat_nat @ P @ Ys @ Vs2 ) ) ) ) ).

% list_all2_append1
thf(fact_828_list__all2__append,axiom,
    ! [Xs2: list_nat,Ys: list_nat,P: nat > nat > $o,Us2: list_nat,Vs: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( list_all2_nat_nat @ P @ ( append_nat @ Xs2 @ Us2 ) @ ( append_nat @ Ys @ Vs ) )
        = ( ( list_all2_nat_nat @ P @ Xs2 @ Ys )
          & ( list_all2_nat_nat @ P @ Us2 @ Vs ) ) ) ) ).

% list_all2_append
thf(fact_829_replicate__app__Cons__same,axiom,
    ! [N: nat,X: bot_bot_term_f,Xs2: list_Bot_bot_term_f] :
      ( ( append3968237287470817971term_f @ ( replic6958083601664011982term_f @ N @ X ) @ ( cons_Bot_bot_term_f @ X @ Xs2 ) )
      = ( cons_Bot_bot_term_f @ X @ ( append3968237287470817971term_f @ ( replic6958083601664011982term_f @ N @ X ) @ Xs2 ) ) ) ).

% replicate_app_Cons_same
thf(fact_830_replicate__app__Cons__same,axiom,
    ! [N: nat,X: nat,Xs2: list_nat] :
      ( ( append_nat @ ( replicate_nat @ N @ X ) @ ( cons_nat @ X @ Xs2 ) )
      = ( cons_nat @ X @ ( append_nat @ ( replicate_nat @ N @ X ) @ Xs2 ) ) ) ).

% replicate_app_Cons_same
thf(fact_831_list__induct4,axiom,
    ! [Xs2: list_nat,Ys: list_nat,Zs: list_nat,Ws: list_nat,P: 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 ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
           => ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys2: list_nat,Z3: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_nat @ Ys2 ) )
                 => ( ( ( size_size_list_nat @ Ys2 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys2 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_832_list__induct3,axiom,
    ! [Xs2: list_nat,Ys: list_nat,Zs: list_nat,P: 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 ) )
       => ( ( P @ nil_nat @ nil_nat @ nil_nat )
         => ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys2: list_nat,Z3: nat,Zs2: list_nat] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_size_list_nat @ Ys2 ) )
               => ( ( ( size_size_list_nat @ Ys2 )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys2 @ Zs2 )
                   => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) )
           => ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_833_list__induct2,axiom,
    ! [Xs2: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_nat @ nil_nat )
       => ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys2: list_nat] :
              ( ( ( size_size_list_nat @ Xs3 )
                = ( size_size_list_nat @ Ys2 ) )
             => ( ( P @ Xs3 @ Ys2 )
               => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) )
         => ( P @ Xs2 @ Ys ) ) ) ) ).

% list_induct2
thf(fact_834_replicate__0,axiom,
    ! [X: bot_bot_term_f] :
      ( ( replic6958083601664011982term_f @ zero_zero_nat @ X )
      = nil_Bot_bot_term_f ) ).

% replicate_0
thf(fact_835_replicate__0,axiom,
    ! [X: nat] :
      ( ( replicate_nat @ zero_zero_nat @ X )
      = nil_nat ) ).

% replicate_0
thf(fact_836_List_Oinsert__def,axiom,
    ( insert_nat
    = ( ^ [X2: nat,Xs: list_nat] : ( if_list_nat @ ( member_nat @ X2 @ ( set_nat2 @ Xs ) ) @ Xs @ ( cons_nat @ X2 @ Xs ) ) ) ) ).

% List.insert_def
thf(fact_837_length__pos__if__in__set,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_838_lexord__sufE,axiom,
    ! [Xs2: list_nat,Zs: list_nat,Ys: list_nat,Qs: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Zs ) @ ( append_nat @ Ys @ Qs ) ) @ ( lexord_nat @ R2 ) )
     => ( ( 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 @ R2 ) ) ) ) ) ) ).

% lexord_sufE
thf(fact_839_lex__append__rightI,axiom,
    ! [Xs2: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat,Vs: list_nat,Us2: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( lex_nat @ R2 ) )
     => ( ( ( size_size_list_nat @ Vs )
          = ( size_size_list_nat @ Us2 ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Us2 ) @ ( append_nat @ Ys @ Vs ) ) @ ( lex_nat @ R2 ) ) ) ) ).

% lex_append_rightI
thf(fact_840_lenlex__append1,axiom,
    ! [Us2: list_nat,Xs2: list_nat,R: set_Pr1261947904930325089at_nat,Vs: list_nat,Ys: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us2 @ Xs2 ) @ ( lenlex_nat @ R ) )
     => ( ( ( size_size_list_nat @ Vs )
          = ( size_size_list_nat @ Ys ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Us2 @ Vs ) @ ( append_nat @ Xs2 @ Ys ) ) @ ( lenlex_nat @ R ) ) ) ) ).

% lenlex_append1
thf(fact_841_length__nth__simps_I1_J,axiom,
    ( ( size_size_list_nat @ nil_nat )
    = zero_zero_nat ) ).

% length_nth_simps(1)
thf(fact_842_mergeP_Osimps,axiom,
    ! [A1: bot_bot_term_f,A23: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A1 @ A23 ) @ bot_mergeP_f )
      = ( ? [T2: bot_bot_term_f] :
            ( ( A1 = bot_bot_Bot_f )
            & ( A23 = T2 ) )
        | ? [T2: bot_bot_term_f] :
            ( ( A1 = T2 )
            & ( A23 = bot_bot_Bot_f ) )
        | ? [Ss2: list_Bot_bot_term_f,Ts3: list_Bot_bot_term_f,F3: f] :
            ( ( A1
              = ( bot_bot_BFun_f @ F3 @ Ss2 ) )
            & ( A23
              = ( bot_bot_BFun_f @ F3 @ Ts3 ) )
            & ( ( size_s1458857753309042354term_f @ Ss2 )
              = ( size_s1458857753309042354term_f @ Ts3 ) )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s1458857753309042354term_f @ Ts3 ) )
               => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( nth_Bot_bot_term_f @ Ss2 @ I2 ) @ ( nth_Bot_bot_term_f @ Ts3 @ I2 ) ) @ bot_mergeP_f ) ) ) ) ) ).

% mergeP.simps
thf(fact_843_mergeP_Ocases,axiom,
    ! [A1: bot_bot_term_f,A23: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A1 @ A23 ) @ bot_mergeP_f )
     => ( ( A1 != bot_bot_Bot_f )
       => ( ( A23 != bot_bot_Bot_f )
         => ~ ! [Ss: list_Bot_bot_term_f,Ts: list_Bot_bot_term_f,F: f] :
                ( ( A1
                  = ( bot_bot_BFun_f @ F @ Ss ) )
               => ( ( A23
                    = ( bot_bot_BFun_f @ F @ Ts ) )
                 => ( ( ( size_s1458857753309042354term_f @ Ss )
                      = ( size_s1458857753309042354term_f @ Ts ) )
                   => ~ ! [I3: nat] :
                          ( ( ord_less_nat @ I3 @ ( size_s1458857753309042354term_f @ Ts ) )
                         => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( nth_Bot_bot_term_f @ Ss @ I3 ) @ ( nth_Bot_bot_term_f @ Ts @ I3 ) ) @ bot_mergeP_f ) ) ) ) ) ) ) ) ).

% mergeP.cases
thf(fact_844_nth__Cons__0,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_845_nth__replicate,axiom,
    ! [I: nat,N: nat,X: bot_bot_term_f] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_Bot_bot_term_f @ ( replic6958083601664011982term_f @ N @ X ) @ I )
        = X ) ) ).

% nth_replicate
thf(fact_846_nth__replicate,axiom,
    ! [I: nat,N: nat,X: nat] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_nat @ ( replicate_nat @ N @ X ) @ I )
        = X ) ) ).

% nth_replicate
thf(fact_847_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_848_nth__zip,axiom,
    ! [I: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr7617993195940197384at_nat @ ( zip_nat_nat @ Xs2 @ Ys ) @ I )
          = ( product_Pair_nat_nat @ ( nth_nat @ Xs2 @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_849_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_850_nth__equalityI,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
           => ( ( nth_nat @ Xs2 @ I4 )
              = ( nth_nat @ Ys @ I4 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_851_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X5: nat] : ( P @ I2 @ X5 ) ) )
      = ( ? [Xs: list_nat] :
            ( ( ( size_size_list_nat @ Xs )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_852_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y2: list_nat,Z: list_nat] : ( Y2 = Z ) )
    = ( ^ [Xs: list_nat,Ys6: list_nat] :
          ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys6 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
             => ( ( nth_nat @ Xs @ I2 )
                = ( nth_nat @ Ys6 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_853_Ex__list__of__length__P,axiom,
    ! [N: nat,P: nat > nat > $o] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ N )
         => ? [X6: nat] : ( P @ X6 @ I4 ) )
     => ? [Xs3: list_nat] :
          ( ( ( size_size_list_nat @ Xs3 )
            = N )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ N )
             => ( P @ ( nth_nat @ Xs3 @ I3 ) @ I3 ) ) ) ) ).

% Ex_list_of_length_P
thf(fact_854_ex__set__conv__ex__nth,axiom,
    ! [Xs2: list_nat,P: nat > $o] :
      ( ( ? [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
            & ( P @ X2 ) ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
            & ( P @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ).

% ex_set_conv_ex_nth
thf(fact_855_all__set__conv__all__nth,axiom,
    ! [Xs2: list_nat,P: nat > $o] :
      ( ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
           => ( P @ X2 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
           => ( P @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_856_all__nth__imp__all__set,axiom,
    ! [Xs2: list_nat,P: nat > $o,X: nat] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
         => ( P @ ( nth_nat @ Xs2 @ I4 ) ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_857_in__set__conv__nth,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
            & ( ( nth_nat @ Xs2 @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_858_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_859_nth__mem,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( member_nat @ ( nth_nat @ Xs2 @ N ) @ ( set_nat2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_860_append__Cons__nth__not__middle,axiom,
    ! [I: nat,Xs2: list_nat,U: nat,Ys: list_nat,Z4: nat] :
      ( ( I
       != ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_nat @ ( append_nat @ Xs2 @ ( cons_nat @ U @ Ys ) ) @ I )
        = ( nth_nat @ ( append_nat @ Xs2 @ ( cons_nat @ Z4 @ Ys ) ) @ I ) ) ) ).

% append_Cons_nth_not_middle
thf(fact_861_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_862_P__as__bs__extend,axiom,
    ! [As: list_nat,Bs: list_nat,Cs: list_nat,Ds: list_nat,P: nat > nat > $o] :
      ( ( ( size_size_list_nat @ As )
        = ( size_size_list_nat @ Bs ) )
     => ( ( ( size_size_list_nat @ Cs )
          = ( size_size_list_nat @ Ds ) )
       => ( ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Bs ) )
             => ( P @ ( nth_nat @ As @ I4 ) @ ( nth_nat @ Bs @ I4 ) ) )
         => ( ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ds ) )
               => ( P @ ( nth_nat @ Cs @ I4 ) @ ( nth_nat @ Ds @ I4 ) ) )
           => ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ ( append_nat @ Bs @ Ds ) ) )
               => ( P @ ( nth_nat @ ( append_nat @ As @ Cs ) @ I3 ) @ ( nth_nat @ ( append_nat @ Bs @ Ds ) @ I3 ) ) ) ) ) ) ) ).

% P_as_bs_extend
thf(fact_863_list__all2__conv__all__nth,axiom,
    ( list_all2_nat_nat
    = ( ^ [P6: nat > nat > $o,Xs: list_nat,Ys6: list_nat] :
          ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys6 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
             => ( P6 @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys6 @ I2 ) ) ) ) ) ) ).

% list_all2_conv_all_nth
thf(fact_864_list__all2__all__nthI,axiom,
    ! [A2: list_nat,B: list_nat,P: nat > nat > $o] :
      ( ( ( size_size_list_nat @ A2 )
        = ( size_size_list_nat @ B ) )
     => ( ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ A2 ) )
           => ( P @ ( nth_nat @ A2 @ N3 ) @ ( nth_nat @ B @ N3 ) ) )
       => ( list_all2_nat_nat @ P @ A2 @ B ) ) ) ).

% list_all2_all_nthI
thf(fact_865_list__all2__nthD2,axiom,
    ! [P: nat > nat > $o,Xs2: list_nat,Ys: list_nat,P2: nat] :
      ( ( list_all2_nat_nat @ P @ Xs2 @ Ys )
     => ( ( ord_less_nat @ P2 @ ( size_size_list_nat @ Ys ) )
       => ( P @ ( nth_nat @ Xs2 @ P2 ) @ ( nth_nat @ Ys @ P2 ) ) ) ) ).

% list_all2_nthD2
thf(fact_866_list__all2__nthD,axiom,
    ! [P: nat > nat > $o,Xs2: list_nat,Ys: list_nat,P2: nat] :
      ( ( list_all2_nat_nat @ P @ Xs2 @ Ys )
     => ( ( ord_less_nat @ P2 @ ( size_size_list_nat @ Xs2 ) )
       => ( P @ ( nth_nat @ Xs2 @ P2 ) @ ( nth_nat @ Ys @ P2 ) ) ) ) ).

% list_all2_nthD
thf(fact_867_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_868_append__Cons__nth__right,axiom,
    ! [Xs2: list_nat,I: nat,U: nat,Ys: list_nat,Z4: 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 @ Z4 @ Ys ) ) @ I ) ) ) ).

% append_Cons_nth_right
thf(fact_869_bless__eqp_Ostep,axiom,
    ! [Ss3: list_Bot_bot_term_f,Ts2: list_Bot_bot_term_f,F2: f] :
      ( ( ( size_s1458857753309042354term_f @ Ss3 )
        = ( size_s1458857753309042354term_f @ Ts2 ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s1458857753309042354term_f @ Ts2 ) )
           => ( bot_bless_eqp_f @ ( nth_Bot_bot_term_f @ Ss3 @ I4 ) @ ( nth_Bot_bot_term_f @ Ts2 @ I4 ) ) )
       => ( bot_bless_eqp_f @ ( bot_bot_BFun_f @ F2 @ Ss3 ) @ ( bot_bot_BFun_f @ F2 @ Ts2 ) ) ) ) ).

% bless_eqp.step
thf(fact_870_bless__eq_Ostep,axiom,
    ! [Ss3: list_Bot_bot_term_f,Ts2: list_Bot_bot_term_f,F2: f] :
      ( ( ( size_s1458857753309042354term_f @ Ss3 )
        = ( size_s1458857753309042354term_f @ Ts2 ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s1458857753309042354term_f @ Ts2 ) )
           => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( nth_Bot_bot_term_f @ Ss3 @ I4 ) @ ( nth_Bot_bot_term_f @ Ts2 @ I4 ) ) @ bot_bless_eq_f ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_bot_BFun_f @ F2 @ Ss3 ) @ ( bot_bot_BFun_f @ F2 @ Ts2 ) ) @ bot_bless_eq_f ) ) ) ).

% bless_eq.step
thf(fact_871_mergeP_Ostep,axiom,
    ! [Ss3: list_Bot_bot_term_f,Ts2: list_Bot_bot_term_f,F2: f] :
      ( ( ( size_s1458857753309042354term_f @ Ss3 )
        = ( size_s1458857753309042354term_f @ Ts2 ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s1458857753309042354term_f @ Ts2 ) )
           => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( nth_Bot_bot_term_f @ Ss3 @ I4 ) @ ( nth_Bot_bot_term_f @ Ts2 @ I4 ) ) @ bot_mergeP_f ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_bot_BFun_f @ F2 @ Ss3 ) @ ( bot_bot_BFun_f @ F2 @ Ts2 ) ) @ bot_mergeP_f ) ) ) ).

% mergeP.step
thf(fact_872_bless__eqp_Osimps,axiom,
    ( bot_bless_eqp_f
    = ( ^ [A12: bot_bot_term_f,A24: bot_bot_term_f] :
          ( ? [T2: bot_bot_term_f] :
              ( ( A12 = bot_bot_Bot_f )
              & ( A24 = T2 ) )
          | ? [Ss2: list_Bot_bot_term_f,Ts3: list_Bot_bot_term_f,F3: f] :
              ( ( A12
                = ( bot_bot_BFun_f @ F3 @ Ss2 ) )
              & ( A24
                = ( bot_bot_BFun_f @ F3 @ Ts3 ) )
              & ( ( size_s1458857753309042354term_f @ Ss2 )
                = ( size_s1458857753309042354term_f @ Ts3 ) )
              & ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( size_s1458857753309042354term_f @ Ts3 ) )
                 => ( bot_bless_eqp_f @ ( nth_Bot_bot_term_f @ Ss2 @ I2 ) @ ( nth_Bot_bot_term_f @ Ts3 @ I2 ) ) ) ) ) ) ) ).

% bless_eqp.simps
thf(fact_873_bless__eqp_Ocases,axiom,
    ! [A1: bot_bot_term_f,A23: bot_bot_term_f] :
      ( ( bot_bless_eqp_f @ A1 @ A23 )
     => ( ( A1 != bot_bot_Bot_f )
       => ~ ! [Ss: list_Bot_bot_term_f,Ts: list_Bot_bot_term_f,F: f] :
              ( ( A1
                = ( bot_bot_BFun_f @ F @ Ss ) )
             => ( ( A23
                  = ( bot_bot_BFun_f @ F @ Ts ) )
               => ( ( ( size_s1458857753309042354term_f @ Ss )
                    = ( size_s1458857753309042354term_f @ Ts ) )
                 => ~ ! [I3: nat] :
                        ( ( ord_less_nat @ I3 @ ( size_s1458857753309042354term_f @ Ts ) )
                       => ( bot_bless_eqp_f @ ( nth_Bot_bot_term_f @ Ss @ I3 ) @ ( nth_Bot_bot_term_f @ Ts @ I3 ) ) ) ) ) ) ) ) ).

% bless_eqp.cases
thf(fact_874_in__set__zip,axiom,
    ! [P2: product_prod_nat_nat,Xs2: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ P2 @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs2 @ Ys ) ) )
      = ( ? [N2: nat] :
            ( ( ( nth_nat @ Xs2 @ N2 )
              = ( product_fst_nat_nat @ P2 ) )
            & ( ( nth_nat @ Ys @ N2 )
              = ( product_snd_nat_nat @ P2 ) )
            & ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
            & ( ord_less_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_875_bless__eq_Ocases,axiom,
    ! [A1: bot_bot_term_f,A23: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A1 @ A23 ) @ bot_bless_eq_f )
     => ( ( A1 != bot_bot_Bot_f )
       => ~ ! [Ss: list_Bot_bot_term_f,Ts: list_Bot_bot_term_f,F: f] :
              ( ( A1
                = ( bot_bot_BFun_f @ F @ Ss ) )
             => ( ( A23
                  = ( bot_bot_BFun_f @ F @ Ts ) )
               => ( ( ( size_s1458857753309042354term_f @ Ss )
                    = ( size_s1458857753309042354term_f @ Ts ) )
                 => ~ ! [I3: nat] :
                        ( ( ord_less_nat @ I3 @ ( size_s1458857753309042354term_f @ Ts ) )
                       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( nth_Bot_bot_term_f @ Ss @ I3 ) @ ( nth_Bot_bot_term_f @ Ts @ I3 ) ) @ bot_bless_eq_f ) ) ) ) ) ) ) ).

% bless_eq.cases
thf(fact_876_bless__eq_Osimps,axiom,
    ! [A1: bot_bot_term_f,A23: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A1 @ A23 ) @ bot_bless_eq_f )
      = ( ? [T2: bot_bot_term_f] :
            ( ( A1 = bot_bot_Bot_f )
            & ( A23 = T2 ) )
        | ? [Ss2: list_Bot_bot_term_f,Ts3: list_Bot_bot_term_f,F3: f] :
            ( ( A1
              = ( bot_bot_BFun_f @ F3 @ Ss2 ) )
            & ( A23
              = ( bot_bot_BFun_f @ F3 @ Ts3 ) )
            & ( ( size_s1458857753309042354term_f @ Ss2 )
              = ( size_s1458857753309042354term_f @ Ts3 ) )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s1458857753309042354term_f @ Ts3 ) )
               => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( nth_Bot_bot_term_f @ Ss2 @ I2 ) @ ( nth_Bot_bot_term_f @ Ts3 @ I2 ) ) @ bot_bless_eq_f ) ) ) ) ) ).

% bless_eq.simps
thf(fact_877_in__set__idx,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ? [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
          & ( ( nth_nat @ Xs2 @ I4 )
            = X ) ) ) ).

% in_set_idx
thf(fact_878_nth__equalityE,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( Xs2 = Ys )
     => ~ ( ( ( size_size_list_nat @ Xs2 )
            = ( size_size_list_nat @ Ys ) )
         => ~ ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
               => ( ( nth_nat @ Xs2 @ I3 )
                  = ( nth_nat @ Ys @ I3 ) ) ) ) ) ).

% nth_equalityE
thf(fact_879_permut__sound,axiom,
    ! [I: nat,As: list_nat,F2: nat > nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ As ) )
     => ( ( nth_nat @ ( missing_permut_nat @ As @ F2 ) @ I )
        = ( nth_nat @ As @ ( F2 @ I ) ) ) ) ).

% permut_sound
thf(fact_880_mergePp_Osimps,axiom,
    ( bot_mergePp_f
    = ( ^ [A12: bot_bot_term_f,A24: bot_bot_term_f] :
          ( ? [T2: bot_bot_term_f] :
              ( ( A12 = bot_bot_Bot_f )
              & ( A24 = T2 ) )
          | ? [T2: bot_bot_term_f] :
              ( ( A12 = T2 )
              & ( A24 = bot_bot_Bot_f ) )
          | ? [Ss2: list_Bot_bot_term_f,Ts3: list_Bot_bot_term_f,F3: f] :
              ( ( A12
                = ( bot_bot_BFun_f @ F3 @ Ss2 ) )
              & ( A24
                = ( bot_bot_BFun_f @ F3 @ Ts3 ) )
              & ( ( size_s1458857753309042354term_f @ Ss2 )
                = ( size_s1458857753309042354term_f @ Ts3 ) )
              & ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( size_s1458857753309042354term_f @ Ts3 ) )
                 => ( bot_mergePp_f @ ( nth_Bot_bot_term_f @ Ss2 @ I2 ) @ ( nth_Bot_bot_term_f @ Ts3 @ I2 ) ) ) ) ) ) ) ).

% mergePp.simps
thf(fact_881_mergePp_Obase__r,axiom,
    ! [T: bot_bot_term_f] : ( bot_mergePp_f @ T @ bot_bot_Bot_f ) ).

% mergePp.base_r
thf(fact_882_mergePp_Obase__l,axiom,
    ! [T: bot_bot_term_f] : ( bot_mergePp_f @ bot_bot_Bot_f @ T ) ).

% mergePp.base_l
thf(fact_883_mergePp_Ostep,axiom,
    ! [Ss3: list_Bot_bot_term_f,Ts2: list_Bot_bot_term_f,F2: f] :
      ( ( ( size_s1458857753309042354term_f @ Ss3 )
        = ( size_s1458857753309042354term_f @ Ts2 ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s1458857753309042354term_f @ Ts2 ) )
           => ( bot_mergePp_f @ ( nth_Bot_bot_term_f @ Ss3 @ I4 ) @ ( nth_Bot_bot_term_f @ Ts2 @ I4 ) ) )
       => ( bot_mergePp_f @ ( bot_bot_BFun_f @ F2 @ Ss3 ) @ ( bot_bot_BFun_f @ F2 @ Ts2 ) ) ) ) ).

% mergePp.step
thf(fact_884_mergePp_Ocases,axiom,
    ! [A1: bot_bot_term_f,A23: bot_bot_term_f] :
      ( ( bot_mergePp_f @ A1 @ A23 )
     => ( ( A1 != bot_bot_Bot_f )
       => ( ( A23 != bot_bot_Bot_f )
         => ~ ! [Ss: list_Bot_bot_term_f,Ts: list_Bot_bot_term_f,F: f] :
                ( ( A1
                  = ( bot_bot_BFun_f @ F @ Ss ) )
               => ( ( A23
                    = ( bot_bot_BFun_f @ F @ Ts ) )
                 => ( ( ( size_s1458857753309042354term_f @ Ss )
                      = ( size_s1458857753309042354term_f @ Ts ) )
                   => ~ ! [I3: nat] :
                          ( ( ord_less_nat @ I3 @ ( size_s1458857753309042354term_f @ Ts ) )
                         => ( bot_mergePp_f @ ( nth_Bot_bot_term_f @ Ss @ I3 ) @ ( nth_Bot_bot_term_f @ Ts @ I3 ) ) ) ) ) ) ) ) ) ).

% mergePp.cases
thf(fact_885_trancl__list_Obase,axiom,
    ! [Xs2: list_nat,Ys: list_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs2 @ I4 ) @ ( nth_nat @ Ys @ I4 ) ) @ R5 ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( trancl_list_nat @ R5 ) ) ) ) ).

% trancl_list.base
thf(fact_886_lexord__take__index__conv,axiom,
    ! [X: list_nat,Y: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lexord_nat @ R2 ) )
      = ( ( ( ord_less_nat @ ( size_size_list_nat @ X ) @ ( size_size_list_nat @ Y ) )
          & ( ( take_nat @ ( size_size_list_nat @ X ) @ Y )
            = X ) )
        | ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( ord_min_nat @ ( size_size_list_nat @ X ) @ ( size_size_list_nat @ Y ) ) )
            & ( ( take_nat @ I2 @ X )
              = ( take_nat @ I2 @ Y ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ X @ I2 ) @ ( nth_nat @ Y @ I2 ) ) @ R2 ) ) ) ) ).

% lexord_take_index_conv
thf(fact_887_permut__aux__sound,axiom,
    ! [I: nat,As: list_nat,F2: nat > nat,Bs: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ As ) )
     => ( ( nth_nat @ ( missin1888654203714970382ux_nat @ As @ F2 @ Bs ) @ I )
        = ( nth_nat @ Bs @ ( F2 @ I ) ) ) ) ).

% permut_aux_sound
thf(fact_888_take__take,axiom,
    ! [N: nat,M: nat,Xs2: list_nat] :
      ( ( take_nat @ N @ ( take_nat @ M @ Xs2 ) )
      = ( take_nat @ ( ord_min_nat @ N @ M ) @ Xs2 ) ) ).

% take_take
thf(fact_889_take0,axiom,
    ( ( take_nat @ zero_zero_nat )
    = ( ^ [Xs: list_nat] : nil_nat ) ) ).

% take0
thf(fact_890_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_891_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_892_nth__take,axiom,
    ! [I: nat,N: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_nat @ ( take_nat @ N @ Xs2 ) @ I )
        = ( nth_nat @ Xs2 @ I ) ) ) ).

% nth_take
thf(fact_893_length__take,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( size_size_list_nat @ ( take_nat @ N @ Xs2 ) )
      = ( ord_min_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ).

% length_take
thf(fact_894_take__replicate,axiom,
    ! [I: nat,K: nat,X: bot_bot_term_f] :
      ( ( take_Bot_bot_term_f @ I @ ( replic6958083601664011982term_f @ K @ X ) )
      = ( replic6958083601664011982term_f @ ( ord_min_nat @ I @ K ) @ X ) ) ).

% take_replicate
thf(fact_895_take__replicate,axiom,
    ! [I: nat,K: nat,X: nat] :
      ( ( take_nat @ I @ ( replicate_nat @ K @ X ) )
      = ( replicate_nat @ ( ord_min_nat @ I @ K ) @ X ) ) ).

% take_replicate
thf(fact_896_in__set__takeD,axiom,
    ! [X: nat,N: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( take_nat @ N @ Xs2 ) ) )
     => ( member_nat @ X @ ( set_nat2 @ Xs2 ) ) ) ).

% in_set_takeD
thf(fact_897_take__Nil,axiom,
    ! [N: nat] :
      ( ( take_nat @ N @ nil_nat )
      = nil_nat ) ).

% take_Nil
thf(fact_898_list__all2__takeI,axiom,
    ! [P: nat > nat > $o,Xs2: list_nat,Ys: list_nat,N: nat] :
      ( ( list_all2_nat_nat @ P @ Xs2 @ Ys )
     => ( list_all2_nat_nat @ P @ ( take_nat @ N @ Xs2 ) @ ( take_nat @ N @ Ys ) ) ) ).

% list_all2_takeI
thf(fact_899_take__zip,axiom,
    ! [N: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( take_P2173866234530122223at_nat @ N @ ( zip_nat_nat @ Xs2 @ Ys ) )
      = ( zip_nat_nat @ ( take_nat @ N @ Xs2 ) @ ( take_nat @ N @ Ys ) ) ) ).

% take_zip
thf(fact_900_take__equalityI,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ! [I4: nat] :
          ( ( take_nat @ I4 @ Xs2 )
          = ( take_nat @ I4 @ Ys ) )
     => ( Xs2 = Ys ) ) ).

% take_equalityI
thf(fact_901_take__0,axiom,
    ! [Xs2: list_nat] :
      ( ( take_nat @ zero_zero_nat @ Xs2 )
      = nil_nat ) ).

% take_0
thf(fact_902_zip__obtain__same__length,axiom,
    ! [Xs2: list_nat,Ys: list_nat,P: list_P6011104703257516679at_nat > $o] :
      ( ! [Zs2: list_nat,Ws2: list_nat,N3: nat] :
          ( ( ( size_size_list_nat @ Zs2 )
            = ( size_size_list_nat @ Ws2 ) )
         => ( ( N3
              = ( ord_min_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
           => ( ( Zs2
                = ( take_nat @ N3 @ Xs2 ) )
             => ( ( Ws2
                  = ( take_nat @ N3 @ Ys ) )
               => ( P @ ( zip_nat_nat @ Zs2 @ Ws2 ) ) ) ) ) )
     => ( P @ ( zip_nat_nat @ Xs2 @ Ys ) ) ) ).

% zip_obtain_same_length
thf(fact_903_set__take__nth,axiom,
    ! [X: nat,I: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( take_nat @ I @ Xs2 ) ) )
     => ? [J2: nat] :
          ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
          & ( ord_less_nat @ J2 @ I )
          & ( ( nth_nat @ Xs2 @ J2 )
            = X ) ) ) ).

% set_take_nth
thf(fact_904_in__set__product__lists__length,axiom,
    ! [Xs2: list_nat,Xss: list_list_nat] :
      ( ( member_list_nat @ Xs2 @ ( set_list_nat2 @ ( product_lists_nat @ Xss ) ) )
     => ( ( size_size_list_nat @ Xs2 )
        = ( size_s3023201423986296836st_nat @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_905_lex__take__index,axiom,
    ! [Xs2: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( lex_nat @ R2 ) )
     => ~ ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
           => ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
             => ( ( ( take_nat @ I4 @ Xs2 )
                  = ( take_nat @ I4 @ Ys ) )
               => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs2 @ I4 ) @ ( nth_nat @ Ys @ I4 ) ) @ R2 ) ) ) ) ) ).

% lex_take_index
thf(fact_906_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_907_partition2__sound__P,axiom,
    ! [P: nat > nat > $o,As: list_nat,Bs: list_nat,I3: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ ( produc1382935764643595205st_nat @ ( produc7458156121412276931st_nat @ ( missin5516618432537284226at_nat @ P @ As @ Bs ) ) ) ) )
     => ( P @ ( nth_nat @ ( produc1382935764643595205st_nat @ ( produc7458156121412276931st_nat @ ( missin5516618432537284226at_nat @ P @ As @ Bs ) ) ) @ I3 ) @ ( nth_nat @ ( produc5865812112468994567st_nat @ ( produc7458156121412276931st_nat @ ( missin5516618432537284226at_nat @ P @ As @ Bs ) ) ) @ I3 ) ) ) ).

% partition2_sound_P
thf(fact_908_partition2__sound__nP,axiom,
    ! [P: nat > nat > $o,As: list_nat,Bs: list_nat,I3: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ ( produc1382935764643595205st_nat @ ( produc7615485059283090437st_nat @ ( missin5516618432537284226at_nat @ P @ As @ Bs ) ) ) ) )
     => ~ ( P @ ( nth_nat @ ( produc1382935764643595205st_nat @ ( produc7615485059283090437st_nat @ ( missin5516618432537284226at_nat @ P @ As @ Bs ) ) ) @ I3 ) @ ( nth_nat @ ( produc5865812112468994567st_nat @ ( produc7615485059283090437st_nat @ ( missin5516618432537284226at_nat @ P @ As @ Bs ) ) ) @ I3 ) ) ) ).

% partition2_sound_nP
thf(fact_909_trancl__list_Olist__trancl,axiom,
    ! [Xs2: list_nat,Ys: list_nat,R5: set_Pr1261947904930325089at_nat,I: nat,Z4: nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( trancl_list_nat @ R5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys @ I ) @ Z4 ) @ R5 )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ ( list_update_nat @ Ys @ I @ Z4 ) ) @ ( trancl_list_nat @ R5 ) ) ) ) ) ).

% trancl_list.list_trancl
thf(fact_910_length__list__update,axiom,
    ! [Xs2: list_nat,I: nat,X: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs2 @ I @ X ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_list_update
thf(fact_911_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs2: list_nat,X: nat] :
      ( ( I != J )
     => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J )
        = ( nth_nat @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_912_list__update__id,axiom,
    ! [Xs2: list_nat,I: nat] :
      ( ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_913_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_914_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_915_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_916_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_917_set__swap,axiom,
    ! [I: nat,Xs2: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
       => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I ) ) )
          = ( set_nat2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_918_take__update__swap,axiom,
    ! [M: nat,Xs2: list_nat,N: nat,X: nat] :
      ( ( take_nat @ M @ ( list_update_nat @ Xs2 @ N @ X ) )
      = ( list_update_nat @ ( take_nat @ M @ Xs2 ) @ N @ X ) ) ).

% take_update_swap
thf(fact_919_length__nth__simps_I2_J,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X @ Xs2 ) )
      = ( suc @ ( size_size_list_nat @ Xs2 ) ) ) ).

% length_nth_simps(2)
thf(fact_920_Suc__length__conv,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ( suc @ N )
        = ( size_size_list_nat @ Xs2 ) )
      = ( ? [Y4: nat,Ys6: list_nat] :
            ( ( Xs2
              = ( cons_nat @ Y4 @ Ys6 ) )
            & ( ( size_size_list_nat @ Ys6 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_921_length__Suc__conv,axiom,
    ! [Xs2: list_nat,N: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( suc @ N ) )
      = ( ? [Y4: nat,Ys6: list_nat] :
            ( ( Xs2
              = ( cons_nat @ Y4 @ Ys6 ) )
            & ( ( size_size_list_nat @ Ys6 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_922_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_923_replicate__Suc,axiom,
    ! [N: nat,X: bot_bot_term_f] :
      ( ( replic6958083601664011982term_f @ ( suc @ N ) @ X )
      = ( cons_Bot_bot_term_f @ X @ ( replic6958083601664011982term_f @ N @ X ) ) ) ).

% replicate_Suc
thf(fact_924_replicate__Suc,axiom,
    ! [N: nat,X: nat] :
      ( ( replicate_nat @ ( suc @ N ) @ X )
      = ( cons_nat @ X @ ( replicate_nat @ N @ X ) ) ) ).

% replicate_Suc
thf(fact_925_set__update__memI,axiom,
    ! [N: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_926_nth__list__update,axiom,
    ! [I: nat,Xs2: list_nat,J: nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J )
            = ( nth_nat @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_927_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( list_update_nat @ Xs2 @ I @ X )
          = Xs2 )
        = ( ( nth_nat @ Xs2 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_928_parallel__list__update,axiom,
    ! [N: nat,R2: nat > nat > $o,P2: list_nat > $o,Xs2: list_nat,Ys: list_nat] :
      ( ! [Xs3: list_nat,I4: nat,Y3: nat] :
          ( ( ( size_size_list_nat @ Xs3 )
            = N )
         => ( ( ord_less_nat @ I4 @ N )
           => ( ( R2 @ ( nth_nat @ Xs3 @ I4 ) @ Y3 )
             => ( ( P2 @ Xs3 )
               => ( P2 @ ( list_update_nat @ Xs3 @ I4 @ Y3 ) ) ) ) ) )
     => ( ( ( size_size_list_nat @ Xs2 )
          = N )
       => ( ( P2 @ Xs2 )
         => ( ( ( size_size_list_nat @ Ys )
              = N )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N )
                 => ( R2 @ ( nth_nat @ Xs2 @ I4 ) @ ( nth_nat @ Ys @ I4 ) ) )
             => ( P2 @ Ys ) ) ) ) ) ) ).

% parallel_list_update
thf(fact_929_list__update__append1,axiom,
    ! [I: nat,Xs2: list_nat,Ys: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( list_update_nat @ ( append_nat @ Xs2 @ Ys ) @ I @ X )
        = ( append_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_930_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_931_length__Suc__conv__rev,axiom,
    ! [Xs2: list_nat,N: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( suc @ N ) )
      = ( ? [Y4: nat,Ys6: list_nat] :
            ( ( Xs2
              = ( append_nat @ Ys6 @ ( cons_nat @ Y4 @ nil_nat ) ) )
            & ( ( size_size_list_nat @ Ys6 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_932_trancl__list_Ocases,axiom,
    ! [A1: list_nat,A23: list_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A23 ) @ ( trancl_list_nat @ R5 ) )
     => ( ( ( ( size_size_list_nat @ A1 )
            = ( size_size_list_nat @ A23 ) )
         => ~ ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ A23 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ A1 @ I3 ) @ ( nth_nat @ A23 @ I3 ) ) @ R5 ) ) )
       => ~ ! [Ys2: list_nat,I4: nat,Z3: nat] :
              ( ( A23
                = ( list_update_nat @ Ys2 @ I4 @ Z3 ) )
             => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ Ys2 ) @ ( trancl_list_nat @ R5 ) )
               => ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys2 ) )
                 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys2 @ I4 ) @ Z3 ) @ R5 ) ) ) ) ) ) ).

% trancl_list.cases
thf(fact_933_trancl__list_Osimps,axiom,
    ! [A1: list_nat,A23: list_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A23 ) @ ( trancl_list_nat @ R5 ) )
      = ( ? [Xs: list_nat,Ys6: list_nat] :
            ( ( A1 = Xs )
            & ( A23 = Ys6 )
            & ( ( size_size_list_nat @ Xs )
              = ( size_size_list_nat @ Ys6 ) )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys6 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys6 @ I2 ) ) @ R5 ) ) )
        | ? [Xs: list_nat,Ys6: list_nat,I2: nat,Z5: nat] :
            ( ( A1 = Xs )
            & ( A23
              = ( list_update_nat @ Ys6 @ I2 @ Z5 ) )
            & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys6 ) @ ( trancl_list_nat @ R5 ) )
            & ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys6 ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys6 @ I2 ) @ Z5 ) @ R5 ) ) ) ) ).

% trancl_list.simps
thf(fact_934_filter2__sound,axiom,
    ! [P: nat > nat > $o,As: list_nat,Bs: list_nat,I3: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ ( produc1382935764643595205st_nat @ ( missin7656342814955748836at_nat @ P @ As @ Bs ) ) ) )
     => ( P @ ( nth_nat @ ( produc1382935764643595205st_nat @ ( missin7656342814955748836at_nat @ P @ As @ Bs ) ) @ I3 ) @ ( nth_nat @ ( produc5865812112468994567st_nat @ ( missin7656342814955748836at_nat @ P @ As @ Bs ) ) @ I3 ) ) ) ).

% filter2_sound
thf(fact_935_trancl__listp_Olist__trancl,axiom,
    ! [R5: nat > nat > $o,Xs2: list_nat,Ys: list_nat,I: nat,Z4: nat] :
      ( ( trancl_listp_nat @ R5 @ Xs2 @ Ys )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( R5 @ ( nth_nat @ Ys @ I ) @ Z4 )
         => ( trancl_listp_nat @ R5 @ Xs2 @ ( list_update_nat @ Ys @ I @ Z4 ) ) ) ) ) ).

% trancl_listp.list_trancl
thf(fact_936_trancl__listp_Obase,axiom,
    ! [Xs2: list_nat,Ys: list_nat,R5: nat > nat > $o] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
           => ( R5 @ ( nth_nat @ Xs2 @ I4 ) @ ( nth_nat @ Ys @ I4 ) ) )
       => ( trancl_listp_nat @ R5 @ Xs2 @ Ys ) ) ) ).

% trancl_listp.base
thf(fact_937_filter2__length,axiom,
    ! [P: nat > nat > $o,As: list_nat,Bs: list_nat] :
      ( ( size_size_list_nat @ ( produc1382935764643595205st_nat @ ( missin7656342814955748836at_nat @ P @ As @ Bs ) ) )
      = ( size_size_list_nat @ ( produc5865812112468994567st_nat @ ( missin7656342814955748836at_nat @ P @ As @ Bs ) ) ) ) ).

% filter2_length
thf(fact_938_trancl__listp_Ocases,axiom,
    ! [R5: nat > nat > $o,A1: list_nat,A23: list_nat] :
      ( ( trancl_listp_nat @ R5 @ A1 @ A23 )
     => ( ( ( ( size_size_list_nat @ A1 )
            = ( size_size_list_nat @ A23 ) )
         => ~ ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ A23 ) )
               => ( R5 @ ( nth_nat @ A1 @ I3 ) @ ( nth_nat @ A23 @ I3 ) ) ) )
       => ~ ! [Ys2: list_nat,I4: nat,Z3: nat] :
              ( ( A23
                = ( list_update_nat @ Ys2 @ I4 @ Z3 ) )
             => ( ( trancl_listp_nat @ R5 @ A1 @ Ys2 )
               => ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys2 ) )
                 => ~ ( R5 @ ( nth_nat @ Ys2 @ I4 ) @ Z3 ) ) ) ) ) ) ).

% trancl_listp.cases
thf(fact_939_trancl__listp_Osimps,axiom,
    ( trancl_listp_nat
    = ( ^ [R6: nat > nat > $o,A12: list_nat,A24: list_nat] :
          ( ? [Xs: list_nat,Ys6: list_nat] :
              ( ( A12 = Xs )
              & ( A24 = Ys6 )
              & ( ( size_size_list_nat @ Xs )
                = ( size_size_list_nat @ Ys6 ) )
              & ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys6 ) )
                 => ( R6 @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys6 @ I2 ) ) ) )
          | ? [Xs: list_nat,Ys6: list_nat,I2: nat,Z5: nat] :
              ( ( A12 = Xs )
              & ( A24
                = ( list_update_nat @ Ys6 @ I2 @ Z5 ) )
              & ( trancl_listp_nat @ R6 @ Xs @ Ys6 )
              & ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys6 ) )
              & ( R6 @ ( nth_nat @ Ys6 @ I2 ) @ Z5 ) ) ) ) ) ).

% trancl_listp.simps
thf(fact_940_listrel1__iff__update,axiom,
    ! [Xs2: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( listrel1_nat @ R2 ) )
      = ( ? [Y4: nat,N2: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs2 @ N2 ) @ Y4 ) @ R2 )
            & ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
            & ( Ys
              = ( list_update_nat @ Xs2 @ N2 @ Y4 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_941_take__drop__update__first,axiom,
    ! [J: nat,Ds: list_nat,Cs: list_nat] :
      ( ( ord_less_nat @ J @ ( size_size_list_nat @ Ds ) )
     => ( ( ( size_size_list_nat @ Cs )
          = ( size_size_list_nat @ Ds ) )
       => ( ( list_update_nat @ ( append_nat @ ( take_nat @ J @ Ds ) @ ( drop_nat @ J @ Cs ) ) @ J @ ( nth_nat @ Ds @ J ) )
          = ( append_nat @ ( take_nat @ ( suc @ J ) @ Ds ) @ ( drop_nat @ ( suc @ J ) @ Cs ) ) ) ) ) ).

% take_drop_update_first
thf(fact_942_upd__conv__take__nth__drop,axiom,
    ! [I: nat,Xs2: list_nat,A2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( list_update_nat @ Xs2 @ I @ A2 )
        = ( append_nat @ ( take_nat @ I @ Xs2 ) @ ( cons_nat @ A2 @ ( drop_nat @ ( suc @ I ) @ Xs2 ) ) ) ) ) ).

% upd_conv_take_nth_drop
thf(fact_943_drop0,axiom,
    ( ( drop_nat @ zero_zero_nat )
    = ( ^ [X2: list_nat] : X2 ) ) ).

% drop0
thf(fact_944_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_945_append__take__drop__id,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( append_nat @ ( take_nat @ N @ Xs2 ) @ ( drop_nat @ N @ Xs2 ) )
      = Xs2 ) ).

% append_take_drop_id
thf(fact_946_drop__update__cancel,axiom,
    ! [N: nat,M: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( ( drop_nat @ M @ ( list_update_nat @ Xs2 @ N @ X ) )
        = ( drop_nat @ M @ Xs2 ) ) ) ).

% drop_update_cancel
thf(fact_947_drop__zip,axiom,
    ! [N: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( drop_P8868858903918902087at_nat @ N @ ( zip_nat_nat @ Xs2 @ Ys ) )
      = ( zip_nat_nat @ ( drop_nat @ N @ Xs2 ) @ ( drop_nat @ N @ Ys ) ) ) ).

% drop_zip
thf(fact_948_list__all2__dropI,axiom,
    ! [P: nat > nat > $o,Xs2: list_nat,Ys: list_nat,N: nat] :
      ( ( list_all2_nat_nat @ P @ Xs2 @ Ys )
     => ( list_all2_nat_nat @ P @ ( drop_nat @ N @ Xs2 ) @ ( drop_nat @ N @ Ys ) ) ) ).

% list_all2_dropI
thf(fact_949_drop__0,axiom,
    ! [Xs2: list_nat] :
      ( ( drop_nat @ zero_zero_nat @ Xs2 )
      = Xs2 ) ).

% drop_0
thf(fact_950_drop__Nil,axiom,
    ! [N: nat] :
      ( ( drop_nat @ N @ nil_nat )
      = nil_nat ) ).

% drop_Nil
thf(fact_951_in__set__dropD,axiom,
    ! [X: nat,N: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( drop_nat @ N @ Xs2 ) ) )
     => ( member_nat @ X @ ( set_nat2 @ Xs2 ) ) ) ).

% in_set_dropD
thf(fact_952_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_953_listrel1__eq__len,axiom,
    ! [Xs2: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( listrel1_nat @ R2 ) )
     => ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_954_append__eq__conv__conj,axiom,
    ! [Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Xs2 @ Ys )
        = Zs )
      = ( ( Xs2
          = ( take_nat @ ( size_size_list_nat @ Xs2 ) @ Zs ) )
        & ( Ys
          = ( drop_nat @ ( size_size_list_nat @ Xs2 ) @ Zs ) ) ) ) ).

% append_eq_conv_conj
thf(fact_955_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_956_take__drop__imp__nth,axiom,
    ! [I: nat,Ss3: list_nat,X: nat] :
      ( ( ( append_nat @ ( take_nat @ I @ Ss3 ) @ ( cons_nat @ X @ ( drop_nat @ ( suc @ I ) @ Ss3 ) ) )
        = Ss3 )
     => ( X
        = ( nth_nat @ Ss3 @ I ) ) ) ).

% take_drop_imp_nth
thf(fact_957_zip__append1,axiom,
    ! [Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( zip_nat_nat @ ( append_nat @ Xs2 @ Ys ) @ Zs )
      = ( append985823374593552924at_nat @ ( zip_nat_nat @ Xs2 @ ( take_nat @ ( size_size_list_nat @ Xs2 ) @ Zs ) ) @ ( zip_nat_nat @ Ys @ ( drop_nat @ ( size_size_list_nat @ Xs2 ) @ Zs ) ) ) ) ).

% zip_append1
thf(fact_958_zip__append2,axiom,
    ! [Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( zip_nat_nat @ Xs2 @ ( append_nat @ Ys @ Zs ) )
      = ( append985823374593552924at_nat @ ( zip_nat_nat @ ( take_nat @ ( size_size_list_nat @ Ys ) @ Xs2 ) @ Ys ) @ ( zip_nat_nat @ ( drop_nat @ ( size_size_list_nat @ Ys ) @ Xs2 ) @ Zs ) ) ) ).

% zip_append2
thf(fact_959_nth__drop__0,axiom,
    ! [Ss3: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Ss3 ) )
     => ( ( cons_nat @ ( nth_nat @ Ss3 @ zero_zero_nat ) @ ( drop_nat @ ( suc @ zero_zero_nat ) @ Ss3 ) )
        = Ss3 ) ) ).

% nth_drop_0
thf(fact_960_take__drop__update__second,axiom,
    ! [J: nat,Ds: list_nat,Cs: list_nat] :
      ( ( ord_less_nat @ J @ ( size_size_list_nat @ Ds ) )
     => ( ( ( size_size_list_nat @ Cs )
          = ( size_size_list_nat @ Ds ) )
       => ( ( list_update_nat @ ( append_nat @ ( take_nat @ J @ Ds ) @ ( drop_nat @ J @ Cs ) ) @ J @ ( nth_nat @ Cs @ J ) )
          = ( append_nat @ ( take_nat @ J @ Ds ) @ ( drop_nat @ J @ Cs ) ) ) ) ) ).

% take_drop_update_second
thf(fact_961_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_962_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_963_take__nth__drop__concat,axiom,
    ! [I: nat,Xss: list_list_nat,Ys: list_nat,J: nat,Z4: nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xss ) )
     => ( ( ( nth_list_nat @ Xss @ I )
          = Ys )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Ys ) )
         => ( ( ( nth_nat @ Ys @ J )
              = Z4 )
           => ? [K2: nat] :
                ( ( ord_less_nat @ K2 @ ( size_size_list_nat @ ( concat_nat @ Xss ) ) )
                & ( ( take_nat @ K2 @ ( concat_nat @ Xss ) )
                  = ( append_nat @ ( concat_nat @ ( take_list_nat @ I @ Xss ) ) @ ( take_nat @ J @ Ys ) ) )
                & ( ( nth_nat @ ( concat_nat @ Xss ) @ K2 )
                  = ( nth_nat @ ( nth_list_nat @ Xss @ I ) @ J ) )
                & ( ( drop_nat @ ( suc @ K2 ) @ ( concat_nat @ Xss ) )
                  = ( append_nat @ ( drop_nat @ ( suc @ J ) @ Ys ) @ ( concat_nat @ ( drop_list_nat @ ( suc @ I ) @ Xss ) ) ) ) ) ) ) ) ) ).

% take_nth_drop_concat
thf(fact_964_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_965_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_966_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_967_dual__order_Orefl,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_968_less__eq__prod__simp,axiom,
    ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
      ( ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ X1 @ Y1 ) @ ( product_Pair_nat_nat @ X23 @ Y23 ) )
      = ( ( ord_less_nat @ X1 @ X23 )
        | ( ( ord_less_eq_nat @ X1 @ X23 )
          & ( ord_less_eq_nat @ Y1 @ Y23 ) ) ) ) ).

% less_eq_prod_simp
thf(fact_969_take__all,axiom,
    ! [Xs2: list_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N )
     => ( ( take_nat @ N @ Xs2 )
        = Xs2 ) ) ).

% take_all
thf(fact_970_take__all__iff,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ( take_nat @ N @ Xs2 )
        = Xs2 )
      = ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ).

% take_all_iff
thf(fact_971_list__update__beyond,axiom,
    ! [Xs2: list_nat,I: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I )
     => ( ( list_update_nat @ Xs2 @ I @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_972_take__update__cancel,axiom,
    ! [N: nat,M: nat,Xs2: list_nat,Y: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( take_nat @ N @ ( list_update_nat @ Xs2 @ M @ Y ) )
        = ( take_nat @ N @ Xs2 ) ) ) ).

% take_update_cancel
thf(fact_973_hd__replicate,axiom,
    ! [N: nat,X: bot_bot_term_f] :
      ( ( N != zero_zero_nat )
     => ( ( hd_Bot_bot_term_f @ ( replic6958083601664011982term_f @ N @ X ) )
        = X ) ) ).

% hd_replicate
thf(fact_974_hd__replicate,axiom,
    ! [N: nat,X: nat] :
      ( ( N != zero_zero_nat )
     => ( ( hd_nat @ ( replicate_nat @ N @ X ) )
        = X ) ) ).

% hd_replicate
thf(fact_975_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_976_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_977_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_978_hd__take,axiom,
    ! [J: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ J )
     => ( ( hd_nat @ ( take_nat @ J @ Xs2 ) )
        = ( hd_nat @ Xs2 ) ) ) ).

% hd_take
thf(fact_979_less__prod__simp,axiom,
    ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
      ( ( ord_le1203424502768444845at_nat @ ( product_Pair_nat_nat @ X1 @ Y1 ) @ ( product_Pair_nat_nat @ X23 @ Y23 ) )
      = ( ( ord_less_nat @ X1 @ X23 )
        | ( ( ord_less_eq_nat @ X1 @ X23 )
          & ( ord_less_nat @ Y1 @ Y23 ) ) ) ) ).

% less_prod_simp
thf(fact_980_Product__Lexorder_Oless__eq__prod__def,axiom,
    ( ord_le8460144461188290721at_nat
    = ( ^ [X2: product_prod_nat_nat,Y4: product_prod_nat_nat] :
          ( ( ord_less_nat @ ( product_fst_nat_nat @ X2 ) @ ( product_fst_nat_nat @ Y4 ) )
          | ( ( ord_less_eq_nat @ ( product_fst_nat_nat @ X2 ) @ ( product_fst_nat_nat @ Y4 ) )
            & ( ord_less_eq_nat @ ( product_snd_nat_nat @ X2 ) @ ( product_snd_nat_nat @ Y4 ) ) ) ) ) ) ).

% Product_Lexorder.less_eq_prod_def
thf(fact_981_min__absorb2,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_min_nat @ X @ Y )
        = Y ) ) ).

% min_absorb2
thf(fact_982_min__absorb1,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_min_nat @ X @ Y )
        = X ) ) ).

% min_absorb1
thf(fact_983_min__def,axiom,
    ( ord_min_nat
    = ( ^ [A6: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A6 @ B4 ) @ A6 @ B4 ) ) ) ).

% min_def
thf(fact_984_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_985_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_986_order__less__le__subst2,axiom,
    ! [A2: nat,B: nat,F2: nat > nat,C3: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A2 ) @ C3 ) ) ) ) ).

% order_less_le_subst2
thf(fact_987_order__less__le__subst1,axiom,
    ! [A2: nat,F2: nat > nat,B: nat,C3: nat] :
      ( ( ord_less_nat @ A2 @ ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A2 @ ( F2 @ C3 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_988_order__le__less__subst2,axiom,
    ! [A2: nat,B: nat,F2: nat > nat,C3: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_nat @ ( F2 @ B ) @ C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A2 ) @ C3 ) ) ) ) ).

% order_le_less_subst2
thf(fact_989_order__le__less__subst1,axiom,
    ! [A2: nat,F2: nat > nat,B: nat,C3: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A2 @ ( F2 @ C3 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_990_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z4: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z4 )
       => ( ord_less_nat @ X @ Z4 ) ) ) ).

% order_less_le_trans
thf(fact_991_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z4: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z4 )
       => ( ord_less_nat @ X @ Z4 ) ) ) ).

% order_le_less_trans
thf(fact_992_order__neq__le__trans,axiom,
    ! [A2: nat,B: nat] :
      ( ( A2 != B )
     => ( ( ord_less_eq_nat @ A2 @ B )
       => ( ord_less_nat @ A2 @ B ) ) ) ).

% order_neq_le_trans
thf(fact_993_order__le__neq__trans,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( A2 != B )
       => ( ord_less_nat @ A2 @ B ) ) ) ).

% order_le_neq_trans
thf(fact_994_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_995_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_996_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_997_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X2: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y4 )
          & ( X2 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_998_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y4: nat] :
          ( ( ord_less_nat @ X2 @ Y4 )
          | ( X2 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_999_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A2: nat] :
      ( ( ord_less_nat @ B @ A2 )
     => ( ord_less_eq_nat @ B @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_1000_order_Ostrict__implies__order,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ord_less_eq_nat @ A2 @ B ) ) ).

% order.strict_implies_order
thf(fact_1001_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A6: nat] :
          ( ( ord_less_eq_nat @ B4 @ A6 )
          & ~ ( ord_less_eq_nat @ A6 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1002_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A2: nat,C3: nat] :
      ( ( ord_less_nat @ B @ A2 )
     => ( ( ord_less_eq_nat @ C3 @ B )
       => ( ord_less_nat @ C3 @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_1003_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A2: nat,C3: nat] :
      ( ( ord_less_eq_nat @ B @ A2 )
     => ( ( ord_less_nat @ C3 @ B )
       => ( ord_less_nat @ C3 @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_1004_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A6: nat] :
          ( ( ord_less_eq_nat @ B4 @ A6 )
          & ( A6 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1005_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A6: nat] :
          ( ( ord_less_nat @ B4 @ A6 )
          | ( A6 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1006_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A6: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A6 @ B4 )
          & ~ ( ord_less_eq_nat @ B4 @ A6 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1007_order_Ostrict__trans2,axiom,
    ! [A2: nat,B: nat,C3: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ B @ C3 )
       => ( ord_less_nat @ A2 @ C3 ) ) ) ).

% order.strict_trans2
thf(fact_1008_order_Ostrict__trans1,axiom,
    ! [A2: nat,B: nat,C3: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_nat @ B @ C3 )
       => ( ord_less_nat @ A2 @ C3 ) ) ) ).

% order.strict_trans1
thf(fact_1009_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A6: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A6 @ B4 )
          & ( A6 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1010_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A6: nat,B4: nat] :
          ( ( ord_less_nat @ A6 @ B4 )
          | ( A6 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1011_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_1012_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X2: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y4 )
          & ~ ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_1013_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_1014_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_1015_nless__le,axiom,
    ! [A2: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A2 @ B ) )
      = ( ~ ( ord_less_eq_nat @ A2 @ B )
        | ( A2 = B ) ) ) ).

% nless_le
thf(fact_1016_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_1017_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_1018_top_Oextremum__uniqueI,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
     => ( A2 = top_top_set_nat ) ) ).

% top.extremum_uniqueI
thf(fact_1019_top_Oextremum__unique,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
      = ( A2 = top_top_set_nat ) ) ).

% top.extremum_unique
thf(fact_1020_top__greatest,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).

% top_greatest
thf(fact_1021_nle__le,axiom,
    ! [A2: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
      = ( ( ord_less_eq_nat @ B @ A2 )
        & ( B != A2 ) ) ) ).

% nle_le
thf(fact_1022_le__cases3,axiom,
    ! [X: nat,Y: nat,Z4: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z4 ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z4 ) )
       => ( ( ( ord_less_eq_nat @ X @ Z4 )
           => ~ ( ord_less_eq_nat @ Z4 @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z4 @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z4 )
               => ~ ( ord_less_eq_nat @ Z4 @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z4 @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1023_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
    = ( ^ [X2: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y4 )
          & ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1024_ord__eq__le__trans,axiom,
    ! [A2: nat,B: nat,C3: nat] :
      ( ( A2 = B )
     => ( ( ord_less_eq_nat @ B @ C3 )
       => ( ord_less_eq_nat @ A2 @ C3 ) ) ) ).

% ord_eq_le_trans
thf(fact_1025_ord__le__eq__trans,axiom,
    ! [A2: nat,B: nat,C3: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( B = C3 )
       => ( ord_less_eq_nat @ A2 @ C3 ) ) ) ).

% ord_le_eq_trans
thf(fact_1026_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_1027_order_Otrans,axiom,
    ! [A2: nat,B: nat,C3: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ B @ C3 )
       => ( ord_less_eq_nat @ A2 @ C3 ) ) ) ).

% order.trans
thf(fact_1028_order__trans,axiom,
    ! [X: nat,Y: nat,Z4: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z4 )
       => ( ord_less_eq_nat @ X @ Z4 ) ) ) ).

% order_trans
thf(fact_1029_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B: nat] :
      ( ! [A5: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A5 @ B3 )
         => ( P @ A5 @ B3 ) )
     => ( ! [A5: nat,B3: nat] :
            ( ( P @ B3 @ A5 )
           => ( P @ A5 @ B3 ) )
       => ( P @ A2 @ B ) ) ) ).

% linorder_wlog
thf(fact_1030_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
    = ( ^ [A6: nat,B4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A6 )
          & ( ord_less_eq_nat @ A6 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1031_dual__order_Oantisym,axiom,
    ! [B: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B @ A2 )
     => ( ( ord_less_eq_nat @ A2 @ B )
       => ( A2 = B ) ) ) ).

% dual_order.antisym
thf(fact_1032_dual__order_Otrans,axiom,
    ! [B: nat,A2: nat,C3: nat] :
      ( ( ord_less_eq_nat @ B @ A2 )
     => ( ( ord_less_eq_nat @ C3 @ B )
       => ( ord_less_eq_nat @ C3 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_1033_antisym,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ B @ A2 )
       => ( A2 = B ) ) ) ).

% antisym
thf(fact_1034_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
    = ( ^ [A6: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A6 @ B4 )
          & ( ord_less_eq_nat @ B4 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1035_order__subst1,axiom,
    ! [A2: nat,F2: nat > nat,B: nat,C3: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F2 @ C3 ) ) ) ) ) ).

% order_subst1
thf(fact_1036_order__subst2,axiom,
    ! [A2: nat,B: nat,F2: nat > nat,C3: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A2 ) @ C3 ) ) ) ) ).

% order_subst2
thf(fact_1037_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1038_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_1039_ord__eq__le__subst,axiom,
    ! [A2: nat,F2: nat > nat,B: nat,C3: nat] :
      ( ( A2
        = ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F2 @ C3 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1040_ord__le__eq__subst,axiom,
    ! [A2: nat,B: nat,F2: nat > nat,C3: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ( F2 @ B )
          = C3 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A2 ) @ C3 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1041_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_1042_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_1043_list_Oset__sel_I1_J,axiom,
    ! [A2: list_nat] :
      ( ( A2 != nil_nat )
     => ( member_nat @ ( hd_nat @ A2 ) @ ( set_nat2 @ A2 ) ) ) ).

% list.set_sel(1)
thf(fact_1044_hd__in__set,axiom,
    ! [Xs2: list_nat] :
      ( ( Xs2 != nil_nat )
     => ( member_nat @ ( hd_nat @ Xs2 ) @ ( set_nat2 @ Xs2 ) ) ) ).

% hd_in_set
thf(fact_1045_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_1046_count__le__length,axiom,
    ! [Xs2: list_nat,X: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs2 @ X ) @ ( size_size_list_nat @ Xs2 ) ) ).

% count_le_length
thf(fact_1047_hd__conv__nth,axiom,
    ! [Xs2: list_nat] :
      ( ( Xs2 != nil_nat )
     => ( ( hd_nat @ Xs2 )
        = ( nth_nat @ Xs2 @ zero_zero_nat ) ) ) ).

% hd_conv_nth
thf(fact_1048_eq__length__concat__nth,axiom,
    ! [Xs2: list_list_nat,Ys: list_list_nat] :
      ( ( ( size_s3023201423986296836st_nat @ Xs2 )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Xs2 @ I4 ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I4 ) ) ) )
       => ( ( size_size_list_nat @ ( concat_nat @ Xs2 ) )
          = ( size_size_list_nat @ ( concat_nat @ Ys ) ) ) ) ) ).

% eq_length_concat_nth
thf(fact_1049_Suc__le__length__iff,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs2 ) )
      = ( ? [X2: nat,Ys6: list_nat] :
            ( ( Xs2
              = ( cons_nat @ X2 @ Ys6 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys6 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_1050_lenlex__length,axiom,
    ! [Ms: list_nat,Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R2 ) )
     => ( ord_less_eq_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) ) ) ).

% lenlex_length
thf(fact_1051_concat__all__nth,axiom,
    ! [Xs2: list_list_nat,Ys: list_list_nat,P: nat > nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Xs2 )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Xs2 @ I4 ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I4 ) ) ) )
       => ( ! [I4: nat,J2: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
             => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ ( nth_list_nat @ Xs2 @ I4 ) ) )
               => ( P @ ( nth_nat @ ( nth_list_nat @ Xs2 @ I4 ) @ J2 ) @ ( nth_nat @ ( nth_list_nat @ Ys @ I4 ) @ J2 ) ) ) )
         => ! [K3: nat] :
              ( ( ord_less_nat @ K3 @ ( size_size_list_nat @ ( concat_nat @ Xs2 ) ) )
             => ( P @ ( nth_nat @ ( concat_nat @ Xs2 ) @ K3 ) @ ( nth_nat @ ( concat_nat @ Ys ) @ K3 ) ) ) ) ) ) ).

% concat_all_nth
thf(fact_1052_nth__concat__split,axiom,
    ! [I: nat,Xs2: list_list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs2 ) ) )
     => ? [J2: nat,K2: nat] :
          ( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
          & ( ord_less_nat @ K2 @ ( size_size_list_nat @ ( nth_list_nat @ Xs2 @ J2 ) ) )
          & ( ( nth_nat @ ( concat_nat @ Xs2 ) @ I )
            = ( nth_nat @ ( nth_list_nat @ Xs2 @ J2 ) @ K2 ) ) ) ) ).

% nth_concat_split
thf(fact_1053_nth__concat__two__lists,axiom,
    ! [I: nat,Xs2: list_list_nat,Ys: list_list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs2 ) ) )
     => ( ( ( size_s3023201423986296836st_nat @ Ys )
          = ( size_s3023201423986296836st_nat @ Xs2 ) )
       => ( ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
             => ( ( size_size_list_nat @ ( nth_list_nat @ Ys @ I4 ) )
                = ( size_size_list_nat @ ( nth_list_nat @ Xs2 @ I4 ) ) ) )
         => ? [J2: nat,K2: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
              & ( ord_less_nat @ K2 @ ( size_size_list_nat @ ( nth_list_nat @ Xs2 @ J2 ) ) )
              & ( ( nth_nat @ ( concat_nat @ Xs2 ) @ I )
                = ( nth_nat @ ( nth_list_nat @ Xs2 @ J2 ) @ K2 ) )
              & ( ( nth_nat @ ( concat_nat @ Ys ) @ I )
                = ( nth_nat @ ( nth_list_nat @ Ys @ J2 ) @ K2 ) ) ) ) ) ) ).

% nth_concat_two_lists
thf(fact_1054_nth__concat__diff,axiom,
    ! [I1: nat,Xs2: list_list_nat,I22: nat] :
      ( ( ord_less_nat @ I1 @ ( size_size_list_nat @ ( concat_nat @ Xs2 ) ) )
     => ( ( ord_less_nat @ I22 @ ( size_size_list_nat @ ( concat_nat @ 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_s3023201423986296836st_nat @ Xs2 ) )
              & ( ord_less_nat @ J22 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
              & ( ord_less_nat @ K1 @ ( size_size_list_nat @ ( nth_list_nat @ Xs2 @ J1 ) ) )
              & ( ord_less_nat @ K22 @ ( size_size_list_nat @ ( nth_list_nat @ Xs2 @ J22 ) ) )
              & ( ( nth_nat @ ( concat_nat @ Xs2 ) @ I1 )
                = ( nth_nat @ ( nth_list_nat @ Xs2 @ J1 ) @ K1 ) )
              & ( ( nth_nat @ ( concat_nat @ Xs2 ) @ I22 )
                = ( nth_nat @ ( nth_list_nat @ Xs2 @ J22 ) @ K22 ) ) ) ) ) ) ).

% nth_concat_diff
thf(fact_1055_concat__nth__nthI,axiom,
    ! [Ss3: list_list_nat,Ts2: list_list_nat,P: nat > nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Ss3 )
        = ( size_s3023201423986296836st_nat @ Ts2 ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Ts2 ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Ss3 @ I4 ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ts2 @ I4 ) ) ) )
       => ( ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Ts2 ) )
             => ! [J2: nat] :
                  ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ ( nth_list_nat @ Ts2 @ I4 ) ) )
                 => ( P @ ( nth_nat @ ( nth_list_nat @ Ss3 @ I4 ) @ J2 ) @ ( nth_nat @ ( nth_list_nat @ Ts2 @ I4 ) @ J2 ) ) ) )
         => ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ ( concat_nat @ Ts2 ) ) )
             => ( P @ ( nth_nat @ ( concat_nat @ Ss3 ) @ I3 ) @ ( nth_nat @ ( concat_nat @ Ts2 ) @ I3 ) ) ) ) ) ) ).

% concat_nth_nthI
thf(fact_1056_hd__drop__conv__nth,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( ( hd_nat @ ( drop_nat @ N @ Xs2 ) )
        = ( nth_nat @ Xs2 @ N ) ) ) ).

% hd_drop_conv_nth
thf(fact_1057_nth__take__lemma,axiom,
    ! [K: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Ys ) )
       => ( ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ K )
             => ( ( nth_nat @ Xs2 @ I4 )
                = ( nth_nat @ Ys @ I4 ) ) )
         => ( ( take_nat @ K @ Xs2 )
            = ( take_nat @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_1058_nth__take__prefix,axiom,
    ! [Ys: list_nat,Xs2: list_nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs2 ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
           => ( ( nth_nat @ Xs2 @ I4 )
              = ( nth_nat @ Ys @ I4 ) ) )
       => ( ( take_nat @ ( size_size_list_nat @ Ys ) @ Xs2 )
          = Ys ) ) ) ).

% nth_take_prefix
thf(fact_1059_append__eq__append__conv__if,axiom,
    ! [Xs_1: list_nat,Xs_2: list_nat,Ys_1: list_nat,Ys_2: list_nat] :
      ( ( ( append_nat @ Xs_1 @ Xs_2 )
        = ( append_nat @ Ys_1 @ Ys_2 ) )
      = ( ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_1 ) @ ( size_size_list_nat @ Ys_1 ) )
         => ( ( Xs_1
              = ( take_nat @ ( size_size_list_nat @ Xs_1 ) @ Ys_1 ) )
            & ( Xs_2
              = ( append_nat @ ( drop_nat @ ( size_size_list_nat @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_1 ) @ ( size_size_list_nat @ Ys_1 ) )
         => ( ( ( take_nat @ ( size_size_list_nat @ Ys_1 ) @ Xs_1 )
              = Ys_1 )
            & ( ( append_nat @ ( drop_nat @ ( size_size_list_nat @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
              = Ys_2 ) ) ) ) ) ).

% append_eq_append_conv_if
thf(fact_1060_lexord__sufI,axiom,
    ! [U: list_nat,W2: list_nat,R2: set_Pr1261947904930325089at_nat,V: list_nat,Z4: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ W2 ) @ ( lexord_nat @ R2 ) )
     => ( ( ord_less_eq_nat @ ( size_size_list_nat @ W2 ) @ ( size_size_list_nat @ U ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ U @ V ) @ ( append_nat @ W2 @ Z4 ) ) @ ( lexord_nat @ R2 ) ) ) ) ).

% lexord_sufI
thf(fact_1061_prod__less__def,axiom,
    ( ord_le1203424502768444845at_nat
    = ( ^ [X2: product_prod_nat_nat,Y4: product_prod_nat_nat] :
          ( ( ord_less_nat @ ( product_fst_nat_nat @ X2 ) @ ( product_fst_nat_nat @ Y4 ) )
          | ( ( ord_less_eq_nat @ ( product_fst_nat_nat @ X2 ) @ ( product_fst_nat_nat @ Y4 ) )
            & ( ord_less_nat @ ( product_snd_nat_nat @ X2 ) @ ( product_snd_nat_nat @ Y4 ) ) ) ) ) ) ).

% prod_less_def
thf(fact_1062_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_1063_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_1064_nth__append__take__is__nth__conv,axiom,
    ! [I: nat,J: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ ( size_size_list_nat @ Xs2 ) )
       => ( ( nth_nat @ ( append_nat @ ( take_nat @ J @ Xs2 ) @ Ys ) @ I )
          = ( nth_nat @ Xs2 @ I ) ) ) ) ).

% nth_append_take_is_nth_conv
thf(fact_1065_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_1066_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_1067_concat__index__split__less__length__concat_I4_J,axiom,
    ! [I: nat,Xs2: list_list_nat,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs2 ) ) )
     => ( ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs2 )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ( nth_nat @ ( concat_nat @ Xs2 ) @ I )
          = ( nth_nat @ ( nth_list_nat @ Xs2 @ M ) @ N ) ) ) ) ).

% concat_index_split_less_length_concat(4)
thf(fact_1068_remove__nth__sound__r,axiom,
    ! [N: nat,P2: nat,Xs2: list_nat] :
      ( ( ord_less_eq_nat @ N @ P2 )
     => ( ( ord_less_nat @ P2 @ ( size_size_list_nat @ Xs2 ) )
       => ( ( nth_nat @ ( missin7175274867594579095th_nat @ N @ Xs2 ) @ P2 )
          = ( nth_nat @ Xs2 @ ( suc @ P2 ) ) ) ) ) ).

% remove_nth_sound_r
thf(fact_1069_concat__index__split__larger__lists,axiom,
    ! [I: nat,Xs2: list_list_nat,N: nat,Ys: list_list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs2 ) ) )
     => ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I ) @ ( append_list_nat @ Xs2 @ Ys ) )
        = ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I ) @ Xs2 ) ) ) ).

% concat_index_split_larger_lists
thf(fact_1070_subset__code_I1_J,axiom,
    ! [Xs2: list_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B5 )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
           => ( member_nat @ X2 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_1071_subset__UNIV,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).

% subset_UNIV
thf(fact_1072_set__take__subset,axiom,
    ! [N: nat,Xs2: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ N @ Xs2 ) ) @ ( set_nat2 @ Xs2 ) ) ).

% set_take_subset
thf(fact_1073_set__drop__subset,axiom,
    ! [N: nat,Xs2: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( drop_nat @ N @ Xs2 ) ) @ ( set_nat2 @ Xs2 ) ) ).

% set_drop_subset
thf(fact_1074_set__update__subsetI,axiom,
    ! [Xs2: list_nat,A: set_nat,X: nat,I: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A )
     => ( ( member_nat @ X @ A )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X ) ) @ A ) ) ) ).

% set_update_subsetI
thf(fact_1075_remove__nth__id,axiom,
    ! [Xs2: list_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N )
     => ( ( missin7175274867594579095th_nat @ N @ Xs2 )
        = Xs2 ) ) ).

% remove_nth_id
thf(fact_1076_remove__nth__sound__l,axiom,
    ! [P2: nat,N: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ P2 @ N )
     => ( ( nth_nat @ ( missin7175274867594579095th_nat @ N @ Xs2 ) @ P2 )
        = ( nth_nat @ Xs2 @ P2 ) ) ) ).

% remove_nth_sound_l
thf(fact_1077_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs2: list_nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ M @ Xs2 ) ) @ ( set_nat2 @ ( take_nat @ N @ Xs2 ) ) ) ) ).

% set_take_subset_set_take
thf(fact_1078_set__drop__subset__set__drop,axiom,
    ! [N: nat,M: nat,Xs2: list_nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ord_less_eq_set_nat @ ( set_nat2 @ ( drop_nat @ M @ Xs2 ) ) @ ( set_nat2 @ ( drop_nat @ N @ Xs2 ) ) ) ) ).

% set_drop_subset_set_drop
thf(fact_1079_set__list__subset__eq__nth__conv,axiom,
    ! [Xs2: list_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
           => ( member_nat @ ( nth_nat @ Xs2 @ I2 ) @ A ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_1080_subseteq__set__conv__nth,axiom,
    ! [Ss3: list_nat,T3: set_nat] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ss3 ) )
           => ( member_nat @ ( nth_nat @ Ss3 @ I2 ) @ T3 ) ) )
      = ( ord_less_eq_set_nat @ ( set_nat2 @ Ss3 ) @ T3 ) ) ).

% subseteq_set_conv_nth
thf(fact_1081_remove__nth__len,axiom,
    ! [I: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( size_size_list_nat @ Xs2 )
        = ( suc @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ I @ Xs2 ) ) ) ) ) ).

% remove_nth_len
thf(fact_1082_remove__nth__P__compat,axiom,
    ! [As: list_nat,Bs: list_nat,P: nat > nat > $o,P2: nat] :
      ( ( ( size_size_list_nat @ As )
        = ( size_size_list_nat @ Bs ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ As ) )
           => ( P @ ( nth_nat @ As @ I4 ) @ ( nth_nat @ Bs @ I4 ) ) )
       => ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ P2 @ As ) ) )
           => ( P @ ( nth_nat @ ( missin7175274867594579095th_nat @ P2 @ As ) @ I3 ) @ ( nth_nat @ ( missin7175274867594579095th_nat @ P2 @ Bs ) @ I3 ) ) ) ) ) ).

% remove_nth_P_compat
thf(fact_1083_concat__index__split__mono__first__arg,axiom,
    ! [I: nat,Xs2: list_list_nat,O_idx: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs2 ) ) )
     => ( ord_less_eq_nat @ O_idx @ ( product_fst_nat_nat @ ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ O_idx @ I ) @ Xs2 ) ) ) ) ).

% concat_index_split_mono_first_arg
thf(fact_1084_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_1085_nth__remove__nth__conv,axiom,
    ! [I: nat,N: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ N @ Xs2 ) ) )
     => ( ( nth_nat @ ( missin7175274867594579095th_nat @ N @ Xs2 ) @ I )
        = ( nth_nat @ Xs2 @ ( if_nat @ ( ord_less_nat @ I @ N ) @ I @ ( suc @ I ) ) ) ) ) ).

% nth_remove_nth_conv
thf(fact_1086_remove__nth__def,axiom,
    ( missin7175274867594579095th_nat
    = ( ^ [N2: nat,Xs: list_nat] : ( append_nat @ ( take_nat @ N2 @ Xs ) @ ( drop_nat @ ( suc @ N2 ) @ Xs ) ) ) ) ).

% remove_nth_def
thf(fact_1087_concat__index__split__less__length__concat_I3_J,axiom,
    ! [I: nat,Xs2: list_list_nat,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs2 ) ) )
     => ( ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs2 )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ N @ ( size_size_list_nat @ ( nth_list_nat @ Xs2 @ M ) ) ) ) ) ).

% concat_index_split_less_length_concat(3)
thf(fact_1088_concat__index__split__sound__snd__arg,axiom,
    ! [I: nat,Xs2: list_list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs2 ) ) )
     => ( ord_less_nat @ ( product_snd_nat_nat @ ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs2 ) ) @ ( size_size_list_nat @ ( nth_list_nat @ Xs2 @ ( product_fst_nat_nat @ ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs2 ) ) ) ) ) ) ).

% concat_index_split_sound_snd_arg
thf(fact_1089_concat__index__split__less__length__concat_I2_J,axiom,
    ! [I: nat,Xs2: list_list_nat,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs2 ) ) )
     => ( ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs2 )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ M @ ( size_s3023201423986296836st_nat @ Xs2 ) ) ) ) ).

% concat_index_split_less_length_concat(2)
thf(fact_1090_concat__index__split__sound__fst__arg,axiom,
    ! [I: nat,Xs2: list_list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs2 ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs2 ) ) @ ( size_s3023201423986296836st_nat @ Xs2 ) ) ) ).

% concat_index_split_sound_fst_arg
thf(fact_1091_concat__index__split__unique,axiom,
    ! [I: nat,Xs2: list_list_nat,Ys: list_list_nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs2 ) ) )
     => ( ( ( size_s3023201423986296836st_nat @ Xs2 )
          = ( size_s3023201423986296836st_nat @ Ys ) )
       => ( ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
             => ( ( size_size_list_nat @ ( nth_list_nat @ Xs2 @ I4 ) )
                = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I4 ) ) ) )
         => ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I ) @ Xs2 )
            = ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I ) @ Ys ) ) ) ) ) ).

% concat_index_split_unique
thf(fact_1092_set__list__subset__nth__conv,axiom,
    ! [Xs2: list_nat,A: set_nat,I: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( member_nat @ ( nth_nat @ Xs2 @ I ) @ A ) ) ) ).

% set_list_subset_nth_conv
thf(fact_1093_adjust__idx__rev__nth,axiom,
    ! [I: nat,Xs2: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( J != I )
       => ( ( nth_nat @ ( missin7175274867594579095th_nat @ I @ Xs2 ) @ ( missin3815256168798769645dx_rev @ I @ J ) )
          = ( nth_nat @ Xs2 @ J ) ) ) ) ).

% adjust_idx_rev_nth
thf(fact_1094_adjust__idx__nth,axiom,
    ! [I: nat,Xs2: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_nat @ ( missin7175274867594579095th_nat @ I @ Xs2 ) @ J )
        = ( nth_nat @ Xs2 @ ( missing_adjust_idx @ I @ J ) ) ) ) ).

% adjust_idx_nth
thf(fact_1095_subsetI,axiom,
    ! [A: set_nat,B5: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_nat @ X3 @ B5 ) )
     => ( ord_less_eq_set_nat @ A @ B5 ) ) ).

% subsetI
thf(fact_1096_in__mono,axiom,
    ! [A: set_nat,B5: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A @ B5 )
     => ( ( member_nat @ X @ A )
       => ( member_nat @ X @ B5 ) ) ) ).

% in_mono
thf(fact_1097_subsetD,axiom,
    ! [A: set_nat,B5: set_nat,C3: nat] :
      ( ( ord_less_eq_set_nat @ A @ B5 )
     => ( ( member_nat @ C3 @ A )
       => ( member_nat @ C3 @ B5 ) ) ) ).

% subsetD
thf(fact_1098_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A4: set_nat,B6: set_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A4 )
         => ( member_nat @ X2 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_1099_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A4: set_nat,B6: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A4 )
         => ( member_nat @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_1100_adjust__idx__rev1,axiom,
    ! [I: nat,J: nat] :
      ( ( missin3815256168798769645dx_rev @ I @ ( missing_adjust_idx @ I @ J ) )
      = J ) ).

% adjust_idx_rev1
thf(fact_1101_adjust__idx__rev2,axiom,
    ! [J: nat,I: nat] :
      ( ( J != I )
     => ( ( missing_adjust_idx @ I @ ( missin3815256168798769645dx_rev @ I @ J ) )
        = J ) ) ).

% adjust_idx_rev2
thf(fact_1102_adjust__idx__i,axiom,
    ! [I: nat,J: nat] :
      ( ( missing_adjust_idx @ I @ J )
     != I ) ).

% adjust_idx_i
thf(fact_1103_adjust__idx__def,axiom,
    ( missing_adjust_idx
    = ( ^ [I2: nat,J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ I2 ) @ J3 @ ( suc @ J3 ) ) ) ) ).

% adjust_idx_def
thf(fact_1104_adjust__idx__length,axiom,
    ! [I: nat,Xs2: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ I @ Xs2 ) ) )
       => ( ord_less_nat @ ( missing_adjust_idx @ I @ J ) @ ( size_size_list_nat @ Xs2 ) ) ) ) ).

% adjust_idx_length
thf(fact_1105_adjust__idx__rev__length,axiom,
    ! [I: nat,Xs2: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
       => ( ( J != I )
         => ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I @ J ) @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ I @ Xs2 ) ) ) ) ) ) ).

% adjust_idx_rev_length
thf(fact_1106_concat__index__split__sound__snd__arg__aux,axiom,
    ! [I: nat,Xs2: list_list_nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs2 ) ) )
     => ( ord_less_nat @ ( product_snd_nat_nat @ ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I ) @ Xs2 ) ) @ ( size_size_list_nat @ ( nth_list_nat @ Xs2 @ ( minus_minus_nat @ ( product_fst_nat_nat @ ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I ) @ Xs2 ) ) @ N ) ) ) ) ) ).

% concat_index_split_sound_snd_arg_aux
thf(fact_1107_concat__index__split__sound__fst__arg__aux,axiom,
    ! [I: nat,Xs2: list_list_nat,O_idx: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs2 ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ O_idx @ I ) @ Xs2 ) ) @ ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ O_idx ) ) ) ).

% concat_index_split_sound_fst_arg_aux
thf(fact_1108_drop__drop,axiom,
    ! [N: nat,M: nat,Xs2: list_nat] :
      ( ( drop_nat @ N @ ( drop_nat @ M @ Xs2 ) )
      = ( drop_nat @ ( plus_plus_nat @ N @ M ) @ Xs2 ) ) ).

% drop_drop
thf(fact_1109_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_1110_length__drop,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( size_size_list_nat @ ( drop_nat @ N @ Xs2 ) )
      = ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ).

% length_drop
thf(fact_1111_drop__replicate,axiom,
    ! [I: nat,K: nat,X: bot_bot_term_f] :
      ( ( drop_Bot_bot_term_f @ I @ ( replic6958083601664011982term_f @ K @ X ) )
      = ( replic6958083601664011982term_f @ ( minus_minus_nat @ K @ I ) @ X ) ) ).

% drop_replicate
thf(fact_1112_drop__replicate,axiom,
    ! [I: nat,K: nat,X: nat] :
      ( ( drop_nat @ I @ ( replicate_nat @ K @ X ) )
      = ( replicate_nat @ ( minus_minus_nat @ K @ I ) @ X ) ) ).

% drop_replicate
thf(fact_1113_nth__append__length__plus,axiom,
    ! [Xs2: list_nat,Ys: list_nat,N: nat] :
      ( ( nth_nat @ ( append_nat @ Xs2 @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs2 ) @ N ) )
      = ( nth_nat @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_1114_take__append,axiom,
    ! [N: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( take_nat @ N @ ( append_nat @ Xs2 @ Ys ) )
      = ( append_nat @ ( take_nat @ N @ Xs2 ) @ ( take_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs2 ) ) @ Ys ) ) ) ).

% take_append
thf(fact_1115_drop__append,axiom,
    ! [N: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( drop_nat @ N @ ( append_nat @ Xs2 @ Ys ) )
      = ( append_nat @ ( drop_nat @ N @ Xs2 ) @ ( drop_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs2 ) ) @ Ys ) ) ) ).

% drop_append
thf(fact_1116_nth__drop,axiom,
    ! [N: nat,Xs2: list_nat,I: nat] :
      ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_nat @ ( drop_nat @ N @ Xs2 ) @ I )
        = ( nth_nat @ Xs2 @ ( plus_plus_nat @ N @ I ) ) ) ) ).

% nth_drop
thf(fact_1117_psubsetD,axiom,
    ! [A: set_nat,B5: set_nat,C3: nat] :
      ( ( ord_less_set_nat @ A @ B5 )
     => ( ( member_nat @ C3 @ A )
       => ( member_nat @ C3 @ B5 ) ) ) ).

% psubsetD
thf(fact_1118_replicate__add,axiom,
    ! [N: nat,M: nat,X: bot_bot_term_f] :
      ( ( replic6958083601664011982term_f @ ( plus_plus_nat @ N @ M ) @ X )
      = ( append3968237287470817971term_f @ ( replic6958083601664011982term_f @ N @ X ) @ ( replic6958083601664011982term_f @ M @ X ) ) ) ).

% replicate_add
thf(fact_1119_replicate__add,axiom,
    ! [N: nat,M: nat,X: nat] :
      ( ( replicate_nat @ ( plus_plus_nat @ N @ M ) @ X )
      = ( append_nat @ ( replicate_nat @ N @ X ) @ ( replicate_nat @ M @ X ) ) ) ).

% replicate_add
thf(fact_1120_drop__take,axiom,
    ! [N: nat,M: nat,Xs2: list_nat] :
      ( ( drop_nat @ N @ ( take_nat @ M @ Xs2 ) )
      = ( take_nat @ ( minus_minus_nat @ M @ N ) @ ( drop_nat @ N @ Xs2 ) ) ) ).

% drop_take
thf(fact_1121_take__drop,axiom,
    ! [N: nat,M: nat,Xs2: list_nat] :
      ( ( take_nat @ N @ ( drop_nat @ M @ Xs2 ) )
      = ( drop_nat @ M @ ( take_nat @ ( plus_plus_nat @ N @ M ) @ Xs2 ) ) ) ).

% take_drop
thf(fact_1122_gen__length__def,axiom,
    ( gen_length_nat
    = ( ^ [N2: nat,Xs: list_nat] : ( plus_plus_nat @ N2 @ ( size_size_list_nat @ Xs ) ) ) ) ).

% gen_length_def
thf(fact_1123_drop__update__swap,axiom,
    ! [M: nat,N: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( drop_nat @ M @ ( list_update_nat @ Xs2 @ N @ X ) )
        = ( list_update_nat @ ( drop_nat @ M @ Xs2 ) @ ( minus_minus_nat @ N @ M ) @ X ) ) ) ).

% drop_update_swap
thf(fact_1124_take__add,axiom,
    ! [I: nat,J: nat,Xs2: list_nat] :
      ( ( take_nat @ ( plus_plus_nat @ I @ J ) @ Xs2 )
      = ( append_nat @ ( take_nat @ I @ Xs2 ) @ ( take_nat @ J @ ( drop_nat @ I @ Xs2 ) ) ) ) ).

% take_add
thf(fact_1125_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_1126_nth__append,axiom,
    ! [N: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
       => ( ( nth_nat @ ( append_nat @ Xs2 @ Ys ) @ N )
          = ( nth_nat @ Xs2 @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
       => ( ( nth_nat @ ( append_nat @ Xs2 @ Ys ) @ N )
          = ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs2 ) ) ) ) ) ) ).

% nth_append
thf(fact_1127_list__update__append,axiom,
    ! [N: nat,Xs2: list_nat,Ys: list_nat,X: nat] :
      ( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
       => ( ( list_update_nat @ ( append_nat @ Xs2 @ Ys ) @ N @ X )
          = ( append_nat @ ( list_update_nat @ Xs2 @ N @ X ) @ Ys ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
       => ( ( list_update_nat @ ( append_nat @ Xs2 @ Ys ) @ N @ X )
          = ( append_nat @ Xs2 @ ( list_update_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs2 ) ) @ X ) ) ) ) ) ).

% list_update_append
thf(fact_1128_adjust__idx__rev__def,axiom,
    ( missin3815256168798769645dx_rev
    = ( ^ [I2: nat,J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ I2 ) @ J3 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) ) ).

% adjust_idx_rev_def
thf(fact_1129_nth__append__Cons,axiom,
    ! [I: nat,Xs2: list_nat,Y: nat,Zs: list_nat] :
      ( ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( nth_nat @ ( append_nat @ Xs2 @ ( cons_nat @ Y @ Zs ) ) @ I )
          = ( nth_nat @ Xs2 @ I ) ) )
      & ( ~ ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( ( I
              = ( size_size_list_nat @ Xs2 ) )
           => ( ( nth_nat @ ( append_nat @ Xs2 @ ( cons_nat @ Y @ Zs ) ) @ I )
              = Y ) )
          & ( ( I
             != ( size_size_list_nat @ Xs2 ) )
           => ( ( nth_nat @ ( append_nat @ Xs2 @ ( cons_nat @ Y @ Zs ) ) @ I )
              = ( nth_nat @ Zs @ ( minus_minus_nat @ I @ ( suc @ ( size_size_list_nat @ Xs2 ) ) ) ) ) ) ) ) ) ).

% nth_append_Cons
thf(fact_1130_concat__index__split_Osimps,axiom,
    ! [I_idx: nat,X: list_nat,O_idx: nat,Xs2: list_list_nat] :
      ( ( ( ord_less_nat @ I_idx @ ( size_size_list_nat @ X ) )
       => ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ O_idx @ I_idx ) @ ( cons_list_nat @ X @ Xs2 ) )
          = ( product_Pair_nat_nat @ O_idx @ I_idx ) ) )
      & ( ~ ( ord_less_nat @ I_idx @ ( size_size_list_nat @ X ) )
       => ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ O_idx @ I_idx ) @ ( cons_list_nat @ X @ Xs2 ) )
          = ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ ( suc @ O_idx ) @ ( minus_minus_nat @ I_idx @ ( size_size_list_nat @ X ) ) ) @ Xs2 ) ) ) ) ).

% concat_index_split.simps
thf(fact_1131_in__set__enumerate__eq,axiom,
    ! [P2: product_prod_nat_nat,N: nat,Xs2: list_nat] :
      ( ( member8440522571783428010at_nat @ P2 @ ( set_Pr5648618587558075414at_nat @ ( enumerate_nat @ N @ Xs2 ) ) )
      = ( ( ord_less_eq_nat @ N @ ( product_fst_nat_nat @ P2 ) )
        & ( ord_less_nat @ ( product_fst_nat_nat @ P2 ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs2 ) @ N ) )
        & ( ( nth_nat @ Xs2 @ ( minus_minus_nat @ ( product_fst_nat_nat @ P2 ) @ N ) )
          = ( product_snd_nat_nat @ P2 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_1132_concat__index__split_Oelims,axiom,
    ! [X: product_prod_nat_nat,Xa: list_list_nat,Y: product_prod_nat_nat] :
      ( ( ( concat5568775588498683858it_nat @ X @ Xa )
        = Y )
     => ( ! [O_idx2: nat,I_idx2: nat] :
            ( ( X
              = ( product_Pair_nat_nat @ O_idx2 @ I_idx2 ) )
           => ! [X3: list_nat,Xs3: list_list_nat] :
                ( ( Xa
                  = ( cons_list_nat @ X3 @ Xs3 ) )
               => ~ ( ( ( ord_less_nat @ I_idx2 @ ( size_size_list_nat @ X3 ) )
                     => ( Y
                        = ( product_Pair_nat_nat @ O_idx2 @ I_idx2 ) ) )
                    & ( ~ ( ord_less_nat @ I_idx2 @ ( size_size_list_nat @ X3 ) )
                     => ( Y
                        = ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ ( suc @ O_idx2 ) @ ( minus_minus_nat @ I_idx2 @ ( size_size_list_nat @ X3 ) ) ) @ Xs3 ) ) ) ) ) )
       => ~ ( ( Xa = nil_list_nat )
           => ( Y != undefi3946296454836805481at_nat ) ) ) ) ).

% concat_index_split.elims
thf(fact_1133_length__enumerate,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs2 ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_enumerate
thf(fact_1134_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_1135_nth__enumerate__eq,axiom,
    ! [M: nat,Xs2: list_nat,N: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs2 ) @ M )
        = ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_nat @ Xs2 @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_1136_zip__with__index__from__append,axiom,
    ! [N: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( zip_wi3407404960461264653om_nat @ N @ ( append_nat @ Xs2 @ Ys ) )
      = ( append985823374593552924at_nat @ ( zip_wi3407404960461264653om_nat @ N @ Xs2 ) @ ( zip_wi3407404960461264653om_nat @ ( plus_plus_nat @ N @ ( size_size_list_nat @ Xs2 ) ) @ Ys ) ) ) ).

% zip_with_index_from_append
thf(fact_1137_concat__index__split_Opelims,axiom,
    ! [X: product_prod_nat_nat,Xa: list_list_nat,Y: product_prod_nat_nat] :
      ( ( ( concat5568775588498683858it_nat @ X @ Xa )
        = Y )
     => ( ( accp_P6748712283191960023st_nat @ concat1575402509940520777el_nat @ ( produc8649917717998259994st_nat @ X @ Xa ) )
       => ( ! [O_idx2: nat,I_idx2: nat] :
              ( ( X
                = ( product_Pair_nat_nat @ O_idx2 @ I_idx2 ) )
             => ! [X3: list_nat,Xs3: list_list_nat] :
                  ( ( Xa
                    = ( cons_list_nat @ X3 @ Xs3 ) )
                 => ( ( ( ( ord_less_nat @ I_idx2 @ ( size_size_list_nat @ X3 ) )
                       => ( Y
                          = ( product_Pair_nat_nat @ O_idx2 @ I_idx2 ) ) )
                      & ( ~ ( ord_less_nat @ I_idx2 @ ( size_size_list_nat @ X3 ) )
                       => ( Y
                          = ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ ( suc @ O_idx2 ) @ ( minus_minus_nat @ I_idx2 @ ( size_size_list_nat @ X3 ) ) ) @ Xs3 ) ) ) )
                   => ~ ( accp_P6748712283191960023st_nat @ concat1575402509940520777el_nat @ ( produc8649917717998259994st_nat @ ( product_Pair_nat_nat @ O_idx2 @ I_idx2 ) @ ( cons_list_nat @ X3 @ Xs3 ) ) ) ) ) )
         => ~ ( ( Xa = nil_list_nat )
             => ( ( Y = undefi3946296454836805481at_nat )
               => ~ ( accp_P6748712283191960023st_nat @ concat1575402509940520777el_nat @ ( produc8649917717998259994st_nat @ X @ nil_list_nat ) ) ) ) ) ) ) ).

% concat_index_split.pelims
thf(fact_1138_bot__term_Osize_I4_J,axiom,
    ! [X21: f,X22: list_Bot_bot_term_f] :
      ( ( size_s7982331069984003362term_f @ ( bot_bot_BFun_f @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_l1747024920156826246term_f @ size_s7982331069984003362term_f @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% bot_term.size(4)
thf(fact_1139_size__list__estimation,axiom,
    ! [X: nat,Xs2: list_nat,Y: nat,F2: nat > nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ( ( ord_less_nat @ Y @ ( F2 @ X ) )
       => ( ord_less_nat @ Y @ ( size_list_nat @ F2 @ Xs2 ) ) ) ) ).

% size_list_estimation
thf(fact_1140_size__list__pointwise,axiom,
    ! [Xs2: list_nat,F2: nat > nat,G2: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
     => ( ord_less_eq_nat @ ( size_list_nat @ F2 @ Xs2 ) @ ( size_list_nat @ G2 @ Xs2 ) ) ) ).

% size_list_pointwise
thf(fact_1141_size__list__estimation_H,axiom,
    ! [X: nat,Xs2: list_nat,Y: nat,F2: nat > nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ Y @ ( F2 @ X ) )
       => ( ord_less_eq_nat @ Y @ ( size_list_nat @ F2 @ Xs2 ) ) ) ) ).

% size_list_estimation'
thf(fact_1142_size__simp1,axiom,
    ! [T: nat,Ts2: list_nat] :
      ( ( member_nat @ T @ ( set_nat2 @ Ts2 ) )
     => ( ord_less_nat @ ( size_size_nat @ T ) @ ( suc @ ( size_list_nat @ size_size_nat @ Ts2 ) ) ) ) ).

% size_simp1
thf(fact_1143_size__simp1,axiom,
    ! [T: char,Ts2: list_char] :
      ( ( member_char @ T @ ( set_char2 @ Ts2 ) )
     => ( ord_less_nat @ ( size_size_char @ T ) @ ( suc @ ( size_list_char @ size_size_char @ Ts2 ) ) ) ) ).

% size_simp1
thf(fact_1144_size__simp1,axiom,
    ! [T: list_nat,Ts2: list_list_nat] :
      ( ( member_list_nat @ T @ ( set_list_nat2 @ Ts2 ) )
     => ( ord_less_nat @ ( size_size_list_nat @ T ) @ ( suc @ ( size_list_list_nat @ size_size_list_nat @ Ts2 ) ) ) ) ).

% size_simp1
thf(fact_1145_size__simp2,axiom,
    ! [T: nat,Ts2: list_nat,S2: char] :
      ( ( member_nat @ T @ ( set_nat2 @ Ts2 ) )
     => ( ord_less_nat @ ( size_size_nat @ T ) @ ( suc @ ( suc @ ( plus_plus_nat @ ( size_size_char @ S2 ) @ ( size_list_nat @ size_size_nat @ Ts2 ) ) ) ) ) ) ).

% size_simp2
thf(fact_1146_size__simp2,axiom,
    ! [T: nat,Ts2: list_nat,S2: list_nat] :
      ( ( member_nat @ T @ ( set_nat2 @ Ts2 ) )
     => ( ord_less_nat @ ( size_size_nat @ T ) @ ( suc @ ( suc @ ( plus_plus_nat @ ( size_size_list_nat @ S2 ) @ ( size_list_nat @ size_size_nat @ Ts2 ) ) ) ) ) ) ).

% size_simp2
thf(fact_1147_size__simp2,axiom,
    ! [T: char,Ts2: list_char,S2: char] :
      ( ( member_char @ T @ ( set_char2 @ Ts2 ) )
     => ( ord_less_nat @ ( size_size_char @ T ) @ ( suc @ ( suc @ ( plus_plus_nat @ ( size_size_char @ S2 ) @ ( size_list_char @ size_size_char @ Ts2 ) ) ) ) ) ) ).

% size_simp2
thf(fact_1148_size__simp2,axiom,
    ! [T: char,Ts2: list_char,S2: list_nat] :
      ( ( member_char @ T @ ( set_char2 @ Ts2 ) )
     => ( ord_less_nat @ ( size_size_char @ T ) @ ( suc @ ( suc @ ( plus_plus_nat @ ( size_size_list_nat @ S2 ) @ ( size_list_char @ size_size_char @ Ts2 ) ) ) ) ) ) ).

% size_simp2
thf(fact_1149_size__simp2,axiom,
    ! [T: list_nat,Ts2: list_list_nat,S2: char] :
      ( ( member_list_nat @ T @ ( set_list_nat2 @ Ts2 ) )
     => ( ord_less_nat @ ( size_size_list_nat @ T ) @ ( suc @ ( suc @ ( plus_plus_nat @ ( size_size_char @ S2 ) @ ( size_list_list_nat @ size_size_list_nat @ Ts2 ) ) ) ) ) ) ).

% size_simp2
thf(fact_1150_size__simp2,axiom,
    ! [T: list_nat,Ts2: list_list_nat,S2: list_nat] :
      ( ( member_list_nat @ T @ ( set_list_nat2 @ Ts2 ) )
     => ( ord_less_nat @ ( size_size_list_nat @ T ) @ ( suc @ ( suc @ ( plus_plus_nat @ ( size_size_list_nat @ S2 ) @ ( size_list_list_nat @ size_size_list_nat @ Ts2 ) ) ) ) ) ) ).

% size_simp2
thf(fact_1151_nth__Cons__pos,axiom,
    ! [N: nat,X: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
        = ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_1152_nth__Cons_H,axiom,
    ! [N: nat,X: nat,Xs2: list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
          = X ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
          = ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_1153_drop__Cons_H,axiom,
    ! [N: nat,X: nat,Xs2: list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( drop_nat @ N @ ( cons_nat @ X @ Xs2 ) )
          = ( cons_nat @ X @ Xs2 ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( drop_nat @ N @ ( cons_nat @ X @ Xs2 ) )
          = ( drop_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ) ).

% drop_Cons'
thf(fact_1154_remove__nth__length,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( ( size_size_list_nat @ ( missin7175274867594579095th_nat @ N @ Xs2 ) )
        = ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) ) ) ).

% remove_nth_length
thf(fact_1155_nth__non__equal__first__eq,axiom,
    ! [X: nat,Y: nat,Xs2: list_nat,N: nat] :
      ( ( X != Y )
     => ( ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
          = Y )
        = ( ( ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_1156_take__Cons_H,axiom,
    ! [N: nat,X: nat,Xs2: list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( take_nat @ N @ ( cons_nat @ X @ Xs2 ) )
          = nil_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( take_nat @ N @ ( cons_nat @ X @ Xs2 ) )
          = ( cons_nat @ X @ ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ) ) ).

% take_Cons'
thf(fact_1157_Cons__replicate__eq,axiom,
    ! [X: bot_bot_term_f,Xs2: list_Bot_bot_term_f,N: nat,Y: bot_bot_term_f] :
      ( ( ( cons_Bot_bot_term_f @ X @ Xs2 )
        = ( replic6958083601664011982term_f @ N @ Y ) )
      = ( ( X = Y )
        & ( ord_less_nat @ zero_zero_nat @ N )
        & ( Xs2
          = ( replic6958083601664011982term_f @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ) ) ).

% Cons_replicate_eq
thf(fact_1158_Cons__replicate__eq,axiom,
    ! [X: nat,Xs2: list_nat,N: nat,Y: nat] :
      ( ( ( cons_nat @ X @ Xs2 )
        = ( replicate_nat @ N @ Y ) )
      = ( ( X = Y )
        & ( ord_less_nat @ zero_zero_nat @ N )
        & ( Xs2
          = ( replicate_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ) ) ).

% Cons_replicate_eq
thf(fact_1159_last__nthI,axiom,
    ! [I: nat,Ts2: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ts2 ) )
     => ( ~ ( ord_less_nat @ I @ ( minus_minus_nat @ ( size_size_list_nat @ Ts2 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ( nth_nat @ Ts2 @ I )
          = ( last_nat @ Ts2 ) ) ) ) ).

% last_nthI
thf(fact_1160_remdups__adj__singleton__iff,axiom,
    ! [Xs2: list_Bot_bot_term_f] :
      ( ( ( size_s1458857753309042354term_f @ ( remdup6045283763748034386term_f @ Xs2 ) )
        = ( suc @ zero_zero_nat ) )
      = ( ( Xs2 != nil_Bot_bot_term_f )
        & ( Xs2
          = ( replic6958083601664011982term_f @ ( size_s1458857753309042354term_f @ Xs2 ) @ ( hd_Bot_bot_term_f @ Xs2 ) ) ) ) ) ).

% remdups_adj_singleton_iff
thf(fact_1161_remdups__adj__singleton__iff,axiom,
    ! [Xs2: list_nat] :
      ( ( ( size_size_list_nat @ ( remdups_adj_nat @ Xs2 ) )
        = ( suc @ zero_zero_nat ) )
      = ( ( Xs2 != nil_nat )
        & ( Xs2
          = ( replicate_nat @ ( size_size_list_nat @ Xs2 ) @ ( hd_nat @ Xs2 ) ) ) ) ) ).

% remdups_adj_singleton_iff
thf(fact_1162_last__replicate,axiom,
    ! [N: nat,X: bot_bot_term_f] :
      ( ( N != zero_zero_nat )
     => ( ( last_Bot_bot_term_f @ ( replic6958083601664011982term_f @ N @ X ) )
        = X ) ) ).

% last_replicate
thf(fact_1163_last__replicate,axiom,
    ! [N: nat,X: nat] :
      ( ( N != zero_zero_nat )
     => ( ( last_nat @ ( replicate_nat @ N @ X ) )
        = X ) ) ).

% last_replicate
thf(fact_1164_last__drop,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( ( last_nat @ ( drop_nat @ N @ Xs2 ) )
        = ( last_nat @ Xs2 ) ) ) ).

% last_drop
thf(fact_1165_last__zip,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( Xs2 != nil_nat )
     => ( ( Ys != nil_nat )
       => ( ( ( size_size_list_nat @ Xs2 )
            = ( size_size_list_nat @ Ys ) )
         => ( ( last_P6484183829340986144at_nat @ ( zip_nat_nat @ Xs2 @ Ys ) )
            = ( product_Pair_nat_nat @ ( last_nat @ Xs2 ) @ ( last_nat @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_1166_last__in__set,axiom,
    ! [As: list_nat] :
      ( ( As != nil_nat )
     => ( member_nat @ ( last_nat @ As ) @ ( set_nat2 @ As ) ) ) ).

% last_in_set
thf(fact_1167_remdups__adj__length,axiom,
    ! [Xs2: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ).

% remdups_adj_length
thf(fact_1168_remdups__adj__adjacent,axiom,
    ! [I: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs2 ) ) )
     => ( ( nth_nat @ ( remdups_adj_nat @ Xs2 ) @ I )
       != ( nth_nat @ ( remdups_adj_nat @ Xs2 ) @ ( suc @ I ) ) ) ) ).

% remdups_adj_adjacent
thf(fact_1169_remdups__adj__singleton,axiom,
    ! [Xs2: list_Bot_bot_term_f,X: bot_bot_term_f] :
      ( ( ( remdup6045283763748034386term_f @ Xs2 )
        = ( cons_Bot_bot_term_f @ X @ nil_Bot_bot_term_f ) )
     => ( Xs2
        = ( replic6958083601664011982term_f @ ( size_s1458857753309042354term_f @ Xs2 ) @ X ) ) ) ).

% remdups_adj_singleton
thf(fact_1170_remdups__adj__singleton,axiom,
    ! [Xs2: list_nat,X: nat] :
      ( ( ( remdups_adj_nat @ Xs2 )
        = ( cons_nat @ X @ nil_nat ) )
     => ( Xs2
        = ( replicate_nat @ ( size_size_list_nat @ Xs2 ) @ X ) ) ) ).

% remdups_adj_singleton
thf(fact_1171_remdups__adj__replicate,axiom,
    ! [N: nat,X: bot_bot_term_f] :
      ( ( ( N = zero_zero_nat )
       => ( ( remdup6045283763748034386term_f @ ( replic6958083601664011982term_f @ N @ X ) )
          = nil_Bot_bot_term_f ) )
      & ( ( N != zero_zero_nat )
       => ( ( remdup6045283763748034386term_f @ ( replic6958083601664011982term_f @ N @ X ) )
          = ( cons_Bot_bot_term_f @ X @ nil_Bot_bot_term_f ) ) ) ) ).

% remdups_adj_replicate
thf(fact_1172_remdups__adj__replicate,axiom,
    ! [N: nat,X: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( remdups_adj_nat @ ( replicate_nat @ N @ X ) )
          = nil_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( remdups_adj_nat @ ( replicate_nat @ N @ X ) )
          = ( cons_nat @ X @ nil_nat ) ) ) ) ).

% remdups_adj_replicate
thf(fact_1173_remdups__adj__length__ge1,axiom,
    ! [Xs2: list_nat] :
      ( ( Xs2 != nil_nat )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs2 ) ) ) ) ).

% remdups_adj_length_ge1
thf(fact_1174_last__conv__nth,axiom,
    ! [Xs2: list_nat] :
      ( ( Xs2 != nil_nat )
     => ( ( last_nat @ Xs2 )
        = ( nth_nat @ Xs2 @ ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) ) ) ) ).

% last_conv_nth
thf(fact_1175_last__list__update,axiom,
    ! [Xs2: list_nat,K: nat,X: nat] :
      ( ( Xs2 != nil_nat )
     => ( ( ( K
            = ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) )
         => ( ( last_nat @ ( list_update_nat @ Xs2 @ K @ X ) )
            = X ) )
        & ( ( K
           != ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) )
         => ( ( last_nat @ ( list_update_nat @ Xs2 @ K @ X ) )
            = ( last_nat @ Xs2 ) ) ) ) ) ).

% last_list_update
thf(fact_1176_rtrancl__list__conv,axiom,
    ! [S2: nat,T: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ S2 @ T ) @ ( transi2905341329935302413cl_nat @ R ) )
      = ( ? [Ts3: list_nat] :
            ( ( ( last_nat @ ( cons_nat @ S2 @ Ts3 ) )
              = T )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ts3 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ ( cons_nat @ S2 @ Ts3 ) @ I2 ) @ ( nth_nat @ ( cons_nat @ S2 @ Ts3 ) @ ( suc @ I2 ) ) ) @ R ) ) ) ) ) ).

% rtrancl_list_conv
thf(fact_1177_length__butlast,axiom,
    ! [Xs2: list_nat] :
      ( ( size_size_list_nat @ ( butlast_nat @ Xs2 ) )
      = ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) ) ).

% length_butlast
thf(fact_1178_in__set__butlastD,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Xs2 ) ) )
     => ( member_nat @ X @ ( set_nat2 @ Xs2 ) ) ) ).

% in_set_butlastD
thf(fact_1179_drop__butlast,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( drop_nat @ N @ ( butlast_nat @ Xs2 ) )
      = ( butlast_nat @ ( drop_nat @ N @ Xs2 ) ) ) ).

% drop_butlast
thf(fact_1180_in__set__butlast__appendI,axiom,
    ! [X: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Xs2 ) ) )
        | ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Ys ) ) ) )
     => ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ ( append_nat @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_1181_rtrancl__listrel1__eq__len,axiom,
    ! [X: list_nat,Y: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( transi5285580207609517981st_nat @ ( listrel1_nat @ R2 ) ) )
     => ( ( size_size_list_nat @ X )
        = ( size_size_list_nat @ Y ) ) ) ).

% rtrancl_listrel1_eq_len
thf(fact_1182_nth__butlast,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ ( butlast_nat @ Xs2 ) ) )
     => ( ( nth_nat @ ( butlast_nat @ Xs2 ) @ N )
        = ( nth_nat @ Xs2 @ N ) ) ) ).

% nth_butlast
thf(fact_1183_take__butlast,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( ( take_nat @ N @ ( butlast_nat @ Xs2 ) )
        = ( take_nat @ N @ Xs2 ) ) ) ).

% take_butlast
thf(fact_1184_butlast__conv__take,axiom,
    ( butlast_nat
    = ( ^ [Xs: list_nat] : ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) @ Xs ) ) ) ).

% butlast_conv_take
thf(fact_1185_butlast__list__update,axiom,
    ! [K: nat,Xs2: list_nat,X: nat] :
      ( ( ( K
          = ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) )
       => ( ( butlast_nat @ ( list_update_nat @ Xs2 @ K @ X ) )
          = ( butlast_nat @ Xs2 ) ) )
      & ( ( K
         != ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) )
       => ( ( butlast_nat @ ( list_update_nat @ Xs2 @ K @ X ) )
          = ( list_update_nat @ ( butlast_nat @ Xs2 ) @ K @ X ) ) ) ) ).

% butlast_list_update
thf(fact_1186_butlast__take,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( ( butlast_nat @ ( take_nat @ N @ Xs2 ) )
        = ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ).

% butlast_take
thf(fact_1187_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_1188_min__list__subset,axiom,
    ! [Ys: list_nat,Xs2: list_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Ys ) @ ( set_nat2 @ Xs2 ) )
     => ( ( member_nat @ ( missing_min_list_nat @ Xs2 ) @ ( set_nat2 @ Ys ) )
       => ( ( missing_min_list_nat @ Xs2 )
          = ( missing_min_list_nat @ Ys ) ) ) ) ).

% min_list_subset
thf(fact_1189_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_1190_min__list__nth,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
           => ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I4 ) @ ( nth_nat @ Ys @ I4 ) ) )
       => ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs2 ) @ ( missing_min_list_nat @ Ys ) ) ) ) ).

% min_list_nth
thf(fact_1191_merge__terms_Oelims,axiom,
    ! [X: bot_bot_term_f,Xa: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ( ( bot_merge_terms_f @ X @ Xa )
        = Y )
     => ( ( ( X = bot_bot_Bot_f )
         => ( Y != Xa ) )
       => ( ! [V2: f,Va2: list_Bot_bot_term_f] :
              ( ( X
                = ( bot_bot_BFun_f @ V2 @ Va2 ) )
             => ( ( Xa = bot_bot_Bot_f )
               => ( Y
                 != ( bot_bot_BFun_f @ V2 @ Va2 ) ) ) )
         => ~ ! [F: f,Ss: list_Bot_bot_term_f] :
                ( ( X
                  = ( bot_bot_BFun_f @ F @ Ss ) )
               => ! [G: f,Ts: list_Bot_bot_term_f] :
                    ( ( Xa
                      = ( bot_bot_BFun_f @ G @ Ts ) )
                   => ~ ( ( ( ( F = G )
                            & ( ( size_s1458857753309042354term_f @ Ss )
                              = ( size_s1458857753309042354term_f @ Ts ) ) )
                         => ( Y
                            = ( bot_bot_BFun_f @ F @ ( map_Pr7995175553647191367term_f @ ( produc1677810408741468632term_f @ bot_merge_terms_f ) @ ( zip_Bo7813815397675234645term_f @ Ss @ Ts ) ) ) ) )
                        & ( ~ ( ( F = G )
                              & ( ( size_s1458857753309042354term_f @ Ss )
                                = ( size_s1458857753309042354term_f @ Ts ) ) )
                         => ( Y = undefi3706045505595579942term_f ) ) ) ) ) ) ) ) ).

% merge_terms.elims
thf(fact_1192_length__map,axiom,
    ! [F2: nat > nat,Xs2: list_nat] :
      ( ( size_size_list_nat @ ( map_nat_nat @ F2 @ Xs2 ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_map
thf(fact_1193_map__replicate,axiom,
    ! [F2: bot_bot_term_f > bot_bot_term_f,N: nat,X: bot_bot_term_f] :
      ( ( map_Bo3245817867514345570term_f @ F2 @ ( replic6958083601664011982term_f @ N @ X ) )
      = ( replic6958083601664011982term_f @ N @ ( F2 @ X ) ) ) ).

% map_replicate
thf(fact_1194_map__replicate,axiom,
    ! [F2: bot_bot_term_f > nat,N: nat,X: bot_bot_term_f] :
      ( ( map_Bo1399257005622609828_f_nat @ F2 @ ( replic6958083601664011982term_f @ N @ X ) )
      = ( replicate_nat @ N @ ( F2 @ X ) ) ) ).

% map_replicate
thf(fact_1195_map__replicate,axiom,
    ! [F2: nat > bot_bot_term_f,N: nat,X: nat] :
      ( ( map_na44992194710417700term_f @ F2 @ ( replicate_nat @ N @ X ) )
      = ( replic6958083601664011982term_f @ N @ ( F2 @ X ) ) ) ).

% map_replicate
thf(fact_1196_map__replicate,axiom,
    ! [F2: nat > nat,N: nat,X: nat] :
      ( ( map_nat_nat @ F2 @ ( replicate_nat @ N @ X ) )
      = ( replicate_nat @ N @ ( F2 @ X ) ) ) ).

% map_replicate
thf(fact_1197_nth__map,axiom,
    ! [N: nat,Xs2: list_nat,F2: nat > nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_nat @ ( map_nat_nat @ F2 @ Xs2 ) @ N )
        = ( F2 @ ( nth_nat @ Xs2 @ N ) ) ) ) ).

% nth_map
thf(fact_1198_map__fst__zip,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ ( zip_nat_nat @ Xs2 @ Ys ) )
        = Xs2 ) ) ).

% map_fst_zip
thf(fact_1199_map__snd__zip,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( map_Pr3938374229010428429at_nat @ product_snd_nat_nat @ ( zip_nat_nat @ Xs2 @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_1200_map__nth__eq__conv,axiom,
    ! [Xs2: list_nat,Ys: list_nat,F2: nat > nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( map_nat_nat @ F2 @ Xs2 )
          = Ys )
        = ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
             => ( ( F2 @ ( nth_nat @ Xs2 @ I2 ) )
                = ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).

% map_nth_eq_conv
thf(fact_1201_map__idI,axiom,
    ! [Xs2: list_nat,F2: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ( F2 @ X3 )
            = X3 ) )
     => ( ( map_nat_nat @ F2 @ Xs2 )
        = Xs2 ) ) ).

% map_idI
thf(fact_1202_list_Omap__ident__strong,axiom,
    ! [T: list_nat,F2: nat > nat] :
      ( ! [Z3: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ T ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( map_nat_nat @ F2 @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_1203_drop__map,axiom,
    ! [N: nat,F2: nat > nat,Xs2: list_nat] :
      ( ( drop_nat @ N @ ( map_nat_nat @ F2 @ Xs2 ) )
      = ( map_nat_nat @ F2 @ ( drop_nat @ N @ Xs2 ) ) ) ).

% drop_map
thf(fact_1204_take__map,axiom,
    ! [N: nat,F2: nat > nat,Xs2: list_nat] :
      ( ( take_nat @ N @ ( map_nat_nat @ F2 @ Xs2 ) )
      = ( map_nat_nat @ F2 @ ( take_nat @ N @ Xs2 ) ) ) ).

% take_map
thf(fact_1205_zip__eq__conv,axiom,
    ! [Xs2: list_nat,Ys: list_nat,Zs: list_P6011104703257516679at_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( zip_nat_nat @ Xs2 @ Ys )
          = Zs )
        = ( ( ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ Zs )
            = Xs2 )
          & ( ( map_Pr3938374229010428429at_nat @ product_snd_nat_nat @ Zs )
            = Ys ) ) ) ) ).

% zip_eq_conv
thf(fact_1206_merge__terms_Osimps_I3_J,axiom,
    ! [F2: f,G2: f,Ss3: list_Bot_bot_term_f,Ts2: list_Bot_bot_term_f] :
      ( ( ( ( F2 = G2 )
          & ( ( size_s1458857753309042354term_f @ Ss3 )
            = ( size_s1458857753309042354term_f @ Ts2 ) ) )
       => ( ( bot_merge_terms_f @ ( bot_bot_BFun_f @ F2 @ Ss3 ) @ ( bot_bot_BFun_f @ G2 @ Ts2 ) )
          = ( bot_bot_BFun_f @ F2 @ ( map_Pr7995175553647191367term_f @ ( produc1677810408741468632term_f @ bot_merge_terms_f ) @ ( zip_Bo7813815397675234645term_f @ Ss3 @ Ts2 ) ) ) ) )
      & ( ~ ( ( F2 = G2 )
            & ( ( size_s1458857753309042354term_f @ Ss3 )
              = ( size_s1458857753309042354term_f @ Ts2 ) ) )
       => ( ( bot_merge_terms_f @ ( bot_bot_BFun_f @ F2 @ Ss3 ) @ ( bot_bot_BFun_f @ G2 @ Ts2 ) )
          = undefi3706045505595579942term_f ) ) ) ).

% merge_terms.simps(3)
thf(fact_1207_map__fst__zip__take,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ ( zip_nat_nat @ Xs2 @ Ys ) )
      = ( take_nat @ ( ord_min_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) @ Xs2 ) ) ).

% map_fst_zip_take
thf(fact_1208_map__snd__zip__take,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( map_Pr3938374229010428429at_nat @ product_snd_nat_nat @ ( zip_nat_nat @ Xs2 @ Ys ) )
      = ( take_nat @ ( ord_min_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) @ Ys ) ) ).

% map_snd_zip_take
thf(fact_1209_zip__replicate1,axiom,
    ! [N: nat,X: bot_bot_term_f,Ys: list_nat] :
      ( ( zip_Bo4866563780717559831_f_nat @ ( replic6958083601664011982term_f @ N @ X ) @ Ys )
      = ( map_na489546787450227497_f_nat @ ( produc6944956222323935479_f_nat @ X ) @ ( take_nat @ N @ Ys ) ) ) ).

% zip_replicate1
thf(fact_1210_zip__replicate1,axiom,
    ! [N: nat,X: nat,Ys: list_nat] :
      ( ( zip_nat_nat @ ( replicate_nat @ N @ X ) @ Ys )
      = ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ X ) @ ( take_nat @ N @ Ys ) ) ) ).

% zip_replicate1
thf(fact_1211_merge__terms_Opelims,axiom,
    ! [X: bot_bot_term_f,Xa: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ( ( bot_merge_terms_f @ X @ Xa )
        = Y )
     => ( ( accp_P5091932899567907206term_f @ bot_me5685897738211366532_rel_f @ ( produc2865564700782054965term_f @ X @ Xa ) )
       => ( ( ( X = bot_bot_Bot_f )
           => ( ( Y = Xa )
             => ~ ( accp_P5091932899567907206term_f @ bot_me5685897738211366532_rel_f @ ( produc2865564700782054965term_f @ bot_bot_Bot_f @ Xa ) ) ) )
         => ( ! [V2: f,Va2: list_Bot_bot_term_f] :
                ( ( X
                  = ( bot_bot_BFun_f @ V2 @ Va2 ) )
               => ( ( Xa = bot_bot_Bot_f )
                 => ( ( Y
                      = ( bot_bot_BFun_f @ V2 @ Va2 ) )
                   => ~ ( accp_P5091932899567907206term_f @ bot_me5685897738211366532_rel_f @ ( produc2865564700782054965term_f @ ( bot_bot_BFun_f @ V2 @ Va2 ) @ bot_bot_Bot_f ) ) ) ) )
           => ~ ! [F: f,Ss: list_Bot_bot_term_f] :
                  ( ( X
                    = ( bot_bot_BFun_f @ F @ Ss ) )
                 => ! [G: f,Ts: list_Bot_bot_term_f] :
                      ( ( Xa
                        = ( bot_bot_BFun_f @ G @ Ts ) )
                     => ( ( ( ( ( F = G )
                              & ( ( size_s1458857753309042354term_f @ Ss )
                                = ( size_s1458857753309042354term_f @ Ts ) ) )
                           => ( Y
                              = ( bot_bot_BFun_f @ F @ ( map_Pr7995175553647191367term_f @ ( produc1677810408741468632term_f @ bot_merge_terms_f ) @ ( zip_Bo7813815397675234645term_f @ Ss @ Ts ) ) ) ) )
                          & ( ~ ( ( F = G )
                                & ( ( size_s1458857753309042354term_f @ Ss )
                                  = ( size_s1458857753309042354term_f @ Ts ) ) )
                           => ( Y = undefi3706045505595579942term_f ) ) )
                       => ~ ( accp_P5091932899567907206term_f @ bot_me5685897738211366532_rel_f @ ( produc2865564700782054965term_f @ ( bot_bot_BFun_f @ F @ Ss ) @ ( bot_bot_BFun_f @ G @ Ts ) ) ) ) ) ) ) ) ) ) ).

% merge_terms.pelims
thf(fact_1212_concat__index__split__less__length__concat_I1_J,axiom,
    ! [I: nat,Xs2: list_list_nat,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs2 ) ) )
     => ( ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs2 )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( I
          = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ size_size_list_nat @ ( take_list_nat @ M @ Xs2 ) ) ) @ N ) ) ) ) ).

% concat_index_split_less_length_concat(1)
thf(fact_1213_sum__list__replicate__length_H,axiom,
    ! [N: nat] :
      ( ( groups4561878855575611511st_nat @ ( replicate_nat @ N @ ( suc @ zero_zero_nat ) ) )
      = N ) ).

% sum_list_replicate_length'
thf(fact_1214_sum__list__replicate__length,axiom,
    ! [Xs2: list_nat] :
      ( ( groups4561878855575611511st_nat @ ( replicate_nat @ ( size_size_list_nat @ Xs2 ) @ ( suc @ zero_zero_nat ) ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% sum_list_replicate_length
thf(fact_1215_sum__list__concat,axiom,
    ! [Ls: list_list_nat] :
      ( ( groups4561878855575611511st_nat @ ( concat_nat @ Ls ) )
      = ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ groups4561878855575611511st_nat @ Ls ) ) ) ).

% sum_list_concat
thf(fact_1216_Utility_Osum__list__0,axiom,
    ! [Xs2: list_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( X3 = zero_zero_nat ) )
     => ( ( groups4561878855575611511st_nat @ Xs2 )
        = zero_zero_nat ) ) ).

% Utility.sum_list_0
thf(fact_1217_sum__list__1__E,axiom,
    ! [Xs2: list_nat] :
      ( ( ( groups4561878855575611511st_nat @ Xs2 )
        = ( suc @ zero_zero_nat ) )
     => ~ ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
           => ( ( ( nth_nat @ Xs2 @ I4 )
                = ( suc @ zero_zero_nat ) )
             => ~ ! [J4: nat] :
                    ( ( ord_less_nat @ J4 @ ( size_size_list_nat @ Xs2 ) )
                   => ( ( J4 != I4 )
                     => ( ( nth_nat @ Xs2 @ J4 )
                        = zero_zero_nat ) ) ) ) ) ) ).

% sum_list_1_E
thf(fact_1218_sum__list__replicate__mono,axiom,
    ! [W1: nat,W22: nat,N: nat] :
      ( ( ord_less_eq_nat @ W1 @ W22 )
     => ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( replicate_nat @ N @ W1 ) ) @ ( groups4561878855575611511st_nat @ ( replicate_nat @ N @ W22 ) ) ) ) ).

% sum_list_replicate_mono
thf(fact_1219_nth__sum__listI,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
           => ( ( nth_nat @ Xs2 @ I4 )
              = ( nth_nat @ Ys @ I4 ) ) )
       => ( ( groups4561878855575611511st_nat @ Xs2 )
          = ( groups4561878855575611511st_nat @ Ys ) ) ) ) ).

% nth_sum_listI
thf(fact_1220_sum__list__split,axiom,
    ! [I: nat,Xs2: list_list_nat,J: nat,K: nat,L: nat] :
      ( ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ size_size_list_nat @ ( take_list_nat @ I @ Xs2 ) ) ) @ J )
        = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ size_size_list_nat @ ( take_list_nat @ K @ Xs2 ) ) ) @ L ) )
     => ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs2 ) )
       => ( ( ord_less_nat @ K @ ( size_s3023201423986296836st_nat @ Xs2 ) )
         => ( ( ord_less_nat @ J @ ( size_size_list_nat @ ( nth_list_nat @ Xs2 @ I ) ) )
           => ( ( ord_less_nat @ L @ ( size_size_list_nat @ ( nth_list_nat @ Xs2 @ K ) ) )
             => ( ( I = K )
                & ( J = L ) ) ) ) ) ) ) ).

% sum_list_split
thf(fact_1221_concat__nth__length,axiom,
    ! [I: nat,Uss: list_list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Uss ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ ( nth_list_nat @ Uss @ I ) ) )
       => ( ord_less_nat @ ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ size_size_list_nat @ ( take_list_nat @ I @ Uss ) ) ) @ J ) @ ( size_size_list_nat @ ( concat_nat @ Uss ) ) ) ) ) ).

% concat_nth_length
thf(fact_1222_nth__concat__split_H,axiom,
    ! [I: nat,Xs2: list_list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs2 ) ) )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
           => ! [K2: nat] :
                ( ( ord_less_nat @ K2 @ ( size_size_list_nat @ ( nth_list_nat @ Xs2 @ J2 ) ) )
               => ( ( ( nth_nat @ ( concat_nat @ Xs2 ) @ I )
                    = ( nth_nat @ ( nth_list_nat @ Xs2 @ J2 ) @ K2 ) )
                 => ( I
                   != ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ size_size_list_nat @ ( take_list_nat @ J2 @ Xs2 ) ) ) @ K2 ) ) ) ) ) ) ).

% nth_concat_split'
thf(fact_1223_concat__nth,axiom,
    ! [M: nat,Xs2: list_list_nat,N: nat,I: nat] :
      ( ( ord_less_nat @ M @ ( size_s3023201423986296836st_nat @ Xs2 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_nat @ ( nth_list_nat @ Xs2 @ M ) ) )
       => ( ( I
            = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ size_size_list_nat @ ( take_list_nat @ M @ Xs2 ) ) ) @ N ) )
         => ( ( nth_nat @ ( concat_nat @ Xs2 ) @ I )
            = ( nth_nat @ ( nth_list_nat @ Xs2 @ M ) @ N ) ) ) ) ) ).

% concat_nth
thf(fact_1224_sum__list__eq__0__iff,axiom,
    ! [Ns: list_nat] :
      ( ( ( groups4561878855575611511st_nat @ Ns )
        = zero_zero_nat )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Ns ) )
           => ( X2 = zero_zero_nat ) ) ) ) ).

% sum_list_eq_0_iff
thf(fact_1225_member__le__sum__list,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ( ord_less_eq_nat @ X @ ( groups4561878855575611511st_nat @ Xs2 ) ) ) ).

% member_le_sum_list
thf(fact_1226_sum__list__nonpos,axiom,
    ! [Xs2: list_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ord_less_eq_nat @ X3 @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs2 ) @ zero_zero_nat ) ) ).

% sum_list_nonpos
thf(fact_1227_sum__list__nonneg__eq__0__iff,axiom,
    ! [Xs2: list_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ord_less_eq_nat @ zero_zero_nat @ X3 ) )
     => ( ( ( groups4561878855575611511st_nat @ Xs2 )
          = zero_zero_nat )
        = ( ! [X2: nat] :
              ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
             => ( X2 = zero_zero_nat ) ) ) ) ) ).

% sum_list_nonneg_eq_0_iff
thf(fact_1228_Groups__List_Osum__list__nonneg,axiom,
    ! [Xs2: list_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ord_less_eq_nat @ zero_zero_nat @ X3 ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4561878855575611511st_nat @ Xs2 ) ) ) ).

% Groups_List.sum_list_nonneg
thf(fact_1229_length__concat,axiom,
    ! [Xss: list_list_nat] :
      ( ( size_size_list_nat @ ( concat_nat @ Xss ) )
      = ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ size_size_list_nat @ Xss ) ) ) ).

% length_concat
thf(fact_1230_sum__list__strict__mono,axiom,
    ! [Xs2: list_nat,F2: nat > nat,G2: nat > nat] :
      ( ( Xs2 != nil_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
           => ( ord_less_nat @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
       => ( ord_less_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F2 @ Xs2 ) ) @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ G2 @ Xs2 ) ) ) ) ) ).

% sum_list_strict_mono
thf(fact_1231_elem__le__sum__list,axiom,
    ! [K: nat,Ns: list_nat] :
      ( ( ord_less_nat @ K @ ( size_size_list_nat @ Ns ) )
     => ( ord_less_eq_nat @ ( nth_nat @ Ns @ K ) @ ( groups4561878855575611511st_nat @ Ns ) ) ) ).

% elem_le_sum_list
thf(fact_1232_sum__list__mono2,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
           => ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I4 ) @ ( nth_nat @ Ys @ I4 ) ) )
       => ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs2 ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ) ).

% sum_list_mono2
thf(fact_1233_sum__list__take__eq,axiom,
    ! [K: nat,I: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ K @ I )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( groups4561878855575611511st_nat @ ( take_nat @ I @ Xs2 ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( take_nat @ K @ Xs2 ) ) @ ( nth_nat @ Xs2 @ K ) ) @ ( groups4561878855575611511st_nat @ ( take_nat @ ( minus_minus_nat @ I @ ( suc @ K ) ) @ ( drop_nat @ ( suc @ K ) @ Xs2 ) ) ) ) ) ) ) ).

% sum_list_take_eq
thf(fact_1234_size__list__conv__sum__list,axiom,
    ( size_list_nat
    = ( ^ [F3: nat > nat,Xs: list_nat] : ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F3 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ) ) ).

% size_list_conv_sum_list
thf(fact_1235_sum__list__update,axiom,
    ! [K: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ K @ ( size_size_list_nat @ Xs2 ) )
     => ( ( groups4561878855575611511st_nat @ ( list_update_nat @ Xs2 @ K @ X ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( groups4561878855575611511st_nat @ Xs2 ) @ X ) @ ( nth_nat @ Xs2 @ K ) ) ) ) ).

% sum_list_update
thf(fact_1236_partition__by__nth__nth_I2_J,axiom,
    ! [Ys: list_nat,Xs2: list_nat,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
         => ( ord_less_nat @ ( multih7279076602905066526by_idx @ ( size_size_list_nat @ Xs2 ) @ Ys @ I @ J ) @ ( size_size_list_nat @ Xs2 ) ) ) ) ) ).

% partition_by_nth_nth(2)
thf(fact_1237_partition__by__nth__greater,axiom,
    ! [I: nat,K: nat,Zs: list_nat,J: nat,Xs2: list_nat,Y: nat,Ys: list_nat] :
      ( ( ord_less_nat @ I @ K )
     => ( ( ord_less_nat @ K @ ( size_size_list_nat @ Zs ) )
       => ( ( ord_less_nat @ J @ ( nth_nat @ Zs @ I ) )
         => ( ( ( size_size_list_nat @ Xs2 )
              = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( take_nat @ I @ Zs ) ) @ J ) )
           => ( ( nth_list_nat @ ( multih8544690682651757648by_nat @ ( append_nat @ Xs2 @ ( cons_nat @ Y @ Ys ) ) @ Zs ) @ K )
              = ( take_nat @ ( nth_nat @ Zs @ K ) @ ( drop_nat @ ( minus_minus_nat @ ( groups4561878855575611511st_nat @ ( take_nat @ K @ Zs ) ) @ one_one_nat ) @ ( append_nat @ Xs2 @ Ys ) ) ) ) ) ) ) ) ).

% partition_by_nth_greater
thf(fact_1238_concat__partition__by,axiom,
    ! [Ys: list_nat,Xs2: list_nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_size_list_nat @ Xs2 ) )
     => ( ( concat_nat @ ( multih8544690682651757648by_nat @ Xs2 @ Ys ) )
        = Xs2 ) ) ).

% concat_partition_by
thf(fact_1239_map__length__partition__by,axiom,
    ! [Ys: list_nat,Xs2: list_nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_size_list_nat @ Xs2 ) )
     => ( ( map_list_nat_nat @ size_size_list_nat @ ( multih8544690682651757648by_nat @ Xs2 @ Ys ) )
        = Ys ) ) ).

% map_length_partition_by
thf(fact_1240_partition__by__concat__id,axiom,
    ! [Xss: list_list_nat,Ys: list_nat] :
      ( ( ( size_s3023201423986296836st_nat @ Xss )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Xss @ I4 ) )
              = ( nth_nat @ Ys @ I4 ) ) )
       => ( ( multih8544690682651757648by_nat @ ( concat_nat @ Xss ) @ Ys )
          = Xss ) ) ) ).

% partition_by_concat_id
thf(fact_1241_psubset__imp__ex__mem,axiom,
    ! [A: set_nat,B5: set_nat] :
      ( ( ord_less_set_nat @ A @ B5 )
     => ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1242_Ball__set__partition__by,axiom,
    ! [Ys: list_nat,Xs2: list_nat,P: nat > $o] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_size_list_nat @ Xs2 ) )
     => ( ( ! [X2: list_nat] :
              ( ( member_list_nat @ X2 @ ( set_list_nat2 @ ( multih8544690682651757648by_nat @ Xs2 @ Ys ) ) )
             => ! [Y4: nat] :
                  ( ( member_nat @ Y4 @ ( set_nat2 @ X2 ) )
                 => ( P @ Y4 ) ) ) )
        = ( ! [X2: nat] :
              ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
             => ( P @ X2 ) ) ) ) ) ).

% Ball_set_partition_by
thf(fact_1243_partition__by__nth__nth_I1_J,axiom,
    ! [Ys: list_nat,Xs2: list_nat,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
         => ( ( nth_nat @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs2 @ Ys ) @ I ) @ J )
            = ( nth_nat @ Xs2 @ ( multih7279076602905066526by_idx @ ( size_size_list_nat @ Xs2 ) @ Ys @ I @ J ) ) ) ) ) ) ).

% partition_by_nth_nth(1)
thf(fact_1244_length__partition__by__nth,axiom,
    ! [Ys: list_nat,Xs2: list_nat,I: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( size_size_list_nat @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs2 @ Ys ) @ I ) )
          = ( nth_nat @ Ys @ I ) ) ) ) ).

% length_partition_by_nth
thf(fact_1245_partition__by__nth,axiom,
    ! [I: nat,Ys: list_nat,Xs2: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
     => ( ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs2 @ Ys ) @ I )
        = ( take_nat @ ( nth_nat @ Ys @ I ) @ ( drop_nat @ ( groups4561878855575611511st_nat @ ( take_nat @ I @ Ys ) ) @ Xs2 ) ) ) ) ).

% partition_by_nth
thf(fact_1246_partition__by__nth__nth__elem,axiom,
    ! [Ys: list_nat,Xs2: list_nat,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
         => ( member_nat @ ( nth_nat @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs2 @ Ys ) @ I ) @ J ) @ ( set_nat2 @ Xs2 ) ) ) ) ) ).

% partition_by_nth_nth_elem
thf(fact_1247_partition__by__nth__nth__old,axiom,
    ! [I: nat,Xs2: list_nat,Ys: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ ( multih8544690682651757648by_nat @ Xs2 @ Ys ) ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs2 @ Ys ) @ I ) ) )
       => ( ( ( groups4561878855575611511st_nat @ Ys )
            = ( size_size_list_nat @ Xs2 ) )
         => ( ( nth_nat @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs2 @ Ys ) @ I ) @ J )
            = ( nth_nat @ Xs2 @ ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ size_size_list_nat @ ( take_list_nat @ I @ ( multih8544690682651757648by_nat @ Xs2 @ Ys ) ) ) ) @ J ) ) ) ) ) ) ).

% partition_by_nth_nth_old
thf(fact_1248_partition__by__nth__less,axiom,
    ! [K: nat,I: nat,Zs: list_nat,Xs2: list_nat,J: nat,Y: nat,Ys: list_nat] :
      ( ( ord_less_nat @ K @ I )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Xs2 )
            = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( take_nat @ I @ Zs ) ) @ J ) )
         => ( ( nth_list_nat @ ( multih8544690682651757648by_nat @ ( append_nat @ Xs2 @ ( cons_nat @ Y @ Ys ) ) @ Zs ) @ K )
            = ( take_nat @ ( nth_nat @ Zs @ K ) @ ( drop_nat @ ( groups4561878855575611511st_nat @ ( take_nat @ K @ Zs ) ) @ Xs2 ) ) ) ) ) ) ).

% partition_by_nth_less
thf(fact_1249_DiffI,axiom,
    ! [C3: nat,A: set_nat,B5: set_nat] :
      ( ( member_nat @ C3 @ A )
     => ( ~ ( member_nat @ C3 @ B5 )
       => ( member_nat @ C3 @ ( minus_minus_set_nat @ A @ B5 ) ) ) ) ).

% DiffI
thf(fact_1250_Diff__iff,axiom,
    ! [C3: nat,A: set_nat,B5: set_nat] :
      ( ( member_nat @ C3 @ ( minus_minus_set_nat @ A @ B5 ) )
      = ( ( member_nat @ C3 @ A )
        & ~ ( member_nat @ C3 @ B5 ) ) ) ).

% Diff_iff
thf(fact_1251_DiffE,axiom,
    ! [C3: nat,A: set_nat,B5: set_nat] :
      ( ( member_nat @ C3 @ ( minus_minus_set_nat @ A @ B5 ) )
     => ~ ( ( member_nat @ C3 @ A )
         => ( member_nat @ C3 @ B5 ) ) ) ).

% DiffE
thf(fact_1252_DiffD1,axiom,
    ! [C3: nat,A: set_nat,B5: set_nat] :
      ( ( member_nat @ C3 @ ( minus_minus_set_nat @ A @ B5 ) )
     => ( member_nat @ C3 @ A ) ) ).

% DiffD1
thf(fact_1253_DiffD2,axiom,
    ! [C3: nat,A: set_nat,B5: set_nat] :
      ( ( member_nat @ C3 @ ( minus_minus_set_nat @ A @ B5 ) )
     => ~ ( member_nat @ C3 @ B5 ) ) ).

% DiffD2
thf(fact_1254_distinct__take__drop,axiom,
    ! [Vs: list_nat,I: nat] :
      ( ( distinct_nat @ Vs )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Vs ) )
       => ( distinct_nat @ ( append_nat @ ( take_nat @ I @ Vs ) @ ( drop_nat @ ( suc @ I ) @ Vs ) ) ) ) ) ).

% distinct_take_drop
thf(fact_1255_distinct__swap,axiom,
    ! [I: nat,Xs2: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
       => ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I ) ) )
          = ( distinct_nat @ Xs2 ) ) ) ) ).

% distinct_swap
thf(fact_1256_nth__eq__iff__index__eq,axiom,
    ! [Xs2: list_nat,I: nat,J: nat] :
      ( ( distinct_nat @ Xs2 )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
         => ( ( ( nth_nat @ Xs2 @ I )
              = ( nth_nat @ Xs2 @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_1257_distinct__conv__nth,axiom,
    ( distinct_nat
    = ( ^ [Xs: list_nat] :
        ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
         => ! [J3: nat] :
              ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
             => ( ( I2 != J3 )
               => ( ( nth_nat @ Xs @ I2 )
                 != ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_1258_distinct__drop,axiom,
    ! [Xs2: list_nat,I: nat] :
      ( ( distinct_nat @ Xs2 )
     => ( distinct_nat @ ( drop_nat @ I @ Xs2 ) ) ) ).

% distinct_drop
thf(fact_1259_distinct__take,axiom,
    ! [Xs2: list_nat,I: nat] :
      ( ( distinct_nat @ Xs2 )
     => ( distinct_nat @ ( take_nat @ I @ Xs2 ) ) ) ).

% distinct_take
thf(fact_1260_not__distinct__conv__prefix,axiom,
    ! [As: list_nat] :
      ( ( ~ ( distinct_nat @ As ) )
      = ( ? [Xs: list_nat,Y4: nat,Ys6: list_nat] :
            ( ( member_nat @ Y4 @ ( set_nat2 @ Xs ) )
            & ( distinct_nat @ Xs )
            & ( As
              = ( append_nat @ Xs @ ( cons_nat @ Y4 @ Ys6 ) ) ) ) ) ) ).

% not_distinct_conv_prefix

% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ! [X3: nat] :
      ( ( member_nat @ X3 @ top_top_set_nat )
     => ! [Xa2: nat] :
          ( ( member_nat @ Xa2 @ top_top_set_nat )
         => ( ( ( bot_bot_BFun_f @ f2 @ ( replic6958083601664011982term_f @ X3 @ bot_bot_Bot_f ) )
              = ( bot_bot_BFun_f @ f2 @ ( replic6958083601664011982term_f @ Xa2 @ bot_bot_Bot_f ) ) )
           => ( X3 = Xa2 ) ) ) ) ).

%------------------------------------------------------------------------------