%------------------------------------------------------------------------------
% 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/0068_Ground_MCtxt/prob_00369_013994__18467982_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1762 ( 403 unt; 504 typ;   0 def)
%            Number of atoms       : 4152 (1959 equ;   0 cnn)
%            Maximal formula atoms :   11 (   3 avg)
%            Number of connectives : 19361 ( 245   ~;  34   |; 468   &;16279   @)
%                                         (   0 <=>;2335  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   24 (   8 avg)
%            Number of types       :  115 ( 114 usr)
%            Number of type conns  : 1427 (1427   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  393 ( 390 usr;  10 con; 0-4 aty)
%            Number of variables   : 4780 (  89   ^;4341   !; 350   ?;4780   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 12:01:25.353
%------------------------------------------------------------------------------
% Could-be-implicit typings (114)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__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_J_Mt__List__Olist_It__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_J_J_J,type,
    set_Pr3695809186315947463at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__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_J_Mt__List__Olist_It__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_J_J,type,
    produc3451831593474659303at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__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_Mt__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_J_J,type,
    set_Pr4087777274317423175at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__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_Mt__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_J,type,
    produc289266582803401575at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_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_Mt__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_J,type,
    set_Pr553994874890374343at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_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_Mt__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,
    produc349518998152878311at_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__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_J_J,type,
    list_P272808547438151458at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__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_J_J,type,
    set_Pr5302226732920531656at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__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_J,type,
    produc1035334568315550354at_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__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_J,type,
    list_P2187511852274165404at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
    set_Pr7047708467701517959at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__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_J,type,
    set_Pr938417207581201730at_nat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__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_J_J,type,
    list_l5129023048960694521at_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__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J_Mt__List__Olist_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J_J_J,type,
    set_Pr8517441760511179837ctxt_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    produc492020038885415591at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__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,
    produc2487518378626728076at_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__List__Olist_It__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_J,type,
    list_l3298181151656792051at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__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_J,type,
    set_li3197816953174176717at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J_Mt__List__Olist_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J_J,type,
    produc2442989038347685597ctxt_a: $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__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_Mt__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J_J,type,
    set_Pr8687807964536448157ctxt_a: $tType ).

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

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

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_Mt__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
    produc1017277696136915517ctxt_a: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J_J,type,
    set_Pr302638903384814927ctxt_a: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    set_Pr8511038139508077391st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__b_J_J_Mt__List__Olist_It__List__Olist_Itf__b_J_J_J_J,type,
    set_Pr2215042635423483847list_b: $tType ).

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

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    set_Pr5860164671742971281st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J_J,type,
    set_Pr2972090421851356312ctxt_a: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_Mt__List__Olist_Itf__b_J_J_J,type,
    set_Pr1959420730066914690list_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
    set_Pr8705490787052945405ctxt_a: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
    set_Pr7387263332798406618st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__List__Olist_Itf__b_J_J_J_J,type,
    set_Pr6009741699831149654list_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_Mt__List__Olist_Itf__b_J_J_J,type,
    set_Pr5178944948679147456list_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
    produc335899259021751791ctxt_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc4140329211374400495st_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J_J,type,
    list_P2203693314889663077ctxt_a: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J_J,type,
    set_Pr1203255704990185791ctxt_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__b_J_J_Mt__List__Olist_It__List__Olist_Itf__b_J_J_J,type,
    produc4603089062067288551list_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__List__Olist_It__List__Olist_Itf__b_J_J_J_J,type,
    set_Pr3668070192038973389list_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc3811483368811048881st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
    produc5876917791195906530ctxt_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_Mt__List__Olist_Itf__b_J_J,type,
    produc3551300527538202572list_b: $tType ).

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

thf(ty_n_t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    produc8175205341597284509ctxt_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
    produc2334819108994950564st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__List__Olist_Itf__b_J_J_J,type,
    produc8487164832621770784list_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_Mt__List__Olist_Itf__b_J_J,type,
    produc6204733625450272906list_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    set_Pr1291562142433127754st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_Itf__b_J_J_J,type,
    set_Pr4901976648650938704list_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
    produc4999762543018873695ctxt_a: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
    list_P3876139947223466197ctxt_a: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
    set_Pr1619835084168866991ctxt_a: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Nat__Onat_J_J,type,
    set_Pr9093710254965148079_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__List__Olist_It__List__Olist_Itf__b_J_J_J,type,
    produc3177755504865379949list_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__List__Olist_Itf__b_J_J_J,type,
    set_Pr7665282455119567943list_b: $tType ).

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

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

thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J_J,type,
    list_l3687490334575670478ctxt_a: $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__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
    set_Pr5732450166000137346ctxt_a: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mtf__b_J_J,type,
    set_Pr7985508066815312396xt_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc1099298608082921876st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_Itf__b_J_J,type,
    produc1584444791027398810list_b: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J_J,type,
    list_P5454506040216281530list_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    set_Pr6729162334058022980st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J_J,type,
    set_Pr1349601357184307552list_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mtf__b_J_J,type,
    set_Pr8611280865567961546_nat_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    produc932717863267311695ctxt_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Nat__Onat_J,type,
    produc8497099580089875791_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__List__Olist_Itf__b_J_J,type,
    produc3963297410138542439list_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    produc8199716216217303280at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__List__Olist_Itf__b_J_J_J,type,
    set_Pr3499773787893599181list_b: $tType ).

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

thf(ty_n_t__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__Product____Type__Oprod_Itf__b_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    produc929589974862302156ctxt_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mtf__b_J,type,
    produc7780146698139795030xt_a_b: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
    list_l1578326660578982846ctxt_a: $tType ).

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

thf(ty_n_t__Set__Oset_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
    set_li3075106635084045668ctxt_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc667988024575647758st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J,type,
    produc7811952446676219690list_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mtf__b_J,type,
    produc6459537403673193620_nat_b: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J_J,type,
    list_P2922825790777833268_nat_b: $tType ).

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

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_J,type,
    set_Pr1307281990691478580_b_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J_J,type,
    set_Pr4264375888882495962_nat_b: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_Itf__b_J_J_J,type,
    list_list_list_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
    set_Product_prod_b_b: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_Itf__b_J_J_J,type,
    set_list_list_b: $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__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    list_Ground_gmctxt_a: $tType ).

thf(ty_n_t__Set__Oset_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    set_Ground_gmctxt_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J,type,
    product_prod_b_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J,type,
    product_prod_nat_b: $tType ).

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

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

thf(ty_n_t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    product_prod_b_b: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_Itf__b_J_J,type,
    list_list_b: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_Itf__b_J_J,type,
    set_list_b: $tType ).

thf(ty_n_t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    ground_gmctxt_a: $tType ).

thf(ty_n_t__Ground____Terms__Ogterm_Itf__a_J,type,
    ground_gterm_a: $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__b_J,type,
    list_b: $tType ).

thf(ty_n_t__Set__Oset_Itf__b_J,type,
    set_b: $tType ).

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

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

% Explicit typings (390)
thf(sy_c_Fun__Def_Opair__leq,type,
    fun_pair_leq: set_Pr8693737435421807431at_nat ).

thf(sy_c_Fun__Def_Opair__less,type,
    fun_pair_less: set_Pr8693737435421807431at_nat ).

thf(sy_c_Ground__MCtxt_Ogmctxt__of__gterm_001tf__a,type,
    ground582559344758063963term_a: ground_gterm_a > ground_gmctxt_a ).

thf(sy_c_Ground__MCtxt_Onum__gholes_001tf__a,type,
    ground_num_gholes_a: ground_gmctxt_a > nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: nat > nat > 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_If_001t__Nat__Onat,type,
    if_nat: $o > nat > nat > nat ).

thf(sy_c_List_Oappend_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    append5238750768297430211ctxt_a: list_Ground_gmctxt_a > list_Ground_gmctxt_a > list_Ground_gmctxt_a ).

thf(sy_c_List_Oappend_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    append272212692390751315ctxt_a: list_l1578326660578982846ctxt_a > list_l1578326660578982846ctxt_a > list_l1578326660578982846ctxt_a ).

thf(sy_c_List_Oappend_001t__List__Olist_It__List__Olist_Itf__b_J_J,type,
    append_list_list_b: list_list_list_b > list_list_list_b > list_list_list_b ).

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

thf(sy_c_List_Oappend_001t__List__Olist_It__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,
    append6271109094031990152at_nat: list_l3298181151656792051at_nat > list_l3298181151656792051at_nat > list_l3298181151656792051at_nat ).

thf(sy_c_List_Oappend_001t__List__Olist_Itf__b_J,type,
    append_list_b: list_list_b > list_list_b > list_list_b ).

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

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    append2243830650925735786ctxt_a: list_P3876139947223466197ctxt_a > list_P3876139947223466197ctxt_a > list_P3876139947223466197ctxt_a ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J,type,
    append4873443811822227535list_b: list_P5454506040216281530list_b > list_P5454506040216281530list_b > list_P5454506040216281530list_b ).

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

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    append2382163119426291249at_nat: list_P2187511852274165404at_nat > list_P2187511852274165404at_nat > list_P2187511852274165404at_nat ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J,type,
    append1694031010730255049_nat_b: list_P2922825790777833268_nat_b > list_P2922825790777833268_nat_b > list_P2922825790777833268_nat_b ).

thf(sy_c_List_Oappend_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,
    append8751754712269456642at_nat: list_P8469869581646625389at_nat > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat ).

thf(sy_c_List_Oappend_001tf__b,type,
    append_b: list_b > list_b > list_b ).

thf(sy_c_List_Oconcat_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    concat3993834769610107037ctxt_a: list_l1578326660578982846ctxt_a > list_Ground_gmctxt_a ).

thf(sy_c_List_Oconcat_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    concat6203624070112318253ctxt_a: list_l3687490334575670478ctxt_a > list_l1578326660578982846ctxt_a ).

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

thf(sy_c_List_Oconcat_001t__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,
    concat5897368152400924974at_nat: list_l5129023048960694521at_nat > list_l3298181151656792051at_nat ).

thf(sy_c_List_Oconcat_001t__List__Olist_Itf__b_J,type,
    concat_list_b: list_list_list_b > list_list_b ).

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

thf(sy_c_List_Oconcat_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,
    concat5616020275982107560at_nat: list_l3298181151656792051at_nat > list_P8469869581646625389at_nat ).

thf(sy_c_List_Oconcat_001tf__b,type,
    concat_b: list_list_b > list_b ).

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

thf(sy_c_List_Oenumerate_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    enumer3922341085786344185ctxt_a: nat > list_Ground_gmctxt_a > list_P3876139947223466197ctxt_a ).

thf(sy_c_List_Oenumerate_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    enumer2580734899591370633ctxt_a: nat > list_l1578326660578982846ctxt_a > list_P2203693314889663077ctxt_a ).

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

thf(sy_c_List_Oenumerate_001t__List__Olist_It__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,
    enumer33237656203508690at_nat: nat > list_l3298181151656792051at_nat > list_P272808547438151458at_nat ).

thf(sy_c_List_Oenumerate_001t__List__Olist_Itf__b_J,type,
    enumerate_list_b: nat > list_list_b > list_P5454506040216281530list_b ).

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

thf(sy_c_List_Oenumerate_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    enumer3122081344003035212at_nat: nat > list_P8469869581646625389at_nat > list_P2187511852274165404at_nat ).

thf(sy_c_List_Oenumerate_001tf__b,type,
    enumerate_b: nat > list_b > list_P2922825790777833268_nat_b ).

thf(sy_c_List_Olist_Omap_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    map_Gr980675717119874050ctxt_a: ( ground_gmctxt_a > ground_gmctxt_a ) > list_Ground_gmctxt_a > list_Ground_gmctxt_a ).

thf(sy_c_List_Olist_Omap_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__List__Olist_Itf__b_J,type,
    map_Gr5773545835744863265list_b: ( ground_gmctxt_a > list_b ) > list_Ground_gmctxt_a > list_list_b ).

thf(sy_c_List_Olist_Omap_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Nat__Onat,type,
    map_Gr1679544123147727284_a_nat: ( ground_gmctxt_a > nat ) > list_Ground_gmctxt_a > list_nat ).

thf(sy_c_List_Olist_Omap_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001tf__b,type,
    map_Gr2594013990000539163xt_a_b: ( ground_gmctxt_a > b ) > list_Ground_gmctxt_a > list_b ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    map_li7659068976007724962ctxt_a: ( list_Ground_gmctxt_a > list_Ground_gmctxt_a ) > list_l1578326660578982846ctxt_a > list_l1578326660578982846ctxt_a ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_001t__List__Olist_It__Nat__Onat_J,type,
    map_li7775015282390606420st_nat: ( list_Ground_gmctxt_a > list_nat ) > list_l1578326660578982846ctxt_a > list_list_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_001t__List__Olist_Itf__b_J,type,
    map_li4808364546485981329list_b: ( list_Ground_gmctxt_a > list_b ) > list_l1578326660578982846ctxt_a > list_list_b ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_001t__Nat__Onat,type,
    map_li8618020238875267396_a_nat: ( list_Ground_gmctxt_a > nat ) > list_l1578326660578982846ctxt_a > list_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J_001t__Nat__Onat,type,
    map_li3800085317229163092_a_nat: ( list_l1578326660578982846ctxt_a > nat ) > list_l3687490334575670478ctxt_a > list_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__Nat__Onat,type,
    map_li3275209578935457798at_nat: ( list_list_nat > nat ) > list_list_list_nat > list_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__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_J_001t__Nat__Onat,type,
    map_li611904212741579135at_nat: ( list_l3298181151656792051at_nat > nat ) > list_l5129023048960694521at_nat > list_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__List__Olist_Itf__b_J_J_001t__Nat__Onat,type,
    map_list_list_b_nat: ( list_list_b > nat ) > list_list_list_b > list_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    map_li3283194380818682708ctxt_a: ( list_nat > list_Ground_gmctxt_a ) > list_list_nat > list_l1578326660578982846ctxt_a ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    map_li7225945977422193158st_nat: ( list_nat > list_nat ) > list_list_nat > list_list_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_Itf__b_J,type,
    map_list_nat_list_b: ( list_nat > list_b ) > list_list_nat > list_list_b ).

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__List__Olist_It__Nat__Onat_J_001tf__b,type,
    map_list_nat_b: ( list_nat > b ) > list_list_nat > list_b ).

thf(sy_c_List_Olist_Omap_001t__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_001t__Nat__Onat,type,
    map_li9059624042496273785at_nat: ( list_P8469869581646625389at_nat > nat ) > list_l3298181151656792051at_nat > list_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_Itf__b_J_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    map_li4667870289254436335ctxt_a: ( list_b > list_Ground_gmctxt_a ) > list_list_b > list_l1578326660578982846ctxt_a ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_Itf__b_J_001t__List__Olist_It__List__Olist_Itf__b_J_J,type,
    map_li4464339650582430090list_b: ( list_b > list_list_b ) > list_list_b > list_list_list_b ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_Itf__b_J_001t__List__Olist_It__Nat__Onat_J,type,
    map_list_b_list_nat: ( list_b > list_nat ) > list_list_b > list_list_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_Itf__b_J_001t__List__Olist_Itf__b_J,type,
    map_list_b_list_b: ( list_b > list_b ) > list_list_b > list_list_b ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_Itf__b_J_001t__Nat__Onat,type,
    map_list_b_nat: ( list_b > nat ) > list_list_b > list_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_Itf__b_J_001tf__b,type,
    map_list_b_b: ( list_b > b ) > list_list_b > list_b ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    map_na2753241459123782580ctxt_a: ( nat > ground_gmctxt_a ) > list_nat > list_Ground_gmctxt_a ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_Itf__b_J,type,
    map_nat_list_b: ( nat > list_b ) > list_nat > list_list_b ).

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

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001tf__b,type,
    map_nat_b: ( nat > b ) > list_nat > list_b ).

thf(sy_c_List_Olist_Omap_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_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,
    map_Pr7106542193144342532at_nat: ( produc859450856879609959at_nat > produc859450856879609959at_nat ) > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat ).

thf(sy_c_List_Olist_Omap_001tf__b_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    map_b_3587768286370285017ctxt_a: ( b > ground_gmctxt_a ) > list_b > list_Ground_gmctxt_a ).

thf(sy_c_List_Olist_Omap_001tf__b_001t__List__Olist_It__Nat__Onat_J,type,
    map_b_list_nat: ( b > list_nat ) > list_b > list_list_nat ).

thf(sy_c_List_Olist_Omap_001tf__b_001t__List__Olist_Itf__b_J,type,
    map_b_list_b: ( b > list_b ) > list_b > list_list_b ).

thf(sy_c_List_Olist_Omap_001tf__b_001t__Nat__Onat,type,
    map_b_nat: ( b > nat ) > list_b > list_nat ).

thf(sy_c_List_Olist_Omap_001tf__b_001tf__b,type,
    map_b_b: ( b > b ) > list_b > list_b ).

thf(sy_c_List_Olist_Oset_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    set_Ground_gmctxt_a2: list_Ground_gmctxt_a > set_Ground_gmctxt_a ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    set_li3057194059929822169ctxt_a: list_l1578326660578982846ctxt_a > set_li3075106635084045668ctxt_a ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_Itf__b_J_J,type,
    set_list_list_b2: list_list_list_b > set_list_list_b ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
    set_list_nat2: list_list_nat > set_list_nat ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__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_li7366234323595830146at_nat: list_l3298181151656792051at_nat > set_li3197816953174176717at_nat ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__b_J,type,
    set_list_b2: list_list_b > set_list_b ).

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

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    set_Pr3995427970001988964ctxt_a: list_P3876139947223466197ctxt_a > set_Pr1619835084168866991ctxt_a ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
    set_Pr8908780240170439156ctxt_a: list_P2203693314889663077ctxt_a > set_Pr1203255704990185791ctxt_a ).

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__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_J,type,
    set_Pr6130687943182631741at_nat: list_P272808547438151458at_nat > set_Pr5302226732920531656at_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J,type,
    set_Pr8537765906400867797list_b: list_P5454506040216281530list_b > set_Pr1349601357184307552list_b ).

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__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_Pr2327711193239132855at_nat: list_P2187511852274165404at_nat > set_Pr938417207581201730at_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J,type,
    set_Pr4163146842529940303_nat_b: list_P2922825790777833268_nat_b > set_Pr4264375888882495962_nat_b ).

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_001tf__b,type,
    set_b2: list_b > set_b ).

thf(sy_c_List_Olist__ex_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    list_e8873126888930822106ctxt_a: ( ground_gmctxt_a > $o ) > list_Ground_gmctxt_a > $o ).

thf(sy_c_List_Olist__ex_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    list_e7139191923936702442ctxt_a: ( list_Ground_gmctxt_a > $o ) > list_l1578326660578982846ctxt_a > $o ).

thf(sy_c_List_Olist__ex_001t__List__Olist_It__Nat__Onat_J,type,
    list_ex_list_nat: ( list_nat > $o ) > list_list_nat > $o ).

thf(sy_c_List_Olist__ex_001t__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_e5392882716291896753at_nat: ( list_P8469869581646625389at_nat > $o ) > list_l3298181151656792051at_nat > $o ).

thf(sy_c_List_Olist__ex_001t__List__Olist_Itf__b_J,type,
    list_ex_list_b: ( list_b > $o ) > list_list_b > $o ).

thf(sy_c_List_Olist__ex_001t__Nat__Onat,type,
    list_ex_nat: ( nat > $o ) > list_nat > $o ).

thf(sy_c_List_Olist__ex_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,
    list_e7227117345903044011at_nat: ( produc859450856879609959at_nat > $o ) > list_P8469869581646625389at_nat > $o ).

thf(sy_c_List_Olist__ex_001tf__b,type,
    list_ex_b: ( b > $o ) > list_b > $o ).

thf(sy_c_List_Olist__update_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    list_u7950812632479372336ctxt_a: list_Ground_gmctxt_a > nat > ground_gmctxt_a > list_Ground_gmctxt_a ).

thf(sy_c_List_Olist__update_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    list_u8015824433058371904ctxt_a: list_l1578326660578982846ctxt_a > nat > list_Ground_gmctxt_a > list_l1578326660578982846ctxt_a ).

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

thf(sy_c_List_Olist__update_001t__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_u5583386537922002395at_nat: list_l3298181151656792051at_nat > nat > list_P8469869581646625389at_nat > list_l3298181151656792051at_nat ).

thf(sy_c_List_Olist__update_001t__List__Olist_Itf__b_J,type,
    list_update_list_b: list_list_b > nat > list_b > list_list_b ).

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

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    list_u6180841689913720943at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Olist__update_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,
    list_u5003261594476800725at_nat: list_P8469869581646625389at_nat > nat > produc859450856879609959at_nat > list_P8469869581646625389at_nat ).

thf(sy_c_List_Olist__update_001tf__b,type,
    list_update_b: list_b > nat > b > list_b ).

thf(sy_c_List_Olistrel1_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    listre839271074516595327ctxt_a: set_Pr8705490787052945405ctxt_a > set_Pr8687807964536448157ctxt_a ).

thf(sy_c_List_Olistrel1_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    listre7072643403622106127ctxt_a: set_Pr8687807964536448157ctxt_a > set_Pr8517441760511179837ctxt_a ).

thf(sy_c_List_Olistrel1_001t__List__Olist_It__Nat__Onat_J,type,
    listrel1_list_nat: set_Pr3451248702717554689st_nat > set_Pr1190453367779242145st_nat ).

thf(sy_c_List_Olistrel1_001t__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,
    listre3648893826645724364at_nat: set_Pr4087777274317423175at_nat > set_Pr3695809186315947463at_nat ).

thf(sy_c_List_Olistrel1_001t__List__Olist_Itf__b_J,type,
    listrel1_list_b: set_Pr7665282455119567943list_b > set_Pr2215042635423483847list_b ).

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

thf(sy_c_List_Olistrel1_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    listre4828114922151135584at_nat: set_Pr8693737435421807431at_nat > set_Pr1542805901266377927at_nat ).

thf(sy_c_List_Olistrel1_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,
    listre2293869942453659206at_nat: set_Pr553994874890374343at_nat > set_Pr4087777274317423175at_nat ).

thf(sy_c_List_Olistrel1_001tf__b,type,
    listrel1_b: set_Product_prod_b_b > set_Pr7665282455119567943list_b ).

thf(sy_c_List_Olistrel_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Nat__Onat,type,
    listre5839456775373167073_a_nat: set_Pr9093710254965148079_a_nat > set_Pr8511038139508077391st_nat ).

thf(sy_c_List_Olistrel_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001tf__b,type,
    listre5402109041937956654xt_a_b: set_Pr7985508066815312396xt_a_b > set_Pr1959420730066914690list_b ).

thf(sy_c_List_Olistrel_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
    listrel_list_nat_nat: set_Pr7549493975000385137at_nat > set_Pr5860164671742971281st_nat ).

thf(sy_c_List_Olistrel_001t__List__Olist_It__Nat__Onat_J_001tf__b,type,
    listrel_list_nat_b: set_Pr8611280865567961546_nat_b > set_Pr5178944948679147456list_b ).

thf(sy_c_List_Olistrel_001t__Nat__Onat_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    listre6913154111349222369ctxt_a: set_Pr1619835084168866991ctxt_a > set_Pr302638903384814927ctxt_a ).

thf(sy_c_List_Olistrel_001t__Nat__Onat_001t__List__Olist_Itf__b_J,type,
    listrel_nat_list_b: set_Pr1349601357184307552list_b > set_Pr6009741699831149654list_b ).

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

thf(sy_c_List_Olistrel_001t__Nat__Onat_001tf__b,type,
    listrel_nat_b: set_Pr4264375888882495962_nat_b > set_Pr4901976648650938704list_b ).

thf(sy_c_List_Olistrel_001tf__b_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    listre6395863338307702508ctxt_a: set_Pr5732450166000137346ctxt_a > set_Pr2972090421851356312ctxt_a ).

thf(sy_c_List_Olistrel_001tf__b_001t__List__Olist_It__Nat__Onat_J,type,
    listrel_b_list_nat: set_Pr6729162334058022980st_nat > set_Pr7387263332798406618st_nat ).

thf(sy_c_List_Olistrel_001tf__b_001t__List__Olist_Itf__b_J,type,
    listrel_b_list_b: set_Pr3499773787893599181list_b > set_Pr3668070192038973389list_b ).

thf(sy_c_List_Olistrel_001tf__b_001t__Nat__Onat,type,
    listrel_b_nat: set_Pr1307281990691478580_b_nat > set_Pr1291562142433127754st_nat ).

thf(sy_c_List_Olistrel_001tf__b_001tf__b,type,
    listrel_b_b: set_Product_prod_b_b > set_Pr7665282455119567943list_b ).

thf(sy_c_List_Omap__tailrec_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Nat__Onat,type,
    map_ta198793340144898302_a_nat: ( ground_gmctxt_a > nat ) > list_Ground_gmctxt_a > list_nat ).

thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001tf__b,type,
    map_tailrec_nat_b: ( nat > b ) > list_nat > list_b ).

thf(sy_c_List_Omap__tailrec_001tf__b_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    map_ta8077529389956916495ctxt_a: ( b > ground_gmctxt_a ) > list_b > list_Ground_gmctxt_a ).

thf(sy_c_List_Omap__tailrec_001tf__b_001t__List__Olist_Itf__b_J,type,
    map_tailrec_b_list_b: ( b > list_b ) > list_b > list_list_b ).

thf(sy_c_List_Omap__tailrec_001tf__b_001t__Nat__Onat,type,
    map_tailrec_b_nat: ( b > nat ) > list_b > list_nat ).

thf(sy_c_List_Omap__tailrec_001tf__b_001tf__b,type,
    map_tailrec_b_b: ( b > b ) > list_b > list_b ).

thf(sy_c_List_Omaps_001tf__b_001tf__b,type,
    maps_b_b: ( b > list_b ) > list_b > list_b ).

thf(sy_c_List_Onth_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    nth_Ground_gmctxt_a: list_Ground_gmctxt_a > nat > ground_gmctxt_a ).

thf(sy_c_List_Onth_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    nth_li6678718213161987943ctxt_a: list_l1578326660578982846ctxt_a > nat > list_Ground_gmctxt_a ).

thf(sy_c_List_Onth_001t__List__Olist_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
    nth_li3520809774789768823ctxt_a: list_l3687490334575670478ctxt_a > nat > list_l1578326660578982846ctxt_a ).

thf(sy_c_List_Onth_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    nth_list_list_nat: list_list_list_nat > nat > list_list_nat ).

thf(sy_c_List_Onth_001t__List__Olist_It__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_J,type,
    nth_li2593672534642986362at_nat: list_l5129023048960694521at_nat > nat > list_l3298181151656792051at_nat ).

thf(sy_c_List_Onth_001t__List__Olist_It__List__Olist_Itf__b_J_J,type,
    nth_list_list_b: list_list_list_b > nat > list_list_b ).

thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
    nth_list_nat: list_list_nat > nat > list_nat ).

thf(sy_c_List_Onth_001t__List__Olist_It__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,
    nth_li8805938130461222772at_nat: list_l3298181151656792051at_nat > nat > list_P8469869581646625389at_nat ).

thf(sy_c_List_Onth_001t__List__Olist_Itf__b_J,type,
    nth_list_b: list_list_b > nat > list_b ).

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    nth_Pr3641570550162790742ctxt_a: list_P3876139947223466197ctxt_a > nat > produc932717863267311695ctxt_a ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J,type,
    nth_Pr6103996347157616483list_b: list_P5454506040216281530list_b > nat > produc7811952446676219690list_b ).

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_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__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,
    nth_Pr4155663282896371525at_nat: list_P2187511852274165404at_nat > nat > produc2487518378626728076at_nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J,type,
    nth_Pr2476257085692543965_nat_b: list_P2922825790777833268_nat_b > nat > product_prod_nat_b ).

thf(sy_c_List_Onth_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,
    nth_Pr6744343527793145070at_nat: list_P8469869581646625389at_nat > nat > produc859450856879609959at_nat ).

thf(sy_c_List_Onth_001tf__b,type,
    nth_b: list_b > nat > b ).

thf(sy_c_List_Oproduct__lists_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    produc6102553363301462639ctxt_a: list_l1578326660578982846ctxt_a > list_l1578326660578982846ctxt_a ).

thf(sy_c_List_Oproduct__lists_001t__List__Olist_Itf__b_J,type,
    product_lists_list_b: list_list_list_b > list_list_list_b ).

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

thf(sy_c_List_Oproduct__lists_001t__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,
    produc1375813136864985686at_nat: list_l3298181151656792051at_nat > list_l3298181151656792051at_nat ).

thf(sy_c_List_Oproduct__lists_001tf__b,type,
    product_lists_b: list_list_b > list_list_b ).

thf(sy_c_List_Osubseqs_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    subseq4078942962620312969ctxt_a: list_Ground_gmctxt_a > list_l1578326660578982846ctxt_a ).

thf(sy_c_List_Osubseqs_001t__List__Olist_Itf__b_J,type,
    subseqs_list_b: list_list_b > list_list_list_b ).

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

thf(sy_c_List_Osubseqs_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,
    subseq4464984717476426172at_nat: list_P8469869581646625389at_nat > list_l3298181151656792051at_nat ).

thf(sy_c_List_Osubseqs_001tf__b,type,
    subseqs_b: list_b > list_list_b ).

thf(sy_c_List_Otake_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    take_Ground_gmctxt_a: nat > list_Ground_gmctxt_a > list_Ground_gmctxt_a ).

thf(sy_c_List_Otake_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    take_l4544854310384971712ctxt_a: nat > list_l1578326660578982846ctxt_a > list_l1578326660578982846ctxt_a ).

thf(sy_c_List_Otake_001t__List__Olist_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
    take_l7489526141625251408ctxt_a: nat > list_l3687490334575670478ctxt_a > list_l3687490334575670478ctxt_a ).

thf(sy_c_List_Otake_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    take_list_list_nat: nat > list_list_list_nat > list_list_list_nat ).

thf(sy_c_List_Otake_001t__List__Olist_It__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_J,type,
    take_l5453269369450450401at_nat: nat > list_l5129023048960694521at_nat > list_l5129023048960694521at_nat ).

thf(sy_c_List_Otake_001t__List__Olist_It__List__Olist_Itf__b_J_J,type,
    take_list_list_b: nat > list_list_list_b > list_list_list_b ).

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

thf(sy_c_List_Otake_001t__List__Olist_It__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,
    take_l4110974213330421595at_nat: nat > list_l3298181151656792051at_nat > list_l3298181151656792051at_nat ).

thf(sy_c_List_Otake_001t__List__Olist_Itf__b_J,type,
    take_list_b: nat > list_list_b > list_list_b ).

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

thf(sy_c_List_Otake_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    take_P5254422574997664853at_nat: nat > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat ).

thf(sy_c_List_Otake_001tf__b,type,
    take_b: nat > list_b > list_b ).

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_Opermut_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    missin4491563000646178374ctxt_a: list_Ground_gmctxt_a > ( nat > nat ) > list_Ground_gmctxt_a ).

thf(sy_c_Missing__List_Opermut_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    missin7467934043202574422ctxt_a: list_l1578326660578982846ctxt_a > ( nat > nat ) > list_l1578326660578982846ctxt_a ).

thf(sy_c_Missing__List_Opermut_001t__List__Olist_It__Nat__Onat_J,type,
    missin2960327169605006088st_nat: list_list_nat > ( nat > nat ) > list_list_nat ).

thf(sy_c_Missing__List_Opermut_001t__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,
    missin2985456621902692421at_nat: list_l3298181151656792051at_nat > ( nat > nat ) > list_l3298181151656792051at_nat ).

thf(sy_c_Missing__List_Opermut_001t__List__Olist_Itf__b_J,type,
    missin8258838938715587805list_b: list_list_b > ( nat > nat ) > list_list_b ).

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_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,
    missin1395697627405724223at_nat: list_P8469869581646625389at_nat > ( nat > nat ) > list_P8469869581646625389at_nat ).

thf(sy_c_Missing__List_Opermut_001tf__b,type,
    missing_permut_b: list_b > ( nat > nat ) > list_b ).

thf(sy_c_Missing__List_Opermut__aux_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    missin1343833451449182556ctxt_a: list_Ground_gmctxt_a > ( nat > nat ) > list_Ground_gmctxt_a > list_Ground_gmctxt_a ).

thf(sy_c_Missing__List_Opermut__aux_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    missin8538012249342617708ctxt_a: list_l1578326660578982846ctxt_a > ( nat > nat ) > list_l1578326660578982846ctxt_a > list_l1578326660578982846ctxt_a ).

thf(sy_c_Missing__List_Opermut__aux_001t__List__Olist_It__Nat__Onat_J,type,
    missin3204601078308456222st_nat: list_list_nat > ( nat > nat ) > list_list_nat > list_list_nat ).

thf(sy_c_Missing__List_Opermut__aux_001t__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,
    missin3013719938226745775at_nat: list_l3298181151656792051at_nat > ( nat > nat ) > list_l3298181151656792051at_nat > list_l3298181151656792051at_nat ).

thf(sy_c_Missing__List_Opermut__aux_001t__List__Olist_Itf__b_J,type,
    missin5389789488504408647list_b: list_list_b > ( nat > nat ) > list_list_b > list_list_b ).

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_Opermut__aux_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,
    missin7899315033207363753at_nat: list_P8469869581646625389at_nat > ( nat > nat ) > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat ).

thf(sy_c_Missing__List_Opermut__aux_001tf__b,type,
    missing_permut_aux_b: list_b > ( nat > nat ) > list_b > list_b ).

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

thf(sy_c_Missing__List_Oremove__nth_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    missin7823264313509747941ctxt_a: nat > list_Ground_gmctxt_a > list_Ground_gmctxt_a ).

thf(sy_c_Missing__List_Oremove__nth_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    missin1875899094540944245ctxt_a: nat > list_l1578326660578982846ctxt_a > list_l1578326660578982846ctxt_a ).

thf(sy_c_Missing__List_Oremove__nth_001t__List__Olist_It__Nat__Onat_J,type,
    missin5133812807938250791st_nat: nat > list_list_nat > list_list_nat ).

thf(sy_c_Missing__List_Oremove__nth_001t__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,
    missin7550826715306853094at_nat: nat > list_l3298181151656792051at_nat > list_l3298181151656792051at_nat ).

thf(sy_c_Missing__List_Oremove__nth_001t__List__Olist_Itf__b_J,type,
    missin8041159180453292414list_b: nat > list_list_b > list_list_b ).

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

thf(sy_c_Missing__List_Oremove__nth_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,
    missin1453772821081764192at_nat: nat > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat ).

thf(sy_c_Missing__List_Oremove__nth_001tf__b,type,
    missing_remove_nth_b: nat > list_b > list_b ).

thf(sy_c_Multihole__Context_Opartition__by_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    multih5172578161653122718ctxt_a: list_Ground_gmctxt_a > list_nat > list_l1578326660578982846ctxt_a ).

thf(sy_c_Multihole__Context_Opartition__by_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    multih1205877868680985774ctxt_a: list_l1578326660578982846ctxt_a > list_nat > list_l3687490334575670478ctxt_a ).

thf(sy_c_Multihole__Context_Opartition__by_001t__List__Olist_It__Nat__Onat_J,type,
    multih5630945896903731040st_nat: list_list_nat > list_nat > list_list_list_nat ).

thf(sy_c_Multihole__Context_Opartition__by_001t__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,
    multih560249069488021997at_nat: list_l3298181151656792051at_nat > list_nat > list_l5129023048960694521at_nat ).

thf(sy_c_Multihole__Context_Opartition__by_001t__List__Olist_Itf__b_J,type,
    multih2267195492336891525list_b: list_list_b > list_nat > list_list_list_b ).

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_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,
    multih7555706055023738855at_nat: list_P8469869581646625389at_nat > list_nat > list_l3298181151656792051at_nat ).

thf(sy_c_Multihole__Context_Opartition__by_001tf__b,type,
    multih3537974503344769663n_by_b: list_b > list_nat > list_list_b ).

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__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    size_s962548663205356226ctxt_a: list_Ground_gmctxt_a > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
    size_s6994882312647770194ctxt_a: list_l1578326660578982846ctxt_a > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J_J,type,
    size_s2801996683354709858ctxt_a: list_l3687490334575670478ctxt_a > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
    size_s6248950052170075156st_nat: list_list_list_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__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_J_J,type,
    size_s7371574421788826469at_nat: list_l5129023048960694521at_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__b_J_J_J,type,
    size_s9106009106869804029list_b: list_list_list_b > nat ).

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__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_J,type,
    size_s6878757106947045471at_nat: list_l3298181151656792051at_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_Itf__b_J_J,type,
    size_s420531428170919671list_b: list_list_b > nat ).

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
    size_s4225619979278260545ctxt_a: list_P3876139947223466197ctxt_a > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J_J,type,
    size_s8332131329787750990list_b: list_P5454506040216281530list_b > nat ).

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__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_J,type,
    size_s2512440580853752112at_nat: list_P2187511852274165404at_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J_J,type,
    size_s314938103728740808_nat_b: list_P2922825790777833268_nat_b > 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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    size_s3679842834875189465at_nat: list_P8469869581646625389at_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__b_J,type,
    size_size_list_b: list_b > 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__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    ord_le4032861497522155828ctxt_a: set_Ground_gmctxt_a > set_Ground_gmctxt_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
    ord_le9074214841948581316ctxt_a: set_li3075106635084045668ctxt_a > set_li3075106635084045668ctxt_a > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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_J,type,
    ord_le5602213209262422893at_nat: set_li3197816953174176717at_nat > set_li3197816953174176717at_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__b_J_J,type,
    ord_le8932221534207217157list_b: set_list_b > set_list_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_eq_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_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,
    ord_le3000389064537975527at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o ).

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

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

thf(sy_c_Product__Type_OPair_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    produc6347872285926497493ctxt_a: ground_gmctxt_a > ground_gmctxt_a > produc8175205341597284509ctxt_a ).

thf(sy_c_Product__Type_OPair_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Nat__Onat,type,
    produc2030840239341956231_a_nat: ground_gmctxt_a > nat > produc8497099580089875791_a_nat ).

thf(sy_c_Product__Type_OPair_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001tf__b,type,
    produc239971217247291912xt_a_b: ground_gmctxt_a > b > produc7780146698139795030xt_a_b ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    produc4772472886035905909ctxt_a: list_Ground_gmctxt_a > list_Ground_gmctxt_a > produc1017277696136915517ctxt_a ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc5273301679375258151st_nat: list_Ground_gmctxt_a > list_nat > produc4140329211374400495st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_001t__List__Olist_Itf__b_J,type,
    produc578867730268368766list_b: list_Ground_gmctxt_a > list_b > produc3551300527538202572list_b ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J_001t__List__Olist_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
    produc6157733871278919957ctxt_a: list_l1578326660578982846ctxt_a > list_l1578326660578982846ctxt_a > produc2442989038347685597ctxt_a ).

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

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc8303078633012690025st_nat: list_list_nat > list_nat > produc3811483368811048881st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_Itf__b_J,type,
    produc1825125551900217788list_b: list_list_nat > list_b > produc6204733625450272906list_b ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__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_J_001t__List__Olist_It__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_J,type,
    produc1890770240352350807at_nat: list_l3298181151656792051at_nat > list_l3298181151656792051at_nat > produc3451831593474659303at_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_Itf__b_J_J_001t__List__Olist_It__List__Olist_Itf__b_J_J,type,
    produc8181704880241203287list_b: list_list_b > list_list_b > produc4603089062067288551list_b ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    produc781480777803334439ctxt_a: list_nat > list_Ground_gmctxt_a > produc335899259021751791ctxt_a ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__List__Olist_Itf__b_J_J,type,
    produc2084500703101687378list_b: list_nat > list_list_b > produc8487164832621770784list_b ).

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__List__Olist_Itf__b_J,type,
    produc7723716014355252812list_b: list_nat > list_b > produc1584444791027398810list_b ).

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_001tf__b,type,
    produc1615968307924265414_nat_b: list_nat > b > produc6459537403673193620_nat_b ).

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

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__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,
    produc1338542795132623831at_nat: list_P8469869581646625389at_nat > list_P8469869581646625389at_nat > produc289266582803401575at_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__b_J_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    produc438373473036823772ctxt_a: list_b > list_Ground_gmctxt_a > produc5876917791195906530ctxt_a ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__b_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    produc3393462956800104734st_nat: list_b > list_list_nat > produc2334819108994950564st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__b_J_001t__List__Olist_It__List__Olist_Itf__b_J_J,type,
    produc4417627063519195229list_b: list_b > list_list_b > produc3177755504865379949list_b ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__b_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc5459849259740801934st_nat: list_b > list_nat > produc1099298608082921876st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__b_J_001t__List__Olist_Itf__b_J,type,
    produc1564554178308465111list_b: list_b > list_b > produc3963297410138542439list_b ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    produc3104537575318011527ctxt_a: nat > ground_gmctxt_a > produc932717863267311695ctxt_a ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__List__Olist_Itf__b_J,type,
    produc7903367361620597084list_b: nat > list_b > produc7811952446676219690list_b ).

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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc7482930922861554750at_nat: nat > produc859450856879609959at_nat > produc2487518378626728076at_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001tf__b,type,
    product_Pair_nat_b: nat > b > product_prod_nat_b ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    produc2768974569381949847at_nat: produc8199716216217303280at_nat > produc8199716216217303280at_nat > produc492020038885415591at_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__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_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,
    produc4662710985925991255at_nat: produc859450856879609959at_nat > produc859450856879609959at_nat > produc349518998152878311at_nat ).

thf(sy_c_Product__Type_OPair_001tf__b_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    produc1233725513617037766ctxt_a: b > ground_gmctxt_a > produc929589974862302156ctxt_a ).

thf(sy_c_Product__Type_OPair_001tf__b_001t__List__Olist_It__Nat__Onat_J,type,
    produc8204054404704827016st_nat: b > list_nat > produc667988024575647758st_nat ).

thf(sy_c_Product__Type_OPair_001tf__b_001t__Nat__Onat,type,
    product_Pair_b_nat: b > nat > product_prod_b_nat ).

thf(sy_c_Product__Type_OPair_001tf__b_001tf__b,type,
    product_Pair_b_b: b > b > product_prod_b_b ).

thf(sy_c_Product__Type_Oprod_Ofst_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    produc6156676138143019412at_nat: produc8199716216217303280at_nat > nat > nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    produc8631767124948943475ctxt_a: produc932717863267311695ctxt_a > nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    produc5338924193594264067ctxt_a: produc4999762543018873695ctxt_a > nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    produc8283513326151283381st_nat: produc4575160907756185873st_nat > nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__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,
    produc5598002518098430936at_nat: produc1035334568315550354at_nat > nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__List__Olist_Itf__b_J,type,
    produc4466459399909381744list_b: produc7811952446676219690list_b > 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__Nat__Onat_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,
    produc7178812058934418258at_nat: produc2487518378626728076at_nat > nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001tf__b,type,
    product_fst_nat_b: product_prod_nat_b > nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc3213797794245857475at_nat: produc859450856879609959at_nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    produc1852801350702243542at_nat: produc8199716216217303280at_nat > nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    produc3552744573514591925ctxt_a: produc932717863267311695ctxt_a > ground_gmctxt_a ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    produc1946588146859208005ctxt_a: produc4999762543018873695ctxt_a > list_Ground_gmctxt_a ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    produc1177724860032999927st_nat: produc4575160907756185873st_nat > list_nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__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,
    produc3264766051377360150at_nat: produc1035334568315550354at_nat > list_P8469869581646625389at_nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__List__Olist_Itf__b_J,type,
    produc8425566080785036206list_b: produc7811952446676219690list_b > list_b ).

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__Nat__Onat_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,
    produc3846061323874349968at_nat: produc2487518378626728076at_nat > produc859450856879609959at_nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001tf__b,type,
    product_snd_nat_b: product_prod_nat_b > b ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc6408287024330202629at_nat: produc859450856879609959at_nat > product_prod_nat_nat ).

thf(sy_c_Set_OCollect_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    collec8774204703398931443ctxt_a: ( ground_gmctxt_a > $o ) > set_Ground_gmctxt_a ).

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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    collec7088162979684241874at_nat: ( produc859450856879609959at_nat > $o ) > set_Pr8693737435421807431at_nat ).

thf(sy_c_Set_OCollect_001tf__b,type,
    collect_b: ( b > $o ) > set_b ).

thf(sy_c_Transitive__Closure_Otrancl_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    transi3225167780237802813ctxt_a: set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a ).

thf(sy_c_Transitive__Closure_Otrancl_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    transi8139323374919604173ctxt_a: set_Pr8687807964536448157ctxt_a > set_Pr8687807964536448157ctxt_a ).

thf(sy_c_Transitive__Closure_Otrancl_001t__List__Olist_It__Nat__Onat_J,type,
    transi5465639037617667199st_nat: set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_Transitive__Closure_Otrancl_001t__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,
    transi1440150323567932558at_nat: set_Pr4087777274317423175at_nat > set_Pr4087777274317423175at_nat ).

thf(sy_c_Transitive__Closure_Otrancl_001t__List__Olist_Itf__b_J,type,
    transi1571129030638676774list_b: set_Pr7665282455119567943list_b > set_Pr7665282455119567943list_b ).

thf(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
    transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    transi3146596849628481425at_nat: set_Pr7047708467701517959at_nat > set_Pr7047708467701517959at_nat ).

thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    transi243908449541399842at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).

thf(sy_c_Transitive__Closure_Otrancl_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,
    transi6148169128111658760at_nat: set_Pr553994874890374343at_nat > set_Pr553994874890374343at_nat ).

thf(sy_c_Transitive__Closure_Otrancl_001tf__b,type,
    transitive_trancl_b: set_Product_prod_b_b > set_Product_prod_b_b ).

thf(sy_c_Utils_Oconcat__index__split_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    concat7860870432694265376ctxt_a: product_prod_nat_nat > list_l1578326660578982846ctxt_a > product_prod_nat_nat ).

thf(sy_c_Utils_Oconcat__index__split_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    concat6198194937734591280ctxt_a: product_prod_nat_nat > list_l3687490334575670478ctxt_a > product_prod_nat_nat ).

thf(sy_c_Utils_Oconcat__index__split_001t__List__Olist_It__Nat__Onat_J,type,
    concat5120376260311099874st_nat: product_prod_nat_nat > list_list_list_nat > product_prod_nat_nat ).

thf(sy_c_Utils_Oconcat__index__split_001t__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,
    concat2944144588778160107at_nat: product_prod_nat_nat > list_l5129023048960694521at_nat > product_prod_nat_nat ).

thf(sy_c_Utils_Oconcat__index__split_001t__List__Olist_Itf__b_J,type,
    concat7499549088139563139list_b: product_prod_nat_nat > list_list_list_b > product_prod_nat_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_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,
    concat1027118156116784357at_nat: product_prod_nat_nat > list_l3298181151656792051at_nat > product_prod_nat_nat ).

thf(sy_c_Utils_Oconcat__index__split_001tf__b,type,
    concat_index_split_b: product_prod_nat_nat > list_list_b > product_prod_nat_nat ).

thf(sy_c_Utils_Otrancl__list_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    trancl6028151697025156246ctxt_a: set_Pr8705490787052945405ctxt_a > set_Pr8687807964536448157ctxt_a ).

thf(sy_c_Utils_Otrancl__list_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    trancl5060458363496439462ctxt_a: set_Pr8687807964536448157ctxt_a > set_Pr8517441760511179837ctxt_a ).

thf(sy_c_Utils_Otrancl__list_001t__List__Olist_It__Nat__Onat_J,type,
    trancl_list_list_nat: set_Pr3451248702717554689st_nat > set_Pr1190453367779242145st_nat ).

thf(sy_c_Utils_Otrancl__list_001t__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,
    trancl8580566090571148789at_nat: set_Pr4087777274317423175at_nat > set_Pr3695809186315947463at_nat ).

thf(sy_c_Utils_Otrancl__list_001t__List__Olist_Itf__b_J,type,
    trancl_list_list_b: set_Pr7665282455119567943list_b > set_Pr2215042635423483847list_b ).

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

thf(sy_c_Utils_Otrancl__list_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    trancl3178887147240166537at_nat: set_Pr8693737435421807431at_nat > set_Pr1542805901266377927at_nat ).

thf(sy_c_Utils_Otrancl__list_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,
    trancl3412141547504709615at_nat: set_Pr553994874890374343at_nat > set_Pr4087777274317423175at_nat ).

thf(sy_c_Utils_Otrancl__list_001tf__b,type,
    trancl_list_b: set_Product_prod_b_b > set_Pr7665282455119567943list_b ).

thf(sy_c_Utils_Otrancl__listp_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    trancl3815965565430463256ctxt_a: ( ground_gmctxt_a > ground_gmctxt_a > $o ) > list_Ground_gmctxt_a > list_Ground_gmctxt_a > $o ).

thf(sy_c_Utils_Otrancl__listp_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    trancl4044344332876950568ctxt_a: ( list_Ground_gmctxt_a > list_Ground_gmctxt_a > $o ) > list_l1578326660578982846ctxt_a > list_l1578326660578982846ctxt_a > $o ).

thf(sy_c_Utils_Otrancl__listp_001t__List__Olist_It__Nat__Onat_J,type,
    trancl267208031524270810st_nat: ( list_nat > list_nat > $o ) > list_list_nat > list_list_nat > $o ).

thf(sy_c_Utils_Otrancl__listp_001t__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,
    trancl6797153713963726067at_nat: ( list_P8469869581646625389at_nat > list_P8469869581646625389at_nat > $o ) > list_l3298181151656792051at_nat > list_l3298181151656792051at_nat > $o ).

thf(sy_c_Utils_Otrancl__listp_001t__List__Olist_Itf__b_J,type,
    trancl_listp_list_b: ( list_b > list_b > $o ) > list_list_b > list_list_b > $o ).

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

thf(sy_c_Utils_Otrancl__listp_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,
    trancl2313143553896500717at_nat: ( produc859450856879609959at_nat > produc859450856879609959at_nat > $o ) > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat > $o ).

thf(sy_c_Utils_Otrancl__listp_001tf__b,type,
    trancl_listp_b: ( b > b > $o ) > list_b > list_b > $o ).

thf(sy_c_member_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    member3306386364193653941ctxt_a: ground_gmctxt_a > set_Ground_gmctxt_a > $o ).

thf(sy_c_member_001t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    member2714729351964853061ctxt_a: list_Ground_gmctxt_a > set_li3075106635084045668ctxt_a > $o ).

thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__b_J_J,type,
    member_list_list_b: list_list_b > set_list_list_b > $o ).

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

thf(sy_c_member_001t__List__Olist_It__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,
    member3799944675974059798at_nat: list_P8469869581646625389at_nat > set_li3197816953174176717at_nat > $o ).

thf(sy_c_member_001t__List__Olist_Itf__b_J,type,
    member_list_b: list_b > set_list_b > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    member461722496487281478ctxt_a: produc8175205341597284509ctxt_a > set_Pr8705490787052945405ctxt_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Nat__Onat_J,type,
    member1226896217267473912_a_nat: produc8497099580089875791_a_nat > set_Pr9093710254965148079_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mtf__b_J,type,
    member8304548504578704877xt_a_b: produc7780146698139795030xt_a_b > set_Pr7985508066815312396xt_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_Mt__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
    member6597889099603620838ctxt_a: produc1017277696136915517ctxt_a > set_Pr8687807964536448157ctxt_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member9152701222192952216st_nat: produc4140329211374400495st_nat > set_Pr8511038139508077391st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_Mt__List__Olist_Itf__b_J_J,type,
    member8646710124843958371list_b: produc3551300527538202572list_b > set_Pr1959420730066914690list_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J_Mt__List__Olist_It__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J_J,type,
    member6512746712182228870ctxt_a: produc2442989038347685597ctxt_a > set_Pr8517441760511179837ctxt_a > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member2583041412010425562st_nat: produc3811483368811048881st_nat > set_Pr5860164671742971281st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_Mt__List__Olist_Itf__b_J_J,type,
    member2634019950565409697list_b: produc6204733625450272906list_b > set_Pr5178944948679147456list_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__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_J_Mt__List__Olist_It__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_J_J,type,
    member6399385897798497296at_nat: produc3451831593474659303at_nat > set_Pr3695809186315947463at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__b_J_J_Mt__List__Olist_It__List__Olist_Itf__b_J_J_J,type,
    member7435197128963948048list_b: produc4603089062067288551list_b > set_Pr2215042635423483847list_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
    member5348271269840303512ctxt_a: produc335899259021751791ctxt_a > set_Pr302638903384814927ctxt_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__List__Olist_Itf__b_J_J_J,type,
    member4916451157736907575list_b: produc8487164832621770784list_b > set_Pr6009741699831149654list_b > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_Itf__b_J_J,type,
    member6003184433318419249list_b: produc1584444791027398810list_b > set_Pr4901976648650938704list_b > $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_Mtf__b_J,type,
    member6908590377518958251_nat_b: produc6459537403673193620_nat_b > set_Pr8611280865567961546_nat_b > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__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_J,type,
    member4574794575480667280at_nat: produc289266582803401575at_nat > set_Pr4087777274317423175at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
    member1748955351646886521ctxt_a: produc5876917791195906530ctxt_a > set_Pr2972090421851356312ctxt_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
    member7987477470964863163st_nat: produc2334819108994950564st_nat > set_Pr7387263332798406618st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__List__Olist_It__List__Olist_Itf__b_J_J_J,type,
    member60363355953214230list_b: produc3177755504865379949list_b > set_Pr3668070192038973389list_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member5518038250373942315st_nat: produc1099298608082921876st_nat > set_Pr1291562142433127754st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__List__Olist_Itf__b_J_J,type,
    member2990321877988238992list_b: produc3963297410138542439list_b > set_Pr7665282455119567943list_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    member2885886537299685624ctxt_a: produc932717863267311695ctxt_a > set_Pr1619835084168866991ctxt_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
    member1153228084381711240ctxt_a: produc4999762543018873695ctxt_a > set_Pr1203255704990185791ctxt_a > $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__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_J,type,
    member7629795747344909993at_nat: produc1035334568315550354at_nat > set_Pr5302226732920531656at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J,type,
    member8261005420521984321list_b: produc7811952446676219690list_b > set_Pr1349601357184307552list_b > $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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member5754182594613576739at_nat: produc2487518378626728076at_nat > set_Pr938417207581201730at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J,type,
    member8962352056413324475_nat_b: product_prod_nat_b > set_Pr4264375888882495962_nat_b > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__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,
    member8062223511168850704at_nat: produc349518998152878311at_nat > set_Pr553994874890374343at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    member1453991781301212003ctxt_a: produc929589974862302156ctxt_a > set_Pr5732450166000137346ctxt_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member1117040998421412389st_nat: produc667988024575647758st_nat > set_Pr6729162334058022980st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J,type,
    member6959632917342813205_b_nat: product_prod_b_nat > set_Pr1307281990691478580_b_nat > $o ).

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

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

thf(sy_v_Cs,type,
    cs: list_Ground_gmctxt_a ).

thf(sy_v_thesis,type,
    thesis: $o ).

thf(sy_v_x,type,
    x: b ).

thf(sy_v_xs,type,
    xs: list_b ).

% Relevant facts (1253)
thf(fact_0_assms_I2_J,axiom,
    member_b @ x @ ( set_b2 @ xs ) ).

% assms(2)
thf(fact_1_assms_I1_J,axiom,
    ( ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ cs ) )
    = ( size_size_list_b @ xs ) ) ).

% assms(1)
thf(fact_2_that,axiom,
    ! [I: nat] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ cs ) )
     => ( ( member_b @ x @ ( set_b2 @ ( nth_list_b @ ( multih3537974503344769663n_by_b @ xs @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ cs ) ) @ I ) ) )
       => thesis ) ) ).

% that
thf(fact_3_nth__map,axiom,
    ! [N: nat,Xs: list_b,F: b > nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_b @ Xs ) )
     => ( ( nth_nat @ ( map_b_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_b @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_4_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F: nat > nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( map_nat_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_5_nth__map,axiom,
    ! [N: nat,Xs: list_list_b,F: list_b > nat] :
      ( ( ord_less_nat @ N @ ( size_s420531428170919671list_b @ Xs ) )
     => ( ( nth_nat @ ( map_list_b_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_list_b @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_6_nth__map,axiom,
    ! [N: nat,Xs: list_Ground_gmctxt_a,F: ground_gmctxt_a > nat] :
      ( ( ord_less_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( nth_nat @ ( map_Gr1679544123147727284_a_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_Ground_gmctxt_a @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_7_nth__map,axiom,
    ! [N: nat,Xs: list_b,F: b > list_b] :
      ( ( ord_less_nat @ N @ ( size_size_list_b @ Xs ) )
     => ( ( nth_list_b @ ( map_b_list_b @ F @ Xs ) @ N )
        = ( F @ ( nth_b @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_8_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F: nat > list_b] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_list_b @ ( map_nat_list_b @ F @ Xs ) @ N )
        = ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_9_nth__map,axiom,
    ! [N: nat,Xs: list_list_b,F: list_b > list_b] :
      ( ( ord_less_nat @ N @ ( size_s420531428170919671list_b @ Xs ) )
     => ( ( nth_list_b @ ( map_list_b_list_b @ F @ Xs ) @ N )
        = ( F @ ( nth_list_b @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_10_nth__map,axiom,
    ! [N: nat,Xs: list_Ground_gmctxt_a,F: ground_gmctxt_a > list_b] :
      ( ( ord_less_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( nth_list_b @ ( map_Gr5773545835744863265list_b @ F @ Xs ) @ N )
        = ( F @ ( nth_Ground_gmctxt_a @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_11_nth__map,axiom,
    ! [N: nat,Xs: list_b,F: b > b] :
      ( ( ord_less_nat @ N @ ( size_size_list_b @ Xs ) )
     => ( ( nth_b @ ( map_b_b @ F @ Xs ) @ N )
        = ( F @ ( nth_b @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_12_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F: nat > b] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_b @ ( map_nat_b @ F @ Xs ) @ N )
        = ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_13_partition__gholes__subseteq,axiom,
    ! [Cs: list_Ground_gmctxt_a,Xs: list_list_b,I: nat,X: list_b] :
      ( ( ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs ) )
        = ( size_s420531428170919671list_b @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Cs ) )
       => ( ( member_list_b @ X @ ( set_list_b2 @ ( nth_list_list_b @ ( multih2267195492336891525list_b @ Xs @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs ) ) @ I ) ) )
         => ( member_list_b @ X @ ( set_list_b2 @ Xs ) ) ) ) ) ).

% partition_gholes_subseteq
thf(fact_14_partition__gholes__subseteq,axiom,
    ! [Cs: list_Ground_gmctxt_a,Xs: list_P8469869581646625389at_nat,I: nat,X: produc859450856879609959at_nat] :
      ( ( ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs ) )
        = ( size_s3679842834875189465at_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Cs ) )
       => ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ ( nth_li8805938130461222772at_nat @ ( multih7555706055023738855at_nat @ Xs @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs ) ) @ I ) ) )
         => ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs ) ) ) ) ) ).

% partition_gholes_subseteq
thf(fact_15_partition__gholes__subseteq,axiom,
    ! [Cs: list_Ground_gmctxt_a,Xs: list_Ground_gmctxt_a,I: nat,X: ground_gmctxt_a] :
      ( ( ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs ) )
        = ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Cs ) )
       => ( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ ( nth_li6678718213161987943ctxt_a @ ( multih5172578161653122718ctxt_a @ Xs @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs ) ) @ I ) ) )
         => ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) ) ) ) ) ).

% partition_gholes_subseteq
thf(fact_16_partition__gholes__subseteq,axiom,
    ! [Cs: list_Ground_gmctxt_a,Xs: list_nat,I: nat,X: nat] :
      ( ( ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs ) )
        = ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Cs ) )
       => ( ( member_nat @ X @ ( set_nat2 @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs ) ) @ I ) ) )
         => ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ) ) ).

% partition_gholes_subseteq
thf(fact_17_partition__gholes__subseteq,axiom,
    ! [Cs: list_Ground_gmctxt_a,Xs: list_b,I: nat,X: b] :
      ( ( ( groups4561878855575611511st_nat @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs ) )
        = ( size_size_list_b @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Cs ) )
       => ( ( member_b @ X @ ( set_b2 @ ( nth_list_b @ ( multih3537974503344769663n_by_b @ Xs @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ Cs ) ) @ I ) ) )
         => ( member_b @ X @ ( set_b2 @ Xs ) ) ) ) ) ).

% partition_gholes_subseteq
thf(fact_18_map__eq__conv,axiom,
    ! [F: b > ground_gmctxt_a,Xs: list_b,G: b > ground_gmctxt_a] :
      ( ( ( map_b_3587768286370285017ctxt_a @ F @ Xs )
        = ( map_b_3587768286370285017ctxt_a @ G @ Xs ) )
      = ( ! [X2: b] :
            ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) ) ) ) ).

% map_eq_conv
thf(fact_19_map__eq__conv,axiom,
    ! [F: b > list_b,Xs: list_b,G: b > list_b] :
      ( ( ( map_b_list_b @ F @ Xs )
        = ( map_b_list_b @ G @ Xs ) )
      = ( ! [X2: b] :
            ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) ) ) ) ).

% map_eq_conv
thf(fact_20_map__eq__conv,axiom,
    ! [F: b > nat,Xs: list_b,G: b > nat] :
      ( ( ( map_b_nat @ F @ Xs )
        = ( map_b_nat @ G @ Xs ) )
      = ( ! [X2: b] :
            ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) ) ) ) ).

% map_eq_conv
thf(fact_21_map__eq__conv,axiom,
    ! [F: b > b,Xs: list_b,G: b > b] :
      ( ( ( map_b_b @ F @ Xs )
        = ( map_b_b @ G @ Xs ) )
      = ( ! [X2: b] :
            ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) ) ) ) ).

% map_eq_conv
thf(fact_22_map__eq__conv,axiom,
    ! [F: nat > b,Xs: list_nat,G: nat > b] :
      ( ( ( map_nat_b @ F @ Xs )
        = ( map_nat_b @ G @ Xs ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) ) ) ) ).

% map_eq_conv
thf(fact_23_map__eq__conv,axiom,
    ! [F: ground_gmctxt_a > nat,Xs: list_Ground_gmctxt_a,G: ground_gmctxt_a > nat] :
      ( ( ( map_Gr1679544123147727284_a_nat @ F @ Xs )
        = ( map_Gr1679544123147727284_a_nat @ G @ Xs ) )
      = ( ! [X2: ground_gmctxt_a] :
            ( ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ Xs ) )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) ) ) ) ).

% map_eq_conv
thf(fact_24_length__map,axiom,
    ! [F: b > b,Xs: list_b] :
      ( ( size_size_list_b @ ( map_b_b @ F @ Xs ) )
      = ( size_size_list_b @ Xs ) ) ).

% length_map
thf(fact_25_length__map,axiom,
    ! [F: nat > b,Xs: list_nat] :
      ( ( size_size_list_b @ ( map_nat_b @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_26_length__map,axiom,
    ! [F: b > nat,Xs: list_b] :
      ( ( size_size_list_nat @ ( map_b_nat @ F @ Xs ) )
      = ( size_size_list_b @ Xs ) ) ).

% length_map
thf(fact_27_length__map,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_28_length__map,axiom,
    ! [F: b > ground_gmctxt_a,Xs: list_b] :
      ( ( size_s962548663205356226ctxt_a @ ( map_b_3587768286370285017ctxt_a @ F @ Xs ) )
      = ( size_size_list_b @ Xs ) ) ).

% length_map
thf(fact_29_length__map,axiom,
    ! [F: nat > ground_gmctxt_a,Xs: list_nat] :
      ( ( size_s962548663205356226ctxt_a @ ( map_na2753241459123782580ctxt_a @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_30_length__map,axiom,
    ! [F: ground_gmctxt_a > b,Xs: list_Ground_gmctxt_a] :
      ( ( size_size_list_b @ ( map_Gr2594013990000539163xt_a_b @ F @ Xs ) )
      = ( size_s962548663205356226ctxt_a @ Xs ) ) ).

% length_map
thf(fact_31_length__map,axiom,
    ! [F: ground_gmctxt_a > nat,Xs: list_Ground_gmctxt_a] :
      ( ( size_size_list_nat @ ( map_Gr1679544123147727284_a_nat @ F @ Xs ) )
      = ( size_s962548663205356226ctxt_a @ Xs ) ) ).

% length_map
thf(fact_32_length__map,axiom,
    ! [F: ground_gmctxt_a > ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
      ( ( size_s962548663205356226ctxt_a @ ( map_Gr980675717119874050ctxt_a @ F @ Xs ) )
      = ( size_s962548663205356226ctxt_a @ Xs ) ) ).

% length_map
thf(fact_33_length__map,axiom,
    ! [F: list_b > b,Xs: list_list_b] :
      ( ( size_size_list_b @ ( map_list_b_b @ F @ Xs ) )
      = ( size_s420531428170919671list_b @ Xs ) ) ).

% length_map
thf(fact_34_nth__map__conv,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_b,F: ground_gmctxt_a > nat,G: b > nat] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
           => ( ( F @ ( nth_Ground_gmctxt_a @ Xs @ I2 ) )
              = ( G @ ( nth_b @ Ys @ I2 ) ) ) )
       => ( ( map_Gr1679544123147727284_a_nat @ F @ Xs )
          = ( map_b_nat @ G @ Ys ) ) ) ) ).

% nth_map_conv
thf(fact_35_nth__map__conv,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_nat,F: ground_gmctxt_a > nat,G: nat > nat] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
           => ( ( F @ ( nth_Ground_gmctxt_a @ Xs @ I2 ) )
              = ( G @ ( nth_nat @ Ys @ I2 ) ) ) )
       => ( ( map_Gr1679544123147727284_a_nat @ F @ Xs )
          = ( map_nat_nat @ G @ Ys ) ) ) ) ).

% nth_map_conv
thf(fact_36_nth__map__conv,axiom,
    ! [Xs: list_b,Ys: list_Ground_gmctxt_a,F: b > nat,G: ground_gmctxt_a > nat] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Xs ) )
           => ( ( F @ ( nth_b @ Xs @ I2 ) )
              = ( G @ ( nth_Ground_gmctxt_a @ Ys @ I2 ) ) ) )
       => ( ( map_b_nat @ F @ Xs )
          = ( map_Gr1679544123147727284_a_nat @ G @ Ys ) ) ) ) ).

% nth_map_conv
thf(fact_37_nth__map__conv,axiom,
    ! [Xs: list_nat,Ys: list_Ground_gmctxt_a,F: nat > nat,G: ground_gmctxt_a > nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
           => ( ( F @ ( nth_nat @ Xs @ I2 ) )
              = ( G @ ( nth_Ground_gmctxt_a @ Ys @ I2 ) ) ) )
       => ( ( map_nat_nat @ F @ Xs )
          = ( map_Gr1679544123147727284_a_nat @ G @ Ys ) ) ) ) ).

% nth_map_conv
thf(fact_38_nth__map__conv,axiom,
    ! [Xs: list_list_b,Ys: list_Ground_gmctxt_a,F: list_b > nat,G: ground_gmctxt_a > nat] :
      ( ( ( size_s420531428170919671list_b @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Xs ) )
           => ( ( F @ ( nth_list_b @ Xs @ I2 ) )
              = ( G @ ( nth_Ground_gmctxt_a @ Ys @ I2 ) ) ) )
       => ( ( map_list_b_nat @ F @ Xs )
          = ( map_Gr1679544123147727284_a_nat @ G @ Ys ) ) ) ) ).

% nth_map_conv
thf(fact_39_nth__map__conv,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_list_b,F: ground_gmctxt_a > nat,G: list_b > nat] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_s420531428170919671list_b @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
           => ( ( F @ ( nth_Ground_gmctxt_a @ Xs @ I2 ) )
              = ( G @ ( nth_list_b @ Ys @ I2 ) ) ) )
       => ( ( map_Gr1679544123147727284_a_nat @ F @ Xs )
          = ( map_list_b_nat @ G @ Ys ) ) ) ) ).

% nth_map_conv
thf(fact_40_nth__map__conv,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,F: ground_gmctxt_a > nat,G: ground_gmctxt_a > nat] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
           => ( ( F @ ( nth_Ground_gmctxt_a @ Xs @ I2 ) )
              = ( G @ ( nth_Ground_gmctxt_a @ Ys @ I2 ) ) ) )
       => ( ( map_Gr1679544123147727284_a_nat @ F @ Xs )
          = ( map_Gr1679544123147727284_a_nat @ G @ Ys ) ) ) ) ).

% nth_map_conv
thf(fact_41_nth__map__conv,axiom,
    ! [Xs: list_b,Ys: list_b,F: b > nat,G: b > nat] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Xs ) )
           => ( ( F @ ( nth_b @ Xs @ I2 ) )
              = ( G @ ( nth_b @ Ys @ I2 ) ) ) )
       => ( ( map_b_nat @ F @ Xs )
          = ( map_b_nat @ G @ Ys ) ) ) ) ).

% nth_map_conv
thf(fact_42_nth__map__conv,axiom,
    ! [Xs: list_b,Ys: list_b,F: b > b,G: b > b] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Xs ) )
           => ( ( F @ ( nth_b @ Xs @ I2 ) )
              = ( G @ ( nth_b @ Ys @ I2 ) ) ) )
       => ( ( map_b_b @ F @ Xs )
          = ( map_b_b @ G @ Ys ) ) ) ) ).

% nth_map_conv
thf(fact_43_nth__map__conv,axiom,
    ! [Xs: list_b,Ys: list_nat,F: b > nat,G: nat > nat] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Xs ) )
           => ( ( F @ ( nth_b @ Xs @ I2 ) )
              = ( G @ ( nth_nat @ Ys @ I2 ) ) ) )
       => ( ( map_b_nat @ F @ Xs )
          = ( map_nat_nat @ G @ Ys ) ) ) ) ).

% nth_map_conv
thf(fact_44_map__eq__conv_H,axiom,
    ! [F: ground_gmctxt_a > nat,Xs: list_Ground_gmctxt_a,G: b > nat,Ys: list_b] :
      ( ( ( map_Gr1679544123147727284_a_nat @ F @ Xs )
        = ( map_b_nat @ G @ Ys ) )
      = ( ( ( size_s962548663205356226ctxt_a @ Xs )
          = ( size_size_list_b @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Xs ) )
           => ( ( F @ ( nth_Ground_gmctxt_a @ Xs @ I3 ) )
              = ( G @ ( nth_b @ Ys @ I3 ) ) ) ) ) ) ).

% map_eq_conv'
thf(fact_45_map__eq__conv_H,axiom,
    ! [F: ground_gmctxt_a > nat,Xs: list_Ground_gmctxt_a,G: nat > nat,Ys: list_nat] :
      ( ( ( map_Gr1679544123147727284_a_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
      = ( ( ( size_s962548663205356226ctxt_a @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Xs ) )
           => ( ( F @ ( nth_Ground_gmctxt_a @ Xs @ I3 ) )
              = ( G @ ( nth_nat @ Ys @ I3 ) ) ) ) ) ) ).

% map_eq_conv'
thf(fact_46_map__eq__conv_H,axiom,
    ! [F: b > nat,Xs: list_b,G: ground_gmctxt_a > nat,Ys: list_Ground_gmctxt_a] :
      ( ( ( map_b_nat @ F @ Xs )
        = ( map_Gr1679544123147727284_a_nat @ G @ Ys ) )
      = ( ( ( size_size_list_b @ Xs )
          = ( size_s962548663205356226ctxt_a @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Xs ) )
           => ( ( F @ ( nth_b @ Xs @ I3 ) )
              = ( G @ ( nth_Ground_gmctxt_a @ Ys @ I3 ) ) ) ) ) ) ).

% map_eq_conv'
thf(fact_47_map__eq__conv_H,axiom,
    ! [F: nat > nat,Xs: list_nat,G: ground_gmctxt_a > nat,Ys: list_Ground_gmctxt_a] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_Gr1679544123147727284_a_nat @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_s962548663205356226ctxt_a @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
           => ( ( F @ ( nth_nat @ Xs @ I3 ) )
              = ( G @ ( nth_Ground_gmctxt_a @ Ys @ I3 ) ) ) ) ) ) ).

% map_eq_conv'
thf(fact_48_map__eq__conv_H,axiom,
    ! [F: list_b > nat,Xs: list_list_b,G: ground_gmctxt_a > nat,Ys: list_Ground_gmctxt_a] :
      ( ( ( map_list_b_nat @ F @ Xs )
        = ( map_Gr1679544123147727284_a_nat @ G @ Ys ) )
      = ( ( ( size_s420531428170919671list_b @ Xs )
          = ( size_s962548663205356226ctxt_a @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s420531428170919671list_b @ Xs ) )
           => ( ( F @ ( nth_list_b @ Xs @ I3 ) )
              = ( G @ ( nth_Ground_gmctxt_a @ Ys @ I3 ) ) ) ) ) ) ).

% map_eq_conv'
thf(fact_49_map__eq__conv_H,axiom,
    ! [F: ground_gmctxt_a > nat,Xs: list_Ground_gmctxt_a,G: list_b > nat,Ys: list_list_b] :
      ( ( ( map_Gr1679544123147727284_a_nat @ F @ Xs )
        = ( map_list_b_nat @ G @ Ys ) )
      = ( ( ( size_s962548663205356226ctxt_a @ Xs )
          = ( size_s420531428170919671list_b @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Xs ) )
           => ( ( F @ ( nth_Ground_gmctxt_a @ Xs @ I3 ) )
              = ( G @ ( nth_list_b @ Ys @ I3 ) ) ) ) ) ) ).

% map_eq_conv'
thf(fact_50_map__eq__conv_H,axiom,
    ! [F: ground_gmctxt_a > nat,Xs: list_Ground_gmctxt_a,G: ground_gmctxt_a > nat,Ys: list_Ground_gmctxt_a] :
      ( ( ( map_Gr1679544123147727284_a_nat @ F @ Xs )
        = ( map_Gr1679544123147727284_a_nat @ G @ Ys ) )
      = ( ( ( size_s962548663205356226ctxt_a @ Xs )
          = ( size_s962548663205356226ctxt_a @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Xs ) )
           => ( ( F @ ( nth_Ground_gmctxt_a @ Xs @ I3 ) )
              = ( G @ ( nth_Ground_gmctxt_a @ Ys @ I3 ) ) ) ) ) ) ).

% map_eq_conv'
thf(fact_51_map__eq__conv_H,axiom,
    ! [F: b > nat,Xs: list_b,G: b > nat,Ys: list_b] :
      ( ( ( map_b_nat @ F @ Xs )
        = ( map_b_nat @ G @ Ys ) )
      = ( ( ( size_size_list_b @ Xs )
          = ( size_size_list_b @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Xs ) )
           => ( ( F @ ( nth_b @ Xs @ I3 ) )
              = ( G @ ( nth_b @ Ys @ I3 ) ) ) ) ) ) ).

% map_eq_conv'
thf(fact_52_map__eq__conv_H,axiom,
    ! [F: b > b,Xs: list_b,G: b > b,Ys: list_b] :
      ( ( ( map_b_b @ F @ Xs )
        = ( map_b_b @ G @ Ys ) )
      = ( ( ( size_size_list_b @ Xs )
          = ( size_size_list_b @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Xs ) )
           => ( ( F @ ( nth_b @ Xs @ I3 ) )
              = ( G @ ( nth_b @ Ys @ I3 ) ) ) ) ) ) ).

% map_eq_conv'
thf(fact_53_map__eq__conv_H,axiom,
    ! [F: b > nat,Xs: list_b,G: nat > nat,Ys: list_nat] :
      ( ( ( map_b_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
      = ( ( ( size_size_list_b @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Xs ) )
           => ( ( F @ ( nth_b @ Xs @ I3 ) )
              = ( G @ ( nth_nat @ Ys @ I3 ) ) ) ) ) ) ).

% map_eq_conv'
thf(fact_54_map__nth__conv,axiom,
    ! [F: ground_gmctxt_a > nat,Ss: list_Ground_gmctxt_a,G: nat > nat,Ts: list_nat] :
      ( ( ( map_Gr1679544123147727284_a_nat @ F @ Ss )
        = ( map_nat_nat @ G @ Ts ) )
     => ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Ss ) )
         => ( ( F @ ( nth_Ground_gmctxt_a @ Ss @ I4 ) )
            = ( G @ ( nth_nat @ Ts @ I4 ) ) ) ) ) ).

% map_nth_conv
thf(fact_55_map__nth__conv,axiom,
    ! [F: b > nat,Ss: list_b,G: ground_gmctxt_a > nat,Ts: list_Ground_gmctxt_a] :
      ( ( ( map_b_nat @ F @ Ss )
        = ( map_Gr1679544123147727284_a_nat @ G @ Ts ) )
     => ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_b @ Ss ) )
         => ( ( F @ ( nth_b @ Ss @ I4 ) )
            = ( G @ ( nth_Ground_gmctxt_a @ Ts @ I4 ) ) ) ) ) ).

% map_nth_conv
thf(fact_56_map__nth__conv,axiom,
    ! [F: nat > nat,Ss: list_nat,G: ground_gmctxt_a > nat,Ts: list_Ground_gmctxt_a] :
      ( ( ( map_nat_nat @ F @ Ss )
        = ( map_Gr1679544123147727284_a_nat @ G @ Ts ) )
     => ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ss ) )
         => ( ( F @ ( nth_nat @ Ss @ I4 ) )
            = ( G @ ( nth_Ground_gmctxt_a @ Ts @ I4 ) ) ) ) ) ).

% map_nth_conv
thf(fact_57_map__nth__conv,axiom,
    ! [F: list_b > nat,Ss: list_list_b,G: ground_gmctxt_a > nat,Ts: list_Ground_gmctxt_a] :
      ( ( ( map_list_b_nat @ F @ Ss )
        = ( map_Gr1679544123147727284_a_nat @ G @ Ts ) )
     => ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_s420531428170919671list_b @ Ss ) )
         => ( ( F @ ( nth_list_b @ Ss @ I4 ) )
            = ( G @ ( nth_Ground_gmctxt_a @ Ts @ I4 ) ) ) ) ) ).

% map_nth_conv
thf(fact_58_map__nth__conv,axiom,
    ! [F: ground_gmctxt_a > nat,Ss: list_Ground_gmctxt_a,G: list_b > nat,Ts: list_list_b] :
      ( ( ( map_Gr1679544123147727284_a_nat @ F @ Ss )
        = ( map_list_b_nat @ G @ Ts ) )
     => ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Ss ) )
         => ( ( F @ ( nth_Ground_gmctxt_a @ Ss @ I4 ) )
            = ( G @ ( nth_list_b @ Ts @ I4 ) ) ) ) ) ).

% map_nth_conv
thf(fact_59_map__nth__conv,axiom,
    ! [F: ground_gmctxt_a > nat,Ss: list_Ground_gmctxt_a,G: ground_gmctxt_a > nat,Ts: list_Ground_gmctxt_a] :
      ( ( ( map_Gr1679544123147727284_a_nat @ F @ Ss )
        = ( map_Gr1679544123147727284_a_nat @ G @ Ts ) )
     => ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Ss ) )
         => ( ( F @ ( nth_Ground_gmctxt_a @ Ss @ I4 ) )
            = ( G @ ( nth_Ground_gmctxt_a @ Ts @ I4 ) ) ) ) ) ).

% map_nth_conv
thf(fact_60_map__nth__conv,axiom,
    ! [F: b > nat,Ss: list_b,G: nat > nat,Ts: list_nat] :
      ( ( ( map_b_nat @ F @ Ss )
        = ( map_nat_nat @ G @ Ts ) )
     => ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_b @ Ss ) )
         => ( ( F @ ( nth_b @ Ss @ I4 ) )
            = ( G @ ( nth_nat @ Ts @ I4 ) ) ) ) ) ).

% map_nth_conv
thf(fact_61_map__nth__conv,axiom,
    ! [F: b > nat,Ss: list_b,G: b > nat,Ts: list_b] :
      ( ( ( map_b_nat @ F @ Ss )
        = ( map_b_nat @ G @ Ts ) )
     => ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_b @ Ss ) )
         => ( ( F @ ( nth_b @ Ss @ I4 ) )
            = ( G @ ( nth_b @ Ts @ I4 ) ) ) ) ) ).

% map_nth_conv
thf(fact_62_map__nth__conv,axiom,
    ! [F: b > b,Ss: list_b,G: nat > b,Ts: list_nat] :
      ( ( ( map_b_b @ F @ Ss )
        = ( map_nat_b @ G @ Ts ) )
     => ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_b @ Ss ) )
         => ( ( F @ ( nth_b @ Ss @ I4 ) )
            = ( G @ ( nth_nat @ Ts @ I4 ) ) ) ) ) ).

% map_nth_conv
thf(fact_63_map__nth__conv,axiom,
    ! [F: b > b,Ss: list_b,G: b > b,Ts: list_b] :
      ( ( ( map_b_b @ F @ Ss )
        = ( map_b_b @ G @ Ts ) )
     => ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_b @ Ss ) )
         => ( ( F @ ( nth_b @ Ss @ I4 ) )
            = ( G @ ( nth_b @ Ts @ I4 ) ) ) ) ) ).

% map_nth_conv
thf(fact_64_map__eq__nth__conv,axiom,
    ! [F: b > nat,Xs: list_b,G: b > nat,Ys: list_b] :
      ( ( ( map_b_nat @ F @ Xs )
        = ( map_b_nat @ G @ Ys ) )
      = ( ( ( size_size_list_b @ Xs )
          = ( size_size_list_b @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Ys ) )
           => ( ( F @ ( nth_b @ Xs @ I3 ) )
              = ( G @ ( nth_b @ Ys @ I3 ) ) ) ) ) ) ).

% map_eq_nth_conv
thf(fact_65_map__eq__nth__conv,axiom,
    ! [F: b > b,Xs: list_b,G: b > b,Ys: list_b] :
      ( ( ( map_b_b @ F @ Xs )
        = ( map_b_b @ G @ Ys ) )
      = ( ( ( size_size_list_b @ Xs )
          = ( size_size_list_b @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Ys ) )
           => ( ( F @ ( nth_b @ Xs @ I3 ) )
              = ( G @ ( nth_b @ Ys @ I3 ) ) ) ) ) ) ).

% map_eq_nth_conv
thf(fact_66_map__eq__nth__conv,axiom,
    ! [F: b > nat,Xs: list_b,G: nat > nat,Ys: list_nat] :
      ( ( ( map_b_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
      = ( ( ( size_size_list_b @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_b @ Xs @ I3 ) )
              = ( G @ ( nth_nat @ Ys @ I3 ) ) ) ) ) ) ).

% map_eq_nth_conv
thf(fact_67_map__eq__nth__conv,axiom,
    ! [F: b > b,Xs: list_b,G: nat > b,Ys: list_nat] :
      ( ( ( map_b_b @ F @ Xs )
        = ( map_nat_b @ G @ Ys ) )
      = ( ( ( size_size_list_b @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_b @ Xs @ I3 ) )
              = ( G @ ( nth_nat @ Ys @ I3 ) ) ) ) ) ) ).

% map_eq_nth_conv
thf(fact_68_map__eq__nth__conv,axiom,
    ! [F: nat > nat,Xs: list_nat,G: b > nat,Ys: list_b] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_b_nat @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_b @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I3 ) )
              = ( G @ ( nth_b @ Ys @ I3 ) ) ) ) ) ) ).

% map_eq_nth_conv
thf(fact_69_map__eq__nth__conv,axiom,
    ! [F: nat > b,Xs: list_nat,G: b > b,Ys: list_b] :
      ( ( ( map_nat_b @ F @ Xs )
        = ( map_b_b @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_b @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I3 ) )
              = ( G @ ( nth_b @ Ys @ I3 ) ) ) ) ) ) ).

% map_eq_nth_conv
thf(fact_70_map__eq__nth__conv,axiom,
    ! [F: nat > b,Xs: list_nat,G: nat > b,Ys: list_nat] :
      ( ( ( map_nat_b @ F @ Xs )
        = ( map_nat_b @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I3 ) )
              = ( G @ ( nth_nat @ Ys @ I3 ) ) ) ) ) ) ).

% map_eq_nth_conv
thf(fact_71_map__eq__nth__conv,axiom,
    ! [F: list_nat > nat,Xs: list_list_nat,G: b > nat,Ys: list_b] :
      ( ( ( map_list_nat_nat @ F @ Xs )
        = ( map_b_nat @ G @ Ys ) )
      = ( ( ( size_s3023201423986296836st_nat @ Xs )
          = ( size_size_list_b @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Ys ) )
           => ( ( F @ ( nth_list_nat @ Xs @ I3 ) )
              = ( G @ ( nth_b @ Ys @ I3 ) ) ) ) ) ) ).

% map_eq_nth_conv
thf(fact_72_map__eq__nth__conv,axiom,
    ! [F: list_nat > b,Xs: list_list_nat,G: b > b,Ys: list_b] :
      ( ( ( map_list_nat_b @ F @ Xs )
        = ( map_b_b @ G @ Ys ) )
      = ( ( ( size_s3023201423986296836st_nat @ Xs )
          = ( size_size_list_b @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Ys ) )
           => ( ( F @ ( nth_list_nat @ Xs @ I3 ) )
              = ( G @ ( nth_b @ Ys @ I3 ) ) ) ) ) ) ).

% map_eq_nth_conv
thf(fact_73_map__eq__nth__conv,axiom,
    ! [F: list_nat > b,Xs: list_list_nat,G: nat > b,Ys: list_nat] :
      ( ( ( map_list_nat_b @ F @ Xs )
        = ( map_nat_b @ G @ Ys ) )
      = ( ( ( size_s3023201423986296836st_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_list_nat @ Xs @ I3 ) )
              = ( G @ ( nth_nat @ Ys @ I3 ) ) ) ) ) ) ).

% map_eq_nth_conv
thf(fact_74_map__nth__eq__conv,axiom,
    ! [Xs: list_b,Ys: list_b,F: b > b] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( map_b_b @ F @ Xs )
          = Ys )
        = ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Ys ) )
             => ( ( F @ ( nth_b @ Xs @ I3 ) )
                = ( nth_b @ Ys @ I3 ) ) ) ) ) ) ).

% map_nth_eq_conv
thf(fact_75_map__nth__eq__conv,axiom,
    ! [Xs: list_b,Ys: list_nat,F: b > nat] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( map_b_nat @ F @ Xs )
          = Ys )
        = ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
             => ( ( F @ ( nth_b @ Xs @ I3 ) )
                = ( nth_nat @ Ys @ I3 ) ) ) ) ) ) ).

% map_nth_eq_conv
thf(fact_76_map__nth__eq__conv,axiom,
    ! [Xs: list_nat,Ys: list_b,F: nat > b] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( map_nat_b @ F @ Xs )
          = Ys )
        = ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Ys ) )
             => ( ( F @ ( nth_nat @ Xs @ I3 ) )
                = ( nth_b @ Ys @ I3 ) ) ) ) ) ) ).

% map_nth_eq_conv
thf(fact_77_map__nth__eq__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: nat > nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( map_nat_nat @ F @ Xs )
          = Ys )
        = ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
             => ( ( F @ ( nth_nat @ Xs @ I3 ) )
                = ( nth_nat @ Ys @ I3 ) ) ) ) ) ) ).

% map_nth_eq_conv
thf(fact_78_map__nth__eq__conv,axiom,
    ! [Xs: list_list_nat,Ys: list_b,F: list_nat > b] :
      ( ( ( size_s3023201423986296836st_nat @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( map_list_nat_b @ F @ Xs )
          = Ys )
        = ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Ys ) )
             => ( ( F @ ( nth_list_nat @ Xs @ I3 ) )
                = ( nth_b @ Ys @ I3 ) ) ) ) ) ) ).

% map_nth_eq_conv
thf(fact_79_map__nth__eq__conv,axiom,
    ! [Xs: list_list_nat,Ys: list_nat,F: list_nat > nat] :
      ( ( ( size_s3023201423986296836st_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( map_list_nat_nat @ F @ Xs )
          = Ys )
        = ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
             => ( ( F @ ( nth_list_nat @ Xs @ I3 ) )
                = ( nth_nat @ Ys @ I3 ) ) ) ) ) ) ).

% map_nth_eq_conv
thf(fact_80_map__nth__eq__conv,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_b,F: ground_gmctxt_a > b] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( map_Gr2594013990000539163xt_a_b @ F @ Xs )
          = Ys )
        = ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Ys ) )
             => ( ( F @ ( nth_Ground_gmctxt_a @ Xs @ I3 ) )
                = ( nth_b @ Ys @ I3 ) ) ) ) ) ) ).

% map_nth_eq_conv
thf(fact_81_map__nth__eq__conv,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_nat,F: ground_gmctxt_a > nat] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( map_Gr1679544123147727284_a_nat @ F @ Xs )
          = Ys )
        = ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
             => ( ( F @ ( nth_Ground_gmctxt_a @ Xs @ I3 ) )
                = ( nth_nat @ Ys @ I3 ) ) ) ) ) ) ).

% map_nth_eq_conv
thf(fact_82_map__nth__eq__conv,axiom,
    ! [Xs: list_b,Ys: list_list_nat,F: b > list_nat] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ( ( map_b_list_nat @ F @ Xs )
          = Ys )
        = ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Ys ) )
             => ( ( F @ ( nth_b @ Xs @ I3 ) )
                = ( nth_list_nat @ Ys @ I3 ) ) ) ) ) ) ).

% map_nth_eq_conv
thf(fact_83_map__nth__eq__conv,axiom,
    ! [Xs: list_b,Ys: list_Ground_gmctxt_a,F: b > ground_gmctxt_a] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) )
     => ( ( ( map_b_3587768286370285017ctxt_a @ F @ Xs )
          = Ys )
        = ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Ys ) )
             => ( ( F @ ( nth_b @ Xs @ I3 ) )
                = ( nth_Ground_gmctxt_a @ Ys @ I3 ) ) ) ) ) ) ).

% map_nth_eq_conv
thf(fact_84_nth__mem,axiom,
    ! [N: nat,Xs: list_list_nat] :
      ( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
     => ( member_list_nat @ ( nth_list_nat @ Xs @ N ) @ ( set_list_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_85_nth__mem,axiom,
    ! [N: nat,Xs: list_l1578326660578982846ctxt_a] :
      ( ( ord_less_nat @ N @ ( size_s6994882312647770194ctxt_a @ Xs ) )
     => ( member2714729351964853061ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ N ) @ ( set_li3057194059929822169ctxt_a @ Xs ) ) ) ).

% nth_mem
thf(fact_86_nth__mem,axiom,
    ! [N: nat,Xs: list_l3298181151656792051at_nat] :
      ( ( ord_less_nat @ N @ ( size_s6878757106947045471at_nat @ Xs ) )
     => ( member3799944675974059798at_nat @ ( nth_li8805938130461222772at_nat @ Xs @ N ) @ ( set_li7366234323595830146at_nat @ Xs ) ) ) ).

% nth_mem
thf(fact_87_nth__mem,axiom,
    ! [N: nat,Xs: list_Ground_gmctxt_a] :
      ( ( ord_less_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( member3306386364193653941ctxt_a @ ( nth_Ground_gmctxt_a @ Xs @ N ) @ ( set_Ground_gmctxt_a2 @ Xs ) ) ) ).

% nth_mem
thf(fact_88_nth__mem,axiom,
    ! [N: nat,Xs: list_b] :
      ( ( ord_less_nat @ N @ ( size_size_list_b @ Xs ) )
     => ( member_b @ ( nth_b @ Xs @ N ) @ ( set_b2 @ Xs ) ) ) ).

% nth_mem
thf(fact_89_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_90_nth__mem,axiom,
    ! [N: nat,Xs: list_list_b] :
      ( ( ord_less_nat @ N @ ( size_s420531428170919671list_b @ Xs ) )
     => ( member_list_b @ ( nth_list_b @ Xs @ N ) @ ( set_list_b2 @ Xs ) ) ) ).

% nth_mem
thf(fact_91_nth__mem,axiom,
    ! [N: nat,Xs: list_P8469869581646625389at_nat] :
      ( ( ord_less_nat @ N @ ( size_s3679842834875189465at_nat @ Xs ) )
     => ( member8206827879206165904at_nat @ ( nth_Pr6744343527793145070at_nat @ Xs @ N ) @ ( set_Pr5518436109238095868at_nat @ Xs ) ) ) ).

% nth_mem
thf(fact_92_list__ball__nth,axiom,
    ! [N: nat,Xs: list_list_nat,P: list_nat > $o] :
      ( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
     => ( ! [X3: list_nat] :
            ( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_list_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_93_list__ball__nth,axiom,
    ! [N: nat,Xs: list_l1578326660578982846ctxt_a,P: list_Ground_gmctxt_a > $o] :
      ( ( ord_less_nat @ N @ ( size_s6994882312647770194ctxt_a @ Xs ) )
     => ( ! [X3: list_Ground_gmctxt_a] :
            ( ( member2714729351964853061ctxt_a @ X3 @ ( set_li3057194059929822169ctxt_a @ Xs ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_li6678718213161987943ctxt_a @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_94_list__ball__nth,axiom,
    ! [N: nat,Xs: list_l3298181151656792051at_nat,P: list_P8469869581646625389at_nat > $o] :
      ( ( ord_less_nat @ N @ ( size_s6878757106947045471at_nat @ Xs ) )
     => ( ! [X3: list_P8469869581646625389at_nat] :
            ( ( member3799944675974059798at_nat @ X3 @ ( set_li7366234323595830146at_nat @ Xs ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_li8805938130461222772at_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_95_list__ball__nth,axiom,
    ! [N: nat,Xs: list_Ground_gmctxt_a,P: ground_gmctxt_a > $o] :
      ( ( ord_less_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ! [X3: ground_gmctxt_a] :
            ( ( member3306386364193653941ctxt_a @ X3 @ ( set_Ground_gmctxt_a2 @ Xs ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_Ground_gmctxt_a @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_96_list__ball__nth,axiom,
    ! [N: nat,Xs: list_b,P: b > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_b @ Xs ) )
     => ( ! [X3: b] :
            ( ( member_b @ X3 @ ( set_b2 @ Xs ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_b @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_97_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_98_list__ball__nth,axiom,
    ! [N: nat,Xs: list_list_b,P: list_b > $o] :
      ( ( ord_less_nat @ N @ ( size_s420531428170919671list_b @ Xs ) )
     => ( ! [X3: list_b] :
            ( ( member_list_b @ X3 @ ( set_list_b2 @ Xs ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_list_b @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_99_list__ball__nth,axiom,
    ! [N: nat,Xs: list_P8469869581646625389at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( ord_less_nat @ N @ ( size_s3679842834875189465at_nat @ Xs ) )
     => ( ! [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_Pr6744343527793145070at_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_100_in__set__conv__nth,axiom,
    ! [X: list_nat,Xs: list_list_nat] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
            & ( ( nth_list_nat @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_101_in__set__conv__nth,axiom,
    ! [X: list_Ground_gmctxt_a,Xs: list_l1578326660578982846ctxt_a] :
      ( ( member2714729351964853061ctxt_a @ X @ ( set_li3057194059929822169ctxt_a @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
            & ( ( nth_li6678718213161987943ctxt_a @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_102_in__set__conv__nth,axiom,
    ! [X: list_P8469869581646625389at_nat,Xs: list_l3298181151656792051at_nat] :
      ( ( member3799944675974059798at_nat @ X @ ( set_li7366234323595830146at_nat @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6878757106947045471at_nat @ Xs ) )
            & ( ( nth_li8805938130461222772at_nat @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_103_in__set__conv__nth,axiom,
    ! [X: ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
      ( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Xs ) )
            & ( ( nth_Ground_gmctxt_a @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_104_in__set__conv__nth,axiom,
    ! [X: b,Xs: list_b] :
      ( ( member_b @ X @ ( set_b2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Xs ) )
            & ( ( nth_b @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_105_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_106_in__set__conv__nth,axiom,
    ! [X: list_b,Xs: list_list_b] :
      ( ( member_list_b @ X @ ( set_list_b2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s420531428170919671list_b @ Xs ) )
            & ( ( nth_list_b @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_107_in__set__conv__nth,axiom,
    ! [X: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3679842834875189465at_nat @ Xs ) )
            & ( ( nth_Pr6744343527793145070at_nat @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_108_neq__if__length__neq,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
       != ( size_s962548663205356226ctxt_a @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_109_neq__if__length__neq,axiom,
    ! [Xs: list_b,Ys: list_b] :
      ( ( ( size_size_list_b @ Xs )
       != ( size_size_list_b @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_110_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_111_neq__if__length__neq,axiom,
    ! [Xs: list_list_b,Ys: list_list_b] :
      ( ( ( size_s420531428170919671list_b @ Xs )
       != ( size_s420531428170919671list_b @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_112_neq__if__length__neq,axiom,
    ! [Xs: list_P8469869581646625389at_nat,Ys: list_P8469869581646625389at_nat] :
      ( ( ( size_s3679842834875189465at_nat @ Xs )
       != ( size_s3679842834875189465at_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_113_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_Ground_gmctxt_a] :
      ( ( size_s962548663205356226ctxt_a @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_114_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_b] :
      ( ( size_size_list_b @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_115_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_nat] :
      ( ( size_size_list_nat @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_116_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_list_b] :
      ( ( size_s420531428170919671list_b @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_117_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_P8469869581646625389at_nat] :
      ( ( size_s3679842834875189465at_nat @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_118_length__induct,axiom,
    ! [P: list_Ground_gmctxt_a > $o,Xs: list_Ground_gmctxt_a] :
      ( ! [Xs2: list_Ground_gmctxt_a] :
          ( ! [Ys2: list_Ground_gmctxt_a] :
              ( ( ord_less_nat @ ( size_s962548663205356226ctxt_a @ Ys2 ) @ ( size_s962548663205356226ctxt_a @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_119_length__induct,axiom,
    ! [P: list_b > $o,Xs: list_b] :
      ( ! [Xs2: list_b] :
          ( ! [Ys2: list_b] :
              ( ( ord_less_nat @ ( size_size_list_b @ Ys2 ) @ ( size_size_list_b @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_120_length__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys2: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_121_length__induct,axiom,
    ! [P: list_list_b > $o,Xs: list_list_b] :
      ( ! [Xs2: list_list_b] :
          ( ! [Ys2: list_list_b] :
              ( ( ord_less_nat @ ( size_s420531428170919671list_b @ Ys2 ) @ ( size_s420531428170919671list_b @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_122_length__induct,axiom,
    ! [P: list_P8469869581646625389at_nat > $o,Xs: list_P8469869581646625389at_nat] :
      ( ! [Xs2: list_P8469869581646625389at_nat] :
          ( ! [Ys2: list_P8469869581646625389at_nat] :
              ( ( ord_less_nat @ ( size_s3679842834875189465at_nat @ Ys2 ) @ ( size_s3679842834875189465at_nat @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_123_map__eq__imp__length__eq,axiom,
    ! [F: b > nat,Xs: list_b,G: b > nat,Ys: list_b] :
      ( ( ( map_b_nat @ F @ Xs )
        = ( map_b_nat @ G @ Ys ) )
     => ( ( size_size_list_b @ Xs )
        = ( size_size_list_b @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_124_map__eq__imp__length__eq,axiom,
    ! [F: b > b,Xs: list_b,G: b > b,Ys: list_b] :
      ( ( ( map_b_b @ F @ Xs )
        = ( map_b_b @ G @ Ys ) )
     => ( ( size_size_list_b @ Xs )
        = ( size_size_list_b @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_125_map__eq__imp__length__eq,axiom,
    ! [F: b > nat,Xs: list_b,G: nat > nat,Ys: list_nat] :
      ( ( ( map_b_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_size_list_b @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_126_map__eq__imp__length__eq,axiom,
    ! [F: b > b,Xs: list_b,G: nat > b,Ys: list_nat] :
      ( ( ( map_b_b @ F @ Xs )
        = ( map_nat_b @ G @ Ys ) )
     => ( ( size_size_list_b @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_127_map__eq__imp__length__eq,axiom,
    ! [F: nat > nat,Xs: list_nat,G: b > nat,Ys: list_b] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_b_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_b @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_128_map__eq__imp__length__eq,axiom,
    ! [F: nat > b,Xs: list_nat,G: b > b,Ys: list_b] :
      ( ( ( map_nat_b @ F @ Xs )
        = ( map_b_b @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_b @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_129_map__eq__imp__length__eq,axiom,
    ! [F: nat > b,Xs: list_nat,G: nat > b,Ys: list_nat] :
      ( ( ( map_nat_b @ F @ Xs )
        = ( map_nat_b @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_130_map__eq__imp__length__eq,axiom,
    ! [F: ground_gmctxt_a > b,Xs: list_Ground_gmctxt_a,G: b > b,Ys: list_b] :
      ( ( ( map_Gr2594013990000539163xt_a_b @ F @ Xs )
        = ( map_b_b @ G @ Ys ) )
     => ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_size_list_b @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_131_map__eq__imp__length__eq,axiom,
    ! [F: ground_gmctxt_a > nat,Xs: list_Ground_gmctxt_a,G: b > nat,Ys: list_b] :
      ( ( ( map_Gr1679544123147727284_a_nat @ F @ Xs )
        = ( map_b_nat @ G @ Ys ) )
     => ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_size_list_b @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_132_map__eq__imp__length__eq,axiom,
    ! [F: ground_gmctxt_a > b,Xs: list_Ground_gmctxt_a,G: nat > b,Ys: list_nat] :
      ( ( ( map_Gr2594013990000539163xt_a_b @ F @ Xs )
        = ( map_nat_b @ G @ Ys ) )
     => ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_133_ex__map__conv,axiom,
    ! [Ys: list_list_b,F: b > list_b] :
      ( ( ? [Xs3: list_b] :
            ( Ys
            = ( map_b_list_b @ F @ Xs3 ) ) )
      = ( ! [X2: list_b] :
            ( ( member_list_b @ X2 @ ( set_list_b2 @ Ys ) )
           => ? [Y: b] :
                ( X2
                = ( F @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_134_ex__map__conv,axiom,
    ! [Ys: list_b,F: nat > b] :
      ( ( ? [Xs3: list_nat] :
            ( Ys
            = ( map_nat_b @ F @ Xs3 ) ) )
      = ( ! [X2: b] :
            ( ( member_b @ X2 @ ( set_b2 @ Ys ) )
           => ? [Y: nat] :
                ( X2
                = ( F @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_135_ex__map__conv,axiom,
    ! [Ys: list_b,F: b > b] :
      ( ( ? [Xs3: list_b] :
            ( Ys
            = ( map_b_b @ F @ Xs3 ) ) )
      = ( ! [X2: b] :
            ( ( member_b @ X2 @ ( set_b2 @ Ys ) )
           => ? [Y: b] :
                ( X2
                = ( F @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_136_ex__map__conv,axiom,
    ! [Ys: list_nat,F: ground_gmctxt_a > nat] :
      ( ( ? [Xs3: list_Ground_gmctxt_a] :
            ( Ys
            = ( map_Gr1679544123147727284_a_nat @ F @ Xs3 ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
           => ? [Y: ground_gmctxt_a] :
                ( X2
                = ( F @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_137_ex__map__conv,axiom,
    ! [Ys: list_nat,F: b > nat] :
      ( ( ? [Xs3: list_b] :
            ( Ys
            = ( map_b_nat @ F @ Xs3 ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
           => ? [Y: b] :
                ( X2
                = ( F @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_138_ex__map__conv,axiom,
    ! [Ys: list_Ground_gmctxt_a,F: b > ground_gmctxt_a] :
      ( ( ? [Xs3: list_b] :
            ( Ys
            = ( map_b_3587768286370285017ctxt_a @ F @ Xs3 ) ) )
      = ( ! [X2: ground_gmctxt_a] :
            ( ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ Ys ) )
           => ? [Y: b] :
                ( X2
                = ( F @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_139_map__cong,axiom,
    ! [Xs: list_b,Ys: list_b,F: b > ground_gmctxt_a,G: b > ground_gmctxt_a] :
      ( ( Xs = Ys )
     => ( ! [X3: b] :
            ( ( member_b @ X3 @ ( set_b2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_b_3587768286370285017ctxt_a @ F @ Xs )
          = ( map_b_3587768286370285017ctxt_a @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_140_map__cong,axiom,
    ! [Xs: list_b,Ys: list_b,F: b > list_b,G: b > list_b] :
      ( ( Xs = Ys )
     => ( ! [X3: b] :
            ( ( member_b @ X3 @ ( set_b2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_b_list_b @ F @ Xs )
          = ( map_b_list_b @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_141_map__cong,axiom,
    ! [Xs: list_b,Ys: list_b,F: b > nat,G: b > nat] :
      ( ( Xs = Ys )
     => ( ! [X3: b] :
            ( ( member_b @ X3 @ ( set_b2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_b_nat @ F @ Xs )
          = ( map_b_nat @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_142_map__cong,axiom,
    ! [Xs: list_b,Ys: list_b,F: b > b,G: b > b] :
      ( ( Xs = Ys )
     => ( ! [X3: b] :
            ( ( member_b @ X3 @ ( set_b2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_b_b @ F @ Xs )
          = ( map_b_b @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_143_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: nat > b,G: nat > b] :
      ( ( Xs = Ys )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_nat_b @ F @ Xs )
          = ( map_nat_b @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_144_map__cong,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,F: ground_gmctxt_a > nat,G: ground_gmctxt_a > nat] :
      ( ( Xs = Ys )
     => ( ! [X3: ground_gmctxt_a] :
            ( ( member3306386364193653941ctxt_a @ X3 @ ( set_Ground_gmctxt_a2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_Gr1679544123147727284_a_nat @ F @ Xs )
          = ( map_Gr1679544123147727284_a_nat @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_145_map__idI,axiom,
    ! [Xs: list_b,F: b > b] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ ( set_b2 @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_b_b @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_146_map__idI,axiom,
    ! [Xs: list_nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_nat_nat @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_147_map__idI,axiom,
    ! [Xs: list_Ground_gmctxt_a,F: ground_gmctxt_a > ground_gmctxt_a] :
      ( ! [X3: ground_gmctxt_a] :
          ( ( member3306386364193653941ctxt_a @ X3 @ ( set_Ground_gmctxt_a2 @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_Gr980675717119874050ctxt_a @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_148_map__idI,axiom,
    ! [Xs: list_P8469869581646625389at_nat,F: produc859450856879609959at_nat > produc859450856879609959at_nat] :
      ( ! [X3: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_Pr7106542193144342532at_nat @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_149_map__ext,axiom,
    ! [Xs: list_b,F: b > ground_gmctxt_a,G: b > ground_gmctxt_a] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ ( set_b2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_b_3587768286370285017ctxt_a @ F @ Xs )
        = ( map_b_3587768286370285017ctxt_a @ G @ Xs ) ) ) ).

% map_ext
thf(fact_150_map__ext,axiom,
    ! [Xs: list_b,F: b > list_b,G: b > list_b] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ ( set_b2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_b_list_b @ F @ Xs )
        = ( map_b_list_b @ G @ Xs ) ) ) ).

% map_ext
thf(fact_151_map__ext,axiom,
    ! [Xs: list_b,F: b > nat,G: b > nat] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ ( set_b2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_b_nat @ F @ Xs )
        = ( map_b_nat @ G @ Xs ) ) ) ).

% map_ext
thf(fact_152_map__ext,axiom,
    ! [Xs: list_b,F: b > b,G: b > b] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ ( set_b2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_b_b @ F @ Xs )
        = ( map_b_b @ G @ Xs ) ) ) ).

% map_ext
thf(fact_153_map__ext,axiom,
    ! [Xs: list_nat,F: nat > b,G: nat > b] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_nat_b @ F @ Xs )
        = ( map_nat_b @ G @ Xs ) ) ) ).

% map_ext
thf(fact_154_map__ext,axiom,
    ! [Xs: list_Ground_gmctxt_a,F: ground_gmctxt_a > nat,G: ground_gmctxt_a > nat] :
      ( ! [X3: ground_gmctxt_a] :
          ( ( member3306386364193653941ctxt_a @ X3 @ ( set_Ground_gmctxt_a2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_Gr1679544123147727284_a_nat @ F @ Xs )
        = ( map_Gr1679544123147727284_a_nat @ G @ Xs ) ) ) ).

% map_ext
thf(fact_155_list_Omap__ident__strong,axiom,
    ! [T: list_b,F: b > b] :
      ( ! [Z: b] :
          ( ( member_b @ Z @ ( set_b2 @ T ) )
         => ( ( F @ Z )
            = Z ) )
     => ( ( map_b_b @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_156_list_Omap__ident__strong,axiom,
    ! [T: list_nat,F: nat > nat] :
      ( ! [Z: nat] :
          ( ( member_nat @ Z @ ( set_nat2 @ T ) )
         => ( ( F @ Z )
            = Z ) )
     => ( ( map_nat_nat @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_157_list_Omap__ident__strong,axiom,
    ! [T: list_Ground_gmctxt_a,F: ground_gmctxt_a > ground_gmctxt_a] :
      ( ! [Z: ground_gmctxt_a] :
          ( ( member3306386364193653941ctxt_a @ Z @ ( set_Ground_gmctxt_a2 @ T ) )
         => ( ( F @ Z )
            = Z ) )
     => ( ( map_Gr980675717119874050ctxt_a @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_158_list_Omap__ident__strong,axiom,
    ! [T: list_P8469869581646625389at_nat,F: produc859450856879609959at_nat > produc859450856879609959at_nat] :
      ( ! [Z: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ Z @ ( set_Pr5518436109238095868at_nat @ T ) )
         => ( ( F @ Z )
            = Z ) )
     => ( ( map_Pr7106542193144342532at_nat @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_159_list_Oinj__map__strong,axiom,
    ! [X: list_b,Xa: list_b,F: b > ground_gmctxt_a,Fa: b > ground_gmctxt_a] :
      ( ! [Z: b,Za: b] :
          ( ( member_b @ Z @ ( set_b2 @ X ) )
         => ( ( member_b @ Za @ ( set_b2 @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_b_3587768286370285017ctxt_a @ F @ X )
          = ( map_b_3587768286370285017ctxt_a @ Fa @ Xa ) )
       => ( X = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_160_list_Oinj__map__strong,axiom,
    ! [X: list_b,Xa: list_b,F: b > list_b,Fa: b > list_b] :
      ( ! [Z: b,Za: b] :
          ( ( member_b @ Z @ ( set_b2 @ X ) )
         => ( ( member_b @ Za @ ( set_b2 @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_b_list_b @ F @ X )
          = ( map_b_list_b @ Fa @ Xa ) )
       => ( X = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_161_list_Oinj__map__strong,axiom,
    ! [X: list_b,Xa: list_b,F: b > nat,Fa: b > nat] :
      ( ! [Z: b,Za: b] :
          ( ( member_b @ Z @ ( set_b2 @ X ) )
         => ( ( member_b @ Za @ ( set_b2 @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_b_nat @ F @ X )
          = ( map_b_nat @ Fa @ Xa ) )
       => ( X = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_162_list_Oinj__map__strong,axiom,
    ! [X: list_b,Xa: list_b,F: b > b,Fa: b > b] :
      ( ! [Z: b,Za: b] :
          ( ( member_b @ Z @ ( set_b2 @ X ) )
         => ( ( member_b @ Za @ ( set_b2 @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_b_b @ F @ X )
          = ( map_b_b @ Fa @ Xa ) )
       => ( X = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_163_list_Oinj__map__strong,axiom,
    ! [X: list_nat,Xa: list_nat,F: nat > b,Fa: nat > b] :
      ( ! [Z: nat,Za: nat] :
          ( ( member_nat @ Z @ ( set_nat2 @ X ) )
         => ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_nat_b @ F @ X )
          = ( map_nat_b @ Fa @ Xa ) )
       => ( X = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_164_list_Oinj__map__strong,axiom,
    ! [X: list_Ground_gmctxt_a,Xa: list_Ground_gmctxt_a,F: ground_gmctxt_a > nat,Fa: ground_gmctxt_a > nat] :
      ( ! [Z: ground_gmctxt_a,Za: ground_gmctxt_a] :
          ( ( member3306386364193653941ctxt_a @ Z @ ( set_Ground_gmctxt_a2 @ X ) )
         => ( ( member3306386364193653941ctxt_a @ Za @ ( set_Ground_gmctxt_a2 @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_Gr1679544123147727284_a_nat @ F @ X )
          = ( map_Gr1679544123147727284_a_nat @ Fa @ Xa ) )
       => ( X = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_165_list_Omap__cong0,axiom,
    ! [X: list_b,F: b > ground_gmctxt_a,G: b > ground_gmctxt_a] :
      ( ! [Z: b] :
          ( ( member_b @ Z @ ( set_b2 @ X ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_b_3587768286370285017ctxt_a @ F @ X )
        = ( map_b_3587768286370285017ctxt_a @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_166_list_Omap__cong0,axiom,
    ! [X: list_b,F: b > list_b,G: b > list_b] :
      ( ! [Z: b] :
          ( ( member_b @ Z @ ( set_b2 @ X ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_b_list_b @ F @ X )
        = ( map_b_list_b @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_167_list_Omap__cong0,axiom,
    ! [X: list_b,F: b > nat,G: b > nat] :
      ( ! [Z: b] :
          ( ( member_b @ Z @ ( set_b2 @ X ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_b_nat @ F @ X )
        = ( map_b_nat @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_168_list_Omap__cong0,axiom,
    ! [X: list_b,F: b > b,G: b > b] :
      ( ! [Z: b] :
          ( ( member_b @ Z @ ( set_b2 @ X ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_b_b @ F @ X )
        = ( map_b_b @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_169_list_Omap__cong0,axiom,
    ! [X: list_nat,F: nat > b,G: nat > b] :
      ( ! [Z: nat] :
          ( ( member_nat @ Z @ ( set_nat2 @ X ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_nat_b @ F @ X )
        = ( map_nat_b @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_170_list_Omap__cong0,axiom,
    ! [X: list_Ground_gmctxt_a,F: ground_gmctxt_a > nat,G: ground_gmctxt_a > nat] :
      ( ! [Z: ground_gmctxt_a] :
          ( ( member3306386364193653941ctxt_a @ Z @ ( set_Ground_gmctxt_a2 @ X ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_Gr1679544123147727284_a_nat @ F @ X )
        = ( map_Gr1679544123147727284_a_nat @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_171_list_Omap__cong,axiom,
    ! [X: list_b,Ya: list_b,F: b > ground_gmctxt_a,G: b > ground_gmctxt_a] :
      ( ( X = Ya )
     => ( ! [Z: b] :
            ( ( member_b @ Z @ ( set_b2 @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_b_3587768286370285017ctxt_a @ F @ X )
          = ( map_b_3587768286370285017ctxt_a @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_172_list_Omap__cong,axiom,
    ! [X: list_b,Ya: list_b,F: b > list_b,G: b > list_b] :
      ( ( X = Ya )
     => ( ! [Z: b] :
            ( ( member_b @ Z @ ( set_b2 @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_b_list_b @ F @ X )
          = ( map_b_list_b @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_173_list_Omap__cong,axiom,
    ! [X: list_b,Ya: list_b,F: b > nat,G: b > nat] :
      ( ( X = Ya )
     => ( ! [Z: b] :
            ( ( member_b @ Z @ ( set_b2 @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_b_nat @ F @ X )
          = ( map_b_nat @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_174_list_Omap__cong,axiom,
    ! [X: list_b,Ya: list_b,F: b > b,G: b > b] :
      ( ( X = Ya )
     => ( ! [Z: b] :
            ( ( member_b @ Z @ ( set_b2 @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_b_b @ F @ X )
          = ( map_b_b @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_175_list_Omap__cong,axiom,
    ! [X: list_nat,Ya: list_nat,F: nat > b,G: nat > b] :
      ( ( X = Ya )
     => ( ! [Z: nat] :
            ( ( member_nat @ Z @ ( set_nat2 @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_nat_b @ F @ X )
          = ( map_nat_b @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_176_list_Omap__cong,axiom,
    ! [X: list_Ground_gmctxt_a,Ya: list_Ground_gmctxt_a,F: ground_gmctxt_a > nat,G: ground_gmctxt_a > nat] :
      ( ( X = Ya )
     => ( ! [Z: ground_gmctxt_a] :
            ( ( member3306386364193653941ctxt_a @ Z @ ( set_Ground_gmctxt_a2 @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_Gr1679544123147727284_a_nat @ F @ X )
          = ( map_Gr1679544123147727284_a_nat @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_177_Ex__list__of__length__P,axiom,
    ! [N: nat,P: list_nat > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ? [X4: list_nat] : ( P @ X4 @ I2 ) )
     => ? [Xs2: list_list_nat] :
          ( ( ( size_s3023201423986296836st_nat @ Xs2 )
            = N )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ N )
             => ( P @ ( nth_list_nat @ Xs2 @ I4 ) @ I4 ) ) ) ) ).

% Ex_list_of_length_P
thf(fact_178_Ex__list__of__length__P,axiom,
    ! [N: nat,P: list_Ground_gmctxt_a > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ? [X4: list_Ground_gmctxt_a] : ( P @ X4 @ I2 ) )
     => ? [Xs2: list_l1578326660578982846ctxt_a] :
          ( ( ( size_s6994882312647770194ctxt_a @ Xs2 )
            = N )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ N )
             => ( P @ ( nth_li6678718213161987943ctxt_a @ Xs2 @ I4 ) @ I4 ) ) ) ) ).

% Ex_list_of_length_P
thf(fact_179_Ex__list__of__length__P,axiom,
    ! [N: nat,P: list_P8469869581646625389at_nat > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ? [X4: list_P8469869581646625389at_nat] : ( P @ X4 @ I2 ) )
     => ? [Xs2: list_l3298181151656792051at_nat] :
          ( ( ( size_s6878757106947045471at_nat @ Xs2 )
            = N )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ N )
             => ( P @ ( nth_li8805938130461222772at_nat @ Xs2 @ I4 ) @ I4 ) ) ) ) ).

% Ex_list_of_length_P
thf(fact_180_Ex__list__of__length__P,axiom,
    ! [N: nat,P: ground_gmctxt_a > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ? [X4: ground_gmctxt_a] : ( P @ X4 @ I2 ) )
     => ? [Xs2: list_Ground_gmctxt_a] :
          ( ( ( size_s962548663205356226ctxt_a @ Xs2 )
            = N )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ N )
             => ( P @ ( nth_Ground_gmctxt_a @ Xs2 @ I4 ) @ I4 ) ) ) ) ).

% Ex_list_of_length_P
thf(fact_181_Ex__list__of__length__P,axiom,
    ! [N: nat,P: b > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ? [X4: b] : ( P @ X4 @ I2 ) )
     => ? [Xs2: list_b] :
          ( ( ( size_size_list_b @ Xs2 )
            = N )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ N )
             => ( P @ ( nth_b @ Xs2 @ I4 ) @ I4 ) ) ) ) ).

% Ex_list_of_length_P
thf(fact_182_Ex__list__of__length__P,axiom,
    ! [N: nat,P: nat > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ? [X4: nat] : ( P @ X4 @ I2 ) )
     => ? [Xs2: list_nat] :
          ( ( ( size_size_list_nat @ Xs2 )
            = N )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ N )
             => ( P @ ( nth_nat @ Xs2 @ I4 ) @ I4 ) ) ) ) ).

% Ex_list_of_length_P
thf(fact_183_Ex__list__of__length__P,axiom,
    ! [N: nat,P: list_b > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ? [X4: list_b] : ( P @ X4 @ I2 ) )
     => ? [Xs2: list_list_b] :
          ( ( ( size_s420531428170919671list_b @ Xs2 )
            = N )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ N )
             => ( P @ ( nth_list_b @ Xs2 @ I4 ) @ I4 ) ) ) ) ).

% Ex_list_of_length_P
thf(fact_184_Ex__list__of__length__P,axiom,
    ! [N: nat,P: produc859450856879609959at_nat > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ? [X4: produc859450856879609959at_nat] : ( P @ X4 @ I2 ) )
     => ? [Xs2: list_P8469869581646625389at_nat] :
          ( ( ( size_s3679842834875189465at_nat @ Xs2 )
            = N )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ N )
             => ( P @ ( nth_Pr6744343527793145070at_nat @ Xs2 @ I4 ) @ I4 ) ) ) ) ).

% Ex_list_of_length_P
thf(fact_185_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y2: list_list_nat,Z2: list_list_nat] : ( Y2 = Z2 ) )
    = ( ^ [Xs3: list_list_nat,Ys3: list_list_nat] :
          ( ( ( size_s3023201423986296836st_nat @ Xs3 )
            = ( size_s3023201423986296836st_nat @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs3 ) )
             => ( ( nth_list_nat @ Xs3 @ I3 )
                = ( nth_list_nat @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_186_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y2: list_l1578326660578982846ctxt_a,Z2: list_l1578326660578982846ctxt_a] : ( Y2 = Z2 ) )
    = ( ^ [Xs3: list_l1578326660578982846ctxt_a,Ys3: list_l1578326660578982846ctxt_a] :
          ( ( ( size_s6994882312647770194ctxt_a @ Xs3 )
            = ( size_s6994882312647770194ctxt_a @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s6994882312647770194ctxt_a @ Xs3 ) )
             => ( ( nth_li6678718213161987943ctxt_a @ Xs3 @ I3 )
                = ( nth_li6678718213161987943ctxt_a @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_187_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y2: list_l3298181151656792051at_nat,Z2: list_l3298181151656792051at_nat] : ( Y2 = Z2 ) )
    = ( ^ [Xs3: list_l3298181151656792051at_nat,Ys3: list_l3298181151656792051at_nat] :
          ( ( ( size_s6878757106947045471at_nat @ Xs3 )
            = ( size_s6878757106947045471at_nat @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s6878757106947045471at_nat @ Xs3 ) )
             => ( ( nth_li8805938130461222772at_nat @ Xs3 @ I3 )
                = ( nth_li8805938130461222772at_nat @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_188_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y2: list_Ground_gmctxt_a,Z2: list_Ground_gmctxt_a] : ( Y2 = Z2 ) )
    = ( ^ [Xs3: list_Ground_gmctxt_a,Ys3: list_Ground_gmctxt_a] :
          ( ( ( size_s962548663205356226ctxt_a @ Xs3 )
            = ( size_s962548663205356226ctxt_a @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Xs3 ) )
             => ( ( nth_Ground_gmctxt_a @ Xs3 @ I3 )
                = ( nth_Ground_gmctxt_a @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_189_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y2: list_b,Z2: list_b] : ( Y2 = Z2 ) )
    = ( ^ [Xs3: list_b,Ys3: list_b] :
          ( ( ( size_size_list_b @ Xs3 )
            = ( size_size_list_b @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Xs3 ) )
             => ( ( nth_b @ Xs3 @ I3 )
                = ( nth_b @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_190_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y2: list_nat,Z2: list_nat] : ( Y2 = Z2 ) )
    = ( ^ [Xs3: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs3 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
             => ( ( nth_nat @ Xs3 @ I3 )
                = ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_191_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y2: list_list_b,Z2: list_list_b] : ( Y2 = Z2 ) )
    = ( ^ [Xs3: list_list_b,Ys3: list_list_b] :
          ( ( ( size_s420531428170919671list_b @ Xs3 )
            = ( size_s420531428170919671list_b @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s420531428170919671list_b @ Xs3 ) )
             => ( ( nth_list_b @ Xs3 @ I3 )
                = ( nth_list_b @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_192_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y2: list_P8469869581646625389at_nat,Z2: list_P8469869581646625389at_nat] : ( Y2 = Z2 ) )
    = ( ^ [Xs3: list_P8469869581646625389at_nat,Ys3: list_P8469869581646625389at_nat] :
          ( ( ( size_s3679842834875189465at_nat @ Xs3 )
            = ( size_s3679842834875189465at_nat @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s3679842834875189465at_nat @ Xs3 ) )
             => ( ( nth_Pr6744343527793145070at_nat @ Xs3 @ I3 )
                = ( nth_Pr6744343527793145070at_nat @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_193_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > list_nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X5: list_nat] : ( P @ I3 @ X5 ) ) )
      = ( ? [Xs3: list_list_nat] :
            ( ( ( size_s3023201423986296836st_nat @ Xs3 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_list_nat @ Xs3 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_194_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > list_Ground_gmctxt_a > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X5: list_Ground_gmctxt_a] : ( P @ I3 @ X5 ) ) )
      = ( ? [Xs3: list_l1578326660578982846ctxt_a] :
            ( ( ( size_s6994882312647770194ctxt_a @ Xs3 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_li6678718213161987943ctxt_a @ Xs3 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_195_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > list_P8469869581646625389at_nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X5: list_P8469869581646625389at_nat] : ( P @ I3 @ X5 ) ) )
      = ( ? [Xs3: list_l3298181151656792051at_nat] :
            ( ( ( size_s6878757106947045471at_nat @ Xs3 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_li8805938130461222772at_nat @ Xs3 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_196_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > ground_gmctxt_a > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X5: ground_gmctxt_a] : ( P @ I3 @ X5 ) ) )
      = ( ? [Xs3: list_Ground_gmctxt_a] :
            ( ( ( size_s962548663205356226ctxt_a @ Xs3 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_Ground_gmctxt_a @ Xs3 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_197_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > b > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X5: b] : ( P @ I3 @ X5 ) ) )
      = ( ? [Xs3: list_b] :
            ( ( ( size_size_list_b @ Xs3 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_b @ Xs3 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_198_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X5: nat] : ( P @ I3 @ X5 ) ) )
      = ( ? [Xs3: list_nat] :
            ( ( ( size_size_list_nat @ Xs3 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_199_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > list_b > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X5: list_b] : ( P @ I3 @ X5 ) ) )
      = ( ? [Xs3: list_list_b] :
            ( ( ( size_s420531428170919671list_b @ Xs3 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_list_b @ Xs3 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_200_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > produc859450856879609959at_nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X5: produc859450856879609959at_nat] : ( P @ I3 @ X5 ) ) )
      = ( ? [Xs3: list_P8469869581646625389at_nat] :
            ( ( ( size_s3679842834875189465at_nat @ Xs3 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_Pr6744343527793145070at_nat @ Xs3 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_201_nth__equalityI,axiom,
    ! [Xs: list_list_nat,Ys: list_list_nat] :
      ( ( ( size_s3023201423986296836st_nat @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( ( nth_list_nat @ Xs @ I2 )
              = ( nth_list_nat @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_202_nth__equalityI,axiom,
    ! [Xs: list_l1578326660578982846ctxt_a,Ys: list_l1578326660578982846ctxt_a] :
      ( ( ( size_s6994882312647770194ctxt_a @ Xs )
        = ( size_s6994882312647770194ctxt_a @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
           => ( ( nth_li6678718213161987943ctxt_a @ Xs @ I2 )
              = ( nth_li6678718213161987943ctxt_a @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_203_nth__equalityI,axiom,
    ! [Xs: list_l3298181151656792051at_nat,Ys: list_l3298181151656792051at_nat] :
      ( ( ( size_s6878757106947045471at_nat @ Xs )
        = ( size_s6878757106947045471at_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6878757106947045471at_nat @ Xs ) )
           => ( ( nth_li8805938130461222772at_nat @ Xs @ I2 )
              = ( nth_li8805938130461222772at_nat @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_204_nth__equalityI,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
           => ( ( nth_Ground_gmctxt_a @ Xs @ I2 )
              = ( nth_Ground_gmctxt_a @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_205_nth__equalityI,axiom,
    ! [Xs: list_b,Ys: list_b] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Xs ) )
           => ( ( nth_b @ Xs @ I2 )
              = ( nth_b @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_206_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I2 )
              = ( nth_nat @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_207_nth__equalityI,axiom,
    ! [Xs: list_list_b,Ys: list_list_b] :
      ( ( ( size_s420531428170919671list_b @ Xs )
        = ( size_s420531428170919671list_b @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Xs ) )
           => ( ( nth_list_b @ Xs @ I2 )
              = ( nth_list_b @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_208_nth__equalityI,axiom,
    ! [Xs: list_P8469869581646625389at_nat,Ys: list_P8469869581646625389at_nat] :
      ( ( ( size_s3679842834875189465at_nat @ Xs )
        = ( size_s3679842834875189465at_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3679842834875189465at_nat @ Xs ) )
           => ( ( nth_Pr6744343527793145070at_nat @ Xs @ I2 )
              = ( nth_Pr6744343527793145070at_nat @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_209_ex__set__conv__ex__nth,axiom,
    ! [Xs: list_list_nat,P: list_nat > $o] :
      ( ( ? [X2: list_nat] :
            ( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
            & ( P @ X2 ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
            & ( P @ ( nth_list_nat @ Xs @ I3 ) ) ) ) ) ).

% ex_set_conv_ex_nth
thf(fact_210_ex__set__conv__ex__nth,axiom,
    ! [Xs: list_l1578326660578982846ctxt_a,P: list_Ground_gmctxt_a > $o] :
      ( ( ? [X2: list_Ground_gmctxt_a] :
            ( ( member2714729351964853061ctxt_a @ X2 @ ( set_li3057194059929822169ctxt_a @ Xs ) )
            & ( P @ X2 ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
            & ( P @ ( nth_li6678718213161987943ctxt_a @ Xs @ I3 ) ) ) ) ) ).

% ex_set_conv_ex_nth
thf(fact_211_ex__set__conv__ex__nth,axiom,
    ! [Xs: list_l3298181151656792051at_nat,P: list_P8469869581646625389at_nat > $o] :
      ( ( ? [X2: list_P8469869581646625389at_nat] :
            ( ( member3799944675974059798at_nat @ X2 @ ( set_li7366234323595830146at_nat @ Xs ) )
            & ( P @ X2 ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6878757106947045471at_nat @ Xs ) )
            & ( P @ ( nth_li8805938130461222772at_nat @ Xs @ I3 ) ) ) ) ) ).

% ex_set_conv_ex_nth
thf(fact_212_ex__set__conv__ex__nth,axiom,
    ! [Xs: list_Ground_gmctxt_a,P: ground_gmctxt_a > $o] :
      ( ( ? [X2: ground_gmctxt_a] :
            ( ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ Xs ) )
            & ( P @ X2 ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Xs ) )
            & ( P @ ( nth_Ground_gmctxt_a @ Xs @ I3 ) ) ) ) ) ).

% ex_set_conv_ex_nth
thf(fact_213_ex__set__conv__ex__nth,axiom,
    ! [Xs: list_b,P: b > $o] :
      ( ( ? [X2: b] :
            ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
            & ( P @ X2 ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Xs ) )
            & ( P @ ( nth_b @ Xs @ I3 ) ) ) ) ) ).

% ex_set_conv_ex_nth
thf(fact_214_ex__set__conv__ex__nth,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ? [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
            & ( P @ X2 ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
            & ( P @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).

% ex_set_conv_ex_nth
thf(fact_215_ex__set__conv__ex__nth,axiom,
    ! [Xs: list_list_b,P: list_b > $o] :
      ( ( ? [X2: list_b] :
            ( ( member_list_b @ X2 @ ( set_list_b2 @ Xs ) )
            & ( P @ X2 ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s420531428170919671list_b @ Xs ) )
            & ( P @ ( nth_list_b @ Xs @ I3 ) ) ) ) ) ).

% ex_set_conv_ex_nth
thf(fact_216_ex__set__conv__ex__nth,axiom,
    ! [Xs: list_P8469869581646625389at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( ? [X2: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X2 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
            & ( P @ X2 ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3679842834875189465at_nat @ Xs ) )
            & ( P @ ( nth_Pr6744343527793145070at_nat @ Xs @ I3 ) ) ) ) ) ).

% ex_set_conv_ex_nth
thf(fact_217_all__set__conv__all__nth,axiom,
    ! [Xs: list_list_nat,P: list_nat > $o] :
      ( ( ! [X2: list_nat] :
            ( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
           => ( P @ X2 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( P @ ( nth_list_nat @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_218_all__set__conv__all__nth,axiom,
    ! [Xs: list_l1578326660578982846ctxt_a,P: list_Ground_gmctxt_a > $o] :
      ( ( ! [X2: list_Ground_gmctxt_a] :
            ( ( member2714729351964853061ctxt_a @ X2 @ ( set_li3057194059929822169ctxt_a @ Xs ) )
           => ( P @ X2 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
           => ( P @ ( nth_li6678718213161987943ctxt_a @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_219_all__set__conv__all__nth,axiom,
    ! [Xs: list_l3298181151656792051at_nat,P: list_P8469869581646625389at_nat > $o] :
      ( ( ! [X2: list_P8469869581646625389at_nat] :
            ( ( member3799944675974059798at_nat @ X2 @ ( set_li7366234323595830146at_nat @ Xs ) )
           => ( P @ X2 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6878757106947045471at_nat @ Xs ) )
           => ( P @ ( nth_li8805938130461222772at_nat @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_220_all__set__conv__all__nth,axiom,
    ! [Xs: list_Ground_gmctxt_a,P: ground_gmctxt_a > $o] :
      ( ( ! [X2: ground_gmctxt_a] :
            ( ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ Xs ) )
           => ( P @ X2 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Xs ) )
           => ( P @ ( nth_Ground_gmctxt_a @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_221_all__set__conv__all__nth,axiom,
    ! [Xs: list_b,P: b > $o] :
      ( ( ! [X2: b] :
            ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
           => ( P @ X2 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Xs ) )
           => ( P @ ( nth_b @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_222_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
           => ( P @ X2 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
           => ( P @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_223_all__set__conv__all__nth,axiom,
    ! [Xs: list_list_b,P: list_b > $o] :
      ( ( ! [X2: list_b] :
            ( ( member_list_b @ X2 @ ( set_list_b2 @ Xs ) )
           => ( P @ X2 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s420531428170919671list_b @ Xs ) )
           => ( P @ ( nth_list_b @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_224_all__set__conv__all__nth,axiom,
    ! [Xs: list_P8469869581646625389at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( ! [X2: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X2 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
           => ( P @ X2 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3679842834875189465at_nat @ Xs ) )
           => ( P @ ( nth_Pr6744343527793145070at_nat @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_225_all__nth__imp__all__set,axiom,
    ! [Xs: list_list_nat,P: list_nat > $o,X: list_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
         => ( P @ ( nth_list_nat @ Xs @ I2 ) ) )
     => ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_226_all__nth__imp__all__set,axiom,
    ! [Xs: list_l1578326660578982846ctxt_a,P: list_Ground_gmctxt_a > $o,X: list_Ground_gmctxt_a] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
         => ( P @ ( nth_li6678718213161987943ctxt_a @ Xs @ I2 ) ) )
     => ( ( member2714729351964853061ctxt_a @ X @ ( set_li3057194059929822169ctxt_a @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_227_all__nth__imp__all__set,axiom,
    ! [Xs: list_l3298181151656792051at_nat,P: list_P8469869581646625389at_nat > $o,X: list_P8469869581646625389at_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s6878757106947045471at_nat @ Xs ) )
         => ( P @ ( nth_li8805938130461222772at_nat @ Xs @ I2 ) ) )
     => ( ( member3799944675974059798at_nat @ X @ ( set_li7366234323595830146at_nat @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_228_all__nth__imp__all__set,axiom,
    ! [Xs: list_Ground_gmctxt_a,P: ground_gmctxt_a > $o,X: ground_gmctxt_a] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
         => ( P @ ( nth_Ground_gmctxt_a @ Xs @ I2 ) ) )
     => ( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_229_all__nth__imp__all__set,axiom,
    ! [Xs: list_b,P: b > $o,X: b] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Xs ) )
         => ( P @ ( nth_b @ Xs @ I2 ) ) )
     => ( ( member_b @ X @ ( set_b2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_230_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: nat > $o,X: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
         => ( P @ ( nth_nat @ Xs @ I2 ) ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_231_all__nth__imp__all__set,axiom,
    ! [Xs: list_list_b,P: list_b > $o,X: list_b] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Xs ) )
         => ( P @ ( nth_list_b @ Xs @ I2 ) ) )
     => ( ( member_list_b @ X @ ( set_list_b2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_232_all__nth__imp__all__set,axiom,
    ! [Xs: list_P8469869581646625389at_nat,P: produc859450856879609959at_nat > $o,X: produc859450856879609959at_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3679842834875189465at_nat @ Xs ) )
         => ( P @ ( nth_Pr6744343527793145070at_nat @ Xs @ I2 ) ) )
     => ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_233_in__set__idx,axiom,
    ! [X: list_nat,Xs: list_list_nat] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
     => ? [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
          & ( ( nth_list_nat @ Xs @ I2 )
            = X ) ) ) ).

% in_set_idx
thf(fact_234_in__set__idx,axiom,
    ! [X: list_Ground_gmctxt_a,Xs: list_l1578326660578982846ctxt_a] :
      ( ( member2714729351964853061ctxt_a @ X @ ( set_li3057194059929822169ctxt_a @ Xs ) )
     => ? [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
          & ( ( nth_li6678718213161987943ctxt_a @ Xs @ I2 )
            = X ) ) ) ).

% in_set_idx
thf(fact_235_in__set__idx,axiom,
    ! [X: list_P8469869581646625389at_nat,Xs: list_l3298181151656792051at_nat] :
      ( ( member3799944675974059798at_nat @ X @ ( set_li7366234323595830146at_nat @ Xs ) )
     => ? [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s6878757106947045471at_nat @ Xs ) )
          & ( ( nth_li8805938130461222772at_nat @ Xs @ I2 )
            = X ) ) ) ).

% in_set_idx
thf(fact_236_in__set__idx,axiom,
    ! [X: ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
      ( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
     => ? [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
          & ( ( nth_Ground_gmctxt_a @ Xs @ I2 )
            = X ) ) ) ).

% in_set_idx
thf(fact_237_in__set__idx,axiom,
    ! [X: b,Xs: list_b] :
      ( ( member_b @ X @ ( set_b2 @ Xs ) )
     => ? [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Xs ) )
          & ( ( nth_b @ Xs @ I2 )
            = X ) ) ) ).

% in_set_idx
thf(fact_238_in__set__idx,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ? [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
          & ( ( nth_nat @ Xs @ I2 )
            = X ) ) ) ).

% in_set_idx
thf(fact_239_in__set__idx,axiom,
    ! [X: list_b,Xs: list_list_b] :
      ( ( member_list_b @ X @ ( set_list_b2 @ Xs ) )
     => ? [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Xs ) )
          & ( ( nth_list_b @ Xs @ I2 )
            = X ) ) ) ).

% in_set_idx
thf(fact_240_in__set__idx,axiom,
    ! [X: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs ) )
     => ? [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3679842834875189465at_nat @ Xs ) )
          & ( ( nth_Pr6744343527793145070at_nat @ Xs @ I2 )
            = X ) ) ) ).

% in_set_idx
thf(fact_241_map__partition__by__nth,axiom,
    ! [I: nat,Ys: list_nat,F: b > b,Xs: list_b] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
     => ( ( map_b_b @ F @ ( nth_list_b @ ( multih3537974503344769663n_by_b @ Xs @ Ys ) @ I ) )
        = ( nth_list_b @ ( multih3537974503344769663n_by_b @ ( map_b_b @ F @ Xs ) @ Ys ) @ I ) ) ) ).

% map_partition_by_nth
thf(fact_242_map__partition__by__nth,axiom,
    ! [I: nat,Ys: list_nat,F: b > nat,Xs: list_b] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
     => ( ( map_b_nat @ F @ ( nth_list_b @ ( multih3537974503344769663n_by_b @ Xs @ Ys ) @ I ) )
        = ( nth_list_nat @ ( multih8544690682651757648by_nat @ ( map_b_nat @ F @ Xs ) @ Ys ) @ I ) ) ) ).

% map_partition_by_nth
thf(fact_243_map__partition__by__nth,axiom,
    ! [I: nat,Ys: list_nat,F: nat > b,Xs: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
     => ( ( map_nat_b @ F @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs @ Ys ) @ I ) )
        = ( nth_list_b @ ( multih3537974503344769663n_by_b @ ( map_nat_b @ F @ Xs ) @ Ys ) @ I ) ) ) ).

% map_partition_by_nth
thf(fact_244_map__partition__by__nth,axiom,
    ! [I: nat,Ys: list_nat,F: nat > nat,Xs: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
     => ( ( map_nat_nat @ F @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs @ Ys ) @ I ) )
        = ( nth_list_nat @ ( multih8544690682651757648by_nat @ ( map_nat_nat @ F @ Xs ) @ Ys ) @ I ) ) ) ).

% map_partition_by_nth
thf(fact_245_map__partition__by__nth,axiom,
    ! [I: nat,Ys: list_nat,F: b > list_b,Xs: list_b] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
     => ( ( map_b_list_b @ F @ ( nth_list_b @ ( multih3537974503344769663n_by_b @ Xs @ Ys ) @ I ) )
        = ( nth_list_list_b @ ( multih2267195492336891525list_b @ ( map_b_list_b @ F @ Xs ) @ Ys ) @ I ) ) ) ).

% map_partition_by_nth
thf(fact_246_map__partition__by__nth,axiom,
    ! [I: nat,Ys: list_nat,F: b > ground_gmctxt_a,Xs: list_b] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
     => ( ( map_b_3587768286370285017ctxt_a @ F @ ( nth_list_b @ ( multih3537974503344769663n_by_b @ Xs @ Ys ) @ I ) )
        = ( nth_li6678718213161987943ctxt_a @ ( multih5172578161653122718ctxt_a @ ( map_b_3587768286370285017ctxt_a @ F @ Xs ) @ Ys ) @ I ) ) ) ).

% map_partition_by_nth
thf(fact_247_map__partition__by__nth,axiom,
    ! [I: nat,Ys: list_nat,F: nat > ground_gmctxt_a,Xs: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
     => ( ( map_na2753241459123782580ctxt_a @ F @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs @ Ys ) @ I ) )
        = ( nth_li6678718213161987943ctxt_a @ ( multih5172578161653122718ctxt_a @ ( map_na2753241459123782580ctxt_a @ F @ Xs ) @ Ys ) @ I ) ) ) ).

% map_partition_by_nth
thf(fact_248_map__partition__by__nth,axiom,
    ! [I: nat,Ys: list_nat,F: ground_gmctxt_a > b,Xs: list_Ground_gmctxt_a] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
     => ( ( map_Gr2594013990000539163xt_a_b @ F @ ( nth_li6678718213161987943ctxt_a @ ( multih5172578161653122718ctxt_a @ Xs @ Ys ) @ I ) )
        = ( nth_list_b @ ( multih3537974503344769663n_by_b @ ( map_Gr2594013990000539163xt_a_b @ F @ Xs ) @ Ys ) @ I ) ) ) ).

% map_partition_by_nth
thf(fact_249_map__partition__by__nth,axiom,
    ! [I: nat,Ys: list_nat,F: ground_gmctxt_a > nat,Xs: list_Ground_gmctxt_a] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
     => ( ( map_Gr1679544123147727284_a_nat @ F @ ( nth_li6678718213161987943ctxt_a @ ( multih5172578161653122718ctxt_a @ Xs @ Ys ) @ I ) )
        = ( nth_list_nat @ ( multih8544690682651757648by_nat @ ( map_Gr1679544123147727284_a_nat @ F @ Xs ) @ Ys ) @ I ) ) ) ).

% map_partition_by_nth
thf(fact_250_map__partition__by__nth,axiom,
    ! [I: nat,Ys: list_nat,F: ground_gmctxt_a > ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
     => ( ( map_Gr980675717119874050ctxt_a @ F @ ( nth_li6678718213161987943ctxt_a @ ( multih5172578161653122718ctxt_a @ Xs @ Ys ) @ I ) )
        = ( nth_li6678718213161987943ctxt_a @ ( multih5172578161653122718ctxt_a @ ( map_Gr980675717119874050ctxt_a @ F @ Xs ) @ Ys ) @ I ) ) ) ).

% map_partition_by_nth
thf(fact_251_map__length__partition__by,axiom,
    ! [Ys: list_nat,Xs: list_list_b] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s420531428170919671list_b @ Xs ) )
     => ( ( map_list_list_b_nat @ size_s420531428170919671list_b @ ( multih2267195492336891525list_b @ Xs @ Ys ) )
        = Ys ) ) ).

% map_length_partition_by
thf(fact_252_map__length__partition__by,axiom,
    ! [Ys: list_nat,Xs: list_b] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_size_list_b @ Xs ) )
     => ( ( map_list_b_nat @ size_size_list_b @ ( multih3537974503344769663n_by_b @ Xs @ Ys ) )
        = Ys ) ) ).

% map_length_partition_by
thf(fact_253_map__length__partition__by,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_size_list_nat @ Xs ) )
     => ( ( map_list_nat_nat @ size_size_list_nat @ ( multih8544690682651757648by_nat @ Xs @ Ys ) )
        = Ys ) ) ).

% map_length_partition_by
thf(fact_254_map__length__partition__by,axiom,
    ! [Ys: list_nat,Xs: list_Ground_gmctxt_a] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( map_li8618020238875267396_a_nat @ size_s962548663205356226ctxt_a @ ( multih5172578161653122718ctxt_a @ Xs @ Ys ) )
        = Ys ) ) ).

% map_length_partition_by
thf(fact_255_map__length__partition__by,axiom,
    ! [Ys: list_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s3679842834875189465at_nat @ Xs ) )
     => ( ( map_li9059624042496273785at_nat @ size_s3679842834875189465at_nat @ ( multih7555706055023738855at_nat @ Xs @ Ys ) )
        = Ys ) ) ).

% map_length_partition_by
thf(fact_256_partition__by__nth__nth__elem,axiom,
    ! [Ys: list_nat,Xs: list_list_nat,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s3023201423986296836st_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
         => ( member_list_nat @ ( nth_list_nat @ ( nth_list_list_nat @ ( multih5630945896903731040st_nat @ Xs @ Ys ) @ I ) @ J ) @ ( set_list_nat2 @ Xs ) ) ) ) ) ).

% partition_by_nth_nth_elem
thf(fact_257_partition__by__nth__nth__elem,axiom,
    ! [Ys: list_nat,Xs: list_l1578326660578982846ctxt_a,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s6994882312647770194ctxt_a @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
         => ( member2714729351964853061ctxt_a @ ( nth_li6678718213161987943ctxt_a @ ( nth_li3520809774789768823ctxt_a @ ( multih1205877868680985774ctxt_a @ Xs @ Ys ) @ I ) @ J ) @ ( set_li3057194059929822169ctxt_a @ Xs ) ) ) ) ) ).

% partition_by_nth_nth_elem
thf(fact_258_partition__by__nth__nth__elem,axiom,
    ! [Ys: list_nat,Xs: list_l3298181151656792051at_nat,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s6878757106947045471at_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
         => ( member3799944675974059798at_nat @ ( nth_li8805938130461222772at_nat @ ( nth_li2593672534642986362at_nat @ ( multih560249069488021997at_nat @ Xs @ Ys ) @ I ) @ J ) @ ( set_li7366234323595830146at_nat @ Xs ) ) ) ) ) ).

% partition_by_nth_nth_elem
thf(fact_259_partition__by__nth__nth__elem,axiom,
    ! [Ys: list_nat,Xs: list_list_b,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s420531428170919671list_b @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
         => ( member_list_b @ ( nth_list_b @ ( nth_list_list_b @ ( multih2267195492336891525list_b @ Xs @ Ys ) @ I ) @ J ) @ ( set_list_b2 @ Xs ) ) ) ) ) ).

% partition_by_nth_nth_elem
thf(fact_260_partition__by__nth__nth__elem,axiom,
    ! [Ys: list_nat,Xs: list_b,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_size_list_b @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
         => ( member_b @ ( nth_b @ ( nth_list_b @ ( multih3537974503344769663n_by_b @ Xs @ Ys ) @ I ) @ J ) @ ( set_b2 @ Xs ) ) ) ) ) ).

% partition_by_nth_nth_elem
thf(fact_261_partition__by__nth__nth__elem,axiom,
    ! [Ys: list_nat,Xs: list_nat,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_size_list_nat @ Xs ) )
     => ( ( 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 @ Xs @ Ys ) @ I ) @ J ) @ ( set_nat2 @ Xs ) ) ) ) ) ).

% partition_by_nth_nth_elem
thf(fact_262_partition__by__nth__nth__elem,axiom,
    ! [Ys: list_nat,Xs: list_Ground_gmctxt_a,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
         => ( member3306386364193653941ctxt_a @ ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ ( multih5172578161653122718ctxt_a @ Xs @ Ys ) @ I ) @ J ) @ ( set_Ground_gmctxt_a2 @ Xs ) ) ) ) ) ).

% partition_by_nth_nth_elem
thf(fact_263_partition__by__nth__nth__elem,axiom,
    ! [Ys: list_nat,Xs: list_P8469869581646625389at_nat,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s3679842834875189465at_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
         => ( member8206827879206165904at_nat @ ( nth_Pr6744343527793145070at_nat @ ( nth_li8805938130461222772at_nat @ ( multih7555706055023738855at_nat @ Xs @ Ys ) @ I ) @ J ) @ ( set_Pr5518436109238095868at_nat @ Xs ) ) ) ) ) ).

% partition_by_nth_nth_elem
thf(fact_264_length__partition__by__nth,axiom,
    ! [Ys: list_nat,Xs: list_list_b,I: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s420531428170919671list_b @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( size_s420531428170919671list_b @ ( nth_list_list_b @ ( multih2267195492336891525list_b @ Xs @ Ys ) @ I ) )
          = ( nth_nat @ Ys @ I ) ) ) ) ).

% length_partition_by_nth
thf(fact_265_length__partition__by__nth,axiom,
    ! [Ys: list_nat,Xs: list_b,I: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_size_list_b @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( size_size_list_b @ ( nth_list_b @ ( multih3537974503344769663n_by_b @ Xs @ Ys ) @ I ) )
          = ( nth_nat @ Ys @ I ) ) ) ) ).

% length_partition_by_nth
thf(fact_266_length__partition__by__nth,axiom,
    ! [Ys: list_nat,Xs: list_nat,I: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( size_size_list_nat @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs @ Ys ) @ I ) )
          = ( nth_nat @ Ys @ I ) ) ) ) ).

% length_partition_by_nth
thf(fact_267_length__partition__by__nth,axiom,
    ! [Ys: list_nat,Xs: list_Ground_gmctxt_a,I: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ ( multih5172578161653122718ctxt_a @ Xs @ Ys ) @ I ) )
          = ( nth_nat @ Ys @ I ) ) ) ) ).

% length_partition_by_nth
thf(fact_268_length__partition__by__nth,axiom,
    ! [Ys: list_nat,Xs: list_P8469869581646625389at_nat,I: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s3679842834875189465at_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( size_s3679842834875189465at_nat @ ( nth_li8805938130461222772at_nat @ ( multih7555706055023738855at_nat @ Xs @ Ys ) @ I ) )
          = ( nth_nat @ Ys @ I ) ) ) ) ).

% length_partition_by_nth
thf(fact_269_Ball__set__partition__by,axiom,
    ! [Ys: list_nat,Xs: list_list_b,P: list_b > $o] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s420531428170919671list_b @ Xs ) )
     => ( ( ! [X2: list_list_b] :
              ( ( member_list_list_b @ X2 @ ( set_list_list_b2 @ ( multih2267195492336891525list_b @ Xs @ Ys ) ) )
             => ! [Y: list_b] :
                  ( ( member_list_b @ Y @ ( set_list_b2 @ X2 ) )
                 => ( P @ Y ) ) ) )
        = ( ! [X2: list_b] :
              ( ( member_list_b @ X2 @ ( set_list_b2 @ Xs ) )
             => ( P @ X2 ) ) ) ) ) ).

% Ball_set_partition_by
thf(fact_270_Ball__set__partition__by,axiom,
    ! [Ys: list_nat,Xs: list_b,P: b > $o] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_size_list_b @ Xs ) )
     => ( ( ! [X2: list_b] :
              ( ( member_list_b @ X2 @ ( set_list_b2 @ ( multih3537974503344769663n_by_b @ Xs @ Ys ) ) )
             => ! [Y: b] :
                  ( ( member_b @ Y @ ( set_b2 @ X2 ) )
                 => ( P @ Y ) ) ) )
        = ( ! [X2: b] :
              ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
             => ( P @ X2 ) ) ) ) ) ).

% Ball_set_partition_by
thf(fact_271_Ball__set__partition__by,axiom,
    ! [Ys: list_nat,Xs: list_nat,P: nat > $o] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_size_list_nat @ Xs ) )
     => ( ( ! [X2: list_nat] :
              ( ( member_list_nat @ X2 @ ( set_list_nat2 @ ( multih8544690682651757648by_nat @ Xs @ Ys ) ) )
             => ! [Y: nat] :
                  ( ( member_nat @ Y @ ( set_nat2 @ X2 ) )
                 => ( P @ Y ) ) ) )
        = ( ! [X2: nat] :
              ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
             => ( P @ X2 ) ) ) ) ) ).

% Ball_set_partition_by
thf(fact_272_Ball__set__partition__by,axiom,
    ! [Ys: list_nat,Xs: list_Ground_gmctxt_a,P: ground_gmctxt_a > $o] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( ! [X2: list_Ground_gmctxt_a] :
              ( ( member2714729351964853061ctxt_a @ X2 @ ( set_li3057194059929822169ctxt_a @ ( multih5172578161653122718ctxt_a @ Xs @ Ys ) ) )
             => ! [Y: ground_gmctxt_a] :
                  ( ( member3306386364193653941ctxt_a @ Y @ ( set_Ground_gmctxt_a2 @ X2 ) )
                 => ( P @ Y ) ) ) )
        = ( ! [X2: ground_gmctxt_a] :
              ( ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ Xs ) )
             => ( P @ X2 ) ) ) ) ) ).

% Ball_set_partition_by
thf(fact_273_Ball__set__partition__by,axiom,
    ! [Ys: list_nat,Xs: list_P8469869581646625389at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s3679842834875189465at_nat @ Xs ) )
     => ( ( ! [X2: list_P8469869581646625389at_nat] :
              ( ( member3799944675974059798at_nat @ X2 @ ( set_li7366234323595830146at_nat @ ( multih7555706055023738855at_nat @ Xs @ Ys ) ) )
             => ! [Y: produc859450856879609959at_nat] :
                  ( ( member8206827879206165904at_nat @ Y @ ( set_Pr5518436109238095868at_nat @ X2 ) )
                 => ( P @ Y ) ) ) )
        = ( ! [X2: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X2 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
             => ( P @ X2 ) ) ) ) ) ).

% Ball_set_partition_by
thf(fact_274_nth__sum__listI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I2 )
              = ( nth_nat @ Ys @ I2 ) ) )
       => ( ( groups4561878855575611511st_nat @ Xs )
          = ( groups4561878855575611511st_nat @ Ys ) ) ) ) ).

% nth_sum_listI
thf(fact_275_nth__equalityE,axiom,
    ! [Xs: list_list_nat,Ys: list_list_nat] :
      ( ( Xs = Ys )
     => ~ ( ( ( size_s3023201423986296836st_nat @ Xs )
            = ( size_s3023201423986296836st_nat @ Ys ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs ) )
               => ( ( nth_list_nat @ Xs @ I4 )
                  = ( nth_list_nat @ Ys @ I4 ) ) ) ) ) ).

% nth_equalityE
thf(fact_276_nth__equalityE,axiom,
    ! [Xs: list_l1578326660578982846ctxt_a,Ys: list_l1578326660578982846ctxt_a] :
      ( ( Xs = Ys )
     => ~ ( ( ( size_s6994882312647770194ctxt_a @ Xs )
            = ( size_s6994882312647770194ctxt_a @ Ys ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
               => ( ( nth_li6678718213161987943ctxt_a @ Xs @ I4 )
                  = ( nth_li6678718213161987943ctxt_a @ Ys @ I4 ) ) ) ) ) ).

% nth_equalityE
thf(fact_277_nth__equalityE,axiom,
    ! [Xs: list_l3298181151656792051at_nat,Ys: list_l3298181151656792051at_nat] :
      ( ( Xs = Ys )
     => ~ ( ( ( size_s6878757106947045471at_nat @ Xs )
            = ( size_s6878757106947045471at_nat @ Ys ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_s6878757106947045471at_nat @ Xs ) )
               => ( ( nth_li8805938130461222772at_nat @ Xs @ I4 )
                  = ( nth_li8805938130461222772at_nat @ Ys @ I4 ) ) ) ) ) ).

% nth_equalityE
thf(fact_278_nth__equalityE,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
      ( ( Xs = Ys )
     => ~ ( ( ( size_s962548663205356226ctxt_a @ Xs )
            = ( size_s962548663205356226ctxt_a @ Ys ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Xs ) )
               => ( ( nth_Ground_gmctxt_a @ Xs @ I4 )
                  = ( nth_Ground_gmctxt_a @ Ys @ I4 ) ) ) ) ) ).

% nth_equalityE
thf(fact_279_nth__equalityE,axiom,
    ! [Xs: list_b,Ys: list_b] :
      ( ( Xs = Ys )
     => ~ ( ( ( size_size_list_b @ Xs )
            = ( size_size_list_b @ Ys ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_size_list_b @ Xs ) )
               => ( ( nth_b @ Xs @ I4 )
                  = ( nth_b @ Ys @ I4 ) ) ) ) ) ).

% nth_equalityE
thf(fact_280_nth__equalityE,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
     => ~ ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
               => ( ( nth_nat @ Xs @ I4 )
                  = ( nth_nat @ Ys @ I4 ) ) ) ) ) ).

% nth_equalityE
thf(fact_281_nth__equalityE,axiom,
    ! [Xs: list_list_b,Ys: list_list_b] :
      ( ( Xs = Ys )
     => ~ ( ( ( size_s420531428170919671list_b @ Xs )
            = ( size_s420531428170919671list_b @ Ys ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_s420531428170919671list_b @ Xs ) )
               => ( ( nth_list_b @ Xs @ I4 )
                  = ( nth_list_b @ Ys @ I4 ) ) ) ) ) ).

% nth_equalityE
thf(fact_282_nth__equalityE,axiom,
    ! [Xs: list_P8469869581646625389at_nat,Ys: list_P8469869581646625389at_nat] :
      ( ( Xs = Ys )
     => ~ ( ( ( size_s3679842834875189465at_nat @ Xs )
            = ( size_s3679842834875189465at_nat @ Ys ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_s3679842834875189465at_nat @ Xs ) )
               => ( ( nth_Pr6744343527793145070at_nat @ Xs @ I4 )
                  = ( nth_Pr6744343527793145070at_nat @ Ys @ I4 ) ) ) ) ) ).

% nth_equalityE
thf(fact_283_partition__by__nth__nth_I1_J,axiom,
    ! [Ys: list_nat,Xs: list_list_nat,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s3023201423986296836st_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
         => ( ( nth_list_nat @ ( nth_list_list_nat @ ( multih5630945896903731040st_nat @ Xs @ Ys ) @ I ) @ J )
            = ( nth_list_nat @ Xs @ ( multih7279076602905066526by_idx @ ( size_s3023201423986296836st_nat @ Xs ) @ Ys @ I @ J ) ) ) ) ) ) ).

% partition_by_nth_nth(1)
thf(fact_284_partition__by__nth__nth_I1_J,axiom,
    ! [Ys: list_nat,Xs: list_l1578326660578982846ctxt_a,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s6994882312647770194ctxt_a @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
         => ( ( nth_li6678718213161987943ctxt_a @ ( nth_li3520809774789768823ctxt_a @ ( multih1205877868680985774ctxt_a @ Xs @ Ys ) @ I ) @ J )
            = ( nth_li6678718213161987943ctxt_a @ Xs @ ( multih7279076602905066526by_idx @ ( size_s6994882312647770194ctxt_a @ Xs ) @ Ys @ I @ J ) ) ) ) ) ) ).

% partition_by_nth_nth(1)
thf(fact_285_partition__by__nth__nth_I1_J,axiom,
    ! [Ys: list_nat,Xs: list_l3298181151656792051at_nat,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s6878757106947045471at_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
         => ( ( nth_li8805938130461222772at_nat @ ( nth_li2593672534642986362at_nat @ ( multih560249069488021997at_nat @ Xs @ Ys ) @ I ) @ J )
            = ( nth_li8805938130461222772at_nat @ Xs @ ( multih7279076602905066526by_idx @ ( size_s6878757106947045471at_nat @ Xs ) @ Ys @ I @ J ) ) ) ) ) ) ).

% partition_by_nth_nth(1)
thf(fact_286_partition__by__nth__nth_I1_J,axiom,
    ! [Ys: list_nat,Xs: list_list_b,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s420531428170919671list_b @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
         => ( ( nth_list_b @ ( nth_list_list_b @ ( multih2267195492336891525list_b @ Xs @ Ys ) @ I ) @ J )
            = ( nth_list_b @ Xs @ ( multih7279076602905066526by_idx @ ( size_s420531428170919671list_b @ Xs ) @ Ys @ I @ J ) ) ) ) ) ) ).

% partition_by_nth_nth(1)
thf(fact_287_partition__by__nth__nth_I1_J,axiom,
    ! [Ys: list_nat,Xs: list_b,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_size_list_b @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
         => ( ( nth_b @ ( nth_list_b @ ( multih3537974503344769663n_by_b @ Xs @ Ys ) @ I ) @ J )
            = ( nth_b @ Xs @ ( multih7279076602905066526by_idx @ ( size_size_list_b @ Xs ) @ Ys @ I @ J ) ) ) ) ) ) ).

% partition_by_nth_nth(1)
thf(fact_288_partition__by__nth__nth_I1_J,axiom,
    ! [Ys: list_nat,Xs: list_nat,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
         => ( ( nth_nat @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs @ Ys ) @ I ) @ J )
            = ( nth_nat @ Xs @ ( multih7279076602905066526by_idx @ ( size_size_list_nat @ Xs ) @ Ys @ I @ J ) ) ) ) ) ) ).

% partition_by_nth_nth(1)
thf(fact_289_partition__by__nth__nth_I1_J,axiom,
    ! [Ys: list_nat,Xs: list_Ground_gmctxt_a,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
         => ( ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ ( multih5172578161653122718ctxt_a @ Xs @ Ys ) @ I ) @ J )
            = ( nth_Ground_gmctxt_a @ Xs @ ( multih7279076602905066526by_idx @ ( size_s962548663205356226ctxt_a @ Xs ) @ Ys @ I @ J ) ) ) ) ) ) ).

% partition_by_nth_nth(1)
thf(fact_290_partition__by__nth__nth_I1_J,axiom,
    ! [Ys: list_nat,Xs: list_P8469869581646625389at_nat,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s3679842834875189465at_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
         => ( ( nth_Pr6744343527793145070at_nat @ ( nth_li8805938130461222772at_nat @ ( multih7555706055023738855at_nat @ Xs @ Ys ) @ I ) @ J )
            = ( nth_Pr6744343527793145070at_nat @ Xs @ ( multih7279076602905066526by_idx @ ( size_s3679842834875189465at_nat @ Xs ) @ Ys @ I @ J ) ) ) ) ) ) ).

% partition_by_nth_nth(1)
thf(fact_291_length__partition__by,axiom,
    ! [Xs: list_b,Ys: list_nat] :
      ( ( size_s420531428170919671list_b @ ( multih3537974503344769663n_by_b @ Xs @ Ys ) )
      = ( size_size_list_nat @ Ys ) ) ).

% length_partition_by
thf(fact_292_length__partition__by,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_s3023201423986296836st_nat @ ( multih8544690682651757648by_nat @ Xs @ Ys ) )
      = ( size_size_list_nat @ Ys ) ) ).

% length_partition_by
thf(fact_293_length__partition__by,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_nat] :
      ( ( size_s6994882312647770194ctxt_a @ ( multih5172578161653122718ctxt_a @ Xs @ Ys ) )
      = ( size_size_list_nat @ Ys ) ) ).

% length_partition_by
thf(fact_294_length__partition__by,axiom,
    ! [Xs: list_P8469869581646625389at_nat,Ys: list_nat] :
      ( ( size_s6878757106947045471at_nat @ ( multih7555706055023738855at_nat @ Xs @ Ys ) )
      = ( size_size_list_nat @ Ys ) ) ).

% length_partition_by
thf(fact_295_permut__sound,axiom,
    ! [I: nat,As: list_list_nat,F: nat > nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ As ) )
     => ( ( nth_list_nat @ ( missin2960327169605006088st_nat @ As @ F ) @ I )
        = ( nth_list_nat @ As @ ( F @ I ) ) ) ) ).

% permut_sound
thf(fact_296_permut__sound,axiom,
    ! [I: nat,As: list_l1578326660578982846ctxt_a,F: nat > nat] :
      ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ As ) )
     => ( ( nth_li6678718213161987943ctxt_a @ ( missin7467934043202574422ctxt_a @ As @ F ) @ I )
        = ( nth_li6678718213161987943ctxt_a @ As @ ( F @ I ) ) ) ) ).

% permut_sound
thf(fact_297_permut__sound,axiom,
    ! [I: nat,As: list_l3298181151656792051at_nat,F: nat > nat] :
      ( ( ord_less_nat @ I @ ( size_s6878757106947045471at_nat @ As ) )
     => ( ( nth_li8805938130461222772at_nat @ ( missin2985456621902692421at_nat @ As @ F ) @ I )
        = ( nth_li8805938130461222772at_nat @ As @ ( F @ I ) ) ) ) ).

% permut_sound
thf(fact_298_permut__sound,axiom,
    ! [I: nat,As: list_Ground_gmctxt_a,F: nat > nat] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ As ) )
     => ( ( nth_Ground_gmctxt_a @ ( missin4491563000646178374ctxt_a @ As @ F ) @ I )
        = ( nth_Ground_gmctxt_a @ As @ ( F @ I ) ) ) ) ).

% permut_sound
thf(fact_299_permut__sound,axiom,
    ! [I: nat,As: list_b,F: nat > nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ As ) )
     => ( ( nth_b @ ( missing_permut_b @ As @ F ) @ I )
        = ( nth_b @ As @ ( F @ I ) ) ) ) ).

% permut_sound
thf(fact_300_permut__sound,axiom,
    ! [I: nat,As: list_nat,F: nat > nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ As ) )
     => ( ( nth_nat @ ( missing_permut_nat @ As @ F ) @ I )
        = ( nth_nat @ As @ ( F @ I ) ) ) ) ).

% permut_sound
thf(fact_301_permut__sound,axiom,
    ! [I: nat,As: list_list_b,F: nat > nat] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ As ) )
     => ( ( nth_list_b @ ( missin8258838938715587805list_b @ As @ F ) @ I )
        = ( nth_list_b @ As @ ( F @ I ) ) ) ) ).

% permut_sound
thf(fact_302_permut__sound,axiom,
    ! [I: nat,As: list_P8469869581646625389at_nat,F: nat > nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ As ) )
     => ( ( nth_Pr6744343527793145070at_nat @ ( missin1395697627405724223at_nat @ As @ F ) @ I )
        = ( nth_Pr6744343527793145070at_nat @ As @ ( F @ I ) ) ) ) ).

% permut_sound
thf(fact_303_mem__Collect__eq,axiom,
    ! [A: b,P: b > $o] :
      ( ( member_b @ A @ ( collect_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_304_mem__Collect__eq,axiom,
    ! [A: produc859450856879609959at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ A @ ( collec7088162979684241874at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_305_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_306_mem__Collect__eq,axiom,
    ! [A: ground_gmctxt_a,P: ground_gmctxt_a > $o] :
      ( ( member3306386364193653941ctxt_a @ A @ ( collec8774204703398931443ctxt_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_307_Collect__mem__eq,axiom,
    ! [A2: set_b] :
      ( ( collect_b
        @ ^ [X2: b] : ( member_b @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_308_Collect__mem__eq,axiom,
    ! [A2: set_Pr8693737435421807431at_nat] :
      ( ( collec7088162979684241874at_nat
        @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_309_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_310_Collect__mem__eq,axiom,
    ! [A2: set_Ground_gmctxt_a] :
      ( ( collec8774204703398931443ctxt_a
        @ ^ [X2: ground_gmctxt_a] : ( member3306386364193653941ctxt_a @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_311_map__map__partition__by,axiom,
    ! [F: b > b,Xs: list_b,Ys: list_nat] :
      ( ( map_list_b_list_b @ ( map_b_b @ F ) @ ( multih3537974503344769663n_by_b @ Xs @ Ys ) )
      = ( multih3537974503344769663n_by_b @ ( map_b_b @ F @ Xs ) @ Ys ) ) ).

% map_map_partition_by
thf(fact_312_map__map__partition__by,axiom,
    ! [F: b > nat,Xs: list_b,Ys: list_nat] :
      ( ( map_list_b_list_nat @ ( map_b_nat @ F ) @ ( multih3537974503344769663n_by_b @ Xs @ Ys ) )
      = ( multih8544690682651757648by_nat @ ( map_b_nat @ F @ Xs ) @ Ys ) ) ).

% map_map_partition_by
thf(fact_313_map__map__partition__by,axiom,
    ! [F: nat > b,Xs: list_nat,Ys: list_nat] :
      ( ( map_list_nat_list_b @ ( map_nat_b @ F ) @ ( multih8544690682651757648by_nat @ Xs @ Ys ) )
      = ( multih3537974503344769663n_by_b @ ( map_nat_b @ F @ Xs ) @ Ys ) ) ).

% map_map_partition_by
thf(fact_314_map__map__partition__by,axiom,
    ! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
      ( ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ ( multih8544690682651757648by_nat @ Xs @ Ys ) )
      = ( multih8544690682651757648by_nat @ ( map_nat_nat @ F @ Xs ) @ Ys ) ) ).

% map_map_partition_by
thf(fact_315_map__map__partition__by,axiom,
    ! [F: b > list_b,Xs: list_b,Ys: list_nat] :
      ( ( map_li4464339650582430090list_b @ ( map_b_list_b @ F ) @ ( multih3537974503344769663n_by_b @ Xs @ Ys ) )
      = ( multih2267195492336891525list_b @ ( map_b_list_b @ F @ Xs ) @ Ys ) ) ).

% map_map_partition_by
thf(fact_316_map__map__partition__by,axiom,
    ! [F: b > ground_gmctxt_a,Xs: list_b,Ys: list_nat] :
      ( ( map_li4667870289254436335ctxt_a @ ( map_b_3587768286370285017ctxt_a @ F ) @ ( multih3537974503344769663n_by_b @ Xs @ Ys ) )
      = ( multih5172578161653122718ctxt_a @ ( map_b_3587768286370285017ctxt_a @ F @ Xs ) @ Ys ) ) ).

% map_map_partition_by
thf(fact_317_map__map__partition__by,axiom,
    ! [F: nat > ground_gmctxt_a,Xs: list_nat,Ys: list_nat] :
      ( ( map_li3283194380818682708ctxt_a @ ( map_na2753241459123782580ctxt_a @ F ) @ ( multih8544690682651757648by_nat @ Xs @ Ys ) )
      = ( multih5172578161653122718ctxt_a @ ( map_na2753241459123782580ctxt_a @ F @ Xs ) @ Ys ) ) ).

% map_map_partition_by
thf(fact_318_map__map__partition__by,axiom,
    ! [F: ground_gmctxt_a > b,Xs: list_Ground_gmctxt_a,Ys: list_nat] :
      ( ( map_li4808364546485981329list_b @ ( map_Gr2594013990000539163xt_a_b @ F ) @ ( multih5172578161653122718ctxt_a @ Xs @ Ys ) )
      = ( multih3537974503344769663n_by_b @ ( map_Gr2594013990000539163xt_a_b @ F @ Xs ) @ Ys ) ) ).

% map_map_partition_by
thf(fact_319_map__map__partition__by,axiom,
    ! [F: ground_gmctxt_a > nat,Xs: list_Ground_gmctxt_a,Ys: list_nat] :
      ( ( map_li7775015282390606420st_nat @ ( map_Gr1679544123147727284_a_nat @ F ) @ ( multih5172578161653122718ctxt_a @ Xs @ Ys ) )
      = ( multih8544690682651757648by_nat @ ( map_Gr1679544123147727284_a_nat @ F @ Xs ) @ Ys ) ) ).

% map_map_partition_by
thf(fact_320_map__map__partition__by,axiom,
    ! [F: ground_gmctxt_a > ground_gmctxt_a,Xs: list_Ground_gmctxt_a,Ys: list_nat] :
      ( ( map_li7659068976007724962ctxt_a @ ( map_Gr980675717119874050ctxt_a @ F ) @ ( multih5172578161653122718ctxt_a @ Xs @ Ys ) )
      = ( multih5172578161653122718ctxt_a @ ( map_Gr980675717119874050ctxt_a @ F @ Xs ) @ Ys ) ) ).

% map_map_partition_by
thf(fact_321_permut__aux__sound,axiom,
    ! [I: nat,As: list_list_nat,F: nat > nat,Bs: list_list_nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ As ) )
     => ( ( nth_list_nat @ ( missin3204601078308456222st_nat @ As @ F @ Bs ) @ I )
        = ( nth_list_nat @ Bs @ ( F @ I ) ) ) ) ).

% permut_aux_sound
thf(fact_322_permut__aux__sound,axiom,
    ! [I: nat,As: list_l1578326660578982846ctxt_a,F: nat > nat,Bs: list_l1578326660578982846ctxt_a] :
      ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ As ) )
     => ( ( nth_li6678718213161987943ctxt_a @ ( missin8538012249342617708ctxt_a @ As @ F @ Bs ) @ I )
        = ( nth_li6678718213161987943ctxt_a @ Bs @ ( F @ I ) ) ) ) ).

% permut_aux_sound
thf(fact_323_permut__aux__sound,axiom,
    ! [I: nat,As: list_l3298181151656792051at_nat,F: nat > nat,Bs: list_l3298181151656792051at_nat] :
      ( ( ord_less_nat @ I @ ( size_s6878757106947045471at_nat @ As ) )
     => ( ( nth_li8805938130461222772at_nat @ ( missin3013719938226745775at_nat @ As @ F @ Bs ) @ I )
        = ( nth_li8805938130461222772at_nat @ Bs @ ( F @ I ) ) ) ) ).

% permut_aux_sound
thf(fact_324_permut__aux__sound,axiom,
    ! [I: nat,As: list_Ground_gmctxt_a,F: nat > nat,Bs: list_Ground_gmctxt_a] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ As ) )
     => ( ( nth_Ground_gmctxt_a @ ( missin1343833451449182556ctxt_a @ As @ F @ Bs ) @ I )
        = ( nth_Ground_gmctxt_a @ Bs @ ( F @ I ) ) ) ) ).

% permut_aux_sound
thf(fact_325_permut__aux__sound,axiom,
    ! [I: nat,As: list_b,F: nat > nat,Bs: list_b] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ As ) )
     => ( ( nth_b @ ( missing_permut_aux_b @ As @ F @ Bs ) @ I )
        = ( nth_b @ Bs @ ( F @ I ) ) ) ) ).

% permut_aux_sound
thf(fact_326_permut__aux__sound,axiom,
    ! [I: nat,As: list_nat,F: nat > nat,Bs: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ As ) )
     => ( ( nth_nat @ ( missin1888654203714970382ux_nat @ As @ F @ Bs ) @ I )
        = ( nth_nat @ Bs @ ( F @ I ) ) ) ) ).

% permut_aux_sound
thf(fact_327_permut__aux__sound,axiom,
    ! [I: nat,As: list_list_b,F: nat > nat,Bs: list_list_b] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ As ) )
     => ( ( nth_list_b @ ( missin5389789488504408647list_b @ As @ F @ Bs ) @ I )
        = ( nth_list_b @ Bs @ ( F @ I ) ) ) ) ).

% permut_aux_sound
thf(fact_328_permut__aux__sound,axiom,
    ! [I: nat,As: list_P8469869581646625389at_nat,F: nat > nat,Bs: list_P8469869581646625389at_nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ As ) )
     => ( ( nth_Pr6744343527793145070at_nat @ ( missin7899315033207363753at_nat @ As @ F @ Bs ) @ I )
        = ( nth_Pr6744343527793145070at_nat @ Bs @ ( F @ I ) ) ) ) ).

% permut_aux_sound
thf(fact_329_partition__by__nth__nth_I2_J,axiom,
    ! [Ys: list_nat,Xs: list_Ground_gmctxt_a,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
         => ( ord_less_nat @ ( multih7279076602905066526by_idx @ ( size_s962548663205356226ctxt_a @ Xs ) @ Ys @ I @ J ) @ ( size_s962548663205356226ctxt_a @ Xs ) ) ) ) ) ).

% partition_by_nth_nth(2)
thf(fact_330_partition__by__nth__nth_I2_J,axiom,
    ! [Ys: list_nat,Xs: list_b,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_size_list_b @ Xs ) )
     => ( ( 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_b @ Xs ) @ Ys @ I @ J ) @ ( size_size_list_b @ Xs ) ) ) ) ) ).

% partition_by_nth_nth(2)
thf(fact_331_partition__by__nth__nth_I2_J,axiom,
    ! [Ys: list_nat,Xs: list_nat,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_size_list_nat @ Xs ) )
     => ( ( 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 @ Xs ) @ Ys @ I @ J ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).

% partition_by_nth_nth(2)
thf(fact_332_partition__by__nth__nth_I2_J,axiom,
    ! [Ys: list_nat,Xs: list_list_b,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s420531428170919671list_b @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
         => ( ord_less_nat @ ( multih7279076602905066526by_idx @ ( size_s420531428170919671list_b @ Xs ) @ Ys @ I @ J ) @ ( size_s420531428170919671list_b @ Xs ) ) ) ) ) ).

% partition_by_nth_nth(2)
thf(fact_333_partition__by__nth__nth_I2_J,axiom,
    ! [Ys: list_nat,Xs: list_P8469869581646625389at_nat,I: nat,J: nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s3679842834875189465at_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( ord_less_nat @ J @ ( nth_nat @ Ys @ I ) )
         => ( ord_less_nat @ ( multih7279076602905066526by_idx @ ( size_s3679842834875189465at_nat @ Xs ) @ Ys @ I @ J ) @ ( size_s3679842834875189465at_nat @ Xs ) ) ) ) ) ).

% partition_by_nth_nth(2)
thf(fact_334_partition__by__concat__id,axiom,
    ! [Xss: list_list_list_b,Ys: list_nat] :
      ( ( ( size_s9106009106869804029list_b @ Xss )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
           => ( ( size_s420531428170919671list_b @ ( nth_list_list_b @ Xss @ I2 ) )
              = ( nth_nat @ Ys @ I2 ) ) )
       => ( ( multih2267195492336891525list_b @ ( concat_list_b @ Xss ) @ Ys )
          = Xss ) ) ) ).

% partition_by_concat_id
thf(fact_335_partition__by__concat__id,axiom,
    ! [Xss: list_list_b,Ys: list_nat] :
      ( ( ( size_s420531428170919671list_b @ Xss )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
           => ( ( size_size_list_b @ ( nth_list_b @ Xss @ I2 ) )
              = ( nth_nat @ Ys @ I2 ) ) )
       => ( ( multih3537974503344769663n_by_b @ ( concat_b @ Xss ) @ Ys )
          = Xss ) ) ) ).

% partition_by_concat_id
thf(fact_336_partition__by__concat__id,axiom,
    ! [Xss: list_list_nat,Ys: list_nat] :
      ( ( ( size_s3023201423986296836st_nat @ Xss )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Xss @ I2 ) )
              = ( nth_nat @ Ys @ I2 ) ) )
       => ( ( multih8544690682651757648by_nat @ ( concat_nat @ Xss ) @ Ys )
          = Xss ) ) ) ).

% partition_by_concat_id
thf(fact_337_partition__by__concat__id,axiom,
    ! [Xss: list_l1578326660578982846ctxt_a,Ys: list_nat] :
      ( ( ( size_s6994882312647770194ctxt_a @ Xss )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
           => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xss @ I2 ) )
              = ( nth_nat @ Ys @ I2 ) ) )
       => ( ( multih5172578161653122718ctxt_a @ ( concat3993834769610107037ctxt_a @ Xss ) @ Ys )
          = Xss ) ) ) ).

% partition_by_concat_id
thf(fact_338_partition__by__concat__id,axiom,
    ! [Xss: list_l3298181151656792051at_nat,Ys: list_nat] :
      ( ( ( size_s6878757106947045471at_nat @ Xss )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
           => ( ( size_s3679842834875189465at_nat @ ( nth_li8805938130461222772at_nat @ Xss @ I2 ) )
              = ( nth_nat @ Ys @ I2 ) ) )
       => ( ( multih7555706055023738855at_nat @ ( concat5616020275982107560at_nat @ Xss ) @ Ys )
          = Xss ) ) ) ).

% partition_by_concat_id
thf(fact_339_in__set__product__lists__length,axiom,
    ! [Xs: list_Ground_gmctxt_a,Xss: list_l1578326660578982846ctxt_a] :
      ( ( member2714729351964853061ctxt_a @ Xs @ ( set_li3057194059929822169ctxt_a @ ( produc6102553363301462639ctxt_a @ Xss ) ) )
     => ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_s6994882312647770194ctxt_a @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_340_in__set__product__lists__length,axiom,
    ! [Xs: list_b,Xss: list_list_b] :
      ( ( member_list_b @ Xs @ ( set_list_b2 @ ( product_lists_b @ Xss ) ) )
     => ( ( size_size_list_b @ Xs )
        = ( size_s420531428170919671list_b @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_341_in__set__product__lists__length,axiom,
    ! [Xs: list_nat,Xss: list_list_nat] :
      ( ( member_list_nat @ Xs @ ( set_list_nat2 @ ( product_lists_nat @ Xss ) ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s3023201423986296836st_nat @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_342_in__set__product__lists__length,axiom,
    ! [Xs: list_list_b,Xss: list_list_list_b] :
      ( ( member_list_list_b @ Xs @ ( set_list_list_b2 @ ( product_lists_list_b @ Xss ) ) )
     => ( ( size_s420531428170919671list_b @ Xs )
        = ( size_s9106009106869804029list_b @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_343_in__set__product__lists__length,axiom,
    ! [Xs: list_P8469869581646625389at_nat,Xss: list_l3298181151656792051at_nat] :
      ( ( member3799944675974059798at_nat @ Xs @ ( set_li7366234323595830146at_nat @ ( produc1375813136864985686at_nat @ Xss ) ) )
     => ( ( size_s3679842834875189465at_nat @ Xs )
        = ( size_s6878757106947045471at_nat @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_344_remdups__sort_I2_J,axiom,
    ! [Xs: list_nat] :
      ( ( set_nat2 @ ( missin6101193410121742181rt_nat @ Xs ) )
      = ( set_nat2 @ Xs ) ) ).

% remdups_sort(2)
thf(fact_345_trancl__listp_Obase,axiom,
    ! [Xs: list_list_nat,Ys: list_list_nat,R: list_nat > list_nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ys ) )
           => ( R @ ( nth_list_nat @ Xs @ I2 ) @ ( nth_list_nat @ Ys @ I2 ) ) )
       => ( trancl267208031524270810st_nat @ R @ Xs @ Ys ) ) ) ).

% trancl_listp.base
thf(fact_346_trancl__listp_Obase,axiom,
    ! [Xs: list_l1578326660578982846ctxt_a,Ys: list_l1578326660578982846ctxt_a,R: list_Ground_gmctxt_a > list_Ground_gmctxt_a > $o] :
      ( ( ( size_s6994882312647770194ctxt_a @ Xs )
        = ( size_s6994882312647770194ctxt_a @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6994882312647770194ctxt_a @ Ys ) )
           => ( R @ ( nth_li6678718213161987943ctxt_a @ Xs @ I2 ) @ ( nth_li6678718213161987943ctxt_a @ Ys @ I2 ) ) )
       => ( trancl4044344332876950568ctxt_a @ R @ Xs @ Ys ) ) ) ).

% trancl_listp.base
thf(fact_347_trancl__listp_Obase,axiom,
    ! [Xs: list_l3298181151656792051at_nat,Ys: list_l3298181151656792051at_nat,R: list_P8469869581646625389at_nat > list_P8469869581646625389at_nat > $o] :
      ( ( ( size_s6878757106947045471at_nat @ Xs )
        = ( size_s6878757106947045471at_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6878757106947045471at_nat @ Ys ) )
           => ( R @ ( nth_li8805938130461222772at_nat @ Xs @ I2 ) @ ( nth_li8805938130461222772at_nat @ Ys @ I2 ) ) )
       => ( trancl6797153713963726067at_nat @ R @ Xs @ Ys ) ) ) ).

% trancl_listp.base
thf(fact_348_trancl__listp_Obase,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,R: ground_gmctxt_a > ground_gmctxt_a > $o] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Ys ) )
           => ( R @ ( nth_Ground_gmctxt_a @ Xs @ I2 ) @ ( nth_Ground_gmctxt_a @ Ys @ I2 ) ) )
       => ( trancl3815965565430463256ctxt_a @ R @ Xs @ Ys ) ) ) ).

% trancl_listp.base
thf(fact_349_trancl__listp_Obase,axiom,
    ! [Xs: list_b,Ys: list_b,R: b > b > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Ys ) )
           => ( R @ ( nth_b @ Xs @ I2 ) @ ( nth_b @ Ys @ I2 ) ) )
       => ( trancl_listp_b @ R @ Xs @ Ys ) ) ) ).

% trancl_listp.base
thf(fact_350_trancl__listp_Obase,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: nat > nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
           => ( R @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) )
       => ( trancl_listp_nat @ R @ Xs @ Ys ) ) ) ).

% trancl_listp.base
thf(fact_351_trancl__listp_Obase,axiom,
    ! [Xs: list_list_b,Ys: list_list_b,R: list_b > list_b > $o] :
      ( ( ( size_s420531428170919671list_b @ Xs )
        = ( size_s420531428170919671list_b @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Ys ) )
           => ( R @ ( nth_list_b @ Xs @ I2 ) @ ( nth_list_b @ Ys @ I2 ) ) )
       => ( trancl_listp_list_b @ R @ Xs @ Ys ) ) ) ).

% trancl_listp.base
thf(fact_352_trancl__listp_Obase,axiom,
    ! [Xs: list_P8469869581646625389at_nat,Ys: list_P8469869581646625389at_nat,R: produc859450856879609959at_nat > produc859450856879609959at_nat > $o] :
      ( ( ( size_s3679842834875189465at_nat @ Xs )
        = ( size_s3679842834875189465at_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3679842834875189465at_nat @ Ys ) )
           => ( R @ ( nth_Pr6744343527793145070at_nat @ Xs @ I2 ) @ ( nth_Pr6744343527793145070at_nat @ Ys @ I2 ) ) )
       => ( trancl2313143553896500717at_nat @ R @ Xs @ Ys ) ) ) ).

% trancl_listp.base
thf(fact_353_concat__nth__nthI,axiom,
    ! [Ss: list_list_nat,Ts: list_list_nat,P: nat > nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Ss )
        = ( size_s3023201423986296836st_nat @ Ts ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ts ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Ss @ I2 ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ts @ I2 ) ) ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ts ) )
             => ! [J2: nat] :
                  ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ ( nth_list_nat @ Ts @ I2 ) ) )
                 => ( P @ ( nth_nat @ ( nth_list_nat @ Ss @ I2 ) @ J2 ) @ ( nth_nat @ ( nth_list_nat @ Ts @ I2 ) @ J2 ) ) ) )
         => ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( concat_nat @ Ts ) ) )
             => ( P @ ( nth_nat @ ( concat_nat @ Ss ) @ I4 ) @ ( nth_nat @ ( concat_nat @ Ts ) @ I4 ) ) ) ) ) ) ).

% concat_nth_nthI
thf(fact_354_concat__nth__nthI,axiom,
    ! [Ss: list_list_nat,Ts: list_list_b,P: nat > b > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Ss )
        = ( size_s420531428170919671list_b @ Ts ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Ts ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Ss @ I2 ) )
              = ( size_size_list_b @ ( nth_list_b @ Ts @ I2 ) ) ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Ts ) )
             => ! [J2: nat] :
                  ( ( ord_less_nat @ J2 @ ( size_size_list_b @ ( nth_list_b @ Ts @ I2 ) ) )
                 => ( P @ ( nth_nat @ ( nth_list_nat @ Ss @ I2 ) @ J2 ) @ ( nth_b @ ( nth_list_b @ Ts @ I2 ) @ J2 ) ) ) )
         => ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_b @ ( concat_b @ Ts ) ) )
             => ( P @ ( nth_nat @ ( concat_nat @ Ss ) @ I4 ) @ ( nth_b @ ( concat_b @ Ts ) @ I4 ) ) ) ) ) ) ).

% concat_nth_nthI
thf(fact_355_concat__nth__nthI,axiom,
    ! [Ss: list_list_b,Ts: list_list_nat,P: b > nat > $o] :
      ( ( ( size_s420531428170919671list_b @ Ss )
        = ( size_s3023201423986296836st_nat @ Ts ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ts ) )
           => ( ( size_size_list_b @ ( nth_list_b @ Ss @ I2 ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ts @ I2 ) ) ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ts ) )
             => ! [J2: nat] :
                  ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ ( nth_list_nat @ Ts @ I2 ) ) )
                 => ( P @ ( nth_b @ ( nth_list_b @ Ss @ I2 ) @ J2 ) @ ( nth_nat @ ( nth_list_nat @ Ts @ I2 ) @ J2 ) ) ) )
         => ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( concat_nat @ Ts ) ) )
             => ( P @ ( nth_b @ ( concat_b @ Ss ) @ I4 ) @ ( nth_nat @ ( concat_nat @ Ts ) @ I4 ) ) ) ) ) ) ).

% concat_nth_nthI
thf(fact_356_concat__nth__nthI,axiom,
    ! [Ss: list_list_b,Ts: list_list_b,P: b > b > $o] :
      ( ( ( size_s420531428170919671list_b @ Ss )
        = ( size_s420531428170919671list_b @ Ts ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Ts ) )
           => ( ( size_size_list_b @ ( nth_list_b @ Ss @ I2 ) )
              = ( size_size_list_b @ ( nth_list_b @ Ts @ I2 ) ) ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Ts ) )
             => ! [J2: nat] :
                  ( ( ord_less_nat @ J2 @ ( size_size_list_b @ ( nth_list_b @ Ts @ I2 ) ) )
                 => ( P @ ( nth_b @ ( nth_list_b @ Ss @ I2 ) @ J2 ) @ ( nth_b @ ( nth_list_b @ Ts @ I2 ) @ J2 ) ) ) )
         => ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_b @ ( concat_b @ Ts ) ) )
             => ( P @ ( nth_b @ ( concat_b @ Ss ) @ I4 ) @ ( nth_b @ ( concat_b @ Ts ) @ I4 ) ) ) ) ) ) ).

% concat_nth_nthI
thf(fact_357_concat__nth__nthI,axiom,
    ! [Ss: list_list_list_nat,Ts: list_list_nat,P: list_nat > nat > $o] :
      ( ( ( size_s6248950052170075156st_nat @ Ss )
        = ( size_s3023201423986296836st_nat @ Ts ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ts ) )
           => ( ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Ss @ I2 ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ts @ I2 ) ) ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ts ) )
             => ! [J2: nat] :
                  ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ ( nth_list_nat @ Ts @ I2 ) ) )
                 => ( P @ ( nth_list_nat @ ( nth_list_list_nat @ Ss @ I2 ) @ J2 ) @ ( nth_nat @ ( nth_list_nat @ Ts @ I2 ) @ J2 ) ) ) )
         => ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( concat_nat @ Ts ) ) )
             => ( P @ ( nth_list_nat @ ( concat_list_nat @ Ss ) @ I4 ) @ ( nth_nat @ ( concat_nat @ Ts ) @ I4 ) ) ) ) ) ) ).

% concat_nth_nthI
thf(fact_358_concat__nth__nthI,axiom,
    ! [Ss: list_l1578326660578982846ctxt_a,Ts: list_list_nat,P: ground_gmctxt_a > nat > $o] :
      ( ( ( size_s6994882312647770194ctxt_a @ Ss )
        = ( size_s3023201423986296836st_nat @ Ts ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ts ) )
           => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ss @ I2 ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ts @ I2 ) ) ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ts ) )
             => ! [J2: nat] :
                  ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ ( nth_list_nat @ Ts @ I2 ) ) )
                 => ( P @ ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ss @ I2 ) @ J2 ) @ ( nth_nat @ ( nth_list_nat @ Ts @ I2 ) @ J2 ) ) ) )
         => ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( concat_nat @ Ts ) ) )
             => ( P @ ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Ss ) @ I4 ) @ ( nth_nat @ ( concat_nat @ Ts ) @ I4 ) ) ) ) ) ) ).

% concat_nth_nthI
thf(fact_359_concat__nth__nthI,axiom,
    ! [Ss: list_list_nat,Ts: list_list_list_nat,P: nat > list_nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Ss )
        = ( size_s6248950052170075156st_nat @ Ts ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6248950052170075156st_nat @ Ts ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Ss @ I2 ) )
              = ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Ts @ I2 ) ) ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s6248950052170075156st_nat @ Ts ) )
             => ! [J2: nat] :
                  ( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Ts @ I2 ) ) )
                 => ( P @ ( nth_nat @ ( nth_list_nat @ Ss @ I2 ) @ J2 ) @ ( nth_list_nat @ ( nth_list_list_nat @ Ts @ I2 ) @ J2 ) ) ) )
         => ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ ( concat_list_nat @ Ts ) ) )
             => ( P @ ( nth_nat @ ( concat_nat @ Ss ) @ I4 ) @ ( nth_list_nat @ ( concat_list_nat @ Ts ) @ I4 ) ) ) ) ) ) ).

% concat_nth_nthI
thf(fact_360_concat__nth__nthI,axiom,
    ! [Ss: list_list_nat,Ts: list_l1578326660578982846ctxt_a,P: nat > ground_gmctxt_a > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Ss )
        = ( size_s6994882312647770194ctxt_a @ Ts ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6994882312647770194ctxt_a @ Ts ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Ss @ I2 ) )
              = ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ts @ I2 ) ) ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s6994882312647770194ctxt_a @ Ts ) )
             => ! [J2: nat] :
                  ( ( ord_less_nat @ J2 @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ts @ I2 ) ) )
                 => ( P @ ( nth_nat @ ( nth_list_nat @ Ss @ I2 ) @ J2 ) @ ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ts @ I2 ) @ J2 ) ) ) )
         => ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Ts ) ) )
             => ( P @ ( nth_nat @ ( concat_nat @ Ss ) @ I4 ) @ ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Ts ) @ I4 ) ) ) ) ) ) ).

% concat_nth_nthI
thf(fact_361_concat__nth__nthI,axiom,
    ! [Ss: list_list_nat,Ts: list_list_list_b,P: nat > list_b > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Ss )
        = ( size_s9106009106869804029list_b @ Ts ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s9106009106869804029list_b @ Ts ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Ss @ I2 ) )
              = ( size_s420531428170919671list_b @ ( nth_list_list_b @ Ts @ I2 ) ) ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s9106009106869804029list_b @ Ts ) )
             => ! [J2: nat] :
                  ( ( ord_less_nat @ J2 @ ( size_s420531428170919671list_b @ ( nth_list_list_b @ Ts @ I2 ) ) )
                 => ( P @ ( nth_nat @ ( nth_list_nat @ Ss @ I2 ) @ J2 ) @ ( nth_list_b @ ( nth_list_list_b @ Ts @ I2 ) @ J2 ) ) ) )
         => ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s420531428170919671list_b @ ( concat_list_b @ Ts ) ) )
             => ( P @ ( nth_nat @ ( concat_nat @ Ss ) @ I4 ) @ ( nth_list_b @ ( concat_list_b @ Ts ) @ I4 ) ) ) ) ) ) ).

% concat_nth_nthI
thf(fact_362_concat__nth__nthI,axiom,
    ! [Ss: list_list_list_b,Ts: list_list_nat,P: list_b > nat > $o] :
      ( ( ( size_s9106009106869804029list_b @ Ss )
        = ( size_s3023201423986296836st_nat @ Ts ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ts ) )
           => ( ( size_s420531428170919671list_b @ ( nth_list_list_b @ Ss @ I2 ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ts @ I2 ) ) ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ts ) )
             => ! [J2: nat] :
                  ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ ( nth_list_nat @ Ts @ I2 ) ) )
                 => ( P @ ( nth_list_b @ ( nth_list_list_b @ Ss @ I2 ) @ J2 ) @ ( nth_nat @ ( nth_list_nat @ Ts @ I2 ) @ J2 ) ) ) )
         => ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( concat_nat @ Ts ) ) )
             => ( P @ ( nth_list_b @ ( concat_list_b @ Ss ) @ I4 ) @ ( nth_nat @ ( concat_nat @ Ts ) @ I4 ) ) ) ) ) ) ).

% concat_nth_nthI
thf(fact_363_concat__all__nth,axiom,
    ! [Xs: list_list_nat,Ys: list_list_nat,P: nat > nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I2 ) ) ) )
       => ( ! [I2: nat,J2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
             => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) ) )
               => ( P @ ( nth_nat @ ( nth_list_nat @ Xs @ I2 ) @ J2 ) @ ( nth_nat @ ( nth_list_nat @ Ys @ I2 ) @ J2 ) ) ) )
         => ! [K2: nat] :
              ( ( ord_less_nat @ K2 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
             => ( P @ ( nth_nat @ ( concat_nat @ Xs ) @ K2 ) @ ( nth_nat @ ( concat_nat @ Ys ) @ K2 ) ) ) ) ) ) ).

% concat_all_nth
thf(fact_364_concat__all__nth,axiom,
    ! [Xs: list_list_nat,Ys: list_list_b,P: nat > b > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Xs )
        = ( size_s420531428170919671list_b @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) )
              = ( size_size_list_b @ ( nth_list_b @ Ys @ I2 ) ) ) )
       => ( ! [I2: nat,J2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
             => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) ) )
               => ( P @ ( nth_nat @ ( nth_list_nat @ Xs @ I2 ) @ J2 ) @ ( nth_b @ ( nth_list_b @ Ys @ I2 ) @ J2 ) ) ) )
         => ! [K2: nat] :
              ( ( ord_less_nat @ K2 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
             => ( P @ ( nth_nat @ ( concat_nat @ Xs ) @ K2 ) @ ( nth_b @ ( concat_b @ Ys ) @ K2 ) ) ) ) ) ) ).

% concat_all_nth
thf(fact_365_concat__all__nth,axiom,
    ! [Xs: list_list_b,Ys: list_list_nat,P: b > nat > $o] :
      ( ( ( size_s420531428170919671list_b @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Xs ) )
           => ( ( size_size_list_b @ ( nth_list_b @ Xs @ I2 ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I2 ) ) ) )
       => ( ! [I2: nat,J2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Xs ) )
             => ( ( ord_less_nat @ J2 @ ( size_size_list_b @ ( nth_list_b @ Xs @ I2 ) ) )
               => ( P @ ( nth_b @ ( nth_list_b @ Xs @ I2 ) @ J2 ) @ ( nth_nat @ ( nth_list_nat @ Ys @ I2 ) @ J2 ) ) ) )
         => ! [K2: nat] :
              ( ( ord_less_nat @ K2 @ ( size_size_list_b @ ( concat_b @ Xs ) ) )
             => ( P @ ( nth_b @ ( concat_b @ Xs ) @ K2 ) @ ( nth_nat @ ( concat_nat @ Ys ) @ K2 ) ) ) ) ) ) ).

% concat_all_nth
thf(fact_366_concat__all__nth,axiom,
    ! [Xs: list_list_b,Ys: list_list_b,P: b > b > $o] :
      ( ( ( size_s420531428170919671list_b @ Xs )
        = ( size_s420531428170919671list_b @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Xs ) )
           => ( ( size_size_list_b @ ( nth_list_b @ Xs @ I2 ) )
              = ( size_size_list_b @ ( nth_list_b @ Ys @ I2 ) ) ) )
       => ( ! [I2: nat,J2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Xs ) )
             => ( ( ord_less_nat @ J2 @ ( size_size_list_b @ ( nth_list_b @ Xs @ I2 ) ) )
               => ( P @ ( nth_b @ ( nth_list_b @ Xs @ I2 ) @ J2 ) @ ( nth_b @ ( nth_list_b @ Ys @ I2 ) @ J2 ) ) ) )
         => ! [K2: nat] :
              ( ( ord_less_nat @ K2 @ ( size_size_list_b @ ( concat_b @ Xs ) ) )
             => ( P @ ( nth_b @ ( concat_b @ Xs ) @ K2 ) @ ( nth_b @ ( concat_b @ Ys ) @ K2 ) ) ) ) ) ) ).

% concat_all_nth
thf(fact_367_concat__all__nth,axiom,
    ! [Xs: list_list_list_nat,Ys: list_list_nat,P: list_nat > nat > $o] :
      ( ( ( size_s6248950052170075156st_nat @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6248950052170075156st_nat @ Xs ) )
           => ( ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Xs @ I2 ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I2 ) ) ) )
       => ( ! [I2: nat,J2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s6248950052170075156st_nat @ Xs ) )
             => ( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Xs @ I2 ) ) )
               => ( P @ ( nth_list_nat @ ( nth_list_list_nat @ Xs @ I2 ) @ J2 ) @ ( nth_nat @ ( nth_list_nat @ Ys @ I2 ) @ J2 ) ) ) )
         => ! [K2: nat] :
              ( ( ord_less_nat @ K2 @ ( size_s3023201423986296836st_nat @ ( concat_list_nat @ Xs ) ) )
             => ( P @ ( nth_list_nat @ ( concat_list_nat @ Xs ) @ K2 ) @ ( nth_nat @ ( concat_nat @ Ys ) @ K2 ) ) ) ) ) ) ).

% concat_all_nth
thf(fact_368_concat__all__nth,axiom,
    ! [Xs: list_l1578326660578982846ctxt_a,Ys: list_list_nat,P: ground_gmctxt_a > nat > $o] :
      ( ( ( size_s6994882312647770194ctxt_a @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
           => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I2 ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I2 ) ) ) )
       => ( ! [I2: nat,J2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
             => ( ( ord_less_nat @ J2 @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I2 ) ) )
               => ( P @ ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I2 ) @ J2 ) @ ( nth_nat @ ( nth_list_nat @ Ys @ I2 ) @ J2 ) ) ) )
         => ! [K2: nat] :
              ( ( ord_less_nat @ K2 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
             => ( P @ ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) @ K2 ) @ ( nth_nat @ ( concat_nat @ Ys ) @ K2 ) ) ) ) ) ) ).

% concat_all_nth
thf(fact_369_concat__all__nth,axiom,
    ! [Xs: list_list_nat,Ys: list_list_list_nat,P: nat > list_nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Xs )
        = ( size_s6248950052170075156st_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) )
              = ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Ys @ I2 ) ) ) )
       => ( ! [I2: nat,J2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
             => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) ) )
               => ( P @ ( nth_nat @ ( nth_list_nat @ Xs @ I2 ) @ J2 ) @ ( nth_list_nat @ ( nth_list_list_nat @ Ys @ I2 ) @ J2 ) ) ) )
         => ! [K2: nat] :
              ( ( ord_less_nat @ K2 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
             => ( P @ ( nth_nat @ ( concat_nat @ Xs ) @ K2 ) @ ( nth_list_nat @ ( concat_list_nat @ Ys ) @ K2 ) ) ) ) ) ) ).

% concat_all_nth
thf(fact_370_concat__all__nth,axiom,
    ! [Xs: list_list_nat,Ys: list_l1578326660578982846ctxt_a,P: nat > ground_gmctxt_a > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Xs )
        = ( size_s6994882312647770194ctxt_a @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) )
              = ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ I2 ) ) ) )
       => ( ! [I2: nat,J2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
             => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) ) )
               => ( P @ ( nth_nat @ ( nth_list_nat @ Xs @ I2 ) @ J2 ) @ ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ I2 ) @ J2 ) ) ) )
         => ! [K2: nat] :
              ( ( ord_less_nat @ K2 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
             => ( P @ ( nth_nat @ ( concat_nat @ Xs ) @ K2 ) @ ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Ys ) @ K2 ) ) ) ) ) ) ).

% concat_all_nth
thf(fact_371_concat__all__nth,axiom,
    ! [Xs: list_list_nat,Ys: list_list_list_b,P: nat > list_b > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Xs )
        = ( size_s9106009106869804029list_b @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) )
              = ( size_s420531428170919671list_b @ ( nth_list_list_b @ Ys @ I2 ) ) ) )
       => ( ! [I2: nat,J2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
             => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) ) )
               => ( P @ ( nth_nat @ ( nth_list_nat @ Xs @ I2 ) @ J2 ) @ ( nth_list_b @ ( nth_list_list_b @ Ys @ I2 ) @ J2 ) ) ) )
         => ! [K2: nat] :
              ( ( ord_less_nat @ K2 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
             => ( P @ ( nth_nat @ ( concat_nat @ Xs ) @ K2 ) @ ( nth_list_b @ ( concat_list_b @ Ys ) @ K2 ) ) ) ) ) ) ).

% concat_all_nth
thf(fact_372_concat__all__nth,axiom,
    ! [Xs: list_list_list_b,Ys: list_list_nat,P: list_b > nat > $o] :
      ( ( ( size_s9106009106869804029list_b @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s9106009106869804029list_b @ Xs ) )
           => ( ( size_s420531428170919671list_b @ ( nth_list_list_b @ Xs @ I2 ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I2 ) ) ) )
       => ( ! [I2: nat,J2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s9106009106869804029list_b @ Xs ) )
             => ( ( ord_less_nat @ J2 @ ( size_s420531428170919671list_b @ ( nth_list_list_b @ Xs @ I2 ) ) )
               => ( P @ ( nth_list_b @ ( nth_list_list_b @ Xs @ I2 ) @ J2 ) @ ( nth_nat @ ( nth_list_nat @ Ys @ I2 ) @ J2 ) ) ) )
         => ! [K2: nat] :
              ( ( ord_less_nat @ K2 @ ( size_s420531428170919671list_b @ ( concat_list_b @ Xs ) ) )
             => ( P @ ( nth_list_b @ ( concat_list_b @ Xs ) @ K2 ) @ ( nth_nat @ ( concat_nat @ Ys ) @ K2 ) ) ) ) ) ) ).

% concat_all_nth
thf(fact_373_nth__concat__split,axiom,
    ! [I: nat,Xs: list_list_list_nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ ( concat_list_nat @ Xs ) ) )
     => ? [J2: nat,K3: nat] :
          ( ( ord_less_nat @ J2 @ ( size_s6248950052170075156st_nat @ Xs ) )
          & ( ord_less_nat @ K3 @ ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Xs @ J2 ) ) )
          & ( ( nth_list_nat @ ( concat_list_nat @ Xs ) @ I )
            = ( nth_list_nat @ ( nth_list_list_nat @ Xs @ J2 ) @ K3 ) ) ) ) ).

% nth_concat_split
thf(fact_374_nth__concat__split,axiom,
    ! [I: nat,Xs: list_l3687490334575670478ctxt_a] :
      ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ ( concat6203624070112318253ctxt_a @ Xs ) ) )
     => ? [J2: nat,K3: nat] :
          ( ( ord_less_nat @ J2 @ ( size_s2801996683354709858ctxt_a @ Xs ) )
          & ( ord_less_nat @ K3 @ ( size_s6994882312647770194ctxt_a @ ( nth_li3520809774789768823ctxt_a @ Xs @ J2 ) ) )
          & ( ( nth_li6678718213161987943ctxt_a @ ( concat6203624070112318253ctxt_a @ Xs ) @ I )
            = ( nth_li6678718213161987943ctxt_a @ ( nth_li3520809774789768823ctxt_a @ Xs @ J2 ) @ K3 ) ) ) ) ).

% nth_concat_split
thf(fact_375_nth__concat__split,axiom,
    ! [I: nat,Xs: list_l5129023048960694521at_nat] :
      ( ( ord_less_nat @ I @ ( size_s6878757106947045471at_nat @ ( concat5897368152400924974at_nat @ Xs ) ) )
     => ? [J2: nat,K3: nat] :
          ( ( ord_less_nat @ J2 @ ( size_s7371574421788826469at_nat @ Xs ) )
          & ( ord_less_nat @ K3 @ ( size_s6878757106947045471at_nat @ ( nth_li2593672534642986362at_nat @ Xs @ J2 ) ) )
          & ( ( nth_li8805938130461222772at_nat @ ( concat5897368152400924974at_nat @ Xs ) @ I )
            = ( nth_li8805938130461222772at_nat @ ( nth_li2593672534642986362at_nat @ Xs @ J2 ) @ K3 ) ) ) ) ).

% nth_concat_split
thf(fact_376_nth__concat__split,axiom,
    ! [I: nat,Xs: list_l1578326660578982846ctxt_a] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ? [J2: nat,K3: nat] :
          ( ( ord_less_nat @ J2 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
          & ( ord_less_nat @ K3 @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ J2 ) ) )
          & ( ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) @ I )
            = ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ J2 ) @ K3 ) ) ) ) ).

% nth_concat_split
thf(fact_377_nth__concat__split,axiom,
    ! [I: nat,Xs: list_list_b] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ ( concat_b @ Xs ) ) )
     => ? [J2: nat,K3: nat] :
          ( ( ord_less_nat @ J2 @ ( size_s420531428170919671list_b @ Xs ) )
          & ( ord_less_nat @ K3 @ ( size_size_list_b @ ( nth_list_b @ Xs @ J2 ) ) )
          & ( ( nth_b @ ( concat_b @ Xs ) @ I )
            = ( nth_b @ ( nth_list_b @ Xs @ J2 ) @ K3 ) ) ) ) ).

% nth_concat_split
thf(fact_378_nth__concat__split,axiom,
    ! [I: nat,Xs: list_list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ? [J2: nat,K3: nat] :
          ( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Xs ) )
          & ( ord_less_nat @ K3 @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ J2 ) ) )
          & ( ( nth_nat @ ( concat_nat @ Xs ) @ I )
            = ( nth_nat @ ( nth_list_nat @ Xs @ J2 ) @ K3 ) ) ) ) ).

% nth_concat_split
thf(fact_379_nth__concat__split,axiom,
    ! [I: nat,Xs: list_list_list_b] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ ( concat_list_b @ Xs ) ) )
     => ? [J2: nat,K3: nat] :
          ( ( ord_less_nat @ J2 @ ( size_s9106009106869804029list_b @ Xs ) )
          & ( ord_less_nat @ K3 @ ( size_s420531428170919671list_b @ ( nth_list_list_b @ Xs @ J2 ) ) )
          & ( ( nth_list_b @ ( concat_list_b @ Xs ) @ I )
            = ( nth_list_b @ ( nth_list_list_b @ Xs @ J2 ) @ K3 ) ) ) ) ).

% nth_concat_split
thf(fact_380_nth__concat__split,axiom,
    ! [I: nat,Xs: list_l3298181151656792051at_nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ ( concat5616020275982107560at_nat @ Xs ) ) )
     => ? [J2: nat,K3: nat] :
          ( ( ord_less_nat @ J2 @ ( size_s6878757106947045471at_nat @ Xs ) )
          & ( ord_less_nat @ K3 @ ( size_s3679842834875189465at_nat @ ( nth_li8805938130461222772at_nat @ Xs @ J2 ) ) )
          & ( ( nth_Pr6744343527793145070at_nat @ ( concat5616020275982107560at_nat @ Xs ) @ I )
            = ( nth_Pr6744343527793145070at_nat @ ( nth_li8805938130461222772at_nat @ Xs @ J2 ) @ K3 ) ) ) ) ).

% nth_concat_split
thf(fact_381_nth__concat__two__lists,axiom,
    ! [I: nat,Xs: list_list_b,Ys: list_list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ ( concat_b @ Xs ) ) )
     => ( ( ( size_s3023201423986296836st_nat @ Ys )
          = ( size_s420531428170919671list_b @ Xs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Xs ) )
             => ( ( size_size_list_nat @ ( nth_list_nat @ Ys @ I2 ) )
                = ( size_size_list_b @ ( nth_list_b @ Xs @ I2 ) ) ) )
         => ? [J2: nat,K3: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s420531428170919671list_b @ Xs ) )
              & ( ord_less_nat @ K3 @ ( size_size_list_b @ ( nth_list_b @ Xs @ J2 ) ) )
              & ( ( nth_b @ ( concat_b @ Xs ) @ I )
                = ( nth_b @ ( nth_list_b @ Xs @ J2 ) @ K3 ) )
              & ( ( nth_nat @ ( concat_nat @ Ys ) @ I )
                = ( nth_nat @ ( nth_list_nat @ Ys @ J2 ) @ K3 ) ) ) ) ) ) ).

% nth_concat_two_lists
thf(fact_382_nth__concat__two__lists,axiom,
    ! [I: nat,Xs: list_list_b,Ys: list_list_b] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ ( concat_b @ Xs ) ) )
     => ( ( ( size_s420531428170919671list_b @ Ys )
          = ( size_s420531428170919671list_b @ Xs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Xs ) )
             => ( ( size_size_list_b @ ( nth_list_b @ Ys @ I2 ) )
                = ( size_size_list_b @ ( nth_list_b @ Xs @ I2 ) ) ) )
         => ? [J2: nat,K3: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s420531428170919671list_b @ Xs ) )
              & ( ord_less_nat @ K3 @ ( size_size_list_b @ ( nth_list_b @ Xs @ J2 ) ) )
              & ( ( nth_b @ ( concat_b @ Xs ) @ I )
                = ( nth_b @ ( nth_list_b @ Xs @ J2 ) @ K3 ) )
              & ( ( nth_b @ ( concat_b @ Ys ) @ I )
                = ( nth_b @ ( nth_list_b @ Ys @ J2 ) @ K3 ) ) ) ) ) ) ).

% nth_concat_two_lists
thf(fact_383_nth__concat__two__lists,axiom,
    ! [I: nat,Xs: list_list_nat,Ys: list_list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ( ( size_s3023201423986296836st_nat @ Ys )
          = ( size_s3023201423986296836st_nat @ Xs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
             => ( ( size_size_list_nat @ ( nth_list_nat @ Ys @ I2 ) )
                = ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) ) ) )
         => ? [J2: nat,K3: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Xs ) )
              & ( ord_less_nat @ K3 @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ J2 ) ) )
              & ( ( nth_nat @ ( concat_nat @ Xs ) @ I )
                = ( nth_nat @ ( nth_list_nat @ Xs @ J2 ) @ K3 ) )
              & ( ( nth_nat @ ( concat_nat @ Ys ) @ I )
                = ( nth_nat @ ( nth_list_nat @ Ys @ J2 ) @ K3 ) ) ) ) ) ) ).

% nth_concat_two_lists
thf(fact_384_nth__concat__two__lists,axiom,
    ! [I: nat,Xs: list_list_nat,Ys: list_list_b] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ( ( size_s420531428170919671list_b @ Ys )
          = ( size_s3023201423986296836st_nat @ Xs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
             => ( ( size_size_list_b @ ( nth_list_b @ Ys @ I2 ) )
                = ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) ) ) )
         => ? [J2: nat,K3: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Xs ) )
              & ( ord_less_nat @ K3 @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ J2 ) ) )
              & ( ( nth_nat @ ( concat_nat @ Xs ) @ I )
                = ( nth_nat @ ( nth_list_nat @ Xs @ J2 ) @ K3 ) )
              & ( ( nth_b @ ( concat_b @ Ys ) @ I )
                = ( nth_b @ ( nth_list_b @ Ys @ J2 ) @ K3 ) ) ) ) ) ) ).

% nth_concat_two_lists
thf(fact_385_nth__concat__two__lists,axiom,
    ! [I: nat,Xs: list_list_list_nat,Ys: list_list_nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ ( concat_list_nat @ Xs ) ) )
     => ( ( ( size_s3023201423986296836st_nat @ Ys )
          = ( size_s6248950052170075156st_nat @ Xs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s6248950052170075156st_nat @ Xs ) )
             => ( ( size_size_list_nat @ ( nth_list_nat @ Ys @ I2 ) )
                = ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Xs @ I2 ) ) ) )
         => ? [J2: nat,K3: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s6248950052170075156st_nat @ Xs ) )
              & ( ord_less_nat @ K3 @ ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Xs @ J2 ) ) )
              & ( ( nth_list_nat @ ( concat_list_nat @ Xs ) @ I )
                = ( nth_list_nat @ ( nth_list_list_nat @ Xs @ J2 ) @ K3 ) )
              & ( ( nth_nat @ ( concat_nat @ Ys ) @ I )
                = ( nth_nat @ ( nth_list_nat @ Ys @ J2 ) @ K3 ) ) ) ) ) ) ).

% nth_concat_two_lists
thf(fact_386_nth__concat__two__lists,axiom,
    ! [I: nat,Xs: list_list_list_nat,Ys: list_list_b] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ ( concat_list_nat @ Xs ) ) )
     => ( ( ( size_s420531428170919671list_b @ Ys )
          = ( size_s6248950052170075156st_nat @ Xs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s6248950052170075156st_nat @ Xs ) )
             => ( ( size_size_list_b @ ( nth_list_b @ Ys @ I2 ) )
                = ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Xs @ I2 ) ) ) )
         => ? [J2: nat,K3: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s6248950052170075156st_nat @ Xs ) )
              & ( ord_less_nat @ K3 @ ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Xs @ J2 ) ) )
              & ( ( nth_list_nat @ ( concat_list_nat @ Xs ) @ I )
                = ( nth_list_nat @ ( nth_list_list_nat @ Xs @ J2 ) @ K3 ) )
              & ( ( nth_b @ ( concat_b @ Ys ) @ I )
                = ( nth_b @ ( nth_list_b @ Ys @ J2 ) @ K3 ) ) ) ) ) ) ).

% nth_concat_two_lists
thf(fact_387_nth__concat__two__lists,axiom,
    ! [I: nat,Xs: list_l1578326660578982846ctxt_a,Ys: list_list_nat] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ( ( size_s3023201423986296836st_nat @ Ys )
          = ( size_s6994882312647770194ctxt_a @ Xs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
             => ( ( size_size_list_nat @ ( nth_list_nat @ Ys @ I2 ) )
                = ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I2 ) ) ) )
         => ? [J2: nat,K3: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
              & ( ord_less_nat @ K3 @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ J2 ) ) )
              & ( ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) @ I )
                = ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ J2 ) @ K3 ) )
              & ( ( nth_nat @ ( concat_nat @ Ys ) @ I )
                = ( nth_nat @ ( nth_list_nat @ Ys @ J2 ) @ K3 ) ) ) ) ) ) ).

% nth_concat_two_lists
thf(fact_388_nth__concat__two__lists,axiom,
    ! [I: nat,Xs: list_l1578326660578982846ctxt_a,Ys: list_list_b] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ( ( size_s420531428170919671list_b @ Ys )
          = ( size_s6994882312647770194ctxt_a @ Xs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
             => ( ( size_size_list_b @ ( nth_list_b @ Ys @ I2 ) )
                = ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I2 ) ) ) )
         => ? [J2: nat,K3: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
              & ( ord_less_nat @ K3 @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ J2 ) ) )
              & ( ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) @ I )
                = ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ J2 ) @ K3 ) )
              & ( ( nth_b @ ( concat_b @ Ys ) @ I )
                = ( nth_b @ ( nth_list_b @ Ys @ J2 ) @ K3 ) ) ) ) ) ) ).

% nth_concat_two_lists
thf(fact_389_nth__concat__two__lists,axiom,
    ! [I: nat,Xs: list_list_b,Ys: list_list_list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ ( concat_b @ Xs ) ) )
     => ( ( ( size_s6248950052170075156st_nat @ Ys )
          = ( size_s420531428170919671list_b @ Xs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Xs ) )
             => ( ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Ys @ I2 ) )
                = ( size_size_list_b @ ( nth_list_b @ Xs @ I2 ) ) ) )
         => ? [J2: nat,K3: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s420531428170919671list_b @ Xs ) )
              & ( ord_less_nat @ K3 @ ( size_size_list_b @ ( nth_list_b @ Xs @ J2 ) ) )
              & ( ( nth_b @ ( concat_b @ Xs ) @ I )
                = ( nth_b @ ( nth_list_b @ Xs @ J2 ) @ K3 ) )
              & ( ( nth_list_nat @ ( concat_list_nat @ Ys ) @ I )
                = ( nth_list_nat @ ( nth_list_list_nat @ Ys @ J2 ) @ K3 ) ) ) ) ) ) ).

% nth_concat_two_lists
thf(fact_390_nth__concat__two__lists,axiom,
    ! [I: nat,Xs: list_list_b,Ys: list_l1578326660578982846ctxt_a] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ ( concat_b @ Xs ) ) )
     => ( ( ( size_s6994882312647770194ctxt_a @ Ys )
          = ( size_s420531428170919671list_b @ Xs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Xs ) )
             => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ I2 ) )
                = ( size_size_list_b @ ( nth_list_b @ Xs @ I2 ) ) ) )
         => ? [J2: nat,K3: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s420531428170919671list_b @ Xs ) )
              & ( ord_less_nat @ K3 @ ( size_size_list_b @ ( nth_list_b @ Xs @ J2 ) ) )
              & ( ( nth_b @ ( concat_b @ Xs ) @ I )
                = ( nth_b @ ( nth_list_b @ Xs @ J2 ) @ K3 ) )
              & ( ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Ys ) @ I )
                = ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ J2 ) @ K3 ) ) ) ) ) ) ).

% nth_concat_two_lists
thf(fact_391_concat__partition__by,axiom,
    ! [Ys: list_nat,Xs: list_list_b] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s420531428170919671list_b @ Xs ) )
     => ( ( concat_list_b @ ( multih2267195492336891525list_b @ Xs @ Ys ) )
        = Xs ) ) ).

% concat_partition_by
thf(fact_392_concat__partition__by,axiom,
    ! [Ys: list_nat,Xs: list_b] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_size_list_b @ Xs ) )
     => ( ( concat_b @ ( multih3537974503344769663n_by_b @ Xs @ Ys ) )
        = Xs ) ) ).

% concat_partition_by
thf(fact_393_concat__partition__by,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_size_list_nat @ Xs ) )
     => ( ( concat_nat @ ( multih8544690682651757648by_nat @ Xs @ Ys ) )
        = Xs ) ) ).

% concat_partition_by
thf(fact_394_concat__partition__by,axiom,
    ! [Ys: list_nat,Xs: list_Ground_gmctxt_a] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( concat3993834769610107037ctxt_a @ ( multih5172578161653122718ctxt_a @ Xs @ Ys ) )
        = Xs ) ) ).

% concat_partition_by
thf(fact_395_concat__partition__by,axiom,
    ! [Ys: list_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( ( groups4561878855575611511st_nat @ Ys )
        = ( size_s3679842834875189465at_nat @ Xs ) )
     => ( ( concat5616020275982107560at_nat @ ( multih7555706055023738855at_nat @ Xs @ Ys ) )
        = Xs ) ) ).

% concat_partition_by
thf(fact_396_map__concat,axiom,
    ! [F: ground_gmctxt_a > nat,Xs: list_l1578326660578982846ctxt_a] :
      ( ( map_Gr1679544123147727284_a_nat @ F @ ( concat3993834769610107037ctxt_a @ Xs ) )
      = ( concat_nat @ ( map_li7775015282390606420st_nat @ ( map_Gr1679544123147727284_a_nat @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_397_map__concat,axiom,
    ! [F: nat > b,Xs: list_list_nat] :
      ( ( map_nat_b @ F @ ( concat_nat @ Xs ) )
      = ( concat_b @ ( map_list_nat_list_b @ ( map_nat_b @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_398_map__concat,axiom,
    ! [F: b > ground_gmctxt_a,Xs: list_list_b] :
      ( ( map_b_3587768286370285017ctxt_a @ F @ ( concat_b @ Xs ) )
      = ( concat3993834769610107037ctxt_a @ ( map_li4667870289254436335ctxt_a @ ( map_b_3587768286370285017ctxt_a @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_399_map__concat,axiom,
    ! [F: b > list_b,Xs: list_list_b] :
      ( ( map_b_list_b @ F @ ( concat_b @ Xs ) )
      = ( concat_list_b @ ( map_li4464339650582430090list_b @ ( map_b_list_b @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_400_map__concat,axiom,
    ! [F: b > nat,Xs: list_list_b] :
      ( ( map_b_nat @ F @ ( concat_b @ Xs ) )
      = ( concat_nat @ ( map_list_b_list_nat @ ( map_b_nat @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_401_map__concat,axiom,
    ! [F: b > b,Xs: list_list_b] :
      ( ( map_b_b @ F @ ( concat_b @ Xs ) )
      = ( concat_b @ ( map_list_b_list_b @ ( map_b_b @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_402_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_403_eq__length__concat__nth,axiom,
    ! [Xs: list_list_nat,Ys: list_list_nat] :
      ( ( ( size_s3023201423986296836st_nat @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I2 ) ) ) )
       => ( ( size_size_list_nat @ ( concat_nat @ Xs ) )
          = ( size_size_list_nat @ ( concat_nat @ Ys ) ) ) ) ) ).

% eq_length_concat_nth
thf(fact_404_eq__length__concat__nth,axiom,
    ! [Xs: list_list_nat,Ys: list_list_b] :
      ( ( ( size_s3023201423986296836st_nat @ Xs )
        = ( size_s420531428170919671list_b @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) )
              = ( size_size_list_b @ ( nth_list_b @ Ys @ I2 ) ) ) )
       => ( ( size_size_list_nat @ ( concat_nat @ Xs ) )
          = ( size_size_list_b @ ( concat_b @ Ys ) ) ) ) ) ).

% eq_length_concat_nth
thf(fact_405_eq__length__concat__nth,axiom,
    ! [Xs: list_list_b,Ys: list_list_nat] :
      ( ( ( size_s420531428170919671list_b @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Xs ) )
           => ( ( size_size_list_b @ ( nth_list_b @ Xs @ I2 ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I2 ) ) ) )
       => ( ( size_size_list_b @ ( concat_b @ Xs ) )
          = ( size_size_list_nat @ ( concat_nat @ Ys ) ) ) ) ) ).

% eq_length_concat_nth
thf(fact_406_eq__length__concat__nth,axiom,
    ! [Xs: list_list_b,Ys: list_list_b] :
      ( ( ( size_s420531428170919671list_b @ Xs )
        = ( size_s420531428170919671list_b @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Xs ) )
           => ( ( size_size_list_b @ ( nth_list_b @ Xs @ I2 ) )
              = ( size_size_list_b @ ( nth_list_b @ Ys @ I2 ) ) ) )
       => ( ( size_size_list_b @ ( concat_b @ Xs ) )
          = ( size_size_list_b @ ( concat_b @ Ys ) ) ) ) ) ).

% eq_length_concat_nth
thf(fact_407_eq__length__concat__nth,axiom,
    ! [Xs: list_l1578326660578982846ctxt_a,Ys: list_list_nat] :
      ( ( ( size_s6994882312647770194ctxt_a @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
           => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I2 ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I2 ) ) ) )
       => ( ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) )
          = ( size_size_list_nat @ ( concat_nat @ Ys ) ) ) ) ) ).

% eq_length_concat_nth
thf(fact_408_eq__length__concat__nth,axiom,
    ! [Xs: list_list_nat,Ys: list_l1578326660578982846ctxt_a] :
      ( ( ( size_s3023201423986296836st_nat @ Xs )
        = ( size_s6994882312647770194ctxt_a @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) )
              = ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ I2 ) ) ) )
       => ( ( size_size_list_nat @ ( concat_nat @ Xs ) )
          = ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Ys ) ) ) ) ) ).

% eq_length_concat_nth
thf(fact_409_eq__length__concat__nth,axiom,
    ! [Xs: list_list_nat,Ys: list_list_list_b] :
      ( ( ( size_s3023201423986296836st_nat @ Xs )
        = ( size_s9106009106869804029list_b @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) )
              = ( size_s420531428170919671list_b @ ( nth_list_list_b @ Ys @ I2 ) ) ) )
       => ( ( size_size_list_nat @ ( concat_nat @ Xs ) )
          = ( size_s420531428170919671list_b @ ( concat_list_b @ Ys ) ) ) ) ) ).

% eq_length_concat_nth
thf(fact_410_eq__length__concat__nth,axiom,
    ! [Xs: list_list_list_b,Ys: list_list_nat] :
      ( ( ( size_s9106009106869804029list_b @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s9106009106869804029list_b @ Xs ) )
           => ( ( size_s420531428170919671list_b @ ( nth_list_list_b @ Xs @ I2 ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I2 ) ) ) )
       => ( ( size_s420531428170919671list_b @ ( concat_list_b @ Xs ) )
          = ( size_size_list_nat @ ( concat_nat @ Ys ) ) ) ) ) ).

% eq_length_concat_nth
thf(fact_411_eq__length__concat__nth,axiom,
    ! [Xs: list_l1578326660578982846ctxt_a,Ys: list_list_b] :
      ( ( ( size_s6994882312647770194ctxt_a @ Xs )
        = ( size_s420531428170919671list_b @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
           => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I2 ) )
              = ( size_size_list_b @ ( nth_list_b @ Ys @ I2 ) ) ) )
       => ( ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) )
          = ( size_size_list_b @ ( concat_b @ Ys ) ) ) ) ) ).

% eq_length_concat_nth
thf(fact_412_eq__length__concat__nth,axiom,
    ! [Xs: list_list_list_b,Ys: list_list_b] :
      ( ( ( size_s9106009106869804029list_b @ Xs )
        = ( size_s420531428170919671list_b @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s9106009106869804029list_b @ Xs ) )
           => ( ( size_s420531428170919671list_b @ ( nth_list_list_b @ Xs @ I2 ) )
              = ( size_size_list_b @ ( nth_list_b @ Ys @ I2 ) ) ) )
       => ( ( size_s420531428170919671list_b @ ( concat_list_b @ Xs ) )
          = ( size_size_list_b @ ( concat_b @ Ys ) ) ) ) ) ).

% eq_length_concat_nth
thf(fact_413_length__concat,axiom,
    ! [Xss: list_l1578326660578982846ctxt_a] :
      ( ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xss ) )
      = ( groups4561878855575611511st_nat @ ( map_li8618020238875267396_a_nat @ size_s962548663205356226ctxt_a @ Xss ) ) ) ).

% length_concat
thf(fact_414_length__concat,axiom,
    ! [Xss: list_list_b] :
      ( ( size_size_list_b @ ( concat_b @ Xss ) )
      = ( groups4561878855575611511st_nat @ ( map_list_b_nat @ size_size_list_b @ Xss ) ) ) ).

% length_concat
thf(fact_415_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_416_length__concat,axiom,
    ! [Xss: list_list_list_b] :
      ( ( size_s420531428170919671list_b @ ( concat_list_b @ Xss ) )
      = ( groups4561878855575611511st_nat @ ( map_list_list_b_nat @ size_s420531428170919671list_b @ Xss ) ) ) ).

% length_concat
thf(fact_417_length__concat,axiom,
    ! [Xss: list_l3298181151656792051at_nat] :
      ( ( size_s3679842834875189465at_nat @ ( concat5616020275982107560at_nat @ Xss ) )
      = ( groups4561878855575611511st_nat @ ( map_li9059624042496273785at_nat @ size_s3679842834875189465at_nat @ Xss ) ) ) ).

% length_concat
thf(fact_418_nth__concat__diff,axiom,
    ! [I1: nat,Xs: list_list_list_nat,I22: nat] :
      ( ( ord_less_nat @ I1 @ ( size_s3023201423986296836st_nat @ ( concat_list_nat @ Xs ) ) )
     => ( ( ord_less_nat @ I22 @ ( size_s3023201423986296836st_nat @ ( concat_list_nat @ Xs ) ) )
       => ( ( 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_s6248950052170075156st_nat @ Xs ) )
              & ( ord_less_nat @ J22 @ ( size_s6248950052170075156st_nat @ Xs ) )
              & ( ord_less_nat @ K1 @ ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Xs @ J1 ) ) )
              & ( ord_less_nat @ K22 @ ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Xs @ J22 ) ) )
              & ( ( nth_list_nat @ ( concat_list_nat @ Xs ) @ I1 )
                = ( nth_list_nat @ ( nth_list_list_nat @ Xs @ J1 ) @ K1 ) )
              & ( ( nth_list_nat @ ( concat_list_nat @ Xs ) @ I22 )
                = ( nth_list_nat @ ( nth_list_list_nat @ Xs @ J22 ) @ K22 ) ) ) ) ) ) ).

% nth_concat_diff
thf(fact_419_nth__concat__diff,axiom,
    ! [I1: nat,Xs: list_l3687490334575670478ctxt_a,I22: nat] :
      ( ( ord_less_nat @ I1 @ ( size_s6994882312647770194ctxt_a @ ( concat6203624070112318253ctxt_a @ Xs ) ) )
     => ( ( ord_less_nat @ I22 @ ( size_s6994882312647770194ctxt_a @ ( concat6203624070112318253ctxt_a @ Xs ) ) )
       => ( ( 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_s2801996683354709858ctxt_a @ Xs ) )
              & ( ord_less_nat @ J22 @ ( size_s2801996683354709858ctxt_a @ Xs ) )
              & ( ord_less_nat @ K1 @ ( size_s6994882312647770194ctxt_a @ ( nth_li3520809774789768823ctxt_a @ Xs @ J1 ) ) )
              & ( ord_less_nat @ K22 @ ( size_s6994882312647770194ctxt_a @ ( nth_li3520809774789768823ctxt_a @ Xs @ J22 ) ) )
              & ( ( nth_li6678718213161987943ctxt_a @ ( concat6203624070112318253ctxt_a @ Xs ) @ I1 )
                = ( nth_li6678718213161987943ctxt_a @ ( nth_li3520809774789768823ctxt_a @ Xs @ J1 ) @ K1 ) )
              & ( ( nth_li6678718213161987943ctxt_a @ ( concat6203624070112318253ctxt_a @ Xs ) @ I22 )
                = ( nth_li6678718213161987943ctxt_a @ ( nth_li3520809774789768823ctxt_a @ Xs @ J22 ) @ K22 ) ) ) ) ) ) ).

% nth_concat_diff
thf(fact_420_nth__concat__diff,axiom,
    ! [I1: nat,Xs: list_l5129023048960694521at_nat,I22: nat] :
      ( ( ord_less_nat @ I1 @ ( size_s6878757106947045471at_nat @ ( concat5897368152400924974at_nat @ Xs ) ) )
     => ( ( ord_less_nat @ I22 @ ( size_s6878757106947045471at_nat @ ( concat5897368152400924974at_nat @ Xs ) ) )
       => ( ( 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_s7371574421788826469at_nat @ Xs ) )
              & ( ord_less_nat @ J22 @ ( size_s7371574421788826469at_nat @ Xs ) )
              & ( ord_less_nat @ K1 @ ( size_s6878757106947045471at_nat @ ( nth_li2593672534642986362at_nat @ Xs @ J1 ) ) )
              & ( ord_less_nat @ K22 @ ( size_s6878757106947045471at_nat @ ( nth_li2593672534642986362at_nat @ Xs @ J22 ) ) )
              & ( ( nth_li8805938130461222772at_nat @ ( concat5897368152400924974at_nat @ Xs ) @ I1 )
                = ( nth_li8805938130461222772at_nat @ ( nth_li2593672534642986362at_nat @ Xs @ J1 ) @ K1 ) )
              & ( ( nth_li8805938130461222772at_nat @ ( concat5897368152400924974at_nat @ Xs ) @ I22 )
                = ( nth_li8805938130461222772at_nat @ ( nth_li2593672534642986362at_nat @ Xs @ J22 ) @ K22 ) ) ) ) ) ) ).

% nth_concat_diff
thf(fact_421_nth__concat__diff,axiom,
    ! [I1: nat,Xs: list_l1578326660578982846ctxt_a,I22: nat] :
      ( ( ord_less_nat @ I1 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ( ord_less_nat @ I22 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
       => ( ( 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_s6994882312647770194ctxt_a @ Xs ) )
              & ( ord_less_nat @ J22 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
              & ( ord_less_nat @ K1 @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ J1 ) ) )
              & ( ord_less_nat @ K22 @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ J22 ) ) )
              & ( ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) @ I1 )
                = ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ J1 ) @ K1 ) )
              & ( ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) @ I22 )
                = ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ J22 ) @ K22 ) ) ) ) ) ) ).

% nth_concat_diff
thf(fact_422_nth__concat__diff,axiom,
    ! [I1: nat,Xs: list_list_b,I22: nat] :
      ( ( ord_less_nat @ I1 @ ( size_size_list_b @ ( concat_b @ Xs ) ) )
     => ( ( ord_less_nat @ I22 @ ( size_size_list_b @ ( concat_b @ Xs ) ) )
       => ( ( 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_s420531428170919671list_b @ Xs ) )
              & ( ord_less_nat @ J22 @ ( size_s420531428170919671list_b @ Xs ) )
              & ( ord_less_nat @ K1 @ ( size_size_list_b @ ( nth_list_b @ Xs @ J1 ) ) )
              & ( ord_less_nat @ K22 @ ( size_size_list_b @ ( nth_list_b @ Xs @ J22 ) ) )
              & ( ( nth_b @ ( concat_b @ Xs ) @ I1 )
                = ( nth_b @ ( nth_list_b @ Xs @ J1 ) @ K1 ) )
              & ( ( nth_b @ ( concat_b @ Xs ) @ I22 )
                = ( nth_b @ ( nth_list_b @ Xs @ J22 ) @ K22 ) ) ) ) ) ) ).

% nth_concat_diff
thf(fact_423_nth__concat__diff,axiom,
    ! [I1: nat,Xs: list_list_nat,I22: nat] :
      ( ( ord_less_nat @ I1 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ( ord_less_nat @ I22 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
       => ( ( 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 @ Xs ) )
              & ( ord_less_nat @ J22 @ ( size_s3023201423986296836st_nat @ Xs ) )
              & ( ord_less_nat @ K1 @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ J1 ) ) )
              & ( ord_less_nat @ K22 @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ J22 ) ) )
              & ( ( nth_nat @ ( concat_nat @ Xs ) @ I1 )
                = ( nth_nat @ ( nth_list_nat @ Xs @ J1 ) @ K1 ) )
              & ( ( nth_nat @ ( concat_nat @ Xs ) @ I22 )
                = ( nth_nat @ ( nth_list_nat @ Xs @ J22 ) @ K22 ) ) ) ) ) ) ).

% nth_concat_diff
thf(fact_424_nth__concat__diff,axiom,
    ! [I1: nat,Xs: list_list_list_b,I22: nat] :
      ( ( ord_less_nat @ I1 @ ( size_s420531428170919671list_b @ ( concat_list_b @ Xs ) ) )
     => ( ( ord_less_nat @ I22 @ ( size_s420531428170919671list_b @ ( concat_list_b @ Xs ) ) )
       => ( ( 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_s9106009106869804029list_b @ Xs ) )
              & ( ord_less_nat @ J22 @ ( size_s9106009106869804029list_b @ Xs ) )
              & ( ord_less_nat @ K1 @ ( size_s420531428170919671list_b @ ( nth_list_list_b @ Xs @ J1 ) ) )
              & ( ord_less_nat @ K22 @ ( size_s420531428170919671list_b @ ( nth_list_list_b @ Xs @ J22 ) ) )
              & ( ( nth_list_b @ ( concat_list_b @ Xs ) @ I1 )
                = ( nth_list_b @ ( nth_list_list_b @ Xs @ J1 ) @ K1 ) )
              & ( ( nth_list_b @ ( concat_list_b @ Xs ) @ I22 )
                = ( nth_list_b @ ( nth_list_list_b @ Xs @ J22 ) @ K22 ) ) ) ) ) ) ).

% nth_concat_diff
thf(fact_425_nth__concat__diff,axiom,
    ! [I1: nat,Xs: list_l3298181151656792051at_nat,I22: nat] :
      ( ( ord_less_nat @ I1 @ ( size_s3679842834875189465at_nat @ ( concat5616020275982107560at_nat @ Xs ) ) )
     => ( ( ord_less_nat @ I22 @ ( size_s3679842834875189465at_nat @ ( concat5616020275982107560at_nat @ Xs ) ) )
       => ( ( 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_s6878757106947045471at_nat @ Xs ) )
              & ( ord_less_nat @ J22 @ ( size_s6878757106947045471at_nat @ Xs ) )
              & ( ord_less_nat @ K1 @ ( size_s3679842834875189465at_nat @ ( nth_li8805938130461222772at_nat @ Xs @ J1 ) ) )
              & ( ord_less_nat @ K22 @ ( size_s3679842834875189465at_nat @ ( nth_li8805938130461222772at_nat @ Xs @ J22 ) ) )
              & ( ( nth_Pr6744343527793145070at_nat @ ( concat5616020275982107560at_nat @ Xs ) @ I1 )
                = ( nth_Pr6744343527793145070at_nat @ ( nth_li8805938130461222772at_nat @ Xs @ J1 ) @ K1 ) )
              & ( ( nth_Pr6744343527793145070at_nat @ ( concat5616020275982107560at_nat @ Xs ) @ I22 )
                = ( nth_Pr6744343527793145070at_nat @ ( nth_li8805938130461222772at_nat @ Xs @ J22 ) @ K22 ) ) ) ) ) ) ).

% nth_concat_diff
thf(fact_426_concat__map__maps,axiom,
    ! [F: b > list_b,Xs: list_b] :
      ( ( concat_b @ ( map_b_list_b @ F @ Xs ) )
      = ( maps_b_b @ F @ Xs ) ) ).

% concat_map_maps
thf(fact_427_maps__def,axiom,
    ( maps_b_b
    = ( ^ [F2: b > list_b,Xs3: list_b] : ( concat_b @ ( map_b_list_b @ F2 @ Xs3 ) ) ) ) ).

% maps_def
thf(fact_428_subseqs__length__simple__False,axiom,
    ! [B: list_Ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
      ( ( member2714729351964853061ctxt_a @ B @ ( set_li3057194059929822169ctxt_a @ ( subseq4078942962620312969ctxt_a @ Xs ) ) )
     => ~ ( ord_less_nat @ ( size_s962548663205356226ctxt_a @ Xs ) @ ( size_s962548663205356226ctxt_a @ B ) ) ) ).

% subseqs_length_simple_False
thf(fact_429_subseqs__length__simple__False,axiom,
    ! [B: list_b,Xs: list_b] :
      ( ( member_list_b @ B @ ( set_list_b2 @ ( subseqs_b @ Xs ) ) )
     => ~ ( ord_less_nat @ ( size_size_list_b @ Xs ) @ ( size_size_list_b @ B ) ) ) ).

% subseqs_length_simple_False
thf(fact_430_subseqs__length__simple__False,axiom,
    ! [B: list_nat,Xs: list_nat] :
      ( ( member_list_nat @ B @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
     => ~ ( ord_less_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ B ) ) ) ).

% subseqs_length_simple_False
thf(fact_431_subseqs__length__simple__False,axiom,
    ! [B: list_list_b,Xs: list_list_b] :
      ( ( member_list_list_b @ B @ ( set_list_list_b2 @ ( subseqs_list_b @ Xs ) ) )
     => ~ ( ord_less_nat @ ( size_s420531428170919671list_b @ Xs ) @ ( size_s420531428170919671list_b @ B ) ) ) ).

% subseqs_length_simple_False
thf(fact_432_subseqs__length__simple__False,axiom,
    ! [B: list_P8469869581646625389at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member3799944675974059798at_nat @ B @ ( set_li7366234323595830146at_nat @ ( subseq4464984717476426172at_nat @ Xs ) ) )
     => ~ ( ord_less_nat @ ( size_s3679842834875189465at_nat @ Xs ) @ ( size_s3679842834875189465at_nat @ B ) ) ) ).

% subseqs_length_simple_False
thf(fact_433_remove__nth__P__compat,axiom,
    ! [As: list_b,Bs: list_b,P: b > b > $o,P2: nat] :
      ( ( ( size_size_list_b @ As )
        = ( size_size_list_b @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_b @ As ) )
           => ( P @ ( nth_b @ As @ I2 ) @ ( nth_b @ Bs @ I2 ) ) )
       => ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_b @ ( missing_remove_nth_b @ P2 @ As ) ) )
           => ( P @ ( nth_b @ ( missing_remove_nth_b @ P2 @ As ) @ I4 ) @ ( nth_b @ ( missing_remove_nth_b @ P2 @ Bs ) @ I4 ) ) ) ) ) ).

% remove_nth_P_compat
thf(fact_434_remove__nth__P__compat,axiom,
    ! [As: list_b,Bs: list_nat,P: b > nat > $o,P2: nat] :
      ( ( ( size_size_list_b @ As )
        = ( size_size_list_nat @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_b @ As ) )
           => ( P @ ( nth_b @ As @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) )
       => ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_b @ ( missing_remove_nth_b @ P2 @ As ) ) )
           => ( P @ ( nth_b @ ( missing_remove_nth_b @ P2 @ As ) @ I4 ) @ ( nth_nat @ ( missin7175274867594579095th_nat @ P2 @ Bs ) @ I4 ) ) ) ) ) ).

% remove_nth_P_compat
thf(fact_435_remove__nth__P__compat,axiom,
    ! [As: list_nat,Bs: list_b,P: nat > b > $o,P2: nat] :
      ( ( ( size_size_list_nat @ As )
        = ( size_size_list_b @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ As ) )
           => ( P @ ( nth_nat @ As @ I2 ) @ ( nth_b @ Bs @ I2 ) ) )
       => ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ P2 @ As ) ) )
           => ( P @ ( nth_nat @ ( missin7175274867594579095th_nat @ P2 @ As ) @ I4 ) @ ( nth_b @ ( missing_remove_nth_b @ P2 @ Bs ) @ I4 ) ) ) ) ) ).

% remove_nth_P_compat
thf(fact_436_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 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ As ) )
           => ( P @ ( nth_nat @ As @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) )
       => ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ P2 @ As ) ) )
           => ( P @ ( nth_nat @ ( missin7175274867594579095th_nat @ P2 @ As ) @ I4 ) @ ( nth_nat @ ( missin7175274867594579095th_nat @ P2 @ Bs ) @ I4 ) ) ) ) ) ).

% remove_nth_P_compat
thf(fact_437_remove__nth__P__compat,axiom,
    ! [As: list_list_nat,Bs: list_b,P: list_nat > b > $o,P2: nat] :
      ( ( ( size_s3023201423986296836st_nat @ As )
        = ( size_size_list_b @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ As ) )
           => ( P @ ( nth_list_nat @ As @ I2 ) @ ( nth_b @ Bs @ I2 ) ) )
       => ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ ( missin5133812807938250791st_nat @ P2 @ As ) ) )
           => ( P @ ( nth_list_nat @ ( missin5133812807938250791st_nat @ P2 @ As ) @ I4 ) @ ( nth_b @ ( missing_remove_nth_b @ P2 @ Bs ) @ I4 ) ) ) ) ) ).

% remove_nth_P_compat
thf(fact_438_remove__nth__P__compat,axiom,
    ! [As: list_list_nat,Bs: list_nat,P: list_nat > nat > $o,P2: nat] :
      ( ( ( size_s3023201423986296836st_nat @ As )
        = ( size_size_list_nat @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ As ) )
           => ( P @ ( nth_list_nat @ As @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) )
       => ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ ( missin5133812807938250791st_nat @ P2 @ As ) ) )
           => ( P @ ( nth_list_nat @ ( missin5133812807938250791st_nat @ P2 @ As ) @ I4 ) @ ( nth_nat @ ( missin7175274867594579095th_nat @ P2 @ Bs ) @ I4 ) ) ) ) ) ).

% remove_nth_P_compat
thf(fact_439_remove__nth__P__compat,axiom,
    ! [As: list_Ground_gmctxt_a,Bs: list_b,P: ground_gmctxt_a > b > $o,P2: nat] :
      ( ( ( size_s962548663205356226ctxt_a @ As )
        = ( size_size_list_b @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ As ) )
           => ( P @ ( nth_Ground_gmctxt_a @ As @ I2 ) @ ( nth_b @ Bs @ I2 ) ) )
       => ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ ( missin7823264313509747941ctxt_a @ P2 @ As ) ) )
           => ( P @ ( nth_Ground_gmctxt_a @ ( missin7823264313509747941ctxt_a @ P2 @ As ) @ I4 ) @ ( nth_b @ ( missing_remove_nth_b @ P2 @ Bs ) @ I4 ) ) ) ) ) ).

% remove_nth_P_compat
thf(fact_440_remove__nth__P__compat,axiom,
    ! [As: list_Ground_gmctxt_a,Bs: list_nat,P: ground_gmctxt_a > nat > $o,P2: nat] :
      ( ( ( size_s962548663205356226ctxt_a @ As )
        = ( size_size_list_nat @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ As ) )
           => ( P @ ( nth_Ground_gmctxt_a @ As @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) )
       => ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ ( missin7823264313509747941ctxt_a @ P2 @ As ) ) )
           => ( P @ ( nth_Ground_gmctxt_a @ ( missin7823264313509747941ctxt_a @ P2 @ As ) @ I4 ) @ ( nth_nat @ ( missin7175274867594579095th_nat @ P2 @ Bs ) @ I4 ) ) ) ) ) ).

% remove_nth_P_compat
thf(fact_441_remove__nth__P__compat,axiom,
    ! [As: list_b,Bs: list_list_nat,P: b > list_nat > $o,P2: nat] :
      ( ( ( size_size_list_b @ As )
        = ( size_s3023201423986296836st_nat @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_b @ As ) )
           => ( P @ ( nth_b @ As @ I2 ) @ ( nth_list_nat @ Bs @ I2 ) ) )
       => ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_b @ ( missing_remove_nth_b @ P2 @ As ) ) )
           => ( P @ ( nth_b @ ( missing_remove_nth_b @ P2 @ As ) @ I4 ) @ ( nth_list_nat @ ( missin5133812807938250791st_nat @ P2 @ Bs ) @ I4 ) ) ) ) ) ).

% remove_nth_P_compat
thf(fact_442_remove__nth__P__compat,axiom,
    ! [As: list_b,Bs: list_Ground_gmctxt_a,P: b > ground_gmctxt_a > $o,P2: nat] :
      ( ( ( size_size_list_b @ As )
        = ( size_s962548663205356226ctxt_a @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_b @ As ) )
           => ( P @ ( nth_b @ As @ I2 ) @ ( nth_Ground_gmctxt_a @ Bs @ I2 ) ) )
       => ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_b @ ( missing_remove_nth_b @ P2 @ As ) ) )
           => ( P @ ( nth_b @ ( missing_remove_nth_b @ P2 @ As ) @ I4 ) @ ( nth_Ground_gmctxt_a @ ( missin7823264313509747941ctxt_a @ P2 @ Bs ) @ I4 ) ) ) ) ) ).

% remove_nth_P_compat
thf(fact_443_map__eq__map__tailrec,axiom,
    map_Gr1679544123147727284_a_nat = map_ta198793340144898302_a_nat ).

% map_eq_map_tailrec
thf(fact_444_map__eq__map__tailrec,axiom,
    map_nat_b = map_tailrec_nat_b ).

% map_eq_map_tailrec
thf(fact_445_map__eq__map__tailrec,axiom,
    map_b_3587768286370285017ctxt_a = map_ta8077529389956916495ctxt_a ).

% map_eq_map_tailrec
thf(fact_446_map__eq__map__tailrec,axiom,
    map_b_list_b = map_tailrec_b_list_b ).

% map_eq_map_tailrec
thf(fact_447_map__eq__map__tailrec,axiom,
    map_b_nat = map_tailrec_b_nat ).

% map_eq_map_tailrec
thf(fact_448_map__eq__map__tailrec,axiom,
    map_b_b = map_tailrec_b_b ).

% map_eq_map_tailrec
thf(fact_449_set__swap,axiom,
    ! [I: nat,Xs: list_list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ Xs ) )
       => ( ( set_list_nat2 @ ( list_update_list_nat @ ( list_update_list_nat @ Xs @ I @ ( nth_list_nat @ Xs @ J ) ) @ J @ ( nth_list_nat @ Xs @ I ) ) )
          = ( set_list_nat2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_450_set__swap,axiom,
    ! [I: nat,Xs: list_l1578326660578982846ctxt_a,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s6994882312647770194ctxt_a @ Xs ) )
       => ( ( set_li3057194059929822169ctxt_a @ ( list_u8015824433058371904ctxt_a @ ( list_u8015824433058371904ctxt_a @ Xs @ I @ ( nth_li6678718213161987943ctxt_a @ Xs @ J ) ) @ J @ ( nth_li6678718213161987943ctxt_a @ Xs @ I ) ) )
          = ( set_li3057194059929822169ctxt_a @ Xs ) ) ) ) ).

% set_swap
thf(fact_451_set__swap,axiom,
    ! [I: nat,Xs: list_l3298181151656792051at_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6878757106947045471at_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s6878757106947045471at_nat @ Xs ) )
       => ( ( set_li7366234323595830146at_nat @ ( list_u5583386537922002395at_nat @ ( list_u5583386537922002395at_nat @ Xs @ I @ ( nth_li8805938130461222772at_nat @ Xs @ J ) ) @ J @ ( nth_li8805938130461222772at_nat @ Xs @ I ) ) )
          = ( set_li7366234323595830146at_nat @ Xs ) ) ) ) ).

% set_swap
thf(fact_452_set__swap,axiom,
    ! [I: nat,Xs: list_Ground_gmctxt_a,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s962548663205356226ctxt_a @ Xs ) )
       => ( ( set_Ground_gmctxt_a2 @ ( list_u7950812632479372336ctxt_a @ ( list_u7950812632479372336ctxt_a @ Xs @ I @ ( nth_Ground_gmctxt_a @ Xs @ J ) ) @ J @ ( nth_Ground_gmctxt_a @ Xs @ I ) ) )
          = ( set_Ground_gmctxt_a2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_453_set__swap,axiom,
    ! [I: nat,Xs: list_b,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_b @ Xs ) )
       => ( ( set_b2 @ ( list_update_b @ ( list_update_b @ Xs @ I @ ( nth_b @ Xs @ J ) ) @ J @ ( nth_b @ Xs @ I ) ) )
          = ( set_b2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_454_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
       => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
          = ( set_nat2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_455_set__swap,axiom,
    ! [I: nat,Xs: list_list_b,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s420531428170919671list_b @ Xs ) )
       => ( ( set_list_b2 @ ( list_update_list_b @ ( list_update_list_b @ Xs @ I @ ( nth_list_b @ Xs @ J ) ) @ J @ ( nth_list_b @ Xs @ I ) ) )
          = ( set_list_b2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_456_set__swap,axiom,
    ! [I: nat,Xs: list_P8469869581646625389at_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s3679842834875189465at_nat @ Xs ) )
       => ( ( set_Pr5518436109238095868at_nat @ ( list_u5003261594476800725at_nat @ ( list_u5003261594476800725at_nat @ Xs @ I @ ( nth_Pr6744343527793145070at_nat @ Xs @ J ) ) @ J @ ( nth_Pr6744343527793145070at_nat @ Xs @ I ) ) )
          = ( set_Pr5518436109238095868at_nat @ Xs ) ) ) ) ).

% set_swap
thf(fact_457_list__ex__length,axiom,
    ( list_ex_list_nat
    = ( ^ [P3: list_nat > $o,Xs3: list_list_nat] :
        ? [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( size_s3023201423986296836st_nat @ Xs3 ) )
          & ( P3 @ ( nth_list_nat @ Xs3 @ N2 ) ) ) ) ) ).

% list_ex_length
thf(fact_458_list__ex__length,axiom,
    ( list_e7139191923936702442ctxt_a
    = ( ^ [P3: list_Ground_gmctxt_a > $o,Xs3: list_l1578326660578982846ctxt_a] :
        ? [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( size_s6994882312647770194ctxt_a @ Xs3 ) )
          & ( P3 @ ( nth_li6678718213161987943ctxt_a @ Xs3 @ N2 ) ) ) ) ) ).

% list_ex_length
thf(fact_459_list__ex__length,axiom,
    ( list_e5392882716291896753at_nat
    = ( ^ [P3: list_P8469869581646625389at_nat > $o,Xs3: list_l3298181151656792051at_nat] :
        ? [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( size_s6878757106947045471at_nat @ Xs3 ) )
          & ( P3 @ ( nth_li8805938130461222772at_nat @ Xs3 @ N2 ) ) ) ) ) ).

% list_ex_length
thf(fact_460_list__ex__length,axiom,
    ( list_e8873126888930822106ctxt_a
    = ( ^ [P3: ground_gmctxt_a > $o,Xs3: list_Ground_gmctxt_a] :
        ? [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( size_s962548663205356226ctxt_a @ Xs3 ) )
          & ( P3 @ ( nth_Ground_gmctxt_a @ Xs3 @ N2 ) ) ) ) ) ).

% list_ex_length
thf(fact_461_list__ex__length,axiom,
    ( list_ex_b
    = ( ^ [P3: b > $o,Xs3: list_b] :
        ? [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( size_size_list_b @ Xs3 ) )
          & ( P3 @ ( nth_b @ Xs3 @ N2 ) ) ) ) ) ).

% list_ex_length
thf(fact_462_list__ex__length,axiom,
    ( list_ex_nat
    = ( ^ [P3: nat > $o,Xs3: list_nat] :
        ? [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs3 ) )
          & ( P3 @ ( nth_nat @ Xs3 @ N2 ) ) ) ) ) ).

% list_ex_length
thf(fact_463_list__ex__length,axiom,
    ( list_ex_list_b
    = ( ^ [P3: list_b > $o,Xs3: list_list_b] :
        ? [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( size_s420531428170919671list_b @ Xs3 ) )
          & ( P3 @ ( nth_list_b @ Xs3 @ N2 ) ) ) ) ) ).

% list_ex_length
thf(fact_464_list__ex__length,axiom,
    ( list_e7227117345903044011at_nat
    = ( ^ [P3: produc859450856879609959at_nat > $o,Xs3: list_P8469869581646625389at_nat] :
        ? [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( size_s3679842834875189465at_nat @ Xs3 ) )
          & ( P3 @ ( nth_Pr6744343527793145070at_nat @ Xs3 @ N2 ) ) ) ) ) ).

% list_ex_length
thf(fact_465_length__list__update,axiom,
    ! [Xs: list_Ground_gmctxt_a,I: nat,X: ground_gmctxt_a] :
      ( ( size_s962548663205356226ctxt_a @ ( list_u7950812632479372336ctxt_a @ Xs @ I @ X ) )
      = ( size_s962548663205356226ctxt_a @ Xs ) ) ).

% length_list_update
thf(fact_466_length__list__update,axiom,
    ! [Xs: list_b,I: nat,X: b] :
      ( ( size_size_list_b @ ( list_update_b @ Xs @ I @ X ) )
      = ( size_size_list_b @ Xs ) ) ).

% length_list_update
thf(fact_467_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,X: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_list_update
thf(fact_468_length__list__update,axiom,
    ! [Xs: list_list_b,I: nat,X: list_b] :
      ( ( size_s420531428170919671list_b @ ( list_update_list_b @ Xs @ I @ X ) )
      = ( size_s420531428170919671list_b @ Xs ) ) ).

% length_list_update
thf(fact_469_length__list__update,axiom,
    ! [Xs: list_P8469869581646625389at_nat,I: nat,X: produc859450856879609959at_nat] :
      ( ( size_s3679842834875189465at_nat @ ( list_u5003261594476800725at_nat @ Xs @ I @ X ) )
      = ( size_s3679842834875189465at_nat @ Xs ) ) ).

% length_list_update
thf(fact_470_list__update__id,axiom,
    ! [Xs: list_list_b,I: nat] :
      ( ( list_update_list_b @ Xs @ I @ ( nth_list_b @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_471_list__update__id,axiom,
    ! [Xs: list_nat,I: nat] :
      ( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_472_list__update__id,axiom,
    ! [Xs: list_b,I: nat] :
      ( ( list_update_b @ Xs @ I @ ( nth_b @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_473_list__update__id,axiom,
    ! [Xs: list_Ground_gmctxt_a,I: nat] :
      ( ( list_u7950812632479372336ctxt_a @ Xs @ I @ ( nth_Ground_gmctxt_a @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_474_list__update__id,axiom,
    ! [Xs: list_list_nat,I: nat] :
      ( ( list_update_list_nat @ Xs @ I @ ( nth_list_nat @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_475_list__update__id,axiom,
    ! [Xs: list_l1578326660578982846ctxt_a,I: nat] :
      ( ( list_u8015824433058371904ctxt_a @ Xs @ I @ ( nth_li6678718213161987943ctxt_a @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_476_list__update__id,axiom,
    ! [Xs: list_l3298181151656792051at_nat,I: nat] :
      ( ( list_u5583386537922002395at_nat @ Xs @ I @ ( nth_li8805938130461222772at_nat @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_477_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_list_b,X: list_b] :
      ( ( I != J )
     => ( ( nth_list_b @ ( list_update_list_b @ Xs @ I @ X ) @ J )
        = ( nth_list_b @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_478_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_nat,X: nat] :
      ( ( I != J )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
        = ( nth_nat @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_479_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_b,X: b] :
      ( ( I != J )
     => ( ( nth_b @ ( list_update_b @ Xs @ I @ X ) @ J )
        = ( nth_b @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_480_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_Ground_gmctxt_a,X: ground_gmctxt_a] :
      ( ( I != J )
     => ( ( nth_Ground_gmctxt_a @ ( list_u7950812632479372336ctxt_a @ Xs @ I @ X ) @ J )
        = ( nth_Ground_gmctxt_a @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_481_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_list_nat,X: list_nat] :
      ( ( I != J )
     => ( ( nth_list_nat @ ( list_update_list_nat @ Xs @ I @ X ) @ J )
        = ( nth_list_nat @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_482_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_l1578326660578982846ctxt_a,X: list_Ground_gmctxt_a] :
      ( ( I != J )
     => ( ( nth_li6678718213161987943ctxt_a @ ( list_u8015824433058371904ctxt_a @ Xs @ I @ X ) @ J )
        = ( nth_li6678718213161987943ctxt_a @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_483_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_l3298181151656792051at_nat,X: list_P8469869581646625389at_nat] :
      ( ( I != J )
     => ( ( nth_li8805938130461222772at_nat @ ( list_u5583386537922002395at_nat @ Xs @ I @ X ) @ J )
        = ( nth_li8805938130461222772at_nat @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_484_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_list_nat,X: list_nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
     => ( ( nth_list_nat @ ( list_update_list_nat @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_485_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_l1578326660578982846ctxt_a,X: list_Ground_gmctxt_a] :
      ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Xs ) )
     => ( ( nth_li6678718213161987943ctxt_a @ ( list_u8015824433058371904ctxt_a @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_486_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_l3298181151656792051at_nat,X: list_P8469869581646625389at_nat] :
      ( ( ord_less_nat @ I @ ( size_s6878757106947045471at_nat @ Xs ) )
     => ( ( nth_li8805938130461222772at_nat @ ( list_u5583386537922002395at_nat @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_487_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_Ground_gmctxt_a,X: ground_gmctxt_a] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( nth_Ground_gmctxt_a @ ( list_u7950812632479372336ctxt_a @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_488_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_b,X: b] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
     => ( ( nth_b @ ( list_update_b @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_489_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_490_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_list_b,X: list_b] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ Xs ) )
     => ( ( nth_list_b @ ( list_update_list_b @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_491_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_P8469869581646625389at_nat,X: produc859450856879609959at_nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ Xs ) )
     => ( ( nth_Pr6744343527793145070at_nat @ ( list_u5003261594476800725at_nat @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_492_map__update,axiom,
    ! [F: ground_gmctxt_a > nat,Xs: list_Ground_gmctxt_a,K: nat,Y3: ground_gmctxt_a] :
      ( ( map_Gr1679544123147727284_a_nat @ F @ ( list_u7950812632479372336ctxt_a @ Xs @ K @ Y3 ) )
      = ( list_update_nat @ ( map_Gr1679544123147727284_a_nat @ F @ Xs ) @ K @ ( F @ Y3 ) ) ) ).

% map_update
thf(fact_493_map__update,axiom,
    ! [F: nat > b,Xs: list_nat,K: nat,Y3: nat] :
      ( ( map_nat_b @ F @ ( list_update_nat @ Xs @ K @ Y3 ) )
      = ( list_update_b @ ( map_nat_b @ F @ Xs ) @ K @ ( F @ Y3 ) ) ) ).

% map_update
thf(fact_494_map__update,axiom,
    ! [F: b > ground_gmctxt_a,Xs: list_b,K: nat,Y3: b] :
      ( ( map_b_3587768286370285017ctxt_a @ F @ ( list_update_b @ Xs @ K @ Y3 ) )
      = ( list_u7950812632479372336ctxt_a @ ( map_b_3587768286370285017ctxt_a @ F @ Xs ) @ K @ ( F @ Y3 ) ) ) ).

% map_update
thf(fact_495_map__update,axiom,
    ! [F: b > list_b,Xs: list_b,K: nat,Y3: b] :
      ( ( map_b_list_b @ F @ ( list_update_b @ Xs @ K @ Y3 ) )
      = ( list_update_list_b @ ( map_b_list_b @ F @ Xs ) @ K @ ( F @ Y3 ) ) ) ).

% map_update
thf(fact_496_map__update,axiom,
    ! [F: b > nat,Xs: list_b,K: nat,Y3: b] :
      ( ( map_b_nat @ F @ ( list_update_b @ Xs @ K @ Y3 ) )
      = ( list_update_nat @ ( map_b_nat @ F @ Xs ) @ K @ ( F @ Y3 ) ) ) ).

% map_update
thf(fact_497_map__update,axiom,
    ! [F: b > b,Xs: list_b,K: nat,Y3: b] :
      ( ( map_b_b @ F @ ( list_update_b @ Xs @ K @ Y3 ) )
      = ( list_update_b @ ( map_b_b @ F @ Xs ) @ K @ ( F @ Y3 ) ) ) ).

% map_update
thf(fact_498_list__ex__cong,axiom,
    ! [Xs: list_b,Ys: list_b,F: b > $o,G: b > $o] :
      ( ( Xs = Ys )
     => ( ! [X3: b] :
            ( ( member_b @ X3 @ ( set_b2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( list_ex_b @ F @ Xs )
          = ( list_ex_b @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_499_list__ex__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: nat > $o,G: nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( list_ex_nat @ F @ Xs )
          = ( list_ex_nat @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_500_list__ex__cong,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,F: ground_gmctxt_a > $o,G: ground_gmctxt_a > $o] :
      ( ( Xs = Ys )
     => ( ! [X3: ground_gmctxt_a] :
            ( ( member3306386364193653941ctxt_a @ X3 @ ( set_Ground_gmctxt_a2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( list_e8873126888930822106ctxt_a @ F @ Xs )
          = ( list_e8873126888930822106ctxt_a @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_501_list__ex__cong,axiom,
    ! [Xs: list_P8469869581646625389at_nat,Ys: list_P8469869581646625389at_nat,F: produc859450856879609959at_nat > $o,G: produc859450856879609959at_nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( list_e7227117345903044011at_nat @ F @ Xs )
          = ( list_e7227117345903044011at_nat @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_502_set__update__memI,axiom,
    ! [N: nat,Xs: list_Ground_gmctxt_a,X: ground_gmctxt_a] :
      ( ( ord_less_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ ( list_u7950812632479372336ctxt_a @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_503_set__update__memI,axiom,
    ! [N: nat,Xs: list_b,X: b] :
      ( ( ord_less_nat @ N @ ( size_size_list_b @ Xs ) )
     => ( member_b @ X @ ( set_b2 @ ( list_update_b @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_504_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_505_set__update__memI,axiom,
    ! [N: nat,Xs: list_list_b,X: list_b] :
      ( ( ord_less_nat @ N @ ( size_s420531428170919671list_b @ Xs ) )
     => ( member_list_b @ X @ ( set_list_b2 @ ( list_update_list_b @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_506_set__update__memI,axiom,
    ! [N: nat,Xs: list_P8469869581646625389at_nat,X: produc859450856879609959at_nat] :
      ( ( ord_less_nat @ N @ ( size_s3679842834875189465at_nat @ Xs ) )
     => ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ ( list_u5003261594476800725at_nat @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_507_nth__list__update,axiom,
    ! [I: nat,Xs: list_list_nat,J: nat,X: list_nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_list_nat @ ( list_update_list_nat @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_list_nat @ ( list_update_list_nat @ Xs @ I @ X ) @ J )
            = ( nth_list_nat @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_508_nth__list__update,axiom,
    ! [I: nat,Xs: list_l1578326660578982846ctxt_a,J: nat,X: list_Ground_gmctxt_a] :
      ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_li6678718213161987943ctxt_a @ ( list_u8015824433058371904ctxt_a @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_li6678718213161987943ctxt_a @ ( list_u8015824433058371904ctxt_a @ Xs @ I @ X ) @ J )
            = ( nth_li6678718213161987943ctxt_a @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_509_nth__list__update,axiom,
    ! [I: nat,Xs: list_l3298181151656792051at_nat,J: nat,X: list_P8469869581646625389at_nat] :
      ( ( ord_less_nat @ I @ ( size_s6878757106947045471at_nat @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_li8805938130461222772at_nat @ ( list_u5583386537922002395at_nat @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_li8805938130461222772at_nat @ ( list_u5583386537922002395at_nat @ Xs @ I @ X ) @ J )
            = ( nth_li8805938130461222772at_nat @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_510_nth__list__update,axiom,
    ! [I: nat,Xs: list_Ground_gmctxt_a,J: nat,X: ground_gmctxt_a] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_Ground_gmctxt_a @ ( list_u7950812632479372336ctxt_a @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_Ground_gmctxt_a @ ( list_u7950812632479372336ctxt_a @ Xs @ I @ X ) @ J )
            = ( nth_Ground_gmctxt_a @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_511_nth__list__update,axiom,
    ! [I: nat,Xs: list_b,J: nat,X: b] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_b @ ( list_update_b @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_b @ ( list_update_b @ Xs @ I @ X ) @ J )
            = ( nth_b @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_512_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,J: nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
            = ( nth_nat @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_513_nth__list__update,axiom,
    ! [I: nat,Xs: list_list_b,J: nat,X: list_b] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_list_b @ ( list_update_list_b @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_list_b @ ( list_update_list_b @ Xs @ I @ X ) @ J )
            = ( nth_list_b @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_514_nth__list__update,axiom,
    ! [I: nat,Xs: list_P8469869581646625389at_nat,J: nat,X: produc859450856879609959at_nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_Pr6744343527793145070at_nat @ ( list_u5003261594476800725at_nat @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_Pr6744343527793145070at_nat @ ( list_u5003261594476800725at_nat @ Xs @ I @ X ) @ J )
            = ( nth_Pr6744343527793145070at_nat @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_515_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_list_nat,X: list_nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
     => ( ( ( list_update_list_nat @ Xs @ I @ X )
          = Xs )
        = ( ( nth_list_nat @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_516_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_l1578326660578982846ctxt_a,X: list_Ground_gmctxt_a] :
      ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Xs ) )
     => ( ( ( list_u8015824433058371904ctxt_a @ Xs @ I @ X )
          = Xs )
        = ( ( nth_li6678718213161987943ctxt_a @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_517_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_l3298181151656792051at_nat,X: list_P8469869581646625389at_nat] :
      ( ( ord_less_nat @ I @ ( size_s6878757106947045471at_nat @ Xs ) )
     => ( ( ( list_u5583386537922002395at_nat @ Xs @ I @ X )
          = Xs )
        = ( ( nth_li8805938130461222772at_nat @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_518_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_Ground_gmctxt_a,X: ground_gmctxt_a] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( ( list_u7950812632479372336ctxt_a @ Xs @ I @ X )
          = Xs )
        = ( ( nth_Ground_gmctxt_a @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_519_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_b,X: b] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
     => ( ( ( list_update_b @ Xs @ I @ X )
          = Xs )
        = ( ( nth_b @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_520_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ( list_update_nat @ Xs @ I @ X )
          = Xs )
        = ( ( nth_nat @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_521_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_list_b,X: list_b] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ Xs ) )
     => ( ( ( list_update_list_b @ Xs @ I @ X )
          = Xs )
        = ( ( nth_list_b @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_522_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_P8469869581646625389at_nat,X: produc859450856879609959at_nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ Xs ) )
     => ( ( ( list_u5003261594476800725at_nat @ Xs @ I @ X )
          = Xs )
        = ( ( nth_Pr6744343527793145070at_nat @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_523_parallel__list__update,axiom,
    ! [N: nat,R2: list_nat > list_nat > $o,P2: list_list_nat > $o,Xs: list_list_nat,Ys: list_list_nat] :
      ( ! [Xs2: list_list_nat,I2: nat,Y4: list_nat] :
          ( ( ( size_s3023201423986296836st_nat @ Xs2 )
            = N )
         => ( ( ord_less_nat @ I2 @ N )
           => ( ( R2 @ ( nth_list_nat @ Xs2 @ I2 ) @ Y4 )
             => ( ( P2 @ Xs2 )
               => ( P2 @ ( list_update_list_nat @ Xs2 @ I2 @ Y4 ) ) ) ) ) )
     => ( ( ( size_s3023201423986296836st_nat @ Xs )
          = N )
       => ( ( P2 @ Xs )
         => ( ( ( size_s3023201423986296836st_nat @ Ys )
              = N )
           => ( ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ N )
                 => ( R2 @ ( nth_list_nat @ Xs @ I2 ) @ ( nth_list_nat @ Ys @ I2 ) ) )
             => ( P2 @ Ys ) ) ) ) ) ) ).

% parallel_list_update
thf(fact_524_parallel__list__update,axiom,
    ! [N: nat,R2: list_Ground_gmctxt_a > list_Ground_gmctxt_a > $o,P2: list_l1578326660578982846ctxt_a > $o,Xs: list_l1578326660578982846ctxt_a,Ys: list_l1578326660578982846ctxt_a] :
      ( ! [Xs2: list_l1578326660578982846ctxt_a,I2: nat,Y4: list_Ground_gmctxt_a] :
          ( ( ( size_s6994882312647770194ctxt_a @ Xs2 )
            = N )
         => ( ( ord_less_nat @ I2 @ N )
           => ( ( R2 @ ( nth_li6678718213161987943ctxt_a @ Xs2 @ I2 ) @ Y4 )
             => ( ( P2 @ Xs2 )
               => ( P2 @ ( list_u8015824433058371904ctxt_a @ Xs2 @ I2 @ Y4 ) ) ) ) ) )
     => ( ( ( size_s6994882312647770194ctxt_a @ Xs )
          = N )
       => ( ( P2 @ Xs )
         => ( ( ( size_s6994882312647770194ctxt_a @ Ys )
              = N )
           => ( ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ N )
                 => ( R2 @ ( nth_li6678718213161987943ctxt_a @ Xs @ I2 ) @ ( nth_li6678718213161987943ctxt_a @ Ys @ I2 ) ) )
             => ( P2 @ Ys ) ) ) ) ) ) ).

% parallel_list_update
thf(fact_525_parallel__list__update,axiom,
    ! [N: nat,R2: list_P8469869581646625389at_nat > list_P8469869581646625389at_nat > $o,P2: list_l3298181151656792051at_nat > $o,Xs: list_l3298181151656792051at_nat,Ys: list_l3298181151656792051at_nat] :
      ( ! [Xs2: list_l3298181151656792051at_nat,I2: nat,Y4: list_P8469869581646625389at_nat] :
          ( ( ( size_s6878757106947045471at_nat @ Xs2 )
            = N )
         => ( ( ord_less_nat @ I2 @ N )
           => ( ( R2 @ ( nth_li8805938130461222772at_nat @ Xs2 @ I2 ) @ Y4 )
             => ( ( P2 @ Xs2 )
               => ( P2 @ ( list_u5583386537922002395at_nat @ Xs2 @ I2 @ Y4 ) ) ) ) ) )
     => ( ( ( size_s6878757106947045471at_nat @ Xs )
          = N )
       => ( ( P2 @ Xs )
         => ( ( ( size_s6878757106947045471at_nat @ Ys )
              = N )
           => ( ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ N )
                 => ( R2 @ ( nth_li8805938130461222772at_nat @ Xs @ I2 ) @ ( nth_li8805938130461222772at_nat @ Ys @ I2 ) ) )
             => ( P2 @ Ys ) ) ) ) ) ) ).

% parallel_list_update
thf(fact_526_parallel__list__update,axiom,
    ! [N: nat,R2: ground_gmctxt_a > ground_gmctxt_a > $o,P2: list_Ground_gmctxt_a > $o,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
      ( ! [Xs2: list_Ground_gmctxt_a,I2: nat,Y4: ground_gmctxt_a] :
          ( ( ( size_s962548663205356226ctxt_a @ Xs2 )
            = N )
         => ( ( ord_less_nat @ I2 @ N )
           => ( ( R2 @ ( nth_Ground_gmctxt_a @ Xs2 @ I2 ) @ Y4 )
             => ( ( P2 @ Xs2 )
               => ( P2 @ ( list_u7950812632479372336ctxt_a @ Xs2 @ I2 @ Y4 ) ) ) ) ) )
     => ( ( ( size_s962548663205356226ctxt_a @ Xs )
          = N )
       => ( ( P2 @ Xs )
         => ( ( ( size_s962548663205356226ctxt_a @ Ys )
              = N )
           => ( ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ N )
                 => ( R2 @ ( nth_Ground_gmctxt_a @ Xs @ I2 ) @ ( nth_Ground_gmctxt_a @ Ys @ I2 ) ) )
             => ( P2 @ Ys ) ) ) ) ) ) ).

% parallel_list_update
thf(fact_527_parallel__list__update,axiom,
    ! [N: nat,R2: b > b > $o,P2: list_b > $o,Xs: list_b,Ys: list_b] :
      ( ! [Xs2: list_b,I2: nat,Y4: b] :
          ( ( ( size_size_list_b @ Xs2 )
            = N )
         => ( ( ord_less_nat @ I2 @ N )
           => ( ( R2 @ ( nth_b @ Xs2 @ I2 ) @ Y4 )
             => ( ( P2 @ Xs2 )
               => ( P2 @ ( list_update_b @ Xs2 @ I2 @ Y4 ) ) ) ) ) )
     => ( ( ( size_size_list_b @ Xs )
          = N )
       => ( ( P2 @ Xs )
         => ( ( ( size_size_list_b @ Ys )
              = N )
           => ( ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ N )
                 => ( R2 @ ( nth_b @ Xs @ I2 ) @ ( nth_b @ Ys @ I2 ) ) )
             => ( P2 @ Ys ) ) ) ) ) ) ).

% parallel_list_update
thf(fact_528_parallel__list__update,axiom,
    ! [N: nat,R2: nat > nat > $o,P2: list_nat > $o,Xs: list_nat,Ys: list_nat] :
      ( ! [Xs2: list_nat,I2: nat,Y4: nat] :
          ( ( ( size_size_list_nat @ Xs2 )
            = N )
         => ( ( ord_less_nat @ I2 @ N )
           => ( ( R2 @ ( nth_nat @ Xs2 @ I2 ) @ Y4 )
             => ( ( P2 @ Xs2 )
               => ( P2 @ ( list_update_nat @ Xs2 @ I2 @ Y4 ) ) ) ) ) )
     => ( ( ( size_size_list_nat @ Xs )
          = N )
       => ( ( P2 @ Xs )
         => ( ( ( size_size_list_nat @ Ys )
              = N )
           => ( ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ N )
                 => ( R2 @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) )
             => ( P2 @ Ys ) ) ) ) ) ) ).

% parallel_list_update
thf(fact_529_parallel__list__update,axiom,
    ! [N: nat,R2: list_b > list_b > $o,P2: list_list_b > $o,Xs: list_list_b,Ys: list_list_b] :
      ( ! [Xs2: list_list_b,I2: nat,Y4: list_b] :
          ( ( ( size_s420531428170919671list_b @ Xs2 )
            = N )
         => ( ( ord_less_nat @ I2 @ N )
           => ( ( R2 @ ( nth_list_b @ Xs2 @ I2 ) @ Y4 )
             => ( ( P2 @ Xs2 )
               => ( P2 @ ( list_update_list_b @ Xs2 @ I2 @ Y4 ) ) ) ) ) )
     => ( ( ( size_s420531428170919671list_b @ Xs )
          = N )
       => ( ( P2 @ Xs )
         => ( ( ( size_s420531428170919671list_b @ Ys )
              = N )
           => ( ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ N )
                 => ( R2 @ ( nth_list_b @ Xs @ I2 ) @ ( nth_list_b @ Ys @ I2 ) ) )
             => ( P2 @ Ys ) ) ) ) ) ) ).

% parallel_list_update
thf(fact_530_parallel__list__update,axiom,
    ! [N: nat,R2: produc859450856879609959at_nat > produc859450856879609959at_nat > $o,P2: list_P8469869581646625389at_nat > $o,Xs: list_P8469869581646625389at_nat,Ys: list_P8469869581646625389at_nat] :
      ( ! [Xs2: list_P8469869581646625389at_nat,I2: nat,Y4: produc859450856879609959at_nat] :
          ( ( ( size_s3679842834875189465at_nat @ Xs2 )
            = N )
         => ( ( ord_less_nat @ I2 @ N )
           => ( ( R2 @ ( nth_Pr6744343527793145070at_nat @ Xs2 @ I2 ) @ Y4 )
             => ( ( P2 @ Xs2 )
               => ( P2 @ ( list_u5003261594476800725at_nat @ Xs2 @ I2 @ Y4 ) ) ) ) ) )
     => ( ( ( size_s3679842834875189465at_nat @ Xs )
          = N )
       => ( ( P2 @ Xs )
         => ( ( ( size_s3679842834875189465at_nat @ Ys )
              = N )
           => ( ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ N )
                 => ( R2 @ ( nth_Pr6744343527793145070at_nat @ Xs @ I2 ) @ ( nth_Pr6744343527793145070at_nat @ Ys @ I2 ) ) )
             => ( P2 @ Ys ) ) ) ) ) ) ).

% parallel_list_update
thf(fact_531_remove__nth__sound__l,axiom,
    ! [P2: nat,N: nat,Xs: list_list_b] :
      ( ( ord_less_nat @ P2 @ N )
     => ( ( nth_list_b @ ( missin8041159180453292414list_b @ N @ Xs ) @ P2 )
        = ( nth_list_b @ Xs @ P2 ) ) ) ).

% remove_nth_sound_l
thf(fact_532_remove__nth__sound__l,axiom,
    ! [P2: nat,N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ P2 @ N )
     => ( ( nth_nat @ ( missin7175274867594579095th_nat @ N @ Xs ) @ P2 )
        = ( nth_nat @ Xs @ P2 ) ) ) ).

% remove_nth_sound_l
thf(fact_533_remove__nth__sound__l,axiom,
    ! [P2: nat,N: nat,Xs: list_b] :
      ( ( ord_less_nat @ P2 @ N )
     => ( ( nth_b @ ( missing_remove_nth_b @ N @ Xs ) @ P2 )
        = ( nth_b @ Xs @ P2 ) ) ) ).

% remove_nth_sound_l
thf(fact_534_remove__nth__sound__l,axiom,
    ! [P2: nat,N: nat,Xs: list_Ground_gmctxt_a] :
      ( ( ord_less_nat @ P2 @ N )
     => ( ( nth_Ground_gmctxt_a @ ( missin7823264313509747941ctxt_a @ N @ Xs ) @ P2 )
        = ( nth_Ground_gmctxt_a @ Xs @ P2 ) ) ) ).

% remove_nth_sound_l
thf(fact_535_remove__nth__sound__l,axiom,
    ! [P2: nat,N: nat,Xs: list_list_nat] :
      ( ( ord_less_nat @ P2 @ N )
     => ( ( nth_list_nat @ ( missin5133812807938250791st_nat @ N @ Xs ) @ P2 )
        = ( nth_list_nat @ Xs @ P2 ) ) ) ).

% remove_nth_sound_l
thf(fact_536_remove__nth__sound__l,axiom,
    ! [P2: nat,N: nat,Xs: list_l1578326660578982846ctxt_a] :
      ( ( ord_less_nat @ P2 @ N )
     => ( ( nth_li6678718213161987943ctxt_a @ ( missin1875899094540944245ctxt_a @ N @ Xs ) @ P2 )
        = ( nth_li6678718213161987943ctxt_a @ Xs @ P2 ) ) ) ).

% remove_nth_sound_l
thf(fact_537_remove__nth__sound__l,axiom,
    ! [P2: nat,N: nat,Xs: list_l3298181151656792051at_nat] :
      ( ( ord_less_nat @ P2 @ N )
     => ( ( nth_li8805938130461222772at_nat @ ( missin7550826715306853094at_nat @ N @ Xs ) @ P2 )
        = ( nth_li8805938130461222772at_nat @ Xs @ P2 ) ) ) ).

% remove_nth_sound_l
thf(fact_538_trancl__listp_Ocases,axiom,
    ! [R: list_nat > list_nat > $o,A1: list_list_nat,A22: list_list_nat] :
      ( ( trancl267208031524270810st_nat @ R @ A1 @ A22 )
     => ( ( ( ( size_s3023201423986296836st_nat @ A1 )
            = ( size_s3023201423986296836st_nat @ A22 ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ A22 ) )
               => ( R @ ( nth_list_nat @ A1 @ I4 ) @ ( nth_list_nat @ A22 @ I4 ) ) ) )
       => ~ ! [Ys4: list_list_nat,I2: nat,Z: list_nat] :
              ( ( A22
                = ( list_update_list_nat @ Ys4 @ I2 @ Z ) )
             => ( ( trancl267208031524270810st_nat @ R @ A1 @ Ys4 )
               => ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ys4 ) )
                 => ~ ( R @ ( nth_list_nat @ Ys4 @ I2 ) @ Z ) ) ) ) ) ) ).

% trancl_listp.cases
thf(fact_539_trancl__listp_Ocases,axiom,
    ! [R: list_Ground_gmctxt_a > list_Ground_gmctxt_a > $o,A1: list_l1578326660578982846ctxt_a,A22: list_l1578326660578982846ctxt_a] :
      ( ( trancl4044344332876950568ctxt_a @ R @ A1 @ A22 )
     => ( ( ( ( size_s6994882312647770194ctxt_a @ A1 )
            = ( size_s6994882312647770194ctxt_a @ A22 ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_s6994882312647770194ctxt_a @ A22 ) )
               => ( R @ ( nth_li6678718213161987943ctxt_a @ A1 @ I4 ) @ ( nth_li6678718213161987943ctxt_a @ A22 @ I4 ) ) ) )
       => ~ ! [Ys4: list_l1578326660578982846ctxt_a,I2: nat,Z: list_Ground_gmctxt_a] :
              ( ( A22
                = ( list_u8015824433058371904ctxt_a @ Ys4 @ I2 @ Z ) )
             => ( ( trancl4044344332876950568ctxt_a @ R @ A1 @ Ys4 )
               => ( ( ord_less_nat @ I2 @ ( size_s6994882312647770194ctxt_a @ Ys4 ) )
                 => ~ ( R @ ( nth_li6678718213161987943ctxt_a @ Ys4 @ I2 ) @ Z ) ) ) ) ) ) ).

% trancl_listp.cases
thf(fact_540_trancl__listp_Ocases,axiom,
    ! [R: list_P8469869581646625389at_nat > list_P8469869581646625389at_nat > $o,A1: list_l3298181151656792051at_nat,A22: list_l3298181151656792051at_nat] :
      ( ( trancl6797153713963726067at_nat @ R @ A1 @ A22 )
     => ( ( ( ( size_s6878757106947045471at_nat @ A1 )
            = ( size_s6878757106947045471at_nat @ A22 ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_s6878757106947045471at_nat @ A22 ) )
               => ( R @ ( nth_li8805938130461222772at_nat @ A1 @ I4 ) @ ( nth_li8805938130461222772at_nat @ A22 @ I4 ) ) ) )
       => ~ ! [Ys4: list_l3298181151656792051at_nat,I2: nat,Z: list_P8469869581646625389at_nat] :
              ( ( A22
                = ( list_u5583386537922002395at_nat @ Ys4 @ I2 @ Z ) )
             => ( ( trancl6797153713963726067at_nat @ R @ A1 @ Ys4 )
               => ( ( ord_less_nat @ I2 @ ( size_s6878757106947045471at_nat @ Ys4 ) )
                 => ~ ( R @ ( nth_li8805938130461222772at_nat @ Ys4 @ I2 ) @ Z ) ) ) ) ) ) ).

% trancl_listp.cases
thf(fact_541_trancl__listp_Ocases,axiom,
    ! [R: ground_gmctxt_a > ground_gmctxt_a > $o,A1: list_Ground_gmctxt_a,A22: list_Ground_gmctxt_a] :
      ( ( trancl3815965565430463256ctxt_a @ R @ A1 @ A22 )
     => ( ( ( ( size_s962548663205356226ctxt_a @ A1 )
            = ( size_s962548663205356226ctxt_a @ A22 ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ A22 ) )
               => ( R @ ( nth_Ground_gmctxt_a @ A1 @ I4 ) @ ( nth_Ground_gmctxt_a @ A22 @ I4 ) ) ) )
       => ~ ! [Ys4: list_Ground_gmctxt_a,I2: nat,Z: ground_gmctxt_a] :
              ( ( A22
                = ( list_u7950812632479372336ctxt_a @ Ys4 @ I2 @ Z ) )
             => ( ( trancl3815965565430463256ctxt_a @ R @ A1 @ Ys4 )
               => ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Ys4 ) )
                 => ~ ( R @ ( nth_Ground_gmctxt_a @ Ys4 @ I2 ) @ Z ) ) ) ) ) ) ).

% trancl_listp.cases
thf(fact_542_trancl__listp_Ocases,axiom,
    ! [R: b > b > $o,A1: list_b,A22: list_b] :
      ( ( trancl_listp_b @ R @ A1 @ A22 )
     => ( ( ( ( size_size_list_b @ A1 )
            = ( size_size_list_b @ A22 ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_size_list_b @ A22 ) )
               => ( R @ ( nth_b @ A1 @ I4 ) @ ( nth_b @ A22 @ I4 ) ) ) )
       => ~ ! [Ys4: list_b,I2: nat,Z: b] :
              ( ( A22
                = ( list_update_b @ Ys4 @ I2 @ Z ) )
             => ( ( trancl_listp_b @ R @ A1 @ Ys4 )
               => ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Ys4 ) )
                 => ~ ( R @ ( nth_b @ Ys4 @ I2 ) @ Z ) ) ) ) ) ) ).

% trancl_listp.cases
thf(fact_543_trancl__listp_Ocases,axiom,
    ! [R: nat > nat > $o,A1: list_nat,A22: list_nat] :
      ( ( trancl_listp_nat @ R @ A1 @ A22 )
     => ( ( ( ( size_size_list_nat @ A1 )
            = ( size_size_list_nat @ A22 ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ A22 ) )
               => ( R @ ( nth_nat @ A1 @ I4 ) @ ( nth_nat @ A22 @ I4 ) ) ) )
       => ~ ! [Ys4: list_nat,I2: nat,Z: nat] :
              ( ( A22
                = ( list_update_nat @ Ys4 @ I2 @ Z ) )
             => ( ( trancl_listp_nat @ R @ A1 @ Ys4 )
               => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys4 ) )
                 => ~ ( R @ ( nth_nat @ Ys4 @ I2 ) @ Z ) ) ) ) ) ) ).

% trancl_listp.cases
thf(fact_544_trancl__listp_Ocases,axiom,
    ! [R: list_b > list_b > $o,A1: list_list_b,A22: list_list_b] :
      ( ( trancl_listp_list_b @ R @ A1 @ A22 )
     => ( ( ( ( size_s420531428170919671list_b @ A1 )
            = ( size_s420531428170919671list_b @ A22 ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_s420531428170919671list_b @ A22 ) )
               => ( R @ ( nth_list_b @ A1 @ I4 ) @ ( nth_list_b @ A22 @ I4 ) ) ) )
       => ~ ! [Ys4: list_list_b,I2: nat,Z: list_b] :
              ( ( A22
                = ( list_update_list_b @ Ys4 @ I2 @ Z ) )
             => ( ( trancl_listp_list_b @ R @ A1 @ Ys4 )
               => ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Ys4 ) )
                 => ~ ( R @ ( nth_list_b @ Ys4 @ I2 ) @ Z ) ) ) ) ) ) ).

% trancl_listp.cases
thf(fact_545_trancl__listp_Ocases,axiom,
    ! [R: produc859450856879609959at_nat > produc859450856879609959at_nat > $o,A1: list_P8469869581646625389at_nat,A22: list_P8469869581646625389at_nat] :
      ( ( trancl2313143553896500717at_nat @ R @ A1 @ A22 )
     => ( ( ( ( size_s3679842834875189465at_nat @ A1 )
            = ( size_s3679842834875189465at_nat @ A22 ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_s3679842834875189465at_nat @ A22 ) )
               => ( R @ ( nth_Pr6744343527793145070at_nat @ A1 @ I4 ) @ ( nth_Pr6744343527793145070at_nat @ A22 @ I4 ) ) ) )
       => ~ ! [Ys4: list_P8469869581646625389at_nat,I2: nat,Z: produc859450856879609959at_nat] :
              ( ( A22
                = ( list_u5003261594476800725at_nat @ Ys4 @ I2 @ Z ) )
             => ( ( trancl2313143553896500717at_nat @ R @ A1 @ Ys4 )
               => ( ( ord_less_nat @ I2 @ ( size_s3679842834875189465at_nat @ Ys4 ) )
                 => ~ ( R @ ( nth_Pr6744343527793145070at_nat @ Ys4 @ I2 ) @ Z ) ) ) ) ) ) ).

% trancl_listp.cases
thf(fact_546_trancl__listp_Osimps,axiom,
    ( trancl267208031524270810st_nat
    = ( ^ [R3: list_nat > list_nat > $o,A12: list_list_nat,A23: list_list_nat] :
          ( ? [Xs3: list_list_nat,Ys3: list_list_nat] :
              ( ( A12 = Xs3 )
              & ( A23 = Ys3 )
              & ( ( size_s3023201423986296836st_nat @ Xs3 )
                = ( size_s3023201423986296836st_nat @ Ys3 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Ys3 ) )
                 => ( R3 @ ( nth_list_nat @ Xs3 @ I3 ) @ ( nth_list_nat @ Ys3 @ I3 ) ) ) )
          | ? [Xs3: list_list_nat,Ys3: list_list_nat,I3: nat,Z3: list_nat] :
              ( ( A12 = Xs3 )
              & ( A23
                = ( list_update_list_nat @ Ys3 @ I3 @ Z3 ) )
              & ( trancl267208031524270810st_nat @ R3 @ Xs3 @ Ys3 )
              & ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Ys3 ) )
              & ( R3 @ ( nth_list_nat @ Ys3 @ I3 ) @ Z3 ) ) ) ) ) ).

% trancl_listp.simps
thf(fact_547_trancl__listp_Osimps,axiom,
    ( trancl4044344332876950568ctxt_a
    = ( ^ [R3: list_Ground_gmctxt_a > list_Ground_gmctxt_a > $o,A12: list_l1578326660578982846ctxt_a,A23: list_l1578326660578982846ctxt_a] :
          ( ? [Xs3: list_l1578326660578982846ctxt_a,Ys3: list_l1578326660578982846ctxt_a] :
              ( ( A12 = Xs3 )
              & ( A23 = Ys3 )
              & ( ( size_s6994882312647770194ctxt_a @ Xs3 )
                = ( size_s6994882312647770194ctxt_a @ Ys3 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_s6994882312647770194ctxt_a @ Ys3 ) )
                 => ( R3 @ ( nth_li6678718213161987943ctxt_a @ Xs3 @ I3 ) @ ( nth_li6678718213161987943ctxt_a @ Ys3 @ I3 ) ) ) )
          | ? [Xs3: list_l1578326660578982846ctxt_a,Ys3: list_l1578326660578982846ctxt_a,I3: nat,Z3: list_Ground_gmctxt_a] :
              ( ( A12 = Xs3 )
              & ( A23
                = ( list_u8015824433058371904ctxt_a @ Ys3 @ I3 @ Z3 ) )
              & ( trancl4044344332876950568ctxt_a @ R3 @ Xs3 @ Ys3 )
              & ( ord_less_nat @ I3 @ ( size_s6994882312647770194ctxt_a @ Ys3 ) )
              & ( R3 @ ( nth_li6678718213161987943ctxt_a @ Ys3 @ I3 ) @ Z3 ) ) ) ) ) ).

% trancl_listp.simps
thf(fact_548_trancl__listp_Osimps,axiom,
    ( trancl6797153713963726067at_nat
    = ( ^ [R3: list_P8469869581646625389at_nat > list_P8469869581646625389at_nat > $o,A12: list_l3298181151656792051at_nat,A23: list_l3298181151656792051at_nat] :
          ( ? [Xs3: list_l3298181151656792051at_nat,Ys3: list_l3298181151656792051at_nat] :
              ( ( A12 = Xs3 )
              & ( A23 = Ys3 )
              & ( ( size_s6878757106947045471at_nat @ Xs3 )
                = ( size_s6878757106947045471at_nat @ Ys3 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_s6878757106947045471at_nat @ Ys3 ) )
                 => ( R3 @ ( nth_li8805938130461222772at_nat @ Xs3 @ I3 ) @ ( nth_li8805938130461222772at_nat @ Ys3 @ I3 ) ) ) )
          | ? [Xs3: list_l3298181151656792051at_nat,Ys3: list_l3298181151656792051at_nat,I3: nat,Z3: list_P8469869581646625389at_nat] :
              ( ( A12 = Xs3 )
              & ( A23
                = ( list_u5583386537922002395at_nat @ Ys3 @ I3 @ Z3 ) )
              & ( trancl6797153713963726067at_nat @ R3 @ Xs3 @ Ys3 )
              & ( ord_less_nat @ I3 @ ( size_s6878757106947045471at_nat @ Ys3 ) )
              & ( R3 @ ( nth_li8805938130461222772at_nat @ Ys3 @ I3 ) @ Z3 ) ) ) ) ) ).

% trancl_listp.simps
thf(fact_549_trancl__listp_Osimps,axiom,
    ( trancl3815965565430463256ctxt_a
    = ( ^ [R3: ground_gmctxt_a > ground_gmctxt_a > $o,A12: list_Ground_gmctxt_a,A23: list_Ground_gmctxt_a] :
          ( ? [Xs3: list_Ground_gmctxt_a,Ys3: list_Ground_gmctxt_a] :
              ( ( A12 = Xs3 )
              & ( A23 = Ys3 )
              & ( ( size_s962548663205356226ctxt_a @ Xs3 )
                = ( size_s962548663205356226ctxt_a @ Ys3 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Ys3 ) )
                 => ( R3 @ ( nth_Ground_gmctxt_a @ Xs3 @ I3 ) @ ( nth_Ground_gmctxt_a @ Ys3 @ I3 ) ) ) )
          | ? [Xs3: list_Ground_gmctxt_a,Ys3: list_Ground_gmctxt_a,I3: nat,Z3: ground_gmctxt_a] :
              ( ( A12 = Xs3 )
              & ( A23
                = ( list_u7950812632479372336ctxt_a @ Ys3 @ I3 @ Z3 ) )
              & ( trancl3815965565430463256ctxt_a @ R3 @ Xs3 @ Ys3 )
              & ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Ys3 ) )
              & ( R3 @ ( nth_Ground_gmctxt_a @ Ys3 @ I3 ) @ Z3 ) ) ) ) ) ).

% trancl_listp.simps
thf(fact_550_trancl__listp_Osimps,axiom,
    ( trancl_listp_b
    = ( ^ [R3: b > b > $o,A12: list_b,A23: list_b] :
          ( ? [Xs3: list_b,Ys3: list_b] :
              ( ( A12 = Xs3 )
              & ( A23 = Ys3 )
              & ( ( size_size_list_b @ Xs3 )
                = ( size_size_list_b @ Ys3 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Ys3 ) )
                 => ( R3 @ ( nth_b @ Xs3 @ I3 ) @ ( nth_b @ Ys3 @ I3 ) ) ) )
          | ? [Xs3: list_b,Ys3: list_b,I3: nat,Z3: b] :
              ( ( A12 = Xs3 )
              & ( A23
                = ( list_update_b @ Ys3 @ I3 @ Z3 ) )
              & ( trancl_listp_b @ R3 @ Xs3 @ Ys3 )
              & ( ord_less_nat @ I3 @ ( size_size_list_b @ Ys3 ) )
              & ( R3 @ ( nth_b @ Ys3 @ I3 ) @ Z3 ) ) ) ) ) ).

% trancl_listp.simps
thf(fact_551_trancl__listp_Osimps,axiom,
    ( trancl_listp_nat
    = ( ^ [R3: nat > nat > $o,A12: list_nat,A23: list_nat] :
          ( ? [Xs3: list_nat,Ys3: list_nat] :
              ( ( A12 = Xs3 )
              & ( A23 = Ys3 )
              & ( ( size_size_list_nat @ Xs3 )
                = ( size_size_list_nat @ Ys3 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys3 ) )
                 => ( R3 @ ( nth_nat @ Xs3 @ I3 ) @ ( nth_nat @ Ys3 @ I3 ) ) ) )
          | ? [Xs3: list_nat,Ys3: list_nat,I3: nat,Z3: nat] :
              ( ( A12 = Xs3 )
              & ( A23
                = ( list_update_nat @ Ys3 @ I3 @ Z3 ) )
              & ( trancl_listp_nat @ R3 @ Xs3 @ Ys3 )
              & ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys3 ) )
              & ( R3 @ ( nth_nat @ Ys3 @ I3 ) @ Z3 ) ) ) ) ) ).

% trancl_listp.simps
thf(fact_552_trancl__listp_Osimps,axiom,
    ( trancl_listp_list_b
    = ( ^ [R3: list_b > list_b > $o,A12: list_list_b,A23: list_list_b] :
          ( ? [Xs3: list_list_b,Ys3: list_list_b] :
              ( ( A12 = Xs3 )
              & ( A23 = Ys3 )
              & ( ( size_s420531428170919671list_b @ Xs3 )
                = ( size_s420531428170919671list_b @ Ys3 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_s420531428170919671list_b @ Ys3 ) )
                 => ( R3 @ ( nth_list_b @ Xs3 @ I3 ) @ ( nth_list_b @ Ys3 @ I3 ) ) ) )
          | ? [Xs3: list_list_b,Ys3: list_list_b,I3: nat,Z3: list_b] :
              ( ( A12 = Xs3 )
              & ( A23
                = ( list_update_list_b @ Ys3 @ I3 @ Z3 ) )
              & ( trancl_listp_list_b @ R3 @ Xs3 @ Ys3 )
              & ( ord_less_nat @ I3 @ ( size_s420531428170919671list_b @ Ys3 ) )
              & ( R3 @ ( nth_list_b @ Ys3 @ I3 ) @ Z3 ) ) ) ) ) ).

% trancl_listp.simps
thf(fact_553_trancl__listp_Osimps,axiom,
    ( trancl2313143553896500717at_nat
    = ( ^ [R3: produc859450856879609959at_nat > produc859450856879609959at_nat > $o,A12: list_P8469869581646625389at_nat,A23: list_P8469869581646625389at_nat] :
          ( ? [Xs3: list_P8469869581646625389at_nat,Ys3: list_P8469869581646625389at_nat] :
              ( ( A12 = Xs3 )
              & ( A23 = Ys3 )
              & ( ( size_s3679842834875189465at_nat @ Xs3 )
                = ( size_s3679842834875189465at_nat @ Ys3 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_s3679842834875189465at_nat @ Ys3 ) )
                 => ( R3 @ ( nth_Pr6744343527793145070at_nat @ Xs3 @ I3 ) @ ( nth_Pr6744343527793145070at_nat @ Ys3 @ I3 ) ) ) )
          | ? [Xs3: list_P8469869581646625389at_nat,Ys3: list_P8469869581646625389at_nat,I3: nat,Z3: produc859450856879609959at_nat] :
              ( ( A12 = Xs3 )
              & ( A23
                = ( list_u5003261594476800725at_nat @ Ys3 @ I3 @ Z3 ) )
              & ( trancl2313143553896500717at_nat @ R3 @ Xs3 @ Ys3 )
              & ( ord_less_nat @ I3 @ ( size_s3679842834875189465at_nat @ Ys3 ) )
              & ( R3 @ ( nth_Pr6744343527793145070at_nat @ Ys3 @ I3 ) @ Z3 ) ) ) ) ) ).

% trancl_listp.simps
thf(fact_554_trancl__listp_Olist__trancl,axiom,
    ! [R: list_nat > list_nat > $o,Xs: list_list_nat,Ys: list_list_nat,I: nat,Z4: list_nat] :
      ( ( trancl267208031524270810st_nat @ R @ Xs @ Ys )
     => ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Ys ) )
       => ( ( R @ ( nth_list_nat @ Ys @ I ) @ Z4 )
         => ( trancl267208031524270810st_nat @ R @ Xs @ ( list_update_list_nat @ Ys @ I @ Z4 ) ) ) ) ) ).

% trancl_listp.list_trancl
thf(fact_555_trancl__listp_Olist__trancl,axiom,
    ! [R: list_Ground_gmctxt_a > list_Ground_gmctxt_a > $o,Xs: list_l1578326660578982846ctxt_a,Ys: list_l1578326660578982846ctxt_a,I: nat,Z4: list_Ground_gmctxt_a] :
      ( ( trancl4044344332876950568ctxt_a @ R @ Xs @ Ys )
     => ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Ys ) )
       => ( ( R @ ( nth_li6678718213161987943ctxt_a @ Ys @ I ) @ Z4 )
         => ( trancl4044344332876950568ctxt_a @ R @ Xs @ ( list_u8015824433058371904ctxt_a @ Ys @ I @ Z4 ) ) ) ) ) ).

% trancl_listp.list_trancl
thf(fact_556_trancl__listp_Olist__trancl,axiom,
    ! [R: list_P8469869581646625389at_nat > list_P8469869581646625389at_nat > $o,Xs: list_l3298181151656792051at_nat,Ys: list_l3298181151656792051at_nat,I: nat,Z4: list_P8469869581646625389at_nat] :
      ( ( trancl6797153713963726067at_nat @ R @ Xs @ Ys )
     => ( ( ord_less_nat @ I @ ( size_s6878757106947045471at_nat @ Ys ) )
       => ( ( R @ ( nth_li8805938130461222772at_nat @ Ys @ I ) @ Z4 )
         => ( trancl6797153713963726067at_nat @ R @ Xs @ ( list_u5583386537922002395at_nat @ Ys @ I @ Z4 ) ) ) ) ) ).

% trancl_listp.list_trancl
thf(fact_557_trancl__listp_Olist__trancl,axiom,
    ! [R: ground_gmctxt_a > ground_gmctxt_a > $o,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,I: nat,Z4: ground_gmctxt_a] :
      ( ( trancl3815965565430463256ctxt_a @ R @ Xs @ Ys )
     => ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ys ) )
       => ( ( R @ ( nth_Ground_gmctxt_a @ Ys @ I ) @ Z4 )
         => ( trancl3815965565430463256ctxt_a @ R @ Xs @ ( list_u7950812632479372336ctxt_a @ Ys @ I @ Z4 ) ) ) ) ) ).

% trancl_listp.list_trancl
thf(fact_558_trancl__listp_Olist__trancl,axiom,
    ! [R: b > b > $o,Xs: list_b,Ys: list_b,I: nat,Z4: b] :
      ( ( trancl_listp_b @ R @ Xs @ Ys )
     => ( ( ord_less_nat @ I @ ( size_size_list_b @ Ys ) )
       => ( ( R @ ( nth_b @ Ys @ I ) @ Z4 )
         => ( trancl_listp_b @ R @ Xs @ ( list_update_b @ Ys @ I @ Z4 ) ) ) ) ) ).

% trancl_listp.list_trancl
thf(fact_559_trancl__listp_Olist__trancl,axiom,
    ! [R: nat > nat > $o,Xs: list_nat,Ys: list_nat,I: nat,Z4: nat] :
      ( ( trancl_listp_nat @ R @ Xs @ Ys )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( R @ ( nth_nat @ Ys @ I ) @ Z4 )
         => ( trancl_listp_nat @ R @ Xs @ ( list_update_nat @ Ys @ I @ Z4 ) ) ) ) ) ).

% trancl_listp.list_trancl
thf(fact_560_trancl__listp_Olist__trancl,axiom,
    ! [R: list_b > list_b > $o,Xs: list_list_b,Ys: list_list_b,I: nat,Z4: list_b] :
      ( ( trancl_listp_list_b @ R @ Xs @ Ys )
     => ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ Ys ) )
       => ( ( R @ ( nth_list_b @ Ys @ I ) @ Z4 )
         => ( trancl_listp_list_b @ R @ Xs @ ( list_update_list_b @ Ys @ I @ Z4 ) ) ) ) ) ).

% trancl_listp.list_trancl
thf(fact_561_trancl__listp_Olist__trancl,axiom,
    ! [R: produc859450856879609959at_nat > produc859450856879609959at_nat > $o,Xs: list_P8469869581646625389at_nat,Ys: list_P8469869581646625389at_nat,I: nat,Z4: produc859450856879609959at_nat] :
      ( ( trancl2313143553896500717at_nat @ R @ Xs @ Ys )
     => ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ Ys ) )
       => ( ( R @ ( nth_Pr6744343527793145070at_nat @ Ys @ I ) @ Z4 )
         => ( trancl2313143553896500717at_nat @ R @ Xs @ ( list_u5003261594476800725at_nat @ Ys @ I @ Z4 ) ) ) ) ) ).

% trancl_listp.list_trancl
thf(fact_562_adjust__idx__rev__nth,axiom,
    ! [I: nat,Xs: list_list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
     => ( ( J != I )
       => ( ( nth_list_nat @ ( missin5133812807938250791st_nat @ I @ Xs ) @ ( missin3815256168798769645dx_rev @ I @ J ) )
          = ( nth_list_nat @ Xs @ J ) ) ) ) ).

% adjust_idx_rev_nth
thf(fact_563_adjust__idx__rev__nth,axiom,
    ! [I: nat,Xs: list_l1578326660578982846ctxt_a,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Xs ) )
     => ( ( J != I )
       => ( ( nth_li6678718213161987943ctxt_a @ ( missin1875899094540944245ctxt_a @ I @ Xs ) @ ( missin3815256168798769645dx_rev @ I @ J ) )
          = ( nth_li6678718213161987943ctxt_a @ Xs @ J ) ) ) ) ).

% adjust_idx_rev_nth
thf(fact_564_adjust__idx__rev__nth,axiom,
    ! [I: nat,Xs: list_l3298181151656792051at_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6878757106947045471at_nat @ Xs ) )
     => ( ( J != I )
       => ( ( nth_li8805938130461222772at_nat @ ( missin7550826715306853094at_nat @ I @ Xs ) @ ( missin3815256168798769645dx_rev @ I @ J ) )
          = ( nth_li8805938130461222772at_nat @ Xs @ J ) ) ) ) ).

% adjust_idx_rev_nth
thf(fact_565_adjust__idx__rev__nth,axiom,
    ! [I: nat,Xs: list_Ground_gmctxt_a,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( J != I )
       => ( ( nth_Ground_gmctxt_a @ ( missin7823264313509747941ctxt_a @ I @ Xs ) @ ( missin3815256168798769645dx_rev @ I @ J ) )
          = ( nth_Ground_gmctxt_a @ Xs @ J ) ) ) ) ).

% adjust_idx_rev_nth
thf(fact_566_adjust__idx__rev__nth,axiom,
    ! [I: nat,Xs: list_b,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
     => ( ( J != I )
       => ( ( nth_b @ ( missing_remove_nth_b @ I @ Xs ) @ ( missin3815256168798769645dx_rev @ I @ J ) )
          = ( nth_b @ Xs @ J ) ) ) ) ).

% adjust_idx_rev_nth
thf(fact_567_adjust__idx__rev__nth,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( J != I )
       => ( ( nth_nat @ ( missin7175274867594579095th_nat @ I @ Xs ) @ ( missin3815256168798769645dx_rev @ I @ J ) )
          = ( nth_nat @ Xs @ J ) ) ) ) ).

% adjust_idx_rev_nth
thf(fact_568_adjust__idx__rev__nth,axiom,
    ! [I: nat,Xs: list_list_b,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ Xs ) )
     => ( ( J != I )
       => ( ( nth_list_b @ ( missin8041159180453292414list_b @ I @ Xs ) @ ( missin3815256168798769645dx_rev @ I @ J ) )
          = ( nth_list_b @ Xs @ J ) ) ) ) ).

% adjust_idx_rev_nth
thf(fact_569_adjust__idx__rev__nth,axiom,
    ! [I: nat,Xs: list_P8469869581646625389at_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ Xs ) )
     => ( ( J != I )
       => ( ( nth_Pr6744343527793145070at_nat @ ( missin1453772821081764192at_nat @ I @ Xs ) @ ( missin3815256168798769645dx_rev @ I @ J ) )
          = ( nth_Pr6744343527793145070at_nat @ Xs @ J ) ) ) ) ).

% adjust_idx_rev_nth
thf(fact_570_adjust__idx__nth,axiom,
    ! [I: nat,Xs: list_list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
     => ( ( nth_list_nat @ ( missin5133812807938250791st_nat @ I @ Xs ) @ J )
        = ( nth_list_nat @ Xs @ ( missing_adjust_idx @ I @ J ) ) ) ) ).

% adjust_idx_nth
thf(fact_571_adjust__idx__nth,axiom,
    ! [I: nat,Xs: list_l1578326660578982846ctxt_a,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Xs ) )
     => ( ( nth_li6678718213161987943ctxt_a @ ( missin1875899094540944245ctxt_a @ I @ Xs ) @ J )
        = ( nth_li6678718213161987943ctxt_a @ Xs @ ( missing_adjust_idx @ I @ J ) ) ) ) ).

% adjust_idx_nth
thf(fact_572_adjust__idx__nth,axiom,
    ! [I: nat,Xs: list_l3298181151656792051at_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6878757106947045471at_nat @ Xs ) )
     => ( ( nth_li8805938130461222772at_nat @ ( missin7550826715306853094at_nat @ I @ Xs ) @ J )
        = ( nth_li8805938130461222772at_nat @ Xs @ ( missing_adjust_idx @ I @ J ) ) ) ) ).

% adjust_idx_nth
thf(fact_573_adjust__idx__nth,axiom,
    ! [I: nat,Xs: list_Ground_gmctxt_a,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( nth_Ground_gmctxt_a @ ( missin7823264313509747941ctxt_a @ I @ Xs ) @ J )
        = ( nth_Ground_gmctxt_a @ Xs @ ( missing_adjust_idx @ I @ J ) ) ) ) ).

% adjust_idx_nth
thf(fact_574_adjust__idx__nth,axiom,
    ! [I: nat,Xs: list_b,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
     => ( ( nth_b @ ( missing_remove_nth_b @ I @ Xs ) @ J )
        = ( nth_b @ Xs @ ( missing_adjust_idx @ I @ J ) ) ) ) ).

% adjust_idx_nth
thf(fact_575_adjust__idx__nth,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( missin7175274867594579095th_nat @ I @ Xs ) @ J )
        = ( nth_nat @ Xs @ ( missing_adjust_idx @ I @ J ) ) ) ) ).

% adjust_idx_nth
thf(fact_576_adjust__idx__nth,axiom,
    ! [I: nat,Xs: list_list_b,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ Xs ) )
     => ( ( nth_list_b @ ( missin8041159180453292414list_b @ I @ Xs ) @ J )
        = ( nth_list_b @ Xs @ ( missing_adjust_idx @ I @ J ) ) ) ) ).

% adjust_idx_nth
thf(fact_577_adjust__idx__nth,axiom,
    ! [I: nat,Xs: list_P8469869581646625389at_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ Xs ) )
     => ( ( nth_Pr6744343527793145070at_nat @ ( missin1453772821081764192at_nat @ I @ Xs ) @ J )
        = ( nth_Pr6744343527793145070at_nat @ Xs @ ( missing_adjust_idx @ I @ J ) ) ) ) ).

% adjust_idx_nth
thf(fact_578_old_Oprod_Oinject,axiom,
    ! [A: nat > nat,B: nat,A3: nat > nat,B2: nat] :
      ( ( ( produc72220940542539688at_nat @ A @ B )
        = ( produc72220940542539688at_nat @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_579_old_Oprod_Oinject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A3: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A @ B )
        = ( produc6161850002892822231at_nat @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_580_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A3: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_581_prod_Oinject,axiom,
    ! [X1: nat > nat,X22: nat,Y1: nat > nat,Y22: nat] :
      ( ( ( produc72220940542539688at_nat @ X1 @ X22 )
        = ( produc72220940542539688at_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_582_prod_Oinject,axiom,
    ! [X1: product_prod_nat_nat,X22: product_prod_nat_nat,Y1: product_prod_nat_nat,Y22: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ X1 @ X22 )
        = ( produc6161850002892822231at_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_583_prod_Oinject,axiom,
    ! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X22 )
        = ( product_Pair_nat_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_584_concat__index__split__unique,axiom,
    ! [I: nat,Xs: list_list_b,Ys: list_list_nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ ( concat_b @ Xs ) ) )
     => ( ( ( size_s420531428170919671list_b @ Xs )
          = ( size_s3023201423986296836st_nat @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Xs ) )
             => ( ( size_size_list_b @ ( nth_list_b @ Xs @ I2 ) )
                = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I2 ) ) ) )
         => ( ( concat_index_split_b @ ( product_Pair_nat_nat @ N @ I ) @ Xs )
            = ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I ) @ Ys ) ) ) ) ) ).

% concat_index_split_unique
thf(fact_585_concat__index__split__unique,axiom,
    ! [I: nat,Xs: list_list_b,Ys: list_list_b,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ ( concat_b @ Xs ) ) )
     => ( ( ( size_s420531428170919671list_b @ Xs )
          = ( size_s420531428170919671list_b @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Xs ) )
             => ( ( size_size_list_b @ ( nth_list_b @ Xs @ I2 ) )
                = ( size_size_list_b @ ( nth_list_b @ Ys @ I2 ) ) ) )
         => ( ( concat_index_split_b @ ( product_Pair_nat_nat @ N @ I ) @ Xs )
            = ( concat_index_split_b @ ( product_Pair_nat_nat @ N @ I ) @ Ys ) ) ) ) ) ).

% concat_index_split_unique
thf(fact_586_concat__index__split__unique,axiom,
    ! [I: nat,Xs: list_list_nat,Ys: list_list_nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ( ( size_s3023201423986296836st_nat @ Xs )
          = ( size_s3023201423986296836st_nat @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
             => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) )
                = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I2 ) ) ) )
         => ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I ) @ Xs )
            = ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I ) @ Ys ) ) ) ) ) ).

% concat_index_split_unique
thf(fact_587_concat__index__split__unique,axiom,
    ! [I: nat,Xs: list_list_nat,Ys: list_list_b,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ( ( size_s3023201423986296836st_nat @ Xs )
          = ( size_s420531428170919671list_b @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
             => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) )
                = ( size_size_list_b @ ( nth_list_b @ Ys @ I2 ) ) ) )
         => ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I ) @ Xs )
            = ( concat_index_split_b @ ( product_Pair_nat_nat @ N @ I ) @ Ys ) ) ) ) ) ).

% concat_index_split_unique
thf(fact_588_concat__index__split__unique,axiom,
    ! [I: nat,Xs: list_l1578326660578982846ctxt_a,Ys: list_list_nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ( ( size_s6994882312647770194ctxt_a @ Xs )
          = ( size_s3023201423986296836st_nat @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
             => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I2 ) )
                = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I2 ) ) ) )
         => ( ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ N @ I ) @ Xs )
            = ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I ) @ Ys ) ) ) ) ) ).

% concat_index_split_unique
thf(fact_589_concat__index__split__unique,axiom,
    ! [I: nat,Xs: list_l1578326660578982846ctxt_a,Ys: list_list_b,N: nat] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ( ( size_s6994882312647770194ctxt_a @ Xs )
          = ( size_s420531428170919671list_b @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
             => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I2 ) )
                = ( size_size_list_b @ ( nth_list_b @ Ys @ I2 ) ) ) )
         => ( ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ N @ I ) @ Xs )
            = ( concat_index_split_b @ ( product_Pair_nat_nat @ N @ I ) @ Ys ) ) ) ) ) ).

% concat_index_split_unique
thf(fact_590_concat__index__split__unique,axiom,
    ! [I: nat,Xs: list_list_b,Ys: list_l1578326660578982846ctxt_a,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ ( concat_b @ Xs ) ) )
     => ( ( ( size_s420531428170919671list_b @ Xs )
          = ( size_s6994882312647770194ctxt_a @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Xs ) )
             => ( ( size_size_list_b @ ( nth_list_b @ Xs @ I2 ) )
                = ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ I2 ) ) ) )
         => ( ( concat_index_split_b @ ( product_Pair_nat_nat @ N @ I ) @ Xs )
            = ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ N @ I ) @ Ys ) ) ) ) ) ).

% concat_index_split_unique
thf(fact_591_concat__index__split__unique,axiom,
    ! [I: nat,Xs: list_list_b,Ys: list_list_list_b,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ ( concat_b @ Xs ) ) )
     => ( ( ( size_s420531428170919671list_b @ Xs )
          = ( size_s9106009106869804029list_b @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Xs ) )
             => ( ( size_size_list_b @ ( nth_list_b @ Xs @ I2 ) )
                = ( size_s420531428170919671list_b @ ( nth_list_list_b @ Ys @ I2 ) ) ) )
         => ( ( concat_index_split_b @ ( product_Pair_nat_nat @ N @ I ) @ Xs )
            = ( concat7499549088139563139list_b @ ( product_Pair_nat_nat @ N @ I ) @ Ys ) ) ) ) ) ).

% concat_index_split_unique
thf(fact_592_concat__index__split__unique,axiom,
    ! [I: nat,Xs: list_list_nat,Ys: list_l1578326660578982846ctxt_a,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ( ( size_s3023201423986296836st_nat @ Xs )
          = ( size_s6994882312647770194ctxt_a @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
             => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) )
                = ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ I2 ) ) ) )
         => ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I ) @ Xs )
            = ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ N @ I ) @ Ys ) ) ) ) ) ).

% concat_index_split_unique
thf(fact_593_concat__index__split__unique,axiom,
    ! [I: nat,Xs: list_list_nat,Ys: list_list_list_b,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ( ( size_s3023201423986296836st_nat @ Xs )
          = ( size_s9106009106869804029list_b @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
             => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) )
                = ( size_s420531428170919671list_b @ ( nth_list_list_b @ Ys @ I2 ) ) ) )
         => ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I ) @ Xs )
            = ( concat7499549088139563139list_b @ ( product_Pair_nat_nat @ N @ I ) @ Ys ) ) ) ) ) ).

% concat_index_split_unique
thf(fact_594_adjust__idx__rev__length,axiom,
    ! [I: nat,Xs: list_Ground_gmctxt_a,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s962548663205356226ctxt_a @ Xs ) )
       => ( ( J != I )
         => ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I @ J ) @ ( size_s962548663205356226ctxt_a @ ( missin7823264313509747941ctxt_a @ I @ Xs ) ) ) ) ) ) ).

% adjust_idx_rev_length
thf(fact_595_adjust__idx__rev__length,axiom,
    ! [I: nat,Xs: list_b,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_b @ Xs ) )
       => ( ( J != I )
         => ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I @ J ) @ ( size_size_list_b @ ( missing_remove_nth_b @ I @ Xs ) ) ) ) ) ) ).

% adjust_idx_rev_length
thf(fact_596_adjust__idx__rev__length,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
       => ( ( J != I )
         => ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I @ J ) @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ I @ Xs ) ) ) ) ) ) ).

% adjust_idx_rev_length
thf(fact_597_adjust__idx__rev__length,axiom,
    ! [I: nat,Xs: list_list_b,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s420531428170919671list_b @ Xs ) )
       => ( ( J != I )
         => ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I @ J ) @ ( size_s420531428170919671list_b @ ( missin8041159180453292414list_b @ I @ Xs ) ) ) ) ) ) ).

% adjust_idx_rev_length
thf(fact_598_adjust__idx__rev__length,axiom,
    ! [I: nat,Xs: list_P8469869581646625389at_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s3679842834875189465at_nat @ Xs ) )
       => ( ( J != I )
         => ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I @ J ) @ ( size_s3679842834875189465at_nat @ ( missin1453772821081764192at_nat @ I @ Xs ) ) ) ) ) ) ).

% adjust_idx_rev_length
thf(fact_599_trancl__list_Ocases,axiom,
    ! [A1: list_list_nat,A22: list_list_nat,R: set_Pr3451248702717554689st_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ A1 @ A22 ) @ ( trancl_list_list_nat @ R ) )
     => ( ( ( ( size_s3023201423986296836st_nat @ A1 )
            = ( size_s3023201423986296836st_nat @ A22 ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ A22 ) )
               => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ A1 @ I4 ) @ ( nth_list_nat @ A22 @ I4 ) ) @ R ) ) )
       => ~ ! [Ys4: list_list_nat,I2: nat,Z: list_nat] :
              ( ( A22
                = ( list_update_list_nat @ Ys4 @ I2 @ Z ) )
             => ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ A1 @ Ys4 ) @ ( trancl_list_list_nat @ R ) )
               => ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ys4 ) )
                 => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ Ys4 @ I2 ) @ Z ) @ R ) ) ) ) ) ) ).

% trancl_list.cases
thf(fact_600_trancl__list_Ocases,axiom,
    ! [A1: list_l1578326660578982846ctxt_a,A22: list_l1578326660578982846ctxt_a,R: set_Pr8687807964536448157ctxt_a] :
      ( ( member6512746712182228870ctxt_a @ ( produc6157733871278919957ctxt_a @ A1 @ A22 ) @ ( trancl5060458363496439462ctxt_a @ R ) )
     => ( ( ( ( size_s6994882312647770194ctxt_a @ A1 )
            = ( size_s6994882312647770194ctxt_a @ A22 ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_s6994882312647770194ctxt_a @ A22 ) )
               => ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ ( nth_li6678718213161987943ctxt_a @ A1 @ I4 ) @ ( nth_li6678718213161987943ctxt_a @ A22 @ I4 ) ) @ R ) ) )
       => ~ ! [Ys4: list_l1578326660578982846ctxt_a,I2: nat,Z: list_Ground_gmctxt_a] :
              ( ( A22
                = ( list_u8015824433058371904ctxt_a @ Ys4 @ I2 @ Z ) )
             => ( ( member6512746712182228870ctxt_a @ ( produc6157733871278919957ctxt_a @ A1 @ Ys4 ) @ ( trancl5060458363496439462ctxt_a @ R ) )
               => ( ( ord_less_nat @ I2 @ ( size_s6994882312647770194ctxt_a @ Ys4 ) )
                 => ~ ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys4 @ I2 ) @ Z ) @ R ) ) ) ) ) ) ).

% trancl_list.cases
thf(fact_601_trancl__list_Ocases,axiom,
    ! [A1: list_l3298181151656792051at_nat,A22: list_l3298181151656792051at_nat,R: set_Pr4087777274317423175at_nat] :
      ( ( member6399385897798497296at_nat @ ( produc1890770240352350807at_nat @ A1 @ A22 ) @ ( trancl8580566090571148789at_nat @ R ) )
     => ( ( ( ( size_s6878757106947045471at_nat @ A1 )
            = ( size_s6878757106947045471at_nat @ A22 ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_s6878757106947045471at_nat @ A22 ) )
               => ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ ( nth_li8805938130461222772at_nat @ A1 @ I4 ) @ ( nth_li8805938130461222772at_nat @ A22 @ I4 ) ) @ R ) ) )
       => ~ ! [Ys4: list_l3298181151656792051at_nat,I2: nat,Z: list_P8469869581646625389at_nat] :
              ( ( A22
                = ( list_u5583386537922002395at_nat @ Ys4 @ I2 @ Z ) )
             => ( ( member6399385897798497296at_nat @ ( produc1890770240352350807at_nat @ A1 @ Ys4 ) @ ( trancl8580566090571148789at_nat @ R ) )
               => ( ( ord_less_nat @ I2 @ ( size_s6878757106947045471at_nat @ Ys4 ) )
                 => ~ ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ ( nth_li8805938130461222772at_nat @ Ys4 @ I2 ) @ Z ) @ R ) ) ) ) ) ) ).

% trancl_list.cases
thf(fact_602_trancl__list_Ocases,axiom,
    ! [A1: list_P6011104703257516679at_nat,A22: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ A1 @ A22 ) @ ( trancl3178887147240166537at_nat @ R ) )
     => ( ( ( ( size_s5460976970255530739at_nat @ A1 )
            = ( size_s5460976970255530739at_nat @ A22 ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ A22 ) )
               => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ A1 @ I4 ) @ ( nth_Pr7617993195940197384at_nat @ A22 @ I4 ) ) @ R ) ) )
       => ~ ! [Ys4: list_P6011104703257516679at_nat,I2: nat,Z: product_prod_nat_nat] :
              ( ( A22
                = ( list_u6180841689913720943at_nat @ Ys4 @ I2 @ Z ) )
             => ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ A1 @ Ys4 ) @ ( trancl3178887147240166537at_nat @ R ) )
               => ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Ys4 ) )
                 => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Ys4 @ I2 ) @ Z ) @ R ) ) ) ) ) ) ).

% trancl_list.cases
thf(fact_603_trancl__list_Ocases,axiom,
    ! [A1: list_Ground_gmctxt_a,A22: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
      ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ A1 @ A22 ) @ ( trancl6028151697025156246ctxt_a @ R ) )
     => ( ( ( ( size_s962548663205356226ctxt_a @ A1 )
            = ( size_s962548663205356226ctxt_a @ A22 ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ A22 ) )
               => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ A1 @ I4 ) @ ( nth_Ground_gmctxt_a @ A22 @ I4 ) ) @ R ) ) )
       => ~ ! [Ys4: list_Ground_gmctxt_a,I2: nat,Z: ground_gmctxt_a] :
              ( ( A22
                = ( list_u7950812632479372336ctxt_a @ Ys4 @ I2 @ Z ) )
             => ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ A1 @ Ys4 ) @ ( trancl6028151697025156246ctxt_a @ R ) )
               => ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Ys4 ) )
                 => ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Ys4 @ I2 ) @ Z ) @ R ) ) ) ) ) ) ).

% trancl_list.cases
thf(fact_604_trancl__list_Ocases,axiom,
    ! [A1: list_b,A22: list_b,R: set_Product_prod_b_b] :
      ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ A1 @ A22 ) @ ( trancl_list_b @ R ) )
     => ( ( ( ( size_size_list_b @ A1 )
            = ( size_size_list_b @ A22 ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_size_list_b @ A22 ) )
               => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( nth_b @ A1 @ I4 ) @ ( nth_b @ A22 @ I4 ) ) @ R ) ) )
       => ~ ! [Ys4: list_b,I2: nat,Z: b] :
              ( ( A22
                = ( list_update_b @ Ys4 @ I2 @ Z ) )
             => ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ A1 @ Ys4 ) @ ( trancl_list_b @ R ) )
               => ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Ys4 ) )
                 => ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( nth_b @ Ys4 @ I2 ) @ Z ) @ R ) ) ) ) ) ) ).

% trancl_list.cases
thf(fact_605_trancl__list_Ocases,axiom,
    ! [A1: list_nat,A22: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( trancl_list_nat @ R ) )
     => ( ( ( ( size_size_list_nat @ A1 )
            = ( size_size_list_nat @ A22 ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ A22 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ A1 @ I4 ) @ ( nth_nat @ A22 @ I4 ) ) @ R ) ) )
       => ~ ! [Ys4: list_nat,I2: nat,Z: nat] :
              ( ( A22
                = ( list_update_nat @ Ys4 @ I2 @ Z ) )
             => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ Ys4 ) @ ( trancl_list_nat @ R ) )
               => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys4 ) )
                 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys4 @ I2 ) @ Z ) @ R ) ) ) ) ) ) ).

% trancl_list.cases
thf(fact_606_trancl__list_Ocases,axiom,
    ! [A1: list_list_b,A22: list_list_b,R: set_Pr7665282455119567943list_b] :
      ( ( member7435197128963948048list_b @ ( produc8181704880241203287list_b @ A1 @ A22 ) @ ( trancl_list_list_b @ R ) )
     => ( ( ( ( size_s420531428170919671list_b @ A1 )
            = ( size_s420531428170919671list_b @ A22 ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_s420531428170919671list_b @ A22 ) )
               => ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ ( nth_list_b @ A1 @ I4 ) @ ( nth_list_b @ A22 @ I4 ) ) @ R ) ) )
       => ~ ! [Ys4: list_list_b,I2: nat,Z: list_b] :
              ( ( A22
                = ( list_update_list_b @ Ys4 @ I2 @ Z ) )
             => ( ( member7435197128963948048list_b @ ( produc8181704880241203287list_b @ A1 @ Ys4 ) @ ( trancl_list_list_b @ R ) )
               => ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Ys4 ) )
                 => ~ ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ ( nth_list_b @ Ys4 @ I2 ) @ Z ) @ R ) ) ) ) ) ) ).

% trancl_list.cases
thf(fact_607_trancl__list_Ocases,axiom,
    ! [A1: list_P8469869581646625389at_nat,A22: list_P8469869581646625389at_nat,R: set_Pr553994874890374343at_nat] :
      ( ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ A1 @ A22 ) @ ( trancl3412141547504709615at_nat @ R ) )
     => ( ( ( ( size_s3679842834875189465at_nat @ A1 )
            = ( size_s3679842834875189465at_nat @ A22 ) )
         => ~ ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_s3679842834875189465at_nat @ A22 ) )
               => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( nth_Pr6744343527793145070at_nat @ A1 @ I4 ) @ ( nth_Pr6744343527793145070at_nat @ A22 @ I4 ) ) @ R ) ) )
       => ~ ! [Ys4: list_P8469869581646625389at_nat,I2: nat,Z: produc859450856879609959at_nat] :
              ( ( A22
                = ( list_u5003261594476800725at_nat @ Ys4 @ I2 @ Z ) )
             => ( ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ A1 @ Ys4 ) @ ( trancl3412141547504709615at_nat @ R ) )
               => ( ( ord_less_nat @ I2 @ ( size_s3679842834875189465at_nat @ Ys4 ) )
                 => ~ ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( nth_Pr6744343527793145070at_nat @ Ys4 @ I2 ) @ Z ) @ R ) ) ) ) ) ) ).

% trancl_list.cases
thf(fact_608_trancl__list_Osimps,axiom,
    ! [A1: list_list_nat,A22: list_list_nat,R: set_Pr3451248702717554689st_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ A1 @ A22 ) @ ( trancl_list_list_nat @ R ) )
      = ( ? [Xs3: list_list_nat,Ys3: list_list_nat] :
            ( ( A1 = Xs3 )
            & ( A22 = Ys3 )
            & ( ( size_s3023201423986296836st_nat @ Xs3 )
              = ( size_s3023201423986296836st_nat @ Ys3 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Ys3 ) )
               => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ Xs3 @ I3 ) @ ( nth_list_nat @ Ys3 @ I3 ) ) @ R ) ) )
        | ? [Xs3: list_list_nat,Ys3: list_list_nat,I3: nat,Z3: list_nat] :
            ( ( A1 = Xs3 )
            & ( A22
              = ( list_update_list_nat @ Ys3 @ I3 @ Z3 ) )
            & ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs3 @ Ys3 ) @ ( trancl_list_list_nat @ R ) )
            & ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Ys3 ) )
            & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).

% trancl_list.simps
thf(fact_609_trancl__list_Osimps,axiom,
    ! [A1: list_l1578326660578982846ctxt_a,A22: list_l1578326660578982846ctxt_a,R: set_Pr8687807964536448157ctxt_a] :
      ( ( member6512746712182228870ctxt_a @ ( produc6157733871278919957ctxt_a @ A1 @ A22 ) @ ( trancl5060458363496439462ctxt_a @ R ) )
      = ( ? [Xs3: list_l1578326660578982846ctxt_a,Ys3: list_l1578326660578982846ctxt_a] :
            ( ( A1 = Xs3 )
            & ( A22 = Ys3 )
            & ( ( size_s6994882312647770194ctxt_a @ Xs3 )
              = ( size_s6994882312647770194ctxt_a @ Ys3 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s6994882312647770194ctxt_a @ Ys3 ) )
               => ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs3 @ I3 ) @ ( nth_li6678718213161987943ctxt_a @ Ys3 @ I3 ) ) @ R ) ) )
        | ? [Xs3: list_l1578326660578982846ctxt_a,Ys3: list_l1578326660578982846ctxt_a,I3: nat,Z3: list_Ground_gmctxt_a] :
            ( ( A1 = Xs3 )
            & ( A22
              = ( list_u8015824433058371904ctxt_a @ Ys3 @ I3 @ Z3 ) )
            & ( member6512746712182228870ctxt_a @ ( produc6157733871278919957ctxt_a @ Xs3 @ Ys3 ) @ ( trancl5060458363496439462ctxt_a @ R ) )
            & ( ord_less_nat @ I3 @ ( size_s6994882312647770194ctxt_a @ Ys3 ) )
            & ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).

% trancl_list.simps
thf(fact_610_trancl__list_Osimps,axiom,
    ! [A1: list_l3298181151656792051at_nat,A22: list_l3298181151656792051at_nat,R: set_Pr4087777274317423175at_nat] :
      ( ( member6399385897798497296at_nat @ ( produc1890770240352350807at_nat @ A1 @ A22 ) @ ( trancl8580566090571148789at_nat @ R ) )
      = ( ? [Xs3: list_l3298181151656792051at_nat,Ys3: list_l3298181151656792051at_nat] :
            ( ( A1 = Xs3 )
            & ( A22 = Ys3 )
            & ( ( size_s6878757106947045471at_nat @ Xs3 )
              = ( size_s6878757106947045471at_nat @ Ys3 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s6878757106947045471at_nat @ Ys3 ) )
               => ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ ( nth_li8805938130461222772at_nat @ Xs3 @ I3 ) @ ( nth_li8805938130461222772at_nat @ Ys3 @ I3 ) ) @ R ) ) )
        | ? [Xs3: list_l3298181151656792051at_nat,Ys3: list_l3298181151656792051at_nat,I3: nat,Z3: list_P8469869581646625389at_nat] :
            ( ( A1 = Xs3 )
            & ( A22
              = ( list_u5583386537922002395at_nat @ Ys3 @ I3 @ Z3 ) )
            & ( member6399385897798497296at_nat @ ( produc1890770240352350807at_nat @ Xs3 @ Ys3 ) @ ( trancl8580566090571148789at_nat @ R ) )
            & ( ord_less_nat @ I3 @ ( size_s6878757106947045471at_nat @ Ys3 ) )
            & ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ ( nth_li8805938130461222772at_nat @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).

% trancl_list.simps
thf(fact_611_trancl__list_Osimps,axiom,
    ! [A1: list_P6011104703257516679at_nat,A22: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ A1 @ A22 ) @ ( trancl3178887147240166537at_nat @ R ) )
      = ( ? [Xs3: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat] :
            ( ( A1 = Xs3 )
            & ( A22 = Ys3 )
            & ( ( size_s5460976970255530739at_nat @ Xs3 )
              = ( size_s5460976970255530739at_nat @ Ys3 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Ys3 ) )
               => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs3 @ I3 ) @ ( nth_Pr7617993195940197384at_nat @ Ys3 @ I3 ) ) @ R ) ) )
        | ? [Xs3: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,I3: nat,Z3: product_prod_nat_nat] :
            ( ( A1 = Xs3 )
            & ( A22
              = ( list_u6180841689913720943at_nat @ Ys3 @ I3 @ Z3 ) )
            & ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs3 @ Ys3 ) @ ( trancl3178887147240166537at_nat @ R ) )
            & ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Ys3 ) )
            & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).

% trancl_list.simps
thf(fact_612_trancl__list_Osimps,axiom,
    ! [A1: list_Ground_gmctxt_a,A22: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
      ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ A1 @ A22 ) @ ( trancl6028151697025156246ctxt_a @ R ) )
      = ( ? [Xs3: list_Ground_gmctxt_a,Ys3: list_Ground_gmctxt_a] :
            ( ( A1 = Xs3 )
            & ( A22 = Ys3 )
            & ( ( size_s962548663205356226ctxt_a @ Xs3 )
              = ( size_s962548663205356226ctxt_a @ Ys3 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Ys3 ) )
               => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs3 @ I3 ) @ ( nth_Ground_gmctxt_a @ Ys3 @ I3 ) ) @ R ) ) )
        | ? [Xs3: list_Ground_gmctxt_a,Ys3: list_Ground_gmctxt_a,I3: nat,Z3: ground_gmctxt_a] :
            ( ( A1 = Xs3 )
            & ( A22
              = ( list_u7950812632479372336ctxt_a @ Ys3 @ I3 @ Z3 ) )
            & ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs3 @ Ys3 ) @ ( trancl6028151697025156246ctxt_a @ R ) )
            & ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Ys3 ) )
            & ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).

% trancl_list.simps
thf(fact_613_trancl__list_Osimps,axiom,
    ! [A1: list_b,A22: list_b,R: set_Product_prod_b_b] :
      ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ A1 @ A22 ) @ ( trancl_list_b @ R ) )
      = ( ? [Xs3: list_b,Ys3: list_b] :
            ( ( A1 = Xs3 )
            & ( A22 = Ys3 )
            & ( ( size_size_list_b @ Xs3 )
              = ( size_size_list_b @ Ys3 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Ys3 ) )
               => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( nth_b @ Xs3 @ I3 ) @ ( nth_b @ Ys3 @ I3 ) ) @ R ) ) )
        | ? [Xs3: list_b,Ys3: list_b,I3: nat,Z3: b] :
            ( ( A1 = Xs3 )
            & ( A22
              = ( list_update_b @ Ys3 @ I3 @ Z3 ) )
            & ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Xs3 @ Ys3 ) @ ( trancl_list_b @ R ) )
            & ( ord_less_nat @ I3 @ ( size_size_list_b @ Ys3 ) )
            & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( nth_b @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).

% trancl_list.simps
thf(fact_614_trancl__list_Osimps,axiom,
    ! [A1: list_nat,A22: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( trancl_list_nat @ R ) )
      = ( ? [Xs3: list_nat,Ys3: list_nat] :
            ( ( A1 = Xs3 )
            & ( A22 = Ys3 )
            & ( ( size_size_list_nat @ Xs3 )
              = ( size_size_list_nat @ Ys3 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys3 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs3 @ I3 ) @ ( nth_nat @ Ys3 @ I3 ) ) @ R ) ) )
        | ? [Xs3: list_nat,Ys3: list_nat,I3: nat,Z3: nat] :
            ( ( A1 = Xs3 )
            & ( A22
              = ( list_update_nat @ Ys3 @ I3 @ Z3 ) )
            & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys3 ) @ ( trancl_list_nat @ R ) )
            & ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys3 ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).

% trancl_list.simps
thf(fact_615_trancl__list_Osimps,axiom,
    ! [A1: list_list_b,A22: list_list_b,R: set_Pr7665282455119567943list_b] :
      ( ( member7435197128963948048list_b @ ( produc8181704880241203287list_b @ A1 @ A22 ) @ ( trancl_list_list_b @ R ) )
      = ( ? [Xs3: list_list_b,Ys3: list_list_b] :
            ( ( A1 = Xs3 )
            & ( A22 = Ys3 )
            & ( ( size_s420531428170919671list_b @ Xs3 )
              = ( size_s420531428170919671list_b @ Ys3 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s420531428170919671list_b @ Ys3 ) )
               => ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ ( nth_list_b @ Xs3 @ I3 ) @ ( nth_list_b @ Ys3 @ I3 ) ) @ R ) ) )
        | ? [Xs3: list_list_b,Ys3: list_list_b,I3: nat,Z3: list_b] :
            ( ( A1 = Xs3 )
            & ( A22
              = ( list_update_list_b @ Ys3 @ I3 @ Z3 ) )
            & ( member7435197128963948048list_b @ ( produc8181704880241203287list_b @ Xs3 @ Ys3 ) @ ( trancl_list_list_b @ R ) )
            & ( ord_less_nat @ I3 @ ( size_s420531428170919671list_b @ Ys3 ) )
            & ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ ( nth_list_b @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).

% trancl_list.simps
thf(fact_616_trancl__list_Osimps,axiom,
    ! [A1: list_P8469869581646625389at_nat,A22: list_P8469869581646625389at_nat,R: set_Pr553994874890374343at_nat] :
      ( ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ A1 @ A22 ) @ ( trancl3412141547504709615at_nat @ R ) )
      = ( ? [Xs3: list_P8469869581646625389at_nat,Ys3: list_P8469869581646625389at_nat] :
            ( ( A1 = Xs3 )
            & ( A22 = Ys3 )
            & ( ( size_s3679842834875189465at_nat @ Xs3 )
              = ( size_s3679842834875189465at_nat @ Ys3 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s3679842834875189465at_nat @ Ys3 ) )
               => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( nth_Pr6744343527793145070at_nat @ Xs3 @ I3 ) @ ( nth_Pr6744343527793145070at_nat @ Ys3 @ I3 ) ) @ R ) ) )
        | ? [Xs3: list_P8469869581646625389at_nat,Ys3: list_P8469869581646625389at_nat,I3: nat,Z3: produc859450856879609959at_nat] :
            ( ( A1 = Xs3 )
            & ( A22
              = ( list_u5003261594476800725at_nat @ Ys3 @ I3 @ Z3 ) )
            & ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ Xs3 @ Ys3 ) @ ( trancl3412141547504709615at_nat @ R ) )
            & ( ord_less_nat @ I3 @ ( size_s3679842834875189465at_nat @ Ys3 ) )
            & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( nth_Pr6744343527793145070at_nat @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).

% trancl_list.simps
thf(fact_617_trancl__list_Olist__trancl,axiom,
    ! [Xs: list_list_nat,Ys: list_list_nat,R: set_Pr3451248702717554689st_nat,I: nat,Z4: list_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys ) @ ( trancl_list_list_nat @ R ) )
     => ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Ys ) )
       => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ Ys @ I ) @ Z4 ) @ R )
         => ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ ( list_update_list_nat @ Ys @ I @ Z4 ) ) @ ( trancl_list_list_nat @ R ) ) ) ) ) ).

% trancl_list.list_trancl
thf(fact_618_trancl__list_Olist__trancl,axiom,
    ! [Xs: list_l1578326660578982846ctxt_a,Ys: list_l1578326660578982846ctxt_a,R: set_Pr8687807964536448157ctxt_a,I: nat,Z4: list_Ground_gmctxt_a] :
      ( ( member6512746712182228870ctxt_a @ ( produc6157733871278919957ctxt_a @ Xs @ Ys ) @ ( trancl5060458363496439462ctxt_a @ R ) )
     => ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Ys ) )
       => ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ I ) @ Z4 ) @ R )
         => ( member6512746712182228870ctxt_a @ ( produc6157733871278919957ctxt_a @ Xs @ ( list_u8015824433058371904ctxt_a @ Ys @ I @ Z4 ) ) @ ( trancl5060458363496439462ctxt_a @ R ) ) ) ) ) ).

% trancl_list.list_trancl
thf(fact_619_trancl__list_Olist__trancl,axiom,
    ! [Xs: list_l3298181151656792051at_nat,Ys: list_l3298181151656792051at_nat,R: set_Pr4087777274317423175at_nat,I: nat,Z4: list_P8469869581646625389at_nat] :
      ( ( member6399385897798497296at_nat @ ( produc1890770240352350807at_nat @ Xs @ Ys ) @ ( trancl8580566090571148789at_nat @ R ) )
     => ( ( ord_less_nat @ I @ ( size_s6878757106947045471at_nat @ Ys ) )
       => ( ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ ( nth_li8805938130461222772at_nat @ Ys @ I ) @ Z4 ) @ R )
         => ( member6399385897798497296at_nat @ ( produc1890770240352350807at_nat @ Xs @ ( list_u5583386537922002395at_nat @ Ys @ I @ Z4 ) ) @ ( trancl8580566090571148789at_nat @ R ) ) ) ) ) ).

% trancl_list.list_trancl
thf(fact_620_trancl__list_Olist__trancl,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat,I: nat,Z4: product_prod_nat_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( trancl3178887147240166537at_nat @ R ) )
     => ( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Ys ) )
       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Ys @ I ) @ Z4 ) @ R )
         => ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ ( list_u6180841689913720943at_nat @ Ys @ I @ Z4 ) ) @ ( trancl3178887147240166537at_nat @ R ) ) ) ) ) ).

% trancl_list.list_trancl
thf(fact_621_trancl__list_Olist__trancl,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,I: nat,Z4: ground_gmctxt_a] :
      ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs @ Ys ) @ ( trancl6028151697025156246ctxt_a @ R ) )
     => ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ys ) )
       => ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Ys @ I ) @ Z4 ) @ R )
         => ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs @ ( list_u7950812632479372336ctxt_a @ Ys @ I @ Z4 ) ) @ ( trancl6028151697025156246ctxt_a @ R ) ) ) ) ) ).

% trancl_list.list_trancl
thf(fact_622_trancl__list_Olist__trancl,axiom,
    ! [Xs: list_b,Ys: list_b,R: set_Product_prod_b_b,I: nat,Z4: b] :
      ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Xs @ Ys ) @ ( trancl_list_b @ R ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_b @ Ys ) )
       => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( nth_b @ Ys @ I ) @ Z4 ) @ R )
         => ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Xs @ ( list_update_b @ Ys @ I @ Z4 ) ) @ ( trancl_list_b @ R ) ) ) ) ) ).

% trancl_list.list_trancl
thf(fact_623_trancl__list_Olist__trancl,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,I: nat,Z4: nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( trancl_list_nat @ R ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys @ I ) @ Z4 ) @ R )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( list_update_nat @ Ys @ I @ Z4 ) ) @ ( trancl_list_nat @ R ) ) ) ) ) ).

% trancl_list.list_trancl
thf(fact_624_trancl__list_Olist__trancl,axiom,
    ! [Xs: list_list_b,Ys: list_list_b,R: set_Pr7665282455119567943list_b,I: nat,Z4: list_b] :
      ( ( member7435197128963948048list_b @ ( produc8181704880241203287list_b @ Xs @ Ys ) @ ( trancl_list_list_b @ R ) )
     => ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ Ys ) )
       => ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ ( nth_list_b @ Ys @ I ) @ Z4 ) @ R )
         => ( member7435197128963948048list_b @ ( produc8181704880241203287list_b @ Xs @ ( list_update_list_b @ Ys @ I @ Z4 ) ) @ ( trancl_list_list_b @ R ) ) ) ) ) ).

% trancl_list.list_trancl
thf(fact_625_trancl__list_Olist__trancl,axiom,
    ! [Xs: list_P8469869581646625389at_nat,Ys: list_P8469869581646625389at_nat,R: set_Pr553994874890374343at_nat,I: nat,Z4: produc859450856879609959at_nat] :
      ( ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ Xs @ Ys ) @ ( trancl3412141547504709615at_nat @ R ) )
     => ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ Ys ) )
       => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( nth_Pr6744343527793145070at_nat @ Ys @ I ) @ Z4 ) @ R )
         => ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ Xs @ ( list_u5003261594476800725at_nat @ Ys @ I @ Z4 ) ) @ ( trancl3412141547504709615at_nat @ R ) ) ) ) ) ).

% trancl_list.list_trancl
thf(fact_626_adjust__idx__rev2,axiom,
    ! [J: nat,I: nat] :
      ( ( J != I )
     => ( ( missing_adjust_idx @ I @ ( missin3815256168798769645dx_rev @ I @ J ) )
        = J ) ) ).

% adjust_idx_rev2
thf(fact_627_adjust__idx__rev1,axiom,
    ! [I: nat,J: nat] :
      ( ( missin3815256168798769645dx_rev @ I @ ( missing_adjust_idx @ I @ J ) )
      = J ) ).

% adjust_idx_rev1
thf(fact_628_adjust__idx__i,axiom,
    ! [I: nat,J: nat] :
      ( ( missing_adjust_idx @ I @ J )
     != I ) ).

% adjust_idx_i
thf(fact_629_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc8199716216217303280at_nat] :
      ~ ! [A4: nat > nat,B3: nat] :
          ( Y3
         != ( produc72220940542539688at_nat @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_630_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc859450856879609959at_nat] :
      ~ ! [A4: product_prod_nat_nat,B3: product_prod_nat_nat] :
          ( Y3
         != ( produc6161850002892822231at_nat @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_631_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_nat_nat] :
      ~ ! [A4: nat,B3: nat] :
          ( Y3
         != ( product_Pair_nat_nat @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_632_surj__pair,axiom,
    ! [P2: produc8199716216217303280at_nat] :
    ? [X3: nat > nat,Y4: nat] :
      ( P2
      = ( produc72220940542539688at_nat @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_633_surj__pair,axiom,
    ! [P2: produc859450856879609959at_nat] :
    ? [X3: product_prod_nat_nat,Y4: product_prod_nat_nat] :
      ( P2
      = ( produc6161850002892822231at_nat @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_634_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X3: nat,Y4: nat] :
      ( P2
      = ( product_Pair_nat_nat @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_635_prod__cases,axiom,
    ! [P: produc8199716216217303280at_nat > $o,P2: produc8199716216217303280at_nat] :
      ( ! [A4: nat > nat,B3: nat] : ( P @ ( produc72220940542539688at_nat @ A4 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_636_prod__cases,axiom,
    ! [P: produc859450856879609959at_nat > $o,P2: produc859450856879609959at_nat] :
      ( ! [A4: product_prod_nat_nat,B3: product_prod_nat_nat] : ( P @ ( produc6161850002892822231at_nat @ A4 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_637_prod__cases,axiom,
    ! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
      ( ! [A4: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A4 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_638_Pair__inject,axiom,
    ! [A: nat > nat,B: nat,A3: nat > nat,B2: nat] :
      ( ( ( produc72220940542539688at_nat @ A @ B )
        = ( produc72220940542539688at_nat @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_639_Pair__inject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A3: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A @ B )
        = ( produc6161850002892822231at_nat @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_640_Pair__inject,axiom,
    ! [A: nat,B: nat,A3: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_641_prod__cases3,axiom,
    ! [Y3: produc859450856879609959at_nat] :
      ~ ! [A4: product_prod_nat_nat,B3: nat,C: nat] :
          ( Y3
         != ( produc6161850002892822231at_nat @ A4 @ ( product_Pair_nat_nat @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_642_prod__induct3,axiom,
    ! [P: produc859450856879609959at_nat > $o,X: produc859450856879609959at_nat] :
      ( ! [A4: product_prod_nat_nat,B3: nat,C: nat] : ( P @ ( produc6161850002892822231at_nat @ A4 @ ( product_Pair_nat_nat @ B3 @ C ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_643_trancl__list_Obase,axiom,
    ! [Xs: list_list_nat,Ys: list_list_nat,R: set_Pr3451248702717554689st_nat] :
      ( ( ( size_s3023201423986296836st_nat @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ys ) )
           => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ Xs @ I2 ) @ ( nth_list_nat @ Ys @ I2 ) ) @ R ) )
       => ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys ) @ ( trancl_list_list_nat @ R ) ) ) ) ).

% trancl_list.base
thf(fact_644_trancl__list_Obase,axiom,
    ! [Xs: list_l1578326660578982846ctxt_a,Ys: list_l1578326660578982846ctxt_a,R: set_Pr8687807964536448157ctxt_a] :
      ( ( ( size_s6994882312647770194ctxt_a @ Xs )
        = ( size_s6994882312647770194ctxt_a @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6994882312647770194ctxt_a @ Ys ) )
           => ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I2 ) @ ( nth_li6678718213161987943ctxt_a @ Ys @ I2 ) ) @ R ) )
       => ( member6512746712182228870ctxt_a @ ( produc6157733871278919957ctxt_a @ Xs @ Ys ) @ ( trancl5060458363496439462ctxt_a @ R ) ) ) ) ).

% trancl_list.base
thf(fact_645_trancl__list_Obase,axiom,
    ! [Xs: list_l3298181151656792051at_nat,Ys: list_l3298181151656792051at_nat,R: set_Pr4087777274317423175at_nat] :
      ( ( ( size_s6878757106947045471at_nat @ Xs )
        = ( size_s6878757106947045471at_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6878757106947045471at_nat @ Ys ) )
           => ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ ( nth_li8805938130461222772at_nat @ Xs @ I2 ) @ ( nth_li8805938130461222772at_nat @ Ys @ I2 ) ) @ R ) )
       => ( member6399385897798497296at_nat @ ( produc1890770240352350807at_nat @ Xs @ Ys ) @ ( trancl8580566090571148789at_nat @ R ) ) ) ) ).

% trancl_list.base
thf(fact_646_trancl__list_Obase,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( ( size_s5460976970255530739at_nat @ Xs )
        = ( size_s5460976970255530739at_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Ys ) )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I2 ) @ ( nth_Pr7617993195940197384at_nat @ Ys @ I2 ) ) @ R ) )
       => ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( trancl3178887147240166537at_nat @ R ) ) ) ) ).

% trancl_list.base
thf(fact_647_trancl__list_Obase,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Ys ) )
           => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs @ I2 ) @ ( nth_Ground_gmctxt_a @ Ys @ I2 ) ) @ R ) )
       => ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs @ Ys ) @ ( trancl6028151697025156246ctxt_a @ R ) ) ) ) ).

% trancl_list.base
thf(fact_648_trancl__list_Obase,axiom,
    ! [Xs: list_b,Ys: list_b,R: set_Product_prod_b_b] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Ys ) )
           => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( nth_b @ Xs @ I2 ) @ ( nth_b @ Ys @ I2 ) ) @ R ) )
       => ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Xs @ Ys ) @ ( trancl_list_b @ R ) ) ) ) ).

% trancl_list.base
thf(fact_649_trancl__list_Obase,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) @ R ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( trancl_list_nat @ R ) ) ) ) ).

% trancl_list.base
thf(fact_650_trancl__list_Obase,axiom,
    ! [Xs: list_list_b,Ys: list_list_b,R: set_Pr7665282455119567943list_b] :
      ( ( ( size_s420531428170919671list_b @ Xs )
        = ( size_s420531428170919671list_b @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Ys ) )
           => ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ ( nth_list_b @ Xs @ I2 ) @ ( nth_list_b @ Ys @ I2 ) ) @ R ) )
       => ( member7435197128963948048list_b @ ( produc8181704880241203287list_b @ Xs @ Ys ) @ ( trancl_list_list_b @ R ) ) ) ) ).

% trancl_list.base
thf(fact_651_trancl__list_Obase,axiom,
    ! [Xs: list_P8469869581646625389at_nat,Ys: list_P8469869581646625389at_nat,R: set_Pr553994874890374343at_nat] :
      ( ( ( size_s3679842834875189465at_nat @ Xs )
        = ( size_s3679842834875189465at_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3679842834875189465at_nat @ Ys ) )
           => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( nth_Pr6744343527793145070at_nat @ Xs @ I2 ) @ ( nth_Pr6744343527793145070at_nat @ Ys @ I2 ) ) @ R ) )
       => ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ Xs @ Ys ) @ ( trancl3412141547504709615at_nat @ R ) ) ) ) ).

% trancl_list.base
thf(fact_652_adjust__idx__length,axiom,
    ! [I: nat,Xs: list_Ground_gmctxt_a,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s962548663205356226ctxt_a @ ( missin7823264313509747941ctxt_a @ I @ Xs ) ) )
       => ( ord_less_nat @ ( missing_adjust_idx @ I @ J ) @ ( size_s962548663205356226ctxt_a @ Xs ) ) ) ) ).

% adjust_idx_length
thf(fact_653_adjust__idx__length,axiom,
    ! [I: nat,Xs: list_b,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_b @ ( missing_remove_nth_b @ I @ Xs ) ) )
       => ( ord_less_nat @ ( missing_adjust_idx @ I @ J ) @ ( size_size_list_b @ Xs ) ) ) ) ).

% adjust_idx_length
thf(fact_654_adjust__idx__length,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ I @ Xs ) ) )
       => ( ord_less_nat @ ( missing_adjust_idx @ I @ J ) @ ( size_size_list_nat @ Xs ) ) ) ) ).

% adjust_idx_length
thf(fact_655_adjust__idx__length,axiom,
    ! [I: nat,Xs: list_list_b,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s420531428170919671list_b @ ( missin8041159180453292414list_b @ I @ Xs ) ) )
       => ( ord_less_nat @ ( missing_adjust_idx @ I @ J ) @ ( size_s420531428170919671list_b @ Xs ) ) ) ) ).

% adjust_idx_length
thf(fact_656_adjust__idx__length,axiom,
    ! [I: nat,Xs: list_P8469869581646625389at_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s3679842834875189465at_nat @ ( missin1453772821081764192at_nat @ I @ Xs ) ) )
       => ( ord_less_nat @ ( missing_adjust_idx @ I @ J ) @ ( size_s3679842834875189465at_nat @ Xs ) ) ) ) ).

% adjust_idx_length
thf(fact_657_concat__index__split__less__length__concat_I4_J,axiom,
    ! [I: nat,Xs: list_list_list_nat,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ ( concat_list_nat @ Xs ) ) )
     => ( ( ( concat5120376260311099874st_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ( nth_list_nat @ ( concat_list_nat @ Xs ) @ I )
          = ( nth_list_nat @ ( nth_list_list_nat @ Xs @ M ) @ N ) ) ) ) ).

% concat_index_split_less_length_concat(4)
thf(fact_658_concat__index__split__less__length__concat_I4_J,axiom,
    ! [I: nat,Xs: list_l3687490334575670478ctxt_a,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ ( concat6203624070112318253ctxt_a @ Xs ) ) )
     => ( ( ( concat6198194937734591280ctxt_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ( nth_li6678718213161987943ctxt_a @ ( concat6203624070112318253ctxt_a @ Xs ) @ I )
          = ( nth_li6678718213161987943ctxt_a @ ( nth_li3520809774789768823ctxt_a @ Xs @ M ) @ N ) ) ) ) ).

% concat_index_split_less_length_concat(4)
thf(fact_659_concat__index__split__less__length__concat_I4_J,axiom,
    ! [I: nat,Xs: list_l5129023048960694521at_nat,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_s6878757106947045471at_nat @ ( concat5897368152400924974at_nat @ Xs ) ) )
     => ( ( ( concat2944144588778160107at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ( nth_li8805938130461222772at_nat @ ( concat5897368152400924974at_nat @ Xs ) @ I )
          = ( nth_li8805938130461222772at_nat @ ( nth_li2593672534642986362at_nat @ Xs @ M ) @ N ) ) ) ) ).

% concat_index_split_less_length_concat(4)
thf(fact_660_concat__index__split__less__length__concat_I4_J,axiom,
    ! [I: nat,Xs: list_l1578326660578982846ctxt_a,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ( ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) @ I )
          = ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ M ) @ N ) ) ) ) ).

% concat_index_split_less_length_concat(4)
thf(fact_661_concat__index__split__less__length__concat_I4_J,axiom,
    ! [I: nat,Xs: list_list_b,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ ( concat_b @ Xs ) ) )
     => ( ( ( concat_index_split_b @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ( nth_b @ ( concat_b @ Xs ) @ I )
          = ( nth_b @ ( nth_list_b @ Xs @ M ) @ N ) ) ) ) ).

% concat_index_split_less_length_concat(4)
thf(fact_662_concat__index__split__less__length__concat_I4_J,axiom,
    ! [I: nat,Xs: list_list_nat,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ( nth_nat @ ( concat_nat @ Xs ) @ I )
          = ( nth_nat @ ( nth_list_nat @ Xs @ M ) @ N ) ) ) ) ).

% concat_index_split_less_length_concat(4)
thf(fact_663_concat__index__split__less__length__concat_I4_J,axiom,
    ! [I: nat,Xs: list_list_list_b,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ ( concat_list_b @ Xs ) ) )
     => ( ( ( concat7499549088139563139list_b @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ( nth_list_b @ ( concat_list_b @ Xs ) @ I )
          = ( nth_list_b @ ( nth_list_list_b @ Xs @ M ) @ N ) ) ) ) ).

% concat_index_split_less_length_concat(4)
thf(fact_664_concat__index__split__less__length__concat_I4_J,axiom,
    ! [I: nat,Xs: list_l3298181151656792051at_nat,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ ( concat5616020275982107560at_nat @ Xs ) ) )
     => ( ( ( concat1027118156116784357at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ( nth_Pr6744343527793145070at_nat @ ( concat5616020275982107560at_nat @ Xs ) @ I )
          = ( nth_Pr6744343527793145070at_nat @ ( nth_li8805938130461222772at_nat @ Xs @ M ) @ N ) ) ) ) ).

% concat_index_split_less_length_concat(4)
thf(fact_665_concat__index__split__less__length__concat_I2_J,axiom,
    ! [I: nat,Xs: list_l1578326660578982846ctxt_a,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ( ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ M @ ( size_s6994882312647770194ctxt_a @ Xs ) ) ) ) ).

% concat_index_split_less_length_concat(2)
thf(fact_666_concat__index__split__less__length__concat_I2_J,axiom,
    ! [I: nat,Xs: list_list_b,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ ( concat_b @ Xs ) ) )
     => ( ( ( concat_index_split_b @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ M @ ( size_s420531428170919671list_b @ Xs ) ) ) ) ).

% concat_index_split_less_length_concat(2)
thf(fact_667_concat__index__split__less__length__concat_I2_J,axiom,
    ! [I: nat,Xs: list_list_nat,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ M @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ) ).

% concat_index_split_less_length_concat(2)
thf(fact_668_concat__index__split__less__length__concat_I2_J,axiom,
    ! [I: nat,Xs: list_list_list_b,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ ( concat_list_b @ Xs ) ) )
     => ( ( ( concat7499549088139563139list_b @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ M @ ( size_s9106009106869804029list_b @ Xs ) ) ) ) ).

% concat_index_split_less_length_concat(2)
thf(fact_669_concat__index__split__less__length__concat_I2_J,axiom,
    ! [I: nat,Xs: list_l3298181151656792051at_nat,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ ( concat5616020275982107560at_nat @ Xs ) ) )
     => ( ( ( concat1027118156116784357at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ M @ ( size_s6878757106947045471at_nat @ Xs ) ) ) ) ).

% concat_index_split_less_length_concat(2)
thf(fact_670_concat__index__split__less__length__concat_I3_J,axiom,
    ! [I: nat,Xs: list_l1578326660578982846ctxt_a,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ( ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ N @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ M ) ) ) ) ) ).

% concat_index_split_less_length_concat(3)
thf(fact_671_concat__index__split__less__length__concat_I3_J,axiom,
    ! [I: nat,Xs: list_list_b,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ ( concat_b @ Xs ) ) )
     => ( ( ( concat_index_split_b @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ N @ ( size_size_list_b @ ( nth_list_b @ Xs @ M ) ) ) ) ) ).

% concat_index_split_less_length_concat(3)
thf(fact_672_concat__index__split__less__length__concat_I3_J,axiom,
    ! [I: nat,Xs: list_list_nat,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ N @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ M ) ) ) ) ) ).

% concat_index_split_less_length_concat(3)
thf(fact_673_concat__index__split__less__length__concat_I3_J,axiom,
    ! [I: nat,Xs: list_list_list_b,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ ( concat_list_b @ Xs ) ) )
     => ( ( ( concat7499549088139563139list_b @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ N @ ( size_s420531428170919671list_b @ ( nth_list_list_b @ Xs @ M ) ) ) ) ) ).

% concat_index_split_less_length_concat(3)
thf(fact_674_concat__index__split__less__length__concat_I3_J,axiom,
    ! [I: nat,Xs: list_l3298181151656792051at_nat,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ ( concat5616020275982107560at_nat @ Xs ) ) )
     => ( ( ( concat1027118156116784357at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ N @ ( size_s3679842834875189465at_nat @ ( nth_li8805938130461222772at_nat @ Xs @ M ) ) ) ) ) ).

% concat_index_split_less_length_concat(3)
thf(fact_675_listrel1__iff__update,axiom,
    ! [Xs: list_list_nat,Ys: list_list_nat,R2: set_Pr3451248702717554689st_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys ) @ ( listrel1_list_nat @ R2 ) )
      = ( ? [Y: list_nat,N2: nat] :
            ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ Xs @ N2 ) @ Y ) @ R2 )
            & ( ord_less_nat @ N2 @ ( size_s3023201423986296836st_nat @ Xs ) )
            & ( Ys
              = ( list_update_list_nat @ Xs @ N2 @ Y ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_676_listrel1__iff__update,axiom,
    ! [Xs: list_l1578326660578982846ctxt_a,Ys: list_l1578326660578982846ctxt_a,R2: set_Pr8687807964536448157ctxt_a] :
      ( ( member6512746712182228870ctxt_a @ ( produc6157733871278919957ctxt_a @ Xs @ Ys ) @ ( listre7072643403622106127ctxt_a @ R2 ) )
      = ( ? [Y: list_Ground_gmctxt_a,N2: nat] :
            ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ N2 ) @ Y ) @ R2 )
            & ( ord_less_nat @ N2 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
            & ( Ys
              = ( list_u8015824433058371904ctxt_a @ Xs @ N2 @ Y ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_677_listrel1__iff__update,axiom,
    ! [Xs: list_l3298181151656792051at_nat,Ys: list_l3298181151656792051at_nat,R2: set_Pr4087777274317423175at_nat] :
      ( ( member6399385897798497296at_nat @ ( produc1890770240352350807at_nat @ Xs @ Ys ) @ ( listre3648893826645724364at_nat @ R2 ) )
      = ( ? [Y: list_P8469869581646625389at_nat,N2: nat] :
            ( ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ ( nth_li8805938130461222772at_nat @ Xs @ N2 ) @ Y ) @ R2 )
            & ( ord_less_nat @ N2 @ ( size_s6878757106947045471at_nat @ Xs ) )
            & ( Ys
              = ( list_u5583386537922002395at_nat @ Xs @ N2 @ Y ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_678_listrel1__iff__update,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( listre4828114922151135584at_nat @ R2 ) )
      = ( ? [Y: product_prod_nat_nat,N2: nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N2 ) @ Y ) @ R2 )
            & ( ord_less_nat @ N2 @ ( size_s5460976970255530739at_nat @ Xs ) )
            & ( Ys
              = ( list_u6180841689913720943at_nat @ Xs @ N2 @ Y ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_679_listrel1__iff__update,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a] :
      ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs @ Ys ) @ ( listre839271074516595327ctxt_a @ R2 ) )
      = ( ? [Y: ground_gmctxt_a,N2: nat] :
            ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs @ N2 ) @ Y ) @ R2 )
            & ( ord_less_nat @ N2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
            & ( Ys
              = ( list_u7950812632479372336ctxt_a @ Xs @ N2 @ Y ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_680_listrel1__iff__update,axiom,
    ! [Xs: list_b,Ys: list_b,R2: set_Product_prod_b_b] :
      ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Xs @ Ys ) @ ( listrel1_b @ R2 ) )
      = ( ? [Y: b,N2: nat] :
            ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( nth_b @ Xs @ N2 ) @ Y ) @ R2 )
            & ( ord_less_nat @ N2 @ ( size_size_list_b @ Xs ) )
            & ( Ys
              = ( list_update_b @ Xs @ N2 @ Y ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_681_listrel1__iff__update,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
      = ( ? [Y: nat,N2: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ Y ) @ R2 )
            & ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
            & ( Ys
              = ( list_update_nat @ Xs @ N2 @ Y ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_682_listrel1__iff__update,axiom,
    ! [Xs: list_list_b,Ys: list_list_b,R2: set_Pr7665282455119567943list_b] :
      ( ( member7435197128963948048list_b @ ( produc8181704880241203287list_b @ Xs @ Ys ) @ ( listrel1_list_b @ R2 ) )
      = ( ? [Y: list_b,N2: nat] :
            ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ ( nth_list_b @ Xs @ N2 ) @ Y ) @ R2 )
            & ( ord_less_nat @ N2 @ ( size_s420531428170919671list_b @ Xs ) )
            & ( Ys
              = ( list_update_list_b @ Xs @ N2 @ Y ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_683_listrel1__iff__update,axiom,
    ! [Xs: list_P8469869581646625389at_nat,Ys: list_P8469869581646625389at_nat,R2: set_Pr553994874890374343at_nat] :
      ( ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ Xs @ Ys ) @ ( listre2293869942453659206at_nat @ R2 ) )
      = ( ? [Y: produc859450856879609959at_nat,N2: nat] :
            ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( nth_Pr6744343527793145070at_nat @ Xs @ N2 ) @ Y ) @ R2 )
            & ( ord_less_nat @ N2 @ ( size_s3679842834875189465at_nat @ Xs ) )
            & ( Ys
              = ( list_u5003261594476800725at_nat @ Xs @ N2 @ Y ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_684_concat__index__split__larger__lists,axiom,
    ! [I: nat,Xs: list_l1578326660578982846ctxt_a,N: nat,Ys: list_l1578326660578982846ctxt_a] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ N @ I ) @ ( append272212692390751315ctxt_a @ Xs @ Ys ) )
        = ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ N @ I ) @ Xs ) ) ) ).

% concat_index_split_larger_lists
thf(fact_685_concat__index__split__larger__lists,axiom,
    ! [I: nat,Xs: list_list_b,N: nat,Ys: list_list_b] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ ( concat_b @ Xs ) ) )
     => ( ( concat_index_split_b @ ( product_Pair_nat_nat @ N @ I ) @ ( append_list_b @ Xs @ Ys ) )
        = ( concat_index_split_b @ ( product_Pair_nat_nat @ N @ I ) @ Xs ) ) ) ).

% concat_index_split_larger_lists
thf(fact_686_concat__index__split__larger__lists,axiom,
    ! [I: nat,Xs: list_list_nat,N: nat,Ys: list_list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I ) @ ( append_list_nat @ Xs @ Ys ) )
        = ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I ) @ Xs ) ) ) ).

% concat_index_split_larger_lists
thf(fact_687_concat__index__split__larger__lists,axiom,
    ! [I: nat,Xs: list_list_list_b,N: nat,Ys: list_list_list_b] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ ( concat_list_b @ Xs ) ) )
     => ( ( concat7499549088139563139list_b @ ( product_Pair_nat_nat @ N @ I ) @ ( append_list_list_b @ Xs @ Ys ) )
        = ( concat7499549088139563139list_b @ ( product_Pair_nat_nat @ N @ I ) @ Xs ) ) ) ).

% concat_index_split_larger_lists
thf(fact_688_concat__index__split__larger__lists,axiom,
    ! [I: nat,Xs: list_l3298181151656792051at_nat,N: nat,Ys: list_l3298181151656792051at_nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ ( concat5616020275982107560at_nat @ Xs ) ) )
     => ( ( concat1027118156116784357at_nat @ ( product_Pair_nat_nat @ N @ I ) @ ( append6271109094031990152at_nat @ Xs @ Ys ) )
        = ( concat1027118156116784357at_nat @ ( product_Pair_nat_nat @ N @ I ) @ Xs ) ) ) ).

% concat_index_split_larger_lists
thf(fact_689_trancl__list__conv,axiom,
    ! [Xs: list_list_nat,Ys: list_list_nat,R: set_Pr3451248702717554689st_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys ) @ ( trancl_list_list_nat @ R ) )
      = ( ( ( size_s3023201423986296836st_nat @ Xs )
          = ( size_s3023201423986296836st_nat @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Ys ) )
           => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ Xs @ I3 ) @ ( nth_list_nat @ Ys @ I3 ) ) @ ( transi5465639037617667199st_nat @ R ) ) ) ) ) ).

% trancl_list_conv
thf(fact_690_trancl__list__conv,axiom,
    ! [Xs: list_l1578326660578982846ctxt_a,Ys: list_l1578326660578982846ctxt_a,R: set_Pr8687807964536448157ctxt_a] :
      ( ( member6512746712182228870ctxt_a @ ( produc6157733871278919957ctxt_a @ Xs @ Ys ) @ ( trancl5060458363496439462ctxt_a @ R ) )
      = ( ( ( size_s6994882312647770194ctxt_a @ Xs )
          = ( size_s6994882312647770194ctxt_a @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6994882312647770194ctxt_a @ Ys ) )
           => ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I3 ) @ ( nth_li6678718213161987943ctxt_a @ Ys @ I3 ) ) @ ( transi8139323374919604173ctxt_a @ R ) ) ) ) ) ).

% trancl_list_conv
thf(fact_691_trancl__list__conv,axiom,
    ! [Xs: list_l3298181151656792051at_nat,Ys: list_l3298181151656792051at_nat,R: set_Pr4087777274317423175at_nat] :
      ( ( member6399385897798497296at_nat @ ( produc1890770240352350807at_nat @ Xs @ Ys ) @ ( trancl8580566090571148789at_nat @ R ) )
      = ( ( ( size_s6878757106947045471at_nat @ Xs )
          = ( size_s6878757106947045471at_nat @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6878757106947045471at_nat @ Ys ) )
           => ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ ( nth_li8805938130461222772at_nat @ Xs @ I3 ) @ ( nth_li8805938130461222772at_nat @ Ys @ I3 ) ) @ ( transi1440150323567932558at_nat @ R ) ) ) ) ) ).

% trancl_list_conv
thf(fact_692_trancl__list__conv,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( trancl3178887147240166537at_nat @ R ) )
      = ( ( ( size_s5460976970255530739at_nat @ Xs )
          = ( size_s5460976970255530739at_nat @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Ys ) )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) @ ( nth_Pr7617993195940197384at_nat @ Ys @ I3 ) ) @ ( transi243908449541399842at_nat @ R ) ) ) ) ) ).

% trancl_list_conv
thf(fact_693_trancl__list__conv,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
      ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs @ Ys ) @ ( trancl6028151697025156246ctxt_a @ R ) )
      = ( ( ( size_s962548663205356226ctxt_a @ Xs )
          = ( size_s962548663205356226ctxt_a @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Ys ) )
           => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs @ I3 ) @ ( nth_Ground_gmctxt_a @ Ys @ I3 ) ) @ ( transi3225167780237802813ctxt_a @ R ) ) ) ) ) ).

% trancl_list_conv
thf(fact_694_trancl__list__conv,axiom,
    ! [Xs: list_b,Ys: list_b,R: set_Product_prod_b_b] :
      ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Xs @ Ys ) @ ( trancl_list_b @ R ) )
      = ( ( ( size_size_list_b @ Xs )
          = ( size_size_list_b @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Ys ) )
           => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( nth_b @ Xs @ I3 ) @ ( nth_b @ Ys @ I3 ) ) @ ( transitive_trancl_b @ R ) ) ) ) ) ).

% trancl_list_conv
thf(fact_695_trancl__list__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( trancl_list_nat @ R ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ) ).

% trancl_list_conv
thf(fact_696_trancl__list__conv,axiom,
    ! [Xs: list_list_b,Ys: list_list_b,R: set_Pr7665282455119567943list_b] :
      ( ( member7435197128963948048list_b @ ( produc8181704880241203287list_b @ Xs @ Ys ) @ ( trancl_list_list_b @ R ) )
      = ( ( ( size_s420531428170919671list_b @ Xs )
          = ( size_s420531428170919671list_b @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s420531428170919671list_b @ Ys ) )
           => ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ ( nth_list_b @ Xs @ I3 ) @ ( nth_list_b @ Ys @ I3 ) ) @ ( transi1571129030638676774list_b @ R ) ) ) ) ) ).

% trancl_list_conv
thf(fact_697_trancl__list__conv,axiom,
    ! [Xs: list_P8469869581646625389at_nat,Ys: list_P8469869581646625389at_nat,R: set_Pr553994874890374343at_nat] :
      ( ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ Xs @ Ys ) @ ( trancl3412141547504709615at_nat @ R ) )
      = ( ( ( size_s3679842834875189465at_nat @ Xs )
          = ( size_s3679842834875189465at_nat @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3679842834875189465at_nat @ Ys ) )
           => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( nth_Pr6744343527793145070at_nat @ Xs @ I3 ) @ ( nth_Pr6744343527793145070at_nat @ Ys @ I3 ) ) @ ( transi6148169128111658760at_nat @ R ) ) ) ) ) ).

% trancl_list_conv
thf(fact_698_listrel__iff__nth,axiom,
    ! [Xs: list_b,Ys: list_b,R2: set_Product_prod_b_b] :
      ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Xs @ Ys ) @ ( listrel_b_b @ R2 ) )
      = ( ( ( size_size_list_b @ Xs )
          = ( size_size_list_b @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_b @ Xs ) )
           => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( nth_b @ Xs @ N2 ) @ ( nth_b @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_699_listrel__iff__nth,axiom,
    ! [Xs: list_b,Ys: list_nat,R2: set_Pr1307281990691478580_b_nat] :
      ( ( member5518038250373942315st_nat @ ( produc5459849259740801934st_nat @ Xs @ Ys ) @ ( listrel_b_nat @ R2 ) )
      = ( ( ( size_size_list_b @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_b @ Xs ) )
           => ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ ( nth_b @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_700_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_b,R2: set_Pr4264375888882495962_nat_b] :
      ( ( member6003184433318419249list_b @ ( produc7723716014355252812list_b @ Xs @ Ys ) @ ( listrel_nat_b @ R2 ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_b @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
           => ( member8962352056413324475_nat_b @ ( product_Pair_nat_b @ ( nth_nat @ Xs @ N2 ) @ ( nth_b @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_701_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R2 ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_702_listrel__iff__nth,axiom,
    ! [Xs: list_list_nat,Ys: list_b,R2: set_Pr8611280865567961546_nat_b] :
      ( ( member2634019950565409697list_b @ ( produc1825125551900217788list_b @ Xs @ Ys ) @ ( listrel_list_nat_b @ R2 ) )
      = ( ( ( size_s3023201423986296836st_nat @ Xs )
          = ( size_size_list_b @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( member6908590377518958251_nat_b @ ( produc1615968307924265414_nat_b @ ( nth_list_nat @ Xs @ N2 ) @ ( nth_b @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_703_listrel__iff__nth,axiom,
    ! [Xs: list_list_nat,Ys: list_nat,R2: set_Pr7549493975000385137at_nat] :
      ( ( member2583041412010425562st_nat @ ( produc8303078633012690025st_nat @ Xs @ Ys ) @ ( listrel_list_nat_nat @ R2 ) )
      = ( ( ( size_s3023201423986296836st_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( member7820552544826472890at_nat @ ( produc1298395424260782409at_nat @ ( nth_list_nat @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_704_listrel__iff__nth,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_b,R2: set_Pr7985508066815312396xt_a_b] :
      ( ( member8646710124843958371list_b @ ( produc578867730268368766list_b @ Xs @ Ys ) @ ( listre5402109041937956654xt_a_b @ R2 ) )
      = ( ( ( size_s962548663205356226ctxt_a @ Xs )
          = ( size_size_list_b @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
           => ( member8304548504578704877xt_a_b @ ( produc239971217247291912xt_a_b @ ( nth_Ground_gmctxt_a @ Xs @ N2 ) @ ( nth_b @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_705_listrel__iff__nth,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_nat,R2: set_Pr9093710254965148079_a_nat] :
      ( ( member9152701222192952216st_nat @ ( produc5273301679375258151st_nat @ Xs @ Ys ) @ ( listre5839456775373167073_a_nat @ R2 ) )
      = ( ( ( size_s962548663205356226ctxt_a @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
           => ( member1226896217267473912_a_nat @ ( produc2030840239341956231_a_nat @ ( nth_Ground_gmctxt_a @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_706_listrel__iff__nth,axiom,
    ! [Xs: list_b,Ys: list_list_nat,R2: set_Pr6729162334058022980st_nat] :
      ( ( member7987477470964863163st_nat @ ( produc3393462956800104734st_nat @ Xs @ Ys ) @ ( listrel_b_list_nat @ R2 ) )
      = ( ( ( size_size_list_b @ Xs )
          = ( size_s3023201423986296836st_nat @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_b @ Xs ) )
           => ( member1117040998421412389st_nat @ ( produc8204054404704827016st_nat @ ( nth_b @ Xs @ N2 ) @ ( nth_list_nat @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_707_listrel__iff__nth,axiom,
    ! [Xs: list_b,Ys: list_Ground_gmctxt_a,R2: set_Pr5732450166000137346ctxt_a] :
      ( ( member1748955351646886521ctxt_a @ ( produc438373473036823772ctxt_a @ Xs @ Ys ) @ ( listre6395863338307702508ctxt_a @ R2 ) )
      = ( ( ( size_size_list_b @ Xs )
          = ( size_s962548663205356226ctxt_a @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_b @ Xs ) )
           => ( member1453991781301212003ctxt_a @ ( produc1233725513617037766ctxt_a @ ( nth_b @ Xs @ N2 ) @ ( nth_Ground_gmctxt_a @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_708_trancl__list__induct,axiom,
    ! [Ss: list_list_nat,Ts: list_list_nat,R: set_Pr3451248702717554689st_nat,P: list_list_nat > list_list_nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Ss )
        = ( size_s3023201423986296836st_nat @ Ts ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ts ) )
           => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ Ss @ I2 ) @ ( nth_list_nat @ Ts @ I2 ) ) @ ( transi5465639037617667199st_nat @ R ) ) )
       => ( ! [Xs2: list_list_nat,Ys4: list_list_nat] :
              ( ( ( size_s3023201423986296836st_nat @ Xs2 )
                = ( size_s3023201423986296836st_nat @ Ys4 ) )
             => ( ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Ys4 ) )
                   => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ Xs2 @ I4 ) @ ( nth_list_nat @ Ys4 @ I4 ) ) @ R ) )
               => ( P @ Xs2 @ Ys4 ) ) )
         => ( ! [Xs2: list_list_nat,Ys4: list_list_nat,I2: nat,Z: list_nat] :
                ( ( ( size_s3023201423986296836st_nat @ Xs2 )
                  = ( size_s3023201423986296836st_nat @ Ys4 ) )
               => ( ! [Ia: nat] :
                      ( ( ord_less_nat @ Ia @ ( size_s3023201423986296836st_nat @ Ys4 ) )
                     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ Xs2 @ Ia ) @ ( nth_list_nat @ Ys4 @ Ia ) ) @ ( transi5465639037617667199st_nat @ R ) ) )
                 => ( ( P @ Xs2 @ Ys4 )
                   => ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ys4 ) )
                     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ Ys4 @ I2 ) @ Z ) @ R )
                       => ( P @ Xs2 @ ( list_update_list_nat @ Ys4 @ I2 @ Z ) ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% trancl_list_induct
thf(fact_709_trancl__list__induct,axiom,
    ! [Ss: list_l1578326660578982846ctxt_a,Ts: list_l1578326660578982846ctxt_a,R: set_Pr8687807964536448157ctxt_a,P: list_l1578326660578982846ctxt_a > list_l1578326660578982846ctxt_a > $o] :
      ( ( ( size_s6994882312647770194ctxt_a @ Ss )
        = ( size_s6994882312647770194ctxt_a @ Ts ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6994882312647770194ctxt_a @ Ts ) )
           => ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ss @ I2 ) @ ( nth_li6678718213161987943ctxt_a @ Ts @ I2 ) ) @ ( transi8139323374919604173ctxt_a @ R ) ) )
       => ( ! [Xs2: list_l1578326660578982846ctxt_a,Ys4: list_l1578326660578982846ctxt_a] :
              ( ( ( size_s6994882312647770194ctxt_a @ Xs2 )
                = ( size_s6994882312647770194ctxt_a @ Ys4 ) )
             => ( ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( size_s6994882312647770194ctxt_a @ Ys4 ) )
                   => ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs2 @ I4 ) @ ( nth_li6678718213161987943ctxt_a @ Ys4 @ I4 ) ) @ R ) )
               => ( P @ Xs2 @ Ys4 ) ) )
         => ( ! [Xs2: list_l1578326660578982846ctxt_a,Ys4: list_l1578326660578982846ctxt_a,I2: nat,Z: list_Ground_gmctxt_a] :
                ( ( ( size_s6994882312647770194ctxt_a @ Xs2 )
                  = ( size_s6994882312647770194ctxt_a @ Ys4 ) )
               => ( ! [Ia: nat] :
                      ( ( ord_less_nat @ Ia @ ( size_s6994882312647770194ctxt_a @ Ys4 ) )
                     => ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs2 @ Ia ) @ ( nth_li6678718213161987943ctxt_a @ Ys4 @ Ia ) ) @ ( transi8139323374919604173ctxt_a @ R ) ) )
                 => ( ( P @ Xs2 @ Ys4 )
                   => ( ( ord_less_nat @ I2 @ ( size_s6994882312647770194ctxt_a @ Ys4 ) )
                     => ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys4 @ I2 ) @ Z ) @ R )
                       => ( P @ Xs2 @ ( list_u8015824433058371904ctxt_a @ Ys4 @ I2 @ Z ) ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% trancl_list_induct
thf(fact_710_trancl__list__induct,axiom,
    ! [Ss: list_l3298181151656792051at_nat,Ts: list_l3298181151656792051at_nat,R: set_Pr4087777274317423175at_nat,P: list_l3298181151656792051at_nat > list_l3298181151656792051at_nat > $o] :
      ( ( ( size_s6878757106947045471at_nat @ Ss )
        = ( size_s6878757106947045471at_nat @ Ts ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6878757106947045471at_nat @ Ts ) )
           => ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ ( nth_li8805938130461222772at_nat @ Ss @ I2 ) @ ( nth_li8805938130461222772at_nat @ Ts @ I2 ) ) @ ( transi1440150323567932558at_nat @ R ) ) )
       => ( ! [Xs2: list_l3298181151656792051at_nat,Ys4: list_l3298181151656792051at_nat] :
              ( ( ( size_s6878757106947045471at_nat @ Xs2 )
                = ( size_s6878757106947045471at_nat @ Ys4 ) )
             => ( ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( size_s6878757106947045471at_nat @ Ys4 ) )
                   => ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ ( nth_li8805938130461222772at_nat @ Xs2 @ I4 ) @ ( nth_li8805938130461222772at_nat @ Ys4 @ I4 ) ) @ R ) )
               => ( P @ Xs2 @ Ys4 ) ) )
         => ( ! [Xs2: list_l3298181151656792051at_nat,Ys4: list_l3298181151656792051at_nat,I2: nat,Z: list_P8469869581646625389at_nat] :
                ( ( ( size_s6878757106947045471at_nat @ Xs2 )
                  = ( size_s6878757106947045471at_nat @ Ys4 ) )
               => ( ! [Ia: nat] :
                      ( ( ord_less_nat @ Ia @ ( size_s6878757106947045471at_nat @ Ys4 ) )
                     => ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ ( nth_li8805938130461222772at_nat @ Xs2 @ Ia ) @ ( nth_li8805938130461222772at_nat @ Ys4 @ Ia ) ) @ ( transi1440150323567932558at_nat @ R ) ) )
                 => ( ( P @ Xs2 @ Ys4 )
                   => ( ( ord_less_nat @ I2 @ ( size_s6878757106947045471at_nat @ Ys4 ) )
                     => ( ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ ( nth_li8805938130461222772at_nat @ Ys4 @ I2 ) @ Z ) @ R )
                       => ( P @ Xs2 @ ( list_u5583386537922002395at_nat @ Ys4 @ I2 @ Z ) ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% trancl_list_induct
thf(fact_711_trancl__list__induct,axiom,
    ! [Ss: list_P6011104703257516679at_nat,Ts: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat,P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o] :
      ( ( ( size_s5460976970255530739at_nat @ Ss )
        = ( size_s5460976970255530739at_nat @ Ts ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Ts ) )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Ss @ I2 ) @ ( nth_Pr7617993195940197384at_nat @ Ts @ I2 ) ) @ ( transi243908449541399842at_nat @ R ) ) )
       => ( ! [Xs2: list_P6011104703257516679at_nat,Ys4: list_P6011104703257516679at_nat] :
              ( ( ( size_s5460976970255530739at_nat @ Xs2 )
                = ( size_s5460976970255530739at_nat @ Ys4 ) )
             => ( ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Ys4 ) )
                   => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ I4 ) @ ( nth_Pr7617993195940197384at_nat @ Ys4 @ I4 ) ) @ R ) )
               => ( P @ Xs2 @ Ys4 ) ) )
         => ( ! [Xs2: list_P6011104703257516679at_nat,Ys4: list_P6011104703257516679at_nat,I2: nat,Z: product_prod_nat_nat] :
                ( ( ( size_s5460976970255530739at_nat @ Xs2 )
                  = ( size_s5460976970255530739at_nat @ Ys4 ) )
               => ( ! [Ia: nat] :
                      ( ( ord_less_nat @ Ia @ ( size_s5460976970255530739at_nat @ Ys4 ) )
                     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ Ia ) @ ( nth_Pr7617993195940197384at_nat @ Ys4 @ Ia ) ) @ ( transi243908449541399842at_nat @ R ) ) )
                 => ( ( P @ Xs2 @ Ys4 )
                   => ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Ys4 ) )
                     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Ys4 @ I2 ) @ Z ) @ R )
                       => ( P @ Xs2 @ ( list_u6180841689913720943at_nat @ Ys4 @ I2 @ Z ) ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% trancl_list_induct
thf(fact_712_trancl__list__induct,axiom,
    ! [Ss: list_Ground_gmctxt_a,Ts: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,P: list_Ground_gmctxt_a > list_Ground_gmctxt_a > $o] :
      ( ( ( size_s962548663205356226ctxt_a @ Ss )
        = ( size_s962548663205356226ctxt_a @ Ts ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Ts ) )
           => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Ss @ I2 ) @ ( nth_Ground_gmctxt_a @ Ts @ I2 ) ) @ ( transi3225167780237802813ctxt_a @ R ) ) )
       => ( ! [Xs2: list_Ground_gmctxt_a,Ys4: list_Ground_gmctxt_a] :
              ( ( ( size_s962548663205356226ctxt_a @ Xs2 )
                = ( size_s962548663205356226ctxt_a @ Ys4 ) )
             => ( ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Ys4 ) )
                   => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs2 @ I4 ) @ ( nth_Ground_gmctxt_a @ Ys4 @ I4 ) ) @ R ) )
               => ( P @ Xs2 @ Ys4 ) ) )
         => ( ! [Xs2: list_Ground_gmctxt_a,Ys4: list_Ground_gmctxt_a,I2: nat,Z: ground_gmctxt_a] :
                ( ( ( size_s962548663205356226ctxt_a @ Xs2 )
                  = ( size_s962548663205356226ctxt_a @ Ys4 ) )
               => ( ! [Ia: nat] :
                      ( ( ord_less_nat @ Ia @ ( size_s962548663205356226ctxt_a @ Ys4 ) )
                     => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs2 @ Ia ) @ ( nth_Ground_gmctxt_a @ Ys4 @ Ia ) ) @ ( transi3225167780237802813ctxt_a @ R ) ) )
                 => ( ( P @ Xs2 @ Ys4 )
                   => ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Ys4 ) )
                     => ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Ys4 @ I2 ) @ Z ) @ R )
                       => ( P @ Xs2 @ ( list_u7950812632479372336ctxt_a @ Ys4 @ I2 @ Z ) ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% trancl_list_induct
thf(fact_713_trancl__list__induct,axiom,
    ! [Ss: list_b,Ts: list_b,R: set_Product_prod_b_b,P: list_b > list_b > $o] :
      ( ( ( size_size_list_b @ Ss )
        = ( size_size_list_b @ Ts ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Ts ) )
           => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( nth_b @ Ss @ I2 ) @ ( nth_b @ Ts @ I2 ) ) @ ( transitive_trancl_b @ R ) ) )
       => ( ! [Xs2: list_b,Ys4: list_b] :
              ( ( ( size_size_list_b @ Xs2 )
                = ( size_size_list_b @ Ys4 ) )
             => ( ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( size_size_list_b @ Ys4 ) )
                   => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( nth_b @ Xs2 @ I4 ) @ ( nth_b @ Ys4 @ I4 ) ) @ R ) )
               => ( P @ Xs2 @ Ys4 ) ) )
         => ( ! [Xs2: list_b,Ys4: list_b,I2: nat,Z: b] :
                ( ( ( size_size_list_b @ Xs2 )
                  = ( size_size_list_b @ Ys4 ) )
               => ( ! [Ia: nat] :
                      ( ( ord_less_nat @ Ia @ ( size_size_list_b @ Ys4 ) )
                     => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( nth_b @ Xs2 @ Ia ) @ ( nth_b @ Ys4 @ Ia ) ) @ ( transitive_trancl_b @ R ) ) )
                 => ( ( P @ Xs2 @ Ys4 )
                   => ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Ys4 ) )
                     => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( nth_b @ Ys4 @ I2 ) @ Z ) @ R )
                       => ( P @ Xs2 @ ( list_update_b @ Ys4 @ I2 @ Z ) ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% trancl_list_induct
thf(fact_714_trancl__list__induct,axiom,
    ! [Ss: list_nat,Ts: list_nat,R: set_Pr1261947904930325089at_nat,P: list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Ss )
        = ( size_size_list_nat @ Ts ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ts ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ss @ I2 ) @ ( nth_nat @ Ts @ I2 ) ) @ ( transi6264000038957366511cl_nat @ R ) ) )
       => ( ! [Xs2: list_nat,Ys4: list_nat] :
              ( ( ( size_size_list_nat @ Xs2 )
                = ( size_size_list_nat @ Ys4 ) )
             => ( ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys4 ) )
                   => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs2 @ I4 ) @ ( nth_nat @ Ys4 @ I4 ) ) @ R ) )
               => ( P @ Xs2 @ Ys4 ) ) )
         => ( ! [Xs2: list_nat,Ys4: list_nat,I2: nat,Z: nat] :
                ( ( ( size_size_list_nat @ Xs2 )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ! [Ia: nat] :
                      ( ( ord_less_nat @ Ia @ ( size_size_list_nat @ Ys4 ) )
                     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs2 @ Ia ) @ ( nth_nat @ Ys4 @ Ia ) ) @ ( transi6264000038957366511cl_nat @ R ) ) )
                 => ( ( P @ Xs2 @ Ys4 )
                   => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys4 ) )
                     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys4 @ I2 ) @ Z ) @ R )
                       => ( P @ Xs2 @ ( list_update_nat @ Ys4 @ I2 @ Z ) ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% trancl_list_induct
thf(fact_715_trancl__list__induct,axiom,
    ! [Ss: list_list_b,Ts: list_list_b,R: set_Pr7665282455119567943list_b,P: list_list_b > list_list_b > $o] :
      ( ( ( size_s420531428170919671list_b @ Ss )
        = ( size_s420531428170919671list_b @ Ts ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Ts ) )
           => ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ ( nth_list_b @ Ss @ I2 ) @ ( nth_list_b @ Ts @ I2 ) ) @ ( transi1571129030638676774list_b @ R ) ) )
       => ( ! [Xs2: list_list_b,Ys4: list_list_b] :
              ( ( ( size_s420531428170919671list_b @ Xs2 )
                = ( size_s420531428170919671list_b @ Ys4 ) )
             => ( ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( size_s420531428170919671list_b @ Ys4 ) )
                   => ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ ( nth_list_b @ Xs2 @ I4 ) @ ( nth_list_b @ Ys4 @ I4 ) ) @ R ) )
               => ( P @ Xs2 @ Ys4 ) ) )
         => ( ! [Xs2: list_list_b,Ys4: list_list_b,I2: nat,Z: list_b] :
                ( ( ( size_s420531428170919671list_b @ Xs2 )
                  = ( size_s420531428170919671list_b @ Ys4 ) )
               => ( ! [Ia: nat] :
                      ( ( ord_less_nat @ Ia @ ( size_s420531428170919671list_b @ Ys4 ) )
                     => ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ ( nth_list_b @ Xs2 @ Ia ) @ ( nth_list_b @ Ys4 @ Ia ) ) @ ( transi1571129030638676774list_b @ R ) ) )
                 => ( ( P @ Xs2 @ Ys4 )
                   => ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Ys4 ) )
                     => ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ ( nth_list_b @ Ys4 @ I2 ) @ Z ) @ R )
                       => ( P @ Xs2 @ ( list_update_list_b @ Ys4 @ I2 @ Z ) ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% trancl_list_induct
thf(fact_716_trancl__list__induct,axiom,
    ! [Ss: list_P8469869581646625389at_nat,Ts: list_P8469869581646625389at_nat,R: set_Pr553994874890374343at_nat,P: list_P8469869581646625389at_nat > list_P8469869581646625389at_nat > $o] :
      ( ( ( size_s3679842834875189465at_nat @ Ss )
        = ( size_s3679842834875189465at_nat @ Ts ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3679842834875189465at_nat @ Ts ) )
           => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( nth_Pr6744343527793145070at_nat @ Ss @ I2 ) @ ( nth_Pr6744343527793145070at_nat @ Ts @ I2 ) ) @ ( transi6148169128111658760at_nat @ R ) ) )
       => ( ! [Xs2: list_P8469869581646625389at_nat,Ys4: list_P8469869581646625389at_nat] :
              ( ( ( size_s3679842834875189465at_nat @ Xs2 )
                = ( size_s3679842834875189465at_nat @ Ys4 ) )
             => ( ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( size_s3679842834875189465at_nat @ Ys4 ) )
                   => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( nth_Pr6744343527793145070at_nat @ Xs2 @ I4 ) @ ( nth_Pr6744343527793145070at_nat @ Ys4 @ I4 ) ) @ R ) )
               => ( P @ Xs2 @ Ys4 ) ) )
         => ( ! [Xs2: list_P8469869581646625389at_nat,Ys4: list_P8469869581646625389at_nat,I2: nat,Z: produc859450856879609959at_nat] :
                ( ( ( size_s3679842834875189465at_nat @ Xs2 )
                  = ( size_s3679842834875189465at_nat @ Ys4 ) )
               => ( ! [Ia: nat] :
                      ( ( ord_less_nat @ Ia @ ( size_s3679842834875189465at_nat @ Ys4 ) )
                     => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( nth_Pr6744343527793145070at_nat @ Xs2 @ Ia ) @ ( nth_Pr6744343527793145070at_nat @ Ys4 @ Ia ) ) @ ( transi6148169128111658760at_nat @ R ) ) )
                 => ( ( P @ Xs2 @ Ys4 )
                   => ( ( ord_less_nat @ I2 @ ( size_s3679842834875189465at_nat @ Ys4 ) )
                     => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( nth_Pr6744343527793145070at_nat @ Ys4 @ I2 ) @ Z ) @ R )
                       => ( P @ Xs2 @ ( list_u5003261594476800725at_nat @ Ys4 @ I2 @ Z ) ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% trancl_list_induct
thf(fact_717_append__eq__append__conv,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,Us: list_Ground_gmctxt_a,Vs: list_Ground_gmctxt_a] :
      ( ( ( ( size_s962548663205356226ctxt_a @ Xs )
          = ( size_s962548663205356226ctxt_a @ Ys ) )
        | ( ( size_s962548663205356226ctxt_a @ Us )
          = ( size_s962548663205356226ctxt_a @ Vs ) ) )
     => ( ( ( append5238750768297430211ctxt_a @ Xs @ Us )
          = ( append5238750768297430211ctxt_a @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_718_append__eq__append__conv,axiom,
    ! [Xs: list_b,Ys: list_b,Us: list_b,Vs: list_b] :
      ( ( ( ( size_size_list_b @ Xs )
          = ( size_size_list_b @ Ys ) )
        | ( ( size_size_list_b @ Us )
          = ( size_size_list_b @ Vs ) ) )
     => ( ( ( append_b @ Xs @ Us )
          = ( append_b @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_719_append__eq__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
      ( ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        | ( ( size_size_list_nat @ Us )
          = ( size_size_list_nat @ Vs ) ) )
     => ( ( ( append_nat @ Xs @ Us )
          = ( append_nat @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_720_append__eq__append__conv,axiom,
    ! [Xs: list_list_b,Ys: list_list_b,Us: list_list_b,Vs: list_list_b] :
      ( ( ( ( size_s420531428170919671list_b @ Xs )
          = ( size_s420531428170919671list_b @ Ys ) )
        | ( ( size_s420531428170919671list_b @ Us )
          = ( size_s420531428170919671list_b @ Vs ) ) )
     => ( ( ( append_list_b @ Xs @ Us )
          = ( append_list_b @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_721_append__eq__append__conv,axiom,
    ! [Xs: list_P8469869581646625389at_nat,Ys: list_P8469869581646625389at_nat,Us: list_P8469869581646625389at_nat,Vs: list_P8469869581646625389at_nat] :
      ( ( ( ( size_s3679842834875189465at_nat @ Xs )
          = ( size_s3679842834875189465at_nat @ Ys ) )
        | ( ( size_s3679842834875189465at_nat @ Us )
          = ( size_s3679842834875189465at_nat @ Vs ) ) )
     => ( ( ( append8751754712269456642at_nat @ Xs @ Us )
          = ( append8751754712269456642at_nat @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_722_map__append,axiom,
    ! [F: ground_gmctxt_a > nat,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
      ( ( map_Gr1679544123147727284_a_nat @ F @ ( append5238750768297430211ctxt_a @ Xs @ Ys ) )
      = ( append_nat @ ( map_Gr1679544123147727284_a_nat @ F @ Xs ) @ ( map_Gr1679544123147727284_a_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_723_map__append,axiom,
    ! [F: nat > b,Xs: list_nat,Ys: list_nat] :
      ( ( map_nat_b @ F @ ( append_nat @ Xs @ Ys ) )
      = ( append_b @ ( map_nat_b @ F @ Xs ) @ ( map_nat_b @ F @ Ys ) ) ) ).

% map_append
thf(fact_724_map__append,axiom,
    ! [F: b > ground_gmctxt_a,Xs: list_b,Ys: list_b] :
      ( ( map_b_3587768286370285017ctxt_a @ F @ ( append_b @ Xs @ Ys ) )
      = ( append5238750768297430211ctxt_a @ ( map_b_3587768286370285017ctxt_a @ F @ Xs ) @ ( map_b_3587768286370285017ctxt_a @ F @ Ys ) ) ) ).

% map_append
thf(fact_725_map__append,axiom,
    ! [F: b > list_b,Xs: list_b,Ys: list_b] :
      ( ( map_b_list_b @ F @ ( append_b @ Xs @ Ys ) )
      = ( append_list_b @ ( map_b_list_b @ F @ Xs ) @ ( map_b_list_b @ F @ Ys ) ) ) ).

% map_append
thf(fact_726_map__append,axiom,
    ! [F: b > nat,Xs: list_b,Ys: list_b] :
      ( ( map_b_nat @ F @ ( append_b @ Xs @ Ys ) )
      = ( append_nat @ ( map_b_nat @ F @ Xs ) @ ( map_b_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_727_map__append,axiom,
    ! [F: b > b,Xs: list_b,Ys: list_b] :
      ( ( map_b_b @ F @ ( append_b @ Xs @ Ys ) )
      = ( append_b @ ( map_b_b @ F @ Xs ) @ ( map_b_b @ F @ Ys ) ) ) ).

% map_append
thf(fact_728_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_729_trancl__map,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,F: product_prod_nat_nat > product_prod_nat_nat,S: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ! [X3: product_prod_nat_nat,Y4: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ R2 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) @ S ) )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ ( transi243908449541399842at_nat @ R2 ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F @ X ) @ ( F @ Y3 ) ) @ ( transi243908449541399842at_nat @ S ) ) ) ) ).

% trancl_map
thf(fact_730_trancl__map,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,F: product_prod_nat_nat > nat,S: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ! [X3: product_prod_nat_nat,Y4: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) @ S ) )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ ( transi243908449541399842at_nat @ R2 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y3 ) ) @ ( transi6264000038957366511cl_nat @ S ) ) ) ) ).

% trancl_map
thf(fact_731_trancl__map,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,F: nat > product_prod_nat_nat,S: set_Pr8693737435421807431at_nat,X: nat,Y3: nat] :
      ( ! [X3: nat,Y4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R2 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) @ S ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F @ X ) @ ( F @ Y3 ) ) @ ( transi243908449541399842at_nat @ S ) ) ) ) ).

% trancl_map
thf(fact_732_trancl__map,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,F: nat > nat,S: set_Pr1261947904930325089at_nat,X: nat,Y3: nat] :
      ( ! [X3: nat,Y4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) @ S ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y3 ) ) @ ( transi6264000038957366511cl_nat @ S ) ) ) ) ).

% trancl_map
thf(fact_733_append__eq__map__conv,axiom,
    ! [Ys: list_nat,Zs: list_nat,F: ground_gmctxt_a > nat,Xs: list_Ground_gmctxt_a] :
      ( ( ( append_nat @ Ys @ Zs )
        = ( map_Gr1679544123147727284_a_nat @ F @ Xs ) )
      = ( ? [Us2: list_Ground_gmctxt_a,Vs2: list_Ground_gmctxt_a] :
            ( ( Xs
              = ( append5238750768297430211ctxt_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Gr1679544123147727284_a_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_Gr1679544123147727284_a_nat @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_734_append__eq__map__conv,axiom,
    ! [Ys: list_b,Zs: list_b,F: nat > b,Xs: list_nat] :
      ( ( ( append_b @ Ys @ Zs )
        = ( map_nat_b @ F @ Xs ) )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_nat_b @ F @ Us2 ) )
            & ( Zs
              = ( map_nat_b @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_735_append__eq__map__conv,axiom,
    ! [Ys: list_Ground_gmctxt_a,Zs: list_Ground_gmctxt_a,F: b > ground_gmctxt_a,Xs: list_b] :
      ( ( ( append5238750768297430211ctxt_a @ Ys @ Zs )
        = ( map_b_3587768286370285017ctxt_a @ F @ Xs ) )
      = ( ? [Us2: list_b,Vs2: list_b] :
            ( ( Xs
              = ( append_b @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_b_3587768286370285017ctxt_a @ F @ Us2 ) )
            & ( Zs
              = ( map_b_3587768286370285017ctxt_a @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_736_append__eq__map__conv,axiom,
    ! [Ys: list_list_b,Zs: list_list_b,F: b > list_b,Xs: list_b] :
      ( ( ( append_list_b @ Ys @ Zs )
        = ( map_b_list_b @ F @ Xs ) )
      = ( ? [Us2: list_b,Vs2: list_b] :
            ( ( Xs
              = ( append_b @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_b_list_b @ F @ Us2 ) )
            & ( Zs
              = ( map_b_list_b @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_737_append__eq__map__conv,axiom,
    ! [Ys: list_nat,Zs: list_nat,F: b > nat,Xs: list_b] :
      ( ( ( append_nat @ Ys @ Zs )
        = ( map_b_nat @ F @ Xs ) )
      = ( ? [Us2: list_b,Vs2: list_b] :
            ( ( Xs
              = ( append_b @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_b_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_b_nat @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_738_append__eq__map__conv,axiom,
    ! [Ys: list_b,Zs: list_b,F: b > b,Xs: list_b] :
      ( ( ( append_b @ Ys @ Zs )
        = ( map_b_b @ F @ Xs ) )
      = ( ? [Us2: list_b,Vs2: list_b] :
            ( ( Xs
              = ( append_b @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_b_b @ F @ Us2 ) )
            & ( Zs
              = ( map_b_b @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_739_map__eq__append__conv,axiom,
    ! [F: ground_gmctxt_a > nat,Xs: list_Ground_gmctxt_a,Ys: list_nat,Zs: list_nat] :
      ( ( ( map_Gr1679544123147727284_a_nat @ F @ Xs )
        = ( append_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_Ground_gmctxt_a,Vs2: list_Ground_gmctxt_a] :
            ( ( Xs
              = ( append5238750768297430211ctxt_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Gr1679544123147727284_a_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_Gr1679544123147727284_a_nat @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_740_map__eq__append__conv,axiom,
    ! [F: nat > b,Xs: list_nat,Ys: list_b,Zs: list_b] :
      ( ( ( map_nat_b @ F @ Xs )
        = ( append_b @ Ys @ Zs ) )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_nat_b @ F @ Us2 ) )
            & ( Zs
              = ( map_nat_b @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_741_map__eq__append__conv,axiom,
    ! [F: b > ground_gmctxt_a,Xs: list_b,Ys: list_Ground_gmctxt_a,Zs: list_Ground_gmctxt_a] :
      ( ( ( map_b_3587768286370285017ctxt_a @ F @ Xs )
        = ( append5238750768297430211ctxt_a @ Ys @ Zs ) )
      = ( ? [Us2: list_b,Vs2: list_b] :
            ( ( Xs
              = ( append_b @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_b_3587768286370285017ctxt_a @ F @ Us2 ) )
            & ( Zs
              = ( map_b_3587768286370285017ctxt_a @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_742_map__eq__append__conv,axiom,
    ! [F: b > list_b,Xs: list_b,Ys: list_list_b,Zs: list_list_b] :
      ( ( ( map_b_list_b @ F @ Xs )
        = ( append_list_b @ Ys @ Zs ) )
      = ( ? [Us2: list_b,Vs2: list_b] :
            ( ( Xs
              = ( append_b @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_b_list_b @ F @ Us2 ) )
            & ( Zs
              = ( map_b_list_b @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_743_map__eq__append__conv,axiom,
    ! [F: b > nat,Xs: list_b,Ys: list_nat,Zs: list_nat] :
      ( ( ( map_b_nat @ F @ Xs )
        = ( append_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_b,Vs2: list_b] :
            ( ( Xs
              = ( append_b @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_b_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_b_nat @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_744_map__eq__append__conv,axiom,
    ! [F: b > b,Xs: list_b,Ys: list_b,Zs: list_b] :
      ( ( ( map_b_b @ F @ Xs )
        = ( append_b @ Ys @ Zs ) )
      = ( ? [Us2: list_b,Vs2: list_b] :
            ( ( Xs
              = ( append_b @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_b_b @ F @ Us2 ) )
            & ( Zs
              = ( map_b_b @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_745_Utility_Osum__list__0,axiom,
    ! [Xs: list_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( X3 = zero_zero_nat ) )
     => ( ( groups4561878855575611511st_nat @ Xs )
        = zero_zero_nat ) ) ).

% Utility.sum_list_0
thf(fact_746_listrel1__eq__len,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a] :
      ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs @ Ys ) @ ( listre839271074516595327ctxt_a @ R2 ) )
     => ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_747_listrel1__eq__len,axiom,
    ! [Xs: list_b,Ys: list_b,R2: set_Product_prod_b_b] :
      ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Xs @ Ys ) @ ( listrel1_b @ R2 ) )
     => ( ( size_size_list_b @ Xs )
        = ( size_size_list_b @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_748_listrel1__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_749_listrel1__eq__len,axiom,
    ! [Xs: list_list_b,Ys: list_list_b,R2: set_Pr7665282455119567943list_b] :
      ( ( member7435197128963948048list_b @ ( produc8181704880241203287list_b @ Xs @ Ys ) @ ( listrel1_list_b @ R2 ) )
     => ( ( size_s420531428170919671list_b @ Xs )
        = ( size_s420531428170919671list_b @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_750_listrel1__eq__len,axiom,
    ! [Xs: list_P8469869581646625389at_nat,Ys: list_P8469869581646625389at_nat,R2: set_Pr553994874890374343at_nat] :
      ( ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ Xs @ Ys ) @ ( listre2293869942453659206at_nat @ R2 ) )
     => ( ( size_s3679842834875189465at_nat @ Xs )
        = ( size_s3679842834875189465at_nat @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_751_listrel__eq__len,axiom,
    ! [Xs: list_b,Ys: list_b,R2: set_Product_prod_b_b] :
      ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Xs @ Ys ) @ ( listrel_b_b @ R2 ) )
     => ( ( size_size_list_b @ Xs )
        = ( size_size_list_b @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_752_listrel__eq__len,axiom,
    ! [Xs: list_b,Ys: list_nat,R2: set_Pr1307281990691478580_b_nat] :
      ( ( member5518038250373942315st_nat @ ( produc5459849259740801934st_nat @ Xs @ Ys ) @ ( listrel_b_nat @ R2 ) )
     => ( ( size_size_list_b @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_753_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_b,R2: set_Pr4264375888882495962_nat_b] :
      ( ( member6003184433318419249list_b @ ( produc7723716014355252812list_b @ Xs @ Ys ) @ ( listrel_nat_b @ R2 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_b @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_754_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R2 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_755_listrel__eq__len,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_b,R2: set_Pr7985508066815312396xt_a_b] :
      ( ( member8646710124843958371list_b @ ( produc578867730268368766list_b @ Xs @ Ys ) @ ( listre5402109041937956654xt_a_b @ R2 ) )
     => ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_size_list_b @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_756_listrel__eq__len,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_nat,R2: set_Pr9093710254965148079_a_nat] :
      ( ( member9152701222192952216st_nat @ ( produc5273301679375258151st_nat @ Xs @ Ys ) @ ( listre5839456775373167073_a_nat @ R2 ) )
     => ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_757_listrel__eq__len,axiom,
    ! [Xs: list_b,Ys: list_Ground_gmctxt_a,R2: set_Pr5732450166000137346ctxt_a] :
      ( ( member1748955351646886521ctxt_a @ ( produc438373473036823772ctxt_a @ Xs @ Ys ) @ ( listre6395863338307702508ctxt_a @ R2 ) )
     => ( ( size_size_list_b @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_758_listrel__eq__len,axiom,
    ! [Xs: list_b,Ys: list_list_b,R2: set_Pr3499773787893599181list_b] :
      ( ( member60363355953214230list_b @ ( produc4417627063519195229list_b @ Xs @ Ys ) @ ( listrel_b_list_b @ R2 ) )
     => ( ( size_size_list_b @ Xs )
        = ( size_s420531428170919671list_b @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_759_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_Ground_gmctxt_a,R2: set_Pr1619835084168866991ctxt_a] :
      ( ( member5348271269840303512ctxt_a @ ( produc781480777803334439ctxt_a @ Xs @ Ys ) @ ( listre6913154111349222369ctxt_a @ R2 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_760_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_list_b,R2: set_Pr1349601357184307552list_b] :
      ( ( member4916451157736907575list_b @ ( produc2084500703101687378list_b @ Xs @ Ys ) @ ( listrel_nat_list_b @ R2 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s420531428170919671list_b @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_761_P__as__bs__extend,axiom,
    ! [As: list_b,Bs: list_b,Cs2: list_b,Ds: list_b,P: b > b > $o] :
      ( ( ( size_size_list_b @ As )
        = ( size_size_list_b @ Bs ) )
     => ( ( ( size_size_list_b @ Cs2 )
          = ( size_size_list_b @ Ds ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Bs ) )
             => ( P @ ( nth_b @ As @ I2 ) @ ( nth_b @ Bs @ I2 ) ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Ds ) )
               => ( P @ ( nth_b @ Cs2 @ I2 ) @ ( nth_b @ Ds @ I2 ) ) )
           => ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_size_list_b @ ( append_b @ Bs @ Ds ) ) )
               => ( P @ ( nth_b @ ( append_b @ As @ Cs2 ) @ I4 ) @ ( nth_b @ ( append_b @ Bs @ Ds ) @ I4 ) ) ) ) ) ) ) ).

% P_as_bs_extend
thf(fact_762_P__as__bs__extend,axiom,
    ! [As: list_b,Bs: list_nat,Cs2: list_b,Ds: list_nat,P: b > nat > $o] :
      ( ( ( size_size_list_b @ As )
        = ( size_size_list_nat @ Bs ) )
     => ( ( ( size_size_list_b @ Cs2 )
          = ( size_size_list_nat @ Ds ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Bs ) )
             => ( P @ ( nth_b @ As @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ds ) )
               => ( P @ ( nth_b @ Cs2 @ I2 ) @ ( nth_nat @ Ds @ I2 ) ) )
           => ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( append_nat @ Bs @ Ds ) ) )
               => ( P @ ( nth_b @ ( append_b @ As @ Cs2 ) @ I4 ) @ ( nth_nat @ ( append_nat @ Bs @ Ds ) @ I4 ) ) ) ) ) ) ) ).

% P_as_bs_extend
thf(fact_763_P__as__bs__extend,axiom,
    ! [As: list_nat,Bs: list_b,Cs2: list_nat,Ds: list_b,P: nat > b > $o] :
      ( ( ( size_size_list_nat @ As )
        = ( size_size_list_b @ Bs ) )
     => ( ( ( size_size_list_nat @ Cs2 )
          = ( size_size_list_b @ Ds ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Bs ) )
             => ( P @ ( nth_nat @ As @ I2 ) @ ( nth_b @ Bs @ I2 ) ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Ds ) )
               => ( P @ ( nth_nat @ Cs2 @ I2 ) @ ( nth_b @ Ds @ I2 ) ) )
           => ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_size_list_b @ ( append_b @ Bs @ Ds ) ) )
               => ( P @ ( nth_nat @ ( append_nat @ As @ Cs2 ) @ I4 ) @ ( nth_b @ ( append_b @ Bs @ Ds ) @ I4 ) ) ) ) ) ) ) ).

% P_as_bs_extend
thf(fact_764_P__as__bs__extend,axiom,
    ! [As: list_nat,Bs: list_nat,Cs2: list_nat,Ds: list_nat,P: nat > nat > $o] :
      ( ( ( size_size_list_nat @ As )
        = ( size_size_list_nat @ Bs ) )
     => ( ( ( size_size_list_nat @ Cs2 )
          = ( size_size_list_nat @ Ds ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Bs ) )
             => ( P @ ( nth_nat @ As @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ds ) )
               => ( P @ ( nth_nat @ Cs2 @ I2 ) @ ( nth_nat @ Ds @ I2 ) ) )
           => ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( append_nat @ Bs @ Ds ) ) )
               => ( P @ ( nth_nat @ ( append_nat @ As @ Cs2 ) @ I4 ) @ ( nth_nat @ ( append_nat @ Bs @ Ds ) @ I4 ) ) ) ) ) ) ) ).

% P_as_bs_extend
thf(fact_765_P__as__bs__extend,axiom,
    ! [As: list_list_nat,Bs: list_b,Cs2: list_list_nat,Ds: list_b,P: list_nat > b > $o] :
      ( ( ( size_s3023201423986296836st_nat @ As )
        = ( size_size_list_b @ Bs ) )
     => ( ( ( size_s3023201423986296836st_nat @ Cs2 )
          = ( size_size_list_b @ Ds ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Bs ) )
             => ( P @ ( nth_list_nat @ As @ I2 ) @ ( nth_b @ Bs @ I2 ) ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Ds ) )
               => ( P @ ( nth_list_nat @ Cs2 @ I2 ) @ ( nth_b @ Ds @ I2 ) ) )
           => ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_size_list_b @ ( append_b @ Bs @ Ds ) ) )
               => ( P @ ( nth_list_nat @ ( append_list_nat @ As @ Cs2 ) @ I4 ) @ ( nth_b @ ( append_b @ Bs @ Ds ) @ I4 ) ) ) ) ) ) ) ).

% P_as_bs_extend
thf(fact_766_P__as__bs__extend,axiom,
    ! [As: list_list_nat,Bs: list_nat,Cs2: list_list_nat,Ds: list_nat,P: list_nat > nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ As )
        = ( size_size_list_nat @ Bs ) )
     => ( ( ( size_s3023201423986296836st_nat @ Cs2 )
          = ( size_size_list_nat @ Ds ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Bs ) )
             => ( P @ ( nth_list_nat @ As @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ds ) )
               => ( P @ ( nth_list_nat @ Cs2 @ I2 ) @ ( nth_nat @ Ds @ I2 ) ) )
           => ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( append_nat @ Bs @ Ds ) ) )
               => ( P @ ( nth_list_nat @ ( append_list_nat @ As @ Cs2 ) @ I4 ) @ ( nth_nat @ ( append_nat @ Bs @ Ds ) @ I4 ) ) ) ) ) ) ) ).

% P_as_bs_extend
thf(fact_767_P__as__bs__extend,axiom,
    ! [As: list_Ground_gmctxt_a,Bs: list_b,Cs2: list_Ground_gmctxt_a,Ds: list_b,P: ground_gmctxt_a > b > $o] :
      ( ( ( size_s962548663205356226ctxt_a @ As )
        = ( size_size_list_b @ Bs ) )
     => ( ( ( size_s962548663205356226ctxt_a @ Cs2 )
          = ( size_size_list_b @ Ds ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Bs ) )
             => ( P @ ( nth_Ground_gmctxt_a @ As @ I2 ) @ ( nth_b @ Bs @ I2 ) ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Ds ) )
               => ( P @ ( nth_Ground_gmctxt_a @ Cs2 @ I2 ) @ ( nth_b @ Ds @ I2 ) ) )
           => ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_size_list_b @ ( append_b @ Bs @ Ds ) ) )
               => ( P @ ( nth_Ground_gmctxt_a @ ( append5238750768297430211ctxt_a @ As @ Cs2 ) @ I4 ) @ ( nth_b @ ( append_b @ Bs @ Ds ) @ I4 ) ) ) ) ) ) ) ).

% P_as_bs_extend
thf(fact_768_P__as__bs__extend,axiom,
    ! [As: list_Ground_gmctxt_a,Bs: list_nat,Cs2: list_Ground_gmctxt_a,Ds: list_nat,P: ground_gmctxt_a > nat > $o] :
      ( ( ( size_s962548663205356226ctxt_a @ As )
        = ( size_size_list_nat @ Bs ) )
     => ( ( ( size_s962548663205356226ctxt_a @ Cs2 )
          = ( size_size_list_nat @ Ds ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Bs ) )
             => ( P @ ( nth_Ground_gmctxt_a @ As @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ds ) )
               => ( P @ ( nth_Ground_gmctxt_a @ Cs2 @ I2 ) @ ( nth_nat @ Ds @ I2 ) ) )
           => ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( append_nat @ Bs @ Ds ) ) )
               => ( P @ ( nth_Ground_gmctxt_a @ ( append5238750768297430211ctxt_a @ As @ Cs2 ) @ I4 ) @ ( nth_nat @ ( append_nat @ Bs @ Ds ) @ I4 ) ) ) ) ) ) ) ).

% P_as_bs_extend
thf(fact_769_P__as__bs__extend,axiom,
    ! [As: list_b,Bs: list_list_nat,Cs2: list_b,Ds: list_list_nat,P: b > list_nat > $o] :
      ( ( ( size_size_list_b @ As )
        = ( size_s3023201423986296836st_nat @ Bs ) )
     => ( ( ( size_size_list_b @ Cs2 )
          = ( size_s3023201423986296836st_nat @ Ds ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Bs ) )
             => ( P @ ( nth_b @ As @ I2 ) @ ( nth_list_nat @ Bs @ I2 ) ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ds ) )
               => ( P @ ( nth_b @ Cs2 @ I2 ) @ ( nth_list_nat @ Ds @ I2 ) ) )
           => ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ ( append_list_nat @ Bs @ Ds ) ) )
               => ( P @ ( nth_b @ ( append_b @ As @ Cs2 ) @ I4 ) @ ( nth_list_nat @ ( append_list_nat @ Bs @ Ds ) @ I4 ) ) ) ) ) ) ) ).

% P_as_bs_extend
thf(fact_770_P__as__bs__extend,axiom,
    ! [As: list_b,Bs: list_Ground_gmctxt_a,Cs2: list_b,Ds: list_Ground_gmctxt_a,P: b > ground_gmctxt_a > $o] :
      ( ( ( size_size_list_b @ As )
        = ( size_s962548663205356226ctxt_a @ Bs ) )
     => ( ( ( size_size_list_b @ Cs2 )
          = ( size_s962548663205356226ctxt_a @ Ds ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Bs ) )
             => ( P @ ( nth_b @ As @ I2 ) @ ( nth_Ground_gmctxt_a @ Bs @ I2 ) ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Ds ) )
               => ( P @ ( nth_b @ Cs2 @ I2 ) @ ( nth_Ground_gmctxt_a @ Ds @ I2 ) ) )
           => ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ ( append5238750768297430211ctxt_a @ Bs @ Ds ) ) )
               => ( P @ ( nth_b @ ( append_b @ As @ Cs2 ) @ I4 ) @ ( nth_Ground_gmctxt_a @ ( append5238750768297430211ctxt_a @ Bs @ Ds ) @ I4 ) ) ) ) ) ) ) ).

% P_as_bs_extend
thf(fact_771_list__update__append1,axiom,
    ! [I: nat,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,X: ground_gmctxt_a] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( list_u7950812632479372336ctxt_a @ ( append5238750768297430211ctxt_a @ Xs @ Ys ) @ I @ X )
        = ( append5238750768297430211ctxt_a @ ( list_u7950812632479372336ctxt_a @ Xs @ I @ X ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_772_list__update__append1,axiom,
    ! [I: nat,Xs: list_b,Ys: list_b,X: b] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
     => ( ( list_update_b @ ( append_b @ Xs @ Ys ) @ I @ X )
        = ( append_b @ ( list_update_b @ Xs @ I @ X ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_773_list__update__append1,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ I @ X )
        = ( append_nat @ ( list_update_nat @ Xs @ I @ X ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_774_list__update__append1,axiom,
    ! [I: nat,Xs: list_list_b,Ys: list_list_b,X: list_b] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ Xs ) )
     => ( ( list_update_list_b @ ( append_list_b @ Xs @ Ys ) @ I @ X )
        = ( append_list_b @ ( list_update_list_b @ Xs @ I @ X ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_775_list__update__append1,axiom,
    ! [I: nat,Xs: list_P8469869581646625389at_nat,Ys: list_P8469869581646625389at_nat,X: produc859450856879609959at_nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ Xs ) )
     => ( ( list_u5003261594476800725at_nat @ ( append8751754712269456642at_nat @ Xs @ Ys ) @ I @ X )
        = ( append8751754712269456642at_nat @ ( list_u5003261594476800725at_nat @ Xs @ I @ X ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_776_length__pos__if__in__set,axiom,
    ! [X: ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
      ( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s962548663205356226ctxt_a @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_777_length__pos__if__in__set,axiom,
    ! [X: b,Xs: list_b] :
      ( ( member_b @ X @ ( set_b2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_b @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_778_length__pos__if__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_779_length__pos__if__in__set,axiom,
    ! [X: list_b,Xs: list_list_b] :
      ( ( member_list_b @ X @ ( set_list_b2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s420531428170919671list_b @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_780_length__pos__if__in__set,axiom,
    ! [X: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3679842834875189465at_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_781_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_782_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_783_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_784_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_785_concat__index__split__sound__fst__arg,axiom,
    ! [I: nat,Xs: list_l1578326660578982846ctxt_a] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs ) ) @ ( size_s6994882312647770194ctxt_a @ Xs ) ) ) ).

% concat_index_split_sound_fst_arg
thf(fact_786_concat__index__split__sound__fst__arg,axiom,
    ! [I: nat,Xs: list_list_b] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ ( concat_b @ Xs ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat_index_split_b @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs ) ) @ ( size_s420531428170919671list_b @ Xs ) ) ) ).

% concat_index_split_sound_fst_arg
thf(fact_787_concat__index__split__sound__fst__arg,axiom,
    ! [I: nat,Xs: list_list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs ) ) @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ).

% concat_index_split_sound_fst_arg
thf(fact_788_concat__index__split__sound__fst__arg,axiom,
    ! [I: nat,Xs: list_list_list_b] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ ( concat_list_b @ Xs ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat7499549088139563139list_b @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs ) ) @ ( size_s9106009106869804029list_b @ Xs ) ) ) ).

% concat_index_split_sound_fst_arg
thf(fact_789_concat__index__split__sound__fst__arg,axiom,
    ! [I: nat,Xs: list_l3298181151656792051at_nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ ( concat5616020275982107560at_nat @ Xs ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat1027118156116784357at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs ) ) @ ( size_s6878757106947045471at_nat @ Xs ) ) ) ).

% concat_index_split_sound_fst_arg
thf(fact_790_trancl_Ocases,axiom,
    ! [A1: product_prod_nat_nat,A22: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A1 @ A22 ) @ ( transi243908449541399842at_nat @ R2 ) )
     => ( ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A1 @ A22 ) @ R2 )
       => ~ ! [B3: product_prod_nat_nat] :
              ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A1 @ B3 ) @ ( transi243908449541399842at_nat @ R2 ) )
             => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B3 @ A22 ) @ R2 ) ) ) ) ).

% trancl.cases
thf(fact_791_trancl_Ocases,axiom,
    ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ R2 )
       => ~ ! [B3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ R2 ) ) ) ) ).

% trancl.cases
thf(fact_792_trancl_Osimps,axiom,
    ! [A1: product_prod_nat_nat,A22: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A1 @ A22 ) @ ( transi243908449541399842at_nat @ R2 ) )
      = ( ? [A5: product_prod_nat_nat,B4: product_prod_nat_nat] :
            ( ( A1 = A5 )
            & ( A22 = B4 )
            & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A5 @ B4 ) @ R2 ) )
        | ? [A5: product_prod_nat_nat,B4: product_prod_nat_nat,C2: product_prod_nat_nat] :
            ( ( A1 = A5 )
            & ( A22 = C2 )
            & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A5 @ B4 ) @ ( transi243908449541399842at_nat @ R2 ) )
            & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B4 @ C2 ) @ R2 ) ) ) ) ).

% trancl.simps
thf(fact_793_trancl_Osimps,axiom,
    ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
      = ( ? [A5: nat,B4: nat] :
            ( ( A1 = A5 )
            & ( A22 = B4 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B4 ) @ R2 ) )
        | ? [A5: nat,B4: nat,C2: nat] :
            ( ( A1 = A5 )
            & ( A22 = C2 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B4 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C2 ) @ R2 ) ) ) ) ).

% trancl.simps
thf(fact_794_trancl_Or__into__trancl,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( transi243908449541399842at_nat @ R2 ) ) ) ).

% trancl.r_into_trancl
thf(fact_795_trancl_Or__into__trancl,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ).

% trancl.r_into_trancl
thf(fact_796_fst__eqD,axiom,
    ! [X: nat > nat,Y3: nat,A: nat > nat] :
      ( ( ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ X @ Y3 ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_797_fst__eqD,axiom,
    ! [X: product_prod_nat_nat,Y3: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_798_fst__eqD,axiom,
    ! [X: nat,Y3: nat,A: nat] :
      ( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y3 ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_799_fst__conv,axiom,
    ! [X1: nat > nat,X22: nat] :
      ( ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_800_fst__conv,axiom,
    ! [X1: product_prod_nat_nat,X22: product_prod_nat_nat] :
      ( ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_801_fst__conv,axiom,
    ! [X1: nat,X22: nat] :
      ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_802_trancl__induct2,axiom,
    ! [Ax: nat > nat,Ay: nat,Bx: nat > nat,By: nat,R2: set_Pr7047708467701517959at_nat,P: ( nat > nat ) > nat > $o] :
      ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ Bx @ By ) ) @ ( transi3146596849628481425at_nat @ R2 ) )
     => ( ! [A4: nat > nat,B3: nat] :
            ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ A4 @ B3 ) ) @ R2 )
           => ( P @ A4 @ B3 ) )
       => ( ! [A4: nat > nat,B3: nat,Aa: nat > nat,Ba: nat] :
              ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ A4 @ B3 ) ) @ ( transi3146596849628481425at_nat @ R2 ) )
             => ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ A4 @ B3 ) @ ( produc72220940542539688at_nat @ Aa @ Ba ) ) @ R2 )
               => ( ( P @ A4 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_803_trancl__induct2,axiom,
    ! [Ax: product_prod_nat_nat,Ay: product_prod_nat_nat,Bx: product_prod_nat_nat,By: product_prod_nat_nat,R2: set_Pr553994874890374343at_nat,P: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ Ax @ Ay ) @ ( produc6161850002892822231at_nat @ Bx @ By ) ) @ ( transi6148169128111658760at_nat @ R2 ) )
     => ( ! [A4: product_prod_nat_nat,B3: product_prod_nat_nat] :
            ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ Ax @ Ay ) @ ( produc6161850002892822231at_nat @ A4 @ B3 ) ) @ R2 )
           => ( P @ A4 @ B3 ) )
       => ( ! [A4: product_prod_nat_nat,B3: product_prod_nat_nat,Aa: product_prod_nat_nat,Ba: product_prod_nat_nat] :
              ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ Ax @ Ay ) @ ( produc6161850002892822231at_nat @ A4 @ B3 ) ) @ ( transi6148169128111658760at_nat @ R2 ) )
             => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ A4 @ B3 ) @ ( produc6161850002892822231at_nat @ Aa @ Ba ) ) @ R2 )
               => ( ( P @ A4 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_804_trancl__induct2,axiom,
    ! [Ax: nat,Ay: nat,Bx: nat,By: nat,R2: set_Pr8693737435421807431at_nat,P: nat > nat > $o] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi243908449541399842at_nat @ R2 ) )
     => ( ! [A4: nat,B3: nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A4 @ B3 ) ) @ R2 )
           => ( P @ A4 @ B3 ) )
       => ( ! [A4: nat,B3: nat,Aa: nat,Ba: nat] :
              ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A4 @ B3 ) ) @ ( transi243908449541399842at_nat @ R2 ) )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R2 )
               => ( ( P @ A4 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_805_size__neq__size__imp__neq,axiom,
    ! [X: list_Ground_gmctxt_a,Y3: list_Ground_gmctxt_a] :
      ( ( ( size_s962548663205356226ctxt_a @ X )
       != ( size_s962548663205356226ctxt_a @ Y3 ) )
     => ( X != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_806_size__neq__size__imp__neq,axiom,
    ! [X: list_b,Y3: list_b] :
      ( ( ( size_size_list_b @ X )
       != ( size_size_list_b @ Y3 ) )
     => ( X != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_807_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y3: list_nat] :
      ( ( ( size_size_list_nat @ X )
       != ( size_size_list_nat @ Y3 ) )
     => ( X != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_808_size__neq__size__imp__neq,axiom,
    ! [X: list_list_b,Y3: list_list_b] :
      ( ( ( size_s420531428170919671list_b @ X )
       != ( size_s420531428170919671list_b @ Y3 ) )
     => ( X != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_809_size__neq__size__imp__neq,axiom,
    ! [X: list_P8469869581646625389at_nat,Y3: list_P8469869581646625389at_nat] :
      ( ( ( size_s3679842834875189465at_nat @ X )
       != ( size_s3679842834875189465at_nat @ Y3 ) )
     => ( X != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_810_linorder__neqE__nat,axiom,
    ! [X: nat,Y3: nat] :
      ( ( X != Y3 )
     => ( ~ ( ord_less_nat @ X @ Y3 )
       => ( ord_less_nat @ Y3 @ X ) ) ) ).

% linorder_neqE_nat
thf(fact_811_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ~ ( P @ N3 )
         => ? [M2: nat] :
              ( ( ord_less_nat @ M2 @ N3 )
              & ~ ( P @ M2 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_812_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N3 )
             => ( P @ M2 ) )
         => ( P @ N3 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_813_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_814_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_815_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_816_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_817_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
      = ( ( ord_less_nat @ M @ N )
        | ( ord_less_nat @ N @ M ) ) ) ).

% nat_neq_iff
thf(fact_818_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_819_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_820_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_821_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_822_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ~ ( P @ N3 )
             => ? [M2: nat] :
                  ( ( ord_less_nat @ M2 @ N3 )
                  & ~ ( P @ M2 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_823_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_824_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_825_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_826_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_827_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_828_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_829_converse__trancl__induct,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,P: product_prod_nat_nat > $o] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( transi243908449541399842at_nat @ R2 ) )
     => ( ! [Y4: product_prod_nat_nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ B ) @ R2 )
           => ( P @ Y4 ) )
       => ( ! [Y4: product_prod_nat_nat,Z: product_prod_nat_nat] :
              ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ Z ) @ R2 )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Z @ B ) @ ( transi243908449541399842at_nat @ R2 ) )
               => ( ( P @ Z )
                 => ( P @ Y4 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_830_converse__trancl__induct,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P: nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ! [Y4: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ B ) @ R2 )
           => ( P @ Y4 ) )
       => ( ! [Y4: nat,Z: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z ) @ R2 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
               => ( ( P @ Z )
                 => ( P @ Y4 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_831_trancl__trans__induct,axiom,
    ! [X: product_prod_nat_nat,Y3: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,P: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ ( transi243908449541399842at_nat @ R2 ) )
     => ( ! [X3: product_prod_nat_nat,Y4: product_prod_nat_nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ R2 )
           => ( P @ X3 @ Y4 ) )
       => ( ! [X3: product_prod_nat_nat,Y4: product_prod_nat_nat,Z: product_prod_nat_nat] :
              ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ ( transi243908449541399842at_nat @ R2 ) )
             => ( ( P @ X3 @ Y4 )
               => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ Z ) @ ( transi243908449541399842at_nat @ R2 ) )
                 => ( ( P @ Y4 @ Z )
                   => ( P @ X3 @ Z ) ) ) ) )
         => ( P @ X @ Y3 ) ) ) ) ).

% trancl_trans_induct
thf(fact_832_trancl__trans__induct,axiom,
    ! [X: nat,Y3: nat,R2: set_Pr1261947904930325089at_nat,P: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ! [X3: nat,Y4: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R2 )
           => ( P @ X3 @ Y4 ) )
       => ( ! [X3: nat,Y4: nat,Z: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
             => ( ( P @ X3 @ Y4 )
               => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) )
                 => ( ( P @ Y4 @ Z )
                   => ( P @ X3 @ Z ) ) ) ) )
         => ( P @ X @ Y3 ) ) ) ) ).

% trancl_trans_induct
thf(fact_833_trancl__into__trancl2,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,C3: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ C3 ) @ ( transi243908449541399842at_nat @ R2 ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ C3 ) @ ( transi243908449541399842at_nat @ R2 ) ) ) ) ).

% trancl_into_trancl2
thf(fact_834_trancl__into__trancl2,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).

% trancl_into_trancl2
thf(fact_835_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,C3: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( transi243908449541399842at_nat @ R2 ) )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ C3 ) @ R2 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ C3 ) @ ( transi243908449541399842at_nat @ R2 ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_836_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ R2 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_837_irrefl__trancl__rD,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ X3 ) @ ( transi243908449541399842at_nat @ R2 ) )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R2 )
       => ( X != Y3 ) ) ) ).

% irrefl_trancl_rD
thf(fact_838_irrefl__trancl__rD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y3: nat] :
      ( ! [X3: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R2 )
       => ( X != Y3 ) ) ) ).

% irrefl_trancl_rD
thf(fact_839_converse__tranclE,axiom,
    ! [X: product_prod_nat_nat,Z4: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Z4 ) @ ( transi243908449541399842at_nat @ R2 ) )
     => ( ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Z4 ) @ R2 )
       => ~ ! [Y4: product_prod_nat_nat] :
              ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y4 ) @ R2 )
             => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ Z4 ) @ ( transi243908449541399842at_nat @ R2 ) ) ) ) ) ).

% converse_tranclE
thf(fact_840_converse__tranclE,axiom,
    ! [X: nat,Z4: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z4 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z4 ) @ R2 )
       => ~ ! [Y4: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R2 )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z4 ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ) ).

% converse_tranclE
thf(fact_841_r__r__into__trancl,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R4: set_Pr8693737435421807431at_nat,C3: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R4 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ C3 ) @ R4 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ C3 ) @ ( transi243908449541399842at_nat @ R4 ) ) ) ) ).

% r_r_into_trancl
thf(fact_842_r__r__into__trancl,axiom,
    ! [A: nat,B: nat,R4: set_Pr1261947904930325089at_nat,C3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ R4 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi6264000038957366511cl_nat @ R4 ) ) ) ) ).

% r_r_into_trancl
thf(fact_843_trancl__induct,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,P: product_prod_nat_nat > $o] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( transi243908449541399842at_nat @ R2 ) )
     => ( ! [Y4: product_prod_nat_nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ Y4 ) @ R2 )
           => ( P @ Y4 ) )
       => ( ! [Y4: product_prod_nat_nat,Z: product_prod_nat_nat] :
              ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ Y4 ) @ ( transi243908449541399842at_nat @ R2 ) )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ Z ) @ R2 )
               => ( ( P @ Y4 )
                 => ( P @ Z ) ) ) )
         => ( P @ B ) ) ) ) ).

% trancl_induct
thf(fact_844_trancl__induct,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P: nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ! [Y4: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y4 ) @ R2 )
           => ( P @ Y4 ) )
       => ( ! [Y4: nat,Z: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y4 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z ) @ R2 )
               => ( ( P @ Y4 )
                 => ( P @ Z ) ) ) )
         => ( P @ B ) ) ) ) ).

% trancl_induct
thf(fact_845_trancl__trans,axiom,
    ! [X: product_prod_nat_nat,Y3: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,Z4: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ ( transi243908449541399842at_nat @ R2 ) )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Z4 ) @ ( transi243908449541399842at_nat @ R2 ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Z4 ) @ ( transi243908449541399842at_nat @ R2 ) ) ) ) ).

% trancl_trans
thf(fact_846_trancl__trans,axiom,
    ! [X: nat,Y3: nat,R2: set_Pr1261947904930325089at_nat,Z4: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z4 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z4 ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).

% trancl_trans
thf(fact_847_tranclE,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( transi243908449541399842at_nat @ R2 ) )
     => ( ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 )
       => ~ ! [C: product_prod_nat_nat] :
              ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ C ) @ ( transi243908449541399842at_nat @ R2 ) )
             => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ C @ B ) @ R2 ) ) ) ) ).

% tranclE
thf(fact_848_tranclE,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
       => ~ ! [C: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C ) @ ( transi6264000038957366511cl_nat @ R2 ) )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ C @ B ) @ R2 ) ) ) ) ).

% tranclE
thf(fact_849_concat__index__split__sound__snd__arg,axiom,
    ! [I: nat,Xs: list_l1578326660578982846ctxt_a] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ord_less_nat @ ( product_snd_nat_nat @ ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs ) ) @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ ( product_fst_nat_nat @ ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs ) ) ) ) ) ) ).

% concat_index_split_sound_snd_arg
thf(fact_850_concat__index__split__sound__snd__arg,axiom,
    ! [I: nat,Xs: list_list_b] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ ( concat_b @ Xs ) ) )
     => ( ord_less_nat @ ( product_snd_nat_nat @ ( concat_index_split_b @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs ) ) @ ( size_size_list_b @ ( nth_list_b @ Xs @ ( product_fst_nat_nat @ ( concat_index_split_b @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs ) ) ) ) ) ) ).

% concat_index_split_sound_snd_arg
thf(fact_851_concat__index__split__sound__snd__arg,axiom,
    ! [I: nat,Xs: list_list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ord_less_nat @ ( product_snd_nat_nat @ ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs ) ) @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ ( product_fst_nat_nat @ ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs ) ) ) ) ) ) ).

% concat_index_split_sound_snd_arg
thf(fact_852_concat__index__split__sound__snd__arg,axiom,
    ! [I: nat,Xs: list_list_list_b] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ ( concat_list_b @ Xs ) ) )
     => ( ord_less_nat @ ( product_snd_nat_nat @ ( concat7499549088139563139list_b @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs ) ) @ ( size_s420531428170919671list_b @ ( nth_list_list_b @ Xs @ ( product_fst_nat_nat @ ( concat7499549088139563139list_b @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs ) ) ) ) ) ) ).

% concat_index_split_sound_snd_arg
thf(fact_853_concat__index__split__sound__snd__arg,axiom,
    ! [I: nat,Xs: list_l3298181151656792051at_nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ ( concat5616020275982107560at_nat @ Xs ) ) )
     => ( ord_less_nat @ ( product_snd_nat_nat @ ( concat1027118156116784357at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs ) ) @ ( size_s3679842834875189465at_nat @ ( nth_li8805938130461222772at_nat @ Xs @ ( product_fst_nat_nat @ ( concat1027118156116784357at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs ) ) ) ) ) ) ).

% concat_index_split_sound_snd_arg
thf(fact_854_concat__index__split__sound__fst__arg__aux,axiom,
    ! [I: nat,Xs: list_l1578326660578982846ctxt_a,O_idx: nat] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ O_idx @ I ) @ Xs ) ) @ ( plus_plus_nat @ ( size_s6994882312647770194ctxt_a @ Xs ) @ O_idx ) ) ) ).

% concat_index_split_sound_fst_arg_aux
thf(fact_855_concat__index__split__sound__fst__arg__aux,axiom,
    ! [I: nat,Xs: list_list_b,O_idx: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ ( concat_b @ Xs ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat_index_split_b @ ( product_Pair_nat_nat @ O_idx @ I ) @ Xs ) ) @ ( plus_plus_nat @ ( size_s420531428170919671list_b @ Xs ) @ O_idx ) ) ) ).

% concat_index_split_sound_fst_arg_aux
thf(fact_856_concat__index__split__sound__fst__arg__aux,axiom,
    ! [I: nat,Xs: list_list_nat,O_idx: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ O_idx @ I ) @ Xs ) ) @ ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ O_idx ) ) ) ).

% concat_index_split_sound_fst_arg_aux
thf(fact_857_concat__index__split__sound__fst__arg__aux,axiom,
    ! [I: nat,Xs: list_list_list_b,O_idx: nat] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ ( concat_list_b @ Xs ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat7499549088139563139list_b @ ( product_Pair_nat_nat @ O_idx @ I ) @ Xs ) ) @ ( plus_plus_nat @ ( size_s9106009106869804029list_b @ Xs ) @ O_idx ) ) ) ).

% concat_index_split_sound_fst_arg_aux
thf(fact_858_concat__index__split__sound__fst__arg__aux,axiom,
    ! [I: nat,Xs: list_l3298181151656792051at_nat,O_idx: nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ ( concat5616020275982107560at_nat @ Xs ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat1027118156116784357at_nat @ ( product_Pair_nat_nat @ O_idx @ I ) @ Xs ) ) @ ( plus_plus_nat @ ( size_s6878757106947045471at_nat @ Xs ) @ O_idx ) ) ) ).

% concat_index_split_sound_fst_arg_aux
thf(fact_859_concat__index__split__mono__first__arg,axiom,
    ! [I: nat,Xs: list_l1578326660578982846ctxt_a,O_idx: nat] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ord_less_eq_nat @ O_idx @ ( product_fst_nat_nat @ ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ O_idx @ I ) @ Xs ) ) ) ) ).

% concat_index_split_mono_first_arg
thf(fact_860_concat__index__split__mono__first__arg,axiom,
    ! [I: nat,Xs: list_list_b,O_idx: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ ( concat_b @ Xs ) ) )
     => ( ord_less_eq_nat @ O_idx @ ( product_fst_nat_nat @ ( concat_index_split_b @ ( product_Pair_nat_nat @ O_idx @ I ) @ Xs ) ) ) ) ).

% concat_index_split_mono_first_arg
thf(fact_861_concat__index__split__mono__first__arg,axiom,
    ! [I: nat,Xs: list_list_nat,O_idx: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ord_less_eq_nat @ O_idx @ ( product_fst_nat_nat @ ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ O_idx @ I ) @ Xs ) ) ) ) ).

% concat_index_split_mono_first_arg
thf(fact_862_concat__index__split__mono__first__arg,axiom,
    ! [I: nat,Xs: list_list_list_b,O_idx: nat] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ ( concat_list_b @ Xs ) ) )
     => ( ord_less_eq_nat @ O_idx @ ( product_fst_nat_nat @ ( concat7499549088139563139list_b @ ( product_Pair_nat_nat @ O_idx @ I ) @ Xs ) ) ) ) ).

% concat_index_split_mono_first_arg
thf(fact_863_concat__index__split__mono__first__arg,axiom,
    ! [I: nat,Xs: list_l3298181151656792051at_nat,O_idx: nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ ( concat5616020275982107560at_nat @ Xs ) ) )
     => ( ord_less_eq_nat @ O_idx @ ( product_fst_nat_nat @ ( concat1027118156116784357at_nat @ ( product_Pair_nat_nat @ O_idx @ I ) @ Xs ) ) ) ) ).

% concat_index_split_mono_first_arg
thf(fact_864_fstI,axiom,
    ! [X: produc8199716216217303280at_nat,Y3: nat > nat,Z4: nat] :
      ( ( X
        = ( produc72220940542539688at_nat @ Y3 @ Z4 ) )
     => ( ( produc6156676138143019412at_nat @ X )
        = Y3 ) ) ).

% fstI
thf(fact_865_fstI,axiom,
    ! [X: produc859450856879609959at_nat,Y3: product_prod_nat_nat,Z4: product_prod_nat_nat] :
      ( ( X
        = ( produc6161850002892822231at_nat @ Y3 @ Z4 ) )
     => ( ( produc3213797794245857475at_nat @ X )
        = Y3 ) ) ).

% fstI
thf(fact_866_fstI,axiom,
    ! [X: product_prod_nat_nat,Y3: nat,Z4: nat] :
      ( ( X
        = ( product_Pair_nat_nat @ Y3 @ Z4 ) )
     => ( ( product_fst_nat_nat @ X )
        = Y3 ) ) ).

% fstI
thf(fact_867_eq__fst__iff,axiom,
    ! [A: nat > nat,P2: produc8199716216217303280at_nat] :
      ( ( A
        = ( produc6156676138143019412at_nat @ P2 ) )
      = ( ? [B4: nat] :
            ( P2
            = ( produc72220940542539688at_nat @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_868_eq__fst__iff,axiom,
    ! [A: product_prod_nat_nat,P2: produc859450856879609959at_nat] :
      ( ( A
        = ( produc3213797794245857475at_nat @ P2 ) )
      = ( ? [B4: product_prod_nat_nat] :
            ( P2
            = ( produc6161850002892822231at_nat @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_869_eq__fst__iff,axiom,
    ! [A: nat,P2: product_prod_nat_nat] :
      ( ( A
        = ( product_fst_nat_nat @ P2 ) )
      = ( ? [B4: nat] :
            ( P2
            = ( product_Pair_nat_nat @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_870_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_871_num__holes__mctxt__of__term,axiom,
    ! [T: ground_gterm_a] :
      ( ( ground_num_gholes_a @ ( ground582559344758063963term_a @ T ) )
      = zero_zero_nat ) ).

% num_holes_mctxt_of_term
thf(fact_872_sum__list__1__E,axiom,
    ! [Xs: list_nat] :
      ( ( ( groups4561878855575611511st_nat @ Xs )
        = ( suc @ zero_zero_nat ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
           => ( ( ( nth_nat @ Xs @ I2 )
                = ( suc @ zero_zero_nat ) )
             => ~ ! [J3: nat] :
                    ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
                   => ( ( J3 != I2 )
                     => ( ( nth_nat @ Xs @ J3 )
                        = zero_zero_nat ) ) ) ) ) ) ).

% sum_list_1_E
thf(fact_873_add__less__cancel__left,axiom,
    ! [C3: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C3 @ A ) @ ( plus_plus_nat @ C3 @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_874_add__less__cancel__right,axiom,
    ! [A: nat,C3: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B @ C3 ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_875_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_876_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_877_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_less_eq
thf(fact_878_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_879_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_880_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_881_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel1
thf(fact_882_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel2
thf(fact_883_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_884_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_885_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_886_length__append,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
      ( ( size_s962548663205356226ctxt_a @ ( append5238750768297430211ctxt_a @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s962548663205356226ctxt_a @ Xs ) @ ( size_s962548663205356226ctxt_a @ Ys ) ) ) ).

% length_append
thf(fact_887_length__append,axiom,
    ! [Xs: list_b,Ys: list_b] :
      ( ( size_size_list_b @ ( append_b @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_b @ Xs ) @ ( size_size_list_b @ Ys ) ) ) ).

% length_append
thf(fact_888_length__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat @ ( append_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_append
thf(fact_889_length__append,axiom,
    ! [Xs: list_list_b,Ys: list_list_b] :
      ( ( size_s420531428170919671list_b @ ( append_list_b @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s420531428170919671list_b @ Xs ) @ ( size_s420531428170919671list_b @ Ys ) ) ) ).

% length_append
thf(fact_890_length__append,axiom,
    ! [Xs: list_P8469869581646625389at_nat,Ys: list_P8469869581646625389at_nat] :
      ( ( size_s3679842834875189465at_nat @ ( append8751754712269456642at_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s3679842834875189465at_nat @ Xs ) @ ( size_s3679842834875189465at_nat @ Ys ) ) ) ).

% length_append
thf(fact_891_prod_Ocollapse,axiom,
    ! [Prod: produc8199716216217303280at_nat] :
      ( ( produc72220940542539688at_nat @ ( produc6156676138143019412at_nat @ Prod ) @ ( produc1852801350702243542at_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_892_prod_Ocollapse,axiom,
    ! [Prod: produc859450856879609959at_nat] :
      ( ( produc6161850002892822231at_nat @ ( produc3213797794245857475at_nat @ Prod ) @ ( produc6408287024330202629at_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_893_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_894_sum__list__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( groups4561878855575611511st_nat @ ( append_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ).

% sum_list_append
thf(fact_895_list__update__beyond,axiom,
    ! [Xs: list_Ground_gmctxt_a,I: nat,X: ground_gmctxt_a] :
      ( ( ord_less_eq_nat @ ( size_s962548663205356226ctxt_a @ Xs ) @ I )
     => ( ( list_u7950812632479372336ctxt_a @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_896_list__update__beyond,axiom,
    ! [Xs: list_b,I: nat,X: b] :
      ( ( ord_less_eq_nat @ ( size_size_list_b @ Xs ) @ I )
     => ( ( list_update_b @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_897_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
     => ( ( list_update_nat @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_898_list__update__beyond,axiom,
    ! [Xs: list_list_b,I: nat,X: list_b] :
      ( ( ord_less_eq_nat @ ( size_s420531428170919671list_b @ Xs ) @ I )
     => ( ( list_update_list_b @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_899_list__update__beyond,axiom,
    ! [Xs: list_P8469869581646625389at_nat,I: nat,X: produc859450856879609959at_nat] :
      ( ( ord_less_eq_nat @ ( size_s3679842834875189465at_nat @ Xs ) @ I )
     => ( ( list_u5003261594476800725at_nat @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_900_nth__append__length__plus,axiom,
    ! [Xs: list_list_nat,Ys: list_list_nat,N: nat] :
      ( ( nth_list_nat @ ( append_list_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ N ) )
      = ( nth_list_nat @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_901_nth__append__length__plus,axiom,
    ! [Xs: list_l1578326660578982846ctxt_a,Ys: list_l1578326660578982846ctxt_a,N: nat] :
      ( ( nth_li6678718213161987943ctxt_a @ ( append272212692390751315ctxt_a @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s6994882312647770194ctxt_a @ Xs ) @ N ) )
      = ( nth_li6678718213161987943ctxt_a @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_902_nth__append__length__plus,axiom,
    ! [Xs: list_l3298181151656792051at_nat,Ys: list_l3298181151656792051at_nat,N: nat] :
      ( ( nth_li8805938130461222772at_nat @ ( append6271109094031990152at_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s6878757106947045471at_nat @ Xs ) @ N ) )
      = ( nth_li8805938130461222772at_nat @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_903_nth__append__length__plus,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,N: nat] :
      ( ( nth_Ground_gmctxt_a @ ( append5238750768297430211ctxt_a @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s962548663205356226ctxt_a @ Xs ) @ N ) )
      = ( nth_Ground_gmctxt_a @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_904_nth__append__length__plus,axiom,
    ! [Xs: list_b,Ys: list_b,N: nat] :
      ( ( nth_b @ ( append_b @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_b @ Xs ) @ N ) )
      = ( nth_b @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_905_nth__append__length__plus,axiom,
    ! [Xs: list_nat,Ys: list_nat,N: nat] :
      ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N ) )
      = ( nth_nat @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_906_nth__append__length__plus,axiom,
    ! [Xs: list_list_b,Ys: list_list_b,N: nat] :
      ( ( nth_list_b @ ( append_list_b @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s420531428170919671list_b @ Xs ) @ N ) )
      = ( nth_list_b @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_907_nth__append__length__plus,axiom,
    ! [Xs: list_P8469869581646625389at_nat,Ys: list_P8469869581646625389at_nat,N: nat] :
      ( ( nth_Pr6744343527793145070at_nat @ ( append8751754712269456642at_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s3679842834875189465at_nat @ Xs ) @ N ) )
      = ( nth_Pr6744343527793145070at_nat @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_908_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_nat @ M3 @ N3 )
         => ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_909_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ? [K3: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_910_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M4: nat,N2: nat] :
        ? [K4: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M4 @ K4 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_911_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_912_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_913_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% less_Suc_eq_le
thf(fact_914_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% le_less_Suc_eq
thf(fact_915_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_916_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_917_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_le_lessD
thf(fact_918_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P @ ( suc @ N3 ) )
                 => ( P @ N3 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_919_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P @ N3 )
                 => ( P @ ( suc @ N3 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_920_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ~ ! [Q: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M @ Q ) ) ) ) ).

% less_natE
thf(fact_921_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_le_eq
thf(fact_922_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_923_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C3: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C3 @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_924_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C3: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C3 @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_925_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_926_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_927_eq__snd__iff,axiom,
    ! [B: nat,P2: produc8199716216217303280at_nat] :
      ( ( B
        = ( produc1852801350702243542at_nat @ P2 ) )
      = ( ? [A5: nat > nat] :
            ( P2
            = ( produc72220940542539688at_nat @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_928_eq__snd__iff,axiom,
    ! [B: product_prod_nat_nat,P2: produc859450856879609959at_nat] :
      ( ( B
        = ( produc6408287024330202629at_nat @ P2 ) )
      = ( ? [A5: product_prod_nat_nat] :
            ( P2
            = ( produc6161850002892822231at_nat @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_929_eq__snd__iff,axiom,
    ! [B: nat,P2: product_prod_nat_nat] :
      ( ( B
        = ( product_snd_nat_nat @ P2 ) )
      = ( ? [A5: nat] :
            ( P2
            = ( product_Pair_nat_nat @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_930_sndI,axiom,
    ! [X: produc8199716216217303280at_nat,Y3: nat > nat,Z4: nat] :
      ( ( X
        = ( produc72220940542539688at_nat @ Y3 @ Z4 ) )
     => ( ( produc1852801350702243542at_nat @ X )
        = Z4 ) ) ).

% sndI
thf(fact_931_sndI,axiom,
    ! [X: produc859450856879609959at_nat,Y3: product_prod_nat_nat,Z4: product_prod_nat_nat] :
      ( ( X
        = ( produc6161850002892822231at_nat @ Y3 @ Z4 ) )
     => ( ( produc6408287024330202629at_nat @ X )
        = Z4 ) ) ).

% sndI
thf(fact_932_sndI,axiom,
    ! [X: product_prod_nat_nat,Y3: nat,Z4: nat] :
      ( ( X
        = ( product_Pair_nat_nat @ Y3 @ Z4 ) )
     => ( ( product_snd_nat_nat @ X )
        = Z4 ) ) ).

% sndI
thf(fact_933_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_934_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_935_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_936_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_937_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C3 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C3 ) ) ) ) ).

% add_strict_increasing
thf(fact_938_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C3 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C3 ) ) ) ) ).

% add_strict_increasing2
thf(fact_939_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K3 )
               => ~ ( P @ I4 ) )
            & ( P @ ( suc @ K3 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_940_snd__conv,axiom,
    ! [X1: nat > nat,X22: nat] :
      ( ( produc1852801350702243542at_nat @ ( produc72220940542539688at_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_941_snd__conv,axiom,
    ! [X1: product_prod_nat_nat,X22: product_prod_nat_nat] :
      ( ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_942_snd__conv,axiom,
    ! [X1: nat,X22: nat] :
      ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_943_snd__eqD,axiom,
    ! [X: nat > nat,Y3: nat,A: nat] :
      ( ( ( produc1852801350702243542at_nat @ ( produc72220940542539688at_nat @ X @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_944_snd__eqD,axiom,
    ! [X: product_prod_nat_nat,Y3: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_945_snd__eqD,axiom,
    ! [X: nat,Y3: nat,A: nat] :
      ( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_946_add__less__imp__less__right,axiom,
    ! [A: nat,C3: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B @ C3 ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_947_add__less__imp__less__left,axiom,
    ! [C3: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C3 @ A ) @ ( plus_plus_nat @ C3 @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_948_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C3: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B @ C3 ) ) ) ).

% add_strict_right_mono
thf(fact_949_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C3: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C3 @ A ) @ ( plus_plus_nat @ C3 @ B ) ) ) ).

% add_strict_left_mono
thf(fact_950_add__strict__mono,axiom,
    ! [A: nat,B: nat,C3: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C3 @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_951_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_952_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_953_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_954_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_955_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] :
            ( ( J
              = ( suc @ I2 ) )
           => ( P @ I2 ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ J )
             => ( ( P @ ( suc @ I2 ) )
               => ( P @ I2 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_956_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
       => ( ! [I2: nat,J2: nat,K3: nat] :
              ( ( ord_less_nat @ I2 @ J2 )
             => ( ( ord_less_nat @ J2 @ K3 )
               => ( ( P @ I2 @ J2 )
                 => ( ( P @ J2 @ K3 )
                   => ( P @ I2 @ K3 ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_957_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_958_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_less_SucD
thf(fact_959_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_960_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M )
      = ( ? [M5: nat] :
            ( ( M
              = ( suc @ M5 ) )
            & ( ord_less_nat @ N @ M5 ) ) ) ) ).

% Suc_less_eq2
thf(fact_961_All__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
           => ( P @ I3 ) ) )
      = ( ( P @ N )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
           => ( P @ I3 ) ) ) ) ).

% All_less_Suc
thf(fact_962_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_963_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( M = N ) ) ) ).

% less_Suc_eq
thf(fact_964_Ex__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
            & ( P @ I3 ) ) )
      = ( ( P @ N )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
            & ( P @ I3 ) ) ) ) ).

% Ex_less_Suc
thf(fact_965_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_966_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_967_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ( suc @ M )
         != N )
       => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).

% Suc_lessI
thf(fact_968_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I @ J2 )
           => ( K
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_969_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_970_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ I @ K )
     => ( ( K
         != ( suc @ I ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I @ J2 )
             => ( K
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_971_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% less_add_eq_less
thf(fact_972_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_less_add2
thf(fact_973_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_less_add1
thf(fact_974_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_975_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_976_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_977_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_978_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
     => ( ord_less_nat @ I @ K ) ) ).

% add_lessD1
thf(fact_979_inf__pigeonhole__principle,axiom,
    ! [N: nat,F: nat > nat > $o] :
      ( ! [K3: nat] :
        ? [I4: nat] :
          ( ( ord_less_nat @ I4 @ N )
          & ( F @ K3 @ I4 ) )
     => ? [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
          & ! [K2: nat] :
            ? [K5: nat] :
              ( ( ord_less_eq_nat @ K2 @ K5 )
              & ( F @ K5 @ I2 ) ) ) ) ).

% inf_pigeonhole_principle
thf(fact_980_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_nat @ I2 @ J2 )
         => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_981_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( M != N )
       => ( ord_less_nat @ M @ N ) ) ) ).

% le_neq_implies_less
thf(fact_982_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_nat @ M @ N )
        | ( M = N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_or_eq_imp_le
thf(fact_983_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M4: nat,N2: nat] :
          ( ( ord_less_nat @ M4 @ N2 )
          | ( M4 = N2 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_984_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_imp_le_nat
thf(fact_985_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M4: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M4 @ N2 )
          & ( M4 != N2 ) ) ) ) ).

% nat_less_le
thf(fact_986_prod_Oexhaust__sel,axiom,
    ! [Prod: produc8199716216217303280at_nat] :
      ( Prod
      = ( produc72220940542539688at_nat @ ( produc6156676138143019412at_nat @ Prod ) @ ( produc1852801350702243542at_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_987_prod_Oexhaust__sel,axiom,
    ! [Prod: produc859450856879609959at_nat] :
      ( Prod
      = ( produc6161850002892822231at_nat @ ( produc3213797794245857475at_nat @ Prod ) @ ( produc6408287024330202629at_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_988_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_989_surjective__pairing,axiom,
    ! [T: produc8199716216217303280at_nat] :
      ( T
      = ( produc72220940542539688at_nat @ ( produc6156676138143019412at_nat @ T ) @ ( produc1852801350702243542at_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_990_surjective__pairing,axiom,
    ! [T: produc859450856879609959at_nat] :
      ( T
      = ( produc6161850002892822231at_nat @ ( produc3213797794245857475at_nat @ T ) @ ( produc6408287024330202629at_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_991_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_992_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_993_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_994_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C ) )
           => ( C = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_995_pos__add__strict,axiom,
    ! [A: nat,B: nat,C3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C3 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C3 ) ) ) ) ).

% pos_add_strict
thf(fact_996_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_997_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_998_ex__Suc__conv,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
            & ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
            & ( P @ ( suc @ I3 ) ) ) ) ) ).

% ex_Suc_conv
thf(fact_999_all__Suc__conv,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
           => ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
           => ( P @ ( suc @ I3 ) ) ) ) ) ).

% all_Suc_conv
thf(fact_1000_all__less__two,axiom,
    ! [P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ ( suc @ zero_zero_nat ) ) )
           => ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        & ( P @ ( suc @ zero_zero_nat ) ) ) ) ).

% all_less_two
thf(fact_1001_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M4: nat] :
            ( N
            = ( suc @ M4 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_1002_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M3: nat] :
          ( N
          = ( suc @ M3 ) ) ) ).

% gr0_implies_Suc
thf(fact_1003_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( M = zero_zero_nat )
        | ? [J4: nat] :
            ( ( M
              = ( suc @ J4 ) )
            & ( ord_less_nat @ J4 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_1004_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K3: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K3 )
          & ( ( plus_plus_nat @ I @ K3 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_1005_remove__nth__sound__r,axiom,
    ! [N: nat,P2: nat,Xs: list_list_nat] :
      ( ( ord_less_eq_nat @ N @ P2 )
     => ( ( ord_less_nat @ P2 @ ( size_s3023201423986296836st_nat @ Xs ) )
       => ( ( nth_list_nat @ ( missin5133812807938250791st_nat @ N @ Xs ) @ P2 )
          = ( nth_list_nat @ Xs @ ( suc @ P2 ) ) ) ) ) ).

% remove_nth_sound_r
thf(fact_1006_remove__nth__sound__r,axiom,
    ! [N: nat,P2: nat,Xs: list_l1578326660578982846ctxt_a] :
      ( ( ord_less_eq_nat @ N @ P2 )
     => ( ( ord_less_nat @ P2 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
       => ( ( nth_li6678718213161987943ctxt_a @ ( missin1875899094540944245ctxt_a @ N @ Xs ) @ P2 )
          = ( nth_li6678718213161987943ctxt_a @ Xs @ ( suc @ P2 ) ) ) ) ) ).

% remove_nth_sound_r
thf(fact_1007_remove__nth__sound__r,axiom,
    ! [N: nat,P2: nat,Xs: list_l3298181151656792051at_nat] :
      ( ( ord_less_eq_nat @ N @ P2 )
     => ( ( ord_less_nat @ P2 @ ( size_s6878757106947045471at_nat @ Xs ) )
       => ( ( nth_li8805938130461222772at_nat @ ( missin7550826715306853094at_nat @ N @ Xs ) @ P2 )
          = ( nth_li8805938130461222772at_nat @ Xs @ ( suc @ P2 ) ) ) ) ) ).

% remove_nth_sound_r
thf(fact_1008_remove__nth__sound__r,axiom,
    ! [N: nat,P2: nat,Xs: list_Ground_gmctxt_a] :
      ( ( ord_less_eq_nat @ N @ P2 )
     => ( ( ord_less_nat @ P2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
       => ( ( nth_Ground_gmctxt_a @ ( missin7823264313509747941ctxt_a @ N @ Xs ) @ P2 )
          = ( nth_Ground_gmctxt_a @ Xs @ ( suc @ P2 ) ) ) ) ) ).

% remove_nth_sound_r
thf(fact_1009_remove__nth__sound__r,axiom,
    ! [N: nat,P2: nat,Xs: list_b] :
      ( ( ord_less_eq_nat @ N @ P2 )
     => ( ( ord_less_nat @ P2 @ ( size_size_list_b @ Xs ) )
       => ( ( nth_b @ ( missing_remove_nth_b @ N @ Xs ) @ P2 )
          = ( nth_b @ Xs @ ( suc @ P2 ) ) ) ) ) ).

% remove_nth_sound_r
thf(fact_1010_remove__nth__sound__r,axiom,
    ! [N: nat,P2: nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ N @ P2 )
     => ( ( ord_less_nat @ P2 @ ( size_size_list_nat @ Xs ) )
       => ( ( nth_nat @ ( missin7175274867594579095th_nat @ N @ Xs ) @ P2 )
          = ( nth_nat @ Xs @ ( suc @ P2 ) ) ) ) ) ).

% remove_nth_sound_r
thf(fact_1011_remove__nth__sound__r,axiom,
    ! [N: nat,P2: nat,Xs: list_list_b] :
      ( ( ord_less_eq_nat @ N @ P2 )
     => ( ( ord_less_nat @ P2 @ ( size_s420531428170919671list_b @ Xs ) )
       => ( ( nth_list_b @ ( missin8041159180453292414list_b @ N @ Xs ) @ P2 )
          = ( nth_list_b @ Xs @ ( suc @ P2 ) ) ) ) ) ).

% remove_nth_sound_r
thf(fact_1012_remove__nth__sound__r,axiom,
    ! [N: nat,P2: nat,Xs: list_P8469869581646625389at_nat] :
      ( ( ord_less_eq_nat @ N @ P2 )
     => ( ( ord_less_nat @ P2 @ ( size_s3679842834875189465at_nat @ Xs ) )
       => ( ( nth_Pr6744343527793145070at_nat @ ( missin1453772821081764192at_nat @ N @ Xs ) @ P2 )
          = ( nth_Pr6744343527793145070at_nat @ Xs @ ( suc @ P2 ) ) ) ) ) ).

% remove_nth_sound_r
thf(fact_1013_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K3 )
               => ~ ( P @ I4 ) )
            & ( P @ K3 ) ) ) ) ).

% ex_least_nat_le
thf(fact_1014_member__le__sum__list,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ord_less_eq_nat @ X @ ( groups4561878855575611511st_nat @ Xs ) ) ) ).

% member_le_sum_list
thf(fact_1015_remove__nth__id,axiom,
    ! [Xs: list_Ground_gmctxt_a,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s962548663205356226ctxt_a @ Xs ) @ N )
     => ( ( missin7823264313509747941ctxt_a @ N @ Xs )
        = Xs ) ) ).

% remove_nth_id
thf(fact_1016_remove__nth__id,axiom,
    ! [Xs: list_b,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_b @ Xs ) @ N )
     => ( ( missing_remove_nth_b @ N @ Xs )
        = Xs ) ) ).

% remove_nth_id
thf(fact_1017_remove__nth__id,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
     => ( ( missin7175274867594579095th_nat @ N @ Xs )
        = Xs ) ) ).

% remove_nth_id
thf(fact_1018_remove__nth__id,axiom,
    ! [Xs: list_list_b,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s420531428170919671list_b @ Xs ) @ N )
     => ( ( missin8041159180453292414list_b @ N @ Xs )
        = Xs ) ) ).

% remove_nth_id
thf(fact_1019_remove__nth__id,axiom,
    ! [Xs: list_P8469869581646625389at_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s3679842834875189465at_nat @ Xs ) @ N )
     => ( ( missin1453772821081764192at_nat @ N @ Xs )
        = Xs ) ) ).

% remove_nth_id
thf(fact_1020_adjust__idx__def,axiom,
    ( missing_adjust_idx
    = ( ^ [I3: nat,J4: nat] : ( if_nat @ ( ord_less_nat @ J4 @ I3 ) @ J4 @ ( suc @ J4 ) ) ) ) ).

% adjust_idx_def
thf(fact_1021_Groups__List_Osum__list__nonneg,axiom,
    ! [Xs: list_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ord_less_eq_nat @ zero_zero_nat @ X3 ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4561878855575611511st_nat @ Xs ) ) ) ).

% Groups_List.sum_list_nonneg
thf(fact_1022_sum__list__nonneg__eq__0__iff,axiom,
    ! [Xs: list_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ord_less_eq_nat @ zero_zero_nat @ X3 ) )
     => ( ( ( groups4561878855575611511st_nat @ Xs )
          = zero_zero_nat )
        = ( ! [X2: nat] :
              ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
             => ( X2 = zero_zero_nat ) ) ) ) ) ).

% sum_list_nonneg_eq_0_iff
thf(fact_1023_sum__list__nonpos,axiom,
    ! [Xs: list_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ord_less_eq_nat @ X3 @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs ) @ zero_zero_nat ) ) ).

% sum_list_nonpos
thf(fact_1024_remove__nth__len,axiom,
    ! [I: nat,Xs: list_Ground_gmctxt_a] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( suc @ ( size_s962548663205356226ctxt_a @ ( missin7823264313509747941ctxt_a @ I @ Xs ) ) ) ) ) ).

% remove_nth_len
thf(fact_1025_remove__nth__len,axiom,
    ! [I: nat,Xs: list_b] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
     => ( ( size_size_list_b @ Xs )
        = ( suc @ ( size_size_list_b @ ( missing_remove_nth_b @ I @ Xs ) ) ) ) ) ).

% remove_nth_len
thf(fact_1026_remove__nth__len,axiom,
    ! [I: nat,Xs: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( size_size_list_nat @ Xs )
        = ( suc @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ I @ Xs ) ) ) ) ) ).

% remove_nth_len
thf(fact_1027_remove__nth__len,axiom,
    ! [I: nat,Xs: list_list_b] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ Xs ) )
     => ( ( size_s420531428170919671list_b @ Xs )
        = ( suc @ ( size_s420531428170919671list_b @ ( missin8041159180453292414list_b @ I @ Xs ) ) ) ) ) ).

% remove_nth_len
thf(fact_1028_remove__nth__len,axiom,
    ! [I: nat,Xs: list_P8469869581646625389at_nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ Xs ) )
     => ( ( size_s3679842834875189465at_nat @ Xs )
        = ( suc @ ( size_s3679842834875189465at_nat @ ( missin1453772821081764192at_nat @ I @ Xs ) ) ) ) ) ).

% remove_nth_len
thf(fact_1029_subseqs__length__simple,axiom,
    ! [B: list_Ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
      ( ( member2714729351964853061ctxt_a @ B @ ( set_li3057194059929822169ctxt_a @ ( subseq4078942962620312969ctxt_a @ Xs ) ) )
     => ( ord_less_eq_nat @ ( size_s962548663205356226ctxt_a @ B ) @ ( size_s962548663205356226ctxt_a @ Xs ) ) ) ).

% subseqs_length_simple
thf(fact_1030_subseqs__length__simple,axiom,
    ! [B: list_b,Xs: list_b] :
      ( ( member_list_b @ B @ ( set_list_b2 @ ( subseqs_b @ Xs ) ) )
     => ( ord_less_eq_nat @ ( size_size_list_b @ B ) @ ( size_size_list_b @ Xs ) ) ) ).

% subseqs_length_simple
thf(fact_1031_subseqs__length__simple,axiom,
    ! [B: list_nat,Xs: list_nat] :
      ( ( member_list_nat @ B @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
     => ( ord_less_eq_nat @ ( size_size_list_nat @ B ) @ ( size_size_list_nat @ Xs ) ) ) ).

% subseqs_length_simple
thf(fact_1032_subseqs__length__simple,axiom,
    ! [B: list_list_b,Xs: list_list_b] :
      ( ( member_list_list_b @ B @ ( set_list_list_b2 @ ( subseqs_list_b @ Xs ) ) )
     => ( ord_less_eq_nat @ ( size_s420531428170919671list_b @ B ) @ ( size_s420531428170919671list_b @ Xs ) ) ) ).

% subseqs_length_simple
thf(fact_1033_subseqs__length__simple,axiom,
    ! [B: list_P8469869581646625389at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member3799944675974059798at_nat @ B @ ( set_li7366234323595830146at_nat @ ( subseq4464984717476426172at_nat @ Xs ) ) )
     => ( ord_less_eq_nat @ ( size_s3679842834875189465at_nat @ B ) @ ( size_s3679842834875189465at_nat @ Xs ) ) ) ).

% subseqs_length_simple
thf(fact_1034_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_1035_sum__list__mono2,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
           => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) )
       => ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ) ).

% sum_list_mono2
thf(fact_1036_nth__remove__nth__conv,axiom,
    ! [I: nat,N: nat,Xs: list_list_nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ ( missin5133812807938250791st_nat @ N @ Xs ) ) )
     => ( ( nth_list_nat @ ( missin5133812807938250791st_nat @ N @ Xs ) @ I )
        = ( nth_list_nat @ Xs @ ( if_nat @ ( ord_less_nat @ I @ N ) @ I @ ( suc @ I ) ) ) ) ) ).

% nth_remove_nth_conv
thf(fact_1037_nth__remove__nth__conv,axiom,
    ! [I: nat,N: nat,Xs: list_l1578326660578982846ctxt_a] :
      ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ ( missin1875899094540944245ctxt_a @ N @ Xs ) ) )
     => ( ( nth_li6678718213161987943ctxt_a @ ( missin1875899094540944245ctxt_a @ N @ Xs ) @ I )
        = ( nth_li6678718213161987943ctxt_a @ Xs @ ( if_nat @ ( ord_less_nat @ I @ N ) @ I @ ( suc @ I ) ) ) ) ) ).

% nth_remove_nth_conv
thf(fact_1038_nth__remove__nth__conv,axiom,
    ! [I: nat,N: nat,Xs: list_l3298181151656792051at_nat] :
      ( ( ord_less_nat @ I @ ( size_s6878757106947045471at_nat @ ( missin7550826715306853094at_nat @ N @ Xs ) ) )
     => ( ( nth_li8805938130461222772at_nat @ ( missin7550826715306853094at_nat @ N @ Xs ) @ I )
        = ( nth_li8805938130461222772at_nat @ Xs @ ( if_nat @ ( ord_less_nat @ I @ N ) @ I @ ( suc @ I ) ) ) ) ) ).

% nth_remove_nth_conv
thf(fact_1039_nth__remove__nth__conv,axiom,
    ! [I: nat,N: nat,Xs: list_Ground_gmctxt_a] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ ( missin7823264313509747941ctxt_a @ N @ Xs ) ) )
     => ( ( nth_Ground_gmctxt_a @ ( missin7823264313509747941ctxt_a @ N @ Xs ) @ I )
        = ( nth_Ground_gmctxt_a @ Xs @ ( if_nat @ ( ord_less_nat @ I @ N ) @ I @ ( suc @ I ) ) ) ) ) ).

% nth_remove_nth_conv
thf(fact_1040_nth__remove__nth__conv,axiom,
    ! [I: nat,N: nat,Xs: list_b] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ ( missing_remove_nth_b @ N @ Xs ) ) )
     => ( ( nth_b @ ( missing_remove_nth_b @ N @ Xs ) @ I )
        = ( nth_b @ Xs @ ( if_nat @ ( ord_less_nat @ I @ N ) @ I @ ( suc @ I ) ) ) ) ) ).

% nth_remove_nth_conv
thf(fact_1041_nth__remove__nth__conv,axiom,
    ! [I: nat,N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ N @ Xs ) ) )
     => ( ( nth_nat @ ( missin7175274867594579095th_nat @ N @ Xs ) @ I )
        = ( nth_nat @ Xs @ ( if_nat @ ( ord_less_nat @ I @ N ) @ I @ ( suc @ I ) ) ) ) ) ).

% nth_remove_nth_conv
thf(fact_1042_nth__remove__nth__conv,axiom,
    ! [I: nat,N: nat,Xs: list_list_b] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ ( missin8041159180453292414list_b @ N @ Xs ) ) )
     => ( ( nth_list_b @ ( missin8041159180453292414list_b @ N @ Xs ) @ I )
        = ( nth_list_b @ Xs @ ( if_nat @ ( ord_less_nat @ I @ N ) @ I @ ( suc @ I ) ) ) ) ) ).

% nth_remove_nth_conv
thf(fact_1043_nth__remove__nth__conv,axiom,
    ! [I: nat,N: nat,Xs: list_P8469869581646625389at_nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ ( missin1453772821081764192at_nat @ N @ Xs ) ) )
     => ( ( nth_Pr6744343527793145070at_nat @ ( missin1453772821081764192at_nat @ N @ Xs ) @ I )
        = ( nth_Pr6744343527793145070at_nat @ Xs @ ( if_nat @ ( ord_less_nat @ I @ N ) @ I @ ( suc @ I ) ) ) ) ) ).

% nth_remove_nth_conv
thf(fact_1044_chain__imp__trancl,axiom,
    ! [S2: nat > product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,I: nat,J: nat] :
      ( ! [I2: nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( S2 @ I2 ) @ ( S2 @ ( suc @ I2 ) ) ) @ R2 )
     => ( ( ord_less_nat @ I @ J )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( S2 @ I ) @ ( S2 @ J ) ) @ ( transi243908449541399842at_nat @ R2 ) ) ) ) ).

% chain_imp_trancl
thf(fact_1045_chain__imp__trancl,axiom,
    ! [S2: nat > nat,R2: set_Pr1261947904930325089at_nat,I: nat,J: nat] :
      ( ! [I2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( S2 @ I2 ) @ ( S2 @ ( suc @ I2 ) ) ) @ R2 )
     => ( ( ord_less_nat @ I @ J )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( S2 @ I ) @ ( S2 @ J ) ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).

% chain_imp_trancl
thf(fact_1046_exI__realizer,axiom,
    ! [P: nat > ( nat > nat ) > $o,Y3: nat,X: nat > nat] :
      ( ( P @ Y3 @ X )
     => ( P @ ( produc1852801350702243542at_nat @ ( produc72220940542539688at_nat @ X @ Y3 ) ) @ ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ X @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_1047_exI__realizer,axiom,
    ! [P: product_prod_nat_nat > product_prod_nat_nat > $o,Y3: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( P @ Y3 @ X )
     => ( P @ ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) ) @ ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_1048_exI__realizer,axiom,
    ! [P: nat > nat > $o,Y3: nat,X: nat] :
      ( ( P @ Y3 @ X )
     => ( P @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y3 ) ) @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_1049_conjI__realizer,axiom,
    ! [P: ( nat > nat ) > $o,P2: nat > nat,Q2: nat > $o,Q3: nat] :
      ( ( P @ P2 )
     => ( ( Q2 @ Q3 )
       => ( ( P @ ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ P2 @ Q3 ) ) )
          & ( Q2 @ ( produc1852801350702243542at_nat @ ( produc72220940542539688at_nat @ P2 @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_1050_conjI__realizer,axiom,
    ! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat,Q2: product_prod_nat_nat > $o,Q3: product_prod_nat_nat] :
      ( ( P @ P2 )
     => ( ( Q2 @ Q3 )
       => ( ( P @ ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ P2 @ Q3 ) ) )
          & ( Q2 @ ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ P2 @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_1051_conjI__realizer,axiom,
    ! [P: nat > $o,P2: nat,Q2: nat > $o,Q3: nat] :
      ( ( P @ P2 )
     => ( ( Q2 @ Q3 )
       => ( ( P @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P2 @ Q3 ) ) )
          & ( Q2 @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P2 @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_1052_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: ( nat > nat ) > nat > $o,X: nat > nat,Y3: nat,A: produc8199716216217303280at_nat] :
      ( ( P @ X @ Y3 )
     => ( ( A
          = ( produc72220940542539688at_nat @ X @ Y3 ) )
       => ( P @ ( produc6156676138143019412at_nat @ A ) @ ( produc1852801350702243542at_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1053_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat,Y3: product_prod_nat_nat,A: produc859450856879609959at_nat] :
      ( ( P @ X @ Y3 )
     => ( ( A
          = ( produc6161850002892822231at_nat @ X @ Y3 ) )
       => ( P @ ( produc3213797794245857475at_nat @ A ) @ ( produc6408287024330202629at_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1054_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: nat > nat > $o,X: nat,Y3: nat,A: product_prod_nat_nat] :
      ( ( P @ X @ Y3 )
     => ( ( A
          = ( product_Pair_nat_nat @ X @ Y3 ) )
       => ( P @ ( product_fst_nat_nat @ A ) @ ( product_snd_nat_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1055_concat__index__split__sound__snd__arg__aux,axiom,
    ! [I: nat,Xs: list_l1578326660578982846ctxt_a,N: nat] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ord_less_nat @ ( product_snd_nat_nat @ ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ N @ I ) @ Xs ) ) @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ ( minus_minus_nat @ ( product_fst_nat_nat @ ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ N @ I ) @ Xs ) ) @ N ) ) ) ) ) ).

% concat_index_split_sound_snd_arg_aux
thf(fact_1056_concat__index__split__sound__snd__arg__aux,axiom,
    ! [I: nat,Xs: list_list_b,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ ( concat_b @ Xs ) ) )
     => ( ord_less_nat @ ( product_snd_nat_nat @ ( concat_index_split_b @ ( product_Pair_nat_nat @ N @ I ) @ Xs ) ) @ ( size_size_list_b @ ( nth_list_b @ Xs @ ( minus_minus_nat @ ( product_fst_nat_nat @ ( concat_index_split_b @ ( product_Pair_nat_nat @ N @ I ) @ Xs ) ) @ N ) ) ) ) ) ).

% concat_index_split_sound_snd_arg_aux
thf(fact_1057_concat__index__split__sound__snd__arg__aux,axiom,
    ! [I: nat,Xs: list_list_nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ord_less_nat @ ( product_snd_nat_nat @ ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I ) @ Xs ) ) @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ ( minus_minus_nat @ ( product_fst_nat_nat @ ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I ) @ Xs ) ) @ N ) ) ) ) ) ).

% concat_index_split_sound_snd_arg_aux
thf(fact_1058_concat__index__split__sound__snd__arg__aux,axiom,
    ! [I: nat,Xs: list_list_list_b,N: nat] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ ( concat_list_b @ Xs ) ) )
     => ( ord_less_nat @ ( product_snd_nat_nat @ ( concat7499549088139563139list_b @ ( product_Pair_nat_nat @ N @ I ) @ Xs ) ) @ ( size_s420531428170919671list_b @ ( nth_list_list_b @ Xs @ ( minus_minus_nat @ ( product_fst_nat_nat @ ( concat7499549088139563139list_b @ ( product_Pair_nat_nat @ N @ I ) @ Xs ) ) @ N ) ) ) ) ) ).

% concat_index_split_sound_snd_arg_aux
thf(fact_1059_concat__index__split__sound__snd__arg__aux,axiom,
    ! [I: nat,Xs: list_l3298181151656792051at_nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ ( concat5616020275982107560at_nat @ Xs ) ) )
     => ( ord_less_nat @ ( product_snd_nat_nat @ ( concat1027118156116784357at_nat @ ( product_Pair_nat_nat @ N @ I ) @ Xs ) ) @ ( size_s3679842834875189465at_nat @ ( nth_li8805938130461222772at_nat @ Xs @ ( minus_minus_nat @ ( product_fst_nat_nat @ ( concat1027118156116784357at_nat @ ( product_Pair_nat_nat @ N @ I ) @ Xs ) ) @ N ) ) ) ) ) ).

% concat_index_split_sound_snd_arg_aux
thf(fact_1060_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
      = ( ord_less_nat @ M @ N ) ) ).

% zero_less_diff
thf(fact_1061_Suc__pred,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% Suc_pred
thf(fact_1062_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ord_less_nat @ M @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_1063_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).

% less_imp_diff_less
thf(fact_1064_subset__code_I1_J,axiom,
    ! [Xs: list_b,B5: set_b] :
      ( ( ord_less_eq_set_b @ ( set_b2 @ Xs ) @ B5 )
      = ( ! [X2: b] :
            ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
           => ( member_b @ X2 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_1065_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B5 )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
           => ( member_nat @ X2 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_1066_subset__code_I1_J,axiom,
    ! [Xs: list_Ground_gmctxt_a,B5: set_Ground_gmctxt_a] :
      ( ( ord_le4032861497522155828ctxt_a @ ( set_Ground_gmctxt_a2 @ Xs ) @ B5 )
      = ( ! [X2: ground_gmctxt_a] :
            ( ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ Xs ) )
           => ( member3306386364193653941ctxt_a @ X2 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_1067_subset__code_I1_J,axiom,
    ! [Xs: list_P8469869581646625389at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( set_Pr5518436109238095868at_nat @ Xs ) @ B5 )
      = ( ! [X2: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X2 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
           => ( member8206827879206165904at_nat @ X2 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_1068_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_1069_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_1070_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).

% diff_less
thf(fact_1071_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M @ N ) ) ) ) ).

% less_diff_iff
thf(fact_1072_diff__less__mono,axiom,
    ! [A: nat,B: nat,C3: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C3 @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C3 ) @ ( minus_minus_nat @ B @ C3 ) ) ) ) ).

% diff_less_mono
thf(fact_1073_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_1074_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_1075_set__update__subsetI,axiom,
    ! [Xs: list_b,A2: set_b,X: b,I: nat] :
      ( ( ord_less_eq_set_b @ ( set_b2 @ Xs ) @ A2 )
     => ( ( member_b @ X @ A2 )
       => ( ord_less_eq_set_b @ ( set_b2 @ ( list_update_b @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_1076_set__update__subsetI,axiom,
    ! [Xs: list_nat,A2: set_nat,X: nat,I: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
     => ( ( member_nat @ X @ A2 )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_1077_set__update__subsetI,axiom,
    ! [Xs: list_Ground_gmctxt_a,A2: set_Ground_gmctxt_a,X: ground_gmctxt_a,I: nat] :
      ( ( ord_le4032861497522155828ctxt_a @ ( set_Ground_gmctxt_a2 @ Xs ) @ A2 )
     => ( ( member3306386364193653941ctxt_a @ X @ A2 )
       => ( ord_le4032861497522155828ctxt_a @ ( set_Ground_gmctxt_a2 @ ( list_u7950812632479372336ctxt_a @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_1078_set__update__subsetI,axiom,
    ! [Xs: list_P8469869581646625389at_nat,A2: set_Pr8693737435421807431at_nat,X: produc859450856879609959at_nat,I: nat] :
      ( ( ord_le3000389064537975527at_nat @ ( set_Pr5518436109238095868at_nat @ Xs ) @ A2 )
     => ( ( member8206827879206165904at_nat @ X @ A2 )
       => ( ord_le3000389064537975527at_nat @ ( set_Pr5518436109238095868at_nat @ ( list_u5003261594476800725at_nat @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_1079_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).

% diff_Suc_less
thf(fact_1080_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D2: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D2 ) )
                & ~ ( P @ D2 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_1081_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P @ zero_zero_nat ) )
        & ! [D2: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D2 ) )
           => ( P @ D2 ) ) ) ) ).

% nat_diff_split
thf(fact_1082_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_1083_nth__append,axiom,
    ! [N: nat,Xs: list_list_nat,Ys: list_list_nat] :
      ( ( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
       => ( ( nth_list_nat @ ( append_list_nat @ Xs @ Ys ) @ N )
          = ( nth_list_nat @ Xs @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
       => ( ( nth_list_nat @ ( append_list_nat @ Xs @ Ys ) @ N )
          = ( nth_list_nat @ Ys @ ( minus_minus_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ) ) ) ).

% nth_append
thf(fact_1084_nth__append,axiom,
    ! [N: nat,Xs: list_l1578326660578982846ctxt_a,Ys: list_l1578326660578982846ctxt_a] :
      ( ( ( ord_less_nat @ N @ ( size_s6994882312647770194ctxt_a @ Xs ) )
       => ( ( nth_li6678718213161987943ctxt_a @ ( append272212692390751315ctxt_a @ Xs @ Ys ) @ N )
          = ( nth_li6678718213161987943ctxt_a @ Xs @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s6994882312647770194ctxt_a @ Xs ) )
       => ( ( nth_li6678718213161987943ctxt_a @ ( append272212692390751315ctxt_a @ Xs @ Ys ) @ N )
          = ( nth_li6678718213161987943ctxt_a @ Ys @ ( minus_minus_nat @ N @ ( size_s6994882312647770194ctxt_a @ Xs ) ) ) ) ) ) ).

% nth_append
thf(fact_1085_nth__append,axiom,
    ! [N: nat,Xs: list_l3298181151656792051at_nat,Ys: list_l3298181151656792051at_nat] :
      ( ( ( ord_less_nat @ N @ ( size_s6878757106947045471at_nat @ Xs ) )
       => ( ( nth_li8805938130461222772at_nat @ ( append6271109094031990152at_nat @ Xs @ Ys ) @ N )
          = ( nth_li8805938130461222772at_nat @ Xs @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s6878757106947045471at_nat @ Xs ) )
       => ( ( nth_li8805938130461222772at_nat @ ( append6271109094031990152at_nat @ Xs @ Ys ) @ N )
          = ( nth_li8805938130461222772at_nat @ Ys @ ( minus_minus_nat @ N @ ( size_s6878757106947045471at_nat @ Xs ) ) ) ) ) ) ).

% nth_append
thf(fact_1086_nth__append,axiom,
    ! [N: nat,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
      ( ( ( ord_less_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) )
       => ( ( nth_Ground_gmctxt_a @ ( append5238750768297430211ctxt_a @ Xs @ Ys ) @ N )
          = ( nth_Ground_gmctxt_a @ Xs @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) )
       => ( ( nth_Ground_gmctxt_a @ ( append5238750768297430211ctxt_a @ Xs @ Ys ) @ N )
          = ( nth_Ground_gmctxt_a @ Ys @ ( minus_minus_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) ) ) ) ) ) ).

% nth_append
thf(fact_1087_nth__append,axiom,
    ! [N: nat,Xs: list_b,Ys: list_b] :
      ( ( ( ord_less_nat @ N @ ( size_size_list_b @ Xs ) )
       => ( ( nth_b @ ( append_b @ Xs @ Ys ) @ N )
          = ( nth_b @ Xs @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_size_list_b @ Xs ) )
       => ( ( nth_b @ ( append_b @ Xs @ Ys ) @ N )
          = ( nth_b @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_b @ Xs ) ) ) ) ) ) ).

% nth_append
thf(fact_1088_nth__append,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
          = ( nth_nat @ Xs @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
          = ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).

% nth_append
thf(fact_1089_nth__append,axiom,
    ! [N: nat,Xs: list_list_b,Ys: list_list_b] :
      ( ( ( ord_less_nat @ N @ ( size_s420531428170919671list_b @ Xs ) )
       => ( ( nth_list_b @ ( append_list_b @ Xs @ Ys ) @ N )
          = ( nth_list_b @ Xs @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s420531428170919671list_b @ Xs ) )
       => ( ( nth_list_b @ ( append_list_b @ Xs @ Ys ) @ N )
          = ( nth_list_b @ Ys @ ( minus_minus_nat @ N @ ( size_s420531428170919671list_b @ Xs ) ) ) ) ) ) ).

% nth_append
thf(fact_1090_nth__append,axiom,
    ! [N: nat,Xs: list_P8469869581646625389at_nat,Ys: list_P8469869581646625389at_nat] :
      ( ( ( ord_less_nat @ N @ ( size_s3679842834875189465at_nat @ Xs ) )
       => ( ( nth_Pr6744343527793145070at_nat @ ( append8751754712269456642at_nat @ Xs @ Ys ) @ N )
          = ( nth_Pr6744343527793145070at_nat @ Xs @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s3679842834875189465at_nat @ Xs ) )
       => ( ( nth_Pr6744343527793145070at_nat @ ( append8751754712269456642at_nat @ Xs @ Ys ) @ N )
          = ( nth_Pr6744343527793145070at_nat @ Ys @ ( minus_minus_nat @ N @ ( size_s3679842834875189465at_nat @ Xs ) ) ) ) ) ) ).

% nth_append
thf(fact_1091_list__update__append,axiom,
    ! [N: nat,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,X: ground_gmctxt_a] :
      ( ( ( ord_less_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) )
       => ( ( list_u7950812632479372336ctxt_a @ ( append5238750768297430211ctxt_a @ Xs @ Ys ) @ N @ X )
          = ( append5238750768297430211ctxt_a @ ( list_u7950812632479372336ctxt_a @ Xs @ N @ X ) @ Ys ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) )
       => ( ( list_u7950812632479372336ctxt_a @ ( append5238750768297430211ctxt_a @ Xs @ Ys ) @ N @ X )
          = ( append5238750768297430211ctxt_a @ Xs @ ( list_u7950812632479372336ctxt_a @ Ys @ ( minus_minus_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) ) @ X ) ) ) ) ) ).

% list_update_append
thf(fact_1092_list__update__append,axiom,
    ! [N: nat,Xs: list_b,Ys: list_b,X: b] :
      ( ( ( ord_less_nat @ N @ ( size_size_list_b @ Xs ) )
       => ( ( list_update_b @ ( append_b @ Xs @ Ys ) @ N @ X )
          = ( append_b @ ( list_update_b @ Xs @ N @ X ) @ Ys ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_size_list_b @ Xs ) )
       => ( ( list_update_b @ ( append_b @ Xs @ Ys ) @ N @ X )
          = ( append_b @ Xs @ ( list_update_b @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_b @ Xs ) ) @ X ) ) ) ) ) ).

% list_update_append
thf(fact_1093_list__update__append,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat,X: nat] :
      ( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X )
          = ( append_nat @ ( list_update_nat @ Xs @ N @ X ) @ Ys ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X )
          = ( append_nat @ Xs @ ( list_update_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ X ) ) ) ) ) ).

% list_update_append
thf(fact_1094_list__update__append,axiom,
    ! [N: nat,Xs: list_list_b,Ys: list_list_b,X: list_b] :
      ( ( ( ord_less_nat @ N @ ( size_s420531428170919671list_b @ Xs ) )
       => ( ( list_update_list_b @ ( append_list_b @ Xs @ Ys ) @ N @ X )
          = ( append_list_b @ ( list_update_list_b @ Xs @ N @ X ) @ Ys ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s420531428170919671list_b @ Xs ) )
       => ( ( list_update_list_b @ ( append_list_b @ Xs @ Ys ) @ N @ X )
          = ( append_list_b @ Xs @ ( list_update_list_b @ Ys @ ( minus_minus_nat @ N @ ( size_s420531428170919671list_b @ Xs ) ) @ X ) ) ) ) ) ).

% list_update_append
thf(fact_1095_list__update__append,axiom,
    ! [N: nat,Xs: list_P8469869581646625389at_nat,Ys: list_P8469869581646625389at_nat,X: produc859450856879609959at_nat] :
      ( ( ( ord_less_nat @ N @ ( size_s3679842834875189465at_nat @ Xs ) )
       => ( ( list_u5003261594476800725at_nat @ ( append8751754712269456642at_nat @ Xs @ Ys ) @ N @ X )
          = ( append8751754712269456642at_nat @ ( list_u5003261594476800725at_nat @ Xs @ N @ X ) @ Ys ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s3679842834875189465at_nat @ Xs ) )
       => ( ( list_u5003261594476800725at_nat @ ( append8751754712269456642at_nat @ Xs @ Ys ) @ N @ X )
          = ( append8751754712269456642at_nat @ Xs @ ( list_u5003261594476800725at_nat @ Ys @ ( minus_minus_nat @ N @ ( size_s3679842834875189465at_nat @ Xs ) ) @ X ) ) ) ) ) ).

% list_update_append
thf(fact_1096_set__list__subset__eq__nth__conv,axiom,
    ! [Xs: list_list_nat,A2: set_list_nat] :
      ( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A2 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( member_list_nat @ ( nth_list_nat @ Xs @ I3 ) @ A2 ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_1097_set__list__subset__eq__nth__conv,axiom,
    ! [Xs: list_l1578326660578982846ctxt_a,A2: set_li3075106635084045668ctxt_a] :
      ( ( ord_le9074214841948581316ctxt_a @ ( set_li3057194059929822169ctxt_a @ Xs ) @ A2 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
           => ( member2714729351964853061ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I3 ) @ A2 ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_1098_set__list__subset__eq__nth__conv,axiom,
    ! [Xs: list_l3298181151656792051at_nat,A2: set_li3197816953174176717at_nat] :
      ( ( ord_le5602213209262422893at_nat @ ( set_li7366234323595830146at_nat @ Xs ) @ A2 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6878757106947045471at_nat @ Xs ) )
           => ( member3799944675974059798at_nat @ ( nth_li8805938130461222772at_nat @ Xs @ I3 ) @ A2 ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_1099_set__list__subset__eq__nth__conv,axiom,
    ! [Xs: list_Ground_gmctxt_a,A2: set_Ground_gmctxt_a] :
      ( ( ord_le4032861497522155828ctxt_a @ ( set_Ground_gmctxt_a2 @ Xs ) @ A2 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Xs ) )
           => ( member3306386364193653941ctxt_a @ ( nth_Ground_gmctxt_a @ Xs @ I3 ) @ A2 ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_1100_set__list__subset__eq__nth__conv,axiom,
    ! [Xs: list_b,A2: set_b] :
      ( ( ord_less_eq_set_b @ ( set_b2 @ Xs ) @ A2 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Xs ) )
           => ( member_b @ ( nth_b @ Xs @ I3 ) @ A2 ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_1101_set__list__subset__eq__nth__conv,axiom,
    ! [Xs: list_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
           => ( member_nat @ ( nth_nat @ Xs @ I3 ) @ A2 ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_1102_set__list__subset__eq__nth__conv,axiom,
    ! [Xs: list_list_b,A2: set_list_b] :
      ( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ Xs ) @ A2 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s420531428170919671list_b @ Xs ) )
           => ( member_list_b @ ( nth_list_b @ Xs @ I3 ) @ A2 ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_1103_set__list__subset__eq__nth__conv,axiom,
    ! [Xs: list_P8469869581646625389at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( set_Pr5518436109238095868at_nat @ Xs ) @ A2 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3679842834875189465at_nat @ Xs ) )
           => ( member8206827879206165904at_nat @ ( nth_Pr6744343527793145070at_nat @ Xs @ I3 ) @ A2 ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_1104_subseteq__set__conv__nth,axiom,
    ! [Ss: list_list_nat,T2: set_list_nat] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Ss ) )
           => ( member_list_nat @ ( nth_list_nat @ Ss @ I3 ) @ T2 ) ) )
      = ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Ss ) @ T2 ) ) ).

% subseteq_set_conv_nth
thf(fact_1105_subseteq__set__conv__nth,axiom,
    ! [Ss: list_l1578326660578982846ctxt_a,T2: set_li3075106635084045668ctxt_a] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6994882312647770194ctxt_a @ Ss ) )
           => ( member2714729351964853061ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ss @ I3 ) @ T2 ) ) )
      = ( ord_le9074214841948581316ctxt_a @ ( set_li3057194059929822169ctxt_a @ Ss ) @ T2 ) ) ).

% subseteq_set_conv_nth
thf(fact_1106_subseteq__set__conv__nth,axiom,
    ! [Ss: list_l3298181151656792051at_nat,T2: set_li3197816953174176717at_nat] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6878757106947045471at_nat @ Ss ) )
           => ( member3799944675974059798at_nat @ ( nth_li8805938130461222772at_nat @ Ss @ I3 ) @ T2 ) ) )
      = ( ord_le5602213209262422893at_nat @ ( set_li7366234323595830146at_nat @ Ss ) @ T2 ) ) ).

% subseteq_set_conv_nth
thf(fact_1107_subseteq__set__conv__nth,axiom,
    ! [Ss: list_Ground_gmctxt_a,T2: set_Ground_gmctxt_a] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Ss ) )
           => ( member3306386364193653941ctxt_a @ ( nth_Ground_gmctxt_a @ Ss @ I3 ) @ T2 ) ) )
      = ( ord_le4032861497522155828ctxt_a @ ( set_Ground_gmctxt_a2 @ Ss ) @ T2 ) ) ).

% subseteq_set_conv_nth
thf(fact_1108_subseteq__set__conv__nth,axiom,
    ! [Ss: list_b,T2: set_b] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Ss ) )
           => ( member_b @ ( nth_b @ Ss @ I3 ) @ T2 ) ) )
      = ( ord_less_eq_set_b @ ( set_b2 @ Ss ) @ T2 ) ) ).

% subseteq_set_conv_nth
thf(fact_1109_subseteq__set__conv__nth,axiom,
    ! [Ss: list_nat,T2: set_nat] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ss ) )
           => ( member_nat @ ( nth_nat @ Ss @ I3 ) @ T2 ) ) )
      = ( ord_less_eq_set_nat @ ( set_nat2 @ Ss ) @ T2 ) ) ).

% subseteq_set_conv_nth
thf(fact_1110_subseteq__set__conv__nth,axiom,
    ! [Ss: list_list_b,T2: set_list_b] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s420531428170919671list_b @ Ss ) )
           => ( member_list_b @ ( nth_list_b @ Ss @ I3 ) @ T2 ) ) )
      = ( ord_le8932221534207217157list_b @ ( set_list_b2 @ Ss ) @ T2 ) ) ).

% subseteq_set_conv_nth
thf(fact_1111_subseteq__set__conv__nth,axiom,
    ! [Ss: list_P8469869581646625389at_nat,T2: set_Pr8693737435421807431at_nat] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3679842834875189465at_nat @ Ss ) )
           => ( member8206827879206165904at_nat @ ( nth_Pr6744343527793145070at_nat @ Ss @ I3 ) @ T2 ) ) )
      = ( ord_le3000389064537975527at_nat @ ( set_Pr5518436109238095868at_nat @ Ss ) @ T2 ) ) ).

% subseteq_set_conv_nth
thf(fact_1112_adjust__idx__rev__def,axiom,
    ( missin3815256168798769645dx_rev
    = ( ^ [I3: nat,J4: nat] : ( if_nat @ ( ord_less_nat @ J4 @ I3 ) @ J4 @ ( minus_minus_nat @ J4 @ ( suc @ zero_zero_nat ) ) ) ) ) ).

% adjust_idx_rev_def
thf(fact_1113_sum__list__update,axiom,
    ! [K: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ K @ ( size_size_list_nat @ Xs ) )
     => ( ( groups4561878855575611511st_nat @ ( list_update_nat @ Xs @ K @ X ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( groups4561878855575611511st_nat @ Xs ) @ X ) @ ( nth_nat @ Xs @ K ) ) ) ) ).

% sum_list_update
thf(fact_1114_inf__concat__simple_Ocases,axiom,
    ! [X: produc8199716216217303280at_nat] :
      ( ! [F3: nat > nat] :
          ( X
         != ( produc72220940542539688at_nat @ F3 @ zero_zero_nat ) )
     => ~ ! [F3: nat > nat,N3: nat] :
            ( X
           != ( produc72220940542539688at_nat @ F3 @ ( suc @ N3 ) ) ) ) ).

% inf_concat_simple.cases
thf(fact_1115_in__set__enumerate__eq,axiom,
    ! [P2: produc4575160907756185873st_nat,N: nat,Xs: list_list_nat] :
      ( ( member1631564025489475386st_nat @ P2 @ ( set_Pr1526418413117253030st_nat @ ( enumerate_list_nat @ N @ Xs ) ) )
      = ( ( ord_less_eq_nat @ N @ ( produc8283513326151283381st_nat @ P2 ) )
        & ( ord_less_nat @ ( produc8283513326151283381st_nat @ P2 ) @ ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ N ) )
        & ( ( nth_list_nat @ Xs @ ( minus_minus_nat @ ( produc8283513326151283381st_nat @ P2 ) @ N ) )
          = ( produc1177724860032999927st_nat @ P2 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_1116_in__set__enumerate__eq,axiom,
    ! [P2: produc4999762543018873695ctxt_a,N: nat,Xs: list_l1578326660578982846ctxt_a] :
      ( ( member1153228084381711240ctxt_a @ P2 @ ( set_Pr8908780240170439156ctxt_a @ ( enumer2580734899591370633ctxt_a @ N @ Xs ) ) )
      = ( ( ord_less_eq_nat @ N @ ( produc5338924193594264067ctxt_a @ P2 ) )
        & ( ord_less_nat @ ( produc5338924193594264067ctxt_a @ P2 ) @ ( plus_plus_nat @ ( size_s6994882312647770194ctxt_a @ Xs ) @ N ) )
        & ( ( nth_li6678718213161987943ctxt_a @ Xs @ ( minus_minus_nat @ ( produc5338924193594264067ctxt_a @ P2 ) @ N ) )
          = ( produc1946588146859208005ctxt_a @ P2 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_1117_in__set__enumerate__eq,axiom,
    ! [P2: produc1035334568315550354at_nat,N: nat,Xs: list_l3298181151656792051at_nat] :
      ( ( member7629795747344909993at_nat @ P2 @ ( set_Pr6130687943182631741at_nat @ ( enumer33237656203508690at_nat @ N @ Xs ) ) )
      = ( ( ord_less_eq_nat @ N @ ( produc5598002518098430936at_nat @ P2 ) )
        & ( ord_less_nat @ ( produc5598002518098430936at_nat @ P2 ) @ ( plus_plus_nat @ ( size_s6878757106947045471at_nat @ Xs ) @ N ) )
        & ( ( nth_li8805938130461222772at_nat @ Xs @ ( minus_minus_nat @ ( produc5598002518098430936at_nat @ P2 ) @ N ) )
          = ( produc3264766051377360150at_nat @ P2 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_1118_in__set__enumerate__eq,axiom,
    ! [P2: produc932717863267311695ctxt_a,N: nat,Xs: list_Ground_gmctxt_a] :
      ( ( member2885886537299685624ctxt_a @ P2 @ ( set_Pr3995427970001988964ctxt_a @ ( enumer3922341085786344185ctxt_a @ N @ Xs ) ) )
      = ( ( ord_less_eq_nat @ N @ ( produc8631767124948943475ctxt_a @ P2 ) )
        & ( ord_less_nat @ ( produc8631767124948943475ctxt_a @ P2 ) @ ( plus_plus_nat @ ( size_s962548663205356226ctxt_a @ Xs ) @ N ) )
        & ( ( nth_Ground_gmctxt_a @ Xs @ ( minus_minus_nat @ ( produc8631767124948943475ctxt_a @ P2 ) @ N ) )
          = ( produc3552744573514591925ctxt_a @ P2 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_1119_in__set__enumerate__eq,axiom,
    ! [P2: product_prod_nat_b,N: nat,Xs: list_b] :
      ( ( member8962352056413324475_nat_b @ P2 @ ( set_Pr4163146842529940303_nat_b @ ( enumerate_b @ N @ Xs ) ) )
      = ( ( ord_less_eq_nat @ N @ ( product_fst_nat_b @ P2 ) )
        & ( ord_less_nat @ ( product_fst_nat_b @ P2 ) @ ( plus_plus_nat @ ( size_size_list_b @ Xs ) @ N ) )
        & ( ( nth_b @ Xs @ ( minus_minus_nat @ ( product_fst_nat_b @ P2 ) @ N ) )
          = ( product_snd_nat_b @ P2 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_1120_in__set__enumerate__eq,axiom,
    ! [P2: product_prod_nat_nat,N: nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ P2 @ ( set_Pr5648618587558075414at_nat @ ( enumerate_nat @ N @ Xs ) ) )
      = ( ( 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 @ Xs ) @ N ) )
        & ( ( nth_nat @ Xs @ ( minus_minus_nat @ ( product_fst_nat_nat @ P2 ) @ N ) )
          = ( product_snd_nat_nat @ P2 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_1121_in__set__enumerate__eq,axiom,
    ! [P2: produc7811952446676219690list_b,N: nat,Xs: list_list_b] :
      ( ( member8261005420521984321list_b @ P2 @ ( set_Pr8537765906400867797list_b @ ( enumerate_list_b @ N @ Xs ) ) )
      = ( ( ord_less_eq_nat @ N @ ( produc4466459399909381744list_b @ P2 ) )
        & ( ord_less_nat @ ( produc4466459399909381744list_b @ P2 ) @ ( plus_plus_nat @ ( size_s420531428170919671list_b @ Xs ) @ N ) )
        & ( ( nth_list_b @ Xs @ ( minus_minus_nat @ ( produc4466459399909381744list_b @ P2 ) @ N ) )
          = ( produc8425566080785036206list_b @ P2 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_1122_in__set__enumerate__eq,axiom,
    ! [P2: produc2487518378626728076at_nat,N: nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member5754182594613576739at_nat @ P2 @ ( set_Pr2327711193239132855at_nat @ ( enumer3122081344003035212at_nat @ N @ Xs ) ) )
      = ( ( ord_less_eq_nat @ N @ ( produc7178812058934418258at_nat @ P2 ) )
        & ( ord_less_nat @ ( produc7178812058934418258at_nat @ P2 ) @ ( plus_plus_nat @ ( size_s3679842834875189465at_nat @ Xs ) @ N ) )
        & ( ( nth_Pr6744343527793145070at_nat @ Xs @ ( minus_minus_nat @ ( produc7178812058934418258at_nat @ P2 ) @ N ) )
          = ( produc3846061323874349968at_nat @ P2 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_1123_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less_nat @ A @ B )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_1124_concat__index__split__less__length__concat_I1_J,axiom,
    ! [I: nat,Xs: list_l1578326660578982846ctxt_a,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ( ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( I
          = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_li8618020238875267396_a_nat @ size_s962548663205356226ctxt_a @ ( take_l4544854310384971712ctxt_a @ M @ Xs ) ) ) @ N ) ) ) ) ).

% concat_index_split_less_length_concat(1)
thf(fact_1125_concat__index__split__less__length__concat_I1_J,axiom,
    ! [I: nat,Xs: list_list_b,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_b @ ( concat_b @ Xs ) ) )
     => ( ( ( concat_index_split_b @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( I
          = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_b_nat @ size_size_list_b @ ( take_list_b @ M @ Xs ) ) ) @ N ) ) ) ) ).

% concat_index_split_less_length_concat(1)
thf(fact_1126_concat__index__split__less__length__concat_I1_J,axiom,
    ! [I: nat,Xs: list_list_nat,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( I
          = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ size_size_list_nat @ ( take_list_nat @ M @ Xs ) ) ) @ N ) ) ) ) ).

% concat_index_split_less_length_concat(1)
thf(fact_1127_concat__index__split__less__length__concat_I1_J,axiom,
    ! [I: nat,Xs: list_list_list_b,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ ( concat_list_b @ Xs ) ) )
     => ( ( ( concat7499549088139563139list_b @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( I
          = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_list_b_nat @ size_s420531428170919671list_b @ ( take_list_list_b @ M @ Xs ) ) ) @ N ) ) ) ) ).

% concat_index_split_less_length_concat(1)
thf(fact_1128_concat__index__split__less__length__concat_I1_J,axiom,
    ! [I: nat,Xs: list_l3298181151656792051at_nat,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ ( concat5616020275982107560at_nat @ Xs ) ) )
     => ( ( ( concat1027118156116784357at_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( I
          = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_li9059624042496273785at_nat @ size_s3679842834875189465at_nat @ ( take_l4110974213330421595at_nat @ M @ Xs ) ) ) @ N ) ) ) ) ).

% concat_index_split_less_length_concat(1)
thf(fact_1129_partition__by__nth__nth__old,axiom,
    ! [I: nat,Xs: list_list_nat,Ys: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6248950052170075156st_nat @ ( multih5630945896903731040st_nat @ Xs @ Ys ) ) )
     => ( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ ( multih5630945896903731040st_nat @ Xs @ Ys ) @ I ) ) )
       => ( ( ( groups4561878855575611511st_nat @ Ys )
            = ( size_s3023201423986296836st_nat @ Xs ) )
         => ( ( nth_list_nat @ ( nth_list_list_nat @ ( multih5630945896903731040st_nat @ Xs @ Ys ) @ I ) @ J )
            = ( nth_list_nat @ Xs @ ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_li3275209578935457798at_nat @ size_s3023201423986296836st_nat @ ( take_list_list_nat @ I @ ( multih5630945896903731040st_nat @ Xs @ Ys ) ) ) ) @ J ) ) ) ) ) ) ).

% partition_by_nth_nth_old
thf(fact_1130_partition__by__nth__nth__old,axiom,
    ! [I: nat,Xs: list_l1578326660578982846ctxt_a,Ys: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s2801996683354709858ctxt_a @ ( multih1205877868680985774ctxt_a @ Xs @ Ys ) ) )
     => ( ( ord_less_nat @ J @ ( size_s6994882312647770194ctxt_a @ ( nth_li3520809774789768823ctxt_a @ ( multih1205877868680985774ctxt_a @ Xs @ Ys ) @ I ) ) )
       => ( ( ( groups4561878855575611511st_nat @ Ys )
            = ( size_s6994882312647770194ctxt_a @ Xs ) )
         => ( ( nth_li6678718213161987943ctxt_a @ ( nth_li3520809774789768823ctxt_a @ ( multih1205877868680985774ctxt_a @ Xs @ Ys ) @ I ) @ J )
            = ( nth_li6678718213161987943ctxt_a @ Xs @ ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_li3800085317229163092_a_nat @ size_s6994882312647770194ctxt_a @ ( take_l7489526141625251408ctxt_a @ I @ ( multih1205877868680985774ctxt_a @ Xs @ Ys ) ) ) ) @ J ) ) ) ) ) ) ).

% partition_by_nth_nth_old
thf(fact_1131_partition__by__nth__nth__old,axiom,
    ! [I: nat,Xs: list_l3298181151656792051at_nat,Ys: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s7371574421788826469at_nat @ ( multih560249069488021997at_nat @ Xs @ Ys ) ) )
     => ( ( ord_less_nat @ J @ ( size_s6878757106947045471at_nat @ ( nth_li2593672534642986362at_nat @ ( multih560249069488021997at_nat @ Xs @ Ys ) @ I ) ) )
       => ( ( ( groups4561878855575611511st_nat @ Ys )
            = ( size_s6878757106947045471at_nat @ Xs ) )
         => ( ( nth_li8805938130461222772at_nat @ ( nth_li2593672534642986362at_nat @ ( multih560249069488021997at_nat @ Xs @ Ys ) @ I ) @ J )
            = ( nth_li8805938130461222772at_nat @ Xs @ ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_li611904212741579135at_nat @ size_s6878757106947045471at_nat @ ( take_l5453269369450450401at_nat @ I @ ( multih560249069488021997at_nat @ Xs @ Ys ) ) ) ) @ J ) ) ) ) ) ) ).

% partition_by_nth_nth_old
thf(fact_1132_partition__by__nth__nth__old,axiom,
    ! [I: nat,Xs: list_list_b,Ys: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s9106009106869804029list_b @ ( multih2267195492336891525list_b @ Xs @ Ys ) ) )
     => ( ( ord_less_nat @ J @ ( size_s420531428170919671list_b @ ( nth_list_list_b @ ( multih2267195492336891525list_b @ Xs @ Ys ) @ I ) ) )
       => ( ( ( groups4561878855575611511st_nat @ Ys )
            = ( size_s420531428170919671list_b @ Xs ) )
         => ( ( nth_list_b @ ( nth_list_list_b @ ( multih2267195492336891525list_b @ Xs @ Ys ) @ I ) @ J )
            = ( nth_list_b @ Xs @ ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_list_b_nat @ size_s420531428170919671list_b @ ( take_list_list_b @ I @ ( multih2267195492336891525list_b @ Xs @ Ys ) ) ) ) @ J ) ) ) ) ) ) ).

% partition_by_nth_nth_old
thf(fact_1133_partition__by__nth__nth__old,axiom,
    ! [I: nat,Xs: list_b,Ys: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ ( multih3537974503344769663n_by_b @ Xs @ Ys ) ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_b @ ( nth_list_b @ ( multih3537974503344769663n_by_b @ Xs @ Ys ) @ I ) ) )
       => ( ( ( groups4561878855575611511st_nat @ Ys )
            = ( size_size_list_b @ Xs ) )
         => ( ( nth_b @ ( nth_list_b @ ( multih3537974503344769663n_by_b @ Xs @ Ys ) @ I ) @ J )
            = ( nth_b @ Xs @ ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_b_nat @ size_size_list_b @ ( take_list_b @ I @ ( multih3537974503344769663n_by_b @ Xs @ Ys ) ) ) ) @ J ) ) ) ) ) ) ).

% partition_by_nth_nth_old
thf(fact_1134_partition__by__nth__nth__old,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ ( multih8544690682651757648by_nat @ Xs @ Ys ) ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs @ Ys ) @ I ) ) )
       => ( ( ( groups4561878855575611511st_nat @ Ys )
            = ( size_size_list_nat @ Xs ) )
         => ( ( nth_nat @ ( nth_list_nat @ ( multih8544690682651757648by_nat @ Xs @ Ys ) @ I ) @ J )
            = ( nth_nat @ Xs @ ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ size_size_list_nat @ ( take_list_nat @ I @ ( multih8544690682651757648by_nat @ Xs @ Ys ) ) ) ) @ J ) ) ) ) ) ) ).

% partition_by_nth_nth_old
thf(fact_1135_partition__by__nth__nth__old,axiom,
    ! [I: nat,Xs: list_Ground_gmctxt_a,Ys: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ ( multih5172578161653122718ctxt_a @ Xs @ Ys ) ) )
     => ( ( ord_less_nat @ J @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ ( multih5172578161653122718ctxt_a @ Xs @ Ys ) @ I ) ) )
       => ( ( ( groups4561878855575611511st_nat @ Ys )
            = ( size_s962548663205356226ctxt_a @ Xs ) )
         => ( ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ ( multih5172578161653122718ctxt_a @ Xs @ Ys ) @ I ) @ J )
            = ( nth_Ground_gmctxt_a @ Xs @ ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_li8618020238875267396_a_nat @ size_s962548663205356226ctxt_a @ ( take_l4544854310384971712ctxt_a @ I @ ( multih5172578161653122718ctxt_a @ Xs @ Ys ) ) ) ) @ J ) ) ) ) ) ) ).

% partition_by_nth_nth_old
thf(fact_1136_partition__by__nth__nth__old,axiom,
    ! [I: nat,Xs: list_P8469869581646625389at_nat,Ys: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6878757106947045471at_nat @ ( multih7555706055023738855at_nat @ Xs @ Ys ) ) )
     => ( ( ord_less_nat @ J @ ( size_s3679842834875189465at_nat @ ( nth_li8805938130461222772at_nat @ ( multih7555706055023738855at_nat @ Xs @ Ys ) @ I ) ) )
       => ( ( ( groups4561878855575611511st_nat @ Ys )
            = ( size_s3679842834875189465at_nat @ Xs ) )
         => ( ( nth_Pr6744343527793145070at_nat @ ( nth_li8805938130461222772at_nat @ ( multih7555706055023738855at_nat @ Xs @ Ys ) @ I ) @ J )
            = ( nth_Pr6744343527793145070at_nat @ Xs @ ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_li9059624042496273785at_nat @ size_s3679842834875189465at_nat @ ( take_l4110974213330421595at_nat @ I @ ( multih7555706055023738855at_nat @ Xs @ Ys ) ) ) ) @ J ) ) ) ) ) ) ).

% partition_by_nth_nth_old
thf(fact_1137_concat__nth,axiom,
    ! [M: nat,Xs: list_list_list_nat,N: nat,I: nat] :
      ( ( ord_less_nat @ M @ ( size_s6248950052170075156st_nat @ Xs ) )
     => ( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Xs @ M ) ) )
       => ( ( I
            = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_li3275209578935457798at_nat @ size_s3023201423986296836st_nat @ ( take_list_list_nat @ M @ Xs ) ) ) @ N ) )
         => ( ( nth_list_nat @ ( concat_list_nat @ Xs ) @ I )
            = ( nth_list_nat @ ( nth_list_list_nat @ Xs @ M ) @ N ) ) ) ) ) ).

% concat_nth
thf(fact_1138_concat__nth,axiom,
    ! [M: nat,Xs: list_l3687490334575670478ctxt_a,N: nat,I: nat] :
      ( ( ord_less_nat @ M @ ( size_s2801996683354709858ctxt_a @ Xs ) )
     => ( ( ord_less_nat @ N @ ( size_s6994882312647770194ctxt_a @ ( nth_li3520809774789768823ctxt_a @ Xs @ M ) ) )
       => ( ( I
            = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_li3800085317229163092_a_nat @ size_s6994882312647770194ctxt_a @ ( take_l7489526141625251408ctxt_a @ M @ Xs ) ) ) @ N ) )
         => ( ( nth_li6678718213161987943ctxt_a @ ( concat6203624070112318253ctxt_a @ Xs ) @ I )
            = ( nth_li6678718213161987943ctxt_a @ ( nth_li3520809774789768823ctxt_a @ Xs @ M ) @ N ) ) ) ) ) ).

% concat_nth
thf(fact_1139_concat__nth,axiom,
    ! [M: nat,Xs: list_l5129023048960694521at_nat,N: nat,I: nat] :
      ( ( ord_less_nat @ M @ ( size_s7371574421788826469at_nat @ Xs ) )
     => ( ( ord_less_nat @ N @ ( size_s6878757106947045471at_nat @ ( nth_li2593672534642986362at_nat @ Xs @ M ) ) )
       => ( ( I
            = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_li611904212741579135at_nat @ size_s6878757106947045471at_nat @ ( take_l5453269369450450401at_nat @ M @ Xs ) ) ) @ N ) )
         => ( ( nth_li8805938130461222772at_nat @ ( concat5897368152400924974at_nat @ Xs ) @ I )
            = ( nth_li8805938130461222772at_nat @ ( nth_li2593672534642986362at_nat @ Xs @ M ) @ N ) ) ) ) ) ).

% concat_nth
thf(fact_1140_concat__nth,axiom,
    ! [M: nat,Xs: list_l1578326660578982846ctxt_a,N: nat,I: nat] :
      ( ( ord_less_nat @ M @ ( size_s6994882312647770194ctxt_a @ Xs ) )
     => ( ( ord_less_nat @ N @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ M ) ) )
       => ( ( I
            = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_li8618020238875267396_a_nat @ size_s962548663205356226ctxt_a @ ( take_l4544854310384971712ctxt_a @ M @ Xs ) ) ) @ N ) )
         => ( ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) @ I )
            = ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ M ) @ N ) ) ) ) ) ).

% concat_nth
thf(fact_1141_concat__nth,axiom,
    ! [M: nat,Xs: list_list_nat,N: nat,I: nat] :
      ( ( ord_less_nat @ M @ ( size_s3023201423986296836st_nat @ Xs ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ M ) ) )
       => ( ( I
            = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ size_size_list_nat @ ( take_list_nat @ M @ Xs ) ) ) @ N ) )
         => ( ( nth_nat @ ( concat_nat @ Xs ) @ I )
            = ( nth_nat @ ( nth_list_nat @ Xs @ M ) @ N ) ) ) ) ) ).

% concat_nth
thf(fact_1142_concat__nth,axiom,
    ! [M: nat,Xs: list_list_list_b,N: nat,I: nat] :
      ( ( ord_less_nat @ M @ ( size_s9106009106869804029list_b @ Xs ) )
     => ( ( ord_less_nat @ N @ ( size_s420531428170919671list_b @ ( nth_list_list_b @ Xs @ M ) ) )
       => ( ( I
            = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_list_b_nat @ size_s420531428170919671list_b @ ( take_list_list_b @ M @ Xs ) ) ) @ N ) )
         => ( ( nth_list_b @ ( concat_list_b @ Xs ) @ I )
            = ( nth_list_b @ ( nth_list_list_b @ Xs @ M ) @ N ) ) ) ) ) ).

% concat_nth
thf(fact_1143_concat__nth,axiom,
    ! [M: nat,Xs: list_l3298181151656792051at_nat,N: nat,I: nat] :
      ( ( ord_less_nat @ M @ ( size_s6878757106947045471at_nat @ Xs ) )
     => ( ( ord_less_nat @ N @ ( size_s3679842834875189465at_nat @ ( nth_li8805938130461222772at_nat @ Xs @ M ) ) )
       => ( ( I
            = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_li9059624042496273785at_nat @ size_s3679842834875189465at_nat @ ( take_l4110974213330421595at_nat @ M @ Xs ) ) ) @ N ) )
         => ( ( nth_Pr6744343527793145070at_nat @ ( concat5616020275982107560at_nat @ Xs ) @ I )
            = ( nth_Pr6744343527793145070at_nat @ ( nth_li8805938130461222772at_nat @ Xs @ M ) @ N ) ) ) ) ) ).

% concat_nth
thf(fact_1144_concat__nth,axiom,
    ! [M: nat,Xs: list_list_b,N: nat,I: nat] :
      ( ( ord_less_nat @ M @ ( size_s420531428170919671list_b @ Xs ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_b @ ( nth_list_b @ Xs @ M ) ) )
       => ( ( I
            = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_b_nat @ size_size_list_b @ ( take_list_b @ M @ Xs ) ) ) @ N ) )
         => ( ( nth_b @ ( concat_b @ Xs ) @ I )
            = ( nth_b @ ( nth_list_b @ Xs @ M ) @ N ) ) ) ) ) ).

% concat_nth
thf(fact_1145_length__enumerate,axiom,
    ! [N: nat,Xs: list_Ground_gmctxt_a] :
      ( ( size_s4225619979278260545ctxt_a @ ( enumer3922341085786344185ctxt_a @ N @ Xs ) )
      = ( size_s962548663205356226ctxt_a @ Xs ) ) ).

% length_enumerate
thf(fact_1146_length__enumerate,axiom,
    ! [N: nat,Xs: list_b] :
      ( ( size_s314938103728740808_nat_b @ ( enumerate_b @ N @ Xs ) )
      = ( size_size_list_b @ Xs ) ) ).

% length_enumerate
thf(fact_1147_length__enumerate,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_enumerate
thf(fact_1148_length__enumerate,axiom,
    ! [N: nat,Xs: list_list_b] :
      ( ( size_s8332131329787750990list_b @ ( enumerate_list_b @ N @ Xs ) )
      = ( size_s420531428170919671list_b @ Xs ) ) ).

% length_enumerate
thf(fact_1149_length__enumerate,axiom,
    ! [N: nat,Xs: list_P8469869581646625389at_nat] :
      ( ( size_s2512440580853752112at_nat @ ( enumer3122081344003035212at_nat @ N @ Xs ) )
      = ( size_s3679842834875189465at_nat @ Xs ) ) ).

% length_enumerate
thf(fact_1150_take__all,axiom,
    ! [Xs: list_Ground_gmctxt_a,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s962548663205356226ctxt_a @ Xs ) @ N )
     => ( ( take_Ground_gmctxt_a @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_1151_take__all,axiom,
    ! [Xs: list_b,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_b @ Xs ) @ N )
     => ( ( take_b @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_1152_take__all,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
     => ( ( take_nat @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_1153_take__all,axiom,
    ! [Xs: list_list_b,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s420531428170919671list_b @ Xs ) @ N )
     => ( ( take_list_b @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_1154_take__all,axiom,
    ! [Xs: list_P8469869581646625389at_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s3679842834875189465at_nat @ Xs ) @ N )
     => ( ( take_P5254422574997664853at_nat @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_1155_take__all__iff,axiom,
    ! [N: nat,Xs: list_Ground_gmctxt_a] :
      ( ( ( take_Ground_gmctxt_a @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_s962548663205356226ctxt_a @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_1156_take__all__iff,axiom,
    ! [N: nat,Xs: list_b] :
      ( ( ( take_b @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_size_list_b @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_1157_take__all__iff,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ( take_nat @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_1158_take__all__iff,axiom,
    ! [N: nat,Xs: list_list_b] :
      ( ( ( take_list_b @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_s420531428170919671list_b @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_1159_take__all__iff,axiom,
    ! [N: nat,Xs: list_P8469869581646625389at_nat] :
      ( ( ( take_P5254422574997664853at_nat @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_s3679842834875189465at_nat @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_1160_nth__take,axiom,
    ! [I: nat,N: nat,Xs: list_list_b] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_list_b @ ( take_list_b @ N @ Xs ) @ I )
        = ( nth_list_b @ Xs @ I ) ) ) ).

% nth_take
thf(fact_1161_nth__take,axiom,
    ! [I: nat,N: nat,Xs: list_b] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_b @ ( take_b @ N @ Xs ) @ I )
        = ( nth_b @ Xs @ I ) ) ) ).

% nth_take
thf(fact_1162_nth__take,axiom,
    ! [I: nat,N: nat,Xs: list_Ground_gmctxt_a] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_Ground_gmctxt_a @ ( take_Ground_gmctxt_a @ N @ Xs ) @ I )
        = ( nth_Ground_gmctxt_a @ Xs @ I ) ) ) ).

% nth_take
thf(fact_1163_nth__take,axiom,
    ! [I: nat,N: nat,Xs: list_list_nat] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_list_nat @ ( take_list_nat @ N @ Xs ) @ I )
        = ( nth_list_nat @ Xs @ I ) ) ) ).

% nth_take
thf(fact_1164_nth__take,axiom,
    ! [I: nat,N: nat,Xs: list_l1578326660578982846ctxt_a] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_li6678718213161987943ctxt_a @ ( take_l4544854310384971712ctxt_a @ N @ Xs ) @ I )
        = ( nth_li6678718213161987943ctxt_a @ Xs @ I ) ) ) ).

% nth_take
thf(fact_1165_nth__take,axiom,
    ! [I: nat,N: nat,Xs: list_l3298181151656792051at_nat] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_li8805938130461222772at_nat @ ( take_l4110974213330421595at_nat @ N @ Xs ) @ I )
        = ( nth_li8805938130461222772at_nat @ Xs @ I ) ) ) ).

% nth_take
thf(fact_1166_nth__take,axiom,
    ! [I: nat,N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_nat @ ( take_nat @ N @ Xs ) @ I )
        = ( nth_nat @ Xs @ I ) ) ) ).

% nth_take
thf(fact_1167_take__update__cancel,axiom,
    ! [N: nat,M: nat,Xs: list_nat,Y3: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( take_nat @ N @ ( list_update_nat @ Xs @ M @ Y3 ) )
        = ( take_nat @ N @ Xs ) ) ) ).

% take_update_cancel
thf(fact_1168_take__append,axiom,
    ! [N: nat,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
      ( ( take_Ground_gmctxt_a @ N @ ( append5238750768297430211ctxt_a @ Xs @ Ys ) )
      = ( append5238750768297430211ctxt_a @ ( take_Ground_gmctxt_a @ N @ Xs ) @ ( take_Ground_gmctxt_a @ ( minus_minus_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) ) @ Ys ) ) ) ).

% take_append
thf(fact_1169_take__append,axiom,
    ! [N: nat,Xs: list_b,Ys: list_b] :
      ( ( take_b @ N @ ( append_b @ Xs @ Ys ) )
      = ( append_b @ ( take_b @ N @ Xs ) @ ( take_b @ ( minus_minus_nat @ N @ ( size_size_list_b @ Xs ) ) @ Ys ) ) ) ).

% take_append
thf(fact_1170_take__append,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat] :
      ( ( take_nat @ N @ ( append_nat @ Xs @ Ys ) )
      = ( append_nat @ ( take_nat @ N @ Xs ) @ ( take_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).

% take_append
thf(fact_1171_take__append,axiom,
    ! [N: nat,Xs: list_list_b,Ys: list_list_b] :
      ( ( take_list_b @ N @ ( append_list_b @ Xs @ Ys ) )
      = ( append_list_b @ ( take_list_b @ N @ Xs ) @ ( take_list_b @ ( minus_minus_nat @ N @ ( size_s420531428170919671list_b @ Xs ) ) @ Ys ) ) ) ).

% take_append
thf(fact_1172_take__append,axiom,
    ! [N: nat,Xs: list_P8469869581646625389at_nat,Ys: list_P8469869581646625389at_nat] :
      ( ( take_P5254422574997664853at_nat @ N @ ( append8751754712269456642at_nat @ Xs @ Ys ) )
      = ( append8751754712269456642at_nat @ ( take_P5254422574997664853at_nat @ N @ Xs ) @ ( take_P5254422574997664853at_nat @ ( minus_minus_nat @ N @ ( size_s3679842834875189465at_nat @ Xs ) ) @ Ys ) ) ) ).

% take_append
thf(fact_1173_take__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ! [I2: nat] :
          ( ( take_nat @ I2 @ Xs )
          = ( take_nat @ I2 @ Ys ) )
     => ( Xs = Ys ) ) ).

% take_equalityI
thf(fact_1174_in__set__takeD,axiom,
    ! [X: b,N: nat,Xs: list_b] :
      ( ( member_b @ X @ ( set_b2 @ ( take_b @ N @ Xs ) ) )
     => ( member_b @ X @ ( set_b2 @ Xs ) ) ) ).

% in_set_takeD
thf(fact_1175_in__set__takeD,axiom,
    ! [X: nat,N: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) )
     => ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).

% in_set_takeD
thf(fact_1176_in__set__takeD,axiom,
    ! [X: ground_gmctxt_a,N: nat,Xs: list_Ground_gmctxt_a] :
      ( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ ( take_Ground_gmctxt_a @ N @ Xs ) ) )
     => ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) ) ) ).

% in_set_takeD
thf(fact_1177_in__set__takeD,axiom,
    ! [X: produc859450856879609959at_nat,N: nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ ( take_P5254422574997664853at_nat @ N @ Xs ) ) )
     => ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs ) ) ) ).

% in_set_takeD
thf(fact_1178_set__take__subset,axiom,
    ! [N: nat,Xs: list_b] : ( ord_less_eq_set_b @ ( set_b2 @ ( take_b @ N @ Xs ) ) @ ( set_b2 @ Xs ) ) ).

% set_take_subset
thf(fact_1179_set__take__subset,axiom,
    ! [N: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).

% set_take_subset
thf(fact_1180_set__take__subset,axiom,
    ! [N: nat,Xs: list_Ground_gmctxt_a] : ( ord_le4032861497522155828ctxt_a @ ( set_Ground_gmctxt_a2 @ ( take_Ground_gmctxt_a @ N @ Xs ) ) @ ( set_Ground_gmctxt_a2 @ Xs ) ) ).

% set_take_subset
thf(fact_1181_set__take__subset,axiom,
    ! [N: nat,Xs: list_P8469869581646625389at_nat] : ( ord_le3000389064537975527at_nat @ ( set_Pr5518436109238095868at_nat @ ( take_P5254422574997664853at_nat @ N @ Xs ) ) @ ( set_Pr5518436109238095868at_nat @ Xs ) ) ).

% set_take_subset
thf(fact_1182_take__map,axiom,
    ! [N: nat,F: nat > nat,Xs: list_nat] :
      ( ( take_nat @ N @ ( map_nat_nat @ F @ Xs ) )
      = ( map_nat_nat @ F @ ( take_nat @ N @ Xs ) ) ) ).

% take_map
thf(fact_1183_take__map,axiom,
    ! [N: nat,F: ground_gmctxt_a > nat,Xs: list_Ground_gmctxt_a] :
      ( ( take_nat @ N @ ( map_Gr1679544123147727284_a_nat @ F @ Xs ) )
      = ( map_Gr1679544123147727284_a_nat @ F @ ( take_Ground_gmctxt_a @ N @ Xs ) ) ) ).

% take_map
thf(fact_1184_take__map,axiom,
    ! [N: nat,F: nat > b,Xs: list_nat] :
      ( ( take_b @ N @ ( map_nat_b @ F @ Xs ) )
      = ( map_nat_b @ F @ ( take_nat @ N @ Xs ) ) ) ).

% take_map
thf(fact_1185_take__map,axiom,
    ! [N: nat,F: b > ground_gmctxt_a,Xs: list_b] :
      ( ( take_Ground_gmctxt_a @ N @ ( map_b_3587768286370285017ctxt_a @ F @ Xs ) )
      = ( map_b_3587768286370285017ctxt_a @ F @ ( take_b @ N @ Xs ) ) ) ).

% take_map
thf(fact_1186_take__map,axiom,
    ! [N: nat,F: b > list_b,Xs: list_b] :
      ( ( take_list_b @ N @ ( map_b_list_b @ F @ Xs ) )
      = ( map_b_list_b @ F @ ( take_b @ N @ Xs ) ) ) ).

% take_map
thf(fact_1187_take__map,axiom,
    ! [N: nat,F: b > nat,Xs: list_b] :
      ( ( take_nat @ N @ ( map_b_nat @ F @ Xs ) )
      = ( map_b_nat @ F @ ( take_b @ N @ Xs ) ) ) ).

% take_map
thf(fact_1188_take__map,axiom,
    ! [N: nat,F: b > b,Xs: list_b] :
      ( ( take_b @ N @ ( map_b_b @ F @ Xs ) )
      = ( map_b_b @ F @ ( take_b @ N @ Xs ) ) ) ).

% take_map
thf(fact_1189_take__update__swap,axiom,
    ! [M: nat,Xs: list_nat,N: nat,X: nat] :
      ( ( take_nat @ M @ ( list_update_nat @ Xs @ N @ X ) )
      = ( list_update_nat @ ( take_nat @ M @ Xs ) @ N @ X ) ) ).

% take_update_swap
thf(fact_1190_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_b] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_set_b @ ( set_b2 @ ( take_b @ M @ Xs ) ) @ ( set_b2 @ ( take_b @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_1191_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ M @ Xs ) ) @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_1192_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_Ground_gmctxt_a] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_le4032861497522155828ctxt_a @ ( set_Ground_gmctxt_a2 @ ( take_Ground_gmctxt_a @ M @ Xs ) ) @ ( set_Ground_gmctxt_a2 @ ( take_Ground_gmctxt_a @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_1193_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_P8469869581646625389at_nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_le3000389064537975527at_nat @ ( set_Pr5518436109238095868at_nat @ ( take_P5254422574997664853at_nat @ M @ Xs ) ) @ ( set_Pr5518436109238095868at_nat @ ( take_P5254422574997664853at_nat @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_1194_set__take__nth,axiom,
    ! [X: list_nat,I: nat,Xs: list_list_nat] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ ( take_list_nat @ I @ Xs ) ) )
     => ? [J2: nat] :
          ( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Xs ) )
          & ( ord_less_nat @ J2 @ I )
          & ( ( nth_list_nat @ Xs @ J2 )
            = X ) ) ) ).

% set_take_nth
thf(fact_1195_set__take__nth,axiom,
    ! [X: list_Ground_gmctxt_a,I: nat,Xs: list_l1578326660578982846ctxt_a] :
      ( ( member2714729351964853061ctxt_a @ X @ ( set_li3057194059929822169ctxt_a @ ( take_l4544854310384971712ctxt_a @ I @ Xs ) ) )
     => ? [J2: nat] :
          ( ( ord_less_nat @ J2 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
          & ( ord_less_nat @ J2 @ I )
          & ( ( nth_li6678718213161987943ctxt_a @ Xs @ J2 )
            = X ) ) ) ).

% set_take_nth
thf(fact_1196_set__take__nth,axiom,
    ! [X: list_P8469869581646625389at_nat,I: nat,Xs: list_l3298181151656792051at_nat] :
      ( ( member3799944675974059798at_nat @ X @ ( set_li7366234323595830146at_nat @ ( take_l4110974213330421595at_nat @ I @ Xs ) ) )
     => ? [J2: nat] :
          ( ( ord_less_nat @ J2 @ ( size_s6878757106947045471at_nat @ Xs ) )
          & ( ord_less_nat @ J2 @ I )
          & ( ( nth_li8805938130461222772at_nat @ Xs @ J2 )
            = X ) ) ) ).

% set_take_nth
thf(fact_1197_set__take__nth,axiom,
    ! [X: ground_gmctxt_a,I: nat,Xs: list_Ground_gmctxt_a] :
      ( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ ( take_Ground_gmctxt_a @ I @ Xs ) ) )
     => ? [J2: nat] :
          ( ( ord_less_nat @ J2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
          & ( ord_less_nat @ J2 @ I )
          & ( ( nth_Ground_gmctxt_a @ Xs @ J2 )
            = X ) ) ) ).

% set_take_nth
thf(fact_1198_set__take__nth,axiom,
    ! [X: b,I: nat,Xs: list_b] :
      ( ( member_b @ X @ ( set_b2 @ ( take_b @ I @ Xs ) ) )
     => ? [J2: nat] :
          ( ( ord_less_nat @ J2 @ ( size_size_list_b @ Xs ) )
          & ( ord_less_nat @ J2 @ I )
          & ( ( nth_b @ Xs @ J2 )
            = X ) ) ) ).

% set_take_nth
thf(fact_1199_set__take__nth,axiom,
    ! [X: nat,I: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( take_nat @ I @ Xs ) ) )
     => ? [J2: nat] :
          ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
          & ( ord_less_nat @ J2 @ I )
          & ( ( nth_nat @ Xs @ J2 )
            = X ) ) ) ).

% set_take_nth
thf(fact_1200_set__take__nth,axiom,
    ! [X: list_b,I: nat,Xs: list_list_b] :
      ( ( member_list_b @ X @ ( set_list_b2 @ ( take_list_b @ I @ Xs ) ) )
     => ? [J2: nat] :
          ( ( ord_less_nat @ J2 @ ( size_s420531428170919671list_b @ Xs ) )
          & ( ord_less_nat @ J2 @ I )
          & ( ( nth_list_b @ Xs @ J2 )
            = X ) ) ) ).

% set_take_nth
thf(fact_1201_set__take__nth,axiom,
    ! [X: produc859450856879609959at_nat,I: nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ ( take_P5254422574997664853at_nat @ I @ Xs ) ) )
     => ? [J2: nat] :
          ( ( ord_less_nat @ J2 @ ( size_s3679842834875189465at_nat @ Xs ) )
          & ( ord_less_nat @ J2 @ I )
          & ( ( nth_Pr6744343527793145070at_nat @ Xs @ J2 )
            = X ) ) ) ).

% set_take_nth
thf(fact_1202_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_list_nat,Ys: list_list_nat] :
      ( ( ord_less_eq_nat @ K @ ( size_s3023201423986296836st_nat @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s3023201423986296836st_nat @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_list_nat @ Xs @ I2 )
                = ( nth_list_nat @ Ys @ I2 ) ) )
         => ( ( take_list_nat @ K @ Xs )
            = ( take_list_nat @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_1203_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_l1578326660578982846ctxt_a,Ys: list_l1578326660578982846ctxt_a] :
      ( ( ord_less_eq_nat @ K @ ( size_s6994882312647770194ctxt_a @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s6994882312647770194ctxt_a @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_li6678718213161987943ctxt_a @ Xs @ I2 )
                = ( nth_li6678718213161987943ctxt_a @ Ys @ I2 ) ) )
         => ( ( take_l4544854310384971712ctxt_a @ K @ Xs )
            = ( take_l4544854310384971712ctxt_a @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_1204_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_l3298181151656792051at_nat,Ys: list_l3298181151656792051at_nat] :
      ( ( ord_less_eq_nat @ K @ ( size_s6878757106947045471at_nat @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s6878757106947045471at_nat @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_li8805938130461222772at_nat @ Xs @ I2 )
                = ( nth_li8805938130461222772at_nat @ Ys @ I2 ) ) )
         => ( ( take_l4110974213330421595at_nat @ K @ Xs )
            = ( take_l4110974213330421595at_nat @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_1205_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
      ( ( ord_less_eq_nat @ K @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s962548663205356226ctxt_a @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_Ground_gmctxt_a @ Xs @ I2 )
                = ( nth_Ground_gmctxt_a @ Ys @ I2 ) ) )
         => ( ( take_Ground_gmctxt_a @ K @ Xs )
            = ( take_Ground_gmctxt_a @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_1206_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_b,Ys: list_b] :
      ( ( ord_less_eq_nat @ K @ ( size_size_list_b @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_size_list_b @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_b @ Xs @ I2 )
                = ( nth_b @ Ys @ I2 ) ) )
         => ( ( take_b @ K @ Xs )
            = ( take_b @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_1207_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_nat @ Xs @ I2 )
                = ( nth_nat @ Ys @ I2 ) ) )
         => ( ( take_nat @ K @ Xs )
            = ( take_nat @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_1208_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_list_b,Ys: list_list_b] :
      ( ( ord_less_eq_nat @ K @ ( size_s420531428170919671list_b @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s420531428170919671list_b @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_list_b @ Xs @ I2 )
                = ( nth_list_b @ Ys @ I2 ) ) )
         => ( ( take_list_b @ K @ Xs )
            = ( take_list_b @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_1209_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_P8469869581646625389at_nat,Ys: list_P8469869581646625389at_nat] :
      ( ( ord_less_eq_nat @ K @ ( size_s3679842834875189465at_nat @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s3679842834875189465at_nat @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_Pr6744343527793145070at_nat @ Xs @ I2 )
                = ( nth_Pr6744343527793145070at_nat @ Ys @ I2 ) ) )
         => ( ( take_P5254422574997664853at_nat @ K @ Xs )
            = ( take_P5254422574997664853at_nat @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_1210_nth__take__prefix,axiom,
    ! [Ys: list_list_nat,Xs: list_list_nat] :
      ( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Ys ) @ ( size_s3023201423986296836st_nat @ Xs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ys ) )
           => ( ( nth_list_nat @ Xs @ I2 )
              = ( nth_list_nat @ Ys @ I2 ) ) )
       => ( ( take_list_nat @ ( size_s3023201423986296836st_nat @ Ys ) @ Xs )
          = Ys ) ) ) ).

% nth_take_prefix
thf(fact_1211_nth__take__prefix,axiom,
    ! [Ys: list_l1578326660578982846ctxt_a,Xs: list_l1578326660578982846ctxt_a] :
      ( ( ord_less_eq_nat @ ( size_s6994882312647770194ctxt_a @ Ys ) @ ( size_s6994882312647770194ctxt_a @ Xs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6994882312647770194ctxt_a @ Ys ) )
           => ( ( nth_li6678718213161987943ctxt_a @ Xs @ I2 )
              = ( nth_li6678718213161987943ctxt_a @ Ys @ I2 ) ) )
       => ( ( take_l4544854310384971712ctxt_a @ ( size_s6994882312647770194ctxt_a @ Ys ) @ Xs )
          = Ys ) ) ) ).

% nth_take_prefix
thf(fact_1212_nth__take__prefix,axiom,
    ! [Ys: list_l3298181151656792051at_nat,Xs: list_l3298181151656792051at_nat] :
      ( ( ord_less_eq_nat @ ( size_s6878757106947045471at_nat @ Ys ) @ ( size_s6878757106947045471at_nat @ Xs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6878757106947045471at_nat @ Ys ) )
           => ( ( nth_li8805938130461222772at_nat @ Xs @ I2 )
              = ( nth_li8805938130461222772at_nat @ Ys @ I2 ) ) )
       => ( ( take_l4110974213330421595at_nat @ ( size_s6878757106947045471at_nat @ Ys ) @ Xs )
          = Ys ) ) ) ).

% nth_take_prefix
thf(fact_1213_nth__take__prefix,axiom,
    ! [Ys: list_Ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
      ( ( ord_less_eq_nat @ ( size_s962548663205356226ctxt_a @ Ys ) @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Ys ) )
           => ( ( nth_Ground_gmctxt_a @ Xs @ I2 )
              = ( nth_Ground_gmctxt_a @ Ys @ I2 ) ) )
       => ( ( take_Ground_gmctxt_a @ ( size_s962548663205356226ctxt_a @ Ys ) @ Xs )
          = Ys ) ) ) ).

% nth_take_prefix
thf(fact_1214_nth__take__prefix,axiom,
    ! [Ys: list_b,Xs: list_b] :
      ( ( ord_less_eq_nat @ ( size_size_list_b @ Ys ) @ ( size_size_list_b @ Xs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Ys ) )
           => ( ( nth_b @ Xs @ I2 )
              = ( nth_b @ Ys @ I2 ) ) )
       => ( ( take_b @ ( size_size_list_b @ Ys ) @ Xs )
          = Ys ) ) ) ).

% nth_take_prefix
thf(fact_1215_nth__take__prefix,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
           => ( ( nth_nat @ Xs @ I2 )
              = ( nth_nat @ Ys @ I2 ) ) )
       => ( ( take_nat @ ( size_size_list_nat @ Ys ) @ Xs )
          = Ys ) ) ) ).

% nth_take_prefix
thf(fact_1216_nth__take__prefix,axiom,
    ! [Ys: list_list_b,Xs: list_list_b] :
      ( ( ord_less_eq_nat @ ( size_s420531428170919671list_b @ Ys ) @ ( size_s420531428170919671list_b @ Xs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s420531428170919671list_b @ Ys ) )
           => ( ( nth_list_b @ Xs @ I2 )
              = ( nth_list_b @ Ys @ I2 ) ) )
       => ( ( take_list_b @ ( size_s420531428170919671list_b @ Ys ) @ Xs )
          = Ys ) ) ) ).

% nth_take_prefix
thf(fact_1217_nth__take__prefix,axiom,
    ! [Ys: list_P8469869581646625389at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( ord_less_eq_nat @ ( size_s3679842834875189465at_nat @ Ys ) @ ( size_s3679842834875189465at_nat @ Xs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3679842834875189465at_nat @ Ys ) )
           => ( ( nth_Pr6744343527793145070at_nat @ Xs @ I2 )
              = ( nth_Pr6744343527793145070at_nat @ Ys @ I2 ) ) )
       => ( ( take_P5254422574997664853at_nat @ ( size_s3679842834875189465at_nat @ Ys ) @ Xs )
          = Ys ) ) ) ).

% nth_take_prefix
thf(fact_1218_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
      ( ( enumer3922341085786344185ctxt_a @ N @ ( append5238750768297430211ctxt_a @ Xs @ Ys ) )
      = ( append2243830650925735786ctxt_a @ ( enumer3922341085786344185ctxt_a @ N @ Xs ) @ ( enumer3922341085786344185ctxt_a @ ( plus_plus_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_1219_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_b,Ys: list_b] :
      ( ( enumerate_b @ N @ ( append_b @ Xs @ Ys ) )
      = ( append1694031010730255049_nat_b @ ( enumerate_b @ N @ Xs ) @ ( enumerate_b @ ( plus_plus_nat @ N @ ( size_size_list_b @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_1220_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat] :
      ( ( enumerate_nat @ N @ ( append_nat @ Xs @ Ys ) )
      = ( append985823374593552924at_nat @ ( enumerate_nat @ N @ Xs ) @ ( enumerate_nat @ ( plus_plus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_1221_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_list_b,Ys: list_list_b] :
      ( ( enumerate_list_b @ N @ ( append_list_b @ Xs @ Ys ) )
      = ( append4873443811822227535list_b @ ( enumerate_list_b @ N @ Xs ) @ ( enumerate_list_b @ ( plus_plus_nat @ N @ ( size_s420531428170919671list_b @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_1222_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_P8469869581646625389at_nat,Ys: list_P8469869581646625389at_nat] :
      ( ( enumer3122081344003035212at_nat @ N @ ( append8751754712269456642at_nat @ Xs @ Ys ) )
      = ( append2382163119426291249at_nat @ ( enumer3122081344003035212at_nat @ N @ Xs ) @ ( enumer3122081344003035212at_nat @ ( plus_plus_nat @ N @ ( size_s3679842834875189465at_nat @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_1223_nth__append__take__is__nth__conv,axiom,
    ! [I: nat,J: nat,Xs: list_list_nat,Ys: list_list_nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ ( size_s3023201423986296836st_nat @ Xs ) )
       => ( ( nth_list_nat @ ( append_list_nat @ ( take_list_nat @ J @ Xs ) @ Ys ) @ I )
          = ( nth_list_nat @ Xs @ I ) ) ) ) ).

% nth_append_take_is_nth_conv
thf(fact_1224_nth__append__take__is__nth__conv,axiom,
    ! [I: nat,J: nat,Xs: list_l1578326660578982846ctxt_a,Ys: list_l1578326660578982846ctxt_a] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ ( size_s6994882312647770194ctxt_a @ Xs ) )
       => ( ( nth_li6678718213161987943ctxt_a @ ( append272212692390751315ctxt_a @ ( take_l4544854310384971712ctxt_a @ J @ Xs ) @ Ys ) @ I )
          = ( nth_li6678718213161987943ctxt_a @ Xs @ I ) ) ) ) ).

% nth_append_take_is_nth_conv
thf(fact_1225_nth__append__take__is__nth__conv,axiom,
    ! [I: nat,J: nat,Xs: list_l3298181151656792051at_nat,Ys: list_l3298181151656792051at_nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ ( size_s6878757106947045471at_nat @ Xs ) )
       => ( ( nth_li8805938130461222772at_nat @ ( append6271109094031990152at_nat @ ( take_l4110974213330421595at_nat @ J @ Xs ) @ Ys ) @ I )
          = ( nth_li8805938130461222772at_nat @ Xs @ I ) ) ) ) ).

% nth_append_take_is_nth_conv
thf(fact_1226_nth__append__take__is__nth__conv,axiom,
    ! [I: nat,J: nat,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ ( size_s962548663205356226ctxt_a @ Xs ) )
       => ( ( nth_Ground_gmctxt_a @ ( append5238750768297430211ctxt_a @ ( take_Ground_gmctxt_a @ J @ Xs ) @ Ys ) @ I )
          = ( nth_Ground_gmctxt_a @ Xs @ I ) ) ) ) ).

% nth_append_take_is_nth_conv
thf(fact_1227_nth__append__take__is__nth__conv,axiom,
    ! [I: nat,J: nat,Xs: list_b,Ys: list_b] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ ( size_size_list_b @ Xs ) )
       => ( ( nth_b @ ( append_b @ ( take_b @ J @ Xs ) @ Ys ) @ I )
          = ( nth_b @ Xs @ I ) ) ) ) ).

% nth_append_take_is_nth_conv
thf(fact_1228_nth__append__take__is__nth__conv,axiom,
    ! [I: nat,J: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ ( size_size_list_nat @ Xs ) )
       => ( ( nth_nat @ ( append_nat @ ( take_nat @ J @ Xs ) @ Ys ) @ I )
          = ( nth_nat @ Xs @ I ) ) ) ) ).

% nth_append_take_is_nth_conv
thf(fact_1229_nth__append__take__is__nth__conv,axiom,
    ! [I: nat,J: nat,Xs: list_list_b,Ys: list_list_b] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ ( size_s420531428170919671list_b @ Xs ) )
       => ( ( nth_list_b @ ( append_list_b @ ( take_list_b @ J @ Xs ) @ Ys ) @ I )
          = ( nth_list_b @ Xs @ I ) ) ) ) ).

% nth_append_take_is_nth_conv
thf(fact_1230_nth__append__take__is__nth__conv,axiom,
    ! [I: nat,J: nat,Xs: list_P8469869581646625389at_nat,Ys: list_P8469869581646625389at_nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ ( size_s3679842834875189465at_nat @ Xs ) )
       => ( ( nth_Pr6744343527793145070at_nat @ ( append8751754712269456642at_nat @ ( take_P5254422574997664853at_nat @ J @ Xs ) @ Ys ) @ I )
          = ( nth_Pr6744343527793145070at_nat @ Xs @ I ) ) ) ) ).

% nth_append_take_is_nth_conv
thf(fact_1231_sum__list__split,axiom,
    ! [I: nat,Xs: list_l1578326660578982846ctxt_a,J: nat,K: nat,L: nat] :
      ( ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_li8618020238875267396_a_nat @ size_s962548663205356226ctxt_a @ ( take_l4544854310384971712ctxt_a @ I @ Xs ) ) ) @ J )
        = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_li8618020238875267396_a_nat @ size_s962548663205356226ctxt_a @ ( take_l4544854310384971712ctxt_a @ K @ Xs ) ) ) @ L ) )
     => ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Xs ) )
       => ( ( ord_less_nat @ K @ ( size_s6994882312647770194ctxt_a @ Xs ) )
         => ( ( ord_less_nat @ J @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I ) ) )
           => ( ( ord_less_nat @ L @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ K ) ) )
             => ( ( I = K )
                & ( J = L ) ) ) ) ) ) ) ).

% sum_list_split
thf(fact_1232_sum__list__split,axiom,
    ! [I: nat,Xs: list_list_b,J: nat,K: nat,L: nat] :
      ( ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_b_nat @ size_size_list_b @ ( take_list_b @ I @ Xs ) ) ) @ J )
        = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_b_nat @ size_size_list_b @ ( take_list_b @ K @ Xs ) ) ) @ L ) )
     => ( ( ord_less_nat @ I @ ( size_s420531428170919671list_b @ Xs ) )
       => ( ( ord_less_nat @ K @ ( size_s420531428170919671list_b @ Xs ) )
         => ( ( ord_less_nat @ J @ ( size_size_list_b @ ( nth_list_b @ Xs @ I ) ) )
           => ( ( ord_less_nat @ L @ ( size_size_list_b @ ( nth_list_b @ Xs @ K ) ) )
             => ( ( I = K )
                & ( J = L ) ) ) ) ) ) ) ).

% sum_list_split
thf(fact_1233_sum__list__split,axiom,
    ! [I: nat,Xs: 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 @ Xs ) ) ) @ J )
        = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ size_size_list_nat @ ( take_list_nat @ K @ Xs ) ) ) @ L ) )
     => ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
       => ( ( ord_less_nat @ K @ ( size_s3023201423986296836st_nat @ Xs ) )
         => ( ( ord_less_nat @ J @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ I ) ) )
           => ( ( ord_less_nat @ L @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ K ) ) )
             => ( ( I = K )
                & ( J = L ) ) ) ) ) ) ) ).

% sum_list_split
thf(fact_1234_sum__list__split,axiom,
    ! [I: nat,Xs: list_list_list_b,J: nat,K: nat,L: nat] :
      ( ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_list_b_nat @ size_s420531428170919671list_b @ ( take_list_list_b @ I @ Xs ) ) ) @ J )
        = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_list_b_nat @ size_s420531428170919671list_b @ ( take_list_list_b @ K @ Xs ) ) ) @ L ) )
     => ( ( ord_less_nat @ I @ ( size_s9106009106869804029list_b @ Xs ) )
       => ( ( ord_less_nat @ K @ ( size_s9106009106869804029list_b @ Xs ) )
         => ( ( ord_less_nat @ J @ ( size_s420531428170919671list_b @ ( nth_list_list_b @ Xs @ I ) ) )
           => ( ( ord_less_nat @ L @ ( size_s420531428170919671list_b @ ( nth_list_list_b @ Xs @ K ) ) )
             => ( ( I = K )
                & ( J = L ) ) ) ) ) ) ) ).

% sum_list_split
thf(fact_1235_sum__list__split,axiom,
    ! [I: nat,Xs: list_l3298181151656792051at_nat,J: nat,K: nat,L: nat] :
      ( ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_li9059624042496273785at_nat @ size_s3679842834875189465at_nat @ ( take_l4110974213330421595at_nat @ I @ Xs ) ) ) @ J )
        = ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_li9059624042496273785at_nat @ size_s3679842834875189465at_nat @ ( take_l4110974213330421595at_nat @ K @ Xs ) ) ) @ L ) )
     => ( ( ord_less_nat @ I @ ( size_s6878757106947045471at_nat @ Xs ) )
       => ( ( ord_less_nat @ K @ ( size_s6878757106947045471at_nat @ Xs ) )
         => ( ( ord_less_nat @ J @ ( size_s3679842834875189465at_nat @ ( nth_li8805938130461222772at_nat @ Xs @ I ) ) )
           => ( ( ord_less_nat @ L @ ( size_s3679842834875189465at_nat @ ( nth_li8805938130461222772at_nat @ Xs @ K ) ) )
             => ( ( I = K )
                & ( J = L ) ) ) ) ) ) ) ).

% sum_list_split
thf(fact_1236_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_Ground_gmctxt_a,N: nat] :
      ( ( ord_less_nat @ M @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( nth_Pr3641570550162790742ctxt_a @ ( enumer3922341085786344185ctxt_a @ N @ Xs ) @ M )
        = ( produc3104537575318011527ctxt_a @ ( plus_plus_nat @ N @ M ) @ ( nth_Ground_gmctxt_a @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_1237_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_b,N: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_b @ Xs ) )
     => ( ( nth_Pr2476257085692543965_nat_b @ ( enumerate_b @ N @ Xs ) @ M )
        = ( product_Pair_nat_b @ ( plus_plus_nat @ N @ M ) @ ( nth_b @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_1238_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_nat,N: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M )
        = ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_nat @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_1239_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_list_b,N: nat] :
      ( ( ord_less_nat @ M @ ( size_s420531428170919671list_b @ Xs ) )
     => ( ( nth_Pr6103996347157616483list_b @ ( enumerate_list_b @ N @ Xs ) @ M )
        = ( produc7903367361620597084list_b @ ( plus_plus_nat @ N @ M ) @ ( nth_list_b @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_1240_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_P8469869581646625389at_nat,N: nat] :
      ( ( ord_less_nat @ M @ ( size_s3679842834875189465at_nat @ Xs ) )
     => ( ( nth_Pr4155663282896371525at_nat @ ( enumer3122081344003035212at_nat @ N @ Xs ) @ M )
        = ( produc7482930922861554750at_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_Pr6744343527793145070at_nat @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_1241_sum__list__take__eq,axiom,
    ! [K: nat,I: nat,Xs: list_nat] :
      ( ( ord_less_nat @ K @ I )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( groups4561878855575611511st_nat @ ( take_nat @ I @ Xs ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( take_nat @ K @ Xs ) ) @ ( nth_nat @ Xs @ K ) ) @ ( groups4561878855575611511st_nat @ ( take_nat @ ( minus_minus_nat @ I @ ( suc @ K ) ) @ ( drop_nat @ ( suc @ K ) @ Xs ) ) ) ) ) ) ) ).

% sum_list_take_eq
thf(fact_1242_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_1243_Suc__diff__1,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
        = N ) ) ).

% Suc_diff_1
thf(fact_1244_nat__induct__non__zero,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P @ one_one_nat )
       => ( ! [N3: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N3 )
             => ( ( P @ N3 )
               => ( P @ ( suc @ N3 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_1245_Suc__pred_H,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( N
        = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_1246_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_1247_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M: nat] :
      ( ! [K3: nat] :
          ( ( ord_less_nat @ N @ K3 )
         => ( P @ K3 ) )
     => ( ! [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ K3 @ I4 )
                 => ( P @ I4 ) )
             => ( P @ K3 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_1248_pair__lessI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ S @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_1249_pair__less__iff1,axiom,
    ! [X: nat,Y3: nat,Z4: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( product_Pair_nat_nat @ X @ Z4 ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y3 @ Z4 ) ) ).

% pair_less_iff1
thf(fact_1250_pair__lessI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_1251_pair__leqI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ S @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_1252_pair__leqI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).

% pair_leqI1

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y3: nat] :
      ( ( if_nat @ $false @ X @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y3: nat] :
      ( ( if_nat @ $true @ X @ Y3 )
      = X ) ).

% Conjectures (2)
thf(conj_0,hypothesis,
    ! [I4: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ cs ) )
     => ( ( member_b @ x @ ( set_b2 @ ( nth_list_b @ ( multih3537974503344769663n_by_b @ xs @ ( map_Gr1679544123147727284_a_nat @ ground_num_gholes_a @ cs ) ) @ I4 ) ) )
       => thesis ) ) ).

thf(conj_1,conjecture,
    thesis ).

%------------------------------------------------------------------------------