TPTP Problem File: SLH0729^1.p

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

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Eval_FO/0005_Ailamazyan/prob_00988_035628__15658486_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1753 ( 653 unt; 487 typ;   0 def)
%            Number of atoms       : 3361 (1496 equ;   0 cnn)
%            Maximal formula atoms :   22 (   2 avg)
%            Number of connectives : 11682 ( 451   ~;  75   |; 230   &;9405   @)
%                                         (   0 <=>;1521  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   7 avg)
%            Number of types       :   92 (  91 usr)
%            Number of type conns  : 1742 (1742   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  399 ( 396 usr;  21 con; 0-5 aty)
%            Number of variables   : 3947 ( 169   ^;3647   !; 131   ?;3947   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 13:06:12.603
%------------------------------------------------------------------------------
% Could-be-implicit typings (91)
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
    produc6644633505913852071_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J_J_J,type,
    produc1308816576071065077_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    produc1257712949554669095_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
    produc3840784484372613241_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J_J,type,
    produc4888785615639907219_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    produc2645141645350940391_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
    produc2080012076104740651_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_J_J,type,
    list_P6702208479758181172_nat_o: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
    list_P3713345943197369318_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_J_J,type,
    set_Pr2355761042751430670_nat_o: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
    set_Pr8590270543045394624_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_J,type,
    produc2570910961013241518_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    produc1169793092761538656_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
    produc4708264106712766336_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    list_P6037234014419372525_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    produc5706546023945342183_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    produc8636276049245547145_a_nat: $tType ).

thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
    option4961383686904876949_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    produc7989883672844959049_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    produc6993310590455123023_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_J_J,type,
    list_P101710509129098134_nat_o: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    list_P5215039326508400388_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_J_J,type,
    set_Pr1318278509876868720_nat_o: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    set_Pr6431607327256170974_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_Mt__Nat__Onat_J_J,type,
    list_P6081073431594039286at_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
    list_P2367225310644091548_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    produc1342688880896445543st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_Mt__Nat__Onat_J_J,type,
    set_Pr5533812812095091868at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
    set_Pr1819964691145144130_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    list_P6700786065914020821_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    list_P4071774479222187013_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_J,type,
    produc6714841562171439888_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    produc3388037434187558270_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_Mt__Nat__Onat_J,type,
    produc4503689834541559270at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    produc2799874090934363020_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    produc4421259400484624207_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc8736486873979948927_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
    list_P5306481420966312054at_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    list_P6608106274694273645_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    set_Pr4870381170404451655_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    list_P4892687945159610899_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    list_P2238440198752223047_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    list_P314924338204649965_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    produc3560618544861544438st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    produc5001885624171833703_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    list_P8687318089619639155_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    list_P8619360983253792231_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    list_l7568939571203725683_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    set_li24642999971764557_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    list_s8724375208233467597_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    list_P6914728637955986253_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    list_P3065721703829944461_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    set_se7876836687606533031_a_nat: $tType ).

thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    option8121340161786669997_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    produc4502985402200462317_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    list_P1195027771636113901_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    set_Pr7343886759072863943_a_nat: $tType ).

thf(ty_n_t__List__Olist_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_J,type,
    list_S9131871693400613940_nat_o: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_J,type,
    set_Su5390745688266870030_nat_o: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    list_P8484311456091262906_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
    list_P604317786108825364at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc7017002724195966439_a_nat: $tType ).

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

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

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    list_P6164600145584960654at_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    list_P5056861408695629236_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    list_l1586611297644397841_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    set_li8553678471444574443_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    list_s602657755630655083_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    list_l999229665597553643_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    set_se4330304633200676677_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    set_li4726876543167575237_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    list_s8696977637532211781_a_nat: $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__Option__Ooption_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    option2159618704653257803_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    list_l4703314356710769291_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    set_li6526943997496501093_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    list_s7376487411892226277_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    set_se4904748513628223167_a_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__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    option_Sum_sum_a_nat: $tType ).

thf(ty_n_t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    list_Sum_sum_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    set_Sum_sum_a_nat: $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__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    sum_sum_a_nat: $tType ).

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

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

% Explicit typings (396)
thf(sy_c_AList_Oupdate__with__aux_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    update817380752136475202_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat > ( list_Sum_sum_a_nat > list_Sum_sum_a_nat ) > list_P6608106274694273645_a_nat > list_P6608106274694273645_a_nat ).

thf(sy_c_AList_Oupdate__with__aux_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    update5424960881307245506_a_nat: sum_sum_a_nat > sum_sum_a_nat > ( sum_sum_a_nat > sum_sum_a_nat ) > list_P1195027771636113901_a_nat > list_P1195027771636113901_a_nat ).

thf(sy_c_Ailamazyan_Ofo__nmlzd_001tf__a,type,
    fo_nmlzd_a: set_a > list_Sum_sum_a_nat > $o ).

thf(sy_c_Ailamazyan_Orremdups_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    rremdu4777469547168625599_a_nat: list_l4703314356710769291_a_nat > list_l4703314356710769291_a_nat ).

thf(sy_c_Ailamazyan_Orremdups_001t__Nat__Onat,type,
    rremdups_nat: list_nat > list_nat ).

thf(sy_c_Ailamazyan_Orremdups_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    rremdu7275748759011619361_a_nat: list_P1195027771636113901_a_nat > list_P1195027771636113901_a_nat ).

thf(sy_c_Ailamazyan_Orremdups_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    rremdu6376807763710068377_a_nat: list_s7376487411892226277_a_nat > list_s7376487411892226277_a_nat ).

thf(sy_c_Ailamazyan_Orremdups_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    rremdu8304153113908149561_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    bNF_Gr1229660863860170270_a_nat: set_li6526943997496501093_a_nat > sum_sum_a_nat > set_li6526943997496501093_a_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    bNF_Gr5582227268375839130_a_nat: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat > set_Sum_sum_a_nat ).

thf(sy_c_Fun_Ofun__upd_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    fun_up3573206269406175639_a_nat: ( list_Sum_sum_a_nat > set_Sum_sum_a_nat ) > list_Sum_sum_a_nat > set_Sum_sum_a_nat > list_Sum_sum_a_nat > set_Sum_sum_a_nat ).

thf(sy_c_Fun_Ofun__upd_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    fun_up6086130847573437501_a_nat: ( sum_sum_a_nat > sum_sum_a_nat ) > sum_sum_a_nat > sum_sum_a_nat > sum_sum_a_nat > sum_sum_a_nat ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
    times_times_nat: nat > nat > nat ).

thf(sy_c_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    if_lis4685338526944683083_a_nat: $o > list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).

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

thf(sy_c_If_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    if_set_Sum_sum_a_nat: $o > set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).

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

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    sup_su4083067149120280889_a_nat: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    sup_su4655406599365709595_a_nat: set_Pr4870381170404451655_a_nat > set_Pr4870381170404451655_a_nat > set_Pr4870381170404451655_a_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    sup_su2259505869435368603_a_nat: set_Pr7343886759072863943_a_nat > set_Pr7343886759072863943_a_nat > set_Pr7343886759072863943_a_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    sup_su2291686591051470483_a_nat: set_se4904748513628223167_a_nat > set_se4904748513628223167_a_nat > set_se4904748513628223167_a_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    sup_su6804446743777130803_a_nat: set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).

thf(sy_c_List_Oappend_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    append_Sum_sum_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).

thf(sy_c_List_Obind_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    bind_l6437164444685190343_a_nat: list_l4703314356710769291_a_nat > ( list_Sum_sum_a_nat > list_l4703314356710769291_a_nat ) > list_l4703314356710769291_a_nat ).

thf(sy_c_List_Obind_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
    bind_l5927154061965146638at_nat: list_l4703314356710769291_a_nat > ( list_Sum_sum_a_nat > list_nat ) > list_nat ).

thf(sy_c_List_Obind_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    bind_l7097378772172010049_a_nat: list_l4703314356710769291_a_nat > ( list_Sum_sum_a_nat > list_Sum_sum_a_nat ) > list_Sum_sum_a_nat ).

thf(sy_c_List_Obind_001t__Nat__Onat_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    bind_n431031891494352236_a_nat: list_nat > ( nat > list_l4703314356710769291_a_nat ) > list_l4703314356710769291_a_nat ).

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

thf(sy_c_List_Obind_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    bind_n2905796707917019494_a_nat: list_nat > ( nat > list_Sum_sum_a_nat ) > list_Sum_sum_a_nat ).

thf(sy_c_List_Obind_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
    bind_P5957727470335167600at_nat: list_P1195027771636113901_a_nat > ( produc7017002724195966439_a_nat > list_nat ) > list_nat ).

thf(sy_c_List_Obind_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    bind_S7470141391546446541_a_nat: list_Sum_sum_a_nat > ( sum_sum_a_nat > list_l4703314356710769291_a_nat ) > list_l4703314356710769291_a_nat ).

thf(sy_c_List_Obind_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    bind_S7309778641578091144at_nat: list_Sum_sum_a_nat > ( sum_sum_a_nat > list_nat ) > list_nat ).

thf(sy_c_List_Obind_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    bind_S8497345843576593223_a_nat: list_Sum_sum_a_nat > ( sum_sum_a_nat > list_Sum_sum_a_nat ) > list_Sum_sum_a_nat ).

thf(sy_c_List_Ocan__select_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    can_se2197130373557366786_a_nat: ( list_Sum_sum_a_nat > $o ) > set_li6526943997496501093_a_nat > $o ).

thf(sy_c_List_Ocan__select_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    can_se5661427355321125604_a_nat: ( produc5001885624171833703_a_nat > $o ) > set_Pr4870381170404451655_a_nat > $o ).

thf(sy_c_List_Ocan__select_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    can_se8355593179296023652_a_nat: ( produc7017002724195966439_a_nat > $o ) > set_Pr7343886759072863943_a_nat > $o ).

thf(sy_c_List_Ocan__select_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    can_se4650645106769343580_a_nat: ( set_Sum_sum_a_nat > $o ) > set_se4904748513628223167_a_nat > $o ).

thf(sy_c_List_Ocan__select_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    can_se22271275864192764_a_nat: ( sum_sum_a_nat > $o ) > set_Sum_sum_a_nat > $o ).

thf(sy_c_List_Ocoset_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    coset_2342676519794954744_a_nat: list_l4703314356710769291_a_nat > set_li6526943997496501093_a_nat ).

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

thf(sy_c_List_Ocoset_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    coset_3293031159811469914_a_nat: list_P1195027771636113901_a_nat > set_Pr7343886759072863943_a_nat ).

thf(sy_c_List_Ocoset_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    coset_Sum_sum_a_nat: list_Sum_sum_a_nat > set_Sum_sum_a_nat ).

thf(sy_c_List_Odistinct_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    distin5621845738133643396_a_nat: list_l1586611297644397841_a_nat > $o ).

thf(sy_c_List_Odistinct_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    distin5816588314374693350_a_nat: list_l7568939571203725683_a_nat > $o ).

thf(sy_c_List_Odistinct_001t__List__Olist_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    distin7371808003173651294_a_nat: list_l999229665597553643_a_nat > $o ).

thf(sy_c_List_Odistinct_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    distin811021574259663358_a_nat: list_l4703314356710769291_a_nat > $o ).

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

thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    distin3585382044144064736_a_nat: list_P6608106274694273645_a_nat > $o ).

thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    distin2310918592538278330_a_nat: list_P2238440198752223047_a_nat > $o ).

thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    distin6082956270961113178_a_nat: list_P8619360983253792231_a_nat > $o ).

thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    distin2524318136373408390_a_nat: list_P4892687945159610899_a_nat > $o ).

thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    distin2540775250022794848_a_nat: list_P314924338204649965_a_nat > $o ).

thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    distin2369888004842216704_a_nat: list_P3065721703829944461_a_nat > $o ).

thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    distin9124545944939041766_a_nat: list_P8687318089619639155_a_nat > $o ).

thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    distin4917900207559653704_a_nat: list_P6700786065914020821_a_nat > $o ).

thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    distin2701942696825509824_a_nat: list_P6914728637955986253_a_nat > $o ).

thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    distin928153214662094432_a_nat: list_P1195027771636113901_a_nat > $o ).

thf(sy_c_List_Odistinct_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    distin6522264588777926110_a_nat: list_s602657755630655083_a_nat > $o ).

thf(sy_c_List_Odistinct_001t__Set__Oset_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    distin2944342510436172608_a_nat: list_s8724375208233467597_a_nat > $o ).

thf(sy_c_List_Odistinct_001t__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    distin2582252071445032376_a_nat: list_s8696977637532211781_a_nat > $o ).

thf(sy_c_List_Odistinct_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    distin4821671304206132312_a_nat: list_s7376487411892226277_a_nat > $o ).

thf(sy_c_List_Odistinct_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    distin2701893636801681144_a_nat: list_Sum_sum_a_nat > $o ).

thf(sy_c_List_Odrop_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    drop_Sum_sum_a_nat: nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).

thf(sy_c_List_Oextract_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    extrac3289232528736335879_a_nat: ( sum_sum_a_nat > $o ) > list_Sum_sum_a_nat > option4961383686904876949_a_nat ).

thf(sy_c_List_Ofilter_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    filter5373755100216644354_a_nat: ( list_Sum_sum_a_nat > $o ) > list_l4703314356710769291_a_nat > list_l4703314356710769291_a_nat ).

thf(sy_c_List_Ofilter_001t__Nat__Onat,type,
    filter_nat: ( nat > $o ) > list_nat > list_nat ).

thf(sy_c_List_Ofilter_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    filter4328210566177025508_a_nat: ( produc5001885624171833703_a_nat > $o ) > list_P6608106274694273645_a_nat > list_P6608106274694273645_a_nat ).

thf(sy_c_List_Ofilter_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    filter6896953120613689188_a_nat: ( produc7017002724195966439_a_nat > $o ) > list_P1195027771636113901_a_nat > list_P1195027771636113901_a_nat ).

thf(sy_c_List_Ofilter_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    filter4447861443987396444_a_nat: ( set_Sum_sum_a_nat > $o ) > list_s7376487411892226277_a_nat > list_s7376487411892226277_a_nat ).

thf(sy_c_List_Ofilter_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    filter_Sum_sum_a_nat: ( sum_sum_a_nat > $o ) > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).

thf(sy_c_List_Olenlex_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    lenlex_Sum_sum_a_nat: set_Pr7343886759072863943_a_nat > set_Pr4870381170404451655_a_nat ).

thf(sy_c_List_Olex_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    lex_Sum_sum_a_nat: set_Pr7343886759072863943_a_nat > set_Pr4870381170404451655_a_nat ).

thf(sy_c_List_Olexord_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    lexord_Sum_sum_a_nat: set_Pr7343886759072863943_a_nat > set_Pr4870381170404451655_a_nat ).

thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    cons_l6604326339930385211_a_nat: list_Sum_sum_a_nat > list_l4703314356710769291_a_nat > list_l4703314356710769291_a_nat ).

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

thf(sy_c_List_Olist_OCons_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    cons_Sum_sum_a_nat: sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).

thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    nil_li1906260230833442699_a_nat: list_l4703314356710769291_a_nat ).

thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    nil_Pr1141218757497453421_a_nat: list_P6608106274694273645_a_nat ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
    nil_Pr2978926869271283070at_nat: list_P604317786108825364at_nat ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    nil_Pr4306293334262433127_a_nat: list_P8619360983253792231_a_nat ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    nil_Pr2751885571400186148_a_nat: list_P8484311456091262906_a_nat ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    nil_Pr5478986624290739719at_nat: list_P6011104703257516679at_nat ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    nil_Pr237480997409426078_a_nat: list_P5056861408695629236_a_nat ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
    nil_Pr1739053331407100384at_nat: list_P5306481420966312054at_nat ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    nil_Pr7347883008240361715_a_nat: list_P8687318089619639155_a_nat ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    nil_Pr2041209113074292344at_nat: list_P6164600145584960654at_nat ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    nil_Pr6585251977456444909_a_nat: list_P1195027771636113901_a_nat ).

thf(sy_c_List_Olist_ONil_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    nil_se616183255201024101_a_nat: list_s7376487411892226277_a_nat ).

thf(sy_c_List_Olist_ONil_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    nil_Sum_sum_a_nat: list_Sum_sum_a_nat ).

thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    hd_Pro3965030861105696466_a_nat: list_P1195027771636113901_a_nat > produc7017002724195966439_a_nat ).

thf(sy_c_List_Olist_Ohd_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    hd_Sum_sum_a_nat: list_Sum_sum_a_nat > sum_sum_a_nat ).

thf(sy_c_List_Olist_Olist__all2_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    list_a5617550422536820423_a_nat: ( list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o ) > list_l4703314356710769291_a_nat > list_l4703314356710769291_a_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
    list_a1471712951254312974at_nat: ( list_Sum_sum_a_nat > nat > $o ) > list_l4703314356710769291_a_nat > list_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    list_a7096074725797064097_a_nat: ( list_Sum_sum_a_nat > set_Sum_sum_a_nat > $o ) > list_l4703314356710769291_a_nat > list_s7376487411892226277_a_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    list_a8267565288689615937_a_nat: ( list_Sum_sum_a_nat > sum_sum_a_nat > $o ) > list_l4703314356710769291_a_nat > list_Sum_sum_a_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Nat__Onat_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    list_a5198962817638294380_a_nat: ( nat > list_Sum_sum_a_nat > $o ) > list_nat > list_l4703314356710769291_a_nat > $o ).

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

thf(sy_c_List_Olist_Olist__all2_001t__Nat__Onat_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    list_a6225475761577929678_a_nat: ( nat > produc7017002724195966439_a_nat > $o ) > list_nat > list_P1195027771636113901_a_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    list_a8277918560160606054_a_nat: ( nat > sum_sum_a_nat > $o ) > list_nat > list_Sum_sum_a_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    list_a2005310791389460295_a_nat: ( produc5001885624171833703_a_nat > produc5001885624171833703_a_nat > $o ) > list_P6608106274694273645_a_nat > list_P6608106274694273645_a_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
    list_a7879879654818888304at_nat: ( produc7017002724195966439_a_nat > nat > $o ) > list_P1195027771636113901_a_nat > list_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    list_a2823864037932130119_a_nat: ( produc7017002724195966439_a_nat > produc7017002724195966439_a_nat > $o ) > list_P1195027771636113901_a_nat > list_P1195027771636113901_a_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    list_a746113844164839775_a_nat: ( produc7017002724195966439_a_nat > sum_sum_a_nat > $o ) > list_P1195027771636113901_a_nat > list_Sum_sum_a_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    list_a2188187147496901485_a_nat: ( set_Sum_sum_a_nat > list_Sum_sum_a_nat > $o ) > list_s7376487411892226277_a_nat > list_l4703314356710769291_a_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    list_a4614116639835821383_a_nat: ( set_Sum_sum_a_nat > set_Sum_sum_a_nat > $o ) > list_s7376487411892226277_a_nat > list_s7376487411892226277_a_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    list_a8920083903403951079_a_nat: ( set_Sum_sum_a_nat > sum_sum_a_nat > $o ) > list_s7376487411892226277_a_nat > list_Sum_sum_a_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    list_a8640327908064052429_a_nat: ( sum_sum_a_nat > list_Sum_sum_a_nat > $o ) > list_Sum_sum_a_nat > list_l4703314356710769291_a_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    list_a3458528456966901896at_nat: ( sum_sum_a_nat > nat > $o ) > list_Sum_sum_a_nat > list_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    list_a1440509178630569775_a_nat: ( sum_sum_a_nat > produc7017002724195966439_a_nat > $o ) > list_Sum_sum_a_nat > list_P1195027771636113901_a_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    list_a2481226626064268967_a_nat: ( sum_sum_a_nat > set_Sum_sum_a_nat > $o ) > list_Sum_sum_a_nat > list_s7376487411892226277_a_nat > $o ).

thf(sy_c_List_Olist_Olist__all2_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    list_a8491001234060422471_a_nat: ( sum_sum_a_nat > sum_sum_a_nat > $o ) > list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    map_li2834150261122285406_a_nat: ( list_l4703314356710769291_a_nat > set_li6526943997496501093_a_nat ) > list_l1586611297644397841_a_nat > list_s602657755630655083_a_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    map_li6209216568146725854_a_nat: ( list_P1195027771636113901_a_nat > set_Pr7343886759072863943_a_nat ) > list_l7568939571203725683_a_nat > list_s8724375208233467597_a_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    map_li3657301428870794462_a_nat: ( list_s7376487411892226277_a_nat > set_se4904748513628223167_a_nat ) > list_l999229665597553643_a_nat > list_s8696977637532211781_a_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    map_li6507455427659069316_a_nat: ( list_Sum_sum_a_nat > list_Sum_sum_a_nat ) > list_l4703314356710769291_a_nat > list_l4703314356710769291_a_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
    map_li1827815449145584657at_nat: ( list_Sum_sum_a_nat > nat ) > list_l4703314356710769291_a_nat > list_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    map_li6727107299570610150_a_nat: ( list_Sum_sum_a_nat > produc7017002724195966439_a_nat ) > list_l4703314356710769291_a_nat > list_P1195027771636113901_a_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    map_li9079814933684165854_a_nat: ( list_Sum_sum_a_nat > set_Sum_sum_a_nat ) > list_l4703314356710769291_a_nat > list_s7376487411892226277_a_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    map_li3106614296470775678_a_nat: ( list_Sum_sum_a_nat > sum_sum_a_nat ) > list_l4703314356710769291_a_nat > list_Sum_sum_a_nat ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    map_na5555065315529566063_a_nat: ( nat > list_Sum_sum_a_nat ) > list_nat > list_l4703314356710769291_a_nat ).

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

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    map_na7890605889587578321_a_nat: ( nat > produc7017002724195966439_a_nat ) > list_nat > list_P1195027771636113901_a_nat ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    map_na823391071729141993_a_nat: ( nat > sum_sum_a_nat ) > list_nat > list_Sum_sum_a_nat ).

thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    map_Pr5241799845241595682_a_nat: ( produc5001885624171833703_a_nat > list_Sum_sum_a_nat ) > list_P6608106274694273645_a_nat > list_l4703314356710769291_a_nat ).

thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    map_Pr3377394081699160580_a_nat: ( produc5001885624171833703_a_nat > produc5001885624171833703_a_nat ) > list_P6608106274694273645_a_nat > list_P6608106274694273645_a_nat ).

thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    map_Pr9108395128902024172_a_nat: ( produc8736486873979948927_a_nat > produc7017002724195966439_a_nat ) > list_P4071774479222187013_a_nat > list_P1195027771636113901_a_nat ).

thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    map_Pr1987915761617520668_a_nat: ( produc4421259400484624207_a_nat > produc7017002724195966439_a_nat ) > list_P6700786065914020821_a_nat > list_P1195027771636113901_a_nat ).

thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    map_Pr5309716638702485410_a_nat: ( produc7017002724195966439_a_nat > list_Sum_sum_a_nat ) > list_P1195027771636113901_a_nat > list_l4703314356710769291_a_nat ).

thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
    map_Pr321637745973761139at_nat: ( produc7017002724195966439_a_nat > nat ) > list_P1195027771636113901_a_nat > list_nat ).

thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    map_Pr3484656804399879172_a_nat: ( produc7017002724195966439_a_nat > produc7017002724195966439_a_nat ) > list_P1195027771636113901_a_nat > list_P1195027771636113901_a_nat ).

thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    map_Pr3527087583225239196_a_nat: ( produc7017002724195966439_a_nat > sum_sum_a_nat ) > list_P1195027771636113901_a_nat > list_Sum_sum_a_nat ).

thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    map_se1607064023862629124_a_nat: ( set_Sum_sum_a_nat > set_Sum_sum_a_nat ) > list_s7376487411892226277_a_nat > list_s7376487411892226277_a_nat ).

thf(sy_c_List_Olist_Omap_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    map_Su3479376915845212170_a_nat: ( sum_sum_a_nat > list_Sum_sum_a_nat ) > list_Sum_sum_a_nat > list_l4703314356710769291_a_nat ).

thf(sy_c_List_Olist_Omap_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    map_Su5227373005390213643at_nat: ( sum_sum_a_nat > nat ) > list_Sum_sum_a_nat > list_nat ).

thf(sy_c_List_Olist_Omap_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    map_Su4221482917690969196_a_nat: ( sum_sum_a_nat > produc7017002724195966439_a_nat ) > list_Sum_sum_a_nat > list_P1195027771636113901_a_nat ).

thf(sy_c_List_Olist_Omap_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    map_Su6628313897983864932_a_nat: ( sum_sum_a_nat > set_Sum_sum_a_nat ) > list_Sum_sum_a_nat > list_s7376487411892226277_a_nat ).

thf(sy_c_List_Olist_Omap_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    map_Su2790769393171190532_a_nat: ( sum_sum_a_nat > sum_sum_a_nat ) > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).

thf(sy_c_List_Olist_Oset_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J,type,
    set_Su8658726457838599491_nat_o: list_S9131871693400613940_nat_o > set_Su5390745688266870030_nat_o ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    set_li7766024463445446560_a_nat: list_l1586611297644397841_a_nat > set_li8553678471444574443_a_nat ).

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

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    set_li1234883600314943746_a_nat: list_l7568939571203725683_a_nat > set_li24642999971764557_a_nat ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    set_li1117937094484414586_a_nat: list_l999229665597553643_a_nat > set_li4726876543167575237_a_nat ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    set_li2392974972034027290_a_nat: list_l4703314356710769291_a_nat > set_li6526943997496501093_a_nat ).

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

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    set_Pr2608475206242241043_a_nat: list_P5215039326508400388_a_nat > set_Pr6431607327256170974_a_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    set_Pr3652278763065731573_a_nat: list_P3713345943197369318_a_nat > set_Pr8590270543045394624_a_nat ).

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

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_J,type,
    set_Pr5935279334226122661_nat_o: list_P101710509129098134_nat_o > set_Pr1318278509876868720_nat_o ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    set_Pr808418073638175228_a_nat: list_P6608106274694273645_a_nat > set_Pr4870381170404451655_a_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    set_Pr7465079174729833655_a_nat: list_P2367225310644091548_a_nat > set_Pr1819964691145144130_a_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_J,type,
    set_Pr5053396631317434435_nat_o: list_P6702208479758181172_nat_o > set_Pr2355761042751430670_nat_o ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_Mt__Nat__Onat_J,type,
    set_Pr9168894918337029905at_nat: list_P6081073431594039286at_nat > set_Pr5533812812095091868at_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    set_Pr2931239450594890620_a_nat: list_P1195027771636113901_a_nat > set_Pr7343886759072863943_a_nat ).

thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    set_se6249060928767827828_a_nat: list_s7376487411892226277_a_nat > set_se4904748513628223167_a_nat ).

thf(sy_c_List_Olist_Oset_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    set_Sum_sum_a_nat2: list_Sum_sum_a_nat > set_Sum_sum_a_nat ).

thf(sy_c_List_Olist__ex1_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    list_e4824308470127291086_a_nat: ( list_Sum_sum_a_nat > $o ) > list_l4703314356710769291_a_nat > $o ).

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

thf(sy_c_List_Olist__ex1_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    list_e2762905939734966704_a_nat: ( produc5001885624171833703_a_nat > $o ) > list_P6608106274694273645_a_nat > $o ).

thf(sy_c_List_Olist__ex1_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    list_e7186075253971769136_a_nat: ( produc7017002724195966439_a_nat > $o ) > list_P1195027771636113901_a_nat > $o ).

thf(sy_c_List_Olist__ex1_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    list_e4825210516842321192_a_nat: ( set_Sum_sum_a_nat > $o ) > list_s7376487411892226277_a_nat > $o ).

thf(sy_c_List_Olist__ex1_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    list_e7903279875808954312_a_nat: ( sum_sum_a_nat > $o ) > list_Sum_sum_a_nat > $o ).

thf(sy_c_List_Olist__update_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J,type,
    list_u3817880519379542044_nat_o: list_S9131871693400613940_nat_o > nat > ( sum_sum_a_nat > sum_sum_a_nat > $o ) > list_S9131871693400613940_nat_o ).

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__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    list_u3340303885232119283_a_nat: list_l4703314356710769291_a_nat > nat > list_Sum_sum_a_nat > list_l4703314356710769291_a_nat ).

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

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    list_u6567445468097547756_a_nat: list_P5215039326508400388_a_nat > nat > produc3388037434187558270_a_nat > list_P5215039326508400388_a_nat ).

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    list_u560528484306587342_a_nat: list_P3713345943197369318_a_nat > nat > produc1169793092761538656_a_nat > list_P3713345943197369318_a_nat ).

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    list_u5682793034032774415st_nat: list_P7940050157051400743st_nat > nat > produc1828647624359046049st_nat > list_P7940050157051400743st_nat ).

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    list_u5062466531432706517_a_nat: list_P6608106274694273645_a_nat > nat > produc5001885624171833703_a_nat > list_P6608106274694273645_a_nat ).

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    list_u662661324514154910_a_nat: list_P2367225310644091548_a_nat > nat > produc2799874090934363020_a_nat > list_P2367225310644091548_a_nat ).

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    list_u932594531332099413_a_nat: list_P6037234014419372525_a_nat > nat > produc5706546023945342183_a_nat > list_P6037234014419372525_a_nat ).

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    list_u1716540500728718061_a_nat: list_P4071774479222187013_a_nat > nat > produc8736486873979948927_a_nat > list_P4071774479222187013_a_nat ).

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    list_u6624685064088169149_a_nat: list_P6700786065914020821_a_nat > nat > produc4421259400484624207_a_nat > list_P6700786065914020821_a_nat ).

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    list_u4149385051645232213_a_nat: list_P1195027771636113901_a_nat > nat > produc7017002724195966439_a_nat > list_P1195027771636113901_a_nat ).

thf(sy_c_List_Olist__update_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    list_u7655389300282739405_a_nat: list_s7376487411892226277_a_nat > nat > set_Sum_sum_a_nat > list_s7376487411892226277_a_nat ).

thf(sy_c_List_Olist__update_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    list_u9138855634547462509_a_nat: list_Sum_sum_a_nat > nat > sum_sum_a_nat > list_Sum_sum_a_nat ).

thf(sy_c_List_Olistrel1_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    listre5822253806113410398_a_nat: set_Pr7343886759072863943_a_nat > set_Pr4870381170404451655_a_nat ).

thf(sy_c_List_Omap__filter_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    map_fi8339409367572549514_a_nat: ( list_Sum_sum_a_nat > option2159618704653257803_a_nat ) > list_l4703314356710769291_a_nat > list_l4703314356710769291_a_nat ).

thf(sy_c_List_Omap__filter_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
    map_fi7901273347920489099at_nat: ( list_Sum_sum_a_nat > option_nat ) > list_l4703314356710769291_a_nat > list_nat ).

thf(sy_c_List_Omap__filter_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    map_fi7130843104821525124_a_nat: ( list_Sum_sum_a_nat > option_Sum_sum_a_nat ) > list_l4703314356710769291_a_nat > list_Sum_sum_a_nat ).

thf(sy_c_List_Omap__filter_001t__Nat__Onat_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    map_fi2405151177449694697_a_nat: ( nat > option2159618704653257803_a_nat ) > list_nat > list_l4703314356710769291_a_nat ).

thf(sy_c_List_Omap__filter_001t__Nat__Onat_001t__Nat__Onat,type,
    map_filter_nat_nat: ( nat > option_nat ) > list_nat > list_nat ).

thf(sy_c_List_Omap__filter_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    map_fi8306527579551992931_a_nat: ( nat > option_Sum_sum_a_nat ) > list_nat > list_Sum_sum_a_nat ).

thf(sy_c_List_Omap__filter_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
    map_fi4553214607747458797at_nat: ( produc7017002724195966439_a_nat > option_nat ) > list_P1195027771636113901_a_nat > list_nat ).

thf(sy_c_List_Omap__filter_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    map_fi8991591133371761674_a_nat: ( produc7017002724195966439_a_nat > option8121340161786669997_a_nat ) > list_P1195027771636113901_a_nat > list_P1195027771636113901_a_nat ).

thf(sy_c_List_Omap__filter_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    map_fi4286012129131091362_a_nat: ( produc7017002724195966439_a_nat > option_Sum_sum_a_nat ) > list_P1195027771636113901_a_nat > list_Sum_sum_a_nat ).

thf(sy_c_List_Omap__filter_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    map_fi7503605724195961616_a_nat: ( sum_sum_a_nat > option2159618704653257803_a_nat ) > list_Sum_sum_a_nat > list_l4703314356710769291_a_nat ).

thf(sy_c_List_Omap__filter_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    map_fi3487137476358288773at_nat: ( sum_sum_a_nat > option_nat ) > list_Sum_sum_a_nat > list_nat ).

thf(sy_c_List_Omap__filter_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    map_fi4980407463596821362_a_nat: ( sum_sum_a_nat > option8121340161786669997_a_nat ) > list_Sum_sum_a_nat > list_P1195027771636113901_a_nat ).

thf(sy_c_List_Omap__filter_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    map_fi2176729826146509066_a_nat: ( sum_sum_a_nat > option_Sum_sum_a_nat ) > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).

thf(sy_c_List_Omap__tailrec_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    map_ta4970547734792538152_a_nat: ( list_Sum_sum_a_nat > set_Sum_sum_a_nat ) > list_l4703314356710769291_a_nat > list_s7376487411892226277_a_nat ).

thf(sy_c_List_Omap__tailrec_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    map_ta197872344851190606_a_nat: ( sum_sum_a_nat > sum_sum_a_nat ) > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).

thf(sy_c_List_Onth_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    nth_Sum_sum_a_nat: list_Sum_sum_a_nat > nat > sum_sum_a_nat ).

thf(sy_c_List_Opartition_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    partit6043679984613064702_a_nat: ( sum_sum_a_nat > $o ) > list_Sum_sum_a_nat > produc5001885624171833703_a_nat ).

thf(sy_c_List_Oproduct_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc8487947894020673669_a_nat: list_l4703314356710769291_a_nat > list_l4703314356710769291_a_nat > list_P6608106274694273645_a_nat ).

thf(sy_c_List_Oproduct_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc4585785479280518751_a_nat: list_l4703314356710769291_a_nat > list_s7376487411892226277_a_nat > list_P2238440198752223047_a_nat ).

thf(sy_c_List_Oproduct_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    produc3075939308942243071_a_nat: list_l4703314356710769291_a_nat > list_Sum_sum_a_nat > list_P8619360983253792231_a_nat ).

thf(sy_c_List_Oproduct_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc8901269937835131947_a_nat: list_s7376487411892226277_a_nat > list_l4703314356710769291_a_nat > list_P4892687945159610899_a_nat ).

thf(sy_c_List_Oproduct_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc706862384706489605_a_nat: list_s7376487411892226277_a_nat > list_s7376487411892226277_a_nat > list_P314924338204649965_a_nat ).

thf(sy_c_List_Oproduct_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    produc4421902774560781221_a_nat: list_s7376487411892226277_a_nat > list_Sum_sum_a_nat > list_P3065721703829944461_a_nat ).

thf(sy_c_List_Oproduct_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc3448701928316679563_a_nat: list_Sum_sum_a_nat > list_l4703314356710769291_a_nat > list_P8687318089619639155_a_nat ).

thf(sy_c_List_Oproduct_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc2458227688186579949_a_nat: list_Sum_sum_a_nat > list_P1195027771636113901_a_nat > list_P6700786065914020821_a_nat ).

thf(sy_c_List_Oproduct_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc7206417534075874917_a_nat: list_Sum_sum_a_nat > list_s7376487411892226277_a_nat > list_P6914728637955986253_a_nat ).

thf(sy_c_List_Oproduct_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    produc7134955936706270469_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_P1195027771636113901_a_nat ).

thf(sy_c_List_Oproduct__lists_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc4105601411378645364_a_nat: list_l1586611297644397841_a_nat > list_l1586611297644397841_a_nat ).

thf(sy_c_List_Oproduct__lists_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc3960677393309065686_a_nat: list_l7568939571203725683_a_nat > list_l7568939571203725683_a_nat ).

thf(sy_c_List_Oproduct__lists_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc5572861305105196750_a_nat: list_l999229665597553643_a_nat > list_l999229665597553643_a_nat ).

thf(sy_c_List_Oproduct__lists_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    produc2893206433618375022_a_nat: list_l4703314356710769291_a_nat > list_l4703314356710769291_a_nat ).

thf(sy_c_List_Oshuffles__rel_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    shuffl6050921173287438825_a_nat: produc5001885624171833703_a_nat > produc5001885624171833703_a_nat > $o ).

thf(sy_c_List_Osplice_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    splice_Sum_sum_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).

thf(sy_c_List_Osplice__rel_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    splice6660502696798462123_a_nat: produc5001885624171833703_a_nat > produc5001885624171833703_a_nat > $o ).

thf(sy_c_List_Osubseqs_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    subseq4596077244143087834_a_nat: list_l4703314356710769291_a_nat > list_l1586611297644397841_a_nat ).

thf(sy_c_List_Osubseqs_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    subseq5198752961258583356_a_nat: list_P1195027771636113901_a_nat > list_l7568939571203725683_a_nat ).

thf(sy_c_List_Osubseqs_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    subseq1669510427954520372_a_nat: list_s7376487411892226277_a_nat > list_l999229665597553643_a_nat ).

thf(sy_c_List_Osubseqs_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    subseq8414445098004693972_a_nat: list_Sum_sum_a_nat > list_l4703314356710769291_a_nat ).

thf(sy_c_List_Otake_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    take_l4056659018708330355_a_nat: nat > list_l4703314356710769291_a_nat > list_l4703314356710769291_a_nat ).

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

thf(sy_c_List_Otake_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    take_P1460166718689098581_a_nat: nat > list_P6608106274694273645_a_nat > list_P6608106274694273645_a_nat ).

thf(sy_c_List_Otake_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    take_P6053833779430135509_a_nat: nat > list_P6037234014419372525_a_nat > list_P6037234014419372525_a_nat ).

thf(sy_c_List_Otake_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    take_P5314707920032984173_a_nat: nat > list_P4071774479222187013_a_nat > list_P4071774479222187013_a_nat ).

thf(sy_c_List_Otake_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    take_P999480446537659453_a_nat: nat > list_P6700786065914020821_a_nat > list_P6700786065914020821_a_nat ).

thf(sy_c_List_Otake_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    take_P3324508860426538453_a_nat: nat > list_P1195027771636113901_a_nat > list_P1195027771636113901_a_nat ).

thf(sy_c_List_Otake_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    take_s3782404380517143629_a_nat: nat > list_s7376487411892226277_a_nat > list_s7376487411892226277_a_nat ).

thf(sy_c_List_Otake_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    take_Sum_sum_a_nat: nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).

thf(sy_c_List_Ounion_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    union_2406054330973375359_a_nat: list_l4703314356710769291_a_nat > list_l4703314356710769291_a_nat > list_l4703314356710769291_a_nat ).

thf(sy_c_List_Ounion_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    union_7179124240371534305_a_nat: list_P1195027771636113901_a_nat > list_P1195027771636113901_a_nat > list_P1195027771636113901_a_nat ).

thf(sy_c_List_Ounion_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    union_185282995832094297_a_nat: list_s7376487411892226277_a_nat > list_s7376487411892226277_a_nat > list_s7376487411892226277_a_nat ).

thf(sy_c_List_Ounion_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    union_Sum_sum_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).

thf(sy_c_List_Ozip_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    zip_Su2412782442871466510_a_nat: list_S9131871693400613940_nat_o > list_l4703314356710769291_a_nat > list_P5215039326508400388_a_nat ).

thf(sy_c_List_Ozip_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    zip_Su7083767565316534512_a_nat: list_S9131871693400613940_nat_o > list_P6608106274694273645_a_nat > list_P3713345943197369318_a_nat ).

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

thf(sy_c_List_Ozip_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J,type,
    zip_li6912039999028636704_nat_o: list_l4703314356710769291_a_nat > list_S9131871693400613940_nat_o > list_P101710509129098134_nat_o ).

thf(sy_c_List_Ozip_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    zip_li7023018480688994807_a_nat: list_l4703314356710769291_a_nat > list_l4703314356710769291_a_nat > list_P6608106274694273645_a_nat ).

thf(sy_c_List_Ozip_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
    zip_li3415139562815969502at_nat: list_l4703314356710769291_a_nat > list_nat > list_P604317786108825364at_nat ).

thf(sy_c_List_Ozip_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    zip_li3824709458655879025_a_nat: list_l4703314356710769291_a_nat > list_Sum_sum_a_nat > list_P8619360983253792231_a_nat ).

thf(sy_c_List_Ozip_001t__Nat__Onat_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    zip_na7142389429199950908_a_nat: list_nat > list_l4703314356710769291_a_nat > list_P8484311456091262906_a_nat ).

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

thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    zip_na3853549618184943902_a_nat: list_nat > list_P6608106274694273645_a_nat > list_P2367225310644091548_a_nat ).

thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    zip_na2013496608136855606_a_nat: list_nat > list_Sum_sum_a_nat > list_P5056861408695629236_a_nat ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J,type,
    zip_Pr1675919120921984702_nat_o: list_P6608106274694273645_a_nat > list_S9131871693400613940_nat_o > list_P6702208479758181172_nat_o ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Nat__Onat,type,
    zip_Pr2845026493572296384at_nat: list_P6608106274694273645_a_nat > list_nat > list_P6081073431594039286at_nat ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
    zip_Pr5895826916983950144at_nat: list_P1195027771636113901_a_nat > list_nat > list_P5306481420966312054at_nat ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    zip_Pr9202122185580356215_a_nat: list_P1195027771636113901_a_nat > list_P1195027771636113901_a_nat > list_P6037234014419372525_a_nat ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    zip_Pr3707921967074149519_a_nat: list_P1195027771636113901_a_nat > list_Sum_sum_a_nat > list_P4071774479222187013_a_nat ).

thf(sy_c_List_Ozip_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    zip_Su4197472078030315517_a_nat: list_Sum_sum_a_nat > list_l4703314356710769291_a_nat > list_P8687318089619639155_a_nat ).

thf(sy_c_List_Ozip_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    zip_Su6417478541797927256at_nat: list_Sum_sum_a_nat > list_nat > list_P6164600145584960654at_nat ).

thf(sy_c_List_Ozip_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    zip_Su4402317301539879519_a_nat: list_Sum_sum_a_nat > list_P1195027771636113901_a_nat > list_P6700786065914020821_a_nat ).

thf(sy_c_List_Ozip_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    zip_Su7355543910597222519_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_P1195027771636113901_a_nat ).

thf(sy_c_Map_Omap__of_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    map_of7357225391242338723_a_nat: list_P1195027771636113901_a_nat > sum_sum_a_nat > option_Sum_sum_a_nat ).

thf(sy_c_Map_Oran_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    ran_Su6606007654835531183_a_nat: ( sum_sum_a_nat > option_Sum_sum_a_nat ) > set_Sum_sum_a_nat ).

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    size_s5212483967078203639_a_nat: list_l4703314356710769291_a_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    size_s7247017532682008665_a_nat: list_P1195027771636113901_a_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    size_s2619173126757776721_a_nat: list_s7376487411892226277_a_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    size_s5283204784079214577_a_nat: list_Sum_sum_a_nat > nat ).

thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    none_P2628347790636752432_a_nat: option4961383686904876949_a_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
    some_nat: nat > option_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    some_P7299253002876453812_a_nat: produc6993310590455123023_a_nat > option4961383686904876949_a_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    some_Sum_sum_a_nat: sum_sum_a_nat > option_Sum_sum_a_nat ).

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    ord_le4638968407124865547_a_nat: set_Sum_sum_a_nat > set_Sum_sum_a_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_M_062_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
    ord_le6948912460583348046_nat_o: ( list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o ) > ( list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J,type,
    ord_le5975651393758295374_nat_o: ( sum_sum_a_nat > sum_sum_a_nat > $o ) > ( sum_sum_a_nat > sum_sum_a_nat > $o ) > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    ord_le1147066620699065093_a_nat: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    ord_le4837030659158318311_a_nat: set_Pr4870381170404451655_a_nat > set_Pr4870381170404451655_a_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    ord_le7501488065512009319_a_nat: set_Pr7343886759072863943_a_nat > set_Pr7343886759072863943_a_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    ord_le7974500612278410847_a_nat: set_se4904748513628223167_a_nat > set_se4904748513628223167_a_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    ord_le1325389633284124927_a_nat: set_Sum_sum_a_nat > set_Sum_sum_a_nat > $o ).

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

thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    ord_mi4660535444180972358_a_nat: set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc8857357843056227054_a_nat: ( sum_sum_a_nat > sum_sum_a_nat > $o ) > list_Sum_sum_a_nat > produc3388037434187558270_a_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    produc2770702104727902672_a_nat: ( sum_sum_a_nat > sum_sum_a_nat > $o ) > produc5001885624171833703_a_nat > produc1169793092761538656_a_nat ).

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__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J,type,
    produc4133243362358621440_nat_o: list_Sum_sum_a_nat > ( sum_sum_a_nat > sum_sum_a_nat > $o ) > produc6714841562171439888_nat_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc7990843422341522135_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat > produc5001885624171833703_a_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    produc3577755783922481849_a_nat: list_Sum_sum_a_nat > produc5001885624171833703_a_nat > produc7989883672844959049_a_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    produc3915550414976331931_a_nat: list_Sum_sum_a_nat > produc7989883672844959049_a_nat > produc2080012076104740651_a_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    produc7578773175992006207_a_nat: list_Sum_sum_a_nat > produc4502985402200462317_a_nat > produc6993310590455123023_a_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J_J,type,
    produc2222311365791241189_a_nat: list_Sum_sum_a_nat > produc4888785615639907219_a_nat > produc1308816576071065077_a_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    produc5571422593549375550_a_nat: nat > produc5001885624171833703_a_nat > produc2799874090934363020_a_nat ).

thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc6570249034887359912st_nat: option_nat > produc1828647624359046049st_nat > produc3560618544861544438st_nat ).

thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    produc785221303313254649_a_nat: option_Sum_sum_a_nat > produc5001885624171833703_a_nat > produc8636276049245547145_a_nat ).

thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    produc2856885992714847610_a_nat: option_Sum_sum_a_nat > produc2799874090934363020_a_nat > produc4708264106712766336_a_nat ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    produc3127615363426500119_a_nat: produc3388037434187558270_a_nat > produc3388037434187558270_a_nat > produc1257712949554669095_a_nat ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_001t__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    produc7511706038033277847_a_nat: produc1169793092761538656_a_nat > produc1169793092761538656_a_nat > produc6644633505913852071_a_nat ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc988773521843941591st_nat: produc1828647624359046049st_nat > produc1828647624359046049st_nat > produc1342688880896445543st_nat ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J,type,
    produc6586225697188128670_nat_o: produc5001885624171833703_a_nat > ( sum_sum_a_nat > sum_sum_a_nat > $o ) > produc2570910961013241518_nat_o ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Nat__Onat,type,
    produc4562899468936728032at_nat: produc5001885624171833703_a_nat > nat > produc4503689834541559270at_nat ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    produc3311116555204928343_a_nat: produc5001885624171833703_a_nat > produc5001885624171833703_a_nat > produc2645141645350940391_a_nat ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    produc5196189046490067889_a_nat: produc2799874090934363020_a_nat > produc2799874090934363020_a_nat > produc3840784484372613241_a_nat ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc4162992740250938199_a_nat: produc7017002724195966439_a_nat > produc7017002724195966439_a_nat > produc5706546023945342183_a_nat ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    produc8967895823682010479_a_nat: produc7017002724195966439_a_nat > sum_sum_a_nat > produc8736486873979948927_a_nat ).

thf(sy_c_Product__Type_OPair_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc6350064662657521885_a_nat: sum_sum_a_nat > list_Sum_sum_a_nat > produc4502985402200462317_a_nat ).

thf(sy_c_Product__Type_OPair_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
    produc5024437050894261379_a_nat: sum_sum_a_nat > produc2080012076104740651_a_nat > produc4888785615639907219_a_nat ).

thf(sy_c_Product__Type_OPair_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc438919121292964671_a_nat: sum_sum_a_nat > produc7017002724195966439_a_nat > produc4421259400484624207_a_nat ).

thf(sy_c_Product__Type_OPair_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    produc1212125651291703639_a_nat: sum_sum_a_nat > sum_sum_a_nat > produc7017002724195966439_a_nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc8258299166050244826_a_nat: produc3388037434187558270_a_nat > sum_sum_a_nat > sum_sum_a_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ofst_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    produc3757488583116884924_a_nat: produc1169793092761538656_a_nat > sum_sum_a_nat > sum_sum_a_nat > $o ).

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

thf(sy_c_Product__Type_Oprod_Ofst_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J,type,
    produc3534184685352639212_nat_o: produc6714841562171439888_nat_o > list_Sum_sum_a_nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc2015473822650161347_a_nat: produc5001885624171833703_a_nat > list_Sum_sum_a_nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    produc7060131065833098066_a_nat: produc2799874090934363020_a_nat > nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J,type,
    produc7573012175577110922_nat_o: produc2570910961013241518_nat_o > produc5001885624171833703_a_nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Nat__Onat,type,
    produc6051607941220450548at_nat: produc4503689834541559270at_nat > produc5001885624171833703_a_nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc2561508259746451779_a_nat: produc5706546023945342183_a_nat > produc7017002724195966439_a_nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    produc7342064319418986843_a_nat: produc8736486873979948927_a_nat > produc7017002724195966439_a_nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc8036459653884716843_a_nat: produc4421259400484624207_a_nat > sum_sum_a_nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    produc3940427395981429571_a_nat: produc7017002724195966439_a_nat > sum_sum_a_nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc6825104080635414556_a_nat: produc3388037434187558270_a_nat > list_Sum_sum_a_nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    produc7398779253711719678_a_nat: produc1169793092761538656_a_nat > produc5001885624171833703_a_nat ).

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

thf(sy_c_Product__Type_Oprod_Osnd_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J,type,
    produc2100989599937808942_nat_o: produc6714841562171439888_nat_o > sum_sum_a_nat > sum_sum_a_nat > $o ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc2553995981594484741_a_nat: produc5001885624171833703_a_nat > list_Sum_sum_a_nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    produc4900202433808968080_a_nat: produc2799874090934363020_a_nat > produc5001885624171833703_a_nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J,type,
    produc1990930809317169868_nat_o: produc2570910961013241518_nat_o > sum_sum_a_nat > sum_sum_a_nat > $o ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Nat__Onat,type,
    produc3891679309196320562at_nat: produc4503689834541559270at_nat > nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc4069943850158121093_a_nat: produc5706546023945342183_a_nat > produc7017002724195966439_a_nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    produc6099106989504882077_a_nat: produc8736486873979948927_a_nat > sum_sum_a_nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc6793502323970612077_a_nat: produc4421259400484624207_a_nat > produc7017002724195966439_a_nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    produc8028804885397038213_a_nat: produc7017002724195966439_a_nat > sum_sum_a_nat ).

thf(sy_c_Product__Type_Oprod_Oswap_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc2389921742016000142_a_nat: produc3388037434187558270_a_nat > produc6714841562171439888_nat_o ).

thf(sy_c_Product__Type_Oprod_Oswap_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    produc596726598905640816_a_nat: produc1169793092761538656_a_nat > produc2570910961013241518_nat_o ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc7439058410299273593st_nat: produc1828647624359046049st_nat > produc1828647624359046049st_nat ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J,type,
    produc6889179298173170336_nat_o: produc6714841562171439888_nat_o > produc3388037434187558270_a_nat ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    produc4286111038383333495_a_nat: produc5001885624171833703_a_nat > produc5001885624171833703_a_nat ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__Nat__Onat_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    produc5249648814965379742_a_nat: produc2799874090934363020_a_nat > produc4503689834541559270at_nat ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J,type,
    produc4412250191365866814_nat_o: produc2570910961013241518_nat_o > produc1169793092761538656_a_nat ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Nat__Onat,type,
    produc4241125690352732224at_nat: produc4503689834541559270at_nat > produc2799874090934363020_a_nat ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    produc33134186151372535_a_nat: produc7017002724195966439_a_nat > produc7017002724195966439_a_nat ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    collec7555443234367654128_a_nat: ( list_Sum_sum_a_nat > $o ) > set_li6526943997496501093_a_nat ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    collec3394043145073115602_a_nat: ( produc5001885624171833703_a_nat > $o ) > set_Pr4870381170404451655_a_nat ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    collec1294315627089844562_a_nat: ( produc7017002724195966439_a_nat > $o ) > set_Pr7343886759072863943_a_nat ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    collec4049389696321283146_a_nat: ( set_Sum_sum_a_nat > $o ) > set_se4904748513628223167_a_nat ).

thf(sy_c_Set_OCollect_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    collec7073057861543223018_a_nat: ( sum_sum_a_nat > $o ) > set_Sum_sum_a_nat ).

thf(sy_c_Set_OPow_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    pow_Sum_sum_a_nat: set_Sum_sum_a_nat > set_se4904748513628223167_a_nat ).

thf(sy_c_Set_Ofilter_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    filter4744287584154367926_a_nat: ( list_Sum_sum_a_nat > $o ) > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ).

thf(sy_c_Set_Ofilter_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    filter3383343615081260696_a_nat: ( produc5001885624171833703_a_nat > $o ) > set_Pr4870381170404451655_a_nat > set_Pr4870381170404451655_a_nat ).

thf(sy_c_Set_Ofilter_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    filter417278669945599000_a_nat: ( produc7017002724195966439_a_nat > $o ) > set_Pr7343886759072863943_a_nat > set_Pr7343886759072863943_a_nat ).

thf(sy_c_Set_Ofilter_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    filter6315372162244433424_a_nat: ( set_Sum_sum_a_nat > $o ) > set_se4904748513628223167_a_nat > set_se4904748513628223167_a_nat ).

thf(sy_c_Set_Ofilter_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    filter_Sum_sum_a_nat2: ( sum_sum_a_nat > $o ) > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).

thf(sy_c_Set_Oimage_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    image_2800575032594406047_a_nat: ( list_l4703314356710769291_a_nat > set_li6526943997496501093_a_nat ) > set_li8553678471444574443_a_nat > set_se4330304633200676677_a_nat ).

thf(sy_c_Set_Oimage_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    image_8418046036402785055_a_nat: ( list_P1195027771636113901_a_nat > set_Pr7343886759072863943_a_nat ) > set_li24642999971764557_a_nat > set_se7876836687606533031_a_nat ).

thf(sy_c_Set_Oimage_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    image_5081948215111134021_a_nat: ( list_Sum_sum_a_nat > list_Sum_sum_a_nat ) > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ).

thf(sy_c_Set_Oimage_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    image_188849635640366503_a_nat: ( list_Sum_sum_a_nat > produc7017002724195966439_a_nat ) > set_li6526943997496501093_a_nat > set_Pr7343886759072863943_a_nat ).

thf(sy_c_Set_Oimage_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    image_3940260845811589407_a_nat: ( list_Sum_sum_a_nat > set_Sum_sum_a_nat ) > set_li6526943997496501093_a_nat > set_se4904748513628223167_a_nat ).

thf(sy_c_Set_Oimage_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    image_6705138127771971519_a_nat: ( list_Sum_sum_a_nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat > set_Sum_sum_a_nat ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    image_7994831011627017571_a_nat: ( produc7017002724195966439_a_nat > list_Sum_sum_a_nat ) > set_Pr7343886759072863943_a_nat > set_li6526943997496501093_a_nat ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    image_4446780897320315845_a_nat: ( produc7017002724195966439_a_nat > produc7017002724195966439_a_nat ) > set_Pr7343886759072863943_a_nat > set_Pr7343886759072863943_a_nat ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    image_7319751417386094813_a_nat: ( produc7017002724195966439_a_nat > sum_sum_a_nat ) > set_Pr7343886759072863943_a_nat > set_Sum_sum_a_nat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    image_8255745304366202603_a_nat: ( set_Sum_sum_a_nat > list_Sum_sum_a_nat ) > set_se4904748513628223167_a_nat > set_li6526943997496501093_a_nat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    image_5599399343138760645_a_nat: ( set_Sum_sum_a_nat > set_Sum_sum_a_nat ) > set_se4904748513628223167_a_nat > set_se4904748513628223167_a_nat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    image_5092943885087329893_a_nat: ( set_Sum_sum_a_nat > sum_sum_a_nat ) > set_se4904748513628223167_a_nat > set_Sum_sum_a_nat ).

thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    image_7077900747146408011_a_nat: ( sum_sum_a_nat > list_Sum_sum_a_nat ) > set_Sum_sum_a_nat > set_li6526943997496501093_a_nat ).

thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    image_8014146751851824813_a_nat: ( sum_sum_a_nat > produc7017002724195966439_a_nat ) > set_Sum_sum_a_nat > set_Pr7343886759072863943_a_nat ).

thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    image_7877458644602423589_a_nat: ( sum_sum_a_nat > set_Sum_sum_a_nat ) > set_Sum_sum_a_nat > set_se4904748513628223167_a_nat ).

thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    image_7142520692256960453_a_nat: ( sum_sum_a_nat > sum_sum_a_nat ) > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).

thf(sy_c_Set__Linorder_Oproper__intrvl_OCompl__set__less__aux_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    set_pr2532843088427607396_a_nat: ( sum_sum_a_nat > sum_sum_a_nat > $o ) > ( option_Sum_sum_a_nat > option_Sum_sum_a_nat > $o ) > option_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o ).

thf(sy_c_Set__Linorder_Oproper__intrvl_OCompl__set__less__aux__rel_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    set_pr2097269997492014367_a_nat: ( sum_sum_a_nat > sum_sum_a_nat > $o ) > produc8636276049245547145_a_nat > produc8636276049245547145_a_nat > $o ).

thf(sy_c_Set__Linorder_Oproper__intrvl_Oset__less__eq__aux__Compl_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    set_pr1411927237749039722_a_nat: ( sum_sum_a_nat > sum_sum_a_nat > $o ) > ( option_Sum_sum_a_nat > option_Sum_sum_a_nat > $o ) > option_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o ).

thf(sy_c_Set__Linorder_Oproper__intrvl_Oset__less__eq__aux__Compl__rel_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    set_pr7360018828099327897_a_nat: ( sum_sum_a_nat > sum_sum_a_nat > $o ) > produc8636276049245547145_a_nat > produc8636276049245547145_a_nat > $o ).

thf(sy_c_Set__Linorder_Oproper__intrvl__class_OCompl__set__less__aux_001t__Nat__Onat,type,
    set_pr2156344338276335947ux_nat: option_nat > list_nat > list_nat > $o ).

thf(sy_c_Set__Linorder_Oproper__intrvl__class_OCompl__set__less__aux__rel_001t__Nat__Onat,type,
    set_pr6598557696555394320el_nat: produc3560618544861544438st_nat > produc3560618544861544438st_nat > $o ).

thf(sy_c_Set__Linorder_Oproper__intrvl__class_Oproper__interval_001t__Nat__Onat,type,
    set_pr3848668467136637714al_nat: option_nat > option_nat > $o ).

thf(sy_c_Set__Linorder_Oproper__intrvl__class_Oset__less__aux__Compl_001t__Nat__Onat,type,
    set_pr3298563537802266553pl_nat: option_nat > list_nat > list_nat > $o ).

thf(sy_c_Set__Linorder_Oproper__intrvl__class_Oset__less__eq__aux__Compl_001t__Nat__Onat,type,
    set_pr4650252130045998533pl_nat: option_nat > list_nat > list_nat > $o ).

thf(sy_c_Set__Linorder_Oproper__intrvl__class_Oset__less__eq__aux__Compl__rel_001t__Nat__Onat,type,
    set_pr9200262421968554262el_nat: produc3560618544861544438st_nat > produc3560618544861544438st_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    accp_P2437373109160157296_a_nat: ( produc5001885624171833703_a_nat > produc5001885624171833703_a_nat > $o ) > produc5001885624171833703_a_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    accp_P5867667535934726061st_nat: ( produc3560618544861544438st_nat > produc3560618544861544438st_nat > $o ) > produc3560618544861544438st_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    accp_P5320691669302659474_a_nat: ( produc8636276049245547145_a_nat > produc8636276049245547145_a_nat > $o ) > produc8636276049245547145_a_nat > $o ).

thf(sy_c_member_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J,type,
    member8178426116768754391_nat_o: ( sum_sum_a_nat > sum_sum_a_nat > $o ) > set_Su5390745688266870030_nat_o > $o ).

thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    member6656483030743143220_a_nat: list_l4703314356710769291_a_nat > set_li8553678471444574443_a_nat > $o ).

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

thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    member6227626580160590998_a_nat: list_P1195027771636113901_a_nat > set_li24642999971764557_a_nat > $o ).

thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    member6403511879741922830_a_nat: list_s7376487411892226277_a_nat > set_li4726876543167575237_a_nat > $o ).

thf(sy_c_member_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    member408289922725080238_a_nat: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    member3173438207993588647_a_nat: produc3388037434187558270_a_nat > set_Pr6431607327256170974_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    member6733730998074818441_a_nat: produc1169793092761538656_a_nat > set_Pr8590270543045394624_a_nat > $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__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_J,type,
    member6500242335977470265_nat_o: produc6714841562171439888_nat_o > set_Pr1318278509876868720_nat_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    member7457213283480048528_a_nat: produc5001885624171833703_a_nat > set_Pr4870381170404451655_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    member1670656326792299043_a_nat: produc2799874090934363020_a_nat > set_Pr1819964691145144130_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_J,type,
    member8134848866326521303_nat_o: produc2570910961013241518_nat_o > set_Pr2355761042751430670_nat_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_Mt__Nat__Onat_J,type,
    member3374472070399495293at_nat: produc4503689834541559270at_nat > set_Pr5533812812095091868at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    member3723442691059620112_a_nat: produc7017002724195966439_a_nat > set_Pr7343886759072863943_a_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    member5553968465346197646_a_nat: set_li6526943997496501093_a_nat > set_se4330304633200676677_a_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    member6181414433633234416_a_nat: set_Pr7343886759072863943_a_nat > set_se7876836687606533031_a_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    member8098812455498974984_a_nat: set_Sum_sum_a_nat > set_se4904748513628223167_a_nat > $o ).

thf(sy_c_member_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
    member_Sum_sum_a_nat: sum_sum_a_nat > set_Sum_sum_a_nat > $o ).

thf(sy_v_AD,type,
    ad: set_a ).

thf(sy_v_i,type,
    i: nat ).

thf(sy_v_xs,type,
    xs: list_Sum_sum_a_nat ).

% Relevant facts (1258)
thf(fact_0_assms,axiom,
    fo_nmlzd_a @ ad @ xs ).

% assms
thf(fact_1_take__equalityI,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,Ys: list_P1195027771636113901_a_nat] :
      ( ! [I: nat] :
          ( ( take_P3324508860426538453_a_nat @ I @ Xs )
          = ( take_P3324508860426538453_a_nat @ I @ Ys ) )
     => ( Xs = Ys ) ) ).

% take_equalityI
thf(fact_2_take__equalityI,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ! [I: nat] :
          ( ( take_Sum_sum_a_nat @ I @ Xs )
          = ( take_Sum_sum_a_nat @ I @ Ys ) )
     => ( Xs = Ys ) ) ).

% take_equalityI
thf(fact_3_map__filter__take,axiom,
    ! [F: sum_sum_a_nat > option8121340161786669997_a_nat,I2: nat,Xs: list_Sum_sum_a_nat] :
    ? [J: nat] :
      ( ( map_fi4980407463596821362_a_nat @ F @ ( take_Sum_sum_a_nat @ I2 @ Xs ) )
      = ( take_P3324508860426538453_a_nat @ J @ ( map_fi4980407463596821362_a_nat @ F @ Xs ) ) ) ).

% map_filter_take
thf(fact_4_map__filter__take,axiom,
    ! [F: produc7017002724195966439_a_nat > option_Sum_sum_a_nat,I2: nat,Xs: list_P1195027771636113901_a_nat] :
    ? [J: nat] :
      ( ( map_fi4286012129131091362_a_nat @ F @ ( take_P3324508860426538453_a_nat @ I2 @ Xs ) )
      = ( take_Sum_sum_a_nat @ J @ ( map_fi4286012129131091362_a_nat @ F @ Xs ) ) ) ).

% map_filter_take
thf(fact_5_map__filter__take,axiom,
    ! [F: produc7017002724195966439_a_nat > option8121340161786669997_a_nat,I2: nat,Xs: list_P1195027771636113901_a_nat] :
    ? [J: nat] :
      ( ( map_fi8991591133371761674_a_nat @ F @ ( take_P3324508860426538453_a_nat @ I2 @ Xs ) )
      = ( take_P3324508860426538453_a_nat @ J @ ( map_fi8991591133371761674_a_nat @ F @ Xs ) ) ) ).

% map_filter_take
thf(fact_6_map__filter__take,axiom,
    ! [F: sum_sum_a_nat > option_Sum_sum_a_nat,I2: nat,Xs: list_Sum_sum_a_nat] :
    ? [J: nat] :
      ( ( map_fi2176729826146509066_a_nat @ F @ ( take_Sum_sum_a_nat @ I2 @ Xs ) )
      = ( take_Sum_sum_a_nat @ J @ ( map_fi2176729826146509066_a_nat @ F @ Xs ) ) ) ).

% map_filter_take
thf(fact_7_rremdups__take,axiom,
    ! [I2: nat,Xs: list_P1195027771636113901_a_nat] :
    ? [J: nat] :
      ( ( rremdu7275748759011619361_a_nat @ ( take_P3324508860426538453_a_nat @ I2 @ Xs ) )
      = ( take_P3324508860426538453_a_nat @ J @ ( rremdu7275748759011619361_a_nat @ Xs ) ) ) ).

% rremdups_take
thf(fact_8_rremdups__take,axiom,
    ! [I2: nat,Xs: list_Sum_sum_a_nat] :
    ? [J: nat] :
      ( ( rremdu8304153113908149561_a_nat @ ( take_Sum_sum_a_nat @ I2 @ Xs ) )
      = ( take_Sum_sum_a_nat @ J @ ( rremdu8304153113908149561_a_nat @ Xs ) ) ) ).

% rremdups_take
thf(fact_9_filter__take,axiom,
    ! [P: produc7017002724195966439_a_nat > $o,I2: nat,Xs: list_P1195027771636113901_a_nat] :
    ? [J: nat] :
      ( ( filter6896953120613689188_a_nat @ P @ ( take_P3324508860426538453_a_nat @ I2 @ Xs ) )
      = ( take_P3324508860426538453_a_nat @ J @ ( filter6896953120613689188_a_nat @ P @ Xs ) ) ) ).

% filter_take
thf(fact_10_filter__take,axiom,
    ! [P: sum_sum_a_nat > $o,I2: nat,Xs: list_Sum_sum_a_nat] :
    ? [J: nat] :
      ( ( filter_Sum_sum_a_nat @ P @ ( take_Sum_sum_a_nat @ I2 @ Xs ) )
      = ( take_Sum_sum_a_nat @ J @ ( filter_Sum_sum_a_nat @ P @ Xs ) ) ) ).

% filter_take
thf(fact_11_take__take,axiom,
    ! [N: nat,M: nat,Xs: list_P1195027771636113901_a_nat] :
      ( ( take_P3324508860426538453_a_nat @ N @ ( take_P3324508860426538453_a_nat @ M @ Xs ) )
      = ( take_P3324508860426538453_a_nat @ ( ord_min_nat @ N @ M ) @ Xs ) ) ).

% take_take
thf(fact_12_take__take,axiom,
    ! [N: nat,M: nat,Xs: list_Sum_sum_a_nat] :
      ( ( take_Sum_sum_a_nat @ N @ ( take_Sum_sum_a_nat @ M @ Xs ) )
      = ( take_Sum_sum_a_nat @ ( ord_min_nat @ N @ M ) @ Xs ) ) ).

% take_take
thf(fact_13_take__Nil,axiom,
    ! [N: nat] :
      ( ( take_P3324508860426538453_a_nat @ N @ nil_Pr6585251977456444909_a_nat )
      = nil_Pr6585251977456444909_a_nat ) ).

% take_Nil
thf(fact_14_take__Nil,axiom,
    ! [N: nat] :
      ( ( take_l4056659018708330355_a_nat @ N @ nil_li1906260230833442699_a_nat )
      = nil_li1906260230833442699_a_nat ) ).

% take_Nil
thf(fact_15_take__Nil,axiom,
    ! [N: nat] :
      ( ( take_nat @ N @ nil_nat )
      = nil_nat ) ).

% take_Nil
thf(fact_16_take__Nil,axiom,
    ! [N: nat] :
      ( ( take_Sum_sum_a_nat @ N @ nil_Sum_sum_a_nat )
      = nil_Sum_sum_a_nat ) ).

% take_Nil
thf(fact_17_take__update__swap,axiom,
    ! [M: nat,Xs: list_P1195027771636113901_a_nat,N: nat,X: produc7017002724195966439_a_nat] :
      ( ( take_P3324508860426538453_a_nat @ M @ ( list_u4149385051645232213_a_nat @ Xs @ N @ X ) )
      = ( list_u4149385051645232213_a_nat @ ( take_P3324508860426538453_a_nat @ M @ Xs ) @ N @ X ) ) ).

% take_update_swap
thf(fact_18_take__update__swap,axiom,
    ! [M: nat,Xs: list_Sum_sum_a_nat,N: nat,X: sum_sum_a_nat] :
      ( ( take_Sum_sum_a_nat @ M @ ( list_u9138855634547462509_a_nat @ Xs @ N @ X ) )
      = ( list_u9138855634547462509_a_nat @ ( take_Sum_sum_a_nat @ M @ Xs ) @ N @ X ) ) ).

% take_update_swap
thf(fact_19_in__set__takeD,axiom,
    ! [X: set_Sum_sum_a_nat,N: nat,Xs: list_s7376487411892226277_a_nat] :
      ( ( member8098812455498974984_a_nat @ X @ ( set_se6249060928767827828_a_nat @ ( take_s3782404380517143629_a_nat @ N @ Xs ) ) )
     => ( member8098812455498974984_a_nat @ X @ ( set_se6249060928767827828_a_nat @ Xs ) ) ) ).

% in_set_takeD
thf(fact_20_in__set__takeD,axiom,
    ! [X: produc5001885624171833703_a_nat,N: nat,Xs: list_P6608106274694273645_a_nat] :
      ( ( member7457213283480048528_a_nat @ X @ ( set_Pr808418073638175228_a_nat @ ( take_P1460166718689098581_a_nat @ N @ Xs ) ) )
     => ( member7457213283480048528_a_nat @ X @ ( set_Pr808418073638175228_a_nat @ Xs ) ) ) ).

% in_set_takeD
thf(fact_21_in__set__takeD,axiom,
    ! [X: list_Sum_sum_a_nat,N: nat,Xs: list_l4703314356710769291_a_nat] :
      ( ( member408289922725080238_a_nat @ X @ ( set_li2392974972034027290_a_nat @ ( take_l4056659018708330355_a_nat @ N @ Xs ) ) )
     => ( member408289922725080238_a_nat @ X @ ( set_li2392974972034027290_a_nat @ Xs ) ) ) ).

% in_set_takeD
thf(fact_22_in__set__takeD,axiom,
    ! [X: produc7017002724195966439_a_nat,N: nat,Xs: list_P1195027771636113901_a_nat] :
      ( ( member3723442691059620112_a_nat @ X @ ( set_Pr2931239450594890620_a_nat @ ( take_P3324508860426538453_a_nat @ N @ Xs ) ) )
     => ( member3723442691059620112_a_nat @ X @ ( set_Pr2931239450594890620_a_nat @ Xs ) ) ) ).

% in_set_takeD
thf(fact_23_in__set__takeD,axiom,
    ! [X: sum_sum_a_nat,N: nat,Xs: list_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ ( take_Sum_sum_a_nat @ N @ Xs ) ) )
     => ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ).

% in_set_takeD
thf(fact_24_distinct__take,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,I2: nat] :
      ( ( distin928153214662094432_a_nat @ Xs )
     => ( distin928153214662094432_a_nat @ ( take_P3324508860426538453_a_nat @ I2 @ Xs ) ) ) ).

% distinct_take
thf(fact_25_distinct__take,axiom,
    ! [Xs: list_s7376487411892226277_a_nat,I2: nat] :
      ( ( distin4821671304206132312_a_nat @ Xs )
     => ( distin4821671304206132312_a_nat @ ( take_s3782404380517143629_a_nat @ I2 @ Xs ) ) ) ).

% distinct_take
thf(fact_26_distinct__take,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,I2: nat] :
      ( ( distin811021574259663358_a_nat @ Xs )
     => ( distin811021574259663358_a_nat @ ( take_l4056659018708330355_a_nat @ I2 @ Xs ) ) ) ).

% distinct_take
thf(fact_27_distinct__take,axiom,
    ! [Xs: list_Sum_sum_a_nat,I2: nat] :
      ( ( distin2701893636801681144_a_nat @ Xs )
     => ( distin2701893636801681144_a_nat @ ( take_Sum_sum_a_nat @ I2 @ Xs ) ) ) ).

% distinct_take
thf(fact_28_take__zip,axiom,
    ! [N: nat,Xs: list_Sum_sum_a_nat,Ys: list_P1195027771636113901_a_nat] :
      ( ( take_P999480446537659453_a_nat @ N @ ( zip_Su4402317301539879519_a_nat @ Xs @ Ys ) )
      = ( zip_Su4402317301539879519_a_nat @ ( take_Sum_sum_a_nat @ N @ Xs ) @ ( take_P3324508860426538453_a_nat @ N @ Ys ) ) ) ).

% take_zip
thf(fact_29_take__zip,axiom,
    ! [N: nat,Xs: list_P1195027771636113901_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( take_P5314707920032984173_a_nat @ N @ ( zip_Pr3707921967074149519_a_nat @ Xs @ Ys ) )
      = ( zip_Pr3707921967074149519_a_nat @ ( take_P3324508860426538453_a_nat @ N @ Xs ) @ ( take_Sum_sum_a_nat @ N @ Ys ) ) ) ).

% take_zip
thf(fact_30_take__zip,axiom,
    ! [N: nat,Xs: list_P1195027771636113901_a_nat,Ys: list_P1195027771636113901_a_nat] :
      ( ( take_P6053833779430135509_a_nat @ N @ ( zip_Pr9202122185580356215_a_nat @ Xs @ Ys ) )
      = ( zip_Pr9202122185580356215_a_nat @ ( take_P3324508860426538453_a_nat @ N @ Xs ) @ ( take_P3324508860426538453_a_nat @ N @ Ys ) ) ) ).

% take_zip
thf(fact_31_take__zip,axiom,
    ! [N: nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( take_P3324508860426538453_a_nat @ N @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) )
      = ( zip_Su7355543910597222519_a_nat @ ( take_Sum_sum_a_nat @ N @ Xs ) @ ( take_Sum_sum_a_nat @ N @ Ys ) ) ) ).

% take_zip
thf(fact_32_take__map,axiom,
    ! [N: nat,F: produc7017002724195966439_a_nat > sum_sum_a_nat,Xs: list_P1195027771636113901_a_nat] :
      ( ( take_Sum_sum_a_nat @ N @ ( map_Pr3527087583225239196_a_nat @ F @ Xs ) )
      = ( map_Pr3527087583225239196_a_nat @ F @ ( take_P3324508860426538453_a_nat @ N @ Xs ) ) ) ).

% take_map
thf(fact_33_take__map,axiom,
    ! [N: nat,F: sum_sum_a_nat > produc7017002724195966439_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( take_P3324508860426538453_a_nat @ N @ ( map_Su4221482917690969196_a_nat @ F @ Xs ) )
      = ( map_Su4221482917690969196_a_nat @ F @ ( take_Sum_sum_a_nat @ N @ Xs ) ) ) ).

% take_map
thf(fact_34_take__map,axiom,
    ! [N: nat,F: produc7017002724195966439_a_nat > produc7017002724195966439_a_nat,Xs: list_P1195027771636113901_a_nat] :
      ( ( take_P3324508860426538453_a_nat @ N @ ( map_Pr3484656804399879172_a_nat @ F @ Xs ) )
      = ( map_Pr3484656804399879172_a_nat @ F @ ( take_P3324508860426538453_a_nat @ N @ Xs ) ) ) ).

% take_map
thf(fact_35_take__map,axiom,
    ! [N: nat,F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat] :
      ( ( take_s3782404380517143629_a_nat @ N @ ( map_li9079814933684165854_a_nat @ F @ Xs ) )
      = ( map_li9079814933684165854_a_nat @ F @ ( take_l4056659018708330355_a_nat @ N @ Xs ) ) ) ).

% take_map
thf(fact_36_take__map,axiom,
    ! [N: nat,F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( take_Sum_sum_a_nat @ N @ ( map_Su2790769393171190532_a_nat @ F @ Xs ) )
      = ( map_Su2790769393171190532_a_nat @ F @ ( take_Sum_sum_a_nat @ N @ Xs ) ) ) ).

% take_map
thf(fact_37_list__all2__takeI,axiom,
    ! [P: sum_sum_a_nat > produc7017002724195966439_a_nat > $o,Xs: list_Sum_sum_a_nat,Ys: list_P1195027771636113901_a_nat,N: nat] :
      ( ( list_a1440509178630569775_a_nat @ P @ Xs @ Ys )
     => ( list_a1440509178630569775_a_nat @ P @ ( take_Sum_sum_a_nat @ N @ Xs ) @ ( take_P3324508860426538453_a_nat @ N @ Ys ) ) ) ).

% list_all2_takeI
thf(fact_38_list__all2__takeI,axiom,
    ! [P: produc7017002724195966439_a_nat > sum_sum_a_nat > $o,Xs: list_P1195027771636113901_a_nat,Ys: list_Sum_sum_a_nat,N: nat] :
      ( ( list_a746113844164839775_a_nat @ P @ Xs @ Ys )
     => ( list_a746113844164839775_a_nat @ P @ ( take_P3324508860426538453_a_nat @ N @ Xs ) @ ( take_Sum_sum_a_nat @ N @ Ys ) ) ) ).

% list_all2_takeI
thf(fact_39_list__all2__takeI,axiom,
    ! [P: produc7017002724195966439_a_nat > produc7017002724195966439_a_nat > $o,Xs: list_P1195027771636113901_a_nat,Ys: list_P1195027771636113901_a_nat,N: nat] :
      ( ( list_a2823864037932130119_a_nat @ P @ Xs @ Ys )
     => ( list_a2823864037932130119_a_nat @ P @ ( take_P3324508860426538453_a_nat @ N @ Xs ) @ ( take_P3324508860426538453_a_nat @ N @ Ys ) ) ) ).

% list_all2_takeI
thf(fact_40_list__all2__takeI,axiom,
    ! [P: sum_sum_a_nat > sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,N: nat] :
      ( ( list_a8491001234060422471_a_nat @ P @ Xs @ Ys )
     => ( list_a8491001234060422471_a_nat @ P @ ( take_Sum_sum_a_nat @ N @ Xs ) @ ( take_Sum_sum_a_nat @ N @ Ys ) ) ) ).

% list_all2_takeI
thf(fact_41_list__update__overwrite,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,I2: nat,X: produc7017002724195966439_a_nat,Y: produc7017002724195966439_a_nat] :
      ( ( list_u4149385051645232213_a_nat @ ( list_u4149385051645232213_a_nat @ Xs @ I2 @ X ) @ I2 @ Y )
      = ( list_u4149385051645232213_a_nat @ Xs @ I2 @ Y ) ) ).

% list_update_overwrite
thf(fact_42_list__update__overwrite,axiom,
    ! [Xs: list_Sum_sum_a_nat,I2: nat,X: sum_sum_a_nat,Y: sum_sum_a_nat] :
      ( ( list_u9138855634547462509_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I2 @ X ) @ I2 @ Y )
      = ( list_u9138855634547462509_a_nat @ Xs @ I2 @ Y ) ) ).

% list_update_overwrite
thf(fact_43_list_Omap__disc__iff,axiom,
    ! [F: sum_sum_a_nat > sum_sum_a_nat,A: list_Sum_sum_a_nat] :
      ( ( ( map_Su2790769393171190532_a_nat @ F @ A )
        = nil_Sum_sum_a_nat )
      = ( A = nil_Sum_sum_a_nat ) ) ).

% list.map_disc_iff
thf(fact_44_list_Omap__disc__iff,axiom,
    ! [F: nat > nat,A: list_nat] :
      ( ( ( map_nat_nat @ F @ A )
        = nil_nat )
      = ( A = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_45_list_Omap__disc__iff,axiom,
    ! [F: nat > sum_sum_a_nat,A: list_nat] :
      ( ( ( map_na823391071729141993_a_nat @ F @ A )
        = nil_Sum_sum_a_nat )
      = ( A = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_46_list_Omap__disc__iff,axiom,
    ! [F: sum_sum_a_nat > nat,A: list_Sum_sum_a_nat] :
      ( ( ( map_Su5227373005390213643at_nat @ F @ A )
        = nil_nat )
      = ( A = nil_Sum_sum_a_nat ) ) ).

% list.map_disc_iff
thf(fact_47_list_Omap__disc__iff,axiom,
    ! [F: nat > list_Sum_sum_a_nat,A: list_nat] :
      ( ( ( map_na5555065315529566063_a_nat @ F @ A )
        = nil_li1906260230833442699_a_nat )
      = ( A = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_48_list_Omap__disc__iff,axiom,
    ! [F: list_Sum_sum_a_nat > nat,A: list_l4703314356710769291_a_nat] :
      ( ( ( map_li1827815449145584657at_nat @ F @ A )
        = nil_nat )
      = ( A = nil_li1906260230833442699_a_nat ) ) ).

% list.map_disc_iff
thf(fact_49_list_Omap__disc__iff,axiom,
    ! [F: list_Sum_sum_a_nat > sum_sum_a_nat,A: list_l4703314356710769291_a_nat] :
      ( ( ( map_li3106614296470775678_a_nat @ F @ A )
        = nil_Sum_sum_a_nat )
      = ( A = nil_li1906260230833442699_a_nat ) ) ).

% list.map_disc_iff
thf(fact_50_list_Omap__disc__iff,axiom,
    ! [F: sum_sum_a_nat > list_Sum_sum_a_nat,A: list_Sum_sum_a_nat] :
      ( ( ( map_Su3479376915845212170_a_nat @ F @ A )
        = nil_li1906260230833442699_a_nat )
      = ( A = nil_Sum_sum_a_nat ) ) ).

% list.map_disc_iff
thf(fact_51_list_Omap__disc__iff,axiom,
    ! [F: nat > produc7017002724195966439_a_nat,A: list_nat] :
      ( ( ( map_na7890605889587578321_a_nat @ F @ A )
        = nil_Pr6585251977456444909_a_nat )
      = ( A = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_52_list_Omap__disc__iff,axiom,
    ! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: list_l4703314356710769291_a_nat] :
      ( ( ( map_li6507455427659069316_a_nat @ F @ A )
        = nil_li1906260230833442699_a_nat )
      = ( A = nil_li1906260230833442699_a_nat ) ) ).

% list.map_disc_iff
thf(fact_53_Nil__is__map__conv,axiom,
    ! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( nil_Sum_sum_a_nat
        = ( map_Su2790769393171190532_a_nat @ F @ Xs ) )
      = ( Xs = nil_Sum_sum_a_nat ) ) ).

% Nil_is_map_conv
thf(fact_54_Nil__is__map__conv,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( nil_nat
        = ( map_nat_nat @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_55_Nil__is__map__conv,axiom,
    ! [F: nat > sum_sum_a_nat,Xs: list_nat] :
      ( ( nil_Sum_sum_a_nat
        = ( map_na823391071729141993_a_nat @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_56_Nil__is__map__conv,axiom,
    ! [F: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat] :
      ( ( nil_nat
        = ( map_Su5227373005390213643at_nat @ F @ Xs ) )
      = ( Xs = nil_Sum_sum_a_nat ) ) ).

% Nil_is_map_conv
thf(fact_57_Nil__is__map__conv,axiom,
    ! [F: nat > list_Sum_sum_a_nat,Xs: list_nat] :
      ( ( nil_li1906260230833442699_a_nat
        = ( map_na5555065315529566063_a_nat @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_58_Nil__is__map__conv,axiom,
    ! [F: list_Sum_sum_a_nat > nat,Xs: list_l4703314356710769291_a_nat] :
      ( ( nil_nat
        = ( map_li1827815449145584657at_nat @ F @ Xs ) )
      = ( Xs = nil_li1906260230833442699_a_nat ) ) ).

% Nil_is_map_conv
thf(fact_59_Nil__is__map__conv,axiom,
    ! [F: list_Sum_sum_a_nat > sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat] :
      ( ( nil_Sum_sum_a_nat
        = ( map_li3106614296470775678_a_nat @ F @ Xs ) )
      = ( Xs = nil_li1906260230833442699_a_nat ) ) ).

% Nil_is_map_conv
thf(fact_60_Nil__is__map__conv,axiom,
    ! [F: sum_sum_a_nat > list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( nil_li1906260230833442699_a_nat
        = ( map_Su3479376915845212170_a_nat @ F @ Xs ) )
      = ( Xs = nil_Sum_sum_a_nat ) ) ).

% Nil_is_map_conv
thf(fact_61_Nil__is__map__conv,axiom,
    ! [F: nat > produc7017002724195966439_a_nat,Xs: list_nat] :
      ( ( nil_Pr6585251977456444909_a_nat
        = ( map_na7890605889587578321_a_nat @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_62_Nil__is__map__conv,axiom,
    ! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat] :
      ( ( nil_li1906260230833442699_a_nat
        = ( map_li6507455427659069316_a_nat @ F @ Xs ) )
      = ( Xs = nil_li1906260230833442699_a_nat ) ) ).

% Nil_is_map_conv
thf(fact_63_map__is__Nil__conv,axiom,
    ! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
        = nil_Sum_sum_a_nat )
      = ( Xs = nil_Sum_sum_a_nat ) ) ).

% map_is_Nil_conv
thf(fact_64_map__is__Nil__conv,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_65_map__is__Nil__conv,axiom,
    ! [F: nat > sum_sum_a_nat,Xs: list_nat] :
      ( ( ( map_na823391071729141993_a_nat @ F @ Xs )
        = nil_Sum_sum_a_nat )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_66_map__is__Nil__conv,axiom,
    ! [F: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat] :
      ( ( ( map_Su5227373005390213643at_nat @ F @ Xs )
        = nil_nat )
      = ( Xs = nil_Sum_sum_a_nat ) ) ).

% map_is_Nil_conv
thf(fact_67_map__is__Nil__conv,axiom,
    ! [F: nat > list_Sum_sum_a_nat,Xs: list_nat] :
      ( ( ( map_na5555065315529566063_a_nat @ F @ Xs )
        = nil_li1906260230833442699_a_nat )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_68_map__is__Nil__conv,axiom,
    ! [F: list_Sum_sum_a_nat > nat,Xs: list_l4703314356710769291_a_nat] :
      ( ( ( map_li1827815449145584657at_nat @ F @ Xs )
        = nil_nat )
      = ( Xs = nil_li1906260230833442699_a_nat ) ) ).

% map_is_Nil_conv
thf(fact_69_map__is__Nil__conv,axiom,
    ! [F: list_Sum_sum_a_nat > sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat] :
      ( ( ( map_li3106614296470775678_a_nat @ F @ Xs )
        = nil_Sum_sum_a_nat )
      = ( Xs = nil_li1906260230833442699_a_nat ) ) ).

% map_is_Nil_conv
thf(fact_70_map__is__Nil__conv,axiom,
    ! [F: sum_sum_a_nat > list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( ( map_Su3479376915845212170_a_nat @ F @ Xs )
        = nil_li1906260230833442699_a_nat )
      = ( Xs = nil_Sum_sum_a_nat ) ) ).

% map_is_Nil_conv
thf(fact_71_map__is__Nil__conv,axiom,
    ! [F: nat > produc7017002724195966439_a_nat,Xs: list_nat] :
      ( ( ( map_na7890605889587578321_a_nat @ F @ Xs )
        = nil_Pr6585251977456444909_a_nat )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_72_map__is__Nil__conv,axiom,
    ! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat] :
      ( ( ( map_li6507455427659069316_a_nat @ F @ Xs )
        = nil_li1906260230833442699_a_nat )
      = ( Xs = nil_li1906260230833442699_a_nat ) ) ).

% map_is_Nil_conv
thf(fact_73_map__eq__conv,axiom,
    ! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat] :
      ( ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
        = ( map_Su2790769393171190532_a_nat @ G @ Xs ) )
      = ( ! [X2: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) ) ) ) ).

% map_eq_conv
thf(fact_74_map__eq__conv,axiom,
    ! [F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat,G: list_Sum_sum_a_nat > set_Sum_sum_a_nat] :
      ( ( ( map_li9079814933684165854_a_nat @ F @ Xs )
        = ( map_li9079814933684165854_a_nat @ G @ Xs ) )
      = ( ! [X2: list_Sum_sum_a_nat] :
            ( ( member408289922725080238_a_nat @ X2 @ ( set_li2392974972034027290_a_nat @ Xs ) )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) ) ) ) ).

% map_eq_conv
thf(fact_75_list__all2__Nil,axiom,
    ! [P: sum_sum_a_nat > sum_sum_a_nat > $o,Ys: list_Sum_sum_a_nat] :
      ( ( list_a8491001234060422471_a_nat @ P @ nil_Sum_sum_a_nat @ Ys )
      = ( Ys = nil_Sum_sum_a_nat ) ) ).

% list_all2_Nil
thf(fact_76_list__all2__Nil,axiom,
    ! [P: nat > nat > $o,Ys: list_nat] :
      ( ( list_all2_nat_nat @ P @ nil_nat @ Ys )
      = ( Ys = nil_nat ) ) ).

% list_all2_Nil
thf(fact_77_list__all2__Nil,axiom,
    ! [P: sum_sum_a_nat > nat > $o,Ys: list_nat] :
      ( ( list_a3458528456966901896at_nat @ P @ nil_Sum_sum_a_nat @ Ys )
      = ( Ys = nil_nat ) ) ).

% list_all2_Nil
thf(fact_78_list__all2__Nil,axiom,
    ! [P: nat > sum_sum_a_nat > $o,Ys: list_Sum_sum_a_nat] :
      ( ( list_a8277918560160606054_a_nat @ P @ nil_nat @ Ys )
      = ( Ys = nil_Sum_sum_a_nat ) ) ).

% list_all2_Nil
thf(fact_79_list__all2__Nil,axiom,
    ! [P: list_Sum_sum_a_nat > nat > $o,Ys: list_nat] :
      ( ( list_a1471712951254312974at_nat @ P @ nil_li1906260230833442699_a_nat @ Ys )
      = ( Ys = nil_nat ) ) ).

% list_all2_Nil
thf(fact_80_list__all2__Nil,axiom,
    ! [P: nat > list_Sum_sum_a_nat > $o,Ys: list_l4703314356710769291_a_nat] :
      ( ( list_a5198962817638294380_a_nat @ P @ nil_nat @ Ys )
      = ( Ys = nil_li1906260230833442699_a_nat ) ) ).

% list_all2_Nil
thf(fact_81_list__all2__Nil,axiom,
    ! [P: sum_sum_a_nat > list_Sum_sum_a_nat > $o,Ys: list_l4703314356710769291_a_nat] :
      ( ( list_a8640327908064052429_a_nat @ P @ nil_Sum_sum_a_nat @ Ys )
      = ( Ys = nil_li1906260230833442699_a_nat ) ) ).

% list_all2_Nil
thf(fact_82_list__all2__Nil,axiom,
    ! [P: list_Sum_sum_a_nat > sum_sum_a_nat > $o,Ys: list_Sum_sum_a_nat] :
      ( ( list_a8267565288689615937_a_nat @ P @ nil_li1906260230833442699_a_nat @ Ys )
      = ( Ys = nil_Sum_sum_a_nat ) ) ).

% list_all2_Nil
thf(fact_83_list__all2__Nil,axiom,
    ! [P: produc7017002724195966439_a_nat > nat > $o,Ys: list_nat] :
      ( ( list_a7879879654818888304at_nat @ P @ nil_Pr6585251977456444909_a_nat @ Ys )
      = ( Ys = nil_nat ) ) ).

% list_all2_Nil
thf(fact_84_list__all2__Nil,axiom,
    ! [P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,Ys: list_l4703314356710769291_a_nat] :
      ( ( list_a5617550422536820423_a_nat @ P @ nil_li1906260230833442699_a_nat @ Ys )
      = ( Ys = nil_li1906260230833442699_a_nat ) ) ).

% list_all2_Nil
thf(fact_85_list__all2__Nil2,axiom,
    ! [P: sum_sum_a_nat > sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] :
      ( ( list_a8491001234060422471_a_nat @ P @ Xs @ nil_Sum_sum_a_nat )
      = ( Xs = nil_Sum_sum_a_nat ) ) ).

% list_all2_Nil2
thf(fact_86_list__all2__Nil2,axiom,
    ! [P: nat > nat > $o,Xs: list_nat] :
      ( ( list_all2_nat_nat @ P @ Xs @ nil_nat )
      = ( Xs = nil_nat ) ) ).

% list_all2_Nil2
thf(fact_87_list__all2__Nil2,axiom,
    ! [P: nat > sum_sum_a_nat > $o,Xs: list_nat] :
      ( ( list_a8277918560160606054_a_nat @ P @ Xs @ nil_Sum_sum_a_nat )
      = ( Xs = nil_nat ) ) ).

% list_all2_Nil2
thf(fact_88_list__all2__Nil2,axiom,
    ! [P: sum_sum_a_nat > nat > $o,Xs: list_Sum_sum_a_nat] :
      ( ( list_a3458528456966901896at_nat @ P @ Xs @ nil_nat )
      = ( Xs = nil_Sum_sum_a_nat ) ) ).

% list_all2_Nil2
thf(fact_89_list__all2__Nil2,axiom,
    ! [P: nat > list_Sum_sum_a_nat > $o,Xs: list_nat] :
      ( ( list_a5198962817638294380_a_nat @ P @ Xs @ nil_li1906260230833442699_a_nat )
      = ( Xs = nil_nat ) ) ).

% list_all2_Nil2
thf(fact_90_list__all2__Nil2,axiom,
    ! [P: list_Sum_sum_a_nat > nat > $o,Xs: list_l4703314356710769291_a_nat] :
      ( ( list_a1471712951254312974at_nat @ P @ Xs @ nil_nat )
      = ( Xs = nil_li1906260230833442699_a_nat ) ) ).

% list_all2_Nil2
thf(fact_91_list__all2__Nil2,axiom,
    ! [P: list_Sum_sum_a_nat > sum_sum_a_nat > $o,Xs: list_l4703314356710769291_a_nat] :
      ( ( list_a8267565288689615937_a_nat @ P @ Xs @ nil_Sum_sum_a_nat )
      = ( Xs = nil_li1906260230833442699_a_nat ) ) ).

% list_all2_Nil2
thf(fact_92_list__all2__Nil2,axiom,
    ! [P: sum_sum_a_nat > list_Sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] :
      ( ( list_a8640327908064052429_a_nat @ P @ Xs @ nil_li1906260230833442699_a_nat )
      = ( Xs = nil_Sum_sum_a_nat ) ) ).

% list_all2_Nil2
thf(fact_93_list__all2__Nil2,axiom,
    ! [P: nat > produc7017002724195966439_a_nat > $o,Xs: list_nat] :
      ( ( list_a6225475761577929678_a_nat @ P @ Xs @ nil_Pr6585251977456444909_a_nat )
      = ( Xs = nil_nat ) ) ).

% list_all2_Nil2
thf(fact_94_list__all2__Nil2,axiom,
    ! [P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,Xs: list_l4703314356710769291_a_nat] :
      ( ( list_a5617550422536820423_a_nat @ P @ Xs @ nil_li1906260230833442699_a_nat )
      = ( Xs = nil_li1906260230833442699_a_nat ) ) ).

% list_all2_Nil2
thf(fact_95_filter__True,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,P: list_Sum_sum_a_nat > $o] :
      ( ! [X3: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ X3 @ ( set_li2392974972034027290_a_nat @ Xs ) )
         => ( P @ X3 ) )
     => ( ( filter5373755100216644354_a_nat @ P @ Xs )
        = Xs ) ) ).

% filter_True
thf(fact_96_filter__True,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,P: produc7017002724195966439_a_nat > $o] :
      ( ! [X3: produc7017002724195966439_a_nat] :
          ( ( member3723442691059620112_a_nat @ X3 @ ( set_Pr2931239450594890620_a_nat @ Xs ) )
         => ( P @ X3 ) )
     => ( ( filter6896953120613689188_a_nat @ P @ Xs )
        = Xs ) ) ).

% filter_True
thf(fact_97_filter__True,axiom,
    ! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
      ( ! [X3: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Xs ) )
         => ( P @ X3 ) )
     => ( ( filter_Sum_sum_a_nat @ P @ Xs )
        = Xs ) ) ).

% filter_True
thf(fact_98_zip__Nil,axiom,
    ! [Ys: list_Sum_sum_a_nat] :
      ( ( zip_Su7355543910597222519_a_nat @ nil_Sum_sum_a_nat @ Ys )
      = nil_Pr6585251977456444909_a_nat ) ).

% zip_Nil
thf(fact_99_Nil__eq__zip__iff,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( nil_Pr6585251977456444909_a_nat
        = ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_Sum_sum_a_nat )
        | ( Ys = nil_Sum_sum_a_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_100_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( nil_Pr5478986624290739719at_nat
        = ( zip_nat_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_101_Nil__eq__zip__iff,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_nat] :
      ( ( nil_Pr2041209113074292344at_nat
        = ( zip_Su6417478541797927256at_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_Sum_sum_a_nat )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_102_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_Sum_sum_a_nat] :
      ( ( nil_Pr237480997409426078_a_nat
        = ( zip_na2013496608136855606_a_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_Sum_sum_a_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_103_Nil__eq__zip__iff,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,Ys: list_nat] :
      ( ( nil_Pr2978926869271283070at_nat
        = ( zip_li3415139562815969502at_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_li1906260230833442699_a_nat )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_104_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_l4703314356710769291_a_nat] :
      ( ( nil_Pr2751885571400186148_a_nat
        = ( zip_na7142389429199950908_a_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_li1906260230833442699_a_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_105_Nil__eq__zip__iff,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_l4703314356710769291_a_nat] :
      ( ( nil_Pr7347883008240361715_a_nat
        = ( zip_Su4197472078030315517_a_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_Sum_sum_a_nat )
        | ( Ys = nil_li1906260230833442699_a_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_106_Nil__eq__zip__iff,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( nil_Pr4306293334262433127_a_nat
        = ( zip_li3824709458655879025_a_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_li1906260230833442699_a_nat )
        | ( Ys = nil_Sum_sum_a_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_107_Nil__eq__zip__iff,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,Ys: list_nat] :
      ( ( nil_Pr1739053331407100384at_nat
        = ( zip_Pr5895826916983950144at_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr6585251977456444909_a_nat )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_108_Nil__eq__zip__iff,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat] :
      ( ( nil_Pr1141218757497453421_a_nat
        = ( zip_li7023018480688994807_a_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_li1906260230833442699_a_nat )
        | ( Ys = nil_li1906260230833442699_a_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_109_zip__eq__Nil__iff,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( zip_Su7355543910597222519_a_nat @ Xs @ Ys )
        = nil_Pr6585251977456444909_a_nat )
      = ( ( Xs = nil_Sum_sum_a_nat )
        | ( Ys = nil_Sum_sum_a_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_110_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( zip_nat_nat @ Xs @ Ys )
        = nil_Pr5478986624290739719at_nat )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_111_zip__eq__Nil__iff,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_nat] :
      ( ( ( zip_Su6417478541797927256at_nat @ Xs @ Ys )
        = nil_Pr2041209113074292344at_nat )
      = ( ( Xs = nil_Sum_sum_a_nat )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_112_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( zip_na2013496608136855606_a_nat @ Xs @ Ys )
        = nil_Pr237480997409426078_a_nat )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_Sum_sum_a_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_113_zip__eq__Nil__iff,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,Ys: list_nat] :
      ( ( ( zip_li3415139562815969502at_nat @ Xs @ Ys )
        = nil_Pr2978926869271283070at_nat )
      = ( ( Xs = nil_li1906260230833442699_a_nat )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_114_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_l4703314356710769291_a_nat] :
      ( ( ( zip_na7142389429199950908_a_nat @ Xs @ Ys )
        = nil_Pr2751885571400186148_a_nat )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_li1906260230833442699_a_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_115_zip__eq__Nil__iff,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_l4703314356710769291_a_nat] :
      ( ( ( zip_Su4197472078030315517_a_nat @ Xs @ Ys )
        = nil_Pr7347883008240361715_a_nat )
      = ( ( Xs = nil_Sum_sum_a_nat )
        | ( Ys = nil_li1906260230833442699_a_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_116_zip__eq__Nil__iff,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( zip_li3824709458655879025_a_nat @ Xs @ Ys )
        = nil_Pr4306293334262433127_a_nat )
      = ( ( Xs = nil_li1906260230833442699_a_nat )
        | ( Ys = nil_Sum_sum_a_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_117_zip__eq__Nil__iff,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,Ys: list_nat] :
      ( ( ( zip_Pr5895826916983950144at_nat @ Xs @ Ys )
        = nil_Pr1739053331407100384at_nat )
      = ( ( Xs = nil_Pr6585251977456444909_a_nat )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_118_zip__eq__Nil__iff,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat] :
      ( ( ( zip_li7023018480688994807_a_nat @ Xs @ Ys )
        = nil_Pr1141218757497453421_a_nat )
      = ( ( Xs = nil_li1906260230833442699_a_nat )
        | ( Ys = nil_li1906260230833442699_a_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_119_list__update__nonempty,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,K: nat,X: list_Sum_sum_a_nat] :
      ( ( ( list_u3340303885232119283_a_nat @ Xs @ K @ X )
        = nil_li1906260230833442699_a_nat )
      = ( Xs = nil_li1906260230833442699_a_nat ) ) ).

% list_update_nonempty
thf(fact_120_list__update__nonempty,axiom,
    ! [Xs: list_nat,K: nat,X: nat] :
      ( ( ( list_update_nat @ Xs @ K @ X )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% list_update_nonempty
thf(fact_121_list__update__nonempty,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,K: nat,X: produc7017002724195966439_a_nat] :
      ( ( ( list_u4149385051645232213_a_nat @ Xs @ K @ X )
        = nil_Pr6585251977456444909_a_nat )
      = ( Xs = nil_Pr6585251977456444909_a_nat ) ) ).

% list_update_nonempty
thf(fact_122_list__update__nonempty,axiom,
    ! [Xs: list_Sum_sum_a_nat,K: nat,X: sum_sum_a_nat] :
      ( ( ( list_u9138855634547462509_a_nat @ Xs @ K @ X )
        = nil_Sum_sum_a_nat )
      = ( Xs = nil_Sum_sum_a_nat ) ) ).

% list_update_nonempty
thf(fact_123_filter__False,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ~ ( P @ X3 ) )
     => ( ( filter_nat @ P @ Xs )
        = nil_nat ) ) ).

% filter_False
thf(fact_124_filter__False,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,P: list_Sum_sum_a_nat > $o] :
      ( ! [X3: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ X3 @ ( set_li2392974972034027290_a_nat @ Xs ) )
         => ~ ( P @ X3 ) )
     => ( ( filter5373755100216644354_a_nat @ P @ Xs )
        = nil_li1906260230833442699_a_nat ) ) ).

% filter_False
thf(fact_125_filter__False,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,P: produc7017002724195966439_a_nat > $o] :
      ( ! [X3: produc7017002724195966439_a_nat] :
          ( ( member3723442691059620112_a_nat @ X3 @ ( set_Pr2931239450594890620_a_nat @ Xs ) )
         => ~ ( P @ X3 ) )
     => ( ( filter6896953120613689188_a_nat @ P @ Xs )
        = nil_Pr6585251977456444909_a_nat ) ) ).

% filter_False
thf(fact_126_filter__False,axiom,
    ! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
      ( ! [X3: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Xs ) )
         => ~ ( P @ X3 ) )
     => ( ( filter_Sum_sum_a_nat @ P @ Xs )
        = nil_Sum_sum_a_nat ) ) ).

% filter_False
thf(fact_127_zip_Osimps_I1_J,axiom,
    ! [Xs: list_Sum_sum_a_nat] :
      ( ( zip_Su7355543910597222519_a_nat @ Xs @ nil_Sum_sum_a_nat )
      = nil_Pr6585251977456444909_a_nat ) ).

% zip.simps(1)
thf(fact_128_list_Osimps_I8_J,axiom,
    ! [F: sum_sum_a_nat > sum_sum_a_nat] :
      ( ( map_Su2790769393171190532_a_nat @ F @ nil_Sum_sum_a_nat )
      = nil_Sum_sum_a_nat ) ).

% list.simps(8)
thf(fact_129_list_Osimps_I8_J,axiom,
    ! [F: nat > nat] :
      ( ( map_nat_nat @ F @ nil_nat )
      = nil_nat ) ).

% list.simps(8)
thf(fact_130_list_Osimps_I8_J,axiom,
    ! [F: sum_sum_a_nat > nat] :
      ( ( map_Su5227373005390213643at_nat @ F @ nil_Sum_sum_a_nat )
      = nil_nat ) ).

% list.simps(8)
thf(fact_131_list_Osimps_I8_J,axiom,
    ! [F: nat > sum_sum_a_nat] :
      ( ( map_na823391071729141993_a_nat @ F @ nil_nat )
      = nil_Sum_sum_a_nat ) ).

% list.simps(8)
thf(fact_132_list_Osimps_I8_J,axiom,
    ! [F: list_Sum_sum_a_nat > nat] :
      ( ( map_li1827815449145584657at_nat @ F @ nil_li1906260230833442699_a_nat )
      = nil_nat ) ).

% list.simps(8)
thf(fact_133_list_Osimps_I8_J,axiom,
    ! [F: nat > list_Sum_sum_a_nat] :
      ( ( map_na5555065315529566063_a_nat @ F @ nil_nat )
      = nil_li1906260230833442699_a_nat ) ).

% list.simps(8)
thf(fact_134_list_Osimps_I8_J,axiom,
    ! [F: sum_sum_a_nat > list_Sum_sum_a_nat] :
      ( ( map_Su3479376915845212170_a_nat @ F @ nil_Sum_sum_a_nat )
      = nil_li1906260230833442699_a_nat ) ).

% list.simps(8)
thf(fact_135_list_Osimps_I8_J,axiom,
    ! [F: list_Sum_sum_a_nat > sum_sum_a_nat] :
      ( ( map_li3106614296470775678_a_nat @ F @ nil_li1906260230833442699_a_nat )
      = nil_Sum_sum_a_nat ) ).

% list.simps(8)
thf(fact_136_list_Osimps_I8_J,axiom,
    ! [F: produc7017002724195966439_a_nat > nat] :
      ( ( map_Pr321637745973761139at_nat @ F @ nil_Pr6585251977456444909_a_nat )
      = nil_nat ) ).

% list.simps(8)
thf(fact_137_list_Osimps_I8_J,axiom,
    ! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
      ( ( map_li6507455427659069316_a_nat @ F @ nil_li1906260230833442699_a_nat )
      = nil_li1906260230833442699_a_nat ) ).

% list.simps(8)
thf(fact_138_filter_Osimps_I1_J,axiom,
    ! [P: produc7017002724195966439_a_nat > $o] :
      ( ( filter6896953120613689188_a_nat @ P @ nil_Pr6585251977456444909_a_nat )
      = nil_Pr6585251977456444909_a_nat ) ).

% filter.simps(1)
thf(fact_139_filter_Osimps_I1_J,axiom,
    ! [P: list_Sum_sum_a_nat > $o] :
      ( ( filter5373755100216644354_a_nat @ P @ nil_li1906260230833442699_a_nat )
      = nil_li1906260230833442699_a_nat ) ).

% filter.simps(1)
thf(fact_140_filter_Osimps_I1_J,axiom,
    ! [P: nat > $o] :
      ( ( filter_nat @ P @ nil_nat )
      = nil_nat ) ).

% filter.simps(1)
thf(fact_141_filter_Osimps_I1_J,axiom,
    ! [P: sum_sum_a_nat > $o] :
      ( ( filter_Sum_sum_a_nat @ P @ nil_Sum_sum_a_nat )
      = nil_Sum_sum_a_nat ) ).

% filter.simps(1)
thf(fact_142_distinct_Osimps_I1_J,axiom,
    distin4821671304206132312_a_nat @ nil_se616183255201024101_a_nat ).

% distinct.simps(1)
thf(fact_143_distinct_Osimps_I1_J,axiom,
    distin928153214662094432_a_nat @ nil_Pr6585251977456444909_a_nat ).

% distinct.simps(1)
thf(fact_144_distinct_Osimps_I1_J,axiom,
    distin811021574259663358_a_nat @ nil_li1906260230833442699_a_nat ).

% distinct.simps(1)
thf(fact_145_distinct_Osimps_I1_J,axiom,
    distinct_nat @ nil_nat ).

% distinct.simps(1)
thf(fact_146_distinct_Osimps_I1_J,axiom,
    distin2701893636801681144_a_nat @ nil_Sum_sum_a_nat ).

% distinct.simps(1)
thf(fact_147_list__update__code_I1_J,axiom,
    ! [I2: nat,Y: list_Sum_sum_a_nat] :
      ( ( list_u3340303885232119283_a_nat @ nil_li1906260230833442699_a_nat @ I2 @ Y )
      = nil_li1906260230833442699_a_nat ) ).

% list_update_code(1)
thf(fact_148_list__update__code_I1_J,axiom,
    ! [I2: nat,Y: nat] :
      ( ( list_update_nat @ nil_nat @ I2 @ Y )
      = nil_nat ) ).

% list_update_code(1)
thf(fact_149_list__update__code_I1_J,axiom,
    ! [I2: nat,Y: produc7017002724195966439_a_nat] :
      ( ( list_u4149385051645232213_a_nat @ nil_Pr6585251977456444909_a_nat @ I2 @ Y )
      = nil_Pr6585251977456444909_a_nat ) ).

% list_update_code(1)
thf(fact_150_list__update__code_I1_J,axiom,
    ! [I2: nat,Y: sum_sum_a_nat] :
      ( ( list_u9138855634547462509_a_nat @ nil_Sum_sum_a_nat @ I2 @ Y )
      = nil_Sum_sum_a_nat ) ).

% list_update_code(1)
thf(fact_151_map__filter__simps_I2_J,axiom,
    ! [F: sum_sum_a_nat > option_Sum_sum_a_nat] :
      ( ( map_fi2176729826146509066_a_nat @ F @ nil_Sum_sum_a_nat )
      = nil_Sum_sum_a_nat ) ).

% map_filter_simps(2)
thf(fact_152_map__filter__simps_I2_J,axiom,
    ! [F: nat > option_nat] :
      ( ( map_filter_nat_nat @ F @ nil_nat )
      = nil_nat ) ).

% map_filter_simps(2)
thf(fact_153_map__filter__simps_I2_J,axiom,
    ! [F: sum_sum_a_nat > option_nat] :
      ( ( map_fi3487137476358288773at_nat @ F @ nil_Sum_sum_a_nat )
      = nil_nat ) ).

% map_filter_simps(2)
thf(fact_154_map__filter__simps_I2_J,axiom,
    ! [F: nat > option_Sum_sum_a_nat] :
      ( ( map_fi8306527579551992931_a_nat @ F @ nil_nat )
      = nil_Sum_sum_a_nat ) ).

% map_filter_simps(2)
thf(fact_155_map__filter__simps_I2_J,axiom,
    ! [F: list_Sum_sum_a_nat > option_nat] :
      ( ( map_fi7901273347920489099at_nat @ F @ nil_li1906260230833442699_a_nat )
      = nil_nat ) ).

% map_filter_simps(2)
thf(fact_156_map__filter__simps_I2_J,axiom,
    ! [F: nat > option2159618704653257803_a_nat] :
      ( ( map_fi2405151177449694697_a_nat @ F @ nil_nat )
      = nil_li1906260230833442699_a_nat ) ).

% map_filter_simps(2)
thf(fact_157_map__filter__simps_I2_J,axiom,
    ! [F: sum_sum_a_nat > option2159618704653257803_a_nat] :
      ( ( map_fi7503605724195961616_a_nat @ F @ nil_Sum_sum_a_nat )
      = nil_li1906260230833442699_a_nat ) ).

% map_filter_simps(2)
thf(fact_158_map__filter__simps_I2_J,axiom,
    ! [F: list_Sum_sum_a_nat > option_Sum_sum_a_nat] :
      ( ( map_fi7130843104821525124_a_nat @ F @ nil_li1906260230833442699_a_nat )
      = nil_Sum_sum_a_nat ) ).

% map_filter_simps(2)
thf(fact_159_map__filter__simps_I2_J,axiom,
    ! [F: produc7017002724195966439_a_nat > option_nat] :
      ( ( map_fi4553214607747458797at_nat @ F @ nil_Pr6585251977456444909_a_nat )
      = nil_nat ) ).

% map_filter_simps(2)
thf(fact_160_map__filter__simps_I2_J,axiom,
    ! [F: list_Sum_sum_a_nat > option2159618704653257803_a_nat] :
      ( ( map_fi8339409367572549514_a_nat @ F @ nil_li1906260230833442699_a_nat )
      = nil_li1906260230833442699_a_nat ) ).

% map_filter_simps(2)
thf(fact_161_list_Octr__transfer_I1_J,axiom,
    ! [R: sum_sum_a_nat > sum_sum_a_nat > $o] : ( list_a8491001234060422471_a_nat @ R @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat ) ).

% list.ctr_transfer(1)
thf(fact_162_list_Octr__transfer_I1_J,axiom,
    ! [R: nat > nat > $o] : ( list_all2_nat_nat @ R @ nil_nat @ nil_nat ) ).

% list.ctr_transfer(1)
thf(fact_163_list_Octr__transfer_I1_J,axiom,
    ! [R: sum_sum_a_nat > nat > $o] : ( list_a3458528456966901896at_nat @ R @ nil_Sum_sum_a_nat @ nil_nat ) ).

% list.ctr_transfer(1)
thf(fact_164_list_Octr__transfer_I1_J,axiom,
    ! [R: nat > sum_sum_a_nat > $o] : ( list_a8277918560160606054_a_nat @ R @ nil_nat @ nil_Sum_sum_a_nat ) ).

% list.ctr_transfer(1)
thf(fact_165_list_Octr__transfer_I1_J,axiom,
    ! [R: list_Sum_sum_a_nat > nat > $o] : ( list_a1471712951254312974at_nat @ R @ nil_li1906260230833442699_a_nat @ nil_nat ) ).

% list.ctr_transfer(1)
thf(fact_166_list_Octr__transfer_I1_J,axiom,
    ! [R: nat > list_Sum_sum_a_nat > $o] : ( list_a5198962817638294380_a_nat @ R @ nil_nat @ nil_li1906260230833442699_a_nat ) ).

% list.ctr_transfer(1)
thf(fact_167_list_Octr__transfer_I1_J,axiom,
    ! [R: sum_sum_a_nat > list_Sum_sum_a_nat > $o] : ( list_a8640327908064052429_a_nat @ R @ nil_Sum_sum_a_nat @ nil_li1906260230833442699_a_nat ) ).

% list.ctr_transfer(1)
thf(fact_168_list_Octr__transfer_I1_J,axiom,
    ! [R: list_Sum_sum_a_nat > sum_sum_a_nat > $o] : ( list_a8267565288689615937_a_nat @ R @ nil_li1906260230833442699_a_nat @ nil_Sum_sum_a_nat ) ).

% list.ctr_transfer(1)
thf(fact_169_list_Octr__transfer_I1_J,axiom,
    ! [R: produc7017002724195966439_a_nat > nat > $o] : ( list_a7879879654818888304at_nat @ R @ nil_Pr6585251977456444909_a_nat @ nil_nat ) ).

% list.ctr_transfer(1)
thf(fact_170_list_Octr__transfer_I1_J,axiom,
    ! [R: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] : ( list_a5617550422536820423_a_nat @ R @ nil_li1906260230833442699_a_nat @ nil_li1906260230833442699_a_nat ) ).

% list.ctr_transfer(1)
thf(fact_171_list__update_Osimps_I1_J,axiom,
    ! [I2: nat,V: list_Sum_sum_a_nat] :
      ( ( list_u3340303885232119283_a_nat @ nil_li1906260230833442699_a_nat @ I2 @ V )
      = nil_li1906260230833442699_a_nat ) ).

% list_update.simps(1)
thf(fact_172_list__update_Osimps_I1_J,axiom,
    ! [I2: nat,V: nat] :
      ( ( list_update_nat @ nil_nat @ I2 @ V )
      = nil_nat ) ).

% list_update.simps(1)
thf(fact_173_list__update_Osimps_I1_J,axiom,
    ! [I2: nat,V: produc7017002724195966439_a_nat] :
      ( ( list_u4149385051645232213_a_nat @ nil_Pr6585251977456444909_a_nat @ I2 @ V )
      = nil_Pr6585251977456444909_a_nat ) ).

% list_update.simps(1)
thf(fact_174_list__update_Osimps_I1_J,axiom,
    ! [I2: nat,V: sum_sum_a_nat] :
      ( ( list_u9138855634547462509_a_nat @ nil_Sum_sum_a_nat @ I2 @ V )
      = nil_Sum_sum_a_nat ) ).

% list_update.simps(1)
thf(fact_175_rremdups_Osimps_I1_J,axiom,
    ( ( rremdu7275748759011619361_a_nat @ nil_Pr6585251977456444909_a_nat )
    = nil_Pr6585251977456444909_a_nat ) ).

% rremdups.simps(1)
thf(fact_176_rremdups_Osimps_I1_J,axiom,
    ( ( rremdu4777469547168625599_a_nat @ nil_li1906260230833442699_a_nat )
    = nil_li1906260230833442699_a_nat ) ).

% rremdups.simps(1)
thf(fact_177_rremdups_Osimps_I1_J,axiom,
    ( ( rremdups_nat @ nil_nat )
    = nil_nat ) ).

% rremdups.simps(1)
thf(fact_178_rremdups_Osimps_I1_J,axiom,
    ( ( rremdu8304153113908149561_a_nat @ nil_Sum_sum_a_nat )
    = nil_Sum_sum_a_nat ) ).

% rremdups.simps(1)
thf(fact_179_list_Orel__eq,axiom,
    ( ( list_a8491001234060422471_a_nat
      @ ^ [Y2: sum_sum_a_nat,Z: sum_sum_a_nat] : ( Y2 = Z ) )
    = ( ^ [Y2: list_Sum_sum_a_nat,Z: list_Sum_sum_a_nat] : ( Y2 = Z ) ) ) ).

% list.rel_eq
thf(fact_180_list_Omap__cong,axiom,
    ! [X: list_Sum_sum_a_nat,Ya: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat] :
      ( ( X = Ya )
     => ( ! [Z2: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ Z2 @ ( set_Sum_sum_a_nat2 @ Ya ) )
           => ( ( F @ Z2 )
              = ( G @ Z2 ) ) )
       => ( ( map_Su2790769393171190532_a_nat @ F @ X )
          = ( map_Su2790769393171190532_a_nat @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_181_list_Omap__cong,axiom,
    ! [X: list_l4703314356710769291_a_nat,Ya: list_l4703314356710769291_a_nat,F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,G: list_Sum_sum_a_nat > set_Sum_sum_a_nat] :
      ( ( X = Ya )
     => ( ! [Z2: list_Sum_sum_a_nat] :
            ( ( member408289922725080238_a_nat @ Z2 @ ( set_li2392974972034027290_a_nat @ Ya ) )
           => ( ( F @ Z2 )
              = ( G @ Z2 ) ) )
       => ( ( map_li9079814933684165854_a_nat @ F @ X )
          = ( map_li9079814933684165854_a_nat @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_182_list_Orel__cong,axiom,
    ! [X: list_Sum_sum_a_nat,Ya: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat,Xa: list_Sum_sum_a_nat,R: sum_sum_a_nat > sum_sum_a_nat > $o,Ra: sum_sum_a_nat > sum_sum_a_nat > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z2: sum_sum_a_nat,Yb: sum_sum_a_nat] :
              ( ( member_Sum_sum_a_nat @ Z2 @ ( set_Sum_sum_a_nat2 @ Ya ) )
             => ( ( member_Sum_sum_a_nat @ Yb @ ( set_Sum_sum_a_nat2 @ Xa ) )
               => ( ( R @ Z2 @ Yb )
                  = ( Ra @ Z2 @ Yb ) ) ) )
         => ( ( list_a8491001234060422471_a_nat @ R @ X @ Y )
            = ( list_a8491001234060422471_a_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_183_list_Orel__cong,axiom,
    ! [X: list_s7376487411892226277_a_nat,Ya: list_s7376487411892226277_a_nat,Y: list_Sum_sum_a_nat,Xa: list_Sum_sum_a_nat,R: set_Sum_sum_a_nat > sum_sum_a_nat > $o,Ra: set_Sum_sum_a_nat > sum_sum_a_nat > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z2: set_Sum_sum_a_nat,Yb: sum_sum_a_nat] :
              ( ( member8098812455498974984_a_nat @ Z2 @ ( set_se6249060928767827828_a_nat @ Ya ) )
             => ( ( member_Sum_sum_a_nat @ Yb @ ( set_Sum_sum_a_nat2 @ Xa ) )
               => ( ( R @ Z2 @ Yb )
                  = ( Ra @ Z2 @ Yb ) ) ) )
         => ( ( list_a8920083903403951079_a_nat @ R @ X @ Y )
            = ( list_a8920083903403951079_a_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_184_list_Orel__cong,axiom,
    ! [X: list_Sum_sum_a_nat,Ya: list_Sum_sum_a_nat,Y: list_s7376487411892226277_a_nat,Xa: list_s7376487411892226277_a_nat,R: sum_sum_a_nat > set_Sum_sum_a_nat > $o,Ra: sum_sum_a_nat > set_Sum_sum_a_nat > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z2: sum_sum_a_nat,Yb: set_Sum_sum_a_nat] :
              ( ( member_Sum_sum_a_nat @ Z2 @ ( set_Sum_sum_a_nat2 @ Ya ) )
             => ( ( member8098812455498974984_a_nat @ Yb @ ( set_se6249060928767827828_a_nat @ Xa ) )
               => ( ( R @ Z2 @ Yb )
                  = ( Ra @ Z2 @ Yb ) ) ) )
         => ( ( list_a2481226626064268967_a_nat @ R @ X @ Y )
            = ( list_a2481226626064268967_a_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_185_list_Orel__cong,axiom,
    ! [X: list_Sum_sum_a_nat,Ya: list_Sum_sum_a_nat,Y: list_l4703314356710769291_a_nat,Xa: list_l4703314356710769291_a_nat,R: sum_sum_a_nat > list_Sum_sum_a_nat > $o,Ra: sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z2: sum_sum_a_nat,Yb: list_Sum_sum_a_nat] :
              ( ( member_Sum_sum_a_nat @ Z2 @ ( set_Sum_sum_a_nat2 @ Ya ) )
             => ( ( member408289922725080238_a_nat @ Yb @ ( set_li2392974972034027290_a_nat @ Xa ) )
               => ( ( R @ Z2 @ Yb )
                  = ( Ra @ Z2 @ Yb ) ) ) )
         => ( ( list_a8640327908064052429_a_nat @ R @ X @ Y )
            = ( list_a8640327908064052429_a_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_186_list_Orel__cong,axiom,
    ! [X: list_l4703314356710769291_a_nat,Ya: list_l4703314356710769291_a_nat,Y: list_Sum_sum_a_nat,Xa: list_Sum_sum_a_nat,R: list_Sum_sum_a_nat > sum_sum_a_nat > $o,Ra: list_Sum_sum_a_nat > sum_sum_a_nat > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z2: list_Sum_sum_a_nat,Yb: sum_sum_a_nat] :
              ( ( member408289922725080238_a_nat @ Z2 @ ( set_li2392974972034027290_a_nat @ Ya ) )
             => ( ( member_Sum_sum_a_nat @ Yb @ ( set_Sum_sum_a_nat2 @ Xa ) )
               => ( ( R @ Z2 @ Yb )
                  = ( Ra @ Z2 @ Yb ) ) ) )
         => ( ( list_a8267565288689615937_a_nat @ R @ X @ Y )
            = ( list_a8267565288689615937_a_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_187_list_Orel__cong,axiom,
    ! [X: list_s7376487411892226277_a_nat,Ya: list_s7376487411892226277_a_nat,Y: list_s7376487411892226277_a_nat,Xa: list_s7376487411892226277_a_nat,R: set_Sum_sum_a_nat > set_Sum_sum_a_nat > $o,Ra: set_Sum_sum_a_nat > set_Sum_sum_a_nat > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z2: set_Sum_sum_a_nat,Yb: set_Sum_sum_a_nat] :
              ( ( member8098812455498974984_a_nat @ Z2 @ ( set_se6249060928767827828_a_nat @ Ya ) )
             => ( ( member8098812455498974984_a_nat @ Yb @ ( set_se6249060928767827828_a_nat @ Xa ) )
               => ( ( R @ Z2 @ Yb )
                  = ( Ra @ Z2 @ Yb ) ) ) )
         => ( ( list_a4614116639835821383_a_nat @ R @ X @ Y )
            = ( list_a4614116639835821383_a_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_188_list_Orel__cong,axiom,
    ! [X: list_s7376487411892226277_a_nat,Ya: list_s7376487411892226277_a_nat,Y: list_l4703314356710769291_a_nat,Xa: list_l4703314356710769291_a_nat,R: set_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,Ra: set_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z2: set_Sum_sum_a_nat,Yb: list_Sum_sum_a_nat] :
              ( ( member8098812455498974984_a_nat @ Z2 @ ( set_se6249060928767827828_a_nat @ Ya ) )
             => ( ( member408289922725080238_a_nat @ Yb @ ( set_li2392974972034027290_a_nat @ Xa ) )
               => ( ( R @ Z2 @ Yb )
                  = ( Ra @ Z2 @ Yb ) ) ) )
         => ( ( list_a2188187147496901485_a_nat @ R @ X @ Y )
            = ( list_a2188187147496901485_a_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_189_list_Orel__cong,axiom,
    ! [X: list_l4703314356710769291_a_nat,Ya: list_l4703314356710769291_a_nat,Y: list_s7376487411892226277_a_nat,Xa: list_s7376487411892226277_a_nat,R: list_Sum_sum_a_nat > set_Sum_sum_a_nat > $o,Ra: list_Sum_sum_a_nat > set_Sum_sum_a_nat > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z2: list_Sum_sum_a_nat,Yb: set_Sum_sum_a_nat] :
              ( ( member408289922725080238_a_nat @ Z2 @ ( set_li2392974972034027290_a_nat @ Ya ) )
             => ( ( member8098812455498974984_a_nat @ Yb @ ( set_se6249060928767827828_a_nat @ Xa ) )
               => ( ( R @ Z2 @ Yb )
                  = ( Ra @ Z2 @ Yb ) ) ) )
         => ( ( list_a7096074725797064097_a_nat @ R @ X @ Y )
            = ( list_a7096074725797064097_a_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_190_list_Orel__cong,axiom,
    ! [X: list_l4703314356710769291_a_nat,Ya: list_l4703314356710769291_a_nat,Y: list_l4703314356710769291_a_nat,Xa: list_l4703314356710769291_a_nat,R: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,Ra: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z2: list_Sum_sum_a_nat,Yb: list_Sum_sum_a_nat] :
              ( ( member408289922725080238_a_nat @ Z2 @ ( set_li2392974972034027290_a_nat @ Ya ) )
             => ( ( member408289922725080238_a_nat @ Yb @ ( set_li2392974972034027290_a_nat @ Xa ) )
               => ( ( R @ Z2 @ Yb )
                  = ( Ra @ Z2 @ Yb ) ) ) )
         => ( ( list_a5617550422536820423_a_nat @ R @ X @ Y )
            = ( list_a5617550422536820423_a_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_191_list_Orel__cong,axiom,
    ! [X: list_Sum_sum_a_nat,Ya: list_Sum_sum_a_nat,Y: list_P1195027771636113901_a_nat,Xa: list_P1195027771636113901_a_nat,R: sum_sum_a_nat > produc7017002724195966439_a_nat > $o,Ra: sum_sum_a_nat > produc7017002724195966439_a_nat > $o] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z2: sum_sum_a_nat,Yb: produc7017002724195966439_a_nat] :
              ( ( member_Sum_sum_a_nat @ Z2 @ ( set_Sum_sum_a_nat2 @ Ya ) )
             => ( ( member3723442691059620112_a_nat @ Yb @ ( set_Pr2931239450594890620_a_nat @ Xa ) )
               => ( ( R @ Z2 @ Yb )
                  = ( Ra @ Z2 @ Yb ) ) ) )
         => ( ( list_a1440509178630569775_a_nat @ R @ X @ Y )
            = ( list_a1440509178630569775_a_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_192_list_Orel__refl,axiom,
    ! [Ra: sum_sum_a_nat > sum_sum_a_nat > $o,X: list_Sum_sum_a_nat] :
      ( ! [X3: sum_sum_a_nat] : ( Ra @ X3 @ X3 )
     => ( list_a8491001234060422471_a_nat @ Ra @ X @ X ) ) ).

% list.rel_refl
thf(fact_193_list_Omap__cong0,axiom,
    ! [X: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat] :
      ( ! [Z2: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ Z2 @ ( set_Sum_sum_a_nat2 @ X ) )
         => ( ( F @ Z2 )
            = ( G @ Z2 ) ) )
     => ( ( map_Su2790769393171190532_a_nat @ F @ X )
        = ( map_Su2790769393171190532_a_nat @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_194_list_Omap__cong0,axiom,
    ! [X: list_l4703314356710769291_a_nat,F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,G: list_Sum_sum_a_nat > set_Sum_sum_a_nat] :
      ( ! [Z2: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ Z2 @ ( set_li2392974972034027290_a_nat @ X ) )
         => ( ( F @ Z2 )
            = ( G @ Z2 ) ) )
     => ( ( map_li9079814933684165854_a_nat @ F @ X )
        = ( map_li9079814933684165854_a_nat @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_195_list_Oinj__map__strong,axiom,
    ! [X: list_Sum_sum_a_nat,Xa: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,Fa: sum_sum_a_nat > sum_sum_a_nat] :
      ( ! [Z2: sum_sum_a_nat,Za: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ Z2 @ ( set_Sum_sum_a_nat2 @ X ) )
         => ( ( member_Sum_sum_a_nat @ Za @ ( set_Sum_sum_a_nat2 @ Xa ) )
           => ( ( ( F @ Z2 )
                = ( Fa @ Za ) )
             => ( Z2 = Za ) ) ) )
     => ( ( ( map_Su2790769393171190532_a_nat @ F @ X )
          = ( map_Su2790769393171190532_a_nat @ Fa @ Xa ) )
       => ( X = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_196_list_Oinj__map__strong,axiom,
    ! [X: list_l4703314356710769291_a_nat,Xa: list_l4703314356710769291_a_nat,F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,Fa: list_Sum_sum_a_nat > set_Sum_sum_a_nat] :
      ( ! [Z2: list_Sum_sum_a_nat,Za: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ Z2 @ ( set_li2392974972034027290_a_nat @ X ) )
         => ( ( member408289922725080238_a_nat @ Za @ ( set_li2392974972034027290_a_nat @ Xa ) )
           => ( ( ( F @ Z2 )
                = ( Fa @ Za ) )
             => ( Z2 = Za ) ) ) )
     => ( ( ( map_li9079814933684165854_a_nat @ F @ X )
          = ( map_li9079814933684165854_a_nat @ Fa @ Xa ) )
       => ( X = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_197_list_Orel__mono__strong,axiom,
    ! [R: sum_sum_a_nat > sum_sum_a_nat > $o,X: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat,Ra: sum_sum_a_nat > sum_sum_a_nat > $o] :
      ( ( list_a8491001234060422471_a_nat @ R @ X @ Y )
     => ( ! [Z2: sum_sum_a_nat,Yb: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ Z2 @ ( set_Sum_sum_a_nat2 @ X ) )
           => ( ( member_Sum_sum_a_nat @ Yb @ ( set_Sum_sum_a_nat2 @ Y ) )
             => ( ( R @ Z2 @ Yb )
               => ( Ra @ Z2 @ Yb ) ) ) )
       => ( list_a8491001234060422471_a_nat @ Ra @ X @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_198_list_Orel__mono__strong,axiom,
    ! [R: set_Sum_sum_a_nat > sum_sum_a_nat > $o,X: list_s7376487411892226277_a_nat,Y: list_Sum_sum_a_nat,Ra: set_Sum_sum_a_nat > sum_sum_a_nat > $o] :
      ( ( list_a8920083903403951079_a_nat @ R @ X @ Y )
     => ( ! [Z2: set_Sum_sum_a_nat,Yb: sum_sum_a_nat] :
            ( ( member8098812455498974984_a_nat @ Z2 @ ( set_se6249060928767827828_a_nat @ X ) )
           => ( ( member_Sum_sum_a_nat @ Yb @ ( set_Sum_sum_a_nat2 @ Y ) )
             => ( ( R @ Z2 @ Yb )
               => ( Ra @ Z2 @ Yb ) ) ) )
       => ( list_a8920083903403951079_a_nat @ Ra @ X @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_199_list_Orel__mono__strong,axiom,
    ! [R: sum_sum_a_nat > set_Sum_sum_a_nat > $o,X: list_Sum_sum_a_nat,Y: list_s7376487411892226277_a_nat,Ra: sum_sum_a_nat > set_Sum_sum_a_nat > $o] :
      ( ( list_a2481226626064268967_a_nat @ R @ X @ Y )
     => ( ! [Z2: sum_sum_a_nat,Yb: set_Sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ Z2 @ ( set_Sum_sum_a_nat2 @ X ) )
           => ( ( member8098812455498974984_a_nat @ Yb @ ( set_se6249060928767827828_a_nat @ Y ) )
             => ( ( R @ Z2 @ Yb )
               => ( Ra @ Z2 @ Yb ) ) ) )
       => ( list_a2481226626064268967_a_nat @ Ra @ X @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_200_list_Orel__mono__strong,axiom,
    ! [R: sum_sum_a_nat > list_Sum_sum_a_nat > $o,X: list_Sum_sum_a_nat,Y: list_l4703314356710769291_a_nat,Ra: sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
      ( ( list_a8640327908064052429_a_nat @ R @ X @ Y )
     => ( ! [Z2: sum_sum_a_nat,Yb: list_Sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ Z2 @ ( set_Sum_sum_a_nat2 @ X ) )
           => ( ( member408289922725080238_a_nat @ Yb @ ( set_li2392974972034027290_a_nat @ Y ) )
             => ( ( R @ Z2 @ Yb )
               => ( Ra @ Z2 @ Yb ) ) ) )
       => ( list_a8640327908064052429_a_nat @ Ra @ X @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_201_list_Orel__mono__strong,axiom,
    ! [R: list_Sum_sum_a_nat > sum_sum_a_nat > $o,X: list_l4703314356710769291_a_nat,Y: list_Sum_sum_a_nat,Ra: list_Sum_sum_a_nat > sum_sum_a_nat > $o] :
      ( ( list_a8267565288689615937_a_nat @ R @ X @ Y )
     => ( ! [Z2: list_Sum_sum_a_nat,Yb: sum_sum_a_nat] :
            ( ( member408289922725080238_a_nat @ Z2 @ ( set_li2392974972034027290_a_nat @ X ) )
           => ( ( member_Sum_sum_a_nat @ Yb @ ( set_Sum_sum_a_nat2 @ Y ) )
             => ( ( R @ Z2 @ Yb )
               => ( Ra @ Z2 @ Yb ) ) ) )
       => ( list_a8267565288689615937_a_nat @ Ra @ X @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_202_list_Orel__mono__strong,axiom,
    ! [R: set_Sum_sum_a_nat > set_Sum_sum_a_nat > $o,X: list_s7376487411892226277_a_nat,Y: list_s7376487411892226277_a_nat,Ra: set_Sum_sum_a_nat > set_Sum_sum_a_nat > $o] :
      ( ( list_a4614116639835821383_a_nat @ R @ X @ Y )
     => ( ! [Z2: set_Sum_sum_a_nat,Yb: set_Sum_sum_a_nat] :
            ( ( member8098812455498974984_a_nat @ Z2 @ ( set_se6249060928767827828_a_nat @ X ) )
           => ( ( member8098812455498974984_a_nat @ Yb @ ( set_se6249060928767827828_a_nat @ Y ) )
             => ( ( R @ Z2 @ Yb )
               => ( Ra @ Z2 @ Yb ) ) ) )
       => ( list_a4614116639835821383_a_nat @ Ra @ X @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_203_list_Orel__mono__strong,axiom,
    ! [R: set_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,X: list_s7376487411892226277_a_nat,Y: list_l4703314356710769291_a_nat,Ra: set_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
      ( ( list_a2188187147496901485_a_nat @ R @ X @ Y )
     => ( ! [Z2: set_Sum_sum_a_nat,Yb: list_Sum_sum_a_nat] :
            ( ( member8098812455498974984_a_nat @ Z2 @ ( set_se6249060928767827828_a_nat @ X ) )
           => ( ( member408289922725080238_a_nat @ Yb @ ( set_li2392974972034027290_a_nat @ Y ) )
             => ( ( R @ Z2 @ Yb )
               => ( Ra @ Z2 @ Yb ) ) ) )
       => ( list_a2188187147496901485_a_nat @ Ra @ X @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_204_list_Orel__mono__strong,axiom,
    ! [R: list_Sum_sum_a_nat > set_Sum_sum_a_nat > $o,X: list_l4703314356710769291_a_nat,Y: list_s7376487411892226277_a_nat,Ra: list_Sum_sum_a_nat > set_Sum_sum_a_nat > $o] :
      ( ( list_a7096074725797064097_a_nat @ R @ X @ Y )
     => ( ! [Z2: list_Sum_sum_a_nat,Yb: set_Sum_sum_a_nat] :
            ( ( member408289922725080238_a_nat @ Z2 @ ( set_li2392974972034027290_a_nat @ X ) )
           => ( ( member8098812455498974984_a_nat @ Yb @ ( set_se6249060928767827828_a_nat @ Y ) )
             => ( ( R @ Z2 @ Yb )
               => ( Ra @ Z2 @ Yb ) ) ) )
       => ( list_a7096074725797064097_a_nat @ Ra @ X @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_205_list_Orel__mono__strong,axiom,
    ! [R: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,X: list_l4703314356710769291_a_nat,Y: list_l4703314356710769291_a_nat,Ra: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
      ( ( list_a5617550422536820423_a_nat @ R @ X @ Y )
     => ( ! [Z2: list_Sum_sum_a_nat,Yb: list_Sum_sum_a_nat] :
            ( ( member408289922725080238_a_nat @ Z2 @ ( set_li2392974972034027290_a_nat @ X ) )
           => ( ( member408289922725080238_a_nat @ Yb @ ( set_li2392974972034027290_a_nat @ Y ) )
             => ( ( R @ Z2 @ Yb )
               => ( Ra @ Z2 @ Yb ) ) ) )
       => ( list_a5617550422536820423_a_nat @ Ra @ X @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_206_list_Orel__mono__strong,axiom,
    ! [R: sum_sum_a_nat > produc7017002724195966439_a_nat > $o,X: list_Sum_sum_a_nat,Y: list_P1195027771636113901_a_nat,Ra: sum_sum_a_nat > produc7017002724195966439_a_nat > $o] :
      ( ( list_a1440509178630569775_a_nat @ R @ X @ Y )
     => ( ! [Z2: sum_sum_a_nat,Yb: produc7017002724195966439_a_nat] :
            ( ( member_Sum_sum_a_nat @ Z2 @ ( set_Sum_sum_a_nat2 @ X ) )
           => ( ( member3723442691059620112_a_nat @ Yb @ ( set_Pr2931239450594890620_a_nat @ Y ) )
             => ( ( R @ Z2 @ Yb )
               => ( Ra @ Z2 @ Yb ) ) ) )
       => ( list_a1440509178630569775_a_nat @ Ra @ X @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_207_list_Orel__refl__strong,axiom,
    ! [X: list_s7376487411892226277_a_nat,Ra: set_Sum_sum_a_nat > set_Sum_sum_a_nat > $o] :
      ( ! [Z2: set_Sum_sum_a_nat] :
          ( ( member8098812455498974984_a_nat @ Z2 @ ( set_se6249060928767827828_a_nat @ X ) )
         => ( Ra @ Z2 @ Z2 ) )
     => ( list_a4614116639835821383_a_nat @ Ra @ X @ X ) ) ).

% list.rel_refl_strong
thf(fact_208_list_Orel__refl__strong,axiom,
    ! [X: list_P6608106274694273645_a_nat,Ra: produc5001885624171833703_a_nat > produc5001885624171833703_a_nat > $o] :
      ( ! [Z2: produc5001885624171833703_a_nat] :
          ( ( member7457213283480048528_a_nat @ Z2 @ ( set_Pr808418073638175228_a_nat @ X ) )
         => ( Ra @ Z2 @ Z2 ) )
     => ( list_a2005310791389460295_a_nat @ Ra @ X @ X ) ) ).

% list.rel_refl_strong
thf(fact_209_list_Orel__refl__strong,axiom,
    ! [X: list_l4703314356710769291_a_nat,Ra: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
      ( ! [Z2: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ Z2 @ ( set_li2392974972034027290_a_nat @ X ) )
         => ( Ra @ Z2 @ Z2 ) )
     => ( list_a5617550422536820423_a_nat @ Ra @ X @ X ) ) ).

% list.rel_refl_strong
thf(fact_210_list_Orel__refl__strong,axiom,
    ! [X: list_P1195027771636113901_a_nat,Ra: produc7017002724195966439_a_nat > produc7017002724195966439_a_nat > $o] :
      ( ! [Z2: produc7017002724195966439_a_nat] :
          ( ( member3723442691059620112_a_nat @ Z2 @ ( set_Pr2931239450594890620_a_nat @ X ) )
         => ( Ra @ Z2 @ Z2 ) )
     => ( list_a2823864037932130119_a_nat @ Ra @ X @ X ) ) ).

% list.rel_refl_strong
thf(fact_211_list_Orel__refl__strong,axiom,
    ! [X: list_Sum_sum_a_nat,Ra: sum_sum_a_nat > sum_sum_a_nat > $o] :
      ( ! [Z2: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ Z2 @ ( set_Sum_sum_a_nat2 @ X ) )
         => ( Ra @ Z2 @ Z2 ) )
     => ( list_a8491001234060422471_a_nat @ Ra @ X @ X ) ) ).

% list.rel_refl_strong
thf(fact_212_list_Omap__ident__strong,axiom,
    ! [T: list_s7376487411892226277_a_nat,F: set_Sum_sum_a_nat > set_Sum_sum_a_nat] :
      ( ! [Z2: set_Sum_sum_a_nat] :
          ( ( member8098812455498974984_a_nat @ Z2 @ ( set_se6249060928767827828_a_nat @ T ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( map_se1607064023862629124_a_nat @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_213_list_Omap__ident__strong,axiom,
    ! [T: list_P6608106274694273645_a_nat,F: produc5001885624171833703_a_nat > produc5001885624171833703_a_nat] :
      ( ! [Z2: produc5001885624171833703_a_nat] :
          ( ( member7457213283480048528_a_nat @ Z2 @ ( set_Pr808418073638175228_a_nat @ T ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( map_Pr3377394081699160580_a_nat @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_214_list_Omap__ident__strong,axiom,
    ! [T: list_l4703314356710769291_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
      ( ! [Z2: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ Z2 @ ( set_li2392974972034027290_a_nat @ T ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( map_li6507455427659069316_a_nat @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_215_list_Omap__ident__strong,axiom,
    ! [T: list_P1195027771636113901_a_nat,F: produc7017002724195966439_a_nat > produc7017002724195966439_a_nat] :
      ( ! [Z2: produc7017002724195966439_a_nat] :
          ( ( member3723442691059620112_a_nat @ Z2 @ ( set_Pr2931239450594890620_a_nat @ T ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( map_Pr3484656804399879172_a_nat @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_216_list_Omap__ident__strong,axiom,
    ! [T: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
      ( ! [Z2: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ Z2 @ ( set_Sum_sum_a_nat2 @ T ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( map_Su2790769393171190532_a_nat @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_217_mem__Collect__eq,axiom,
    ! [A: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > $o] :
      ( ( member408289922725080238_a_nat @ A @ ( collec7555443234367654128_a_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_218_mem__Collect__eq,axiom,
    ! [A: set_Sum_sum_a_nat,P: set_Sum_sum_a_nat > $o] :
      ( ( member8098812455498974984_a_nat @ A @ ( collec4049389696321283146_a_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_219_mem__Collect__eq,axiom,
    ! [A: produc7017002724195966439_a_nat,P: produc7017002724195966439_a_nat > $o] :
      ( ( member3723442691059620112_a_nat @ A @ ( collec1294315627089844562_a_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_220_mem__Collect__eq,axiom,
    ! [A: produc5001885624171833703_a_nat,P: produc5001885624171833703_a_nat > $o] :
      ( ( member7457213283480048528_a_nat @ A @ ( collec3394043145073115602_a_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_221_mem__Collect__eq,axiom,
    ! [A: sum_sum_a_nat,P: sum_sum_a_nat > $o] :
      ( ( member_Sum_sum_a_nat @ A @ ( collec7073057861543223018_a_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_222_Collect__mem__eq,axiom,
    ! [A2: set_li6526943997496501093_a_nat] :
      ( ( collec7555443234367654128_a_nat
        @ ^ [X2: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_223_Collect__mem__eq,axiom,
    ! [A2: set_se4904748513628223167_a_nat] :
      ( ( collec4049389696321283146_a_nat
        @ ^ [X2: set_Sum_sum_a_nat] : ( member8098812455498974984_a_nat @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_224_Collect__mem__eq,axiom,
    ! [A2: set_Pr7343886759072863943_a_nat] :
      ( ( collec1294315627089844562_a_nat
        @ ^ [X2: produc7017002724195966439_a_nat] : ( member3723442691059620112_a_nat @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_225_Collect__mem__eq,axiom,
    ! [A2: set_Pr4870381170404451655_a_nat] :
      ( ( collec3394043145073115602_a_nat
        @ ^ [X2: produc5001885624171833703_a_nat] : ( member7457213283480048528_a_nat @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_226_Collect__mem__eq,axiom,
    ! [A2: set_Sum_sum_a_nat] :
      ( ( collec7073057861543223018_a_nat
        @ ^ [X2: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_227_Collect__cong,axiom,
    ! [P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
      ( ! [X3: sum_sum_a_nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec7073057861543223018_a_nat @ P )
        = ( collec7073057861543223018_a_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_228_map__ext,axiom,
    ! [Xs: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat] :
      ( ! [X3: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
        = ( map_Su2790769393171190532_a_nat @ G @ Xs ) ) ) ).

% map_ext
thf(fact_229_map__ext,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,G: list_Sum_sum_a_nat > set_Sum_sum_a_nat] :
      ( ! [X3: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ X3 @ ( set_li2392974972034027290_a_nat @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_li9079814933684165854_a_nat @ F @ Xs )
        = ( map_li9079814933684165854_a_nat @ G @ Xs ) ) ) ).

% map_ext
thf(fact_230_map__idI,axiom,
    ! [Xs: list_s7376487411892226277_a_nat,F: set_Sum_sum_a_nat > set_Sum_sum_a_nat] :
      ( ! [X3: set_Sum_sum_a_nat] :
          ( ( member8098812455498974984_a_nat @ X3 @ ( set_se6249060928767827828_a_nat @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_se1607064023862629124_a_nat @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_231_map__idI,axiom,
    ! [Xs: list_P6608106274694273645_a_nat,F: produc5001885624171833703_a_nat > produc5001885624171833703_a_nat] :
      ( ! [X3: produc5001885624171833703_a_nat] :
          ( ( member7457213283480048528_a_nat @ X3 @ ( set_Pr808418073638175228_a_nat @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_Pr3377394081699160580_a_nat @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_232_map__idI,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
      ( ! [X3: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ X3 @ ( set_li2392974972034027290_a_nat @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_li6507455427659069316_a_nat @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_233_map__idI,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,F: produc7017002724195966439_a_nat > produc7017002724195966439_a_nat] :
      ( ! [X3: produc7017002724195966439_a_nat] :
          ( ( member3723442691059620112_a_nat @ X3 @ ( set_Pr2931239450594890620_a_nat @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_Pr3484656804399879172_a_nat @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_234_map__idI,axiom,
    ! [Xs: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
      ( ! [X3: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_235_map__cong,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat] :
      ( ( Xs = Ys )
     => ( ! [X3: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
          = ( map_Su2790769393171190532_a_nat @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_236_map__cong,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat,F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,G: list_Sum_sum_a_nat > set_Sum_sum_a_nat] :
      ( ( Xs = Ys )
     => ( ! [X3: list_Sum_sum_a_nat] :
            ( ( member408289922725080238_a_nat @ X3 @ ( set_li2392974972034027290_a_nat @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_li9079814933684165854_a_nat @ F @ Xs )
          = ( map_li9079814933684165854_a_nat @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_237_map__update,axiom,
    ! [F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat,K: nat,Y: list_Sum_sum_a_nat] :
      ( ( map_li9079814933684165854_a_nat @ F @ ( list_u3340303885232119283_a_nat @ Xs @ K @ Y ) )
      = ( list_u7655389300282739405_a_nat @ ( map_li9079814933684165854_a_nat @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).

% map_update
thf(fact_238_map__update,axiom,
    ! [F: sum_sum_a_nat > produc7017002724195966439_a_nat,Xs: list_Sum_sum_a_nat,K: nat,Y: sum_sum_a_nat] :
      ( ( map_Su4221482917690969196_a_nat @ F @ ( list_u9138855634547462509_a_nat @ Xs @ K @ Y ) )
      = ( list_u4149385051645232213_a_nat @ ( map_Su4221482917690969196_a_nat @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).

% map_update
thf(fact_239_map__update,axiom,
    ! [F: produc7017002724195966439_a_nat > sum_sum_a_nat,Xs: list_P1195027771636113901_a_nat,K: nat,Y: produc7017002724195966439_a_nat] :
      ( ( map_Pr3527087583225239196_a_nat @ F @ ( list_u4149385051645232213_a_nat @ Xs @ K @ Y ) )
      = ( list_u9138855634547462509_a_nat @ ( map_Pr3527087583225239196_a_nat @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).

% map_update
thf(fact_240_map__update,axiom,
    ! [F: produc7017002724195966439_a_nat > produc7017002724195966439_a_nat,Xs: list_P1195027771636113901_a_nat,K: nat,Y: produc7017002724195966439_a_nat] :
      ( ( map_Pr3484656804399879172_a_nat @ F @ ( list_u4149385051645232213_a_nat @ Xs @ K @ Y ) )
      = ( list_u4149385051645232213_a_nat @ ( map_Pr3484656804399879172_a_nat @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).

% map_update
thf(fact_241_map__update,axiom,
    ! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat,K: nat,Y: sum_sum_a_nat] :
      ( ( map_Su2790769393171190532_a_nat @ F @ ( list_u9138855634547462509_a_nat @ Xs @ K @ Y ) )
      = ( list_u9138855634547462509_a_nat @ ( map_Su2790769393171190532_a_nat @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).

% map_update
thf(fact_242_ex__map__conv,axiom,
    ! [Ys: list_s7376487411892226277_a_nat,F: list_Sum_sum_a_nat > set_Sum_sum_a_nat] :
      ( ( ? [Xs2: list_l4703314356710769291_a_nat] :
            ( Ys
            = ( map_li9079814933684165854_a_nat @ F @ Xs2 ) ) )
      = ( ! [X2: set_Sum_sum_a_nat] :
            ( ( member8098812455498974984_a_nat @ X2 @ ( set_se6249060928767827828_a_nat @ Ys ) )
           => ? [Y3: list_Sum_sum_a_nat] :
                ( X2
                = ( F @ Y3 ) ) ) ) ) ).

% ex_map_conv
thf(fact_243_ex__map__conv,axiom,
    ! [Ys: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
      ( ( ? [Xs2: list_Sum_sum_a_nat] :
            ( Ys
            = ( map_Su2790769393171190532_a_nat @ F @ Xs2 ) ) )
      = ( ! [X2: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Ys ) )
           => ? [Y3: sum_sum_a_nat] :
                ( X2
                = ( F @ Y3 ) ) ) ) ) ).

% ex_map_conv
thf(fact_244_filter__cong,axiom,
    ! [Xs: list_s7376487411892226277_a_nat,Ys: list_s7376487411892226277_a_nat,P: set_Sum_sum_a_nat > $o,Q: set_Sum_sum_a_nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X3: set_Sum_sum_a_nat] :
            ( ( member8098812455498974984_a_nat @ X3 @ ( set_se6249060928767827828_a_nat @ Ys ) )
           => ( ( P @ X3 )
              = ( Q @ X3 ) ) )
       => ( ( filter4447861443987396444_a_nat @ P @ Xs )
          = ( filter4447861443987396444_a_nat @ Q @ Ys ) ) ) ) ).

% filter_cong
thf(fact_245_filter__cong,axiom,
    ! [Xs: list_P6608106274694273645_a_nat,Ys: list_P6608106274694273645_a_nat,P: produc5001885624171833703_a_nat > $o,Q: produc5001885624171833703_a_nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X3: produc5001885624171833703_a_nat] :
            ( ( member7457213283480048528_a_nat @ X3 @ ( set_Pr808418073638175228_a_nat @ Ys ) )
           => ( ( P @ X3 )
              = ( Q @ X3 ) ) )
       => ( ( filter4328210566177025508_a_nat @ P @ Xs )
          = ( filter4328210566177025508_a_nat @ Q @ Ys ) ) ) ) ).

% filter_cong
thf(fact_246_filter__cong,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat,P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X3: list_Sum_sum_a_nat] :
            ( ( member408289922725080238_a_nat @ X3 @ ( set_li2392974972034027290_a_nat @ Ys ) )
           => ( ( P @ X3 )
              = ( Q @ X3 ) ) )
       => ( ( filter5373755100216644354_a_nat @ P @ Xs )
          = ( filter5373755100216644354_a_nat @ Q @ Ys ) ) ) ) ).

% filter_cong
thf(fact_247_filter__cong,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,Ys: list_P1195027771636113901_a_nat,P: produc7017002724195966439_a_nat > $o,Q: produc7017002724195966439_a_nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X3: produc7017002724195966439_a_nat] :
            ( ( member3723442691059620112_a_nat @ X3 @ ( set_Pr2931239450594890620_a_nat @ Ys ) )
           => ( ( P @ X3 )
              = ( Q @ X3 ) ) )
       => ( ( filter6896953120613689188_a_nat @ P @ Xs )
          = ( filter6896953120613689188_a_nat @ Q @ Ys ) ) ) ) ).

% filter_cong
thf(fact_248_filter__cong,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X3: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Ys ) )
           => ( ( P @ X3 )
              = ( Q @ X3 ) ) )
       => ( ( filter_Sum_sum_a_nat @ P @ Xs )
          = ( filter_Sum_sum_a_nat @ Q @ Ys ) ) ) ) ).

% filter_cong
thf(fact_249_list__all2__eq,axiom,
    ( ( ^ [Y2: list_Sum_sum_a_nat,Z: list_Sum_sum_a_nat] : ( Y2 = Z ) )
    = ( list_a8491001234060422471_a_nat
      @ ^ [Y2: sum_sum_a_nat,Z: sum_sum_a_nat] : ( Y2 = Z ) ) ) ).

% list_all2_eq
thf(fact_250_distinct__zipI1,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( distin2701893636801681144_a_nat @ Xs )
     => ( distin928153214662094432_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) ) ) ).

% distinct_zipI1
thf(fact_251_distinct__zipI2,axiom,
    ! [Ys: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( distin2701893636801681144_a_nat @ Ys )
     => ( distin928153214662094432_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) ) ) ).

% distinct_zipI2
thf(fact_252_filter__id__conv,axiom,
    ! [P: list_Sum_sum_a_nat > $o,Xs: list_l4703314356710769291_a_nat] :
      ( ( ( filter5373755100216644354_a_nat @ P @ Xs )
        = Xs )
      = ( ! [X2: list_Sum_sum_a_nat] :
            ( ( member408289922725080238_a_nat @ X2 @ ( set_li2392974972034027290_a_nat @ Xs ) )
           => ( P @ X2 ) ) ) ) ).

% filter_id_conv
thf(fact_253_filter__id__conv,axiom,
    ! [P: produc7017002724195966439_a_nat > $o,Xs: list_P1195027771636113901_a_nat] :
      ( ( ( filter6896953120613689188_a_nat @ P @ Xs )
        = Xs )
      = ( ! [X2: produc7017002724195966439_a_nat] :
            ( ( member3723442691059620112_a_nat @ X2 @ ( set_Pr2931239450594890620_a_nat @ Xs ) )
           => ( P @ X2 ) ) ) ) ).

% filter_id_conv
thf(fact_254_filter__id__conv,axiom,
    ! [P: sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] :
      ( ( ( filter_Sum_sum_a_nat @ P @ Xs )
        = Xs )
      = ( ! [X2: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
           => ( P @ X2 ) ) ) ) ).

% filter_id_conv
thf(fact_255_list__all2__mono,axiom,
    ! [P: sum_sum_a_nat > sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Q: sum_sum_a_nat > sum_sum_a_nat > $o] :
      ( ( list_a8491001234060422471_a_nat @ P @ Xs @ Ys )
     => ( ! [Xs3: sum_sum_a_nat,Ys2: sum_sum_a_nat] :
            ( ( P @ Xs3 @ Ys2 )
           => ( Q @ Xs3 @ Ys2 ) )
       => ( list_a8491001234060422471_a_nat @ Q @ Xs @ Ys ) ) ) ).

% list_all2_mono
thf(fact_256_list__all2__refl,axiom,
    ! [P: sum_sum_a_nat > sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] :
      ( ! [X3: sum_sum_a_nat] : ( P @ X3 @ X3 )
     => ( list_a8491001234060422471_a_nat @ P @ Xs @ Xs ) ) ).

% list_all2_refl
thf(fact_257_list__all2__same,axiom,
    ! [P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,Xs: list_l4703314356710769291_a_nat] :
      ( ( list_a5617550422536820423_a_nat @ P @ Xs @ Xs )
      = ( ! [X2: list_Sum_sum_a_nat] :
            ( ( member408289922725080238_a_nat @ X2 @ ( set_li2392974972034027290_a_nat @ Xs ) )
           => ( P @ X2 @ X2 ) ) ) ) ).

% list_all2_same
thf(fact_258_list__all2__same,axiom,
    ! [P: produc7017002724195966439_a_nat > produc7017002724195966439_a_nat > $o,Xs: list_P1195027771636113901_a_nat] :
      ( ( list_a2823864037932130119_a_nat @ P @ Xs @ Xs )
      = ( ! [X2: produc7017002724195966439_a_nat] :
            ( ( member3723442691059620112_a_nat @ X2 @ ( set_Pr2931239450594890620_a_nat @ Xs ) )
           => ( P @ X2 @ X2 ) ) ) ) ).

% list_all2_same
thf(fact_259_list__all2__same,axiom,
    ! [P: sum_sum_a_nat > sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] :
      ( ( list_a8491001234060422471_a_nat @ P @ Xs @ Xs )
      = ( ! [X2: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
           => ( P @ X2 @ X2 ) ) ) ) ).

% list_all2_same
thf(fact_260_distinct__filter,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,P: produc7017002724195966439_a_nat > $o] :
      ( ( distin928153214662094432_a_nat @ Xs )
     => ( distin928153214662094432_a_nat @ ( filter6896953120613689188_a_nat @ P @ Xs ) ) ) ).

% distinct_filter
thf(fact_261_distinct__filter,axiom,
    ! [Xs: list_s7376487411892226277_a_nat,P: set_Sum_sum_a_nat > $o] :
      ( ( distin4821671304206132312_a_nat @ Xs )
     => ( distin4821671304206132312_a_nat @ ( filter4447861443987396444_a_nat @ P @ Xs ) ) ) ).

% distinct_filter
thf(fact_262_distinct__filter,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,P: list_Sum_sum_a_nat > $o] :
      ( ( distin811021574259663358_a_nat @ Xs )
     => ( distin811021574259663358_a_nat @ ( filter5373755100216644354_a_nat @ P @ Xs ) ) ) ).

% distinct_filter
thf(fact_263_distinct__filter,axiom,
    ! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
      ( ( distin2701893636801681144_a_nat @ Xs )
     => ( distin2701893636801681144_a_nat @ ( filter_Sum_sum_a_nat @ P @ Xs ) ) ) ).

% distinct_filter
thf(fact_264_list__all2__trans,axiom,
    ! [P1: sum_sum_a_nat > sum_sum_a_nat > $o,P2: sum_sum_a_nat > sum_sum_a_nat > $o,P3: sum_sum_a_nat > sum_sum_a_nat > $o,As: list_Sum_sum_a_nat,Bs: list_Sum_sum_a_nat,Cs: list_Sum_sum_a_nat] :
      ( ! [A3: sum_sum_a_nat,B: sum_sum_a_nat,C: sum_sum_a_nat] :
          ( ( P1 @ A3 @ B )
         => ( ( P2 @ B @ C )
           => ( P3 @ A3 @ C ) ) )
     => ( ( list_a8491001234060422471_a_nat @ P1 @ As @ Bs )
       => ( ( list_a8491001234060422471_a_nat @ P2 @ Bs @ Cs )
         => ( list_a8491001234060422471_a_nat @ P3 @ As @ Cs ) ) ) ) ).

% list_all2_trans
thf(fact_265_list__update__swap,axiom,
    ! [I2: nat,I3: nat,Xs: list_P1195027771636113901_a_nat,X: produc7017002724195966439_a_nat,X4: produc7017002724195966439_a_nat] :
      ( ( I2 != I3 )
     => ( ( list_u4149385051645232213_a_nat @ ( list_u4149385051645232213_a_nat @ Xs @ I2 @ X ) @ I3 @ X4 )
        = ( list_u4149385051645232213_a_nat @ ( list_u4149385051645232213_a_nat @ Xs @ I3 @ X4 ) @ I2 @ X ) ) ) ).

% list_update_swap
thf(fact_266_list__update__swap,axiom,
    ! [I2: nat,I3: nat,Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat,X4: sum_sum_a_nat] :
      ( ( I2 != I3 )
     => ( ( list_u9138855634547462509_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I2 @ X ) @ I3 @ X4 )
        = ( list_u9138855634547462509_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I3 @ X4 ) @ I2 @ X ) ) ) ).

% list_update_swap
thf(fact_267_filter__empty__conv,axiom,
    ! [P: nat > $o,Xs: list_nat] :
      ( ( ( filter_nat @ P @ Xs )
        = nil_nat )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
           => ~ ( P @ X2 ) ) ) ) ).

% filter_empty_conv
thf(fact_268_filter__empty__conv,axiom,
    ! [P: list_Sum_sum_a_nat > $o,Xs: list_l4703314356710769291_a_nat] :
      ( ( ( filter5373755100216644354_a_nat @ P @ Xs )
        = nil_li1906260230833442699_a_nat )
      = ( ! [X2: list_Sum_sum_a_nat] :
            ( ( member408289922725080238_a_nat @ X2 @ ( set_li2392974972034027290_a_nat @ Xs ) )
           => ~ ( P @ X2 ) ) ) ) ).

% filter_empty_conv
thf(fact_269_filter__empty__conv,axiom,
    ! [P: produc7017002724195966439_a_nat > $o,Xs: list_P1195027771636113901_a_nat] :
      ( ( ( filter6896953120613689188_a_nat @ P @ Xs )
        = nil_Pr6585251977456444909_a_nat )
      = ( ! [X2: produc7017002724195966439_a_nat] :
            ( ( member3723442691059620112_a_nat @ X2 @ ( set_Pr2931239450594890620_a_nat @ Xs ) )
           => ~ ( P @ X2 ) ) ) ) ).

% filter_empty_conv
thf(fact_270_filter__empty__conv,axiom,
    ! [P: sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] :
      ( ( ( filter_Sum_sum_a_nat @ P @ Xs )
        = nil_Sum_sum_a_nat )
      = ( ! [X2: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
           => ~ ( P @ X2 ) ) ) ) ).

% filter_empty_conv
thf(fact_271_list__all2__antisym,axiom,
    ! [P: sum_sum_a_nat > sum_sum_a_nat > $o,Q: sum_sum_a_nat > sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ! [X3: sum_sum_a_nat,Y4: sum_sum_a_nat] :
          ( ( P @ X3 @ Y4 )
         => ( ( Q @ Y4 @ X3 )
           => ( X3 = Y4 ) ) )
     => ( ( list_a8491001234060422471_a_nat @ P @ Xs @ Ys )
       => ( ( list_a8491001234060422471_a_nat @ Q @ Ys @ Xs )
         => ( Xs = Ys ) ) ) ) ).

% list_all2_antisym
thf(fact_272_rremdups__set,axiom,
    ! [Xs: list_l4703314356710769291_a_nat] :
      ( ( set_li2392974972034027290_a_nat @ ( rremdu4777469547168625599_a_nat @ Xs ) )
      = ( set_li2392974972034027290_a_nat @ Xs ) ) ).

% rremdups_set
thf(fact_273_rremdups__set,axiom,
    ! [Xs: list_P1195027771636113901_a_nat] :
      ( ( set_Pr2931239450594890620_a_nat @ ( rremdu7275748759011619361_a_nat @ Xs ) )
      = ( set_Pr2931239450594890620_a_nat @ Xs ) ) ).

% rremdups_set
thf(fact_274_rremdups__set,axiom,
    ! [Xs: list_Sum_sum_a_nat] :
      ( ( set_Sum_sum_a_nat2 @ ( rremdu8304153113908149561_a_nat @ Xs ) )
      = ( set_Sum_sum_a_nat2 @ Xs ) ) ).

% rremdups_set
thf(fact_275_filter__rremdups,axiom,
    ! [P: sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] :
      ( ( filter_Sum_sum_a_nat @ P @ ( rremdu8304153113908149561_a_nat @ Xs ) )
      = ( rremdu8304153113908149561_a_nat @ ( filter_Sum_sum_a_nat @ P @ Xs ) ) ) ).

% filter_rremdups
thf(fact_276_list__all2__update__cong,axiom,
    ! [P: sum_sum_a_nat > produc7017002724195966439_a_nat > $o,Xs: list_Sum_sum_a_nat,Ys: list_P1195027771636113901_a_nat,X: sum_sum_a_nat,Y: produc7017002724195966439_a_nat,I2: nat] :
      ( ( list_a1440509178630569775_a_nat @ P @ Xs @ Ys )
     => ( ( P @ X @ Y )
       => ( list_a1440509178630569775_a_nat @ P @ ( list_u9138855634547462509_a_nat @ Xs @ I2 @ X ) @ ( list_u4149385051645232213_a_nat @ Ys @ I2 @ Y ) ) ) ) ).

% list_all2_update_cong
thf(fact_277_list__all2__update__cong,axiom,
    ! [P: produc7017002724195966439_a_nat > sum_sum_a_nat > $o,Xs: list_P1195027771636113901_a_nat,Ys: list_Sum_sum_a_nat,X: produc7017002724195966439_a_nat,Y: sum_sum_a_nat,I2: nat] :
      ( ( list_a746113844164839775_a_nat @ P @ Xs @ Ys )
     => ( ( P @ X @ Y )
       => ( list_a746113844164839775_a_nat @ P @ ( list_u4149385051645232213_a_nat @ Xs @ I2 @ X ) @ ( list_u9138855634547462509_a_nat @ Ys @ I2 @ Y ) ) ) ) ).

% list_all2_update_cong
thf(fact_278_list__all2__update__cong,axiom,
    ! [P: produc7017002724195966439_a_nat > produc7017002724195966439_a_nat > $o,Xs: list_P1195027771636113901_a_nat,Ys: list_P1195027771636113901_a_nat,X: produc7017002724195966439_a_nat,Y: produc7017002724195966439_a_nat,I2: nat] :
      ( ( list_a2823864037932130119_a_nat @ P @ Xs @ Ys )
     => ( ( P @ X @ Y )
       => ( list_a2823864037932130119_a_nat @ P @ ( list_u4149385051645232213_a_nat @ Xs @ I2 @ X ) @ ( list_u4149385051645232213_a_nat @ Ys @ I2 @ Y ) ) ) ) ).

% list_all2_update_cong
thf(fact_279_list__all2__update__cong,axiom,
    ! [P: sum_sum_a_nat > sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,X: sum_sum_a_nat,Y: sum_sum_a_nat,I2: nat] :
      ( ( list_a8491001234060422471_a_nat @ P @ Xs @ Ys )
     => ( ( P @ X @ Y )
       => ( list_a8491001234060422471_a_nat @ P @ ( list_u9138855634547462509_a_nat @ Xs @ I2 @ X ) @ ( list_u9138855634547462509_a_nat @ Ys @ I2 @ Y ) ) ) ) ).

% list_all2_update_cong
thf(fact_280_distinct__rremdups,axiom,
    ! [Xs: list_P1195027771636113901_a_nat] : ( distin928153214662094432_a_nat @ ( rremdu7275748759011619361_a_nat @ Xs ) ) ).

% distinct_rremdups
thf(fact_281_distinct__rremdups,axiom,
    ! [Xs: list_s7376487411892226277_a_nat] : ( distin4821671304206132312_a_nat @ ( rremdu6376807763710068377_a_nat @ Xs ) ) ).

% distinct_rremdups
thf(fact_282_distinct__rremdups,axiom,
    ! [Xs: list_l4703314356710769291_a_nat] : ( distin811021574259663358_a_nat @ ( rremdu4777469547168625599_a_nat @ Xs ) ) ).

% distinct_rremdups
thf(fact_283_distinct__rremdups,axiom,
    ! [Xs: list_Sum_sum_a_nat] : ( distin2701893636801681144_a_nat @ ( rremdu8304153113908149561_a_nat @ Xs ) ) ).

% distinct_rremdups
thf(fact_284_distinct__map__filterI,axiom,
    ! [F: sum_sum_a_nat > produc7017002724195966439_a_nat,Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
      ( ( distin928153214662094432_a_nat @ ( map_Su4221482917690969196_a_nat @ F @ Xs ) )
     => ( distin928153214662094432_a_nat @ ( map_Su4221482917690969196_a_nat @ F @ ( filter_Sum_sum_a_nat @ P @ Xs ) ) ) ) ).

% distinct_map_filterI
thf(fact_285_distinct__map__filterI,axiom,
    ! [F: sum_sum_a_nat > set_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
      ( ( distin4821671304206132312_a_nat @ ( map_Su6628313897983864932_a_nat @ F @ Xs ) )
     => ( distin4821671304206132312_a_nat @ ( map_Su6628313897983864932_a_nat @ F @ ( filter_Sum_sum_a_nat @ P @ Xs ) ) ) ) ).

% distinct_map_filterI
thf(fact_286_distinct__map__filterI,axiom,
    ! [F: sum_sum_a_nat > list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
      ( ( distin811021574259663358_a_nat @ ( map_Su3479376915845212170_a_nat @ F @ Xs ) )
     => ( distin811021574259663358_a_nat @ ( map_Su3479376915845212170_a_nat @ F @ ( filter_Sum_sum_a_nat @ P @ Xs ) ) ) ) ).

% distinct_map_filterI
thf(fact_287_distinct__map__filterI,axiom,
    ! [F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat,P: list_Sum_sum_a_nat > $o] :
      ( ( distin4821671304206132312_a_nat @ ( map_li9079814933684165854_a_nat @ F @ Xs ) )
     => ( distin4821671304206132312_a_nat @ ( map_li9079814933684165854_a_nat @ F @ ( filter5373755100216644354_a_nat @ P @ Xs ) ) ) ) ).

% distinct_map_filterI
thf(fact_288_distinct__map__filterI,axiom,
    ! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
      ( ( distin2701893636801681144_a_nat @ ( map_Su2790769393171190532_a_nat @ F @ Xs ) )
     => ( distin2701893636801681144_a_nat @ ( map_Su2790769393171190532_a_nat @ F @ ( filter_Sum_sum_a_nat @ P @ Xs ) ) ) ) ).

% distinct_map_filterI
thf(fact_289_Containers__Auxiliary_Oempty__filter__conv,axiom,
    ! [P: nat > $o,Xs: list_nat] :
      ( ( nil_nat
        = ( filter_nat @ P @ Xs ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
           => ~ ( P @ X2 ) ) ) ) ).

% Containers_Auxiliary.empty_filter_conv
thf(fact_290_Containers__Auxiliary_Oempty__filter__conv,axiom,
    ! [P: list_Sum_sum_a_nat > $o,Xs: list_l4703314356710769291_a_nat] :
      ( ( nil_li1906260230833442699_a_nat
        = ( filter5373755100216644354_a_nat @ P @ Xs ) )
      = ( ! [X2: list_Sum_sum_a_nat] :
            ( ( member408289922725080238_a_nat @ X2 @ ( set_li2392974972034027290_a_nat @ Xs ) )
           => ~ ( P @ X2 ) ) ) ) ).

% Containers_Auxiliary.empty_filter_conv
thf(fact_291_Containers__Auxiliary_Oempty__filter__conv,axiom,
    ! [P: produc7017002724195966439_a_nat > $o,Xs: list_P1195027771636113901_a_nat] :
      ( ( nil_Pr6585251977456444909_a_nat
        = ( filter6896953120613689188_a_nat @ P @ Xs ) )
      = ( ! [X2: produc7017002724195966439_a_nat] :
            ( ( member3723442691059620112_a_nat @ X2 @ ( set_Pr2931239450594890620_a_nat @ Xs ) )
           => ~ ( P @ X2 ) ) ) ) ).

% Containers_Auxiliary.empty_filter_conv
thf(fact_292_Containers__Auxiliary_Oempty__filter__conv,axiom,
    ! [P: sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] :
      ( ( nil_Sum_sum_a_nat
        = ( filter_Sum_sum_a_nat @ P @ Xs ) )
      = ( ! [X2: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
           => ~ ( P @ X2 ) ) ) ) ).

% Containers_Auxiliary.empty_filter_conv
thf(fact_293_min_Oidem,axiom,
    ! [A: nat] :
      ( ( ord_min_nat @ A @ A )
      = A ) ).

% min.idem
thf(fact_294_min_Oleft__idem,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_min_nat @ A @ ( ord_min_nat @ A @ B2 ) )
      = ( ord_min_nat @ A @ B2 ) ) ).

% min.left_idem
thf(fact_295_min_Oright__idem,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_min_nat @ ( ord_min_nat @ A @ B2 ) @ B2 )
      = ( ord_min_nat @ A @ B2 ) ) ).

% min.right_idem
thf(fact_296_distinct__union,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,Ys: list_P1195027771636113901_a_nat] :
      ( ( distin928153214662094432_a_nat @ ( union_7179124240371534305_a_nat @ Xs @ Ys ) )
      = ( distin928153214662094432_a_nat @ Ys ) ) ).

% distinct_union
thf(fact_297_distinct__union,axiom,
    ! [Xs: list_s7376487411892226277_a_nat,Ys: list_s7376487411892226277_a_nat] :
      ( ( distin4821671304206132312_a_nat @ ( union_185282995832094297_a_nat @ Xs @ Ys ) )
      = ( distin4821671304206132312_a_nat @ Ys ) ) ).

% distinct_union
thf(fact_298_distinct__union,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat] :
      ( ( distin811021574259663358_a_nat @ ( union_2406054330973375359_a_nat @ Xs @ Ys ) )
      = ( distin811021574259663358_a_nat @ Ys ) ) ).

% distinct_union
thf(fact_299_distinct__union,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( distin2701893636801681144_a_nat @ ( union_Sum_sum_a_nat @ Xs @ Ys ) )
      = ( distin2701893636801681144_a_nat @ Ys ) ) ).

% distinct_union
thf(fact_300_in__set__simps_I3_J,axiom,
    ! [X: set_Sum_sum_a_nat] :
      ~ ( member8098812455498974984_a_nat @ X @ ( set_se6249060928767827828_a_nat @ nil_se616183255201024101_a_nat ) ) ).

% in_set_simps(3)
thf(fact_301_in__set__simps_I3_J,axiom,
    ! [X: produc5001885624171833703_a_nat] :
      ~ ( member7457213283480048528_a_nat @ X @ ( set_Pr808418073638175228_a_nat @ nil_Pr1141218757497453421_a_nat ) ) ).

% in_set_simps(3)
thf(fact_302_in__set__simps_I3_J,axiom,
    ! [X: nat] :
      ~ ( member_nat @ X @ ( set_nat2 @ nil_nat ) ) ).

% in_set_simps(3)
thf(fact_303_in__set__simps_I3_J,axiom,
    ! [X: list_Sum_sum_a_nat] :
      ~ ( member408289922725080238_a_nat @ X @ ( set_li2392974972034027290_a_nat @ nil_li1906260230833442699_a_nat ) ) ).

% in_set_simps(3)
thf(fact_304_in__set__simps_I3_J,axiom,
    ! [X: produc7017002724195966439_a_nat] :
      ~ ( member3723442691059620112_a_nat @ X @ ( set_Pr2931239450594890620_a_nat @ nil_Pr6585251977456444909_a_nat ) ) ).

% in_set_simps(3)
thf(fact_305_in__set__simps_I3_J,axiom,
    ! [X: sum_sum_a_nat] :
      ~ ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ nil_Sum_sum_a_nat ) ) ).

% in_set_simps(3)
thf(fact_306_map__eq__map__tailrec,axiom,
    map_Su2790769393171190532_a_nat = map_ta197872344851190606_a_nat ).

% map_eq_map_tailrec
thf(fact_307_map__eq__map__tailrec,axiom,
    map_li9079814933684165854_a_nat = map_ta4970547734792538152_a_nat ).

% map_eq_map_tailrec
thf(fact_308_list__ex1__simps_I1_J,axiom,
    ! [P: produc7017002724195966439_a_nat > $o] :
      ~ ( list_e7186075253971769136_a_nat @ P @ nil_Pr6585251977456444909_a_nat ) ).

% list_ex1_simps(1)
thf(fact_309_list__ex1__simps_I1_J,axiom,
    ! [P: list_Sum_sum_a_nat > $o] :
      ~ ( list_e4824308470127291086_a_nat @ P @ nil_li1906260230833442699_a_nat ) ).

% list_ex1_simps(1)
thf(fact_310_list__ex1__simps_I1_J,axiom,
    ! [P: nat > $o] :
      ~ ( list_ex1_nat @ P @ nil_nat ) ).

% list_ex1_simps(1)
thf(fact_311_list__ex1__simps_I1_J,axiom,
    ! [P: sum_sum_a_nat > $o] :
      ~ ( list_e7903279875808954312_a_nat @ P @ nil_Sum_sum_a_nat ) ).

% list_ex1_simps(1)
thf(fact_312_filter__set,axiom,
    ! [P: list_Sum_sum_a_nat > $o,Xs: list_l4703314356710769291_a_nat] :
      ( ( filter4744287584154367926_a_nat @ P @ ( set_li2392974972034027290_a_nat @ Xs ) )
      = ( set_li2392974972034027290_a_nat @ ( filter5373755100216644354_a_nat @ P @ Xs ) ) ) ).

% filter_set
thf(fact_313_filter__set,axiom,
    ! [P: produc7017002724195966439_a_nat > $o,Xs: list_P1195027771636113901_a_nat] :
      ( ( filter417278669945599000_a_nat @ P @ ( set_Pr2931239450594890620_a_nat @ Xs ) )
      = ( set_Pr2931239450594890620_a_nat @ ( filter6896953120613689188_a_nat @ P @ Xs ) ) ) ).

% filter_set
thf(fact_314_filter__set,axiom,
    ! [P: sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] :
      ( ( filter_Sum_sum_a_nat2 @ P @ ( set_Sum_sum_a_nat2 @ Xs ) )
      = ( set_Sum_sum_a_nat2 @ ( filter_Sum_sum_a_nat @ P @ Xs ) ) ) ).

% filter_set
thf(fact_315_min_Oassoc,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( ord_min_nat @ ( ord_min_nat @ A @ B2 ) @ C2 )
      = ( ord_min_nat @ A @ ( ord_min_nat @ B2 @ C2 ) ) ) ).

% min.assoc
thf(fact_316_list__ex1__iff,axiom,
    ( list_e4825210516842321192_a_nat
    = ( ^ [P4: set_Sum_sum_a_nat > $o,Xs2: list_s7376487411892226277_a_nat] :
        ? [X2: set_Sum_sum_a_nat] :
          ( ( member8098812455498974984_a_nat @ X2 @ ( set_se6249060928767827828_a_nat @ Xs2 ) )
          & ( P4 @ X2 )
          & ! [Y3: set_Sum_sum_a_nat] :
              ( ( ( member8098812455498974984_a_nat @ Y3 @ ( set_se6249060928767827828_a_nat @ Xs2 ) )
                & ( P4 @ Y3 ) )
             => ( Y3 = X2 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_317_list__ex1__iff,axiom,
    ( list_e2762905939734966704_a_nat
    = ( ^ [P4: produc5001885624171833703_a_nat > $o,Xs2: list_P6608106274694273645_a_nat] :
        ? [X2: produc5001885624171833703_a_nat] :
          ( ( member7457213283480048528_a_nat @ X2 @ ( set_Pr808418073638175228_a_nat @ Xs2 ) )
          & ( P4 @ X2 )
          & ! [Y3: produc5001885624171833703_a_nat] :
              ( ( ( member7457213283480048528_a_nat @ Y3 @ ( set_Pr808418073638175228_a_nat @ Xs2 ) )
                & ( P4 @ Y3 ) )
             => ( Y3 = X2 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_318_list__ex1__iff,axiom,
    ( list_e4824308470127291086_a_nat
    = ( ^ [P4: list_Sum_sum_a_nat > $o,Xs2: list_l4703314356710769291_a_nat] :
        ? [X2: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ X2 @ ( set_li2392974972034027290_a_nat @ Xs2 ) )
          & ( P4 @ X2 )
          & ! [Y3: list_Sum_sum_a_nat] :
              ( ( ( member408289922725080238_a_nat @ Y3 @ ( set_li2392974972034027290_a_nat @ Xs2 ) )
                & ( P4 @ Y3 ) )
             => ( Y3 = X2 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_319_list__ex1__iff,axiom,
    ( list_e7186075253971769136_a_nat
    = ( ^ [P4: produc7017002724195966439_a_nat > $o,Xs2: list_P1195027771636113901_a_nat] :
        ? [X2: produc7017002724195966439_a_nat] :
          ( ( member3723442691059620112_a_nat @ X2 @ ( set_Pr2931239450594890620_a_nat @ Xs2 ) )
          & ( P4 @ X2 )
          & ! [Y3: produc7017002724195966439_a_nat] :
              ( ( ( member3723442691059620112_a_nat @ Y3 @ ( set_Pr2931239450594890620_a_nat @ Xs2 ) )
                & ( P4 @ Y3 ) )
             => ( Y3 = X2 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_320_list__ex1__iff,axiom,
    ( list_e7903279875808954312_a_nat
    = ( ^ [P4: sum_sum_a_nat > $o,Xs2: list_Sum_sum_a_nat] :
        ? [X2: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs2 ) )
          & ( P4 @ X2 )
          & ! [Y3: sum_sum_a_nat] :
              ( ( ( member_Sum_sum_a_nat @ Y3 @ ( set_Sum_sum_a_nat2 @ Xs2 ) )
                & ( P4 @ Y3 ) )
             => ( Y3 = X2 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_321_min_Oleft__commute,axiom,
    ! [B2: nat,A: nat,C2: nat] :
      ( ( ord_min_nat @ B2 @ ( ord_min_nat @ A @ C2 ) )
      = ( ord_min_nat @ A @ ( ord_min_nat @ B2 @ C2 ) ) ) ).

% min.left_commute
thf(fact_322_min_Ocommute,axiom,
    ( ord_min_nat
    = ( ^ [A4: nat,B3: nat] : ( ord_min_nat @ B3 @ A4 ) ) ) ).

% min.commute
thf(fact_323_Set_Omember__filter,axiom,
    ! [X: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > $o,A2: set_li6526943997496501093_a_nat] :
      ( ( member408289922725080238_a_nat @ X @ ( filter4744287584154367926_a_nat @ P @ A2 ) )
      = ( ( member408289922725080238_a_nat @ X @ A2 )
        & ( P @ X ) ) ) ).

% Set.member_filter
thf(fact_324_Set_Omember__filter,axiom,
    ! [X: set_Sum_sum_a_nat,P: set_Sum_sum_a_nat > $o,A2: set_se4904748513628223167_a_nat] :
      ( ( member8098812455498974984_a_nat @ X @ ( filter6315372162244433424_a_nat @ P @ A2 ) )
      = ( ( member8098812455498974984_a_nat @ X @ A2 )
        & ( P @ X ) ) ) ).

% Set.member_filter
thf(fact_325_Set_Omember__filter,axiom,
    ! [X: produc7017002724195966439_a_nat,P: produc7017002724195966439_a_nat > $o,A2: set_Pr7343886759072863943_a_nat] :
      ( ( member3723442691059620112_a_nat @ X @ ( filter417278669945599000_a_nat @ P @ A2 ) )
      = ( ( member3723442691059620112_a_nat @ X @ A2 )
        & ( P @ X ) ) ) ).

% Set.member_filter
thf(fact_326_Set_Omember__filter,axiom,
    ! [X: produc5001885624171833703_a_nat,P: produc5001885624171833703_a_nat > $o,A2: set_Pr4870381170404451655_a_nat] :
      ( ( member7457213283480048528_a_nat @ X @ ( filter3383343615081260696_a_nat @ P @ A2 ) )
      = ( ( member7457213283480048528_a_nat @ X @ A2 )
        & ( P @ X ) ) ) ).

% Set.member_filter
thf(fact_327_Set_Omember__filter,axiom,
    ! [X: sum_sum_a_nat,P: sum_sum_a_nat > $o,A2: set_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ X @ ( filter_Sum_sum_a_nat2 @ P @ A2 ) )
      = ( ( member_Sum_sum_a_nat @ X @ A2 )
        & ( P @ X ) ) ) ).

% Set.member_filter
thf(fact_328_can__select__set__list__ex1,axiom,
    ! [P: list_Sum_sum_a_nat > $o,A2: list_l4703314356710769291_a_nat] :
      ( ( can_se2197130373557366786_a_nat @ P @ ( set_li2392974972034027290_a_nat @ A2 ) )
      = ( list_e4824308470127291086_a_nat @ P @ A2 ) ) ).

% can_select_set_list_ex1
thf(fact_329_can__select__set__list__ex1,axiom,
    ! [P: produc7017002724195966439_a_nat > $o,A2: list_P1195027771636113901_a_nat] :
      ( ( can_se8355593179296023652_a_nat @ P @ ( set_Pr2931239450594890620_a_nat @ A2 ) )
      = ( list_e7186075253971769136_a_nat @ P @ A2 ) ) ).

% can_select_set_list_ex1
thf(fact_330_can__select__set__list__ex1,axiom,
    ! [P: sum_sum_a_nat > $o,A2: list_Sum_sum_a_nat] :
      ( ( can_se22271275864192764_a_nat @ P @ ( set_Sum_sum_a_nat2 @ A2 ) )
      = ( list_e7903279875808954312_a_nat @ P @ A2 ) ) ).

% can_select_set_list_ex1
thf(fact_331_distinct__product,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( distin2701893636801681144_a_nat @ Xs )
     => ( ( distin2701893636801681144_a_nat @ Ys )
       => ( distin928153214662094432_a_nat @ ( produc7134955936706270469_a_nat @ Xs @ Ys ) ) ) ) ).

% distinct_product
thf(fact_332_distinct__product,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_s7376487411892226277_a_nat] :
      ( ( distin2701893636801681144_a_nat @ Xs )
     => ( ( distin4821671304206132312_a_nat @ Ys )
       => ( distin2701942696825509824_a_nat @ ( produc7206417534075874917_a_nat @ Xs @ Ys ) ) ) ) ).

% distinct_product
thf(fact_333_distinct__product,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_l4703314356710769291_a_nat] :
      ( ( distin2701893636801681144_a_nat @ Xs )
     => ( ( distin811021574259663358_a_nat @ Ys )
       => ( distin9124545944939041766_a_nat @ ( produc3448701928316679563_a_nat @ Xs @ Ys ) ) ) ) ).

% distinct_product
thf(fact_334_distinct__product,axiom,
    ! [Xs: list_s7376487411892226277_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( distin4821671304206132312_a_nat @ Xs )
     => ( ( distin2701893636801681144_a_nat @ Ys )
       => ( distin2369888004842216704_a_nat @ ( produc4421902774560781221_a_nat @ Xs @ Ys ) ) ) ) ).

% distinct_product
thf(fact_335_distinct__product,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( distin811021574259663358_a_nat @ Xs )
     => ( ( distin2701893636801681144_a_nat @ Ys )
       => ( distin6082956270961113178_a_nat @ ( produc3075939308942243071_a_nat @ Xs @ Ys ) ) ) ) ).

% distinct_product
thf(fact_336_distinct__product,axiom,
    ! [Xs: list_s7376487411892226277_a_nat,Ys: list_s7376487411892226277_a_nat] :
      ( ( distin4821671304206132312_a_nat @ Xs )
     => ( ( distin4821671304206132312_a_nat @ Ys )
       => ( distin2540775250022794848_a_nat @ ( produc706862384706489605_a_nat @ Xs @ Ys ) ) ) ) ).

% distinct_product
thf(fact_337_distinct__product,axiom,
    ! [Xs: list_s7376487411892226277_a_nat,Ys: list_l4703314356710769291_a_nat] :
      ( ( distin4821671304206132312_a_nat @ Xs )
     => ( ( distin811021574259663358_a_nat @ Ys )
       => ( distin2524318136373408390_a_nat @ ( produc8901269937835131947_a_nat @ Xs @ Ys ) ) ) ) ).

% distinct_product
thf(fact_338_distinct__product,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,Ys: list_s7376487411892226277_a_nat] :
      ( ( distin811021574259663358_a_nat @ Xs )
     => ( ( distin4821671304206132312_a_nat @ Ys )
       => ( distin2310918592538278330_a_nat @ ( produc4585785479280518751_a_nat @ Xs @ Ys ) ) ) ) ).

% distinct_product
thf(fact_339_distinct__product,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat] :
      ( ( distin811021574259663358_a_nat @ Xs )
     => ( ( distin811021574259663358_a_nat @ Ys )
       => ( distin3585382044144064736_a_nat @ ( produc8487947894020673669_a_nat @ Xs @ Ys ) ) ) ) ).

% distinct_product
thf(fact_340_distinct__product,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_P1195027771636113901_a_nat] :
      ( ( distin2701893636801681144_a_nat @ Xs )
     => ( ( distin928153214662094432_a_nat @ Ys )
       => ( distin4917900207559653704_a_nat @ ( produc2458227688186579949_a_nat @ Xs @ Ys ) ) ) ) ).

% distinct_product
thf(fact_341_product_Osimps_I1_J,axiom,
    ! [Uu: list_Sum_sum_a_nat] :
      ( ( produc7134955936706270469_a_nat @ nil_Sum_sum_a_nat @ Uu )
      = nil_Pr6585251977456444909_a_nat ) ).

% product.simps(1)
thf(fact_342_set__union,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat] :
      ( ( set_li2392974972034027290_a_nat @ ( union_2406054330973375359_a_nat @ Xs @ Ys ) )
      = ( sup_su4083067149120280889_a_nat @ ( set_li2392974972034027290_a_nat @ Xs ) @ ( set_li2392974972034027290_a_nat @ Ys ) ) ) ).

% set_union
thf(fact_343_set__union,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,Ys: list_P1195027771636113901_a_nat] :
      ( ( set_Pr2931239450594890620_a_nat @ ( union_7179124240371534305_a_nat @ Xs @ Ys ) )
      = ( sup_su2259505869435368603_a_nat @ ( set_Pr2931239450594890620_a_nat @ Xs ) @ ( set_Pr2931239450594890620_a_nat @ Ys ) ) ) ).

% set_union
thf(fact_344_set__union,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( set_Sum_sum_a_nat2 @ ( union_Sum_sum_a_nat @ Xs @ Ys ) )
      = ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( set_Sum_sum_a_nat2 @ Ys ) ) ) ).

% set_union
thf(fact_345_zip__obtain__same__length,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,Ys: list_P1195027771636113901_a_nat,P: list_P6037234014419372525_a_nat > $o] :
      ( ! [Zs: list_P1195027771636113901_a_nat,Ws: list_P1195027771636113901_a_nat,N2: nat] :
          ( ( ( size_s7247017532682008665_a_nat @ Zs )
            = ( size_s7247017532682008665_a_nat @ Ws ) )
         => ( ( N2
              = ( ord_min_nat @ ( size_s7247017532682008665_a_nat @ Xs ) @ ( size_s7247017532682008665_a_nat @ Ys ) ) )
           => ( ( Zs
                = ( take_P3324508860426538453_a_nat @ N2 @ Xs ) )
             => ( ( Ws
                  = ( take_P3324508860426538453_a_nat @ N2 @ Ys ) )
               => ( P @ ( zip_Pr9202122185580356215_a_nat @ Zs @ Ws ) ) ) ) ) )
     => ( P @ ( zip_Pr9202122185580356215_a_nat @ Xs @ Ys ) ) ) ).

% zip_obtain_same_length
thf(fact_346_zip__obtain__same__length,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,Ys: list_Sum_sum_a_nat,P: list_P4071774479222187013_a_nat > $o] :
      ( ! [Zs: list_P1195027771636113901_a_nat,Ws: list_Sum_sum_a_nat,N2: nat] :
          ( ( ( size_s7247017532682008665_a_nat @ Zs )
            = ( size_s5283204784079214577_a_nat @ Ws ) )
         => ( ( N2
              = ( ord_min_nat @ ( size_s7247017532682008665_a_nat @ Xs ) @ ( size_s5283204784079214577_a_nat @ Ys ) ) )
           => ( ( Zs
                = ( take_P3324508860426538453_a_nat @ N2 @ Xs ) )
             => ( ( Ws
                  = ( take_Sum_sum_a_nat @ N2 @ Ys ) )
               => ( P @ ( zip_Pr3707921967074149519_a_nat @ Zs @ Ws ) ) ) ) ) )
     => ( P @ ( zip_Pr3707921967074149519_a_nat @ Xs @ Ys ) ) ) ).

% zip_obtain_same_length
thf(fact_347_zip__obtain__same__length,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_P1195027771636113901_a_nat,P: list_P6700786065914020821_a_nat > $o] :
      ( ! [Zs: list_Sum_sum_a_nat,Ws: list_P1195027771636113901_a_nat,N2: nat] :
          ( ( ( size_s5283204784079214577_a_nat @ Zs )
            = ( size_s7247017532682008665_a_nat @ Ws ) )
         => ( ( N2
              = ( ord_min_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ ( size_s7247017532682008665_a_nat @ Ys ) ) )
           => ( ( Zs
                = ( take_Sum_sum_a_nat @ N2 @ Xs ) )
             => ( ( Ws
                  = ( take_P3324508860426538453_a_nat @ N2 @ Ys ) )
               => ( P @ ( zip_Su4402317301539879519_a_nat @ Zs @ Ws ) ) ) ) ) )
     => ( P @ ( zip_Su4402317301539879519_a_nat @ Xs @ Ys ) ) ) ).

% zip_obtain_same_length
thf(fact_348_zip__obtain__same__length,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,P: list_P1195027771636113901_a_nat > $o] :
      ( ! [Zs: list_Sum_sum_a_nat,Ws: list_Sum_sum_a_nat,N2: nat] :
          ( ( ( size_s5283204784079214577_a_nat @ Zs )
            = ( size_s5283204784079214577_a_nat @ Ws ) )
         => ( ( N2
              = ( ord_min_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ ( size_s5283204784079214577_a_nat @ Ys ) ) )
           => ( ( Zs
                = ( take_Sum_sum_a_nat @ N2 @ Xs ) )
             => ( ( Ws
                  = ( take_Sum_sum_a_nat @ N2 @ Ys ) )
               => ( P @ ( zip_Su7355543910597222519_a_nat @ Zs @ Ws ) ) ) ) ) )
     => ( P @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) ) ) ).

% zip_obtain_same_length
thf(fact_349_distinct__set__subseqs,axiom,
    ! [Xs: list_s7376487411892226277_a_nat] :
      ( ( distin4821671304206132312_a_nat @ Xs )
     => ( distin2582252071445032376_a_nat @ ( map_li3657301428870794462_a_nat @ set_se6249060928767827828_a_nat @ ( subseq1669510427954520372_a_nat @ Xs ) ) ) ) ).

% distinct_set_subseqs
thf(fact_350_distinct__set__subseqs,axiom,
    ! [Xs: list_l4703314356710769291_a_nat] :
      ( ( distin811021574259663358_a_nat @ Xs )
     => ( distin6522264588777926110_a_nat @ ( map_li2834150261122285406_a_nat @ set_li2392974972034027290_a_nat @ ( subseq4596077244143087834_a_nat @ Xs ) ) ) ) ).

% distinct_set_subseqs
thf(fact_351_distinct__set__subseqs,axiom,
    ! [Xs: list_P1195027771636113901_a_nat] :
      ( ( distin928153214662094432_a_nat @ Xs )
     => ( distin2944342510436172608_a_nat @ ( map_li6209216568146725854_a_nat @ set_Pr2931239450594890620_a_nat @ ( subseq5198752961258583356_a_nat @ Xs ) ) ) ) ).

% distinct_set_subseqs
thf(fact_352_distinct__set__subseqs,axiom,
    ! [Xs: list_Sum_sum_a_nat] :
      ( ( distin2701893636801681144_a_nat @ Xs )
     => ( distin4821671304206132312_a_nat @ ( map_li9079814933684165854_a_nat @ set_Sum_sum_a_nat2 @ ( subseq8414445098004693972_a_nat @ Xs ) ) ) ) ).

% distinct_set_subseqs
thf(fact_353_map__fun__upd,axiom,
    ! [Y: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,V: sum_sum_a_nat] :
      ( ~ ( member_Sum_sum_a_nat @ Y @ ( set_Sum_sum_a_nat2 @ Xs ) )
     => ( ( map_Su2790769393171190532_a_nat @ ( fun_up6086130847573437501_a_nat @ F @ Y @ V ) @ Xs )
        = ( map_Su2790769393171190532_a_nat @ F @ Xs ) ) ) ).

% map_fun_upd
thf(fact_354_map__fun__upd,axiom,
    ! [Y: list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat,F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,V: set_Sum_sum_a_nat] :
      ( ~ ( member408289922725080238_a_nat @ Y @ ( set_li2392974972034027290_a_nat @ Xs ) )
     => ( ( map_li9079814933684165854_a_nat @ ( fun_up3573206269406175639_a_nat @ F @ Y @ V ) @ Xs )
        = ( map_li9079814933684165854_a_nat @ F @ Xs ) ) ) ).

% map_fun_upd
thf(fact_355_bind__simps_I1_J,axiom,
    ! [F: sum_sum_a_nat > list_Sum_sum_a_nat] :
      ( ( bind_S8497345843576593223_a_nat @ nil_Sum_sum_a_nat @ F )
      = nil_Sum_sum_a_nat ) ).

% bind_simps(1)
thf(fact_356_bind__simps_I1_J,axiom,
    ! [F: nat > list_nat] :
      ( ( bind_nat_nat @ nil_nat @ F )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_357_bind__simps_I1_J,axiom,
    ! [F: sum_sum_a_nat > list_nat] :
      ( ( bind_S7309778641578091144at_nat @ nil_Sum_sum_a_nat @ F )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_358_bind__simps_I1_J,axiom,
    ! [F: nat > list_Sum_sum_a_nat] :
      ( ( bind_n2905796707917019494_a_nat @ nil_nat @ F )
      = nil_Sum_sum_a_nat ) ).

% bind_simps(1)
thf(fact_359_bind__simps_I1_J,axiom,
    ! [F: list_Sum_sum_a_nat > list_nat] :
      ( ( bind_l5927154061965146638at_nat @ nil_li1906260230833442699_a_nat @ F )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_360_bind__simps_I1_J,axiom,
    ! [F: nat > list_l4703314356710769291_a_nat] :
      ( ( bind_n431031891494352236_a_nat @ nil_nat @ F )
      = nil_li1906260230833442699_a_nat ) ).

% bind_simps(1)
thf(fact_361_bind__simps_I1_J,axiom,
    ! [F: sum_sum_a_nat > list_l4703314356710769291_a_nat] :
      ( ( bind_S7470141391546446541_a_nat @ nil_Sum_sum_a_nat @ F )
      = nil_li1906260230833442699_a_nat ) ).

% bind_simps(1)
thf(fact_362_bind__simps_I1_J,axiom,
    ! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
      ( ( bind_l7097378772172010049_a_nat @ nil_li1906260230833442699_a_nat @ F )
      = nil_Sum_sum_a_nat ) ).

% bind_simps(1)
thf(fact_363_bind__simps_I1_J,axiom,
    ! [F: produc7017002724195966439_a_nat > list_nat] :
      ( ( bind_P5957727470335167600at_nat @ nil_Pr6585251977456444909_a_nat @ F )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_364_bind__simps_I1_J,axiom,
    ! [F: list_Sum_sum_a_nat > list_l4703314356710769291_a_nat] :
      ( ( bind_l6437164444685190343_a_nat @ nil_li1906260230833442699_a_nat @ F )
      = nil_li1906260230833442699_a_nat ) ).

% bind_simps(1)
thf(fact_365_take__update__cancel,axiom,
    ! [N: nat,M: nat,Xs: list_P1195027771636113901_a_nat,Y: produc7017002724195966439_a_nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( take_P3324508860426538453_a_nat @ N @ ( list_u4149385051645232213_a_nat @ Xs @ M @ Y ) )
        = ( take_P3324508860426538453_a_nat @ N @ Xs ) ) ) ).

% take_update_cancel
thf(fact_366_take__update__cancel,axiom,
    ! [N: nat,M: nat,Xs: list_Sum_sum_a_nat,Y: sum_sum_a_nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( take_Sum_sum_a_nat @ N @ ( list_u9138855634547462509_a_nat @ Xs @ M @ Y ) )
        = ( take_Sum_sum_a_nat @ N @ Xs ) ) ) ).

% take_update_cancel
thf(fact_367_sup_Oright__idem,axiom,
    ! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
      ( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) @ B2 )
      = ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ).

% sup.right_idem
thf(fact_368_sup_Oright__idem,axiom,
    ! [A: nat,B2: nat] :
      ( ( sup_sup_nat @ ( sup_sup_nat @ A @ B2 ) @ B2 )
      = ( sup_sup_nat @ A @ B2 ) ) ).

% sup.right_idem
thf(fact_369_sup__left__idem,axiom,
    ! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
      ( ( sup_su6804446743777130803_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ X @ Y ) )
      = ( sup_su6804446743777130803_a_nat @ X @ Y ) ) ).

% sup_left_idem
thf(fact_370_sup__left__idem,axiom,
    ! [X: nat,Y: nat] :
      ( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
      = ( sup_sup_nat @ X @ Y ) ) ).

% sup_left_idem
thf(fact_371_sup_Oleft__idem,axiom,
    ! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
      ( ( sup_su6804446743777130803_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) )
      = ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ).

% sup.left_idem
thf(fact_372_sup_Oleft__idem,axiom,
    ! [A: nat,B2: nat] :
      ( ( sup_sup_nat @ A @ ( sup_sup_nat @ A @ B2 ) )
      = ( sup_sup_nat @ A @ B2 ) ) ).

% sup.left_idem
thf(fact_373_sup__idem,axiom,
    ! [X: set_Sum_sum_a_nat] :
      ( ( sup_su6804446743777130803_a_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_374_sup__idem,axiom,
    ! [X: nat] :
      ( ( sup_sup_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_375_sup_Oidem,axiom,
    ! [A: set_Sum_sum_a_nat] :
      ( ( sup_su6804446743777130803_a_nat @ A @ A )
      = A ) ).

% sup.idem
thf(fact_376_sup_Oidem,axiom,
    ! [A: nat] :
      ( ( sup_sup_nat @ A @ A )
      = A ) ).

% sup.idem
thf(fact_377_UnCI,axiom,
    ! [C2: list_Sum_sum_a_nat,B4: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
      ( ( ~ ( member408289922725080238_a_nat @ C2 @ B4 )
       => ( member408289922725080238_a_nat @ C2 @ A2 ) )
     => ( member408289922725080238_a_nat @ C2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B4 ) ) ) ).

% UnCI
thf(fact_378_UnCI,axiom,
    ! [C2: set_Sum_sum_a_nat,B4: set_se4904748513628223167_a_nat,A2: set_se4904748513628223167_a_nat] :
      ( ( ~ ( member8098812455498974984_a_nat @ C2 @ B4 )
       => ( member8098812455498974984_a_nat @ C2 @ A2 ) )
     => ( member8098812455498974984_a_nat @ C2 @ ( sup_su2291686591051470483_a_nat @ A2 @ B4 ) ) ) ).

% UnCI
thf(fact_379_UnCI,axiom,
    ! [C2: produc7017002724195966439_a_nat,B4: set_Pr7343886759072863943_a_nat,A2: set_Pr7343886759072863943_a_nat] :
      ( ( ~ ( member3723442691059620112_a_nat @ C2 @ B4 )
       => ( member3723442691059620112_a_nat @ C2 @ A2 ) )
     => ( member3723442691059620112_a_nat @ C2 @ ( sup_su2259505869435368603_a_nat @ A2 @ B4 ) ) ) ).

% UnCI
thf(fact_380_UnCI,axiom,
    ! [C2: produc5001885624171833703_a_nat,B4: set_Pr4870381170404451655_a_nat,A2: set_Pr4870381170404451655_a_nat] :
      ( ( ~ ( member7457213283480048528_a_nat @ C2 @ B4 )
       => ( member7457213283480048528_a_nat @ C2 @ A2 ) )
     => ( member7457213283480048528_a_nat @ C2 @ ( sup_su4655406599365709595_a_nat @ A2 @ B4 ) ) ) ).

% UnCI
thf(fact_381_UnCI,axiom,
    ! [C2: sum_sum_a_nat,B4: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
      ( ( ~ ( member_Sum_sum_a_nat @ C2 @ B4 )
       => ( member_Sum_sum_a_nat @ C2 @ A2 ) )
     => ( member_Sum_sum_a_nat @ C2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B4 ) ) ) ).

% UnCI
thf(fact_382_Un__iff,axiom,
    ! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
      ( ( member408289922725080238_a_nat @ C2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B4 ) )
      = ( ( member408289922725080238_a_nat @ C2 @ A2 )
        | ( member408289922725080238_a_nat @ C2 @ B4 ) ) ) ).

% Un_iff
thf(fact_383_Un__iff,axiom,
    ! [C2: set_Sum_sum_a_nat,A2: set_se4904748513628223167_a_nat,B4: set_se4904748513628223167_a_nat] :
      ( ( member8098812455498974984_a_nat @ C2 @ ( sup_su2291686591051470483_a_nat @ A2 @ B4 ) )
      = ( ( member8098812455498974984_a_nat @ C2 @ A2 )
        | ( member8098812455498974984_a_nat @ C2 @ B4 ) ) ) ).

% Un_iff
thf(fact_384_Un__iff,axiom,
    ! [C2: produc7017002724195966439_a_nat,A2: set_Pr7343886759072863943_a_nat,B4: set_Pr7343886759072863943_a_nat] :
      ( ( member3723442691059620112_a_nat @ C2 @ ( sup_su2259505869435368603_a_nat @ A2 @ B4 ) )
      = ( ( member3723442691059620112_a_nat @ C2 @ A2 )
        | ( member3723442691059620112_a_nat @ C2 @ B4 ) ) ) ).

% Un_iff
thf(fact_385_Un__iff,axiom,
    ! [C2: produc5001885624171833703_a_nat,A2: set_Pr4870381170404451655_a_nat,B4: set_Pr4870381170404451655_a_nat] :
      ( ( member7457213283480048528_a_nat @ C2 @ ( sup_su4655406599365709595_a_nat @ A2 @ B4 ) )
      = ( ( member7457213283480048528_a_nat @ C2 @ A2 )
        | ( member7457213283480048528_a_nat @ C2 @ B4 ) ) ) ).

% Un_iff
thf(fact_386_Un__iff,axiom,
    ! [C2: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ C2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B4 ) )
      = ( ( member_Sum_sum_a_nat @ C2 @ A2 )
        | ( member_Sum_sum_a_nat @ C2 @ B4 ) ) ) ).

% Un_iff
thf(fact_387_sup_Obounded__iff,axiom,
    ! [B2: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ B2 @ C2 ) @ A )
      = ( ( ord_le1325389633284124927_a_nat @ B2 @ A )
        & ( ord_le1325389633284124927_a_nat @ C2 @ A ) ) ) ).

% sup.bounded_iff
thf(fact_388_sup_Obounded__iff,axiom,
    ! [B2: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A )
      = ( ( ord_less_eq_nat @ B2 @ A )
        & ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% sup.bounded_iff
thf(fact_389_le__sup__iff,axiom,
    ! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z3: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ X @ Y ) @ Z3 )
      = ( ( ord_le1325389633284124927_a_nat @ X @ Z3 )
        & ( ord_le1325389633284124927_a_nat @ Y @ Z3 ) ) ) ).

% le_sup_iff
thf(fact_390_le__sup__iff,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z3 )
      = ( ( ord_less_eq_nat @ X @ Z3 )
        & ( ord_less_eq_nat @ Y @ Z3 ) ) ) ).

% le_sup_iff
thf(fact_391_length__map,axiom,
    ! [F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat] :
      ( ( size_s2619173126757776721_a_nat @ ( map_li9079814933684165854_a_nat @ F @ Xs ) )
      = ( size_s5212483967078203639_a_nat @ Xs ) ) ).

% length_map
thf(fact_392_length__map,axiom,
    ! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( size_s5283204784079214577_a_nat @ ( map_Su2790769393171190532_a_nat @ F @ Xs ) )
      = ( size_s5283204784079214577_a_nat @ Xs ) ) ).

% length_map
thf(fact_393_min_Oabsorb1,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_min_nat @ A @ B2 )
        = A ) ) ).

% min.absorb1
thf(fact_394_min_Oabsorb2,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_min_nat @ A @ B2 )
        = B2 ) ) ).

% min.absorb2
thf(fact_395_min_Obounded__iff,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B2 @ C2 ) )
      = ( ( ord_less_eq_nat @ A @ B2 )
        & ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% min.bounded_iff
thf(fact_396_length__list__update,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,I2: nat,X: produc7017002724195966439_a_nat] :
      ( ( size_s7247017532682008665_a_nat @ ( list_u4149385051645232213_a_nat @ Xs @ I2 @ X ) )
      = ( size_s7247017532682008665_a_nat @ Xs ) ) ).

% length_list_update
thf(fact_397_length__list__update,axiom,
    ! [Xs: list_Sum_sum_a_nat,I2: nat,X: sum_sum_a_nat] :
      ( ( size_s5283204784079214577_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I2 @ X ) )
      = ( size_s5283204784079214577_a_nat @ Xs ) ) ).

% length_list_update
thf(fact_398_take__all,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s7247017532682008665_a_nat @ Xs ) @ N )
     => ( ( take_P3324508860426538453_a_nat @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_399_take__all,axiom,
    ! [Xs: list_Sum_sum_a_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ N )
     => ( ( take_Sum_sum_a_nat @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_400_take__all__iff,axiom,
    ! [N: nat,Xs: list_P1195027771636113901_a_nat] :
      ( ( ( take_P3324508860426538453_a_nat @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_s7247017532682008665_a_nat @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_401_take__all__iff,axiom,
    ! [N: nat,Xs: list_Sum_sum_a_nat] :
      ( ( ( take_Sum_sum_a_nat @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_402_list__update__beyond,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,I2: nat,X: produc7017002724195966439_a_nat] :
      ( ( ord_less_eq_nat @ ( size_s7247017532682008665_a_nat @ Xs ) @ I2 )
     => ( ( list_u4149385051645232213_a_nat @ Xs @ I2 @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_403_list__update__beyond,axiom,
    ! [Xs: list_Sum_sum_a_nat,I2: nat,X: sum_sum_a_nat] :
      ( ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ I2 )
     => ( ( list_u9138855634547462509_a_nat @ Xs @ I2 @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_404_length__zip,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( size_s7247017532682008665_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) )
      = ( ord_min_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).

% length_zip
thf(fact_405_length__take,axiom,
    ! [N: nat,Xs: list_P1195027771636113901_a_nat] :
      ( ( size_s7247017532682008665_a_nat @ ( take_P3324508860426538453_a_nat @ N @ Xs ) )
      = ( ord_min_nat @ ( size_s7247017532682008665_a_nat @ Xs ) @ N ) ) ).

% length_take
thf(fact_406_length__take,axiom,
    ! [N: nat,Xs: list_Sum_sum_a_nat] :
      ( ( size_s5283204784079214577_a_nat @ ( take_Sum_sum_a_nat @ N @ Xs ) )
      = ( ord_min_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ N ) ) ).

% length_take
thf(fact_407_sup__left__commute,axiom,
    ! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z3: set_Sum_sum_a_nat] :
      ( ( sup_su6804446743777130803_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ Y @ Z3 ) )
      = ( sup_su6804446743777130803_a_nat @ Y @ ( sup_su6804446743777130803_a_nat @ X @ Z3 ) ) ) ).

% sup_left_commute
thf(fact_408_sup__left__commute,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z3 ) )
      = ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z3 ) ) ) ).

% sup_left_commute
thf(fact_409_sup_Oleft__commute,axiom,
    ! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
      ( ( sup_su6804446743777130803_a_nat @ B2 @ ( sup_su6804446743777130803_a_nat @ A @ C2 ) )
      = ( sup_su6804446743777130803_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ B2 @ C2 ) ) ) ).

% sup.left_commute
thf(fact_410_sup_Oleft__commute,axiom,
    ! [B2: nat,A: nat,C2: nat] :
      ( ( sup_sup_nat @ B2 @ ( sup_sup_nat @ A @ C2 ) )
      = ( sup_sup_nat @ A @ ( sup_sup_nat @ B2 @ C2 ) ) ) ).

% sup.left_commute
thf(fact_411_sup_OcoboundedI2,axiom,
    ! [C2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ C2 @ B2 )
     => ( ord_le1325389633284124927_a_nat @ C2 @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_412_sup_OcoboundedI2,axiom,
    ! [C2: nat,B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ C2 @ B2 )
     => ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_413_sup_OcoboundedI1,axiom,
    ! [C2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ C2 @ A )
     => ( ord_le1325389633284124927_a_nat @ C2 @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_414_sup_OcoboundedI1,axiom,
    ! [C2: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C2 @ A )
     => ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_415_sup_Oabsorb__iff2,axiom,
    ( ord_le1325389633284124927_a_nat
    = ( ^ [A4: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
          ( ( sup_su6804446743777130803_a_nat @ A4 @ B3 )
          = B3 ) ) ) ).

% sup.absorb_iff2
thf(fact_416_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( sup_sup_nat @ A4 @ B3 )
          = B3 ) ) ) ).

% sup.absorb_iff2
thf(fact_417_sup_Oabsorb__iff1,axiom,
    ( ord_le1325389633284124927_a_nat
    = ( ^ [B3: set_Sum_sum_a_nat,A4: set_Sum_sum_a_nat] :
          ( ( sup_su6804446743777130803_a_nat @ A4 @ B3 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_418_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( sup_sup_nat @ A4 @ B3 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_419_sup_Ocobounded2,axiom,
    ! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ B2 @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ).

% sup.cobounded2
thf(fact_420_sup_Ocobounded2,axiom,
    ! [B2: nat,A: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A @ B2 ) ) ).

% sup.cobounded2
thf(fact_421_sup_Ocobounded1,axiom,
    ! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ).

% sup.cobounded1
thf(fact_422_sup_Ocobounded1,axiom,
    ! [A: nat,B2: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B2 ) ) ).

% sup.cobounded1
thf(fact_423_sup_Oorder__iff,axiom,
    ( ord_le1325389633284124927_a_nat
    = ( ^ [B3: set_Sum_sum_a_nat,A4: set_Sum_sum_a_nat] :
          ( A4
          = ( sup_su6804446743777130803_a_nat @ A4 @ B3 ) ) ) ) ).

% sup.order_iff
thf(fact_424_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A4: nat] :
          ( A4
          = ( sup_sup_nat @ A4 @ B3 ) ) ) ) ).

% sup.order_iff
thf(fact_425_sup_OboundedI,axiom,
    ! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ B2 @ A )
     => ( ( ord_le1325389633284124927_a_nat @ C2 @ A )
       => ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ B2 @ C2 ) @ A ) ) ) ).

% sup.boundedI
thf(fact_426_sup_OboundedI,axiom,
    ! [B2: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_less_eq_nat @ C2 @ A )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A ) ) ) ).

% sup.boundedI
thf(fact_427_sup_OboundedE,axiom,
    ! [B2: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ B2 @ C2 ) @ A )
     => ~ ( ( ord_le1325389633284124927_a_nat @ B2 @ A )
         => ~ ( ord_le1325389633284124927_a_nat @ C2 @ A ) ) ) ).

% sup.boundedE
thf(fact_428_sup_OboundedE,axiom,
    ! [B2: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A )
     => ~ ( ( ord_less_eq_nat @ B2 @ A )
         => ~ ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% sup.boundedE
thf(fact_429_sup__commute,axiom,
    ( sup_su6804446743777130803_a_nat
    = ( ^ [X2: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ Y3 @ X2 ) ) ) ).

% sup_commute
thf(fact_430_sup__commute,axiom,
    ( sup_sup_nat
    = ( ^ [X2: nat,Y3: nat] : ( sup_sup_nat @ Y3 @ X2 ) ) ) ).

% sup_commute
thf(fact_431_sup__absorb2,axiom,
    ! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ X @ Y )
     => ( ( sup_su6804446743777130803_a_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_432_sup__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( sup_sup_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_433_sup__absorb1,axiom,
    ! [Y: set_Sum_sum_a_nat,X: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ Y @ X )
     => ( ( sup_su6804446743777130803_a_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_434_sup__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( sup_sup_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_435_sup_Ocommute,axiom,
    ( sup_su6804446743777130803_a_nat
    = ( ^ [A4: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ B3 @ A4 ) ) ) ).

% sup.commute
thf(fact_436_sup_Ocommute,axiom,
    ( sup_sup_nat
    = ( ^ [A4: nat,B3: nat] : ( sup_sup_nat @ B3 @ A4 ) ) ) ).

% sup.commute
thf(fact_437_sup_Oabsorb2,axiom,
    ! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ A @ B2 )
     => ( ( sup_su6804446743777130803_a_nat @ A @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_438_sup_Oabsorb2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( sup_sup_nat @ A @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_439_sup_Oabsorb1,axiom,
    ! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ B2 @ A )
     => ( ( sup_su6804446743777130803_a_nat @ A @ B2 )
        = A ) ) ).

% sup.absorb1
thf(fact_440_sup_Oabsorb1,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( sup_sup_nat @ A @ B2 )
        = A ) ) ).

% sup.absorb1
thf(fact_441_sup__unique,axiom,
    ! [F: set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat,X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
      ( ! [X3: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ X3 @ ( F @ X3 @ Y4 ) )
     => ( ! [X3: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ Y4 @ ( F @ X3 @ Y4 ) )
       => ( ! [X3: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] :
              ( ( ord_le1325389633284124927_a_nat @ Y4 @ X3 )
             => ( ( ord_le1325389633284124927_a_nat @ Z2 @ X3 )
               => ( ord_le1325389633284124927_a_nat @ ( F @ Y4 @ Z2 ) @ X3 ) ) )
         => ( ( sup_su6804446743777130803_a_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_442_sup__unique,axiom,
    ! [F: nat > nat > nat,X: nat,Y: nat] :
      ( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y4 ) )
     => ( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X3 @ Y4 ) )
       => ( ! [X3: nat,Y4: nat,Z2: nat] :
              ( ( ord_less_eq_nat @ Y4 @ X3 )
             => ( ( ord_less_eq_nat @ Z2 @ X3 )
               => ( ord_less_eq_nat @ ( F @ Y4 @ Z2 ) @ X3 ) ) )
         => ( ( sup_sup_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_443_sup_OorderI,axiom,
    ! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
      ( ( A
        = ( sup_su6804446743777130803_a_nat @ A @ B2 ) )
     => ( ord_le1325389633284124927_a_nat @ B2 @ A ) ) ).

% sup.orderI
thf(fact_444_sup_OorderI,axiom,
    ! [A: nat,B2: nat] :
      ( ( A
        = ( sup_sup_nat @ A @ B2 ) )
     => ( ord_less_eq_nat @ B2 @ A ) ) ).

% sup.orderI
thf(fact_445_sup_OorderE,axiom,
    ! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ B2 @ A )
     => ( A
        = ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ) ).

% sup.orderE
thf(fact_446_sup_OorderE,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( A
        = ( sup_sup_nat @ A @ B2 ) ) ) ).

% sup.orderE
thf(fact_447_le__iff__sup,axiom,
    ( ord_le1325389633284124927_a_nat
    = ( ^ [X2: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
          ( ( sup_su6804446743777130803_a_nat @ X2 @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_448_le__iff__sup,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y3: nat] :
          ( ( sup_sup_nat @ X2 @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_449_sup__least,axiom,
    ! [Y: set_Sum_sum_a_nat,X: set_Sum_sum_a_nat,Z3: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ Y @ X )
     => ( ( ord_le1325389633284124927_a_nat @ Z3 @ X )
       => ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ Y @ Z3 ) @ X ) ) ) ).

% sup_least
thf(fact_450_sup__least,axiom,
    ! [Y: nat,X: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ Z3 @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z3 ) @ X ) ) ) ).

% sup_least
thf(fact_451_sup__assoc,axiom,
    ! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z3: set_Sum_sum_a_nat] :
      ( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ X @ Y ) @ Z3 )
      = ( sup_su6804446743777130803_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ Y @ Z3 ) ) ) ).

% sup_assoc
thf(fact_452_sup__assoc,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z3 )
      = ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z3 ) ) ) ).

% sup_assoc
thf(fact_453_sup_Oassoc,axiom,
    ! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
      ( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) @ C2 )
      = ( sup_su6804446743777130803_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ B2 @ C2 ) ) ) ).

% sup.assoc
thf(fact_454_sup_Oassoc,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( sup_sup_nat @ ( sup_sup_nat @ A @ B2 ) @ C2 )
      = ( sup_sup_nat @ A @ ( sup_sup_nat @ B2 @ C2 ) ) ) ).

% sup.assoc
thf(fact_455_sup__mono,axiom,
    ! [A: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,D: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ A @ C2 )
     => ( ( ord_le1325389633284124927_a_nat @ B2 @ D )
       => ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) @ ( sup_su6804446743777130803_a_nat @ C2 @ D ) ) ) ) ).

% sup_mono
thf(fact_456_sup__mono,axiom,
    ! [A: nat,C2: nat,B2: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ C2 )
     => ( ( ord_less_eq_nat @ B2 @ D )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ ( sup_sup_nat @ C2 @ D ) ) ) ) ).

% sup_mono
thf(fact_457_sup_Omono,axiom,
    ! [C2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,D: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ C2 @ A )
     => ( ( ord_le1325389633284124927_a_nat @ D @ B2 )
       => ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ C2 @ D ) @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ) ) ).

% sup.mono
thf(fact_458_sup_Omono,axiom,
    ! [C2: nat,A: nat,D: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C2 @ A )
     => ( ( ord_less_eq_nat @ D @ B2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D ) @ ( sup_sup_nat @ A @ B2 ) ) ) ) ).

% sup.mono
thf(fact_459_le__supI2,axiom,
    ! [X: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ X @ B2 )
     => ( ord_le1325389633284124927_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ) ).

% le_supI2
thf(fact_460_le__supI2,axiom,
    ! [X: nat,B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ X @ B2 )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B2 ) ) ) ).

% le_supI2
thf(fact_461_le__supI1,axiom,
    ! [X: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ X @ A )
     => ( ord_le1325389633284124927_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ) ).

% le_supI1
thf(fact_462_le__supI1,axiom,
    ! [X: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ X @ A )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B2 ) ) ) ).

% le_supI1
thf(fact_463_sup__ge2,axiom,
    ! [Y: set_Sum_sum_a_nat,X: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ Y @ ( sup_su6804446743777130803_a_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_464_sup__ge2,axiom,
    ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_465_sup__ge1,axiom,
    ! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_466_sup__ge1,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_467_le__supI,axiom,
    ! [A: set_Sum_sum_a_nat,X: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ A @ X )
     => ( ( ord_le1325389633284124927_a_nat @ B2 @ X )
       => ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_468_le__supI,axiom,
    ! [A: nat,X: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ X )
     => ( ( ord_less_eq_nat @ B2 @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_469_le__supE,axiom,
    ! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,X: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) @ X )
     => ~ ( ( ord_le1325389633284124927_a_nat @ A @ X )
         => ~ ( ord_le1325389633284124927_a_nat @ B2 @ X ) ) ) ).

% le_supE
thf(fact_470_le__supE,axiom,
    ! [A: nat,B2: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ X )
     => ~ ( ( ord_less_eq_nat @ A @ X )
         => ~ ( ord_less_eq_nat @ B2 @ X ) ) ) ).

% le_supE
thf(fact_471_inf__sup__ord_I3_J,axiom,
    ! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_472_inf__sup__ord_I3_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_473_inf__sup__ord_I4_J,axiom,
    ! [Y: set_Sum_sum_a_nat,X: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ Y @ ( sup_su6804446743777130803_a_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_474_inf__sup__ord_I4_J,axiom,
    ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_475_inf__sup__aci_I5_J,axiom,
    ( sup_su6804446743777130803_a_nat
    = ( ^ [X2: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ Y3 @ X2 ) ) ) ).

% inf_sup_aci(5)
thf(fact_476_inf__sup__aci_I5_J,axiom,
    ( sup_sup_nat
    = ( ^ [X2: nat,Y3: nat] : ( sup_sup_nat @ Y3 @ X2 ) ) ) ).

% inf_sup_aci(5)
thf(fact_477_inf__sup__aci_I6_J,axiom,
    ! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z3: set_Sum_sum_a_nat] :
      ( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ X @ Y ) @ Z3 )
      = ( sup_su6804446743777130803_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ Y @ Z3 ) ) ) ).

% inf_sup_aci(6)
thf(fact_478_inf__sup__aci_I6_J,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z3 )
      = ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z3 ) ) ) ).

% inf_sup_aci(6)
thf(fact_479_inf__sup__aci_I7_J,axiom,
    ! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z3: set_Sum_sum_a_nat] :
      ( ( sup_su6804446743777130803_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ Y @ Z3 ) )
      = ( sup_su6804446743777130803_a_nat @ Y @ ( sup_su6804446743777130803_a_nat @ X @ Z3 ) ) ) ).

% inf_sup_aci(7)
thf(fact_480_inf__sup__aci_I7_J,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z3 ) )
      = ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z3 ) ) ) ).

% inf_sup_aci(7)
thf(fact_481_inf__sup__aci_I8_J,axiom,
    ! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
      ( ( sup_su6804446743777130803_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ X @ Y ) )
      = ( sup_su6804446743777130803_a_nat @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_482_inf__sup__aci_I8_J,axiom,
    ! [X: nat,Y: nat] :
      ( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
      = ( sup_sup_nat @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_483_can__select__def,axiom,
    ( can_se2197130373557366786_a_nat
    = ( ^ [P4: list_Sum_sum_a_nat > $o,A5: set_li6526943997496501093_a_nat] :
        ? [X2: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ X2 @ A5 )
          & ( P4 @ X2 )
          & ! [Y3: list_Sum_sum_a_nat] :
              ( ( ( member408289922725080238_a_nat @ Y3 @ A5 )
                & ( P4 @ Y3 ) )
             => ( Y3 = X2 ) ) ) ) ) ).

% can_select_def
thf(fact_484_can__select__def,axiom,
    ( can_se4650645106769343580_a_nat
    = ( ^ [P4: set_Sum_sum_a_nat > $o,A5: set_se4904748513628223167_a_nat] :
        ? [X2: set_Sum_sum_a_nat] :
          ( ( member8098812455498974984_a_nat @ X2 @ A5 )
          & ( P4 @ X2 )
          & ! [Y3: set_Sum_sum_a_nat] :
              ( ( ( member8098812455498974984_a_nat @ Y3 @ A5 )
                & ( P4 @ Y3 ) )
             => ( Y3 = X2 ) ) ) ) ) ).

% can_select_def
thf(fact_485_can__select__def,axiom,
    ( can_se8355593179296023652_a_nat
    = ( ^ [P4: produc7017002724195966439_a_nat > $o,A5: set_Pr7343886759072863943_a_nat] :
        ? [X2: produc7017002724195966439_a_nat] :
          ( ( member3723442691059620112_a_nat @ X2 @ A5 )
          & ( P4 @ X2 )
          & ! [Y3: produc7017002724195966439_a_nat] :
              ( ( ( member3723442691059620112_a_nat @ Y3 @ A5 )
                & ( P4 @ Y3 ) )
             => ( Y3 = X2 ) ) ) ) ) ).

% can_select_def
thf(fact_486_can__select__def,axiom,
    ( can_se5661427355321125604_a_nat
    = ( ^ [P4: produc5001885624171833703_a_nat > $o,A5: set_Pr4870381170404451655_a_nat] :
        ? [X2: produc5001885624171833703_a_nat] :
          ( ( member7457213283480048528_a_nat @ X2 @ A5 )
          & ( P4 @ X2 )
          & ! [Y3: produc5001885624171833703_a_nat] :
              ( ( ( member7457213283480048528_a_nat @ Y3 @ A5 )
                & ( P4 @ Y3 ) )
             => ( Y3 = X2 ) ) ) ) ) ).

% can_select_def
thf(fact_487_can__select__def,axiom,
    ( can_se22271275864192764_a_nat
    = ( ^ [P4: sum_sum_a_nat > $o,A5: set_Sum_sum_a_nat] :
        ? [X2: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ X2 @ A5 )
          & ( P4 @ X2 )
          & ! [Y3: sum_sum_a_nat] :
              ( ( ( member_Sum_sum_a_nat @ Y3 @ A5 )
                & ( P4 @ Y3 ) )
             => ( Y3 = X2 ) ) ) ) ) ).

% can_select_def
thf(fact_488_UnE,axiom,
    ! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
      ( ( member408289922725080238_a_nat @ C2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B4 ) )
     => ( ~ ( member408289922725080238_a_nat @ C2 @ A2 )
       => ( member408289922725080238_a_nat @ C2 @ B4 ) ) ) ).

% UnE
thf(fact_489_UnE,axiom,
    ! [C2: set_Sum_sum_a_nat,A2: set_se4904748513628223167_a_nat,B4: set_se4904748513628223167_a_nat] :
      ( ( member8098812455498974984_a_nat @ C2 @ ( sup_su2291686591051470483_a_nat @ A2 @ B4 ) )
     => ( ~ ( member8098812455498974984_a_nat @ C2 @ A2 )
       => ( member8098812455498974984_a_nat @ C2 @ B4 ) ) ) ).

% UnE
thf(fact_490_UnE,axiom,
    ! [C2: produc7017002724195966439_a_nat,A2: set_Pr7343886759072863943_a_nat,B4: set_Pr7343886759072863943_a_nat] :
      ( ( member3723442691059620112_a_nat @ C2 @ ( sup_su2259505869435368603_a_nat @ A2 @ B4 ) )
     => ( ~ ( member3723442691059620112_a_nat @ C2 @ A2 )
       => ( member3723442691059620112_a_nat @ C2 @ B4 ) ) ) ).

% UnE
thf(fact_491_UnE,axiom,
    ! [C2: produc5001885624171833703_a_nat,A2: set_Pr4870381170404451655_a_nat,B4: set_Pr4870381170404451655_a_nat] :
      ( ( member7457213283480048528_a_nat @ C2 @ ( sup_su4655406599365709595_a_nat @ A2 @ B4 ) )
     => ( ~ ( member7457213283480048528_a_nat @ C2 @ A2 )
       => ( member7457213283480048528_a_nat @ C2 @ B4 ) ) ) ).

% UnE
thf(fact_492_UnE,axiom,
    ! [C2: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ C2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B4 ) )
     => ( ~ ( member_Sum_sum_a_nat @ C2 @ A2 )
       => ( member_Sum_sum_a_nat @ C2 @ B4 ) ) ) ).

% UnE
thf(fact_493_UnI1,axiom,
    ! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
      ( ( member408289922725080238_a_nat @ C2 @ A2 )
     => ( member408289922725080238_a_nat @ C2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B4 ) ) ) ).

% UnI1
thf(fact_494_UnI1,axiom,
    ! [C2: set_Sum_sum_a_nat,A2: set_se4904748513628223167_a_nat,B4: set_se4904748513628223167_a_nat] :
      ( ( member8098812455498974984_a_nat @ C2 @ A2 )
     => ( member8098812455498974984_a_nat @ C2 @ ( sup_su2291686591051470483_a_nat @ A2 @ B4 ) ) ) ).

% UnI1
thf(fact_495_UnI1,axiom,
    ! [C2: produc7017002724195966439_a_nat,A2: set_Pr7343886759072863943_a_nat,B4: set_Pr7343886759072863943_a_nat] :
      ( ( member3723442691059620112_a_nat @ C2 @ A2 )
     => ( member3723442691059620112_a_nat @ C2 @ ( sup_su2259505869435368603_a_nat @ A2 @ B4 ) ) ) ).

% UnI1
thf(fact_496_UnI1,axiom,
    ! [C2: produc5001885624171833703_a_nat,A2: set_Pr4870381170404451655_a_nat,B4: set_Pr4870381170404451655_a_nat] :
      ( ( member7457213283480048528_a_nat @ C2 @ A2 )
     => ( member7457213283480048528_a_nat @ C2 @ ( sup_su4655406599365709595_a_nat @ A2 @ B4 ) ) ) ).

% UnI1
thf(fact_497_UnI1,axiom,
    ! [C2: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ C2 @ A2 )
     => ( member_Sum_sum_a_nat @ C2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B4 ) ) ) ).

% UnI1
thf(fact_498_UnI2,axiom,
    ! [C2: list_Sum_sum_a_nat,B4: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
      ( ( member408289922725080238_a_nat @ C2 @ B4 )
     => ( member408289922725080238_a_nat @ C2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B4 ) ) ) ).

% UnI2
thf(fact_499_UnI2,axiom,
    ! [C2: set_Sum_sum_a_nat,B4: set_se4904748513628223167_a_nat,A2: set_se4904748513628223167_a_nat] :
      ( ( member8098812455498974984_a_nat @ C2 @ B4 )
     => ( member8098812455498974984_a_nat @ C2 @ ( sup_su2291686591051470483_a_nat @ A2 @ B4 ) ) ) ).

% UnI2
thf(fact_500_UnI2,axiom,
    ! [C2: produc7017002724195966439_a_nat,B4: set_Pr7343886759072863943_a_nat,A2: set_Pr7343886759072863943_a_nat] :
      ( ( member3723442691059620112_a_nat @ C2 @ B4 )
     => ( member3723442691059620112_a_nat @ C2 @ ( sup_su2259505869435368603_a_nat @ A2 @ B4 ) ) ) ).

% UnI2
thf(fact_501_UnI2,axiom,
    ! [C2: produc5001885624171833703_a_nat,B4: set_Pr4870381170404451655_a_nat,A2: set_Pr4870381170404451655_a_nat] :
      ( ( member7457213283480048528_a_nat @ C2 @ B4 )
     => ( member7457213283480048528_a_nat @ C2 @ ( sup_su4655406599365709595_a_nat @ A2 @ B4 ) ) ) ).

% UnI2
thf(fact_502_UnI2,axiom,
    ! [C2: sum_sum_a_nat,B4: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ C2 @ B4 )
     => ( member_Sum_sum_a_nat @ C2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B4 ) ) ) ).

% UnI2
thf(fact_503_bex__Un,axiom,
    ! [A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
      ( ( ? [X2: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B4 ) )
            & ( P @ X2 ) ) )
      = ( ? [X2: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X2 @ A2 )
            & ( P @ X2 ) )
        | ? [X2: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X2 @ B4 )
            & ( P @ X2 ) ) ) ) ).

% bex_Un
thf(fact_504_ball__Un,axiom,
    ! [A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
      ( ( ! [X2: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B4 ) )
           => ( P @ X2 ) ) )
      = ( ! [X2: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X2 @ A2 )
           => ( P @ X2 ) )
        & ! [X2: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X2 @ B4 )
           => ( P @ X2 ) ) ) ) ).

% ball_Un
thf(fact_505_Un__assoc,axiom,
    ! [A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat,C3: set_Sum_sum_a_nat] :
      ( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B4 ) @ C3 )
      = ( sup_su6804446743777130803_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ B4 @ C3 ) ) ) ).

% Un_assoc
thf(fact_506_Un__absorb,axiom,
    ! [A2: set_Sum_sum_a_nat] :
      ( ( sup_su6804446743777130803_a_nat @ A2 @ A2 )
      = A2 ) ).

% Un_absorb
thf(fact_507_Un__commute,axiom,
    ( sup_su6804446743777130803_a_nat
    = ( ^ [A5: set_Sum_sum_a_nat,B5: set_Sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ B5 @ A5 ) ) ) ).

% Un_commute
thf(fact_508_Un__left__absorb,axiom,
    ! [A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
      ( ( sup_su6804446743777130803_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B4 ) )
      = ( sup_su6804446743777130803_a_nat @ A2 @ B4 ) ) ).

% Un_left_absorb
thf(fact_509_Un__left__commute,axiom,
    ! [A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat,C3: set_Sum_sum_a_nat] :
      ( ( sup_su6804446743777130803_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ B4 @ C3 ) )
      = ( sup_su6804446743777130803_a_nat @ B4 @ ( sup_su6804446743777130803_a_nat @ A2 @ C3 ) ) ) ).

% Un_left_commute
thf(fact_510_neq__if__length__neq,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
       != ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_511_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_Sum_sum_a_nat] :
      ( ( size_s5283204784079214577_a_nat @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_512_subseqs__refl,axiom,
    ! [Xs: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ Xs @ ( set_li2392974972034027290_a_nat @ ( subseq8414445098004693972_a_nat @ Xs ) ) ) ).

% subseqs_refl
thf(fact_513_length__filter__le,axiom,
    ! [P: sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] : ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ ( filter_Sum_sum_a_nat @ P @ Xs ) ) @ ( size_s5283204784079214577_a_nat @ Xs ) ) ).

% length_filter_le
thf(fact_514_map__eq__imp__length__eq,axiom,
    ! [F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat,G: list_Sum_sum_a_nat > set_Sum_sum_a_nat,Ys: list_l4703314356710769291_a_nat] :
      ( ( ( map_li9079814933684165854_a_nat @ F @ Xs )
        = ( map_li9079814933684165854_a_nat @ G @ Ys ) )
     => ( ( size_s5212483967078203639_a_nat @ Xs )
        = ( size_s5212483967078203639_a_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_515_map__eq__imp__length__eq,axiom,
    ! [F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat,G: sum_sum_a_nat > set_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( map_li9079814933684165854_a_nat @ F @ Xs )
        = ( map_Su6628313897983864932_a_nat @ G @ Ys ) )
     => ( ( size_s5212483967078203639_a_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_516_map__eq__imp__length__eq,axiom,
    ! [F: sum_sum_a_nat > set_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > set_Sum_sum_a_nat,Ys: list_l4703314356710769291_a_nat] :
      ( ( ( map_Su6628313897983864932_a_nat @ F @ Xs )
        = ( map_li9079814933684165854_a_nat @ G @ Ys ) )
     => ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_s5212483967078203639_a_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_517_map__eq__imp__length__eq,axiom,
    ! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
        = ( map_Su2790769393171190532_a_nat @ G @ Ys ) )
     => ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_518_list__all2__lengthD,axiom,
    ! [P: sum_sum_a_nat > sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( list_a8491001234060422471_a_nat @ P @ Xs @ Ys )
     => ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).

% list_all2_lengthD
thf(fact_519_min_Omono,axiom,
    ! [A: nat,C2: nat,B2: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ C2 )
     => ( ( ord_less_eq_nat @ B2 @ D )
       => ( ord_less_eq_nat @ ( ord_min_nat @ A @ B2 ) @ ( ord_min_nat @ C2 @ D ) ) ) ) ).

% min.mono
thf(fact_520_min_OorderE,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( A
        = ( ord_min_nat @ A @ B2 ) ) ) ).

% min.orderE
thf(fact_521_min_OorderI,axiom,
    ! [A: nat,B2: nat] :
      ( ( A
        = ( ord_min_nat @ A @ B2 ) )
     => ( ord_less_eq_nat @ A @ B2 ) ) ).

% min.orderI
thf(fact_522_min_OboundedE,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B2 @ C2 ) )
     => ~ ( ( ord_less_eq_nat @ A @ B2 )
         => ~ ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% min.boundedE
thf(fact_523_min_OboundedI,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ A @ C2 )
       => ( ord_less_eq_nat @ A @ ( ord_min_nat @ B2 @ C2 ) ) ) ) ).

% min.boundedI
thf(fact_524_min_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B3: nat] :
          ( A4
          = ( ord_min_nat @ A4 @ B3 ) ) ) ) ).

% min.order_iff
thf(fact_525_min_Ocobounded1,axiom,
    ! [A: nat,B2: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B2 ) @ A ) ).

% min.cobounded1
thf(fact_526_min_Ocobounded2,axiom,
    ! [A: nat,B2: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B2 ) @ B2 ) ).

% min.cobounded2
thf(fact_527_min_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_min_nat @ A4 @ B3 )
          = A4 ) ) ) ).

% min.absorb_iff1
thf(fact_528_min_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( ord_min_nat @ A4 @ B3 )
          = B3 ) ) ) ).

% min.absorb_iff2
thf(fact_529_min_OcoboundedI1,axiom,
    ! [A: nat,C2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ C2 )
     => ( ord_less_eq_nat @ ( ord_min_nat @ A @ B2 ) @ C2 ) ) ).

% min.coboundedI1
thf(fact_530_min_OcoboundedI2,axiom,
    ! [B2: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ C2 )
     => ( ord_less_eq_nat @ ( ord_min_nat @ A @ B2 ) @ C2 ) ) ).

% min.coboundedI2
thf(fact_531_min__le__iff__disj,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ ( ord_min_nat @ X @ Y ) @ Z3 )
      = ( ( ord_less_eq_nat @ X @ Z3 )
        | ( ord_less_eq_nat @ Y @ Z3 ) ) ) ).

% min_le_iff_disj
thf(fact_532_subseqs__distinctD,axiom,
    ! [Ys: list_P1195027771636113901_a_nat,Xs: list_P1195027771636113901_a_nat] :
      ( ( member6227626580160590998_a_nat @ Ys @ ( set_li1234883600314943746_a_nat @ ( subseq5198752961258583356_a_nat @ Xs ) ) )
     => ( ( distin928153214662094432_a_nat @ Xs )
       => ( distin928153214662094432_a_nat @ Ys ) ) ) ).

% subseqs_distinctD
thf(fact_533_subseqs__distinctD,axiom,
    ! [Ys: list_s7376487411892226277_a_nat,Xs: list_s7376487411892226277_a_nat] :
      ( ( member6403511879741922830_a_nat @ Ys @ ( set_li1117937094484414586_a_nat @ ( subseq1669510427954520372_a_nat @ Xs ) ) )
     => ( ( distin4821671304206132312_a_nat @ Xs )
       => ( distin4821671304206132312_a_nat @ Ys ) ) ) ).

% subseqs_distinctD
thf(fact_534_subseqs__distinctD,axiom,
    ! [Ys: list_l4703314356710769291_a_nat,Xs: list_l4703314356710769291_a_nat] :
      ( ( member6656483030743143220_a_nat @ Ys @ ( set_li7766024463445446560_a_nat @ ( subseq4596077244143087834_a_nat @ Xs ) ) )
     => ( ( distin811021574259663358_a_nat @ Xs )
       => ( distin811021574259663358_a_nat @ Ys ) ) ) ).

% subseqs_distinctD
thf(fact_535_subseqs__distinctD,axiom,
    ! [Ys: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( member408289922725080238_a_nat @ Ys @ ( set_li2392974972034027290_a_nat @ ( subseq8414445098004693972_a_nat @ Xs ) ) )
     => ( ( distin2701893636801681144_a_nat @ Xs )
       => ( distin2701893636801681144_a_nat @ Ys ) ) ) ).

% subseqs_distinctD
thf(fact_536_list__bind__cong,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,F: sum_sum_a_nat > list_Sum_sum_a_nat,G: sum_sum_a_nat > list_Sum_sum_a_nat] :
      ( ( Xs = Ys )
     => ( ! [X3: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Xs ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( bind_S8497345843576593223_a_nat @ Xs @ F )
          = ( bind_S8497345843576593223_a_nat @ Ys @ G ) ) ) ) ).

% list_bind_cong
thf(fact_537_order__refl,axiom,
    ! [X: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ X @ X ) ).

% order_refl
thf(fact_538_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_539_dual__order_Orefl,axiom,
    ! [A: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A @ A ) ).

% dual_order.refl
thf(fact_540_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_541_min__def,axiom,
    ( ord_mi4660535444180972358_a_nat
    = ( ^ [A4: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] : ( if_set_Sum_sum_a_nat @ ( ord_le1325389633284124927_a_nat @ A4 @ B3 ) @ A4 @ B3 ) ) ) ).

% min_def
thf(fact_542_min__def,axiom,
    ( ord_min_nat
    = ( ^ [A4: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B3 ) @ A4 @ B3 ) ) ) ).

% min_def
thf(fact_543_min__absorb1,axiom,
    ! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ X @ Y )
     => ( ( ord_mi4660535444180972358_a_nat @ X @ Y )
        = X ) ) ).

% min_absorb1
thf(fact_544_min__absorb1,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_min_nat @ X @ Y )
        = X ) ) ).

% min_absorb1
thf(fact_545_min__absorb2,axiom,
    ! [Y: set_Sum_sum_a_nat,X: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ Y @ X )
     => ( ( ord_mi4660535444180972358_a_nat @ X @ Y )
        = Y ) ) ).

% min_absorb2
thf(fact_546_min__absorb2,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_min_nat @ X @ Y )
        = Y ) ) ).

% min_absorb2
thf(fact_547_map__fst__zip__take,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( map_Pr9108395128902024172_a_nat @ produc7342064319418986843_a_nat @ ( zip_Pr3707921967074149519_a_nat @ Xs @ Ys ) )
      = ( take_P3324508860426538453_a_nat @ ( ord_min_nat @ ( size_s7247017532682008665_a_nat @ Xs ) @ ( size_s5283204784079214577_a_nat @ Ys ) ) @ Xs ) ) ).

% map_fst_zip_take
thf(fact_548_map__fst__zip__take,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( map_Pr3527087583225239196_a_nat @ produc3940427395981429571_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) )
      = ( take_Sum_sum_a_nat @ ( ord_min_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ ( size_s5283204784079214577_a_nat @ Ys ) ) @ Xs ) ) ).

% map_fst_zip_take
thf(fact_549_map__fst__zip__take,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat] :
      ( ( map_Pr5241799845241595682_a_nat @ produc2015473822650161347_a_nat @ ( zip_li7023018480688994807_a_nat @ Xs @ Ys ) )
      = ( take_l4056659018708330355_a_nat @ ( ord_min_nat @ ( size_s5212483967078203639_a_nat @ Xs ) @ ( size_s5212483967078203639_a_nat @ Ys ) ) @ Xs ) ) ).

% map_fst_zip_take
thf(fact_550_map__snd__zip__take,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_P1195027771636113901_a_nat] :
      ( ( map_Pr1987915761617520668_a_nat @ produc6793502323970612077_a_nat @ ( zip_Su4402317301539879519_a_nat @ Xs @ Ys ) )
      = ( take_P3324508860426538453_a_nat @ ( ord_min_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ ( size_s7247017532682008665_a_nat @ Ys ) ) @ Ys ) ) ).

% map_snd_zip_take
thf(fact_551_map__snd__zip__take,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( map_Pr3527087583225239196_a_nat @ produc8028804885397038213_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) )
      = ( take_Sum_sum_a_nat @ ( ord_min_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ ( size_s5283204784079214577_a_nat @ Ys ) ) @ Ys ) ) ).

% map_snd_zip_take
thf(fact_552_Un__subset__iff,axiom,
    ! [A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat,C3: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B4 ) @ C3 )
      = ( ( ord_le1325389633284124927_a_nat @ A2 @ C3 )
        & ( ord_le1325389633284124927_a_nat @ B4 @ C3 ) ) ) ).

% Un_subset_iff
thf(fact_553_map__fst__zip,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( ( map_Pr3527087583225239196_a_nat @ produc3940427395981429571_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_554_map__fst__zip,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat] :
      ( ( ( size_s5212483967078203639_a_nat @ Xs )
        = ( size_s5212483967078203639_a_nat @ Ys ) )
     => ( ( map_Pr5241799845241595682_a_nat @ produc2015473822650161347_a_nat @ ( zip_li7023018480688994807_a_nat @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_555_map__snd__zip,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( ( map_Pr3527087583225239196_a_nat @ produc8028804885397038213_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_556_pair__list__eqI,axiom,
    ! [Xs: list_P6608106274694273645_a_nat,Ys: list_P6608106274694273645_a_nat] :
      ( ( ( map_Pr5241799845241595682_a_nat @ produc2015473822650161347_a_nat @ Xs )
        = ( map_Pr5241799845241595682_a_nat @ produc2015473822650161347_a_nat @ Ys ) )
     => ( ( ( map_Pr5241799845241595682_a_nat @ produc2553995981594484741_a_nat @ Xs )
          = ( map_Pr5241799845241595682_a_nat @ produc2553995981594484741_a_nat @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_557_pair__list__eqI,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,Ys: list_P1195027771636113901_a_nat] :
      ( ( ( map_Pr3527087583225239196_a_nat @ produc3940427395981429571_a_nat @ Xs )
        = ( map_Pr3527087583225239196_a_nat @ produc3940427395981429571_a_nat @ Ys ) )
     => ( ( ( map_Pr3527087583225239196_a_nat @ produc8028804885397038213_a_nat @ Xs )
          = ( map_Pr3527087583225239196_a_nat @ produc8028804885397038213_a_nat @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_558_list_Orel__mono,axiom,
    ! [R: sum_sum_a_nat > sum_sum_a_nat > $o,Ra: sum_sum_a_nat > sum_sum_a_nat > $o] :
      ( ( ord_le5975651393758295374_nat_o @ R @ Ra )
     => ( ord_le6948912460583348046_nat_o @ ( list_a8491001234060422471_a_nat @ R ) @ ( list_a8491001234060422471_a_nat @ Ra ) ) ) ).

% list.rel_mono
thf(fact_559_zip__map__fst__snd,axiom,
    ! [Zs2: list_P6608106274694273645_a_nat] :
      ( ( zip_li7023018480688994807_a_nat @ ( map_Pr5241799845241595682_a_nat @ produc2015473822650161347_a_nat @ Zs2 ) @ ( map_Pr5241799845241595682_a_nat @ produc2553995981594484741_a_nat @ Zs2 ) )
      = Zs2 ) ).

% zip_map_fst_snd
thf(fact_560_zip__map__fst__snd,axiom,
    ! [Zs2: list_P1195027771636113901_a_nat] :
      ( ( zip_Su7355543910597222519_a_nat @ ( map_Pr3527087583225239196_a_nat @ produc3940427395981429571_a_nat @ Zs2 ) @ ( map_Pr3527087583225239196_a_nat @ produc8028804885397038213_a_nat @ Zs2 ) )
      = Zs2 ) ).

% zip_map_fst_snd
thf(fact_561_zip__eq__conv,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat,Zs2: list_P6608106274694273645_a_nat] :
      ( ( ( size_s5212483967078203639_a_nat @ Xs )
        = ( size_s5212483967078203639_a_nat @ Ys ) )
     => ( ( ( zip_li7023018480688994807_a_nat @ Xs @ Ys )
          = Zs2 )
        = ( ( ( map_Pr5241799845241595682_a_nat @ produc2015473822650161347_a_nat @ Zs2 )
            = Xs )
          & ( ( map_Pr5241799845241595682_a_nat @ produc2553995981594484741_a_nat @ Zs2 )
            = Ys ) ) ) ) ).

% zip_eq_conv
thf(fact_562_zip__eq__conv,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs2: list_P1195027771636113901_a_nat] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( ( ( zip_Su7355543910597222519_a_nat @ Xs @ Ys )
          = Zs2 )
        = ( ( ( map_Pr3527087583225239196_a_nat @ produc3940427395981429571_a_nat @ Zs2 )
            = Xs )
          & ( ( map_Pr3527087583225239196_a_nat @ produc8028804885397038213_a_nat @ Zs2 )
            = Ys ) ) ) ) ).

% zip_eq_conv
thf(fact_563_update__zip,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_P1195027771636113901_a_nat,I2: nat,Xy: produc4421259400484624207_a_nat] :
      ( ( list_u6624685064088169149_a_nat @ ( zip_Su4402317301539879519_a_nat @ Xs @ Ys ) @ I2 @ Xy )
      = ( zip_Su4402317301539879519_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I2 @ ( produc8036459653884716843_a_nat @ Xy ) ) @ ( list_u4149385051645232213_a_nat @ Ys @ I2 @ ( produc6793502323970612077_a_nat @ Xy ) ) ) ) ).

% update_zip
thf(fact_564_update__zip,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,Ys: list_Sum_sum_a_nat,I2: nat,Xy: produc8736486873979948927_a_nat] :
      ( ( list_u1716540500728718061_a_nat @ ( zip_Pr3707921967074149519_a_nat @ Xs @ Ys ) @ I2 @ Xy )
      = ( zip_Pr3707921967074149519_a_nat @ ( list_u4149385051645232213_a_nat @ Xs @ I2 @ ( produc7342064319418986843_a_nat @ Xy ) ) @ ( list_u9138855634547462509_a_nat @ Ys @ I2 @ ( produc6099106989504882077_a_nat @ Xy ) ) ) ) ).

% update_zip
thf(fact_565_update__zip,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,Ys: list_P1195027771636113901_a_nat,I2: nat,Xy: produc5706546023945342183_a_nat] :
      ( ( list_u932594531332099413_a_nat @ ( zip_Pr9202122185580356215_a_nat @ Xs @ Ys ) @ I2 @ Xy )
      = ( zip_Pr9202122185580356215_a_nat @ ( list_u4149385051645232213_a_nat @ Xs @ I2 @ ( produc2561508259746451779_a_nat @ Xy ) ) @ ( list_u4149385051645232213_a_nat @ Ys @ I2 @ ( produc4069943850158121093_a_nat @ Xy ) ) ) ) ).

% update_zip
thf(fact_566_update__zip,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat,I2: nat,Xy: produc5001885624171833703_a_nat] :
      ( ( list_u5062466531432706517_a_nat @ ( zip_li7023018480688994807_a_nat @ Xs @ Ys ) @ I2 @ Xy )
      = ( zip_li7023018480688994807_a_nat @ ( list_u3340303885232119283_a_nat @ Xs @ I2 @ ( produc2015473822650161347_a_nat @ Xy ) ) @ ( list_u3340303885232119283_a_nat @ Ys @ I2 @ ( produc2553995981594484741_a_nat @ Xy ) ) ) ) ).

% update_zip
thf(fact_567_update__zip,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,I2: nat,Xy: produc7017002724195966439_a_nat] :
      ( ( list_u4149385051645232213_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) @ I2 @ Xy )
      = ( zip_Su7355543910597222519_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I2 @ ( produc3940427395981429571_a_nat @ Xy ) ) @ ( list_u9138855634547462509_a_nat @ Ys @ I2 @ ( produc8028804885397038213_a_nat @ Xy ) ) ) ) ).

% update_zip
thf(fact_568_subset__code_I1_J,axiom,
    ! [Xs: list_s7376487411892226277_a_nat,B4: set_se4904748513628223167_a_nat] :
      ( ( ord_le7974500612278410847_a_nat @ ( set_se6249060928767827828_a_nat @ Xs ) @ B4 )
      = ( ! [X2: set_Sum_sum_a_nat] :
            ( ( member8098812455498974984_a_nat @ X2 @ ( set_se6249060928767827828_a_nat @ Xs ) )
           => ( member8098812455498974984_a_nat @ X2 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_569_subset__code_I1_J,axiom,
    ! [Xs: list_P6608106274694273645_a_nat,B4: set_Pr4870381170404451655_a_nat] :
      ( ( ord_le4837030659158318311_a_nat @ ( set_Pr808418073638175228_a_nat @ Xs ) @ B4 )
      = ( ! [X2: produc5001885624171833703_a_nat] :
            ( ( member7457213283480048528_a_nat @ X2 @ ( set_Pr808418073638175228_a_nat @ Xs ) )
           => ( member7457213283480048528_a_nat @ X2 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_570_subset__code_I1_J,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,B4: set_li6526943997496501093_a_nat] :
      ( ( ord_le1147066620699065093_a_nat @ ( set_li2392974972034027290_a_nat @ Xs ) @ B4 )
      = ( ! [X2: list_Sum_sum_a_nat] :
            ( ( member408289922725080238_a_nat @ X2 @ ( set_li2392974972034027290_a_nat @ Xs ) )
           => ( member408289922725080238_a_nat @ X2 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_571_subset__code_I1_J,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,B4: set_Pr7343886759072863943_a_nat] :
      ( ( ord_le7501488065512009319_a_nat @ ( set_Pr2931239450594890620_a_nat @ Xs ) @ B4 )
      = ( ! [X2: produc7017002724195966439_a_nat] :
            ( ( member3723442691059620112_a_nat @ X2 @ ( set_Pr2931239450594890620_a_nat @ Xs ) )
           => ( member3723442691059620112_a_nat @ X2 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_572_subset__code_I1_J,axiom,
    ! [Xs: list_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ B4 )
      = ( ! [X2: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
           => ( member_Sum_sum_a_nat @ X2 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_573_subset__Un__eq,axiom,
    ( ord_le1325389633284124927_a_nat
    = ( ^ [A5: set_Sum_sum_a_nat,B5: set_Sum_sum_a_nat] :
          ( ( sup_su6804446743777130803_a_nat @ A5 @ B5 )
          = B5 ) ) ) ).

% subset_Un_eq
thf(fact_574_subset__UnE,axiom,
    ! [C3: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ C3 @ ( sup_su6804446743777130803_a_nat @ A2 @ B4 ) )
     => ~ ! [A6: set_Sum_sum_a_nat] :
            ( ( ord_le1325389633284124927_a_nat @ A6 @ A2 )
           => ! [B6: set_Sum_sum_a_nat] :
                ( ( ord_le1325389633284124927_a_nat @ B6 @ B4 )
               => ( C3
                 != ( sup_su6804446743777130803_a_nat @ A6 @ B6 ) ) ) ) ) ).

% subset_UnE
thf(fact_575_Un__absorb2,axiom,
    ! [B4: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ B4 @ A2 )
     => ( ( sup_su6804446743777130803_a_nat @ A2 @ B4 )
        = A2 ) ) ).

% Un_absorb2
thf(fact_576_Un__absorb1,axiom,
    ! [A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ A2 @ B4 )
     => ( ( sup_su6804446743777130803_a_nat @ A2 @ B4 )
        = B4 ) ) ).

% Un_absorb1
thf(fact_577_Un__upper2,axiom,
    ! [B4: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ B4 @ ( sup_su6804446743777130803_a_nat @ A2 @ B4 ) ) ).

% Un_upper2
thf(fact_578_Un__upper1,axiom,
    ! [A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B4 ) ) ).

% Un_upper1
thf(fact_579_Un__least,axiom,
    ! [A2: set_Sum_sum_a_nat,C3: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ A2 @ C3 )
     => ( ( ord_le1325389633284124927_a_nat @ B4 @ C3 )
       => ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B4 ) @ C3 ) ) ) ).

% Un_least
thf(fact_580_Un__mono,axiom,
    ! [A2: set_Sum_sum_a_nat,C3: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat,D2: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ A2 @ C3 )
     => ( ( ord_le1325389633284124927_a_nat @ B4 @ D2 )
       => ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B4 ) @ ( sup_su6804446743777130803_a_nat @ C3 @ D2 ) ) ) ) ).

% Un_mono
thf(fact_581_filter__is__subset,axiom,
    ! [P: list_Sum_sum_a_nat > $o,Xs: list_l4703314356710769291_a_nat] : ( ord_le1147066620699065093_a_nat @ ( set_li2392974972034027290_a_nat @ ( filter5373755100216644354_a_nat @ P @ Xs ) ) @ ( set_li2392974972034027290_a_nat @ Xs ) ) ).

% filter_is_subset
thf(fact_582_filter__is__subset,axiom,
    ! [P: produc7017002724195966439_a_nat > $o,Xs: list_P1195027771636113901_a_nat] : ( ord_le7501488065512009319_a_nat @ ( set_Pr2931239450594890620_a_nat @ ( filter6896953120613689188_a_nat @ P @ Xs ) ) @ ( set_Pr2931239450594890620_a_nat @ Xs ) ) ).

% filter_is_subset
thf(fact_583_filter__is__subset,axiom,
    ! [P: sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ ( filter_Sum_sum_a_nat @ P @ Xs ) ) @ ( set_Sum_sum_a_nat2 @ Xs ) ) ).

% filter_is_subset
thf(fact_584_set__take__subset,axiom,
    ! [N: nat,Xs: list_l4703314356710769291_a_nat] : ( ord_le1147066620699065093_a_nat @ ( set_li2392974972034027290_a_nat @ ( take_l4056659018708330355_a_nat @ N @ Xs ) ) @ ( set_li2392974972034027290_a_nat @ Xs ) ) ).

% set_take_subset
thf(fact_585_set__take__subset,axiom,
    ! [N: nat,Xs: list_P1195027771636113901_a_nat] : ( ord_le7501488065512009319_a_nat @ ( set_Pr2931239450594890620_a_nat @ ( take_P3324508860426538453_a_nat @ N @ Xs ) ) @ ( set_Pr2931239450594890620_a_nat @ Xs ) ) ).

% set_take_subset
thf(fact_586_set__take__subset,axiom,
    ! [N: nat,Xs: list_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ ( take_Sum_sum_a_nat @ N @ Xs ) ) @ ( set_Sum_sum_a_nat2 @ Xs ) ) ).

% set_take_subset
thf(fact_587_set__update__subsetI,axiom,
    ! [Xs: list_s7376487411892226277_a_nat,A2: set_se4904748513628223167_a_nat,X: set_Sum_sum_a_nat,I2: nat] :
      ( ( ord_le7974500612278410847_a_nat @ ( set_se6249060928767827828_a_nat @ Xs ) @ A2 )
     => ( ( member8098812455498974984_a_nat @ X @ A2 )
       => ( ord_le7974500612278410847_a_nat @ ( set_se6249060928767827828_a_nat @ ( list_u7655389300282739405_a_nat @ Xs @ I2 @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_588_set__update__subsetI,axiom,
    ! [Xs: list_P6608106274694273645_a_nat,A2: set_Pr4870381170404451655_a_nat,X: produc5001885624171833703_a_nat,I2: nat] :
      ( ( ord_le4837030659158318311_a_nat @ ( set_Pr808418073638175228_a_nat @ Xs ) @ A2 )
     => ( ( member7457213283480048528_a_nat @ X @ A2 )
       => ( ord_le4837030659158318311_a_nat @ ( set_Pr808418073638175228_a_nat @ ( list_u5062466531432706517_a_nat @ Xs @ I2 @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_589_set__update__subsetI,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,A2: set_li6526943997496501093_a_nat,X: list_Sum_sum_a_nat,I2: nat] :
      ( ( ord_le1147066620699065093_a_nat @ ( set_li2392974972034027290_a_nat @ Xs ) @ A2 )
     => ( ( member408289922725080238_a_nat @ X @ A2 )
       => ( ord_le1147066620699065093_a_nat @ ( set_li2392974972034027290_a_nat @ ( list_u3340303885232119283_a_nat @ Xs @ I2 @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_590_set__update__subsetI,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,A2: set_Pr7343886759072863943_a_nat,X: produc7017002724195966439_a_nat,I2: nat] :
      ( ( ord_le7501488065512009319_a_nat @ ( set_Pr2931239450594890620_a_nat @ Xs ) @ A2 )
     => ( ( member3723442691059620112_a_nat @ X @ A2 )
       => ( ord_le7501488065512009319_a_nat @ ( set_Pr2931239450594890620_a_nat @ ( list_u4149385051645232213_a_nat @ Xs @ I2 @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_591_set__update__subsetI,axiom,
    ! [Xs: list_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,X: sum_sum_a_nat,I2: nat] :
      ( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ A2 )
     => ( ( member_Sum_sum_a_nat @ X @ A2 )
       => ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ ( list_u9138855634547462509_a_nat @ Xs @ I2 @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_592_order__antisym__conv,axiom,
    ! [Y: set_Sum_sum_a_nat,X: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ Y @ X )
     => ( ( ord_le1325389633284124927_a_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_593_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_594_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_595_ord__le__eq__subst,axiom,
    ! [A: nat,B2: nat,F: nat > set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_le1325389633284124927_a_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le1325389633284124927_a_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_596_ord__le__eq__subst,axiom,
    ! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > nat,C2: nat] :
      ( ( ord_le1325389633284124927_a_nat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] :
              ( ( ord_le1325389633284124927_a_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_597_ord__le__eq__subst,axiom,
    ! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] :
              ( ( ord_le1325389633284124927_a_nat @ X3 @ Y4 )
             => ( ord_le1325389633284124927_a_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le1325389633284124927_a_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_598_ord__le__eq__subst,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_599_ord__eq__le__subst,axiom,
    ! [A: set_Sum_sum_a_nat,F: nat > set_Sum_sum_a_nat,B2: nat,C2: nat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_le1325389633284124927_a_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le1325389633284124927_a_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_600_ord__eq__le__subst,axiom,
    ! [A: nat,F: set_Sum_sum_a_nat > nat,B2: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_le1325389633284124927_a_nat @ B2 @ C2 )
       => ( ! [X3: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] :
              ( ( ord_le1325389633284124927_a_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_601_ord__eq__le__subst,axiom,
    ! [A: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_le1325389633284124927_a_nat @ B2 @ C2 )
       => ( ! [X3: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] :
              ( ( ord_le1325389633284124927_a_nat @ X3 @ Y4 )
             => ( ord_le1325389633284124927_a_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le1325389633284124927_a_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_602_ord__eq__le__subst,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C2: nat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_603_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_604_order__eq__refl,axiom,
    ! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
      ( ( X = Y )
     => ( ord_le1325389633284124927_a_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_605_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_606_order__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_le1325389633284124927_a_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_le1325389633284124927_a_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le1325389633284124927_a_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_607_order__subst2,axiom,
    ! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > nat,C2: nat] :
      ( ( ord_le1325389633284124927_a_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X3: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] :
              ( ( ord_le1325389633284124927_a_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_608_order__subst2,axiom,
    ! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ A @ B2 )
     => ( ( ord_le1325389633284124927_a_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X3: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] :
              ( ( ord_le1325389633284124927_a_nat @ X3 @ Y4 )
             => ( ord_le1325389633284124927_a_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le1325389633284124927_a_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_609_order__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_610_order__subst1,axiom,
    ! [A: nat,F: set_Sum_sum_a_nat > nat,B2: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
     => ( ( ord_le1325389633284124927_a_nat @ B2 @ C2 )
       => ( ! [X3: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] :
              ( ( ord_le1325389633284124927_a_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_611_order__subst1,axiom,
    ! [A: set_Sum_sum_a_nat,F: nat > set_Sum_sum_a_nat,B2: nat,C2: nat] :
      ( ( ord_le1325389633284124927_a_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_le1325389633284124927_a_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le1325389633284124927_a_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_612_order__subst1,axiom,
    ! [A: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ A @ ( F @ B2 ) )
     => ( ( ord_le1325389633284124927_a_nat @ B2 @ C2 )
       => ( ! [X3: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] :
              ( ( ord_le1325389633284124927_a_nat @ X3 @ Y4 )
             => ( ord_le1325389633284124927_a_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le1325389633284124927_a_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_613_order__subst1,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_614_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] : ( Y2 = Z ) )
    = ( ^ [A4: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
          ( ( ord_le1325389633284124927_a_nat @ A4 @ B3 )
          & ( ord_le1325389633284124927_a_nat @ B3 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_615_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
          & ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_616_antisym,axiom,
    ! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ A @ B2 )
     => ( ( ord_le1325389633284124927_a_nat @ B2 @ A )
       => ( A = B2 ) ) ) ).

% antisym
thf(fact_617_antisym,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ A )
       => ( A = B2 ) ) ) ).

% antisym
thf(fact_618_dual__order_Otrans,axiom,
    ! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ B2 @ A )
     => ( ( ord_le1325389633284124927_a_nat @ C2 @ B2 )
       => ( ord_le1325389633284124927_a_nat @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_619_dual__order_Otrans,axiom,
    ! [B2: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_less_eq_nat @ C2 @ B2 )
       => ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_620_dual__order_Oantisym,axiom,
    ! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ B2 @ A )
     => ( ( ord_le1325389633284124927_a_nat @ A @ B2 )
       => ( A = B2 ) ) ) ).

% dual_order.antisym
thf(fact_621_dual__order_Oantisym,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_less_eq_nat @ A @ B2 )
       => ( A = B2 ) ) ) ).

% dual_order.antisym
thf(fact_622_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] : ( Y2 = Z ) )
    = ( ^ [A4: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
          ( ( ord_le1325389633284124927_a_nat @ B3 @ A4 )
          & ( ord_le1325389633284124927_a_nat @ A4 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_623_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A4 )
          & ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_624_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B2: nat] :
      ( ! [A3: nat,B: nat] :
          ( ( ord_less_eq_nat @ A3 @ B )
         => ( P @ A3 @ B ) )
     => ( ! [A3: nat,B: nat] :
            ( ( P @ B @ A3 )
           => ( P @ A3 @ B ) )
       => ( P @ A @ B2 ) ) ) ).

% linorder_wlog
thf(fact_625_order__trans,axiom,
    ! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z3: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ X @ Y )
     => ( ( ord_le1325389633284124927_a_nat @ Y @ Z3 )
       => ( ord_le1325389633284124927_a_nat @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_626_order__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z3 )
       => ( ord_less_eq_nat @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_627_order_Otrans,axiom,
    ! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ A @ B2 )
     => ( ( ord_le1325389633284124927_a_nat @ B2 @ C2 )
       => ( ord_le1325389633284124927_a_nat @ A @ C2 ) ) ) ).

% order.trans
thf(fact_628_order_Otrans,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% order.trans
thf(fact_629_order__antisym,axiom,
    ! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ X @ Y )
     => ( ( ord_le1325389633284124927_a_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_630_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_631_ord__le__eq__trans,axiom,
    ! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ A @ B2 )
     => ( ( B2 = C2 )
       => ( ord_le1325389633284124927_a_nat @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_632_ord__le__eq__trans,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_633_ord__eq__le__trans,axiom,
    ! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
      ( ( A = B2 )
     => ( ( ord_le1325389633284124927_a_nat @ B2 @ C2 )
       => ( ord_le1325389633284124927_a_nat @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_634_ord__eq__le__trans,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( A = B2 )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_635_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] : ( Y2 = Z ) )
    = ( ^ [X2: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
          ( ( ord_le1325389633284124927_a_nat @ X2 @ Y3 )
          & ( ord_le1325389633284124927_a_nat @ Y3 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_636_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
    = ( ^ [X2: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y3 )
          & ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_637_le__cases3,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z3 ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z3 ) )
       => ( ( ( ord_less_eq_nat @ X @ Z3 )
           => ~ ( ord_less_eq_nat @ Z3 @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z3 @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z3 )
               => ~ ( ord_less_eq_nat @ Z3 @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z3 @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_638_nle__le,axiom,
    ! [A: nat,B2: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
      = ( ( ord_less_eq_nat @ B2 @ A )
        & ( B2 != A ) ) ) ).

% nle_le
thf(fact_639_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_l4703314356710769291_a_nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_le1147066620699065093_a_nat @ ( set_li2392974972034027290_a_nat @ ( take_l4056659018708330355_a_nat @ M @ Xs ) ) @ ( set_li2392974972034027290_a_nat @ ( take_l4056659018708330355_a_nat @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_640_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_P1195027771636113901_a_nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_le7501488065512009319_a_nat @ ( set_Pr2931239450594890620_a_nat @ ( take_P3324508860426538453_a_nat @ M @ Xs ) ) @ ( set_Pr2931239450594890620_a_nat @ ( take_P3324508860426538453_a_nat @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_641_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_Sum_sum_a_nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ ( take_Sum_sum_a_nat @ M @ Xs ) ) @ ( set_Sum_sum_a_nat2 @ ( take_Sum_sum_a_nat @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_642_distinct__map__fst__filter,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,P: produc7017002724195966439_a_nat > $o] :
      ( ( distin2701893636801681144_a_nat @ ( map_Pr3527087583225239196_a_nat @ produc3940427395981429571_a_nat @ Xs ) )
     => ( distin2701893636801681144_a_nat @ ( map_Pr3527087583225239196_a_nat @ produc3940427395981429571_a_nat @ ( filter6896953120613689188_a_nat @ P @ Xs ) ) ) ) ).

% distinct_map_fst_filter
thf(fact_643_distinct__map__fst__filter,axiom,
    ! [Xs: list_P6608106274694273645_a_nat,P: produc5001885624171833703_a_nat > $o] :
      ( ( distin811021574259663358_a_nat @ ( map_Pr5241799845241595682_a_nat @ produc2015473822650161347_a_nat @ Xs ) )
     => ( distin811021574259663358_a_nat @ ( map_Pr5241799845241595682_a_nat @ produc2015473822650161347_a_nat @ ( filter4328210566177025508_a_nat @ P @ Xs ) ) ) ) ).

% distinct_map_fst_filter
thf(fact_644_prod__eq__iff,axiom,
    ( ( ^ [Y2: produc5001885624171833703_a_nat,Z: produc5001885624171833703_a_nat] : ( Y2 = Z ) )
    = ( ^ [S: produc5001885624171833703_a_nat,T2: produc5001885624171833703_a_nat] :
          ( ( ( produc2015473822650161347_a_nat @ S )
            = ( produc2015473822650161347_a_nat @ T2 ) )
          & ( ( produc2553995981594484741_a_nat @ S )
            = ( produc2553995981594484741_a_nat @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_645_prod__eq__iff,axiom,
    ( ( ^ [Y2: produc7017002724195966439_a_nat,Z: produc7017002724195966439_a_nat] : ( Y2 = Z ) )
    = ( ^ [S: produc7017002724195966439_a_nat,T2: produc7017002724195966439_a_nat] :
          ( ( ( produc3940427395981429571_a_nat @ S )
            = ( produc3940427395981429571_a_nat @ T2 ) )
          & ( ( produc8028804885397038213_a_nat @ S )
            = ( produc8028804885397038213_a_nat @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_646_exE__realizer_H,axiom,
    ! [P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,P5: produc5001885624171833703_a_nat] :
      ( ( P @ ( produc2553995981594484741_a_nat @ P5 ) @ ( produc2015473822650161347_a_nat @ P5 ) )
     => ~ ! [X3: list_Sum_sum_a_nat,Y4: list_Sum_sum_a_nat] :
            ~ ( P @ Y4 @ X3 ) ) ).

% exE_realizer'
thf(fact_647_exE__realizer_H,axiom,
    ! [P: sum_sum_a_nat > sum_sum_a_nat > $o,P5: produc7017002724195966439_a_nat] :
      ( ( P @ ( produc8028804885397038213_a_nat @ P5 ) @ ( produc3940427395981429571_a_nat @ P5 ) )
     => ~ ! [X3: sum_sum_a_nat,Y4: sum_sum_a_nat] :
            ~ ( P @ Y4 @ X3 ) ) ).

% exE_realizer'
thf(fact_648_prod__eqI,axiom,
    ! [P5: produc5001885624171833703_a_nat,Q2: produc5001885624171833703_a_nat] :
      ( ( ( produc2015473822650161347_a_nat @ P5 )
        = ( produc2015473822650161347_a_nat @ Q2 ) )
     => ( ( ( produc2553995981594484741_a_nat @ P5 )
          = ( produc2553995981594484741_a_nat @ Q2 ) )
       => ( P5 = Q2 ) ) ) ).

% prod_eqI
thf(fact_649_prod__eqI,axiom,
    ! [P5: produc7017002724195966439_a_nat,Q2: produc7017002724195966439_a_nat] :
      ( ( ( produc3940427395981429571_a_nat @ P5 )
        = ( produc3940427395981429571_a_nat @ Q2 ) )
     => ( ( ( produc8028804885397038213_a_nat @ P5 )
          = ( produc8028804885397038213_a_nat @ Q2 ) )
       => ( P5 = Q2 ) ) ) ).

% prod_eqI
thf(fact_650_prod_Oexpand,axiom,
    ! [Prod: produc5001885624171833703_a_nat,Prod2: produc5001885624171833703_a_nat] :
      ( ( ( ( produc2015473822650161347_a_nat @ Prod )
          = ( produc2015473822650161347_a_nat @ Prod2 ) )
        & ( ( produc2553995981594484741_a_nat @ Prod )
          = ( produc2553995981594484741_a_nat @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_651_prod_Oexpand,axiom,
    ! [Prod: produc7017002724195966439_a_nat,Prod2: produc7017002724195966439_a_nat] :
      ( ( ( ( produc3940427395981429571_a_nat @ Prod )
          = ( produc3940427395981429571_a_nat @ Prod2 ) )
        & ( ( produc8028804885397038213_a_nat @ Prod )
          = ( produc8028804885397038213_a_nat @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_652_distinct__product__lists,axiom,
    ! [Xss: list_l7568939571203725683_a_nat] :
      ( ! [X3: list_P1195027771636113901_a_nat] :
          ( ( member6227626580160590998_a_nat @ X3 @ ( set_li1234883600314943746_a_nat @ Xss ) )
         => ( distin928153214662094432_a_nat @ X3 ) )
     => ( distin5816588314374693350_a_nat @ ( produc3960677393309065686_a_nat @ Xss ) ) ) ).

% distinct_product_lists
thf(fact_653_distinct__product__lists,axiom,
    ! [Xss: list_l999229665597553643_a_nat] :
      ( ! [X3: list_s7376487411892226277_a_nat] :
          ( ( member6403511879741922830_a_nat @ X3 @ ( set_li1117937094484414586_a_nat @ Xss ) )
         => ( distin4821671304206132312_a_nat @ X3 ) )
     => ( distin7371808003173651294_a_nat @ ( produc5572861305105196750_a_nat @ Xss ) ) ) ).

% distinct_product_lists
thf(fact_654_distinct__product__lists,axiom,
    ! [Xss: list_l1586611297644397841_a_nat] :
      ( ! [X3: list_l4703314356710769291_a_nat] :
          ( ( member6656483030743143220_a_nat @ X3 @ ( set_li7766024463445446560_a_nat @ Xss ) )
         => ( distin811021574259663358_a_nat @ X3 ) )
     => ( distin5621845738133643396_a_nat @ ( produc4105601411378645364_a_nat @ Xss ) ) ) ).

% distinct_product_lists
thf(fact_655_distinct__product__lists,axiom,
    ! [Xss: list_l4703314356710769291_a_nat] :
      ( ! [X3: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ X3 @ ( set_li2392974972034027290_a_nat @ Xss ) )
         => ( distin2701893636801681144_a_nat @ X3 ) )
     => ( distin811021574259663358_a_nat @ ( produc2893206433618375022_a_nat @ Xss ) ) ) ).

% distinct_product_lists
thf(fact_656_subsetI,axiom,
    ! [A2: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
      ( ! [X3: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ X3 @ A2 )
         => ( member408289922725080238_a_nat @ X3 @ B4 ) )
     => ( ord_le1147066620699065093_a_nat @ A2 @ B4 ) ) ).

% subsetI
thf(fact_657_subsetI,axiom,
    ! [A2: set_se4904748513628223167_a_nat,B4: set_se4904748513628223167_a_nat] :
      ( ! [X3: set_Sum_sum_a_nat] :
          ( ( member8098812455498974984_a_nat @ X3 @ A2 )
         => ( member8098812455498974984_a_nat @ X3 @ B4 ) )
     => ( ord_le7974500612278410847_a_nat @ A2 @ B4 ) ) ).

% subsetI
thf(fact_658_subsetI,axiom,
    ! [A2: set_Pr7343886759072863943_a_nat,B4: set_Pr7343886759072863943_a_nat] :
      ( ! [X3: produc7017002724195966439_a_nat] :
          ( ( member3723442691059620112_a_nat @ X3 @ A2 )
         => ( member3723442691059620112_a_nat @ X3 @ B4 ) )
     => ( ord_le7501488065512009319_a_nat @ A2 @ B4 ) ) ).

% subsetI
thf(fact_659_subsetI,axiom,
    ! [A2: set_Pr4870381170404451655_a_nat,B4: set_Pr4870381170404451655_a_nat] :
      ( ! [X3: produc5001885624171833703_a_nat] :
          ( ( member7457213283480048528_a_nat @ X3 @ A2 )
         => ( member7457213283480048528_a_nat @ X3 @ B4 ) )
     => ( ord_le4837030659158318311_a_nat @ A2 @ B4 ) ) ).

% subsetI
thf(fact_660_subsetI,axiom,
    ! [A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
      ( ! [X3: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ X3 @ A2 )
         => ( member_Sum_sum_a_nat @ X3 @ B4 ) )
     => ( ord_le1325389633284124927_a_nat @ A2 @ B4 ) ) ).

% subsetI
thf(fact_661_subset__antisym,axiom,
    ! [A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ A2 @ B4 )
     => ( ( ord_le1325389633284124927_a_nat @ B4 @ A2 )
       => ( A2 = B4 ) ) ) ).

% subset_antisym
thf(fact_662_in__mono,axiom,
    ! [A2: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat,X: list_Sum_sum_a_nat] :
      ( ( ord_le1147066620699065093_a_nat @ A2 @ B4 )
     => ( ( member408289922725080238_a_nat @ X @ A2 )
       => ( member408289922725080238_a_nat @ X @ B4 ) ) ) ).

% in_mono
thf(fact_663_in__mono,axiom,
    ! [A2: set_se4904748513628223167_a_nat,B4: set_se4904748513628223167_a_nat,X: set_Sum_sum_a_nat] :
      ( ( ord_le7974500612278410847_a_nat @ A2 @ B4 )
     => ( ( member8098812455498974984_a_nat @ X @ A2 )
       => ( member8098812455498974984_a_nat @ X @ B4 ) ) ) ).

% in_mono
thf(fact_664_in__mono,axiom,
    ! [A2: set_Pr7343886759072863943_a_nat,B4: set_Pr7343886759072863943_a_nat,X: produc7017002724195966439_a_nat] :
      ( ( ord_le7501488065512009319_a_nat @ A2 @ B4 )
     => ( ( member3723442691059620112_a_nat @ X @ A2 )
       => ( member3723442691059620112_a_nat @ X @ B4 ) ) ) ).

% in_mono
thf(fact_665_in__mono,axiom,
    ! [A2: set_Pr4870381170404451655_a_nat,B4: set_Pr4870381170404451655_a_nat,X: produc5001885624171833703_a_nat] :
      ( ( ord_le4837030659158318311_a_nat @ A2 @ B4 )
     => ( ( member7457213283480048528_a_nat @ X @ A2 )
       => ( member7457213283480048528_a_nat @ X @ B4 ) ) ) ).

% in_mono
thf(fact_666_in__mono,axiom,
    ! [A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat,X: sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ A2 @ B4 )
     => ( ( member_Sum_sum_a_nat @ X @ A2 )
       => ( member_Sum_sum_a_nat @ X @ B4 ) ) ) ).

% in_mono
thf(fact_667_subsetD,axiom,
    ! [A2: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat,C2: list_Sum_sum_a_nat] :
      ( ( ord_le1147066620699065093_a_nat @ A2 @ B4 )
     => ( ( member408289922725080238_a_nat @ C2 @ A2 )
       => ( member408289922725080238_a_nat @ C2 @ B4 ) ) ) ).

% subsetD
thf(fact_668_subsetD,axiom,
    ! [A2: set_se4904748513628223167_a_nat,B4: set_se4904748513628223167_a_nat,C2: set_Sum_sum_a_nat] :
      ( ( ord_le7974500612278410847_a_nat @ A2 @ B4 )
     => ( ( member8098812455498974984_a_nat @ C2 @ A2 )
       => ( member8098812455498974984_a_nat @ C2 @ B4 ) ) ) ).

% subsetD
thf(fact_669_subsetD,axiom,
    ! [A2: set_Pr7343886759072863943_a_nat,B4: set_Pr7343886759072863943_a_nat,C2: produc7017002724195966439_a_nat] :
      ( ( ord_le7501488065512009319_a_nat @ A2 @ B4 )
     => ( ( member3723442691059620112_a_nat @ C2 @ A2 )
       => ( member3723442691059620112_a_nat @ C2 @ B4 ) ) ) ).

% subsetD
thf(fact_670_subsetD,axiom,
    ! [A2: set_Pr4870381170404451655_a_nat,B4: set_Pr4870381170404451655_a_nat,C2: produc5001885624171833703_a_nat] :
      ( ( ord_le4837030659158318311_a_nat @ A2 @ B4 )
     => ( ( member7457213283480048528_a_nat @ C2 @ A2 )
       => ( member7457213283480048528_a_nat @ C2 @ B4 ) ) ) ).

% subsetD
thf(fact_671_subsetD,axiom,
    ! [A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat,C2: sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ A2 @ B4 )
     => ( ( member_Sum_sum_a_nat @ C2 @ A2 )
       => ( member_Sum_sum_a_nat @ C2 @ B4 ) ) ) ).

% subsetD
thf(fact_672_equalityE,axiom,
    ! [A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
      ( ( A2 = B4 )
     => ~ ( ( ord_le1325389633284124927_a_nat @ A2 @ B4 )
         => ~ ( ord_le1325389633284124927_a_nat @ B4 @ A2 ) ) ) ).

% equalityE
thf(fact_673_subset__eq,axiom,
    ( ord_le1147066620699065093_a_nat
    = ( ^ [A5: set_li6526943997496501093_a_nat,B5: set_li6526943997496501093_a_nat] :
        ! [X2: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ X2 @ A5 )
         => ( member408289922725080238_a_nat @ X2 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_674_subset__eq,axiom,
    ( ord_le7974500612278410847_a_nat
    = ( ^ [A5: set_se4904748513628223167_a_nat,B5: set_se4904748513628223167_a_nat] :
        ! [X2: set_Sum_sum_a_nat] :
          ( ( member8098812455498974984_a_nat @ X2 @ A5 )
         => ( member8098812455498974984_a_nat @ X2 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_675_subset__eq,axiom,
    ( ord_le7501488065512009319_a_nat
    = ( ^ [A5: set_Pr7343886759072863943_a_nat,B5: set_Pr7343886759072863943_a_nat] :
        ! [X2: produc7017002724195966439_a_nat] :
          ( ( member3723442691059620112_a_nat @ X2 @ A5 )
         => ( member3723442691059620112_a_nat @ X2 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_676_subset__eq,axiom,
    ( ord_le4837030659158318311_a_nat
    = ( ^ [A5: set_Pr4870381170404451655_a_nat,B5: set_Pr4870381170404451655_a_nat] :
        ! [X2: produc5001885624171833703_a_nat] :
          ( ( member7457213283480048528_a_nat @ X2 @ A5 )
         => ( member7457213283480048528_a_nat @ X2 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_677_subset__eq,axiom,
    ( ord_le1325389633284124927_a_nat
    = ( ^ [A5: set_Sum_sum_a_nat,B5: set_Sum_sum_a_nat] :
        ! [X2: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ X2 @ A5 )
         => ( member_Sum_sum_a_nat @ X2 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_678_equalityD1,axiom,
    ! [A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
      ( ( A2 = B4 )
     => ( ord_le1325389633284124927_a_nat @ A2 @ B4 ) ) ).

% equalityD1
thf(fact_679_Set_OequalityD2,axiom,
    ! [A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
      ( ( A2 = B4 )
     => ( ord_le1325389633284124927_a_nat @ B4 @ A2 ) ) ).

% Set.equalityD2
thf(fact_680_subset__iff,axiom,
    ( ord_le1147066620699065093_a_nat
    = ( ^ [A5: set_li6526943997496501093_a_nat,B5: set_li6526943997496501093_a_nat] :
        ! [T2: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ T2 @ A5 )
         => ( member408289922725080238_a_nat @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_681_subset__iff,axiom,
    ( ord_le7974500612278410847_a_nat
    = ( ^ [A5: set_se4904748513628223167_a_nat,B5: set_se4904748513628223167_a_nat] :
        ! [T2: set_Sum_sum_a_nat] :
          ( ( member8098812455498974984_a_nat @ T2 @ A5 )
         => ( member8098812455498974984_a_nat @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_682_subset__iff,axiom,
    ( ord_le7501488065512009319_a_nat
    = ( ^ [A5: set_Pr7343886759072863943_a_nat,B5: set_Pr7343886759072863943_a_nat] :
        ! [T2: produc7017002724195966439_a_nat] :
          ( ( member3723442691059620112_a_nat @ T2 @ A5 )
         => ( member3723442691059620112_a_nat @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_683_subset__iff,axiom,
    ( ord_le4837030659158318311_a_nat
    = ( ^ [A5: set_Pr4870381170404451655_a_nat,B5: set_Pr4870381170404451655_a_nat] :
        ! [T2: produc5001885624171833703_a_nat] :
          ( ( member7457213283480048528_a_nat @ T2 @ A5 )
         => ( member7457213283480048528_a_nat @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_684_subset__iff,axiom,
    ( ord_le1325389633284124927_a_nat
    = ( ^ [A5: set_Sum_sum_a_nat,B5: set_Sum_sum_a_nat] :
        ! [T2: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ T2 @ A5 )
         => ( member_Sum_sum_a_nat @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_685_subset__refl,axiom,
    ! [A2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A2 @ A2 ) ).

% subset_refl
thf(fact_686_Collect__mono,axiom,
    ! [P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
      ( ! [X3: sum_sum_a_nat] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le1325389633284124927_a_nat @ ( collec7073057861543223018_a_nat @ P ) @ ( collec7073057861543223018_a_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_687_subset__trans,axiom,
    ! [A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat,C3: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ A2 @ B4 )
     => ( ( ord_le1325389633284124927_a_nat @ B4 @ C3 )
       => ( ord_le1325389633284124927_a_nat @ A2 @ C3 ) ) ) ).

% subset_trans
thf(fact_688_set__eq__subset,axiom,
    ( ( ^ [Y2: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] : ( Y2 = Z ) )
    = ( ^ [A5: set_Sum_sum_a_nat,B5: set_Sum_sum_a_nat] :
          ( ( ord_le1325389633284124927_a_nat @ A5 @ B5 )
          & ( ord_le1325389633284124927_a_nat @ B5 @ A5 ) ) ) ) ).

% set_eq_subset
thf(fact_689_Collect__mono__iff,axiom,
    ! [P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
      ( ( ord_le1325389633284124927_a_nat @ ( collec7073057861543223018_a_nat @ P ) @ ( collec7073057861543223018_a_nat @ Q ) )
      = ( ! [X2: sum_sum_a_nat] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_690_in__set__product__lists__length,axiom,
    ! [Xs: list_Sum_sum_a_nat,Xss: list_l4703314356710769291_a_nat] :
      ( ( member408289922725080238_a_nat @ Xs @ ( set_li2392974972034027290_a_nat @ ( produc2893206433618375022_a_nat @ Xss ) ) )
     => ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_s5212483967078203639_a_nat @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_691_subset__code_I3_J,axiom,
    ~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).

% subset_code(3)
thf(fact_692_subset__code_I3_J,axiom,
    ~ ( ord_le1147066620699065093_a_nat @ ( coset_2342676519794954744_a_nat @ nil_li1906260230833442699_a_nat ) @ ( set_li2392974972034027290_a_nat @ nil_li1906260230833442699_a_nat ) ) ).

% subset_code(3)
thf(fact_693_subset__code_I3_J,axiom,
    ~ ( ord_le7501488065512009319_a_nat @ ( coset_3293031159811469914_a_nat @ nil_Pr6585251977456444909_a_nat ) @ ( set_Pr2931239450594890620_a_nat @ nil_Pr6585251977456444909_a_nat ) ) ).

% subset_code(3)
thf(fact_694_subset__code_I3_J,axiom,
    ~ ( ord_le1325389633284124927_a_nat @ ( coset_Sum_sum_a_nat @ nil_Sum_sum_a_nat ) @ ( set_Sum_sum_a_nat2 @ nil_Sum_sum_a_nat ) ) ).

% subset_code(3)
thf(fact_695_zip__update,axiom,
    ! [Xs: list_Sum_sum_a_nat,I2: nat,X: sum_sum_a_nat,Ys: list_P1195027771636113901_a_nat,Y: produc7017002724195966439_a_nat] :
      ( ( zip_Su4402317301539879519_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I2 @ X ) @ ( list_u4149385051645232213_a_nat @ Ys @ I2 @ Y ) )
      = ( list_u6624685064088169149_a_nat @ ( zip_Su4402317301539879519_a_nat @ Xs @ Ys ) @ I2 @ ( produc438919121292964671_a_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_696_zip__update,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,I2: nat,X: produc7017002724195966439_a_nat,Ys: list_Sum_sum_a_nat,Y: sum_sum_a_nat] :
      ( ( zip_Pr3707921967074149519_a_nat @ ( list_u4149385051645232213_a_nat @ Xs @ I2 @ X ) @ ( list_u9138855634547462509_a_nat @ Ys @ I2 @ Y ) )
      = ( list_u1716540500728718061_a_nat @ ( zip_Pr3707921967074149519_a_nat @ Xs @ Ys ) @ I2 @ ( produc8967895823682010479_a_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_697_zip__update,axiom,
    ! [Xs: list_P1195027771636113901_a_nat,I2: nat,X: produc7017002724195966439_a_nat,Ys: list_P1195027771636113901_a_nat,Y: produc7017002724195966439_a_nat] :
      ( ( zip_Pr9202122185580356215_a_nat @ ( list_u4149385051645232213_a_nat @ Xs @ I2 @ X ) @ ( list_u4149385051645232213_a_nat @ Ys @ I2 @ Y ) )
      = ( list_u932594531332099413_a_nat @ ( zip_Pr9202122185580356215_a_nat @ Xs @ Ys ) @ I2 @ ( produc4162992740250938199_a_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_698_zip__update,axiom,
    ! [Xs: list_l4703314356710769291_a_nat,I2: nat,X: list_Sum_sum_a_nat,Ys: list_l4703314356710769291_a_nat,Y: list_Sum_sum_a_nat] :
      ( ( zip_li7023018480688994807_a_nat @ ( list_u3340303885232119283_a_nat @ Xs @ I2 @ X ) @ ( list_u3340303885232119283_a_nat @ Ys @ I2 @ Y ) )
      = ( list_u5062466531432706517_a_nat @ ( zip_li7023018480688994807_a_nat @ Xs @ Ys ) @ I2 @ ( produc7990843422341522135_a_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_699_zip__update,axiom,
    ! [Xs: list_list_nat,I2: nat,X: list_nat,Ys: list_list_nat,Y: list_nat] :
      ( ( zip_li7157463729305086713st_nat @ ( list_update_list_nat @ Xs @ I2 @ X ) @ ( list_update_list_nat @ Ys @ I2 @ Y ) )
      = ( list_u5682793034032774415st_nat @ ( zip_li7157463729305086713st_nat @ Xs @ Ys ) @ I2 @ ( produc2694037385005941721st_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_700_zip__update,axiom,
    ! [Xs: list_nat,I2: nat,X: nat,Ys: list_P6608106274694273645_a_nat,Y: produc5001885624171833703_a_nat] :
      ( ( zip_na3853549618184943902_a_nat @ ( list_update_nat @ Xs @ I2 @ X ) @ ( list_u5062466531432706517_a_nat @ Ys @ I2 @ Y ) )
      = ( list_u662661324514154910_a_nat @ ( zip_na3853549618184943902_a_nat @ Xs @ Ys ) @ I2 @ ( produc5571422593549375550_a_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_701_zip__update,axiom,
    ! [Xs: list_S9131871693400613940_nat_o,I2: nat,X: sum_sum_a_nat > sum_sum_a_nat > $o,Ys: list_P6608106274694273645_a_nat,Y: produc5001885624171833703_a_nat] :
      ( ( zip_Su7083767565316534512_a_nat @ ( list_u3817880519379542044_nat_o @ Xs @ I2 @ X ) @ ( list_u5062466531432706517_a_nat @ Ys @ I2 @ Y ) )
      = ( list_u560528484306587342_a_nat @ ( zip_Su7083767565316534512_a_nat @ Xs @ Ys ) @ I2 @ ( produc2770702104727902672_a_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_702_zip__update,axiom,
    ! [Xs: list_S9131871693400613940_nat_o,I2: nat,X: sum_sum_a_nat > sum_sum_a_nat > $o,Ys: list_l4703314356710769291_a_nat,Y: list_Sum_sum_a_nat] :
      ( ( zip_Su2412782442871466510_a_nat @ ( list_u3817880519379542044_nat_o @ Xs @ I2 @ X ) @ ( list_u3340303885232119283_a_nat @ Ys @ I2 @ Y ) )
      = ( list_u6567445468097547756_a_nat @ ( zip_Su2412782442871466510_a_nat @ Xs @ Ys ) @ I2 @ ( produc8857357843056227054_a_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_703_zip__update,axiom,
    ! [Xs: list_Sum_sum_a_nat,I2: nat,X: sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Y: sum_sum_a_nat] :
      ( ( zip_Su7355543910597222519_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I2 @ X ) @ ( list_u9138855634547462509_a_nat @ Ys @ I2 @ Y ) )
      = ( list_u4149385051645232213_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) @ I2 @ ( produc1212125651291703639_a_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_704_length__product,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( size_s7247017532682008665_a_nat @ ( produc7134955936706270469_a_nat @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).

% length_product
thf(fact_705_subset__subseqs,axiom,
    ! [X5: set_li6526943997496501093_a_nat,Xs: list_l4703314356710769291_a_nat] :
      ( ( ord_le1147066620699065093_a_nat @ X5 @ ( set_li2392974972034027290_a_nat @ Xs ) )
     => ( member5553968465346197646_a_nat @ X5 @ ( image_2800575032594406047_a_nat @ set_li2392974972034027290_a_nat @ ( set_li7766024463445446560_a_nat @ ( subseq4596077244143087834_a_nat @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_706_subset__subseqs,axiom,
    ! [X5: set_Pr7343886759072863943_a_nat,Xs: list_P1195027771636113901_a_nat] :
      ( ( ord_le7501488065512009319_a_nat @ X5 @ ( set_Pr2931239450594890620_a_nat @ Xs ) )
     => ( member6181414433633234416_a_nat @ X5 @ ( image_8418046036402785055_a_nat @ set_Pr2931239450594890620_a_nat @ ( set_li1234883600314943746_a_nat @ ( subseq5198752961258583356_a_nat @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_707_subset__subseqs,axiom,
    ! [X5: set_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ X5 @ ( set_Sum_sum_a_nat2 @ Xs ) )
     => ( member8098812455498974984_a_nat @ X5 @ ( image_3940260845811589407_a_nat @ set_Sum_sum_a_nat2 @ ( set_li2392974972034027290_a_nat @ ( subseq8414445098004693972_a_nat @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_708_AList_Odistinct__update__with__aux,axiom,
    ! [V: sum_sum_a_nat,K: sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,Ps: list_P1195027771636113901_a_nat] :
      ( ( distin2701893636801681144_a_nat @ ( map_Pr3527087583225239196_a_nat @ produc3940427395981429571_a_nat @ ( update5424960881307245506_a_nat @ V @ K @ F @ Ps ) ) )
      = ( distin2701893636801681144_a_nat @ ( map_Pr3527087583225239196_a_nat @ produc3940427395981429571_a_nat @ Ps ) ) ) ).

% AList.distinct_update_with_aux
thf(fact_709_AList_Odistinct__update__with__aux,axiom,
    ! [V: list_Sum_sum_a_nat,K: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,Ps: list_P6608106274694273645_a_nat] :
      ( ( distin811021574259663358_a_nat @ ( map_Pr5241799845241595682_a_nat @ produc2015473822650161347_a_nat @ ( update817380752136475202_a_nat @ V @ K @ F @ Ps ) ) )
      = ( distin811021574259663358_a_nat @ ( map_Pr5241799845241595682_a_nat @ produc2015473822650161347_a_nat @ Ps ) ) ) ).

% AList.distinct_update_with_aux
thf(fact_710_fst__swap,axiom,
    ! [X: produc5001885624171833703_a_nat] :
      ( ( produc2015473822650161347_a_nat @ ( produc4286111038383333495_a_nat @ X ) )
      = ( produc2553995981594484741_a_nat @ X ) ) ).

% fst_swap
thf(fact_711_fst__swap,axiom,
    ! [X: produc7017002724195966439_a_nat] :
      ( ( produc3940427395981429571_a_nat @ ( produc33134186151372535_a_nat @ X ) )
      = ( produc8028804885397038213_a_nat @ X ) ) ).

% fst_swap
thf(fact_712_prod_Oinject,axiom,
    ! [X1: list_Sum_sum_a_nat,X22: list_Sum_sum_a_nat,Y1: list_Sum_sum_a_nat,Y22: list_Sum_sum_a_nat] :
      ( ( ( produc7990843422341522135_a_nat @ X1 @ X22 )
        = ( produc7990843422341522135_a_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_713_prod_Oinject,axiom,
    ! [X1: list_nat,X22: list_nat,Y1: list_nat,Y22: list_nat] :
      ( ( ( produc2694037385005941721st_nat @ X1 @ X22 )
        = ( produc2694037385005941721st_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_714_prod_Oinject,axiom,
    ! [X1: nat,X22: produc5001885624171833703_a_nat,Y1: nat,Y22: produc5001885624171833703_a_nat] :
      ( ( ( produc5571422593549375550_a_nat @ X1 @ X22 )
        = ( produc5571422593549375550_a_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_715_prod_Oinject,axiom,
    ! [X1: sum_sum_a_nat > sum_sum_a_nat > $o,X22: produc5001885624171833703_a_nat,Y1: sum_sum_a_nat > sum_sum_a_nat > $o,Y22: produc5001885624171833703_a_nat] :
      ( ( ( produc2770702104727902672_a_nat @ X1 @ X22 )
        = ( produc2770702104727902672_a_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_716_prod_Oinject,axiom,
    ! [X1: sum_sum_a_nat > sum_sum_a_nat > $o,X22: list_Sum_sum_a_nat,Y1: sum_sum_a_nat > sum_sum_a_nat > $o,Y22: list_Sum_sum_a_nat] :
      ( ( ( produc8857357843056227054_a_nat @ X1 @ X22 )
        = ( produc8857357843056227054_a_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_717_old_Oprod_Oinject,axiom,
    ! [A: list_Sum_sum_a_nat,B2: list_Sum_sum_a_nat,A7: list_Sum_sum_a_nat,B7: list_Sum_sum_a_nat] :
      ( ( ( produc7990843422341522135_a_nat @ A @ B2 )
        = ( produc7990843422341522135_a_nat @ A7 @ B7 ) )
      = ( ( A = A7 )
        & ( B2 = B7 ) ) ) ).

% old.prod.inject
thf(fact_718_old_Oprod_Oinject,axiom,
    ! [A: list_nat,B2: list_nat,A7: list_nat,B7: list_nat] :
      ( ( ( produc2694037385005941721st_nat @ A @ B2 )
        = ( produc2694037385005941721st_nat @ A7 @ B7 ) )
      = ( ( A = A7 )
        & ( B2 = B7 ) ) ) ).

% old.prod.inject
thf(fact_719_old_Oprod_Oinject,axiom,
    ! [A: nat,B2: produc5001885624171833703_a_nat,A7: nat,B7: produc5001885624171833703_a_nat] :
      ( ( ( produc5571422593549375550_a_nat @ A @ B2 )
        = ( produc5571422593549375550_a_nat @ A7 @ B7 ) )
      = ( ( A = A7 )
        & ( B2 = B7 ) ) ) ).

% old.prod.inject
thf(fact_720_old_Oprod_Oinject,axiom,
    ! [A: sum_sum_a_nat > sum_sum_a_nat > $o,B2: produc5001885624171833703_a_nat,A7: sum_sum_a_nat > sum_sum_a_nat > $o,B7: produc5001885624171833703_a_nat] :
      ( ( ( produc2770702104727902672_a_nat @ A @ B2 )
        = ( produc2770702104727902672_a_nat @ A7 @ B7 ) )
      = ( ( A = A7 )
        & ( B2 = B7 ) ) ) ).

% old.prod.inject
thf(fact_721_old_Oprod_Oinject,axiom,
    ! [A: sum_sum_a_nat > sum_sum_a_nat > $o,B2: list_Sum_sum_a_nat,A7: sum_sum_a_nat > sum_sum_a_nat > $o,B7: list_Sum_sum_a_nat] :
      ( ( ( produc8857357843056227054_a_nat @ A @ B2 )
        = ( produc8857357843056227054_a_nat @ A7 @ B7 ) )
      = ( ( A = A7 )
        & ( B2 = B7 ) ) ) ).

% old.prod.inject
thf(fact_722_image__eqI,axiom,
    ! [B2: sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,X: sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
      ( ( B2
        = ( F @ X ) )
     => ( ( member_Sum_sum_a_nat @ X @ A2 )
       => ( member_Sum_sum_a_nat @ B2 @ ( image_7142520692256960453_a_nat @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_723_image__eqI,axiom,
    ! [B2: list_Sum_sum_a_nat,F: sum_sum_a_nat > list_Sum_sum_a_nat,X: sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
      ( ( B2
        = ( F @ X ) )
     => ( ( member_Sum_sum_a_nat @ X @ A2 )
       => ( member408289922725080238_a_nat @ B2 @ ( image_7077900747146408011_a_nat @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_724_image__eqI,axiom,
    ! [B2: set_Sum_sum_a_nat,F: sum_sum_a_nat > set_Sum_sum_a_nat,X: sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
      ( ( B2
        = ( F @ X ) )
     => ( ( member_Sum_sum_a_nat @ X @ A2 )
       => ( member8098812455498974984_a_nat @ B2 @ ( image_7877458644602423589_a_nat @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_725_image__eqI,axiom,
    ! [B2: sum_sum_a_nat,F: list_Sum_sum_a_nat > sum_sum_a_nat,X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
      ( ( B2
        = ( F @ X ) )
     => ( ( member408289922725080238_a_nat @ X @ A2 )
       => ( member_Sum_sum_a_nat @ B2 @ ( image_6705138127771971519_a_nat @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_726_image__eqI,axiom,
    ! [B2: sum_sum_a_nat,F: set_Sum_sum_a_nat > sum_sum_a_nat,X: set_Sum_sum_a_nat,A2: set_se4904748513628223167_a_nat] :
      ( ( B2
        = ( F @ X ) )
     => ( ( member8098812455498974984_a_nat @ X @ A2 )
       => ( member_Sum_sum_a_nat @ B2 @ ( image_5092943885087329893_a_nat @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_727_image__eqI,axiom,
    ! [B2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
      ( ( B2
        = ( F @ X ) )
     => ( ( member408289922725080238_a_nat @ X @ A2 )
       => ( member408289922725080238_a_nat @ B2 @ ( image_5081948215111134021_a_nat @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_728_image__eqI,axiom,
    ! [B2: set_Sum_sum_a_nat,F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
      ( ( B2
        = ( F @ X ) )
     => ( ( member408289922725080238_a_nat @ X @ A2 )
       => ( member8098812455498974984_a_nat @ B2 @ ( image_3940260845811589407_a_nat @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_729_image__eqI,axiom,
    ! [B2: list_Sum_sum_a_nat,F: set_Sum_sum_a_nat > list_Sum_sum_a_nat,X: set_Sum_sum_a_nat,A2: set_se4904748513628223167_a_nat] :
      ( ( B2
        = ( F @ X ) )
     => ( ( member8098812455498974984_a_nat @ X @ A2 )
       => ( member408289922725080238_a_nat @ B2 @ ( image_8255745304366202603_a_nat @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_730_image__eqI,axiom,
    ! [B2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > set_Sum_sum_a_nat,X: set_Sum_sum_a_nat,A2: set_se4904748513628223167_a_nat] :
      ( ( B2
        = ( F @ X ) )
     => ( ( member8098812455498974984_a_nat @ X @ A2 )
       => ( member8098812455498974984_a_nat @ B2 @ ( image_5599399343138760645_a_nat @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_731_image__eqI,axiom,
    ! [B2: produc7017002724195966439_a_nat,F: sum_sum_a_nat > produc7017002724195966439_a_nat,X: sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
      ( ( B2
        = ( F @ X ) )
     => ( ( member_Sum_sum_a_nat @ X @ A2 )
       => ( member3723442691059620112_a_nat @ B2 @ ( image_8014146751851824813_a_nat @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_732_swap__simp,axiom,
    ! [X: produc5001885624171833703_a_nat,Y: nat] :
      ( ( produc4241125690352732224at_nat @ ( produc4562899468936728032at_nat @ X @ Y ) )
      = ( produc5571422593549375550_a_nat @ Y @ X ) ) ).

% swap_simp
thf(fact_733_swap__simp,axiom,
    ! [X: produc5001885624171833703_a_nat,Y: sum_sum_a_nat > sum_sum_a_nat > $o] :
      ( ( produc4412250191365866814_nat_o @ ( produc6586225697188128670_nat_o @ X @ Y ) )
      = ( produc2770702104727902672_a_nat @ Y @ X ) ) ).

% swap_simp
thf(fact_734_swap__simp,axiom,
    ! [X: list_Sum_sum_a_nat,Y: sum_sum_a_nat > sum_sum_a_nat > $o] :
      ( ( produc6889179298173170336_nat_o @ ( produc4133243362358621440_nat_o @ X @ Y ) )
      = ( produc8857357843056227054_a_nat @ Y @ X ) ) ).

% swap_simp
thf(fact_735_swap__simp,axiom,
    ! [X: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
      ( ( produc4286111038383333495_a_nat @ ( produc7990843422341522135_a_nat @ X @ Y ) )
      = ( produc7990843422341522135_a_nat @ Y @ X ) ) ).

% swap_simp
thf(fact_736_swap__simp,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( produc7439058410299273593st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) )
      = ( produc2694037385005941721st_nat @ Y @ X ) ) ).

% swap_simp
thf(fact_737_swap__simp,axiom,
    ! [X: nat,Y: produc5001885624171833703_a_nat] :
      ( ( produc5249648814965379742_a_nat @ ( produc5571422593549375550_a_nat @ X @ Y ) )
      = ( produc4562899468936728032at_nat @ Y @ X ) ) ).

% swap_simp
thf(fact_738_swap__simp,axiom,
    ! [X: sum_sum_a_nat > sum_sum_a_nat > $o,Y: produc5001885624171833703_a_nat] :
      ( ( produc596726598905640816_a_nat @ ( produc2770702104727902672_a_nat @ X @ Y ) )
      = ( produc6586225697188128670_nat_o @ Y @ X ) ) ).

% swap_simp
thf(fact_739_swap__simp,axiom,
    ! [X: sum_sum_a_nat > sum_sum_a_nat > $o,Y: list_Sum_sum_a_nat] :
      ( ( produc2389921742016000142_a_nat @ ( produc8857357843056227054_a_nat @ X @ Y ) )
      = ( produc4133243362358621440_nat_o @ Y @ X ) ) ).

% swap_simp
thf(fact_740_list_Oset__map,axiom,
    ! [F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,V: list_l4703314356710769291_a_nat] :
      ( ( set_se6249060928767827828_a_nat @ ( map_li9079814933684165854_a_nat @ F @ V ) )
      = ( image_3940260845811589407_a_nat @ F @ ( set_li2392974972034027290_a_nat @ V ) ) ) ).

% list.set_map
thf(fact_741_list_Oset__map,axiom,
    ! [F: list_Sum_sum_a_nat > sum_sum_a_nat,V: list_l4703314356710769291_a_nat] :
      ( ( set_Sum_sum_a_nat2 @ ( map_li3106614296470775678_a_nat @ F @ V ) )
      = ( image_6705138127771971519_a_nat @ F @ ( set_li2392974972034027290_a_nat @ V ) ) ) ).

% list.set_map
thf(fact_742_list_Oset__map,axiom,
    ! [F: produc7017002724195966439_a_nat > sum_sum_a_nat,V: list_P1195027771636113901_a_nat] :
      ( ( set_Sum_sum_a_nat2 @ ( map_Pr3527087583225239196_a_nat @ F @ V ) )
      = ( image_7319751417386094813_a_nat @ F @ ( set_Pr2931239450594890620_a_nat @ V ) ) ) ).

% list.set_map
thf(fact_743_list_Oset__map,axiom,
    ! [F: sum_sum_a_nat > list_Sum_sum_a_nat,V: list_Sum_sum_a_nat] :
      ( ( set_li2392974972034027290_a_nat @ ( map_Su3479376915845212170_a_nat @ F @ V ) )
      = ( image_7077900747146408011_a_nat @ F @ ( set_Sum_sum_a_nat2 @ V ) ) ) ).

% list.set_map
thf(fact_744_list_Oset__map,axiom,
    ! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,V: list_l4703314356710769291_a_nat] :
      ( ( set_li2392974972034027290_a_nat @ ( map_li6507455427659069316_a_nat @ F @ V ) )
      = ( image_5081948215111134021_a_nat @ F @ ( set_li2392974972034027290_a_nat @ V ) ) ) ).

% list.set_map
thf(fact_745_list_Oset__map,axiom,
    ! [F: produc7017002724195966439_a_nat > list_Sum_sum_a_nat,V: list_P1195027771636113901_a_nat] :
      ( ( set_li2392974972034027290_a_nat @ ( map_Pr5309716638702485410_a_nat @ F @ V ) )
      = ( image_7994831011627017571_a_nat @ F @ ( set_Pr2931239450594890620_a_nat @ V ) ) ) ).

% list.set_map
thf(fact_746_list_Oset__map,axiom,
    ! [F: sum_sum_a_nat > produc7017002724195966439_a_nat,V: list_Sum_sum_a_nat] :
      ( ( set_Pr2931239450594890620_a_nat @ ( map_Su4221482917690969196_a_nat @ F @ V ) )
      = ( image_8014146751851824813_a_nat @ F @ ( set_Sum_sum_a_nat2 @ V ) ) ) ).

% list.set_map
thf(fact_747_list_Oset__map,axiom,
    ! [F: list_Sum_sum_a_nat > produc7017002724195966439_a_nat,V: list_l4703314356710769291_a_nat] :
      ( ( set_Pr2931239450594890620_a_nat @ ( map_li6727107299570610150_a_nat @ F @ V ) )
      = ( image_188849635640366503_a_nat @ F @ ( set_li2392974972034027290_a_nat @ V ) ) ) ).

% list.set_map
thf(fact_748_list_Oset__map,axiom,
    ! [F: produc7017002724195966439_a_nat > produc7017002724195966439_a_nat,V: list_P1195027771636113901_a_nat] :
      ( ( set_Pr2931239450594890620_a_nat @ ( map_Pr3484656804399879172_a_nat @ F @ V ) )
      = ( image_4446780897320315845_a_nat @ F @ ( set_Pr2931239450594890620_a_nat @ V ) ) ) ).

% list.set_map
thf(fact_749_list_Oset__map,axiom,
    ! [F: sum_sum_a_nat > sum_sum_a_nat,V: list_Sum_sum_a_nat] :
      ( ( set_Sum_sum_a_nat2 @ ( map_Su2790769393171190532_a_nat @ F @ V ) )
      = ( image_7142520692256960453_a_nat @ F @ ( set_Sum_sum_a_nat2 @ V ) ) ) ).

% list.set_map
thf(fact_750_prod_Ocollapse,axiom,
    ! [Prod: produc1828647624359046049st_nat] :
      ( ( produc2694037385005941721st_nat @ ( produc1382935764643595205st_nat @ Prod ) @ ( produc5865812112468994567st_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_751_prod_Ocollapse,axiom,
    ! [Prod: produc2799874090934363020_a_nat] :
      ( ( produc5571422593549375550_a_nat @ ( produc7060131065833098066_a_nat @ Prod ) @ ( produc4900202433808968080_a_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_752_prod_Ocollapse,axiom,
    ! [Prod: produc1169793092761538656_a_nat] :
      ( ( produc2770702104727902672_a_nat @ ( produc3757488583116884924_a_nat @ Prod ) @ ( produc7398779253711719678_a_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_753_prod_Ocollapse,axiom,
    ! [Prod: produc3388037434187558270_a_nat] :
      ( ( produc8857357843056227054_a_nat @ ( produc8258299166050244826_a_nat @ Prod ) @ ( produc6825104080635414556_a_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_754_prod_Ocollapse,axiom,
    ! [Prod: produc5001885624171833703_a_nat] :
      ( ( produc7990843422341522135_a_nat @ ( produc2015473822650161347_a_nat @ Prod ) @ ( produc2553995981594484741_a_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_755_prod_Ocollapse,axiom,
    ! [Prod: produc7017002724195966439_a_nat] :
      ( ( produc1212125651291703639_a_nat @ ( produc3940427395981429571_a_nat @ Prod ) @ ( produc8028804885397038213_a_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_756_snd__swap,axiom,
    ! [X: produc5001885624171833703_a_nat] :
      ( ( produc2553995981594484741_a_nat @ ( produc4286111038383333495_a_nat @ X ) )
      = ( produc2015473822650161347_a_nat @ X ) ) ).

% snd_swap
thf(fact_757_snd__swap,axiom,
    ! [X: produc7017002724195966439_a_nat] :
      ( ( produc8028804885397038213_a_nat @ ( produc33134186151372535_a_nat @ X ) )
      = ( produc3940427395981429571_a_nat @ X ) ) ).

% snd_swap
thf(fact_758_old_Oprod_Oexhaust,axiom,
    ! [Y: produc5001885624171833703_a_nat] :
      ~ ! [A3: list_Sum_sum_a_nat,B: list_Sum_sum_a_nat] :
          ( Y
         != ( produc7990843422341522135_a_nat @ A3 @ B ) ) ).

% old.prod.exhaust
thf(fact_759_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1828647624359046049st_nat] :
      ~ ! [A3: list_nat,B: list_nat] :
          ( Y
         != ( produc2694037385005941721st_nat @ A3 @ B ) ) ).

% old.prod.exhaust
thf(fact_760_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2799874090934363020_a_nat] :
      ~ ! [A3: nat,B: produc5001885624171833703_a_nat] :
          ( Y
         != ( produc5571422593549375550_a_nat @ A3 @ B ) ) ).

% old.prod.exhaust
thf(fact_761_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1169793092761538656_a_nat] :
      ~ ! [A3: sum_sum_a_nat > sum_sum_a_nat > $o,B: produc5001885624171833703_a_nat] :
          ( Y
         != ( produc2770702104727902672_a_nat @ A3 @ B ) ) ).

% old.prod.exhaust
thf(fact_762_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3388037434187558270_a_nat] :
      ~ ! [A3: sum_sum_a_nat > sum_sum_a_nat > $o,B: list_Sum_sum_a_nat] :
          ( Y
         != ( produc8857357843056227054_a_nat @ A3 @ B ) ) ).

% old.prod.exhaust
thf(fact_763_surj__pair,axiom,
    ! [P5: produc5001885624171833703_a_nat] :
    ? [X3: list_Sum_sum_a_nat,Y4: list_Sum_sum_a_nat] :
      ( P5
      = ( produc7990843422341522135_a_nat @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_764_surj__pair,axiom,
    ! [P5: produc1828647624359046049st_nat] :
    ? [X3: list_nat,Y4: list_nat] :
      ( P5
      = ( produc2694037385005941721st_nat @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_765_surj__pair,axiom,
    ! [P5: produc2799874090934363020_a_nat] :
    ? [X3: nat,Y4: produc5001885624171833703_a_nat] :
      ( P5
      = ( produc5571422593549375550_a_nat @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_766_surj__pair,axiom,
    ! [P5: produc1169793092761538656_a_nat] :
    ? [X3: sum_sum_a_nat > sum_sum_a_nat > $o,Y4: produc5001885624171833703_a_nat] :
      ( P5
      = ( produc2770702104727902672_a_nat @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_767_surj__pair,axiom,
    ! [P5: produc3388037434187558270_a_nat] :
    ? [X3: sum_sum_a_nat > sum_sum_a_nat > $o,Y4: list_Sum_sum_a_nat] :
      ( P5
      = ( produc8857357843056227054_a_nat @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_768_prod__cases,axiom,
    ! [P: produc5001885624171833703_a_nat > $o,P5: produc5001885624171833703_a_nat] :
      ( ! [A3: list_Sum_sum_a_nat,B: list_Sum_sum_a_nat] : ( P @ ( produc7990843422341522135_a_nat @ A3 @ B ) )
     => ( P @ P5 ) ) ).

% prod_cases
thf(fact_769_prod__cases,axiom,
    ! [P: produc1828647624359046049st_nat > $o,P5: produc1828647624359046049st_nat] :
      ( ! [A3: list_nat,B: list_nat] : ( P @ ( produc2694037385005941721st_nat @ A3 @ B ) )
     => ( P @ P5 ) ) ).

% prod_cases
thf(fact_770_prod__cases,axiom,
    ! [P: produc2799874090934363020_a_nat > $o,P5: produc2799874090934363020_a_nat] :
      ( ! [A3: nat,B: produc5001885624171833703_a_nat] : ( P @ ( produc5571422593549375550_a_nat @ A3 @ B ) )
     => ( P @ P5 ) ) ).

% prod_cases
thf(fact_771_prod__cases,axiom,
    ! [P: produc1169793092761538656_a_nat > $o,P5: produc1169793092761538656_a_nat] :
      ( ! [A3: sum_sum_a_nat > sum_sum_a_nat > $o,B: produc5001885624171833703_a_nat] : ( P @ ( produc2770702104727902672_a_nat @ A3 @ B ) )
     => ( P @ P5 ) ) ).

% prod_cases
thf(fact_772_prod__cases,axiom,
    ! [P: produc3388037434187558270_a_nat > $o,P5: produc3388037434187558270_a_nat] :
      ( ! [A3: sum_sum_a_nat > sum_sum_a_nat > $o,B: list_Sum_sum_a_nat] : ( P @ ( produc8857357843056227054_a_nat @ A3 @ B ) )
     => ( P @ P5 ) ) ).

% prod_cases
thf(fact_773_Pair__inject,axiom,
    ! [A: list_Sum_sum_a_nat,B2: list_Sum_sum_a_nat,A7: list_Sum_sum_a_nat,B7: list_Sum_sum_a_nat] :
      ( ( ( produc7990843422341522135_a_nat @ A @ B2 )
        = ( produc7990843422341522135_a_nat @ A7 @ B7 ) )
     => ~ ( ( A = A7 )
         => ( B2 != B7 ) ) ) ).

% Pair_inject
thf(fact_774_Pair__inject,axiom,
    ! [A: list_nat,B2: list_nat,A7: list_nat,B7: list_nat] :
      ( ( ( produc2694037385005941721st_nat @ A @ B2 )
        = ( produc2694037385005941721st_nat @ A7 @ B7 ) )
     => ~ ( ( A = A7 )
         => ( B2 != B7 ) ) ) ).

% Pair_inject
thf(fact_775_Pair__inject,axiom,
    ! [A: nat,B2: produc5001885624171833703_a_nat,A7: nat,B7: produc5001885624171833703_a_nat] :
      ( ( ( produc5571422593549375550_a_nat @ A @ B2 )
        = ( produc5571422593549375550_a_nat @ A7 @ B7 ) )
     => ~ ( ( A = A7 )
         => ( B2 != B7 ) ) ) ).

% Pair_inject
thf(fact_776_Pair__inject,axiom,
    ! [A: sum_sum_a_nat > sum_sum_a_nat > $o,B2: produc5001885624171833703_a_nat,A7: sum_sum_a_nat > sum_sum_a_nat > $o,B7: produc5001885624171833703_a_nat] :
      ( ( ( produc2770702104727902672_a_nat @ A @ B2 )
        = ( produc2770702104727902672_a_nat @ A7 @ B7 ) )
     => ~ ( ( A = A7 )
         => ( B2 != B7 ) ) ) ).

% Pair_inject
thf(fact_777_Pair__inject,axiom,
    ! [A: sum_sum_a_nat > sum_sum_a_nat > $o,B2: list_Sum_sum_a_nat,A7: sum_sum_a_nat > sum_sum_a_nat > $o,B7: list_Sum_sum_a_nat] :
      ( ( ( produc8857357843056227054_a_nat @ A @ B2 )
        = ( produc8857357843056227054_a_nat @ A7 @ B7 ) )
     => ~ ( ( A = A7 )
         => ( B2 != B7 ) ) ) ).

% Pair_inject
thf(fact_778_prod__cases3,axiom,
    ! [Y: produc2799874090934363020_a_nat] :
      ~ ! [A3: nat,B: list_Sum_sum_a_nat,C: list_Sum_sum_a_nat] :
          ( Y
         != ( produc5571422593549375550_a_nat @ A3 @ ( produc7990843422341522135_a_nat @ B @ C ) ) ) ).

% prod_cases3
thf(fact_779_prod__cases3,axiom,
    ! [Y: produc1169793092761538656_a_nat] :
      ~ ! [A3: sum_sum_a_nat > sum_sum_a_nat > $o,B: list_Sum_sum_a_nat,C: list_Sum_sum_a_nat] :
          ( Y
         != ( produc2770702104727902672_a_nat @ A3 @ ( produc7990843422341522135_a_nat @ B @ C ) ) ) ).

% prod_cases3
thf(fact_780_prod__induct3,axiom,
    ! [P: produc2799874090934363020_a_nat > $o,X: produc2799874090934363020_a_nat] :
      ( ! [A3: nat,B: list_Sum_sum_a_nat,C: list_Sum_sum_a_nat] : ( P @ ( produc5571422593549375550_a_nat @ A3 @ ( produc7990843422341522135_a_nat @ B @ C ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_781_prod__induct3,axiom,
    ! [P: produc1169793092761538656_a_nat > $o,X: produc1169793092761538656_a_nat] :
      ( ! [A3: sum_sum_a_nat > sum_sum_a_nat > $o,B: list_Sum_sum_a_nat,C: list_Sum_sum_a_nat] : ( P @ ( produc2770702104727902672_a_nat @ A3 @ ( produc7990843422341522135_a_nat @ B @ C ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_782_imageI,axiom,
    ! [X: sum_sum_a_nat,A2: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ X @ A2 )
     => ( member_Sum_sum_a_nat @ ( F @ X ) @ ( image_7142520692256960453_a_nat @ F @ A2 ) ) ) ).

% imageI
thf(fact_783_imageI,axiom,
    ! [X: sum_sum_a_nat,A2: set_Sum_sum_a_nat,F: sum_sum_a_nat > list_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ X @ A2 )
     => ( member408289922725080238_a_nat @ ( F @ X ) @ ( image_7077900747146408011_a_nat @ F @ A2 ) ) ) ).

% imageI
thf(fact_784_imageI,axiom,
    ! [X: sum_sum_a_nat,A2: set_Sum_sum_a_nat,F: sum_sum_a_nat > set_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ X @ A2 )
     => ( member8098812455498974984_a_nat @ ( F @ X ) @ ( image_7877458644602423589_a_nat @ F @ A2 ) ) ) ).

% imageI
thf(fact_785_imageI,axiom,
    ! [X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > sum_sum_a_nat] :
      ( ( member408289922725080238_a_nat @ X @ A2 )
     => ( member_Sum_sum_a_nat @ ( F @ X ) @ ( image_6705138127771971519_a_nat @ F @ A2 ) ) ) ).

% imageI
thf(fact_786_imageI,axiom,
    ! [X: set_Sum_sum_a_nat,A2: set_se4904748513628223167_a_nat,F: set_Sum_sum_a_nat > sum_sum_a_nat] :
      ( ( member8098812455498974984_a_nat @ X @ A2 )
     => ( member_Sum_sum_a_nat @ ( F @ X ) @ ( image_5092943885087329893_a_nat @ F @ A2 ) ) ) ).

% imageI
thf(fact_787_imageI,axiom,
    ! [X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
      ( ( member408289922725080238_a_nat @ X @ A2 )
     => ( member408289922725080238_a_nat @ ( F @ X ) @ ( image_5081948215111134021_a_nat @ F @ A2 ) ) ) ).

% imageI
thf(fact_788_imageI,axiom,
    ! [X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_Sum_sum_a_nat] :
      ( ( member408289922725080238_a_nat @ X @ A2 )
     => ( member8098812455498974984_a_nat @ ( F @ X ) @ ( image_3940260845811589407_a_nat @ F @ A2 ) ) ) ).

% imageI
thf(fact_789_imageI,axiom,
    ! [X: set_Sum_sum_a_nat,A2: set_se4904748513628223167_a_nat,F: set_Sum_sum_a_nat > list_Sum_sum_a_nat] :
      ( ( member8098812455498974984_a_nat @ X @ A2 )
     => ( member408289922725080238_a_nat @ ( F @ X ) @ ( image_8255745304366202603_a_nat @ F @ A2 ) ) ) ).

% imageI
thf(fact_790_imageI,axiom,
    ! [X: set_Sum_sum_a_nat,A2: set_se4904748513628223167_a_nat,F: set_Sum_sum_a_nat > set_Sum_sum_a_nat] :
      ( ( member8098812455498974984_a_nat @ X @ A2 )
     => ( member8098812455498974984_a_nat @ ( F @ X ) @ ( image_5599399343138760645_a_nat @ F @ A2 ) ) ) ).

% imageI
thf(fact_791_imageI,axiom,
    ! [X: sum_sum_a_nat,A2: set_Sum_sum_a_nat,F: sum_sum_a_nat > produc7017002724195966439_a_nat] :
      ( ( member_Sum_sum_a_nat @ X @ A2 )
     => ( member3723442691059620112_a_nat @ ( F @ X ) @ ( image_8014146751851824813_a_nat @ F @ A2 ) ) ) ).

% imageI
thf(fact_792_image__iff,axiom,
    ! [Z3: sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ Z3 @ ( image_7142520692256960453_a_nat @ F @ A2 ) )
      = ( ? [X2: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X2 @ A2 )
            & ( Z3
              = ( F @ X2 ) ) ) ) ) ).

% image_iff
thf(fact_793_image__iff,axiom,
    ! [Z3: set_Sum_sum_a_nat,F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
      ( ( member8098812455498974984_a_nat @ Z3 @ ( image_3940260845811589407_a_nat @ F @ A2 ) )
      = ( ? [X2: list_Sum_sum_a_nat] :
            ( ( member408289922725080238_a_nat @ X2 @ A2 )
            & ( Z3
              = ( F @ X2 ) ) ) ) ) ).

% image_iff
thf(fact_794_bex__imageD,axiom,
    ! [F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,P: set_Sum_sum_a_nat > $o] :
      ( ? [X6: set_Sum_sum_a_nat] :
          ( ( member8098812455498974984_a_nat @ X6 @ ( image_3940260845811589407_a_nat @ F @ A2 ) )
          & ( P @ X6 ) )
     => ? [X3: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ X3 @ A2 )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_795_bex__imageD,axiom,
    ! [F: sum_sum_a_nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
      ( ? [X6: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ X6 @ ( image_7142520692256960453_a_nat @ F @ A2 ) )
          & ( P @ X6 ) )
     => ? [X3: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ X3 @ A2 )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_796_image__cong,axiom,
    ! [M2: set_Sum_sum_a_nat,N3: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat] :
      ( ( M2 = N3 )
     => ( ! [X3: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X3 @ N3 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_7142520692256960453_a_nat @ F @ M2 )
          = ( image_7142520692256960453_a_nat @ G @ N3 ) ) ) ) ).

% image_cong
thf(fact_797_image__cong,axiom,
    ! [M2: set_li6526943997496501093_a_nat,N3: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,G: list_Sum_sum_a_nat > set_Sum_sum_a_nat] :
      ( ( M2 = N3 )
     => ( ! [X3: list_Sum_sum_a_nat] :
            ( ( member408289922725080238_a_nat @ X3 @ N3 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_3940260845811589407_a_nat @ F @ M2 )
          = ( image_3940260845811589407_a_nat @ G @ N3 ) ) ) ) ).

% image_cong
thf(fact_798_ball__imageD,axiom,
    ! [F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,P: set_Sum_sum_a_nat > $o] :
      ( ! [X3: set_Sum_sum_a_nat] :
          ( ( member8098812455498974984_a_nat @ X3 @ ( image_3940260845811589407_a_nat @ F @ A2 ) )
         => ( P @ X3 ) )
     => ! [X6: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ X6 @ A2 )
         => ( P @ ( F @ X6 ) ) ) ) ).

% ball_imageD
thf(fact_799_ball__imageD,axiom,
    ! [F: sum_sum_a_nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
      ( ! [X3: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ X3 @ ( image_7142520692256960453_a_nat @ F @ A2 ) )
         => ( P @ X3 ) )
     => ! [X6: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ X6 @ A2 )
         => ( P @ ( F @ X6 ) ) ) ) ).

% ball_imageD
thf(fact_800_rev__image__eqI,axiom,
    ! [X: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ X @ A2 )
     => ( ( B2
          = ( F @ X ) )
       => ( member_Sum_sum_a_nat @ B2 @ ( image_7142520692256960453_a_nat @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_801_rev__image__eqI,axiom,
    ! [X: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: list_Sum_sum_a_nat,F: sum_sum_a_nat > list_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ X @ A2 )
     => ( ( B2
          = ( F @ X ) )
       => ( member408289922725080238_a_nat @ B2 @ ( image_7077900747146408011_a_nat @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_802_rev__image__eqI,axiom,
    ! [X: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,F: sum_sum_a_nat > set_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ X @ A2 )
     => ( ( B2
          = ( F @ X ) )
       => ( member8098812455498974984_a_nat @ B2 @ ( image_7877458644602423589_a_nat @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_803_rev__image__eqI,axiom,
    ! [X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: sum_sum_a_nat,F: list_Sum_sum_a_nat > sum_sum_a_nat] :
      ( ( member408289922725080238_a_nat @ X @ A2 )
     => ( ( B2
          = ( F @ X ) )
       => ( member_Sum_sum_a_nat @ B2 @ ( image_6705138127771971519_a_nat @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_804_rev__image__eqI,axiom,
    ! [X: set_Sum_sum_a_nat,A2: set_se4904748513628223167_a_nat,B2: sum_sum_a_nat,F: set_Sum_sum_a_nat > sum_sum_a_nat] :
      ( ( member8098812455498974984_a_nat @ X @ A2 )
     => ( ( B2
          = ( F @ X ) )
       => ( member_Sum_sum_a_nat @ B2 @ ( image_5092943885087329893_a_nat @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_805_rev__image__eqI,axiom,
    ! [X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
      ( ( member408289922725080238_a_nat @ X @ A2 )
     => ( ( B2
          = ( F @ X ) )
       => ( member408289922725080238_a_nat @ B2 @ ( image_5081948215111134021_a_nat @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_806_rev__image__eqI,axiom,
    ! [X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_Sum_sum_a_nat,F: list_Sum_sum_a_nat > set_Sum_sum_a_nat] :
      ( ( member408289922725080238_a_nat @ X @ A2 )
     => ( ( B2
          = ( F @ X ) )
       => ( member8098812455498974984_a_nat @ B2 @ ( image_3940260845811589407_a_nat @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_807_rev__image__eqI,axiom,
    ! [X: set_Sum_sum_a_nat,A2: set_se4904748513628223167_a_nat,B2: list_Sum_sum_a_nat,F: set_Sum_sum_a_nat > list_Sum_sum_a_nat] :
      ( ( member8098812455498974984_a_nat @ X @ A2 )
     => ( ( B2
          = ( F @ X ) )
       => ( member408289922725080238_a_nat @ B2 @ ( image_8255745304366202603_a_nat @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_808_rev__image__eqI,axiom,
    ! [X: set_Sum_sum_a_nat,A2: set_se4904748513628223167_a_nat,B2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > set_Sum_sum_a_nat] :
      ( ( member8098812455498974984_a_nat @ X @ A2 )
     => ( ( B2
          = ( F @ X ) )
       => ( member8098812455498974984_a_nat @ B2 @ ( image_5599399343138760645_a_nat @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_809_rev__image__eqI,axiom,
    ! [X: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: produc7017002724195966439_a_nat,F: sum_sum_a_nat > produc7017002724195966439_a_nat] :
      ( ( member_Sum_sum_a_nat @ X @ A2 )
     => ( ( B2
          = ( F @ X ) )
       => ( member3723442691059620112_a_nat @ B2 @ ( image_8014146751851824813_a_nat @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_810_sp__equiv__pair_Ocases,axiom,
    ! [X: produc2645141645350940391_a_nat] :
      ~ ! [A3: list_Sum_sum_a_nat,B: list_Sum_sum_a_nat,A8: list_Sum_sum_a_nat,B8: list_Sum_sum_a_nat] :
          ( X
         != ( produc3311116555204928343_a_nat @ ( produc7990843422341522135_a_nat @ A3 @ B ) @ ( produc7990843422341522135_a_nat @ A8 @ B8 ) ) ) ).

% sp_equiv_pair.cases
thf(fact_811_sp__equiv__pair_Ocases,axiom,
    ! [X: produc1342688880896445543st_nat] :
      ~ ! [A3: list_nat,B: list_nat,A8: list_nat,B8: list_nat] :
          ( X
         != ( produc988773521843941591st_nat @ ( produc2694037385005941721st_nat @ A3 @ B ) @ ( produc2694037385005941721st_nat @ A8 @ B8 ) ) ) ).

% sp_equiv_pair.cases
thf(fact_812_sp__equiv__pair_Ocases,axiom,
    ! [X: produc3840784484372613241_a_nat] :
      ~ ! [A3: nat,B: produc5001885624171833703_a_nat,A8: nat,B8: produc5001885624171833703_a_nat] :
          ( X
         != ( produc5196189046490067889_a_nat @ ( produc5571422593549375550_a_nat @ A3 @ B ) @ ( produc5571422593549375550_a_nat @ A8 @ B8 ) ) ) ).

% sp_equiv_pair.cases
thf(fact_813_sp__equiv__pair_Ocases,axiom,
    ! [X: produc6644633505913852071_a_nat] :
      ~ ! [A3: sum_sum_a_nat > sum_sum_a_nat > $o,B: produc5001885624171833703_a_nat,A8: sum_sum_a_nat > sum_sum_a_nat > $o,B8: produc5001885624171833703_a_nat] :
          ( X
         != ( produc7511706038033277847_a_nat @ ( produc2770702104727902672_a_nat @ A3 @ B ) @ ( produc2770702104727902672_a_nat @ A8 @ B8 ) ) ) ).

% sp_equiv_pair.cases
thf(fact_814_sp__equiv__pair_Ocases,axiom,
    ! [X: produc1257712949554669095_a_nat] :
      ~ ! [A3: sum_sum_a_nat > sum_sum_a_nat > $o,B: list_Sum_sum_a_nat,A8: sum_sum_a_nat > sum_sum_a_nat > $o,B8: list_Sum_sum_a_nat] :
          ( X
         != ( produc3127615363426500119_a_nat @ ( produc8857357843056227054_a_nat @ A3 @ B ) @ ( produc8857357843056227054_a_nat @ A8 @ B8 ) ) ) ).

% sp_equiv_pair.cases
thf(fact_815_subset__image__iff,axiom,
    ! [B4: set_se4904748513628223167_a_nat,F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
      ( ( ord_le7974500612278410847_a_nat @ B4 @ ( image_3940260845811589407_a_nat @ F @ A2 ) )
      = ( ? [AA: set_li6526943997496501093_a_nat] :
            ( ( ord_le1147066620699065093_a_nat @ AA @ A2 )
            & ( B4
              = ( image_3940260845811589407_a_nat @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_816_subset__image__iff,axiom,
    ! [B4: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ B4 @ ( image_7142520692256960453_a_nat @ F @ A2 ) )
      = ( ? [AA: set_Sum_sum_a_nat] :
            ( ( ord_le1325389633284124927_a_nat @ AA @ A2 )
            & ( B4
              = ( image_7142520692256960453_a_nat @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_817_image__subset__iff,axiom,
    ! [F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B4: set_se4904748513628223167_a_nat] :
      ( ( ord_le7974500612278410847_a_nat @ ( image_3940260845811589407_a_nat @ F @ A2 ) @ B4 )
      = ( ! [X2: list_Sum_sum_a_nat] :
            ( ( member408289922725080238_a_nat @ X2 @ A2 )
           => ( member8098812455498974984_a_nat @ ( F @ X2 ) @ B4 ) ) ) ) ).

% image_subset_iff
thf(fact_818_image__subset__iff,axiom,
    ! [F: sum_sum_a_nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ ( image_7142520692256960453_a_nat @ F @ A2 ) @ B4 )
      = ( ! [X2: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X2 @ A2 )
           => ( member_Sum_sum_a_nat @ ( F @ X2 ) @ B4 ) ) ) ) ).

% image_subset_iff
thf(fact_819_subset__imageE,axiom,
    ! [B4: set_se4904748513628223167_a_nat,F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
      ( ( ord_le7974500612278410847_a_nat @ B4 @ ( image_3940260845811589407_a_nat @ F @ A2 ) )
     => ~ ! [C4: set_li6526943997496501093_a_nat] :
            ( ( ord_le1147066620699065093_a_nat @ C4 @ A2 )
           => ( B4
             != ( image_3940260845811589407_a_nat @ F @ C4 ) ) ) ) ).

% subset_imageE
thf(fact_820_subset__imageE,axiom,
    ! [B4: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ B4 @ ( image_7142520692256960453_a_nat @ F @ A2 ) )
     => ~ ! [C4: set_Sum_sum_a_nat] :
            ( ( ord_le1325389633284124927_a_nat @ C4 @ A2 )
           => ( B4
             != ( image_7142520692256960453_a_nat @ F @ C4 ) ) ) ) ).

% subset_imageE
thf(fact_821_image__subsetI,axiom,
    ! [A2: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
      ( ! [X3: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ X3 @ A2 )
         => ( member_Sum_sum_a_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_le1325389633284124927_a_nat @ ( image_7142520692256960453_a_nat @ F @ A2 ) @ B4 ) ) ).

% image_subsetI
thf(fact_822_image__subsetI,axiom,
    ! [A2: set_Sum_sum_a_nat,F: sum_sum_a_nat > list_Sum_sum_a_nat,B4: set_li6526943997496501093_a_nat] :
      ( ! [X3: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ X3 @ A2 )
         => ( member408289922725080238_a_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_le1147066620699065093_a_nat @ ( image_7077900747146408011_a_nat @ F @ A2 ) @ B4 ) ) ).

% image_subsetI
thf(fact_823_image__subsetI,axiom,
    ! [A2: set_Sum_sum_a_nat,F: sum_sum_a_nat > set_Sum_sum_a_nat,B4: set_se4904748513628223167_a_nat] :
      ( ! [X3: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ X3 @ A2 )
         => ( member8098812455498974984_a_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_le7974500612278410847_a_nat @ ( image_7877458644602423589_a_nat @ F @ A2 ) @ B4 ) ) ).

% image_subsetI
thf(fact_824_image__subsetI,axiom,
    ! [A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
      ( ! [X3: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ X3 @ A2 )
         => ( member_Sum_sum_a_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_le1325389633284124927_a_nat @ ( image_6705138127771971519_a_nat @ F @ A2 ) @ B4 ) ) ).

% image_subsetI
thf(fact_825_image__subsetI,axiom,
    ! [A2: set_se4904748513628223167_a_nat,F: set_Sum_sum_a_nat > sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
      ( ! [X3: set_Sum_sum_a_nat] :
          ( ( member8098812455498974984_a_nat @ X3 @ A2 )
         => ( member_Sum_sum_a_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_le1325389633284124927_a_nat @ ( image_5092943885087329893_a_nat @ F @ A2 ) @ B4 ) ) ).

% image_subsetI
thf(fact_826_image__subsetI,axiom,
    ! [A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B4: set_li6526943997496501093_a_nat] :
      ( ! [X3: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ X3 @ A2 )
         => ( member408289922725080238_a_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ A2 ) @ B4 ) ) ).

% image_subsetI
thf(fact_827_image__subsetI,axiom,
    ! [A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,B4: set_se4904748513628223167_a_nat] :
      ( ! [X3: list_Sum_sum_a_nat] :
          ( ( member408289922725080238_a_nat @ X3 @ A2 )
         => ( member8098812455498974984_a_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_le7974500612278410847_a_nat @ ( image_3940260845811589407_a_nat @ F @ A2 ) @ B4 ) ) ).

% image_subsetI
thf(fact_828_image__subsetI,axiom,
    ! [A2: set_se4904748513628223167_a_nat,F: set_Sum_sum_a_nat > list_Sum_sum_a_nat,B4: set_li6526943997496501093_a_nat] :
      ( ! [X3: set_Sum_sum_a_nat] :
          ( ( member8098812455498974984_a_nat @ X3 @ A2 )
         => ( member408289922725080238_a_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_le1147066620699065093_a_nat @ ( image_8255745304366202603_a_nat @ F @ A2 ) @ B4 ) ) ).

% image_subsetI
thf(fact_829_image__subsetI,axiom,
    ! [A2: set_se4904748513628223167_a_nat,F: set_Sum_sum_a_nat > set_Sum_sum_a_nat,B4: set_se4904748513628223167_a_nat] :
      ( ! [X3: set_Sum_sum_a_nat] :
          ( ( member8098812455498974984_a_nat @ X3 @ A2 )
         => ( member8098812455498974984_a_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_le7974500612278410847_a_nat @ ( image_5599399343138760645_a_nat @ F @ A2 ) @ B4 ) ) ).

% image_subsetI
thf(fact_830_image__subsetI,axiom,
    ! [A2: set_Sum_sum_a_nat,F: sum_sum_a_nat > produc7017002724195966439_a_nat,B4: set_Pr7343886759072863943_a_nat] :
      ( ! [X3: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ X3 @ A2 )
         => ( member3723442691059620112_a_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_le7501488065512009319_a_nat @ ( image_8014146751851824813_a_nat @ F @ A2 ) @ B4 ) ) ).

% image_subsetI
thf(fact_831_image__mono,axiom,
    ! [A2: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_Sum_sum_a_nat] :
      ( ( ord_le1147066620699065093_a_nat @ A2 @ B4 )
     => ( ord_le7974500612278410847_a_nat @ ( image_3940260845811589407_a_nat @ F @ A2 ) @ ( image_3940260845811589407_a_nat @ F @ B4 ) ) ) ).

% image_mono
thf(fact_832_image__mono,axiom,
    ! [A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ A2 @ B4 )
     => ( ord_le1325389633284124927_a_nat @ ( image_7142520692256960453_a_nat @ F @ A2 ) @ ( image_7142520692256960453_a_nat @ F @ B4 ) ) ) ).

% image_mono
thf(fact_833_image__Un,axiom,
    ! [F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
      ( ( image_3940260845811589407_a_nat @ F @ ( sup_su4083067149120280889_a_nat @ A2 @ B4 ) )
      = ( sup_su2291686591051470483_a_nat @ ( image_3940260845811589407_a_nat @ F @ A2 ) @ ( image_3940260845811589407_a_nat @ F @ B4 ) ) ) ).

% image_Un
thf(fact_834_image__Un,axiom,
    ! [F: sum_sum_a_nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
      ( ( image_7142520692256960453_a_nat @ F @ ( sup_su6804446743777130803_a_nat @ A2 @ B4 ) )
      = ( sup_su6804446743777130803_a_nat @ ( image_7142520692256960453_a_nat @ F @ A2 ) @ ( image_7142520692256960453_a_nat @ F @ B4 ) ) ) ).

% image_Un
thf(fact_835_fst__eqD,axiom,
    ! [X: list_nat,Y: list_nat,A: list_nat] :
      ( ( ( produc1382935764643595205st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_836_fst__eqD,axiom,
    ! [X: nat,Y: produc5001885624171833703_a_nat,A: nat] :
      ( ( ( produc7060131065833098066_a_nat @ ( produc5571422593549375550_a_nat @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_837_fst__eqD,axiom,
    ! [X: sum_sum_a_nat > sum_sum_a_nat > $o,Y: produc5001885624171833703_a_nat,A: sum_sum_a_nat > sum_sum_a_nat > $o] :
      ( ( ( produc3757488583116884924_a_nat @ ( produc2770702104727902672_a_nat @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_838_fst__eqD,axiom,
    ! [X: sum_sum_a_nat > sum_sum_a_nat > $o,Y: list_Sum_sum_a_nat,A: sum_sum_a_nat > sum_sum_a_nat > $o] :
      ( ( ( produc8258299166050244826_a_nat @ ( produc8857357843056227054_a_nat @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_839_fst__eqD,axiom,
    ! [X: sum_sum_a_nat,Y: sum_sum_a_nat,A: sum_sum_a_nat] :
      ( ( ( produc3940427395981429571_a_nat @ ( produc1212125651291703639_a_nat @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_840_fst__eqD,axiom,
    ! [X: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat,A: list_Sum_sum_a_nat] :
      ( ( ( produc2015473822650161347_a_nat @ ( produc7990843422341522135_a_nat @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_841_fst__conv,axiom,
    ! [X1: list_nat,X22: list_nat] :
      ( ( produc1382935764643595205st_nat @ ( produc2694037385005941721st_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_842_fst__conv,axiom,
    ! [X1: nat,X22: produc5001885624171833703_a_nat] :
      ( ( produc7060131065833098066_a_nat @ ( produc5571422593549375550_a_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_843_fst__conv,axiom,
    ! [X1: sum_sum_a_nat > sum_sum_a_nat > $o,X22: produc5001885624171833703_a_nat] :
      ( ( produc3757488583116884924_a_nat @ ( produc2770702104727902672_a_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_844_fst__conv,axiom,
    ! [X1: sum_sum_a_nat > sum_sum_a_nat > $o,X22: list_Sum_sum_a_nat] :
      ( ( produc8258299166050244826_a_nat @ ( produc8857357843056227054_a_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_845_fst__conv,axiom,
    ! [X1: sum_sum_a_nat,X22: sum_sum_a_nat] :
      ( ( produc3940427395981429571_a_nat @ ( produc1212125651291703639_a_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_846_fst__conv,axiom,
    ! [X1: list_Sum_sum_a_nat,X22: list_Sum_sum_a_nat] :
      ( ( produc2015473822650161347_a_nat @ ( produc7990843422341522135_a_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_847_snd__eqD,axiom,
    ! [X: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat,A: list_Sum_sum_a_nat] :
      ( ( ( produc2553995981594484741_a_nat @ ( produc7990843422341522135_a_nat @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_848_snd__eqD,axiom,
    ! [X: list_nat,Y: list_nat,A: list_nat] :
      ( ( ( produc5865812112468994567st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_849_snd__eqD,axiom,
    ! [X: nat,Y: produc5001885624171833703_a_nat,A: produc5001885624171833703_a_nat] :
      ( ( ( produc4900202433808968080_a_nat @ ( produc5571422593549375550_a_nat @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_850_snd__eqD,axiom,
    ! [X: sum_sum_a_nat > sum_sum_a_nat > $o,Y: produc5001885624171833703_a_nat,A: produc5001885624171833703_a_nat] :
      ( ( ( produc7398779253711719678_a_nat @ ( produc2770702104727902672_a_nat @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_851_snd__eqD,axiom,
    ! [X: sum_sum_a_nat > sum_sum_a_nat > $o,Y: list_Sum_sum_a_nat,A: list_Sum_sum_a_nat] :
      ( ( ( produc6825104080635414556_a_nat @ ( produc8857357843056227054_a_nat @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_852_snd__eqD,axiom,
    ! [X: sum_sum_a_nat,Y: sum_sum_a_nat,A: sum_sum_a_nat] :
      ( ( ( produc8028804885397038213_a_nat @ ( produc1212125651291703639_a_nat @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_853_snd__conv,axiom,
    ! [X1: list_Sum_sum_a_nat,X22: list_Sum_sum_a_nat] :
      ( ( produc2553995981594484741_a_nat @ ( produc7990843422341522135_a_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_854_snd__conv,axiom,
    ! [X1: list_nat,X22: list_nat] :
      ( ( produc5865812112468994567st_nat @ ( produc2694037385005941721st_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_855_snd__conv,axiom,
    ! [X1: nat,X22: produc5001885624171833703_a_nat] :
      ( ( produc4900202433808968080_a_nat @ ( produc5571422593549375550_a_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_856_snd__conv,axiom,
    ! [X1: sum_sum_a_nat > sum_sum_a_nat > $o,X22: produc5001885624171833703_a_nat] :
      ( ( produc7398779253711719678_a_nat @ ( produc2770702104727902672_a_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_857_snd__conv,axiom,
    ! [X1: sum_sum_a_nat > sum_sum_a_nat > $o,X22: list_Sum_sum_a_nat] :
      ( ( produc6825104080635414556_a_nat @ ( produc8857357843056227054_a_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_858_snd__conv,axiom,
    ! [X1: sum_sum_a_nat,X22: sum_sum_a_nat] :
      ( ( produc8028804885397038213_a_nat @ ( produc1212125651291703639_a_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_859_prod_Oswap__def,axiom,
    ( produc7439058410299273593st_nat
    = ( ^ [P6: produc1828647624359046049st_nat] : ( produc2694037385005941721st_nat @ ( produc5865812112468994567st_nat @ P6 ) @ ( produc1382935764643595205st_nat @ P6 ) ) ) ) ).

% prod.swap_def
thf(fact_860_prod_Oswap__def,axiom,
    ( produc4241125690352732224at_nat
    = ( ^ [P6: produc4503689834541559270at_nat] : ( produc5571422593549375550_a_nat @ ( produc3891679309196320562at_nat @ P6 ) @ ( produc6051607941220450548at_nat @ P6 ) ) ) ) ).

% prod.swap_def
thf(fact_861_prod_Oswap__def,axiom,
    ( produc4412250191365866814_nat_o
    = ( ^ [P6: produc2570910961013241518_nat_o] : ( produc2770702104727902672_a_nat @ ( produc1990930809317169868_nat_o @ P6 ) @ ( produc7573012175577110922_nat_o @ P6 ) ) ) ) ).

% prod.swap_def
thf(fact_862_prod_Oswap__def,axiom,
    ( produc6889179298173170336_nat_o
    = ( ^ [P6: produc6714841562171439888_nat_o] : ( produc8857357843056227054_a_nat @ ( produc2100989599937808942_nat_o @ P6 ) @ ( produc3534184685352639212_nat_o @ P6 ) ) ) ) ).

% prod.swap_def
thf(fact_863_prod_Oswap__def,axiom,
    ( produc4286111038383333495_a_nat
    = ( ^ [P6: produc5001885624171833703_a_nat] : ( produc7990843422341522135_a_nat @ ( produc2553995981594484741_a_nat @ P6 ) @ ( produc2015473822650161347_a_nat @ P6 ) ) ) ) ).

% prod.swap_def
thf(fact_864_prod_Oswap__def,axiom,
    ( produc33134186151372535_a_nat
    = ( ^ [P6: produc7017002724195966439_a_nat] : ( produc1212125651291703639_a_nat @ ( produc8028804885397038213_a_nat @ P6 ) @ ( produc3940427395981429571_a_nat @ P6 ) ) ) ) ).

% prod.swap_def
thf(fact_865_set__zip__comm,axiom,
    ! [X: sum_sum_a_nat,Y: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X @ Y ) @ ( set_Pr2931239450594890620_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) ) )
     => ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ Y @ X ) @ ( set_Pr2931239450594890620_a_nat @ ( zip_Su7355543910597222519_a_nat @ Ys @ Xs ) ) ) ) ).

% set_zip_comm
thf(fact_866_set__zip__comm,axiom,
    ! [X: produc5001885624171833703_a_nat,Y: nat,Xs: list_P6608106274694273645_a_nat,Ys: list_nat] :
      ( ( member3374472070399495293at_nat @ ( produc4562899468936728032at_nat @ X @ Y ) @ ( set_Pr9168894918337029905at_nat @ ( zip_Pr2845026493572296384at_nat @ Xs @ Ys ) ) )
     => ( member1670656326792299043_a_nat @ ( produc5571422593549375550_a_nat @ Y @ X ) @ ( set_Pr7465079174729833655_a_nat @ ( zip_na3853549618184943902_a_nat @ Ys @ Xs ) ) ) ) ).

% set_zip_comm
thf(fact_867_set__zip__comm,axiom,
    ! [X: produc5001885624171833703_a_nat,Y: sum_sum_a_nat > sum_sum_a_nat > $o,Xs: list_P6608106274694273645_a_nat,Ys: list_S9131871693400613940_nat_o] :
      ( ( member8134848866326521303_nat_o @ ( produc6586225697188128670_nat_o @ X @ Y ) @ ( set_Pr5053396631317434435_nat_o @ ( zip_Pr1675919120921984702_nat_o @ Xs @ Ys ) ) )
     => ( member6733730998074818441_a_nat @ ( produc2770702104727902672_a_nat @ Y @ X ) @ ( set_Pr3652278763065731573_a_nat @ ( zip_Su7083767565316534512_a_nat @ Ys @ Xs ) ) ) ) ).

% set_zip_comm
thf(fact_868_set__zip__comm,axiom,
    ! [X: list_Sum_sum_a_nat,Y: sum_sum_a_nat > sum_sum_a_nat > $o,Xs: list_l4703314356710769291_a_nat,Ys: list_S9131871693400613940_nat_o] :
      ( ( member6500242335977470265_nat_o @ ( produc4133243362358621440_nat_o @ X @ Y ) @ ( set_Pr5935279334226122661_nat_o @ ( zip_li6912039999028636704_nat_o @ Xs @ Ys ) ) )
     => ( member3173438207993588647_a_nat @ ( produc8857357843056227054_a_nat @ Y @ X ) @ ( set_Pr2608475206242241043_a_nat @ ( zip_Su2412782442871466510_a_nat @ Ys @ Xs ) ) ) ) ).

% set_zip_comm
thf(fact_869_set__zip__comm,axiom,
    ! [X: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat] :
      ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ X @ Y ) @ ( set_Pr808418073638175228_a_nat @ ( zip_li7023018480688994807_a_nat @ Xs @ Ys ) ) )
     => ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Y @ X ) @ ( set_Pr808418073638175228_a_nat @ ( zip_li7023018480688994807_a_nat @ Ys @ Xs ) ) ) ) ).

% set_zip_comm
thf(fact_870_set__zip__comm,axiom,
    ! [X: list_nat,Y: list_nat,Xs: list_list_nat,Ys: list_list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( set_Pr3842133991353686454st_nat @ ( zip_li7157463729305086713st_nat @ Xs @ Ys ) ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y @ X ) @ ( set_Pr3842133991353686454st_nat @ ( zip_li7157463729305086713st_nat @ Ys @ Xs ) ) ) ) ).

% set_zip_comm
thf(fact_871_set__zip__comm,axiom,
    ! [X: nat,Y: produc5001885624171833703_a_nat,Xs: list_nat,Ys: list_P6608106274694273645_a_nat] :
      ( ( member1670656326792299043_a_nat @ ( produc5571422593549375550_a_nat @ X @ Y ) @ ( set_Pr7465079174729833655_a_nat @ ( zip_na3853549618184943902_a_nat @ Xs @ Ys ) ) )
     => ( member3374472070399495293at_nat @ ( produc4562899468936728032at_nat @ Y @ X ) @ ( set_Pr9168894918337029905at_nat @ ( zip_Pr2845026493572296384at_nat @ Ys @ Xs ) ) ) ) ).

% set_zip_comm
thf(fact_872_set__zip__comm,axiom,
    ! [X: sum_sum_a_nat > sum_sum_a_nat > $o,Y: produc5001885624171833703_a_nat,Xs: list_S9131871693400613940_nat_o,Ys: list_P6608106274694273645_a_nat] :
      ( ( member6733730998074818441_a_nat @ ( produc2770702104727902672_a_nat @ X @ Y ) @ ( set_Pr3652278763065731573_a_nat @ ( zip_Su7083767565316534512_a_nat @ Xs @ Ys ) ) )
     => ( member8134848866326521303_nat_o @ ( produc6586225697188128670_nat_o @ Y @ X ) @ ( set_Pr5053396631317434435_nat_o @ ( zip_Pr1675919120921984702_nat_o @ Ys @ Xs ) ) ) ) ).

% set_zip_comm
thf(fact_873_set__zip__comm,axiom,
    ! [X: sum_sum_a_nat > sum_sum_a_nat > $o,Y: list_Sum_sum_a_nat,Xs: list_S9131871693400613940_nat_o,Ys: list_l4703314356710769291_a_nat] :
      ( ( member3173438207993588647_a_nat @ ( produc8857357843056227054_a_nat @ X @ Y ) @ ( set_Pr2608475206242241043_a_nat @ ( zip_Su2412782442871466510_a_nat @ Xs @ Ys ) ) )
     => ( member6500242335977470265_nat_o @ ( produc4133243362358621440_nat_o @ Y @ X ) @ ( set_Pr5935279334226122661_nat_o @ ( zip_li6912039999028636704_nat_o @ Ys @ Xs ) ) ) ) ).

% set_zip_comm
thf(fact_874_image__set,axiom,
    ! [F: sum_sum_a_nat > list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( image_7077900747146408011_a_nat @ F @ ( set_Sum_sum_a_nat2 @ Xs ) )
      = ( set_li2392974972034027290_a_nat @ ( map_Su3479376915845212170_a_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_875_image__set,axiom,
    ! [F: sum_sum_a_nat > produc7017002724195966439_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( image_8014146751851824813_a_nat @ F @ ( set_Sum_sum_a_nat2 @ Xs ) )
      = ( set_Pr2931239450594890620_a_nat @ ( map_Su4221482917690969196_a_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_876_image__set,axiom,
    ! [F: list_Sum_sum_a_nat > set_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat] :
      ( ( image_3940260845811589407_a_nat @ F @ ( set_li2392974972034027290_a_nat @ Xs ) )
      = ( set_se6249060928767827828_a_nat @ ( map_li9079814933684165854_a_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_877_image__set,axiom,
    ! [F: list_Sum_sum_a_nat > sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat] :
      ( ( image_6705138127771971519_a_nat @ F @ ( set_li2392974972034027290_a_nat @ Xs ) )
      = ( set_Sum_sum_a_nat2 @ ( map_li3106614296470775678_a_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_878_image__set,axiom,
    ! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat] :
      ( ( image_5081948215111134021_a_nat @ F @ ( set_li2392974972034027290_a_nat @ Xs ) )
      = ( set_li2392974972034027290_a_nat @ ( map_li6507455427659069316_a_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_879_image__set,axiom,
    ! [F: list_Sum_sum_a_nat > produc7017002724195966439_a_nat,Xs: list_l4703314356710769291_a_nat] :
      ( ( image_188849635640366503_a_nat @ F @ ( set_li2392974972034027290_a_nat @ Xs ) )
      = ( set_Pr2931239450594890620_a_nat @ ( map_li6727107299570610150_a_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_880_image__set,axiom,
    ! [F: produc7017002724195966439_a_nat > sum_sum_a_nat,Xs: list_P1195027771636113901_a_nat] :
      ( ( image_7319751417386094813_a_nat @ F @ ( set_Pr2931239450594890620_a_nat @ Xs ) )
      = ( set_Sum_sum_a_nat2 @ ( map_Pr3527087583225239196_a_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_881_image__set,axiom,
    ! [F: produc7017002724195966439_a_nat > list_Sum_sum_a_nat,Xs: list_P1195027771636113901_a_nat] :
      ( ( image_7994831011627017571_a_nat @ F @ ( set_Pr2931239450594890620_a_nat @ Xs ) )
      = ( set_li2392974972034027290_a_nat @ ( map_Pr5309716638702485410_a_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_882_image__set,axiom,
    ! [F: produc7017002724195966439_a_nat > produc7017002724195966439_a_nat,Xs: list_P1195027771636113901_a_nat] :
      ( ( image_4446780897320315845_a_nat @ F @ ( set_Pr2931239450594890620_a_nat @ Xs ) )
      = ( set_Pr2931239450594890620_a_nat @ ( map_Pr3484656804399879172_a_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_883_image__set,axiom,
    ! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( image_7142520692256960453_a_nat @ F @ ( set_Sum_sum_a_nat2 @ Xs ) )
      = ( set_Sum_sum_a_nat2 @ ( map_Su2790769393171190532_a_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_884_surjective__pairing,axiom,
    ! [T: produc1828647624359046049st_nat] :
      ( T
      = ( produc2694037385005941721st_nat @ ( produc1382935764643595205st_nat @ T ) @ ( produc5865812112468994567st_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_885_surjective__pairing,axiom,
    ! [T: produc2799874090934363020_a_nat] :
      ( T
      = ( produc5571422593549375550_a_nat @ ( produc7060131065833098066_a_nat @ T ) @ ( produc4900202433808968080_a_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_886_surjective__pairing,axiom,
    ! [T: produc1169793092761538656_a_nat] :
      ( T
      = ( produc2770702104727902672_a_nat @ ( produc3757488583116884924_a_nat @ T ) @ ( produc7398779253711719678_a_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_887_surjective__pairing,axiom,
    ! [T: produc3388037434187558270_a_nat] :
      ( T
      = ( produc8857357843056227054_a_nat @ ( produc8258299166050244826_a_nat @ T ) @ ( produc6825104080635414556_a_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_888_surjective__pairing,axiom,
    ! [T: produc5001885624171833703_a_nat] :
      ( T
      = ( produc7990843422341522135_a_nat @ ( produc2015473822650161347_a_nat @ T ) @ ( produc2553995981594484741_a_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_889_surjective__pairing,axiom,
    ! [T: produc7017002724195966439_a_nat] :
      ( T
      = ( produc1212125651291703639_a_nat @ ( produc3940427395981429571_a_nat @ T ) @ ( produc8028804885397038213_a_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_890_conjI__realizer,axiom,
    ! [P: list_nat > $o,P5: list_nat,Q: list_nat > $o,Q2: list_nat] :
      ( ( P @ P5 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc1382935764643595205st_nat @ ( produc2694037385005941721st_nat @ P5 @ Q2 ) ) )
          & ( Q @ ( produc5865812112468994567st_nat @ ( produc2694037385005941721st_nat @ P5 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_891_conjI__realizer,axiom,
    ! [P: nat > $o,P5: nat,Q: produc5001885624171833703_a_nat > $o,Q2: produc5001885624171833703_a_nat] :
      ( ( P @ P5 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc7060131065833098066_a_nat @ ( produc5571422593549375550_a_nat @ P5 @ Q2 ) ) )
          & ( Q @ ( produc4900202433808968080_a_nat @ ( produc5571422593549375550_a_nat @ P5 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_892_conjI__realizer,axiom,
    ! [P: ( sum_sum_a_nat > sum_sum_a_nat > $o ) > $o,P5: sum_sum_a_nat > sum_sum_a_nat > $o,Q: produc5001885624171833703_a_nat > $o,Q2: produc5001885624171833703_a_nat] :
      ( ( P @ P5 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc3757488583116884924_a_nat @ ( produc2770702104727902672_a_nat @ P5 @ Q2 ) ) )
          & ( Q @ ( produc7398779253711719678_a_nat @ ( produc2770702104727902672_a_nat @ P5 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_893_conjI__realizer,axiom,
    ! [P: ( sum_sum_a_nat > sum_sum_a_nat > $o ) > $o,P5: sum_sum_a_nat > sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o,Q2: list_Sum_sum_a_nat] :
      ( ( P @ P5 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc8258299166050244826_a_nat @ ( produc8857357843056227054_a_nat @ P5 @ Q2 ) ) )
          & ( Q @ ( produc6825104080635414556_a_nat @ ( produc8857357843056227054_a_nat @ P5 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_894_conjI__realizer,axiom,
    ! [P: list_Sum_sum_a_nat > $o,P5: list_Sum_sum_a_nat,Q: list_Sum_sum_a_nat > $o,Q2: list_Sum_sum_a_nat] :
      ( ( P @ P5 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc2015473822650161347_a_nat @ ( produc7990843422341522135_a_nat @ P5 @ Q2 ) ) )
          & ( Q @ ( produc2553995981594484741_a_nat @ ( produc7990843422341522135_a_nat @ P5 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_895_conjI__realizer,axiom,
    ! [P: sum_sum_a_nat > $o,P5: sum_sum_a_nat,Q: sum_sum_a_nat > $o,Q2: sum_sum_a_nat] :
      ( ( P @ P5 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc3940427395981429571_a_nat @ ( produc1212125651291703639_a_nat @ P5 @ Q2 ) ) )
          & ( Q @ ( produc8028804885397038213_a_nat @ ( produc1212125651291703639_a_nat @ P5 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_896_exI__realizer,axiom,
    ! [P: list_nat > list_nat > $o,Y: list_nat,X: list_nat] :
      ( ( P @ Y @ X )
     => ( P @ ( produc5865812112468994567st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) ) @ ( produc1382935764643595205st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_897_exI__realizer,axiom,
    ! [P: produc5001885624171833703_a_nat > nat > $o,Y: produc5001885624171833703_a_nat,X: nat] :
      ( ( P @ Y @ X )
     => ( P @ ( produc4900202433808968080_a_nat @ ( produc5571422593549375550_a_nat @ X @ Y ) ) @ ( produc7060131065833098066_a_nat @ ( produc5571422593549375550_a_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_898_exI__realizer,axiom,
    ! [P: produc5001885624171833703_a_nat > ( sum_sum_a_nat > sum_sum_a_nat > $o ) > $o,Y: produc5001885624171833703_a_nat,X: sum_sum_a_nat > sum_sum_a_nat > $o] :
      ( ( P @ Y @ X )
     => ( P @ ( produc7398779253711719678_a_nat @ ( produc2770702104727902672_a_nat @ X @ Y ) ) @ ( produc3757488583116884924_a_nat @ ( produc2770702104727902672_a_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_899_exI__realizer,axiom,
    ! [P: list_Sum_sum_a_nat > ( sum_sum_a_nat > sum_sum_a_nat > $o ) > $o,Y: list_Sum_sum_a_nat,X: sum_sum_a_nat > sum_sum_a_nat > $o] :
      ( ( P @ Y @ X )
     => ( P @ ( produc6825104080635414556_a_nat @ ( produc8857357843056227054_a_nat @ X @ Y ) ) @ ( produc8258299166050244826_a_nat @ ( produc8857357843056227054_a_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_900_exI__realizer,axiom,
    ! [P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,Y: list_Sum_sum_a_nat,X: list_Sum_sum_a_nat] :
      ( ( P @ Y @ X )
     => ( P @ ( produc2553995981594484741_a_nat @ ( produc7990843422341522135_a_nat @ X @ Y ) ) @ ( produc2015473822650161347_a_nat @ ( produc7990843422341522135_a_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_901_exI__realizer,axiom,
    ! [P: sum_sum_a_nat > sum_sum_a_nat > $o,Y: sum_sum_a_nat,X: sum_sum_a_nat] :
      ( ( P @ Y @ X )
     => ( P @ ( produc8028804885397038213_a_nat @ ( produc1212125651291703639_a_nat @ X @ Y ) ) @ ( produc3940427395981429571_a_nat @ ( produc1212125651291703639_a_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_902_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1828647624359046049st_nat] :
      ( Prod
      = ( produc2694037385005941721st_nat @ ( produc1382935764643595205st_nat @ Prod ) @ ( produc5865812112468994567st_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_903_prod_Oexhaust__sel,axiom,
    ! [Prod: produc2799874090934363020_a_nat] :
      ( Prod
      = ( produc5571422593549375550_a_nat @ ( produc7060131065833098066_a_nat @ Prod ) @ ( produc4900202433808968080_a_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_904_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1169793092761538656_a_nat] :
      ( Prod
      = ( produc2770702104727902672_a_nat @ ( produc3757488583116884924_a_nat @ Prod ) @ ( produc7398779253711719678_a_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_905_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3388037434187558270_a_nat] :
      ( Prod
      = ( produc8857357843056227054_a_nat @ ( produc8258299166050244826_a_nat @ Prod ) @ ( produc6825104080635414556_a_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_906_prod_Oexhaust__sel,axiom,
    ! [Prod: produc5001885624171833703_a_nat] :
      ( Prod
      = ( produc7990843422341522135_a_nat @ ( produc2015473822650161347_a_nat @ Prod ) @ ( produc2553995981594484741_a_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_907_prod_Oexhaust__sel,axiom,
    ! [Prod: produc7017002724195966439_a_nat] :
      ( Prod
      = ( produc1212125651291703639_a_nat @ ( produc3940427395981429571_a_nat @ Prod ) @ ( produc8028804885397038213_a_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_908_set__zip__rightD,axiom,
    ! [X: sum_sum_a_nat,Y: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X @ Y ) @ ( set_Pr2931239450594890620_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) ) )
     => ( member_Sum_sum_a_nat @ Y @ ( set_Sum_sum_a_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_909_set__zip__rightD,axiom,
    ! [X: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat] :
      ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ X @ Y ) @ ( set_Pr808418073638175228_a_nat @ ( zip_li7023018480688994807_a_nat @ Xs @ Ys ) ) )
     => ( member408289922725080238_a_nat @ Y @ ( set_li2392974972034027290_a_nat @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_910_set__zip__rightD,axiom,
    ! [X: list_nat,Y: list_nat,Xs: list_list_nat,Ys: list_list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( set_Pr3842133991353686454st_nat @ ( zip_li7157463729305086713st_nat @ Xs @ Ys ) ) )
     => ( member_list_nat @ Y @ ( set_list_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_911_set__zip__rightD,axiom,
    ! [X: nat,Y: produc5001885624171833703_a_nat,Xs: list_nat,Ys: list_P6608106274694273645_a_nat] :
      ( ( member1670656326792299043_a_nat @ ( produc5571422593549375550_a_nat @ X @ Y ) @ ( set_Pr7465079174729833655_a_nat @ ( zip_na3853549618184943902_a_nat @ Xs @ Ys ) ) )
     => ( member7457213283480048528_a_nat @ Y @ ( set_Pr808418073638175228_a_nat @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_912_set__zip__rightD,axiom,
    ! [X: sum_sum_a_nat > sum_sum_a_nat > $o,Y: produc5001885624171833703_a_nat,Xs: list_S9131871693400613940_nat_o,Ys: list_P6608106274694273645_a_nat] :
      ( ( member6733730998074818441_a_nat @ ( produc2770702104727902672_a_nat @ X @ Y ) @ ( set_Pr3652278763065731573_a_nat @ ( zip_Su7083767565316534512_a_nat @ Xs @ Ys ) ) )
     => ( member7457213283480048528_a_nat @ Y @ ( set_Pr808418073638175228_a_nat @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_913_set__zip__rightD,axiom,
    ! [X: sum_sum_a_nat > sum_sum_a_nat > $o,Y: list_Sum_sum_a_nat,Xs: list_S9131871693400613940_nat_o,Ys: list_l4703314356710769291_a_nat] :
      ( ( member3173438207993588647_a_nat @ ( produc8857357843056227054_a_nat @ X @ Y ) @ ( set_Pr2608475206242241043_a_nat @ ( zip_Su2412782442871466510_a_nat @ Xs @ Ys ) ) )
     => ( member408289922725080238_a_nat @ Y @ ( set_li2392974972034027290_a_nat @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_914_set__zip__leftD,axiom,
    ! [X: sum_sum_a_nat,Y: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X @ Y ) @ ( set_Pr2931239450594890620_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) ) )
     => ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_915_set__zip__leftD,axiom,
    ! [X: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat] :
      ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ X @ Y ) @ ( set_Pr808418073638175228_a_nat @ ( zip_li7023018480688994807_a_nat @ Xs @ Ys ) ) )
     => ( member408289922725080238_a_nat @ X @ ( set_li2392974972034027290_a_nat @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_916_set__zip__leftD,axiom,
    ! [X: list_nat,Y: list_nat,Xs: list_list_nat,Ys: list_list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( set_Pr3842133991353686454st_nat @ ( zip_li7157463729305086713st_nat @ Xs @ Ys ) ) )
     => ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_917_set__zip__leftD,axiom,
    ! [X: nat,Y: produc5001885624171833703_a_nat,Xs: list_nat,Ys: list_P6608106274694273645_a_nat] :
      ( ( member1670656326792299043_a_nat @ ( produc5571422593549375550_a_nat @ X @ Y ) @ ( set_Pr7465079174729833655_a_nat @ ( zip_na3853549618184943902_a_nat @ Xs @ Ys ) ) )
     => ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_918_set__zip__leftD,axiom,
    ! [X: sum_sum_a_nat > sum_sum_a_nat > $o,Y: produc5001885624171833703_a_nat,Xs: list_S9131871693400613940_nat_o,Ys: list_P6608106274694273645_a_nat] :
      ( ( member6733730998074818441_a_nat @ ( produc2770702104727902672_a_nat @ X @ Y ) @ ( set_Pr3652278763065731573_a_nat @ ( zip_Su7083767565316534512_a_nat @ Xs @ Ys ) ) )
     => ( member8178426116768754391_nat_o @ X @ ( set_Su8658726457838599491_nat_o @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_919_set__zip__leftD,axiom,
    ! [X: sum_sum_a_nat > sum_sum_a_nat > $o,Y: list_Sum_sum_a_nat,Xs: list_S9131871693400613940_nat_o,Ys: list_l4703314356710769291_a_nat] :
      ( ( member3173438207993588647_a_nat @ ( produc8857357843056227054_a_nat @ X @ Y ) @ ( set_Pr2608475206242241043_a_nat @ ( zip_Su2412782442871466510_a_nat @ Xs @ Ys ) ) )
     => ( member8178426116768754391_nat_o @ X @ ( set_Su8658726457838599491_nat_o @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_920_in__set__zipE,axiom,
    ! [X: list_nat,Y: list_nat,Xs: list_list_nat,Ys: list_list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( set_Pr3842133991353686454st_nat @ ( zip_li7157463729305086713st_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
         => ~ ( member_list_nat @ Y @ ( set_list_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_921_in__set__zipE,axiom,
    ! [X: nat,Y: produc5001885624171833703_a_nat,Xs: list_nat,Ys: list_P6608106274694273645_a_nat] :
      ( ( member1670656326792299043_a_nat @ ( produc5571422593549375550_a_nat @ X @ Y ) @ ( set_Pr7465079174729833655_a_nat @ ( zip_na3853549618184943902_a_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
         => ~ ( member7457213283480048528_a_nat @ Y @ ( set_Pr808418073638175228_a_nat @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_922_in__set__zipE,axiom,
    ! [X: sum_sum_a_nat > sum_sum_a_nat > $o,Y: produc5001885624171833703_a_nat,Xs: list_S9131871693400613940_nat_o,Ys: list_P6608106274694273645_a_nat] :
      ( ( member6733730998074818441_a_nat @ ( produc2770702104727902672_a_nat @ X @ Y ) @ ( set_Pr3652278763065731573_a_nat @ ( zip_Su7083767565316534512_a_nat @ Xs @ Ys ) ) )
     => ~ ( ( member8178426116768754391_nat_o @ X @ ( set_Su8658726457838599491_nat_o @ Xs ) )
         => ~ ( member7457213283480048528_a_nat @ Y @ ( set_Pr808418073638175228_a_nat @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_923_in__set__zipE,axiom,
    ! [X: sum_sum_a_nat > sum_sum_a_nat > $o,Y: list_Sum_sum_a_nat,Xs: list_S9131871693400613940_nat_o,Ys: list_l4703314356710769291_a_nat] :
      ( ( member3173438207993588647_a_nat @ ( produc8857357843056227054_a_nat @ X @ Y ) @ ( set_Pr2608475206242241043_a_nat @ ( zip_Su2412782442871466510_a_nat @ Xs @ Ys ) ) )
     => ~ ( ( member8178426116768754391_nat_o @ X @ ( set_Su8658726457838599491_nat_o @ Xs ) )
         => ~ ( member408289922725080238_a_nat @ Y @ ( set_li2392974972034027290_a_nat @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_924_in__set__zipE,axiom,
    ! [X: sum_sum_a_nat,Y: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X @ Y ) @ ( set_Pr2931239450594890620_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
         => ~ ( member_Sum_sum_a_nat @ Y @ ( set_Sum_sum_a_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_925_zip__same,axiom,
    ! [A: sum_sum_a_nat,B2: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ A @ B2 ) @ ( set_Pr2931239450594890620_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Xs ) ) )
      = ( ( member_Sum_sum_a_nat @ A @ ( set_Sum_sum_a_nat2 @ Xs ) )
        & ( A = B2 ) ) ) ).

% zip_same
thf(fact_926_subset__code_I2_J,axiom,
    ! [A2: set_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ord_le1325389633284124927_a_nat @ A2 @ ( coset_Sum_sum_a_nat @ Ys ) )
      = ( ! [X2: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Ys ) )
           => ~ ( member_Sum_sum_a_nat @ X2 @ A2 ) ) ) ) ).

% subset_code(2)
thf(fact_927_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q2 )
      = ( ord_min_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).

% nat_mult_min_left
thf(fact_928_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q2 ) )
      = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).

% nat_mult_min_right
thf(fact_929_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_930_le__trans,axiom,
    ! [I2: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( ord_less_eq_nat @ J2 @ K )
       => ( ord_less_eq_nat @ I2 @ K ) ) ) ).

% le_trans
thf(fact_931_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_932_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( M = N ) ) ) ).

% le_antisym
thf(fact_933_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
      | ( ord_less_eq_nat @ N @ M ) ) ).

% nat_le_linear
thf(fact_934_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B2: nat] :
      ( ( P @ K )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B2 ) )
       => ? [X3: nat] :
            ( ( P @ X3 )
            & ! [Y5: nat] :
                ( ( P @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_935_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_936_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_937_mult__le__mono,axiom,
    ! [I2: nat,J2: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).

% mult_le_mono
thf(fact_938_mult__le__mono1,axiom,
    ! [I2: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ).

% mult_le_mono1
thf(fact_939_mult__le__mono2,axiom,
    ! [I2: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J2 ) ) ) ).

% mult_le_mono2
thf(fact_940_subseqs__powset,axiom,
    ! [Xs: list_Sum_sum_a_nat] :
      ( ( image_3940260845811589407_a_nat @ set_Sum_sum_a_nat2 @ ( set_li2392974972034027290_a_nat @ ( subseq8414445098004693972_a_nat @ Xs ) ) )
      = ( pow_Sum_sum_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ).

% subseqs_powset
thf(fact_941_transpose_Ocases,axiom,
    ! [X: list_l4703314356710769291_a_nat] :
      ( ( X != nil_li1906260230833442699_a_nat )
     => ( ! [Xss2: list_l4703314356710769291_a_nat] :
            ( X
           != ( cons_l6604326339930385211_a_nat @ nil_Sum_sum_a_nat @ Xss2 ) )
       => ~ ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat,Xss2: list_l4703314356710769291_a_nat] :
              ( X
             != ( cons_l6604326339930385211_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ Xss2 ) ) ) ) ).

% transpose.cases
thf(fact_942_splice_Ocases,axiom,
    ! [X: produc5001885624171833703_a_nat] :
      ( ! [Ys2: list_Sum_sum_a_nat] :
          ( X
         != ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Ys2 ) )
     => ~ ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
            ( X
           != ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ Ys2 ) ) ) ).

% splice.cases
thf(fact_943_shuffles_Ocases,axiom,
    ! [X: produc5001885624171833703_a_nat] :
      ( ! [Ys2: list_Sum_sum_a_nat] :
          ( X
         != ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Ys2 ) )
     => ( ! [Xs3: list_Sum_sum_a_nat] :
            ( X
           != ( produc7990843422341522135_a_nat @ Xs3 @ nil_Sum_sum_a_nat ) )
       => ~ ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat,Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
              ( X
             != ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) ) ) ).

% shuffles.cases
thf(fact_944_sorted__wrt_Ocases,axiom,
    ! [X: produc3388037434187558270_a_nat] :
      ( ! [P7: sum_sum_a_nat > sum_sum_a_nat > $o] :
          ( X
         != ( produc8857357843056227054_a_nat @ P7 @ nil_Sum_sum_a_nat ) )
     => ~ ! [P7: sum_sum_a_nat > sum_sum_a_nat > $o,X3: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
            ( X
           != ( produc8857357843056227054_a_nat @ P7 @ ( cons_Sum_sum_a_nat @ X3 @ Ys2 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_945_successively_Ocases,axiom,
    ! [X: produc3388037434187558270_a_nat] :
      ( ! [P7: sum_sum_a_nat > sum_sum_a_nat > $o] :
          ( X
         != ( produc8857357843056227054_a_nat @ P7 @ nil_Sum_sum_a_nat ) )
     => ( ! [P7: sum_sum_a_nat > sum_sum_a_nat > $o,X3: sum_sum_a_nat] :
            ( X
           != ( produc8857357843056227054_a_nat @ P7 @ ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) ) )
       => ~ ! [P7: sum_sum_a_nat > sum_sum_a_nat > $o,X3: sum_sum_a_nat,Y4: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] :
              ( X
             != ( produc8857357843056227054_a_nat @ P7 @ ( cons_Sum_sum_a_nat @ X3 @ ( cons_Sum_sum_a_nat @ Y4 @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_946_list__nonempty__induct,axiom,
    ! [Xs: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > $o] :
      ( ( Xs != nil_Sum_sum_a_nat )
     => ( ! [X3: sum_sum_a_nat] : ( P @ ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) )
       => ( ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] :
              ( ( Xs3 != nil_Sum_sum_a_nat )
             => ( ( P @ Xs3 )
               => ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_947_list__induct2_H,axiom,
    ! [P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( P @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat )
     => ( ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] : ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ nil_Sum_sum_a_nat )
       => ( ! [Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] : ( P @ nil_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) )
         => ( ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat,Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
                ( ( P @ Xs3 @ Ys2 )
               => ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_948_neq__Nil__conv,axiom,
    ! [Xs: list_Sum_sum_a_nat] :
      ( ( Xs != nil_Sum_sum_a_nat )
      = ( ? [Y3: sum_sum_a_nat,Ys3: list_Sum_sum_a_nat] :
            ( Xs
            = ( cons_Sum_sum_a_nat @ Y3 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_949_list_Oexhaust,axiom,
    ! [Y: list_Sum_sum_a_nat] :
      ( ( Y != nil_Sum_sum_a_nat )
     => ~ ! [X21: sum_sum_a_nat,X222: list_Sum_sum_a_nat] :
            ( Y
           != ( cons_Sum_sum_a_nat @ X21 @ X222 ) ) ) ).

% list.exhaust
thf(fact_950_list_OdiscI,axiom,
    ! [List: list_Sum_sum_a_nat,X212: sum_sum_a_nat,X223: list_Sum_sum_a_nat] :
      ( ( List
        = ( cons_Sum_sum_a_nat @ X212 @ X223 ) )
     => ( List != nil_Sum_sum_a_nat ) ) ).

% list.discI
thf(fact_951_list_Odistinct_I1_J,axiom,
    ! [X212: sum_sum_a_nat,X223: list_Sum_sum_a_nat] :
      ( nil_Sum_sum_a_nat
     != ( cons_Sum_sum_a_nat @ X212 @ X223 ) ) ).

% list.distinct(1)
thf(fact_952_in__set__simps_I1_J,axiom,
    ! [X: sum_sum_a_nat,Y: sum_sum_a_nat,Z3: sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ Y @ ( cons_Sum_sum_a_nat @ Z3 @ Ys ) ) ) )
      = ( ( X = Y )
        | ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ Z3 @ Ys ) ) ) ) ) ).

% in_set_simps(1)
thf(fact_953_set__ConsD,axiom,
    ! [Y: sum_sum_a_nat,X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ Y @ ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member_Sum_sum_a_nat @ Y @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_954_list_Oset__cases,axiom,
    ! [E: sum_sum_a_nat,A: list_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ E @ ( set_Sum_sum_a_nat2 @ A ) )
     => ( ! [Z22: list_Sum_sum_a_nat] :
            ( A
           != ( cons_Sum_sum_a_nat @ E @ Z22 ) )
       => ~ ! [Z1: sum_sum_a_nat,Z22: list_Sum_sum_a_nat] :
              ( ( A
                = ( cons_Sum_sum_a_nat @ Z1 @ Z22 ) )
             => ~ ( member_Sum_sum_a_nat @ E @ ( set_Sum_sum_a_nat2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_955_list_Oset__intros_I1_J,axiom,
    ! [X212: sum_sum_a_nat,X223: list_Sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X212 @ ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ X212 @ X223 ) ) ) ).

% list.set_intros(1)
thf(fact_956_list_Oset__intros_I2_J,axiom,
    ! [Y: sum_sum_a_nat,X223: list_Sum_sum_a_nat,X212: sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ Y @ ( set_Sum_sum_a_nat2 @ X223 ) )
     => ( member_Sum_sum_a_nat @ Y @ ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ X212 @ X223 ) ) ) ) ).

% list.set_intros(2)
thf(fact_957_distinct__length__2__or__more,axiom,
    ! [A: sum_sum_a_nat,B2: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( distin2701893636801681144_a_nat @ ( cons_Sum_sum_a_nat @ A @ ( cons_Sum_sum_a_nat @ B2 @ Xs ) ) )
      = ( ( A != B2 )
        & ( distin2701893636801681144_a_nat @ ( cons_Sum_sum_a_nat @ A @ Xs ) )
        & ( distin2701893636801681144_a_nat @ ( cons_Sum_sum_a_nat @ B2 @ Xs ) ) ) ) ).

% distinct_length_2_or_more
thf(fact_958_filter_Osimps_I2_J,axiom,
    ! [P: sum_sum_a_nat > $o,X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( ( P @ X )
       => ( ( filter_Sum_sum_a_nat @ P @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
          = ( cons_Sum_sum_a_nat @ X @ ( filter_Sum_sum_a_nat @ P @ Xs ) ) ) )
      & ( ~ ( P @ X )
       => ( ( filter_Sum_sum_a_nat @ P @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
          = ( filter_Sum_sum_a_nat @ P @ Xs ) ) ) ) ).

% filter.simps(2)
thf(fact_959_in__set__simps_I2_J,axiom,
    ! [X: sum_sum_a_nat,Y: sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ Y @ nil_Sum_sum_a_nat ) ) )
      = ( X = Y ) ) ).

% in_set_simps(2)
thf(fact_960_list__induct2,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( ( P @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat )
       => ( ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat,Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
              ( ( ( size_s5283204784079214577_a_nat @ Xs3 )
                = ( size_s5283204784079214577_a_nat @ Ys2 ) )
             => ( ( P @ Xs3 @ Ys2 )
               => ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_961_list__induct3,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs2: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( ( ( size_s5283204784079214577_a_nat @ Ys )
          = ( size_s5283204784079214577_a_nat @ Zs2 ) )
       => ( ( P @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat )
         => ( ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat,Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat,Z2: sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
                ( ( ( size_s5283204784079214577_a_nat @ Xs3 )
                  = ( size_s5283204784079214577_a_nat @ Ys2 ) )
               => ( ( ( size_s5283204784079214577_a_nat @ Ys2 )
                    = ( size_s5283204784079214577_a_nat @ Zs ) )
                 => ( ( P @ Xs3 @ Ys2 @ Zs )
                   => ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) @ ( cons_Sum_sum_a_nat @ Z2 @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs2 ) ) ) ) ) ).

% list_induct3
thf(fact_962_list__induct4,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs2: list_Sum_sum_a_nat,Ws2: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( ( ( size_s5283204784079214577_a_nat @ Ys )
          = ( size_s5283204784079214577_a_nat @ Zs2 ) )
       => ( ( ( size_s5283204784079214577_a_nat @ Zs2 )
            = ( size_s5283204784079214577_a_nat @ Ws2 ) )
         => ( ( P @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat )
           => ( ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat,Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat,Z2: sum_sum_a_nat,Zs: list_Sum_sum_a_nat,W: sum_sum_a_nat,Ws: list_Sum_sum_a_nat] :
                  ( ( ( size_s5283204784079214577_a_nat @ Xs3 )
                    = ( size_s5283204784079214577_a_nat @ Ys2 ) )
                 => ( ( ( size_s5283204784079214577_a_nat @ Ys2 )
                      = ( size_s5283204784079214577_a_nat @ Zs ) )
                   => ( ( ( size_s5283204784079214577_a_nat @ Zs )
                        = ( size_s5283204784079214577_a_nat @ Ws ) )
                     => ( ( P @ Xs3 @ Ys2 @ Zs @ Ws )
                       => ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) @ ( cons_Sum_sum_a_nat @ Z2 @ Zs ) @ ( cons_Sum_sum_a_nat @ W @ Ws ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs2 @ Ws2 ) ) ) ) ) ) ).

% list_induct4
thf(fact_963_set__subset__Cons,axiom,
    ! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_964_distinct__singleton,axiom,
    ! [X: sum_sum_a_nat] : ( distin2701893636801681144_a_nat @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) ).

% distinct_singleton
thf(fact_965_list__all2__induct,axiom,
    ! [P: sum_sum_a_nat > sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,R: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
      ( ( list_a8491001234060422471_a_nat @ P @ Xs @ Ys )
     => ( ( R @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat )
       => ( ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat,Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
              ( ( P @ X3 @ Y4 )
             => ( ( list_a8491001234060422471_a_nat @ P @ Xs3 @ Ys2 )
               => ( ( R @ Xs3 @ Ys2 )
                 => ( R @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) ) )
         => ( R @ Xs @ Ys ) ) ) ) ).

% list_all2_induct
thf(fact_966_list_Orel__induct,axiom,
    ! [R: sum_sum_a_nat > sum_sum_a_nat > $o,X: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat,Q: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
      ( ( list_a8491001234060422471_a_nat @ R @ X @ Y )
     => ( ( Q @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat )
       => ( ! [A21: sum_sum_a_nat,A22: list_Sum_sum_a_nat,B21: sum_sum_a_nat,B22: list_Sum_sum_a_nat] :
              ( ( R @ A21 @ B21 )
             => ( ( Q @ A22 @ B22 )
               => ( Q @ ( cons_Sum_sum_a_nat @ A21 @ A22 ) @ ( cons_Sum_sum_a_nat @ B21 @ B22 ) ) ) )
         => ( Q @ X @ Y ) ) ) ) ).

% list.rel_induct
thf(fact_967_list_Orel__cases,axiom,
    ! [R: sum_sum_a_nat > sum_sum_a_nat > $o,A: list_Sum_sum_a_nat,B2: list_Sum_sum_a_nat] :
      ( ( list_a8491001234060422471_a_nat @ R @ A @ B2 )
     => ( ( ( A = nil_Sum_sum_a_nat )
         => ( B2 != nil_Sum_sum_a_nat ) )
       => ~ ! [X12: sum_sum_a_nat,X23: list_Sum_sum_a_nat] :
              ( ( A
                = ( cons_Sum_sum_a_nat @ X12 @ X23 ) )
             => ! [Y12: sum_sum_a_nat,Y23: list_Sum_sum_a_nat] :
                  ( ( B2
                    = ( cons_Sum_sum_a_nat @ Y12 @ Y23 ) )
                 => ( ( R @ X12 @ Y12 )
                   => ~ ( list_a8491001234060422471_a_nat @ R @ X23 @ Y23 ) ) ) ) ) ) ).

% list.rel_cases
thf(fact_968_distinct_Osimps_I2_J,axiom,
    ! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( distin2701893636801681144_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
      = ( ~ ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
        & ( distin2701893636801681144_a_nat @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_969_subseqs_Osimps_I1_J,axiom,
    ( ( subseq8414445098004693972_a_nat @ nil_Sum_sum_a_nat )
    = ( cons_l6604326339930385211_a_nat @ nil_Sum_sum_a_nat @ nil_li1906260230833442699_a_nat ) ) ).

% subseqs.simps(1)
thf(fact_970_product__lists_Osimps_I1_J,axiom,
    ( ( produc2893206433618375022_a_nat @ nil_li1906260230833442699_a_nat )
    = ( cons_l6604326339930385211_a_nat @ nil_Sum_sum_a_nat @ nil_li1906260230833442699_a_nat ) ) ).

% product_lists.simps(1)
thf(fact_971_ord_Oquicksort__acc_Ocases,axiom,
    ! [X: produc5001885624171833703_a_nat] :
      ( ! [Ac: list_Sum_sum_a_nat] :
          ( X
         != ( produc7990843422341522135_a_nat @ Ac @ nil_Sum_sum_a_nat ) )
     => ( ! [Ac: list_Sum_sum_a_nat,X3: sum_sum_a_nat] :
            ( X
           != ( produc7990843422341522135_a_nat @ Ac @ ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) ) )
       => ~ ! [Ac: list_Sum_sum_a_nat,X3: sum_sum_a_nat,V2: sum_sum_a_nat,Va: list_Sum_sum_a_nat] :
              ( X
             != ( produc7990843422341522135_a_nat @ Ac @ ( cons_Sum_sum_a_nat @ X3 @ ( cons_Sum_sum_a_nat @ V2 @ Va ) ) ) ) ) ) ).

% ord.quicksort_acc.cases
thf(fact_972_ord_Oquicksort__part_Ocases,axiom,
    ! [X: produc1308816576071065077_a_nat] :
      ( ! [Ac: list_Sum_sum_a_nat,X3: sum_sum_a_nat,Lts: list_Sum_sum_a_nat,Eqs: list_Sum_sum_a_nat,Gts: list_Sum_sum_a_nat] :
          ( X
         != ( produc2222311365791241189_a_nat @ Ac @ ( produc5024437050894261379_a_nat @ X3 @ ( produc3915550414976331931_a_nat @ Lts @ ( produc3577755783922481849_a_nat @ Eqs @ ( produc7990843422341522135_a_nat @ Gts @ nil_Sum_sum_a_nat ) ) ) ) ) )
     => ~ ! [Ac: list_Sum_sum_a_nat,X3: sum_sum_a_nat,Lts: list_Sum_sum_a_nat,Eqs: list_Sum_sum_a_nat,Gts: list_Sum_sum_a_nat,Z2: sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
            ( X
           != ( produc2222311365791241189_a_nat @ Ac @ ( produc5024437050894261379_a_nat @ X3 @ ( produc3915550414976331931_a_nat @ Lts @ ( produc3577755783922481849_a_nat @ Eqs @ ( produc7990843422341522135_a_nat @ Gts @ ( cons_Sum_sum_a_nat @ Z2 @ Zs ) ) ) ) ) ) ) ) ).

% ord.quicksort_part.cases
thf(fact_973_ord_Osorted__list__subset_Ocases,axiom,
    ! [X: produc1169793092761538656_a_nat] :
      ( ! [Eq: sum_sum_a_nat > sum_sum_a_nat > $o,Ys2: list_Sum_sum_a_nat] :
          ( X
         != ( produc2770702104727902672_a_nat @ Eq @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Ys2 ) ) )
     => ( ! [Eq: sum_sum_a_nat > sum_sum_a_nat > $o,X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] :
            ( X
           != ( produc2770702104727902672_a_nat @ Eq @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ nil_Sum_sum_a_nat ) ) )
       => ~ ! [Eq: sum_sum_a_nat > sum_sum_a_nat > $o,X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat,Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
              ( X
             != ( produc2770702104727902672_a_nat @ Eq @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) ) ) ) ).

% ord.sorted_list_subset.cases
thf(fact_974_ord_Oremdups__sorted_Ocases,axiom,
    ! [X: list_Sum_sum_a_nat] :
      ( ( X != nil_Sum_sum_a_nat )
     => ( ! [X3: sum_sum_a_nat] :
            ( X
           != ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) )
       => ~ ! [X3: sum_sum_a_nat,Y4: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] :
              ( X
             != ( cons_Sum_sum_a_nat @ X3 @ ( cons_Sum_sum_a_nat @ Y4 @ Xs3 ) ) ) ) ) ).

% ord.remdups_sorted.cases
thf(fact_975_proper__intrvl_Oproper__interval__set__Compl__aux_Ocases,axiom,
    ! [X: produc4708264106712766336_a_nat] :
      ( ! [Ao: option_Sum_sum_a_nat,N2: nat] :
          ( X
         != ( produc2856885992714847610_a_nat @ Ao @ ( produc5571422593549375550_a_nat @ N2 @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat ) ) ) )
     => ( ! [Ao: option_Sum_sum_a_nat,N2: nat,Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
            ( X
           != ( produc2856885992714847610_a_nat @ Ao @ ( produc5571422593549375550_a_nat @ N2 @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) ) )
       => ( ! [Ao: option_Sum_sum_a_nat,N2: nat,X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] :
              ( X
             != ( produc2856885992714847610_a_nat @ Ao @ ( produc5571422593549375550_a_nat @ N2 @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ nil_Sum_sum_a_nat ) ) ) )
         => ~ ! [Ao: option_Sum_sum_a_nat,N2: nat,X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat,Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
                ( X
               != ( produc2856885992714847610_a_nat @ Ao @ ( produc5571422593549375550_a_nat @ N2 @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) ) ) ) ) ) ).

% proper_intrvl.proper_interval_set_Compl_aux.cases
thf(fact_976_wlog__le,axiom,
    ! [P: nat > nat > $o,B2: nat,A: nat] :
      ( ! [A3: nat,B: nat] :
          ( ( P @ A3 @ B )
         => ( P @ B @ A3 ) )
     => ( ! [A3: nat,B: nat] :
            ( ( ord_less_eq_nat @ A3 @ B )
           => ( P @ A3 @ B ) )
       => ( P @ B2 @ A ) ) ) ).

% wlog_le
thf(fact_977_proper__intrvl_Oexhaustive_Ocases,axiom,
    ! [X: list_Sum_sum_a_nat] :
      ( ( X != nil_Sum_sum_a_nat )
     => ~ ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] :
            ( X
           != ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) ) ) ).

% proper_intrvl.exhaustive.cases
thf(fact_978_proper__intrvl_Oexhaustive__above_Ocases,axiom,
    ! [X: produc4502985402200462317_a_nat] :
      ( ! [X3: sum_sum_a_nat] :
          ( X
         != ( produc6350064662657521885_a_nat @ X3 @ nil_Sum_sum_a_nat ) )
     => ~ ! [X3: sum_sum_a_nat,Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
            ( X
           != ( produc6350064662657521885_a_nat @ X3 @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) ) ).

% proper_intrvl.exhaustive_above.cases
thf(fact_979_proper__intrvl_OCompl__set__less__eq__aux_Ocases,axiom,
    ! [X: produc8636276049245547145_a_nat] :
      ( ! [Ao: option_Sum_sum_a_nat] :
          ( X
         != ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat ) ) )
     => ( ! [Ao: option_Sum_sum_a_nat,Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
            ( X
           != ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) )
       => ( ! [Ao: option_Sum_sum_a_nat,X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] :
              ( X
             != ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ nil_Sum_sum_a_nat ) ) )
         => ~ ! [Ao: option_Sum_sum_a_nat,X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat,Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
                ( X
               != ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) ) ) ) ) ).

% proper_intrvl.Compl_set_less_eq_aux.cases
thf(fact_980_proper__intrvl_Oset__less__eq__aux__Compl_Ocases,axiom,
    ! [X: produc8636276049245547145_a_nat] :
      ( ! [Ao: option_Sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
          ( X
         != ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Ys2 ) ) )
     => ( ! [Ao: option_Sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] :
            ( X
           != ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ Xs3 @ nil_Sum_sum_a_nat ) ) )
       => ~ ! [Ao: option_Sum_sum_a_nat,X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat,Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
              ( X
             != ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) ) ) ) ).

% proper_intrvl.set_less_eq_aux_Compl.cases
thf(fact_981_proper__intrvl_Oproper__interval__set__aux_Ocases,axiom,
    ! [X: produc5001885624171833703_a_nat] :
      ( ! [Xs3: list_Sum_sum_a_nat] :
          ( X
         != ( produc7990843422341522135_a_nat @ Xs3 @ nil_Sum_sum_a_nat ) )
     => ( ! [Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
            ( X
           != ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) )
       => ~ ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat,Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
              ( X
             != ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) ) ) ).

% proper_intrvl.proper_interval_set_aux.cases
thf(fact_982_proper__intrvl_Oproper__interval__Compl__set__aux_Ocases,axiom,
    ! [X: produc8636276049245547145_a_nat] :
      ( ! [Ao: option_Sum_sum_a_nat,X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat,Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
          ( X
         != ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) )
     => ( ! [Ao: option_Sum_sum_a_nat,Uv: list_Sum_sum_a_nat] :
            ( X
           != ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Uv ) ) )
       => ~ ! [Ao: option_Sum_sum_a_nat,Uu2: list_Sum_sum_a_nat] :
              ( X
             != ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ Uu2 @ nil_Sum_sum_a_nat ) ) ) ) ) ).

% proper_intrvl.proper_interval_Compl_set_aux.cases
thf(fact_983_Nil2__notin__lex,axiom,
    ! [Xs: list_Sum_sum_a_nat,R2: set_Pr7343886759072863943_a_nat] :
      ~ ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ nil_Sum_sum_a_nat ) @ ( lex_Sum_sum_a_nat @ R2 ) ) ).

% Nil2_notin_lex
thf(fact_984_Nil__notin__lex,axiom,
    ! [Ys: list_Sum_sum_a_nat,R2: set_Pr7343886759072863943_a_nat] :
      ~ ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Ys ) @ ( lex_Sum_sum_a_nat @ R2 ) ) ).

% Nil_notin_lex
thf(fact_985_splice_Opinduct,axiom,
    ! [A0: list_Sum_sum_a_nat,A1: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
      ( ( accp_P2437373109160157296_a_nat @ splice6660502696798462123_a_nat @ ( produc7990843422341522135_a_nat @ A0 @ A1 ) )
     => ( ! [Ys2: list_Sum_sum_a_nat] :
            ( ( accp_P2437373109160157296_a_nat @ splice6660502696798462123_a_nat @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Ys2 ) )
           => ( P @ nil_Sum_sum_a_nat @ Ys2 ) )
       => ( ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
              ( ( accp_P2437373109160157296_a_nat @ splice6660502696798462123_a_nat @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ Ys2 ) )
             => ( ( P @ Ys2 @ Xs3 )
               => ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ Ys2 ) ) )
         => ( P @ A0 @ A1 ) ) ) ) ).

% splice.pinduct
thf(fact_986_splice_Opelims,axiom,
    ! [X: list_Sum_sum_a_nat,Xa: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
      ( ( ( splice_Sum_sum_a_nat @ X @ Xa )
        = Y )
     => ( ( accp_P2437373109160157296_a_nat @ splice6660502696798462123_a_nat @ ( produc7990843422341522135_a_nat @ X @ Xa ) )
       => ( ( ( X = nil_Sum_sum_a_nat )
           => ( ( Y = Xa )
             => ~ ( accp_P2437373109160157296_a_nat @ splice6660502696798462123_a_nat @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Xa ) ) ) )
         => ~ ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] :
                ( ( X
                  = ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) )
               => ( ( Y
                    = ( cons_Sum_sum_a_nat @ X3 @ ( splice_Sum_sum_a_nat @ Xa @ Xs3 ) ) )
                 => ~ ( accp_P2437373109160157296_a_nat @ splice6660502696798462123_a_nat @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ Xa ) ) ) ) ) ) ) ).

% splice.pelims
thf(fact_987_shuffles_Opinduct,axiom,
    ! [A0: list_Sum_sum_a_nat,A1: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
      ( ( accp_P2437373109160157296_a_nat @ shuffl6050921173287438825_a_nat @ ( produc7990843422341522135_a_nat @ A0 @ A1 ) )
     => ( ! [Ys2: list_Sum_sum_a_nat] :
            ( ( accp_P2437373109160157296_a_nat @ shuffl6050921173287438825_a_nat @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Ys2 ) )
           => ( P @ nil_Sum_sum_a_nat @ Ys2 ) )
       => ( ! [Xs3: list_Sum_sum_a_nat] :
              ( ( accp_P2437373109160157296_a_nat @ shuffl6050921173287438825_a_nat @ ( produc7990843422341522135_a_nat @ Xs3 @ nil_Sum_sum_a_nat ) )
             => ( P @ Xs3 @ nil_Sum_sum_a_nat ) )
         => ( ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat,Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
                ( ( accp_P2437373109160157296_a_nat @ shuffl6050921173287438825_a_nat @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) )
               => ( ( P @ Xs3 @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) )
                 => ( ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ Ys2 )
                   => ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) ) )
           => ( P @ A0 @ A1 ) ) ) ) ) ).

% shuffles.pinduct
thf(fact_988_splice_Opsimps_I1_J,axiom,
    ! [Ys: list_Sum_sum_a_nat] :
      ( ( accp_P2437373109160157296_a_nat @ splice6660502696798462123_a_nat @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Ys ) )
     => ( ( splice_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ Ys )
        = Ys ) ) ).

% splice.psimps(1)
thf(fact_989_split__Nil__iff,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( splice_Sum_sum_a_nat @ Xs @ Ys )
        = nil_Sum_sum_a_nat )
      = ( ( Xs = nil_Sum_sum_a_nat )
        & ( Ys = nil_Sum_sum_a_nat ) ) ) ).

% split_Nil_iff
thf(fact_990_splice__Nil2,axiom,
    ! [Xs: list_Sum_sum_a_nat] :
      ( ( splice_Sum_sum_a_nat @ Xs @ nil_Sum_sum_a_nat )
      = Xs ) ).

% splice_Nil2
thf(fact_991_splice_Osimps_I1_J,axiom,
    ! [Ys: list_Sum_sum_a_nat] :
      ( ( splice_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ Ys )
      = Ys ) ).

% splice.simps(1)
thf(fact_992_splice_Oelims,axiom,
    ! [X: list_Sum_sum_a_nat,Xa: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
      ( ( ( splice_Sum_sum_a_nat @ X @ Xa )
        = Y )
     => ( ( ( X = nil_Sum_sum_a_nat )
         => ( Y != Xa ) )
       => ~ ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] :
              ( ( X
                = ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) )
             => ( Y
               != ( cons_Sum_sum_a_nat @ X3 @ ( splice_Sum_sum_a_nat @ Xa @ Xs3 ) ) ) ) ) ) ).

% splice.elims
thf(fact_993_partition__set,axiom,
    ! [P: sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat,Yes: list_Sum_sum_a_nat,No: list_Sum_sum_a_nat] :
      ( ( ( partit6043679984613064702_a_nat @ P @ Xs )
        = ( produc7990843422341522135_a_nat @ Yes @ No ) )
     => ( ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ Yes ) @ ( set_Sum_sum_a_nat2 @ No ) )
        = ( set_Sum_sum_a_nat2 @ Xs ) ) ) ).

% partition_set
thf(fact_994_append_Oright__neutral,axiom,
    ! [A: list_Sum_sum_a_nat] :
      ( ( append_Sum_sum_a_nat @ A @ nil_Sum_sum_a_nat )
      = A ) ).

% append.right_neutral
thf(fact_995_append__Nil2,axiom,
    ! [Xs: list_Sum_sum_a_nat] :
      ( ( append_Sum_sum_a_nat @ Xs @ nil_Sum_sum_a_nat )
      = Xs ) ).

% append_Nil2
thf(fact_996_append__self__conv,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( append_Sum_sum_a_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_Sum_sum_a_nat ) ) ).

% append_self_conv
thf(fact_997_self__append__conv,axiom,
    ! [Y: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( Y
        = ( append_Sum_sum_a_nat @ Y @ Ys ) )
      = ( Ys = nil_Sum_sum_a_nat ) ) ).

% self_append_conv
thf(fact_998_append__self__conv2,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( append_Sum_sum_a_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_Sum_sum_a_nat ) ) ).

% append_self_conv2
thf(fact_999_self__append__conv2,axiom,
    ! [Y: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( Y
        = ( append_Sum_sum_a_nat @ Xs @ Y ) )
      = ( Xs = nil_Sum_sum_a_nat ) ) ).

% self_append_conv2
thf(fact_1000_Nil__is__append__conv,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( nil_Sum_sum_a_nat
        = ( append_Sum_sum_a_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_Sum_sum_a_nat )
        & ( Ys = nil_Sum_sum_a_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_1001_append__is__Nil__conv,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( append_Sum_sum_a_nat @ Xs @ Ys )
        = nil_Sum_sum_a_nat )
      = ( ( Xs = nil_Sum_sum_a_nat )
        & ( Ys = nil_Sum_sum_a_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_1002_filter__append,axiom,
    ! [P: sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( filter_Sum_sum_a_nat @ P @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
      = ( append_Sum_sum_a_nat @ ( filter_Sum_sum_a_nat @ P @ Xs ) @ ( filter_Sum_sum_a_nat @ P @ Ys ) ) ) ).

% filter_append
thf(fact_1003_append1__eq__conv,axiom,
    ! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Y: sum_sum_a_nat] :
      ( ( ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) )
        = ( append_Sum_sum_a_nat @ Ys @ ( cons_Sum_sum_a_nat @ Y @ nil_Sum_sum_a_nat ) ) )
      = ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
thf(fact_1004_set__append,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( set_Sum_sum_a_nat2 @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
      = ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( set_Sum_sum_a_nat2 @ Ys ) ) ) ).

% set_append
thf(fact_1005_list__update__length,axiom,
    ! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Y: sum_sum_a_nat] :
      ( ( list_u9138855634547462509_a_nat @ ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ X @ Ys ) ) @ ( size_s5283204784079214577_a_nat @ Xs ) @ Y )
      = ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ Y @ Ys ) ) ) ).

% list_update_length
thf(fact_1006_eq__Nil__appendI,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_1007_append_Oleft__neutral,axiom,
    ! [A: list_Sum_sum_a_nat] :
      ( ( append_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_1008_append__Nil,axiom,
    ! [Ys: list_Sum_sum_a_nat] :
      ( ( append_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_1009_rev__nonempty__induct,axiom,
    ! [Xs: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > $o] :
      ( ( Xs != nil_Sum_sum_a_nat )
     => ( ! [X3: sum_sum_a_nat] : ( P @ ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) )
       => ( ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] :
              ( ( Xs3 != nil_Sum_sum_a_nat )
             => ( ( P @ Xs3 )
               => ( P @ ( append_Sum_sum_a_nat @ Xs3 @ ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_1010_append__eq__Cons__conv,axiom,
    ! [Ys: list_Sum_sum_a_nat,Zs2: list_Sum_sum_a_nat,X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( ( append_Sum_sum_a_nat @ Ys @ Zs2 )
        = ( cons_Sum_sum_a_nat @ X @ Xs ) )
      = ( ( ( Ys = nil_Sum_sum_a_nat )
          & ( Zs2
            = ( cons_Sum_sum_a_nat @ X @ Xs ) ) )
        | ? [Ys4: list_Sum_sum_a_nat] :
            ( ( Ys
              = ( cons_Sum_sum_a_nat @ X @ Ys4 ) )
            & ( ( append_Sum_sum_a_nat @ Ys4 @ Zs2 )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_1011_Cons__eq__append__conv,axiom,
    ! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
      ( ( ( cons_Sum_sum_a_nat @ X @ Xs )
        = ( append_Sum_sum_a_nat @ Ys @ Zs2 ) )
      = ( ( ( Ys = nil_Sum_sum_a_nat )
          & ( ( cons_Sum_sum_a_nat @ X @ Xs )
            = Zs2 ) )
        | ? [Ys4: list_Sum_sum_a_nat] :
            ( ( ( cons_Sum_sum_a_nat @ X @ Ys4 )
              = Ys )
            & ( Xs
              = ( append_Sum_sum_a_nat @ Ys4 @ Zs2 ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_1012_rev__exhaust,axiom,
    ! [Xs: list_Sum_sum_a_nat] :
      ( ( Xs != nil_Sum_sum_a_nat )
     => ~ ! [Ys2: list_Sum_sum_a_nat,Y4: sum_sum_a_nat] :
            ( Xs
           != ( append_Sum_sum_a_nat @ Ys2 @ ( cons_Sum_sum_a_nat @ Y4 @ nil_Sum_sum_a_nat ) ) ) ) ).

% rev_exhaust
thf(fact_1013_rev__induct,axiom,
    ! [P: list_Sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] :
      ( ( P @ nil_Sum_sum_a_nat )
     => ( ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] :
            ( ( P @ Xs3 )
           => ( P @ ( append_Sum_sum_a_nat @ Xs3 @ ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_1014_split__list,axiom,
    ! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
     => ? [Ys2: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
          ( Xs
          = ( append_Sum_sum_a_nat @ Ys2 @ ( cons_Sum_sum_a_nat @ X @ Zs ) ) ) ) ).

% split_list
thf(fact_1015_split__list__last,axiom,
    ! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
     => ? [Ys2: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
          ( ( Xs
            = ( append_Sum_sum_a_nat @ Ys2 @ ( cons_Sum_sum_a_nat @ X @ Zs ) ) )
          & ~ ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Zs ) ) ) ) ).

% split_list_last
thf(fact_1016_split__list__prop,axiom,
    ! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
      ( ? [X6: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ X6 @ ( set_Sum_sum_a_nat2 @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys2: list_Sum_sum_a_nat,X3: sum_sum_a_nat] :
          ( ? [Zs: list_Sum_sum_a_nat] :
              ( Xs
              = ( append_Sum_sum_a_nat @ Ys2 @ ( cons_Sum_sum_a_nat @ X3 @ Zs ) ) )
          & ( P @ X3 ) ) ) ).

% split_list_prop
thf(fact_1017_split__list__first,axiom,
    ! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
     => ? [Ys2: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
          ( ( Xs
            = ( append_Sum_sum_a_nat @ Ys2 @ ( cons_Sum_sum_a_nat @ X @ Zs ) ) )
          & ~ ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_1018_split__list__propE,axiom,
    ! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
      ( ? [X6: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ X6 @ ( set_Sum_sum_a_nat2 @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys2: list_Sum_sum_a_nat,X3: sum_sum_a_nat] :
            ( ? [Zs: list_Sum_sum_a_nat] :
                ( Xs
                = ( append_Sum_sum_a_nat @ Ys2 @ ( cons_Sum_sum_a_nat @ X3 @ Zs ) ) )
           => ~ ( P @ X3 ) ) ) ).

% split_list_propE
thf(fact_1019_append__Cons__eq__iff,axiom,
    ! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Xs4: list_Sum_sum_a_nat,Ys5: list_Sum_sum_a_nat] :
      ( ~ ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
     => ( ~ ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Ys ) )
       => ( ( ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ X @ Ys ) )
            = ( append_Sum_sum_a_nat @ Xs4 @ ( cons_Sum_sum_a_nat @ X @ Ys5 ) ) )
          = ( ( Xs = Xs4 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_1020_in__set__conv__decomp,axiom,
    ! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
      = ( ? [Ys3: list_Sum_sum_a_nat,Zs3: list_Sum_sum_a_nat] :
            ( Xs
            = ( append_Sum_sum_a_nat @ Ys3 @ ( cons_Sum_sum_a_nat @ X @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_1021_split__list__last__prop,axiom,
    ! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
      ( ? [X6: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ X6 @ ( set_Sum_sum_a_nat2 @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys2: list_Sum_sum_a_nat,X3: sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
          ( ( Xs
            = ( append_Sum_sum_a_nat @ Ys2 @ ( cons_Sum_sum_a_nat @ X3 @ Zs ) ) )
          & ( P @ X3 )
          & ! [Xa2: sum_sum_a_nat] :
              ( ( member_Sum_sum_a_nat @ Xa2 @ ( set_Sum_sum_a_nat2 @ Zs ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_1022_split__list__first__prop,axiom,
    ! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
      ( ? [X6: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ X6 @ ( set_Sum_sum_a_nat2 @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys2: list_Sum_sum_a_nat,X3: sum_sum_a_nat] :
          ( ? [Zs: list_Sum_sum_a_nat] :
              ( Xs
              = ( append_Sum_sum_a_nat @ Ys2 @ ( cons_Sum_sum_a_nat @ X3 @ Zs ) ) )
          & ( P @ X3 )
          & ! [Xa2: sum_sum_a_nat] :
              ( ( member_Sum_sum_a_nat @ Xa2 @ ( set_Sum_sum_a_nat2 @ Ys2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_1023_split__list__last__propE,axiom,
    ! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
      ( ? [X6: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ X6 @ ( set_Sum_sum_a_nat2 @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys2: list_Sum_sum_a_nat,X3: sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
            ( ( Xs
              = ( append_Sum_sum_a_nat @ Ys2 @ ( cons_Sum_sum_a_nat @ X3 @ Zs ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: sum_sum_a_nat] :
                    ( ( member_Sum_sum_a_nat @ Xa2 @ ( set_Sum_sum_a_nat2 @ Zs ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_1024_split__list__first__propE,axiom,
    ! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
      ( ? [X6: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ X6 @ ( set_Sum_sum_a_nat2 @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys2: list_Sum_sum_a_nat,X3: sum_sum_a_nat] :
            ( ? [Zs: list_Sum_sum_a_nat] :
                ( Xs
                = ( append_Sum_sum_a_nat @ Ys2 @ ( cons_Sum_sum_a_nat @ X3 @ Zs ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: sum_sum_a_nat] :
                    ( ( member_Sum_sum_a_nat @ Xa2 @ ( set_Sum_sum_a_nat2 @ Ys2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_1025_in__set__conv__decomp__last,axiom,
    ! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
      = ( ? [Ys3: list_Sum_sum_a_nat,Zs3: list_Sum_sum_a_nat] :
            ( ( Xs
              = ( append_Sum_sum_a_nat @ Ys3 @ ( cons_Sum_sum_a_nat @ X @ Zs3 ) ) )
            & ~ ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_1026_in__set__conv__decomp__first,axiom,
    ! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
      = ( ? [Ys3: list_Sum_sum_a_nat,Zs3: list_Sum_sum_a_nat] :
            ( ( Xs
              = ( append_Sum_sum_a_nat @ Ys3 @ ( cons_Sum_sum_a_nat @ X @ Zs3 ) ) )
            & ~ ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_1027_split__list__last__prop__iff,axiom,
    ! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
      ( ( ? [X2: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
            & ( P @ X2 ) ) )
      = ( ? [Ys3: list_Sum_sum_a_nat,X2: sum_sum_a_nat,Zs3: list_Sum_sum_a_nat] :
            ( ( Xs
              = ( append_Sum_sum_a_nat @ Ys3 @ ( cons_Sum_sum_a_nat @ X2 @ Zs3 ) ) )
            & ( P @ X2 )
            & ! [Y3: sum_sum_a_nat] :
                ( ( member_Sum_sum_a_nat @ Y3 @ ( set_Sum_sum_a_nat2 @ Zs3 ) )
               => ~ ( P @ Y3 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_1028_split__list__first__prop__iff,axiom,
    ! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
      ( ( ? [X2: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
            & ( P @ X2 ) ) )
      = ( ? [Ys3: list_Sum_sum_a_nat,X2: sum_sum_a_nat] :
            ( ? [Zs3: list_Sum_sum_a_nat] :
                ( Xs
                = ( append_Sum_sum_a_nat @ Ys3 @ ( cons_Sum_sum_a_nat @ X2 @ Zs3 ) ) )
            & ( P @ X2 )
            & ! [Y3: sum_sum_a_nat] :
                ( ( member_Sum_sum_a_nat @ Y3 @ ( set_Sum_sum_a_nat2 @ Ys3 ) )
               => ~ ( P @ Y3 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_1029_partition__filter1,axiom,
    ! [P: sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] :
      ( ( produc2015473822650161347_a_nat @ ( partit6043679984613064702_a_nat @ P @ Xs ) )
      = ( filter_Sum_sum_a_nat @ P @ Xs ) ) ).

% partition_filter1
thf(fact_1030_same__length__different,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( Xs != Ys )
     => ( ( ( size_s5283204784079214577_a_nat @ Xs )
          = ( size_s5283204784079214577_a_nat @ Ys ) )
       => ? [Pre: list_Sum_sum_a_nat,X3: sum_sum_a_nat,Xs5: list_Sum_sum_a_nat,Y4: sum_sum_a_nat,Ys6: list_Sum_sum_a_nat] :
            ( ( X3 != Y4 )
            & ( Xs
              = ( append_Sum_sum_a_nat @ Pre @ ( append_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) @ Xs5 ) ) )
            & ( Ys
              = ( append_Sum_sum_a_nat @ Pre @ ( append_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ Y4 @ nil_Sum_sum_a_nat ) @ Ys6 ) ) ) ) ) ) ).

% same_length_different
thf(fact_1031_not__distinct__decomp,axiom,
    ! [Ws2: list_Sum_sum_a_nat] :
      ( ~ ( distin2701893636801681144_a_nat @ Ws2 )
     => ? [Xs3: list_Sum_sum_a_nat,Ys2: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat,Y4: sum_sum_a_nat] :
          ( Ws2
          = ( append_Sum_sum_a_nat @ Xs3 @ ( append_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ Y4 @ nil_Sum_sum_a_nat ) @ ( append_Sum_sum_a_nat @ Ys2 @ ( append_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ Y4 @ nil_Sum_sum_a_nat ) @ Zs ) ) ) ) ) ) ).

% not_distinct_decomp
thf(fact_1032_not__distinct__conv__prefix,axiom,
    ! [As: list_Sum_sum_a_nat] :
      ( ( ~ ( distin2701893636801681144_a_nat @ As ) )
      = ( ? [Xs2: list_Sum_sum_a_nat,Y3: sum_sum_a_nat,Ys3: list_Sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ Y3 @ ( set_Sum_sum_a_nat2 @ Xs2 ) )
            & ( distin2701893636801681144_a_nat @ Xs2 )
            & ( As
              = ( append_Sum_sum_a_nat @ Xs2 @ ( cons_Sum_sum_a_nat @ Y3 @ Ys3 ) ) ) ) ) ) ).

% not_distinct_conv_prefix
thf(fact_1033_Cons__eq__filterD,axiom,
    ! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o,Ys: list_Sum_sum_a_nat] :
      ( ( ( cons_Sum_sum_a_nat @ X @ Xs )
        = ( filter_Sum_sum_a_nat @ P @ Ys ) )
     => ? [Us: list_Sum_sum_a_nat,Vs: list_Sum_sum_a_nat] :
          ( ( Ys
            = ( append_Sum_sum_a_nat @ Us @ ( cons_Sum_sum_a_nat @ X @ Vs ) ) )
          & ! [X6: sum_sum_a_nat] :
              ( ( member_Sum_sum_a_nat @ X6 @ ( set_Sum_sum_a_nat2 @ Us ) )
             => ~ ( P @ X6 ) )
          & ( P @ X )
          & ( Xs
            = ( filter_Sum_sum_a_nat @ P @ Vs ) ) ) ) ).

% Cons_eq_filterD
thf(fact_1034_filter__eq__ConsD,axiom,
    ! [P: sum_sum_a_nat > $o,Ys: list_Sum_sum_a_nat,X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( ( filter_Sum_sum_a_nat @ P @ Ys )
        = ( cons_Sum_sum_a_nat @ X @ Xs ) )
     => ? [Us: list_Sum_sum_a_nat,Vs: list_Sum_sum_a_nat] :
          ( ( Ys
            = ( append_Sum_sum_a_nat @ Us @ ( cons_Sum_sum_a_nat @ X @ Vs ) ) )
          & ! [X6: sum_sum_a_nat] :
              ( ( member_Sum_sum_a_nat @ X6 @ ( set_Sum_sum_a_nat2 @ Us ) )
             => ~ ( P @ X6 ) )
          & ( P @ X )
          & ( Xs
            = ( filter_Sum_sum_a_nat @ P @ Vs ) ) ) ) ).

% filter_eq_ConsD
thf(fact_1035_Cons__eq__filter__iff,axiom,
    ! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o,Ys: list_Sum_sum_a_nat] :
      ( ( ( cons_Sum_sum_a_nat @ X @ Xs )
        = ( filter_Sum_sum_a_nat @ P @ Ys ) )
      = ( ? [Us2: list_Sum_sum_a_nat,Vs2: list_Sum_sum_a_nat] :
            ( ( Ys
              = ( append_Sum_sum_a_nat @ Us2 @ ( cons_Sum_sum_a_nat @ X @ Vs2 ) ) )
            & ! [X2: sum_sum_a_nat] :
                ( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Us2 ) )
               => ~ ( P @ X2 ) )
            & ( P @ X )
            & ( Xs
              = ( filter_Sum_sum_a_nat @ P @ Vs2 ) ) ) ) ) ).

% Cons_eq_filter_iff
thf(fact_1036_filter__eq__Cons__iff,axiom,
    ! [P: sum_sum_a_nat > $o,Ys: list_Sum_sum_a_nat,X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( ( filter_Sum_sum_a_nat @ P @ Ys )
        = ( cons_Sum_sum_a_nat @ X @ Xs ) )
      = ( ? [Us2: list_Sum_sum_a_nat,Vs2: list_Sum_sum_a_nat] :
            ( ( Ys
              = ( append_Sum_sum_a_nat @ Us2 @ ( cons_Sum_sum_a_nat @ X @ Vs2 ) ) )
            & ! [X2: sum_sum_a_nat] :
                ( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Us2 ) )
               => ~ ( P @ X2 ) )
            & ( P @ X )
            & ( Xs
              = ( filter_Sum_sum_a_nat @ P @ Vs2 ) ) ) ) ) ).

% filter_eq_Cons_iff
thf(fact_1037_partition_Osimps_I1_J,axiom,
    ! [P: sum_sum_a_nat > $o] :
      ( ( partit6043679984613064702_a_nat @ P @ nil_Sum_sum_a_nat )
      = ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat ) ) ).

% partition.simps(1)
thf(fact_1038_partition__P,axiom,
    ! [P: sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat,Yes: list_Sum_sum_a_nat,No: list_Sum_sum_a_nat] :
      ( ( ( partit6043679984613064702_a_nat @ P @ Xs )
        = ( produc7990843422341522135_a_nat @ Yes @ No ) )
     => ( ! [X6: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X6 @ ( set_Sum_sum_a_nat2 @ Yes ) )
           => ( P @ X6 ) )
        & ! [X6: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X6 @ ( set_Sum_sum_a_nat2 @ No ) )
           => ~ ( P @ X6 ) ) ) ) ).

% partition_P
thf(fact_1039_rremdups__app,axiom,
    ! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
      ( ( rremdu8304153113908149561_a_nat @ ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) )
      = ( append_Sum_sum_a_nat @ ( rremdu8304153113908149561_a_nat @ Xs ) @ ( if_lis4685338526944683083_a_nat @ ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) ) @ nil_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) ) ) ).

% rremdups_app
thf(fact_1040_SuccD,axiom,
    ! [K: sum_sum_a_nat,Kl: set_li6526943997496501093_a_nat,Kl2: list_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ K @ ( bNF_Gr5582227268375839130_a_nat @ Kl @ Kl2 ) )
     => ( member408289922725080238_a_nat @ ( append_Sum_sum_a_nat @ Kl2 @ ( cons_Sum_sum_a_nat @ K @ nil_Sum_sum_a_nat ) ) @ Kl ) ) ).

% SuccD
thf(fact_1041_SuccI,axiom,
    ! [Kl2: list_Sum_sum_a_nat,K: sum_sum_a_nat,Kl: set_li6526943997496501093_a_nat] :
      ( ( member408289922725080238_a_nat @ ( append_Sum_sum_a_nat @ Kl2 @ ( cons_Sum_sum_a_nat @ K @ nil_Sum_sum_a_nat ) ) @ Kl )
     => ( member_Sum_sum_a_nat @ K @ ( bNF_Gr5582227268375839130_a_nat @ Kl @ Kl2 ) ) ) ).

% SuccI
thf(fact_1042_empty__Shift,axiom,
    ! [Kl: set_li6526943997496501093_a_nat,K: sum_sum_a_nat] :
      ( ( member408289922725080238_a_nat @ nil_Sum_sum_a_nat @ Kl )
     => ( ( member_Sum_sum_a_nat @ K @ ( bNF_Gr5582227268375839130_a_nat @ Kl @ nil_Sum_sum_a_nat ) )
       => ( member408289922725080238_a_nat @ nil_Sum_sum_a_nat @ ( bNF_Gr1229660863860170270_a_nat @ Kl @ K ) ) ) ) ).

% empty_Shift
thf(fact_1043_min_Oabsorb3,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_min_nat @ A @ B2 )
        = A ) ) ).

% min.absorb3
thf(fact_1044_min_Oabsorb4,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ( ord_min_nat @ A @ B2 )
        = B2 ) ) ).

% min.absorb4
thf(fact_1045_min__less__iff__conj,axiom,
    ! [Z3: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ Z3 @ ( ord_min_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Z3 @ X )
        & ( ord_less_nat @ Z3 @ Y ) ) ) ).

% min_less_iff_conj
thf(fact_1046_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I2: nat,J2: nat] :
      ( ! [I: nat,J: nat] :
          ( ( ord_less_nat @ I @ J )
         => ( ord_less_nat @ ( F @ I ) @ ( F @ J ) ) )
     => ( ( ord_less_eq_nat @ I2 @ J2 )
       => ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J2 ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_1047_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_1048_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_1049_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N4: nat] :
          ( ( ord_less_nat @ M3 @ N4 )
          | ( M3 = N4 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_1050_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_1051_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N4: nat] :
          ( ( ord_less_eq_nat @ M3 @ N4 )
          & ( M3 != N4 ) ) ) ) ).

% nat_less_le
thf(fact_1052_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1053_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_1054_order__less__le__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_1055_order__less__le__subst1,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1056_order__le__less__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_1057_order__le__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1058_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z3 )
       => ( ord_less_nat @ X @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_1059_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z3 )
       => ( ord_less_nat @ X @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_1060_order__neq__le__trans,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != B2 )
     => ( ( ord_less_eq_nat @ A @ B2 )
       => ( ord_less_nat @ A @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_1061_order__le__neq__trans,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( A != B2 )
       => ( ord_less_nat @ A @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_1062_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1063_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_1064_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_1065_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X2: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y3 )
          & ( X2 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_1066_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y3: nat] :
          ( ( ord_less_nat @ X2 @ Y3 )
          | ( X2 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_1067_dual__order_Ostrict__implies__order,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ord_less_eq_nat @ B2 @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1068_order_Ostrict__implies__order,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ord_less_eq_nat @ A @ B2 ) ) ).

% order.strict_implies_order
thf(fact_1069_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B3 @ A4 )
          & ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1070_dual__order_Ostrict__trans2,axiom,
    ! [B2: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ( ord_less_eq_nat @ C2 @ B2 )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1071_dual__order_Ostrict__trans1,axiom,
    ! [B2: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_less_nat @ C2 @ B2 )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1072_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B3 @ A4 )
          & ( A4 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1073_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( ord_less_nat @ B3 @ A4 )
          | ( A4 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1074_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
          & ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1075_order_Ostrict__trans2,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_1076_order_Ostrict__trans1,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_1077_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
          & ( A4 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1078_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_nat @ A4 @ B3 )
          | ( A4 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1079_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X )
     => ( ord_less_nat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_1080_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X2: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y3 )
          & ~ ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_1081_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_1082_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_1083_nless__le,axiom,
    ! [A: nat,B2: nat] :
      ( ( ~ ( ord_less_nat @ A @ B2 ) )
      = ( ~ ( ord_less_eq_nat @ A @ B2 )
        | ( A = B2 ) ) ) ).

% nless_le
thf(fact_1084_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_1085_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_1086_sup_Ostrict__coboundedI2,axiom,
    ! [C2: nat,B2: nat,A: nat] :
      ( ( ord_less_nat @ C2 @ B2 )
     => ( ord_less_nat @ C2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_1087_sup_Ostrict__coboundedI1,axiom,
    ! [C2: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ C2 @ A )
     => ( ord_less_nat @ C2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_1088_sup_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( A4
            = ( sup_sup_nat @ A4 @ B3 ) )
          & ( A4 != B3 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_1089_sup_Ostrict__boundedE,axiom,
    ! [B2: nat,C2: nat,A: nat] :
      ( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A )
     => ~ ( ( ord_less_nat @ B2 @ A )
         => ~ ( ord_less_nat @ C2 @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_1090_sup_Oabsorb4,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( sup_sup_nat @ A @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_1091_sup_Oabsorb3,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ( sup_sup_nat @ A @ B2 )
        = A ) ) ).

% sup.absorb3
thf(fact_1092_less__supI2,axiom,
    ! [X: nat,B2: nat,A: nat] :
      ( ( ord_less_nat @ X @ B2 )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B2 ) ) ) ).

% less_supI2
thf(fact_1093_less__supI1,axiom,
    ! [X: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ X @ A )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B2 ) ) ) ).

% less_supI1
thf(fact_1094_min__less__iff__disj,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_nat @ ( ord_min_nat @ X @ Y ) @ Z3 )
      = ( ( ord_less_nat @ X @ Z3 )
        | ( ord_less_nat @ Y @ Z3 ) ) ) ).

% min_less_iff_disj
thf(fact_1095_min_Ostrict__boundedE,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( ord_min_nat @ B2 @ C2 ) )
     => ~ ( ( ord_less_nat @ A @ B2 )
         => ~ ( ord_less_nat @ A @ C2 ) ) ) ).

% min.strict_boundedE
thf(fact_1096_min_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( A4
            = ( ord_min_nat @ A4 @ B3 ) )
          & ( A4 != B3 ) ) ) ) ).

% min.strict_order_iff
thf(fact_1097_min_Ostrict__coboundedI1,axiom,
    ! [A: nat,C2: nat,B2: nat] :
      ( ( ord_less_nat @ A @ C2 )
     => ( ord_less_nat @ ( ord_min_nat @ A @ B2 ) @ C2 ) ) ).

% min.strict_coboundedI1
thf(fact_1098_min_Ostrict__coboundedI2,axiom,
    ! [B2: nat,C2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ C2 )
     => ( ord_less_nat @ ( ord_min_nat @ A @ B2 ) @ C2 ) ) ).

% min.strict_coboundedI2
thf(fact_1099_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).

% gt_ex
thf(fact_1100_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_1101_order_Oasym,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ~ ( ord_less_nat @ B2 @ A ) ) ).

% order.asym
thf(fact_1102_ord__eq__less__trans,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( A = B2 )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_1103_ord__less__eq__trans,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_1104_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X3 )
             => ( P @ Y5 ) )
         => ( P @ X3 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_1105_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_1106_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_1107_dual__order_Oasym,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ~ ( ord_less_nat @ A @ B2 ) ) ).

% dual_order.asym
thf(fact_1108_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_1109_exists__least__iff,axiom,
    ( ( ^ [P8: nat > $o] :
        ? [X7: nat] : ( P8 @ X7 ) )
    = ( ^ [P4: nat > $o] :
        ? [N4: nat] :
          ( ( P4 @ N4 )
          & ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N4 )
             => ~ ( P4 @ M3 ) ) ) ) ) ).

% exists_least_iff
thf(fact_1110_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B2: nat] :
      ( ! [A3: nat,B: nat] :
          ( ( ord_less_nat @ A3 @ B )
         => ( P @ A3 @ B ) )
     => ( ! [A3: nat] : ( P @ A3 @ A3 )
       => ( ! [A3: nat,B: nat] :
              ( ( P @ B @ A3 )
             => ( P @ A3 @ B ) )
         => ( P @ A @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_1111_order_Ostrict__trans,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_1112_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1113_dual__order_Ostrict__trans,axiom,
    ! [B2: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ( ord_less_nat @ C2 @ B2 )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_1114_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_1115_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_1116_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_1117_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_1118_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      = ( ( ord_less_nat @ X @ Y )
        | ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_1119_order__less__asym_H,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ~ ( ord_less_nat @ B2 @ A ) ) ).

% order_less_asym'
thf(fact_1120_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z3 )
       => ( ord_less_nat @ X @ Z3 ) ) ) ).

% order_less_trans
thf(fact_1121_ord__eq__less__subst,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C2: nat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1122_ord__less__eq__subst,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1123_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_1124_order__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_1125_order__less__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_1126_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_1127_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1128_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
      | ( X = Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_1129_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1130_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_1131_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_1132_length__filter__less,axiom,
    ! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
      ( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
     => ( ~ ( P @ X )
       => ( ord_less_nat @ ( size_s5283204784079214577_a_nat @ ( filter_Sum_sum_a_nat @ P @ Xs ) ) @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ) ).

% length_filter_less
thf(fact_1133_set__update__memI,axiom,
    ! [N: nat,Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
      ( ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ ( list_u9138855634547462509_a_nat @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_1134_list__update__append1,axiom,
    ! [I2: nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( list_u9138855634547462509_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ I2 @ X )
        = ( append_Sum_sum_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I2 @ X ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_1135_complete__interval,axiom,
    ! [A: nat,B2: nat,P: nat > $o] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( P @ A )
       => ( ~ ( P @ B2 )
         => ? [C: nat] :
              ( ( ord_less_eq_nat @ A @ C )
              & ( ord_less_eq_nat @ C @ B2 )
              & ! [X6: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X6 )
                    & ( ord_less_nat @ X6 @ C ) )
                 => ( P @ X6 ) )
              & ! [D3: nat] :
                  ( ! [X3: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X3 )
                        & ( ord_less_nat @ X3 @ D3 ) )
                     => ( P @ X3 ) )
                 => ( ord_less_eq_nat @ D3 @ C ) ) ) ) ) ) ).

% complete_interval
thf(fact_1136_verit__comp__simplify1_I3_J,axiom,
    ! [B7: nat,A7: nat] :
      ( ( ~ ( ord_less_eq_nat @ B7 @ A7 ) )
      = ( ord_less_nat @ A7 @ B7 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1137_psubsetD,axiom,
    ! [A2: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat,C2: sum_sum_a_nat] :
      ( ( ord_le4638968407124865547_a_nat @ A2 @ B4 )
     => ( ( member_Sum_sum_a_nat @ C2 @ A2 )
       => ( member_Sum_sum_a_nat @ C2 @ B4 ) ) ) ).

% psubsetD
thf(fact_1138_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1139_verit__la__disequality,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = B2 )
      | ~ ( ord_less_eq_nat @ A @ B2 )
      | ~ ( ord_less_eq_nat @ B2 @ A ) ) ).

% verit_la_disequality
thf(fact_1140_Nil__lenlex__iff1,axiom,
    ! [Ns: list_Sum_sum_a_nat,R2: set_Pr7343886759072863943_a_nat] :
      ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Ns ) @ ( lenlex_Sum_sum_a_nat @ R2 ) )
      = ( Ns != nil_Sum_sum_a_nat ) ) ).

% Nil_lenlex_iff1
thf(fact_1141_Nil__lenlex__iff2,axiom,
    ! [Ns: list_Sum_sum_a_nat,R2: set_Pr7343886759072863943_a_nat] :
      ~ ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Ns @ nil_Sum_sum_a_nat ) @ ( lenlex_Sum_sum_a_nat @ R2 ) ) ).

% Nil_lenlex_iff2
thf(fact_1142_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N @ K2 )
         => ( P @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ K2 @ I4 )
                 => ( P @ I4 ) )
             => ( P @ K2 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_1143_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z2 )
     => ~ ( ord_less_eq_nat @ T @ X6 ) ) ).

% minf(8)
thf(fact_1144_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z2 @ X6 )
     => ~ ( ord_less_eq_nat @ X6 @ T ) ) ).

% pinf(6)
thf(fact_1145_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z2 @ X6 )
     => ( ord_less_eq_nat @ T @ X6 ) ) ).

% pinf(8)
thf(fact_1146_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z2 )
     => ( ord_less_eq_nat @ X6 @ T ) ) ).

% minf(6)
thf(fact_1147_lex__take__index,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,R2: set_Pr7343886759072863943_a_nat] :
      ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Ys ) @ ( lex_Sum_sum_a_nat @ R2 ) )
     => ~ ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
           => ( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Ys ) )
             => ( ( ( take_Sum_sum_a_nat @ I @ Xs )
                  = ( take_Sum_sum_a_nat @ I @ Ys ) )
               => ~ ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ ( nth_Sum_sum_a_nat @ Xs @ I ) @ ( nth_Sum_sum_a_nat @ Ys @ I ) ) @ R2 ) ) ) ) ) ).

% lex_take_index
thf(fact_1148_nth__list__update__neq,axiom,
    ! [I2: nat,J2: nat,Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
      ( ( I2 != J2 )
     => ( ( nth_Sum_sum_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I2 @ X ) @ J2 )
        = ( nth_Sum_sum_a_nat @ Xs @ J2 ) ) ) ).

% nth_list_update_neq
thf(fact_1149_list__update__id,axiom,
    ! [Xs: list_Sum_sum_a_nat,I2: nat] :
      ( ( list_u9138855634547462509_a_nat @ Xs @ I2 @ ( nth_Sum_sum_a_nat @ Xs @ I2 ) )
      = Xs ) ).

% list_update_id
thf(fact_1150_nth__take,axiom,
    ! [I2: nat,N: nat,Xs: list_Sum_sum_a_nat] :
      ( ( ord_less_nat @ I2 @ N )
     => ( ( nth_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ N @ Xs ) @ I2 )
        = ( nth_Sum_sum_a_nat @ Xs @ I2 ) ) ) ).

% nth_take
thf(fact_1151_nth__list__update__eq,axiom,
    ! [I2: nat,Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( nth_Sum_sum_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I2 @ X ) @ I2 )
        = X ) ) ).

% nth_list_update_eq
thf(fact_1152_set__swap,axiom,
    ! [I2: nat,Xs: list_Sum_sum_a_nat,J2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( ord_less_nat @ J2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
       => ( ( set_Sum_sum_a_nat2 @ ( list_u9138855634547462509_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I2 @ ( nth_Sum_sum_a_nat @ Xs @ J2 ) ) @ J2 @ ( nth_Sum_sum_a_nat @ Xs @ I2 ) ) )
          = ( set_Sum_sum_a_nat2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_1153_distinct__swap,axiom,
    ! [I2: nat,Xs: list_Sum_sum_a_nat,J2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( ord_less_nat @ J2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
       => ( ( distin2701893636801681144_a_nat @ ( list_u9138855634547462509_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I2 @ ( nth_Sum_sum_a_nat @ Xs @ J2 ) ) @ J2 @ ( nth_Sum_sum_a_nat @ Xs @ I2 ) ) )
          = ( distin2701893636801681144_a_nat @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_1154_all__set__conv__all__nth,axiom,
    ! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
      ( ( ! [X2: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
           => ( P @ X2 ) ) )
      = ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_s5283204784079214577_a_nat @ Xs ) )
           => ( P @ ( nth_Sum_sum_a_nat @ Xs @ I5 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1155_all__nth__imp__all__set,axiom,
    ! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o,X: sum_sum_a_nat] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
         => ( P @ ( nth_Sum_sum_a_nat @ Xs @ I ) ) )
     => ( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1156_in__set__conv__nth,axiom,
    ! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_s5283204784079214577_a_nat @ Xs ) )
            & ( ( nth_Sum_sum_a_nat @ Xs @ I5 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1157_list__ball__nth,axiom,
    ! [N: nat,Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
      ( ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ! [X3: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Xs ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_Sum_sum_a_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_1158_nth__mem,axiom,
    ! [N: nat,Xs: list_Sum_sum_a_nat] :
      ( ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( member_Sum_sum_a_nat @ ( nth_Sum_sum_a_nat @ Xs @ N ) @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_1159_distinct__conv__nth,axiom,
    ( distin2701893636801681144_a_nat
    = ( ^ [Xs2: list_Sum_sum_a_nat] :
        ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ ( size_s5283204784079214577_a_nat @ Xs2 ) )
         => ! [J3: nat] :
              ( ( ord_less_nat @ J3 @ ( size_s5283204784079214577_a_nat @ Xs2 ) )
             => ( ( I5 != J3 )
               => ( ( nth_Sum_sum_a_nat @ Xs2 @ I5 )
                 != ( nth_Sum_sum_a_nat @ Xs2 @ J3 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_1160_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_Sum_sum_a_nat,I2: nat,J2: nat] :
      ( ( distin2701893636801681144_a_nat @ Xs )
     => ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
       => ( ( ord_less_nat @ J2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
         => ( ( ( nth_Sum_sum_a_nat @ Xs @ I2 )
              = ( nth_Sum_sum_a_nat @ Xs @ J2 ) )
            = ( I2 = J2 ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_1161_list__update__same__conv,axiom,
    ! [I2: nat,Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( ( list_u9138855634547462509_a_nat @ Xs @ I2 @ X )
          = Xs )
        = ( ( nth_Sum_sum_a_nat @ Xs @ I2 )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_1162_nth__list__update,axiom,
    ! [I2: nat,Xs: list_Sum_sum_a_nat,J2: nat,X: sum_sum_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( ( I2 = J2 )
         => ( ( nth_Sum_sum_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I2 @ X ) @ J2 )
            = X ) )
        & ( ( I2 != J2 )
         => ( ( nth_Sum_sum_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I2 @ X ) @ J2 )
            = ( nth_Sum_sum_a_nat @ Xs @ J2 ) ) ) ) ) ).

% nth_list_update
thf(fact_1163_distinct__Ex1,axiom,
    ! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
      ( ( distin2701893636801681144_a_nat @ Xs )
     => ( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_s5283204784079214577_a_nat @ Xs ) )
            & ( ( nth_Sum_sum_a_nat @ Xs @ X3 )
              = X )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_s5283204784079214577_a_nat @ Xs ) )
                  & ( ( nth_Sum_sum_a_nat @ Xs @ Y5 )
                    = X ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_1164_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ord_less_eq_nat @ K @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s5283204784079214577_a_nat @ Ys ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ K )
             => ( ( nth_Sum_sum_a_nat @ Xs @ I )
                = ( nth_Sum_sum_a_nat @ Ys @ I ) ) )
         => ( ( take_Sum_sum_a_nat @ K @ Xs )
            = ( take_Sum_sum_a_nat @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_1165_lexord__take__index__conv,axiom,
    ! [X: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat,R2: set_Pr7343886759072863943_a_nat] :
      ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ X @ Y ) @ ( lexord_Sum_sum_a_nat @ R2 ) )
      = ( ( ( ord_less_nat @ ( size_s5283204784079214577_a_nat @ X ) @ ( size_s5283204784079214577_a_nat @ Y ) )
          & ( ( take_Sum_sum_a_nat @ ( size_s5283204784079214577_a_nat @ X ) @ Y )
            = X ) )
        | ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( ord_min_nat @ ( size_s5283204784079214577_a_nat @ X ) @ ( size_s5283204784079214577_a_nat @ Y ) ) )
            & ( ( take_Sum_sum_a_nat @ I5 @ X )
              = ( take_Sum_sum_a_nat @ I5 @ Y ) )
            & ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ ( nth_Sum_sum_a_nat @ X @ I5 ) @ ( nth_Sum_sum_a_nat @ Y @ I5 ) ) @ R2 ) ) ) ) ).

% lexord_take_index_conv
thf(fact_1166_lexord__Nil__left,axiom,
    ! [Y: list_Sum_sum_a_nat,R2: set_Pr7343886759072863943_a_nat] :
      ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Y ) @ ( lexord_Sum_sum_a_nat @ R2 ) )
      = ( ? [A4: sum_sum_a_nat,X2: list_Sum_sum_a_nat] :
            ( Y
            = ( cons_Sum_sum_a_nat @ A4 @ X2 ) ) ) ) ).

% lexord_Nil_left
thf(fact_1167_lexord__Nil__right,axiom,
    ! [X: list_Sum_sum_a_nat,R2: set_Pr7343886759072863943_a_nat] :
      ~ ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ X @ nil_Sum_sum_a_nat ) @ ( lexord_Sum_sum_a_nat @ R2 ) ) ).

% lexord_Nil_right
thf(fact_1168_lexord__partial__trans,axiom,
    ! [Xs: list_Sum_sum_a_nat,R2: set_Pr7343886759072863943_a_nat,Ys: list_Sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
      ( ! [X3: sum_sum_a_nat,Y4: sum_sum_a_nat,Z2: sum_sum_a_nat] :
          ( ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Xs ) )
         => ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X3 @ Y4 ) @ R2 )
           => ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ Y4 @ Z2 ) @ R2 )
             => ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X3 @ Z2 ) @ R2 ) ) ) )
     => ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Ys ) @ ( lexord_Sum_sum_a_nat @ R2 ) )
       => ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Ys @ Zs2 ) @ ( lexord_Sum_sum_a_nat @ R2 ) )
         => ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Zs2 ) @ ( lexord_Sum_sum_a_nat @ R2 ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_1169_lexord__same__pref__iff,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs2: list_Sum_sum_a_nat,R2: set_Pr7343886759072863943_a_nat] :
      ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ ( append_Sum_sum_a_nat @ Xs @ Zs2 ) ) @ ( lexord_Sum_sum_a_nat @ R2 ) )
      = ( ? [X2: sum_sum_a_nat] :
            ( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
            & ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X2 @ X2 ) @ R2 ) )
        | ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Ys @ Zs2 ) @ ( lexord_Sum_sum_a_nat @ R2 ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_1170_extract__SomeE,axiom,
    ! [P: sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Y: sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
      ( ( ( extrac3289232528736335879_a_nat @ P @ Xs )
        = ( some_P7299253002876453812_a_nat @ ( produc7578773175992006207_a_nat @ Ys @ ( produc6350064662657521885_a_nat @ Y @ Zs2 ) ) ) )
     => ( ( Xs
          = ( append_Sum_sum_a_nat @ Ys @ ( cons_Sum_sum_a_nat @ Y @ Zs2 ) ) )
        & ( P @ Y )
        & ~ ? [X6: sum_sum_a_nat] :
              ( ( member_Sum_sum_a_nat @ X6 @ ( set_Sum_sum_a_nat2 @ Ys ) )
              & ( P @ X6 ) ) ) ) ).

% extract_SomeE
thf(fact_1171_extract__Some__iff,axiom,
    ! [P: sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Y: sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
      ( ( ( extrac3289232528736335879_a_nat @ P @ Xs )
        = ( some_P7299253002876453812_a_nat @ ( produc7578773175992006207_a_nat @ Ys @ ( produc6350064662657521885_a_nat @ Y @ Zs2 ) ) ) )
      = ( ( Xs
          = ( append_Sum_sum_a_nat @ Ys @ ( cons_Sum_sum_a_nat @ Y @ Zs2 ) ) )
        & ( P @ Y )
        & ~ ? [X2: sum_sum_a_nat] :
              ( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Ys ) )
              & ( P @ X2 ) ) ) ) ).

% extract_Some_iff
thf(fact_1172_set__less__eq__aux__Compl_Opinduct,axiom,
    ! [A0: option_nat,A1: list_nat,A23: list_nat,P: option_nat > list_nat > list_nat > $o] :
      ( ( accp_P5867667535934726061st_nat @ set_pr9200262421968554262el_nat @ ( produc6570249034887359912st_nat @ A0 @ ( produc2694037385005941721st_nat @ A1 @ A23 ) ) )
     => ( ! [Ao: option_nat,Ys2: list_nat] :
            ( ( accp_P5867667535934726061st_nat @ set_pr9200262421968554262el_nat @ ( produc6570249034887359912st_nat @ Ao @ ( produc2694037385005941721st_nat @ nil_nat @ Ys2 ) ) )
           => ( P @ Ao @ nil_nat @ Ys2 ) )
       => ( ! [Ao: option_nat,Xs3: list_nat] :
              ( ( accp_P5867667535934726061st_nat @ set_pr9200262421968554262el_nat @ ( produc6570249034887359912st_nat @ Ao @ ( produc2694037385005941721st_nat @ Xs3 @ nil_nat ) ) )
             => ( P @ Ao @ Xs3 @ nil_nat ) )
         => ( ! [Ao: option_nat,X3: nat,Xs3: list_nat,Y4: nat,Ys2: list_nat] :
                ( ( accp_P5867667535934726061st_nat @ set_pr9200262421968554262el_nat @ ( produc6570249034887359912st_nat @ Ao @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y4 @ Ys2 ) ) ) )
               => ( ( ( ord_less_nat @ X3 @ Y4 )
                   => ( P @ ( some_nat @ X3 ) @ Xs3 @ ( cons_nat @ Y4 @ Ys2 ) ) )
                 => ( ( ~ ( ord_less_nat @ X3 @ Y4 )
                     => ( ( ord_less_nat @ Y4 @ X3 )
                       => ( P @ ( some_nat @ Y4 ) @ ( cons_nat @ X3 @ Xs3 ) @ Ys2 ) ) )
                   => ( P @ Ao @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y4 @ Ys2 ) ) ) ) )
           => ( P @ A0 @ A1 @ A23 ) ) ) ) ) ).

% set_less_eq_aux_Compl.pinduct
thf(fact_1173_Compl__set__less__aux_Opinduct,axiom,
    ! [A0: option_nat,A1: list_nat,A23: list_nat,P: option_nat > list_nat > list_nat > $o] :
      ( ( accp_P5867667535934726061st_nat @ set_pr6598557696555394320el_nat @ ( produc6570249034887359912st_nat @ A0 @ ( produc2694037385005941721st_nat @ A1 @ A23 ) ) )
     => ( ! [Ao: option_nat,Ys2: list_nat] :
            ( ( accp_P5867667535934726061st_nat @ set_pr6598557696555394320el_nat @ ( produc6570249034887359912st_nat @ Ao @ ( produc2694037385005941721st_nat @ nil_nat @ Ys2 ) ) )
           => ( P @ Ao @ nil_nat @ Ys2 ) )
       => ( ! [Ao: option_nat,Xs3: list_nat] :
              ( ( accp_P5867667535934726061st_nat @ set_pr6598557696555394320el_nat @ ( produc6570249034887359912st_nat @ Ao @ ( produc2694037385005941721st_nat @ Xs3 @ nil_nat ) ) )
             => ( P @ Ao @ Xs3 @ nil_nat ) )
         => ( ! [Ao: option_nat,X3: nat,Xs3: list_nat,Y4: nat,Ys2: list_nat] :
                ( ( accp_P5867667535934726061st_nat @ set_pr6598557696555394320el_nat @ ( produc6570249034887359912st_nat @ Ao @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y4 @ Ys2 ) ) ) )
               => ( ( ( ord_less_nat @ X3 @ Y4 )
                   => ( P @ ( some_nat @ X3 ) @ Xs3 @ ( cons_nat @ Y4 @ Ys2 ) ) )
                 => ( ( ~ ( ord_less_nat @ X3 @ Y4 )
                     => ( ( ord_less_nat @ Y4 @ X3 )
                       => ( P @ ( some_nat @ Y4 ) @ ( cons_nat @ X3 @ Xs3 ) @ Ys2 ) ) )
                   => ( P @ Ao @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y4 @ Ys2 ) ) ) ) )
           => ( P @ A0 @ A1 @ A23 ) ) ) ) ) ).

% Compl_set_less_aux.pinduct
thf(fact_1174_proper__intrvl_Oset__less__eq__aux__Compl_Opinduct,axiom,
    ! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,A0: option_Sum_sum_a_nat,A1: list_Sum_sum_a_nat,A23: list_Sum_sum_a_nat,P: option_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
      ( ( accp_P5320691669302659474_a_nat @ ( set_pr7360018828099327897_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ A0 @ ( produc7990843422341522135_a_nat @ A1 @ A23 ) ) )
     => ( ! [Ao: option_Sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
            ( ( accp_P5320691669302659474_a_nat @ ( set_pr7360018828099327897_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Ys2 ) ) )
           => ( P @ Ao @ nil_Sum_sum_a_nat @ Ys2 ) )
       => ( ! [Ao: option_Sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] :
              ( ( accp_P5320691669302659474_a_nat @ ( set_pr7360018828099327897_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ Xs3 @ nil_Sum_sum_a_nat ) ) )
             => ( P @ Ao @ Xs3 @ nil_Sum_sum_a_nat ) )
         => ( ! [Ao: option_Sum_sum_a_nat,X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat,Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
                ( ( accp_P5320691669302659474_a_nat @ ( set_pr7360018828099327897_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) )
               => ( ( ( Less @ X3 @ Y4 )
                   => ( P @ ( some_Sum_sum_a_nat @ X3 ) @ Xs3 @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) )
                 => ( ( ~ ( Less @ X3 @ Y4 )
                     => ( ( Less @ Y4 @ X3 )
                       => ( P @ ( some_Sum_sum_a_nat @ Y4 ) @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ Ys2 ) ) )
                   => ( P @ Ao @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) ) )
           => ( P @ A0 @ A1 @ A23 ) ) ) ) ) ).

% proper_intrvl.set_less_eq_aux_Compl.pinduct
thf(fact_1175_proper__intrvl_OCompl__set__less__aux_Opinduct,axiom,
    ! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,A0: option_Sum_sum_a_nat,A1: list_Sum_sum_a_nat,A23: list_Sum_sum_a_nat,P: option_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
      ( ( accp_P5320691669302659474_a_nat @ ( set_pr2097269997492014367_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ A0 @ ( produc7990843422341522135_a_nat @ A1 @ A23 ) ) )
     => ( ! [Ao: option_Sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
            ( ( accp_P5320691669302659474_a_nat @ ( set_pr2097269997492014367_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Ys2 ) ) )
           => ( P @ Ao @ nil_Sum_sum_a_nat @ Ys2 ) )
       => ( ! [Ao: option_Sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] :
              ( ( accp_P5320691669302659474_a_nat @ ( set_pr2097269997492014367_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ Xs3 @ nil_Sum_sum_a_nat ) ) )
             => ( P @ Ao @ Xs3 @ nil_Sum_sum_a_nat ) )
         => ( ! [Ao: option_Sum_sum_a_nat,X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat,Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
                ( ( accp_P5320691669302659474_a_nat @ ( set_pr2097269997492014367_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) )
               => ( ( ( Less @ X3 @ Y4 )
                   => ( P @ ( some_Sum_sum_a_nat @ X3 ) @ Xs3 @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) )
                 => ( ( ~ ( Less @ X3 @ Y4 )
                     => ( ( Less @ Y4 @ X3 )
                       => ( P @ ( some_Sum_sum_a_nat @ Y4 ) @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ Ys2 ) ) )
                   => ( P @ Ao @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) ) )
           => ( P @ A0 @ A1 @ A23 ) ) ) ) ) ).

% proper_intrvl.Compl_set_less_aux.pinduct
thf(fact_1176_proper__intrvl_Oset__less__eq__aux__Compl_Opelims_I1_J,axiom,
    ! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Proper_interval: option_Sum_sum_a_nat > option_Sum_sum_a_nat > $o,X: option_Sum_sum_a_nat,Xa: list_Sum_sum_a_nat,Xb: list_Sum_sum_a_nat,Y: $o] :
      ( ( ( set_pr1411927237749039722_a_nat @ Less @ Proper_interval @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P5320691669302659474_a_nat @ ( set_pr7360018828099327897_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ X @ ( produc7990843422341522135_a_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_Sum_sum_a_nat )
           => ( Y
             => ~ ( accp_P5320691669302659474_a_nat @ ( set_pr7360018828099327897_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ X @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Xb ) ) ) ) )
         => ( ( ( Xb = nil_Sum_sum_a_nat )
             => ( Y
               => ~ ( accp_P5320691669302659474_a_nat @ ( set_pr7360018828099327897_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ X @ ( produc7990843422341522135_a_nat @ Xa @ nil_Sum_sum_a_nat ) ) ) ) )
           => ~ ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] :
                  ( ( Xa
                    = ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) )
                 => ! [Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
                      ( ( Xb
                        = ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) )
                     => ( ( Y
                          = ( ( ( Less @ X3 @ Y4 )
                             => ( ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ X3 ) )
                                | ( set_pr1411927237749039722_a_nat @ Less @ Proper_interval @ ( some_Sum_sum_a_nat @ X3 ) @ Xs3 @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) )
                            & ( ~ ( Less @ X3 @ Y4 )
                             => ( ( ( Less @ Y4 @ X3 )
                                 => ( ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ Y4 ) )
                                    | ( set_pr1411927237749039722_a_nat @ Less @ Proper_interval @ ( some_Sum_sum_a_nat @ Y4 ) @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ Ys2 ) ) )
                                & ( ~ ( Less @ Y4 @ X3 )
                                 => ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ Y4 ) ) ) ) ) ) )
                       => ~ ( accp_P5320691669302659474_a_nat @ ( set_pr7360018828099327897_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ X @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) ) ) ) ) ) ) ) ) ).

% proper_intrvl.set_less_eq_aux_Compl.pelims(1)
thf(fact_1177_proper__intrvl_Oset__less__eq__aux__Compl_Opelims_I2_J,axiom,
    ! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Proper_interval: option_Sum_sum_a_nat > option_Sum_sum_a_nat > $o,X: option_Sum_sum_a_nat,Xa: list_Sum_sum_a_nat,Xb: list_Sum_sum_a_nat] :
      ( ( set_pr1411927237749039722_a_nat @ Less @ Proper_interval @ X @ Xa @ Xb )
     => ( ( accp_P5320691669302659474_a_nat @ ( set_pr7360018828099327897_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ X @ ( produc7990843422341522135_a_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_Sum_sum_a_nat )
           => ~ ( accp_P5320691669302659474_a_nat @ ( set_pr7360018828099327897_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ X @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Xb ) ) ) )
         => ( ( ( Xb = nil_Sum_sum_a_nat )
             => ~ ( accp_P5320691669302659474_a_nat @ ( set_pr7360018828099327897_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ X @ ( produc7990843422341522135_a_nat @ Xa @ nil_Sum_sum_a_nat ) ) ) )
           => ~ ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] :
                  ( ( Xa
                    = ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) )
                 => ! [Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
                      ( ( Xb
                        = ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) )
                     => ( ( accp_P5320691669302659474_a_nat @ ( set_pr7360018828099327897_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ X @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) )
                       => ~ ( ( ( Less @ X3 @ Y4 )
                             => ( ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ X3 ) )
                                | ( set_pr1411927237749039722_a_nat @ Less @ Proper_interval @ ( some_Sum_sum_a_nat @ X3 ) @ Xs3 @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) )
                            & ( ~ ( Less @ X3 @ Y4 )
                             => ( ( ( Less @ Y4 @ X3 )
                                 => ( ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ Y4 ) )
                                    | ( set_pr1411927237749039722_a_nat @ Less @ Proper_interval @ ( some_Sum_sum_a_nat @ Y4 ) @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ Ys2 ) ) )
                                & ( ~ ( Less @ Y4 @ X3 )
                                 => ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ Y4 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% proper_intrvl.set_less_eq_aux_Compl.pelims(2)
thf(fact_1178_proper__intrvl_Oset__less__eq__aux__Compl_Osimps_I1_J,axiom,
    ! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Proper_interval: option_Sum_sum_a_nat > option_Sum_sum_a_nat > $o,Ao2: option_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] : ( set_pr1411927237749039722_a_nat @ Less @ Proper_interval @ Ao2 @ nil_Sum_sum_a_nat @ Ys ) ).

% proper_intrvl.set_less_eq_aux_Compl.simps(1)
thf(fact_1179_proper__intrvl_Oset__less__eq__aux__Compl_Osimps_I2_J,axiom,
    ! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Proper_interval: option_Sum_sum_a_nat > option_Sum_sum_a_nat > $o,Ao2: option_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] : ( set_pr1411927237749039722_a_nat @ Less @ Proper_interval @ Ao2 @ Xs @ nil_Sum_sum_a_nat ) ).

% proper_intrvl.set_less_eq_aux_Compl.simps(2)
thf(fact_1180_proper__intrvl_Oset__less__eq__aux__Compl_Oelims_I1_J,axiom,
    ! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Proper_interval: option_Sum_sum_a_nat > option_Sum_sum_a_nat > $o,X: option_Sum_sum_a_nat,Xa: list_Sum_sum_a_nat,Xb: list_Sum_sum_a_nat,Y: $o] :
      ( ( ( set_pr1411927237749039722_a_nat @ Less @ Proper_interval @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_Sum_sum_a_nat )
         => ~ Y )
       => ( ( ( Xb = nil_Sum_sum_a_nat )
           => ~ Y )
         => ~ ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] :
                ( ( Xa
                  = ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) )
               => ! [Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
                    ( ( Xb
                      = ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) )
                   => ( Y
                      = ( ~ ( ( ( Less @ X3 @ Y4 )
                             => ( ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ X3 ) )
                                | ( set_pr1411927237749039722_a_nat @ Less @ Proper_interval @ ( some_Sum_sum_a_nat @ X3 ) @ Xs3 @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) )
                            & ( ~ ( Less @ X3 @ Y4 )
                             => ( ( ( Less @ Y4 @ X3 )
                                 => ( ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ Y4 ) )
                                    | ( set_pr1411927237749039722_a_nat @ Less @ Proper_interval @ ( some_Sum_sum_a_nat @ Y4 ) @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ Ys2 ) ) )
                                & ( ~ ( Less @ Y4 @ X3 )
                                 => ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ Y4 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% proper_intrvl.set_less_eq_aux_Compl.elims(1)
thf(fact_1181_proper__intrvl_Oset__less__eq__aux__Compl_Oelims_I2_J,axiom,
    ! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Proper_interval: option_Sum_sum_a_nat > option_Sum_sum_a_nat > $o,X: option_Sum_sum_a_nat,Xa: list_Sum_sum_a_nat,Xb: list_Sum_sum_a_nat] :
      ( ( set_pr1411927237749039722_a_nat @ Less @ Proper_interval @ X @ Xa @ Xb )
     => ( ( Xa != nil_Sum_sum_a_nat )
       => ( ( Xb != nil_Sum_sum_a_nat )
         => ~ ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] :
                ( ( Xa
                  = ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) )
               => ! [Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
                    ( ( Xb
                      = ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) )
                   => ~ ( ( ( Less @ X3 @ Y4 )
                         => ( ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ X3 ) )
                            | ( set_pr1411927237749039722_a_nat @ Less @ Proper_interval @ ( some_Sum_sum_a_nat @ X3 ) @ Xs3 @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) )
                        & ( ~ ( Less @ X3 @ Y4 )
                         => ( ( ( Less @ Y4 @ X3 )
                             => ( ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ Y4 ) )
                                | ( set_pr1411927237749039722_a_nat @ Less @ Proper_interval @ ( some_Sum_sum_a_nat @ Y4 ) @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ Ys2 ) ) )
                            & ( ~ ( Less @ Y4 @ X3 )
                             => ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ Y4 ) ) ) ) ) ) ) ) ) ) ) ).

% proper_intrvl.set_less_eq_aux_Compl.elims(2)
thf(fact_1182_proper__intrvl_Oset__less__eq__aux__Compl_Opsimps_I2_J,axiom,
    ! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Ao2: option_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Proper_interval: option_Sum_sum_a_nat > option_Sum_sum_a_nat > $o] :
      ( ( accp_P5320691669302659474_a_nat @ ( set_pr7360018828099327897_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ Ao2 @ ( produc7990843422341522135_a_nat @ Xs @ nil_Sum_sum_a_nat ) ) )
     => ( set_pr1411927237749039722_a_nat @ Less @ Proper_interval @ Ao2 @ Xs @ nil_Sum_sum_a_nat ) ) ).

% proper_intrvl.set_less_eq_aux_Compl.psimps(2)
thf(fact_1183_proper__intrvl_Oset__less__eq__aux__Compl_Opsimps_I1_J,axiom,
    ! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Ao2: option_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Proper_interval: option_Sum_sum_a_nat > option_Sum_sum_a_nat > $o] :
      ( ( accp_P5320691669302659474_a_nat @ ( set_pr7360018828099327897_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ Ao2 @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Ys ) ) )
     => ( set_pr1411927237749039722_a_nat @ Less @ Proper_interval @ Ao2 @ nil_Sum_sum_a_nat @ Ys ) ) ).

% proper_intrvl.set_less_eq_aux_Compl.psimps(1)
thf(fact_1184_proper__intrvl_OCompl__set__less__aux_Opelims_I3_J,axiom,
    ! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Proper_interval: option_Sum_sum_a_nat > option_Sum_sum_a_nat > $o,X: option_Sum_sum_a_nat,Xa: list_Sum_sum_a_nat,Xb: list_Sum_sum_a_nat] :
      ( ~ ( set_pr2532843088427607396_a_nat @ Less @ Proper_interval @ X @ Xa @ Xb )
     => ( ( accp_P5320691669302659474_a_nat @ ( set_pr2097269997492014367_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ X @ ( produc7990843422341522135_a_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_Sum_sum_a_nat )
           => ~ ( accp_P5320691669302659474_a_nat @ ( set_pr2097269997492014367_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ X @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Xb ) ) ) )
         => ( ( ( Xb = nil_Sum_sum_a_nat )
             => ~ ( accp_P5320691669302659474_a_nat @ ( set_pr2097269997492014367_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ X @ ( produc7990843422341522135_a_nat @ Xa @ nil_Sum_sum_a_nat ) ) ) )
           => ~ ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] :
                  ( ( Xa
                    = ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) )
                 => ! [Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
                      ( ( Xb
                        = ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) )
                     => ( ( accp_P5320691669302659474_a_nat @ ( set_pr2097269997492014367_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ X @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) )
                       => ( ( ( Less @ X3 @ Y4 )
                           => ( ~ ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ X3 ) )
                              & ( set_pr2532843088427607396_a_nat @ Less @ Proper_interval @ ( some_Sum_sum_a_nat @ X3 ) @ Xs3 @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) )
                          & ( ~ ( Less @ X3 @ Y4 )
                           => ( ( ( Less @ Y4 @ X3 )
                               => ( ~ ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ Y4 ) )
                                  & ( set_pr2532843088427607396_a_nat @ Less @ Proper_interval @ ( some_Sum_sum_a_nat @ Y4 ) @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ Ys2 ) ) )
                              & ( ~ ( Less @ Y4 @ X3 )
                               => ~ ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ Y4 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% proper_intrvl.Compl_set_less_aux.pelims(3)
thf(fact_1185_proper__intrvl_OCompl__set__less__aux_Opelims_I1_J,axiom,
    ! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Proper_interval: option_Sum_sum_a_nat > option_Sum_sum_a_nat > $o,X: option_Sum_sum_a_nat,Xa: list_Sum_sum_a_nat,Xb: list_Sum_sum_a_nat,Y: $o] :
      ( ( ( set_pr2532843088427607396_a_nat @ Less @ Proper_interval @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P5320691669302659474_a_nat @ ( set_pr2097269997492014367_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ X @ ( produc7990843422341522135_a_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_Sum_sum_a_nat )
           => ( ~ Y
             => ~ ( accp_P5320691669302659474_a_nat @ ( set_pr2097269997492014367_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ X @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Xb ) ) ) ) )
         => ( ( ( Xb = nil_Sum_sum_a_nat )
             => ( ~ Y
               => ~ ( accp_P5320691669302659474_a_nat @ ( set_pr2097269997492014367_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ X @ ( produc7990843422341522135_a_nat @ Xa @ nil_Sum_sum_a_nat ) ) ) ) )
           => ~ ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] :
                  ( ( Xa
                    = ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) )
                 => ! [Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
                      ( ( Xb
                        = ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) )
                     => ( ( Y
                          = ( ( ( Less @ X3 @ Y4 )
                             => ( ~ ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ X3 ) )
                                & ( set_pr2532843088427607396_a_nat @ Less @ Proper_interval @ ( some_Sum_sum_a_nat @ X3 ) @ Xs3 @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) )
                            & ( ~ ( Less @ X3 @ Y4 )
                             => ( ( ( Less @ Y4 @ X3 )
                                 => ( ~ ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ Y4 ) )
                                    & ( set_pr2532843088427607396_a_nat @ Less @ Proper_interval @ ( some_Sum_sum_a_nat @ Y4 ) @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ Ys2 ) ) )
                                & ( ~ ( Less @ Y4 @ X3 )
                                 => ~ ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ Y4 ) ) ) ) ) ) )
                       => ~ ( accp_P5320691669302659474_a_nat @ ( set_pr2097269997492014367_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ X @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) ) ) ) ) ) ) ) ) ).

% proper_intrvl.Compl_set_less_aux.pelims(1)
thf(fact_1186_proper__intrvl_OCompl__set__less__aux_Osimps_I2_J,axiom,
    ! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Proper_interval: option_Sum_sum_a_nat > option_Sum_sum_a_nat > $o,Ao2: option_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ~ ( set_pr2532843088427607396_a_nat @ Less @ Proper_interval @ Ao2 @ Xs @ nil_Sum_sum_a_nat ) ).

% proper_intrvl.Compl_set_less_aux.simps(2)
thf(fact_1187_proper__intrvl_OCompl__set__less__aux_Osimps_I1_J,axiom,
    ! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Proper_interval: option_Sum_sum_a_nat > option_Sum_sum_a_nat > $o,Ao2: option_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ~ ( set_pr2532843088427607396_a_nat @ Less @ Proper_interval @ Ao2 @ nil_Sum_sum_a_nat @ Ys ) ).

% proper_intrvl.Compl_set_less_aux.simps(1)
thf(fact_1188_proper__intrvl_OCompl__set__less__aux_Oelims_I3_J,axiom,
    ! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Proper_interval: option_Sum_sum_a_nat > option_Sum_sum_a_nat > $o,X: option_Sum_sum_a_nat,Xa: list_Sum_sum_a_nat,Xb: list_Sum_sum_a_nat] :
      ( ~ ( set_pr2532843088427607396_a_nat @ Less @ Proper_interval @ X @ Xa @ Xb )
     => ( ( Xa != nil_Sum_sum_a_nat )
       => ( ( Xb != nil_Sum_sum_a_nat )
         => ~ ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] :
                ( ( Xa
                  = ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) )
               => ! [Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
                    ( ( Xb
                      = ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) )
                   => ( ( ( Less @ X3 @ Y4 )
                       => ( ~ ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ X3 ) )
                          & ( set_pr2532843088427607396_a_nat @ Less @ Proper_interval @ ( some_Sum_sum_a_nat @ X3 ) @ Xs3 @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) )
                      & ( ~ ( Less @ X3 @ Y4 )
                       => ( ( ( Less @ Y4 @ X3 )
                           => ( ~ ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ Y4 ) )
                              & ( set_pr2532843088427607396_a_nat @ Less @ Proper_interval @ ( some_Sum_sum_a_nat @ Y4 ) @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ Ys2 ) ) )
                          & ( ~ ( Less @ Y4 @ X3 )
                           => ~ ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ Y4 ) ) ) ) ) ) ) ) ) ) ) ).

% proper_intrvl.Compl_set_less_aux.elims(3)
thf(fact_1189_proper__intrvl_OCompl__set__less__aux_Oelims_I1_J,axiom,
    ! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Proper_interval: option_Sum_sum_a_nat > option_Sum_sum_a_nat > $o,X: option_Sum_sum_a_nat,Xa: list_Sum_sum_a_nat,Xb: list_Sum_sum_a_nat,Y: $o] :
      ( ( ( set_pr2532843088427607396_a_nat @ Less @ Proper_interval @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_Sum_sum_a_nat )
         => Y )
       => ( ( ( Xb = nil_Sum_sum_a_nat )
           => Y )
         => ~ ! [X3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] :
                ( ( Xa
                  = ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) )
               => ! [Y4: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
                    ( ( Xb
                      = ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) )
                   => ( Y
                      = ( ~ ( ( ( Less @ X3 @ Y4 )
                             => ( ~ ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ X3 ) )
                                & ( set_pr2532843088427607396_a_nat @ Less @ Proper_interval @ ( some_Sum_sum_a_nat @ X3 ) @ Xs3 @ ( cons_Sum_sum_a_nat @ Y4 @ Ys2 ) ) ) )
                            & ( ~ ( Less @ X3 @ Y4 )
                             => ( ( ( Less @ Y4 @ X3 )
                                 => ( ~ ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ Y4 ) )
                                    & ( set_pr2532843088427607396_a_nat @ Less @ Proper_interval @ ( some_Sum_sum_a_nat @ Y4 ) @ ( cons_Sum_sum_a_nat @ X3 @ Xs3 ) @ Ys2 ) ) )
                                & ( ~ ( Less @ Y4 @ X3 )
                                 => ~ ( Proper_interval @ X @ ( some_Sum_sum_a_nat @ Y4 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% proper_intrvl.Compl_set_less_aux.elims(1)
thf(fact_1190_proper__intrvl_OCompl__set__less__aux_Opsimps_I1_J,axiom,
    ! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Ao2: option_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Proper_interval: option_Sum_sum_a_nat > option_Sum_sum_a_nat > $o] :
      ( ( accp_P5320691669302659474_a_nat @ ( set_pr2097269997492014367_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ Ao2 @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Ys ) ) )
     => ~ ( set_pr2532843088427607396_a_nat @ Less @ Proper_interval @ Ao2 @ nil_Sum_sum_a_nat @ Ys ) ) ).

% proper_intrvl.Compl_set_less_aux.psimps(1)
thf(fact_1191_proper__intrvl_OCompl__set__less__aux_Opsimps_I2_J,axiom,
    ! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Ao2: option_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Proper_interval: option_Sum_sum_a_nat > option_Sum_sum_a_nat > $o] :
      ( ( accp_P5320691669302659474_a_nat @ ( set_pr2097269997492014367_a_nat @ Less ) @ ( produc785221303313254649_a_nat @ Ao2 @ ( produc7990843422341522135_a_nat @ Xs @ nil_Sum_sum_a_nat ) ) )
     => ~ ( set_pr2532843088427607396_a_nat @ Less @ Proper_interval @ Ao2 @ Xs @ nil_Sum_sum_a_nat ) ) ).

% proper_intrvl.Compl_set_less_aux.psimps(2)
thf(fact_1192_take__Suc__conv__app__nth,axiom,
    ! [I2: nat,Xs: list_Sum_sum_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( take_Sum_sum_a_nat @ ( suc @ I2 ) @ Xs )
        = ( append_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ I2 @ Xs ) @ ( cons_Sum_sum_a_nat @ ( nth_Sum_sum_a_nat @ Xs @ I2 ) @ nil_Sum_sum_a_nat ) ) ) ) ).

% take_Suc_conv_app_nth
thf(fact_1193_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N @ M ) ) ).

% Suc_le_mono
thf(fact_1194_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).

% min_Suc_Suc
thf(fact_1195_take__Suc__Cons,axiom,
    ! [N: nat,X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
      ( ( take_Sum_sum_a_nat @ ( suc @ N ) @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
      = ( cons_Sum_sum_a_nat @ X @ ( take_Sum_sum_a_nat @ N @ Xs ) ) ) ).

% take_Suc_Cons
thf(fact_1196_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_1197_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1198_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_1199_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_1200_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_1201_inc__induct,axiom,
    ! [I2: nat,J2: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( P @ J2 )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N2 )
             => ( ( ord_less_nat @ N2 @ J2 )
               => ( ( P @ ( suc @ N2 ) )
                 => ( P @ N2 ) ) ) )
         => ( P @ I2 ) ) ) ) ).

% inc_induct
thf(fact_1202_dec__induct,axiom,
    ! [I2: nat,J2: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( P @ I2 )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N2 )
             => ( ( ord_less_nat @ N2 @ J2 )
               => ( ( P @ N2 )
                 => ( P @ ( suc @ N2 ) ) ) ) )
         => ( P @ J2 ) ) ) ) ).

% dec_induct
thf(fact_1203_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_1204_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_1205_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N5: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1206_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N5: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1207_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_mult_le_cancel1
thf(fact_1208_list__update__code_I3_J,axiom,
    ! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,I2: nat,Y: sum_sum_a_nat] :
      ( ( list_u9138855634547462509_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ ( suc @ I2 ) @ Y )
      = ( cons_Sum_sum_a_nat @ X @ ( list_u9138855634547462509_a_nat @ Xs @ I2 @ Y ) ) ) ).

% list_update_code(3)
thf(fact_1209_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X3: nat] : ( R @ X3 @ X3 )
       => ( ! [X3: nat,Y4: nat,Z2: nat] :
              ( ( R @ X3 @ Y4 )
             => ( ( R @ Y4 @ Z2 )
               => ( R @ X3 @ Z2 ) ) )
         => ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
           => ( R @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_1210_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P @ M )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_1211_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
             => ( P @ M4 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_1212_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).

% not_less_eq_eq
thf(fact_1213_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_1214_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M @ N )
        | ( M
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_1215_Suc__le__D,axiom,
    ! [N: nat,M5: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
     => ? [M6: nat] :
          ( M5
          = ( suc @ M6 ) ) ) ).

% Suc_le_D
thf(fact_1216_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_1217_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M @ N )
       => ( M
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_1218_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_1219_length__append__singleton,axiom,
    ! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
      ( ( size_s5283204784079214577_a_nat @ ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) )
      = ( suc @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ).

% length_append_singleton
thf(fact_1220_length__Suc__conv__rev,axiom,
    ! [Xs: list_Sum_sum_a_nat,N: nat] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( suc @ N ) )
      = ( ? [Y3: sum_sum_a_nat,Ys3: list_Sum_sum_a_nat] :
            ( ( Xs
              = ( append_Sum_sum_a_nat @ Ys3 @ ( cons_Sum_sum_a_nat @ Y3 @ nil_Sum_sum_a_nat ) ) )
            & ( ( size_s5283204784079214577_a_nat @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_1221_ran__map__of__zip,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( size_s5283204784079214577_a_nat @ Xs )
        = ( size_s5283204784079214577_a_nat @ Ys ) )
     => ( ( distin2701893636801681144_a_nat @ Xs )
       => ( ( ran_Su6606007654835531183_a_nat @ ( map_of7357225391242338723_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) ) )
          = ( set_Sum_sum_a_nat2 @ Ys ) ) ) ) ).

% ran_map_of_zip
thf(fact_1222_upd__conv__take__nth__drop,axiom,
    ! [I2: nat,Xs: list_Sum_sum_a_nat,A: sum_sum_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( list_u9138855634547462509_a_nat @ Xs @ I2 @ A )
        = ( append_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ I2 @ Xs ) @ ( cons_Sum_sum_a_nat @ A @ ( drop_Sum_sum_a_nat @ ( suc @ I2 ) @ Xs ) ) ) ) ) ).

% upd_conv_take_nth_drop
thf(fact_1223_append__take__drop__id,axiom,
    ! [N: nat,Xs: list_Sum_sum_a_nat] :
      ( ( append_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ N @ Xs ) @ ( drop_Sum_sum_a_nat @ N @ Xs ) )
      = Xs ) ).

% append_take_drop_id
thf(fact_1224_drop__update__cancel,axiom,
    ! [N: nat,M: nat,Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
      ( ( ord_less_nat @ N @ M )
     => ( ( drop_Sum_sum_a_nat @ M @ ( list_u9138855634547462509_a_nat @ Xs @ N @ X ) )
        = ( drop_Sum_sum_a_nat @ M @ Xs ) ) ) ).

% drop_update_cancel
thf(fact_1225_drop__eq__Nil2,axiom,
    ! [N: nat,Xs: list_Sum_sum_a_nat] :
      ( ( nil_Sum_sum_a_nat
        = ( drop_Sum_sum_a_nat @ N @ Xs ) )
      = ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ N ) ) ).

% drop_eq_Nil2
thf(fact_1226_drop__eq__Nil,axiom,
    ! [N: nat,Xs: list_Sum_sum_a_nat] :
      ( ( ( drop_Sum_sum_a_nat @ N @ Xs )
        = nil_Sum_sum_a_nat )
      = ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ N ) ) ).

% drop_eq_Nil
thf(fact_1227_drop__all,axiom,
    ! [Xs: list_Sum_sum_a_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ N )
     => ( ( drop_Sum_sum_a_nat @ N @ Xs )
        = nil_Sum_sum_a_nat ) ) ).

% drop_all
thf(fact_1228_set__drop__subset__set__drop,axiom,
    ! [N: nat,M: nat,Xs: list_Sum_sum_a_nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ ( drop_Sum_sum_a_nat @ M @ Xs ) ) @ ( set_Sum_sum_a_nat2 @ ( drop_Sum_sum_a_nat @ N @ Xs ) ) ) ) ).

% set_drop_subset_set_drop
thf(fact_1229_extract__None__iff,axiom,
    ! [P: sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] :
      ( ( ( extrac3289232528736335879_a_nat @ P @ Xs )
        = none_P2628347790636752432_a_nat )
      = ( ~ ? [X2: sum_sum_a_nat] :
              ( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
              & ( P @ X2 ) ) ) ) ).

% extract_None_iff
thf(fact_1230_extract__Nil__code,axiom,
    ! [P: sum_sum_a_nat > $o] :
      ( ( extrac3289232528736335879_a_nat @ P @ nil_Sum_sum_a_nat )
      = none_P2628347790636752432_a_nat ) ).

% extract_Nil_code
thf(fact_1231_set__drop__subset,axiom,
    ! [N: nat,Xs: list_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ ( drop_Sum_sum_a_nat @ N @ Xs ) ) @ ( set_Sum_sum_a_nat2 @ Xs ) ) ).

% set_drop_subset
thf(fact_1232_drop__Nil,axiom,
    ! [N: nat] :
      ( ( drop_Sum_sum_a_nat @ N @ nil_Sum_sum_a_nat )
      = nil_Sum_sum_a_nat ) ).

% drop_Nil
thf(fact_1233_distinct__drop,axiom,
    ! [Xs: list_Sum_sum_a_nat,I2: nat] :
      ( ( distin2701893636801681144_a_nat @ Xs )
     => ( distin2701893636801681144_a_nat @ ( drop_Sum_sum_a_nat @ I2 @ Xs ) ) ) ).

% distinct_drop
thf(fact_1234_in__set__dropD,axiom,
    ! [X: sum_sum_a_nat,N: nat,Xs: list_Sum_sum_a_nat] :
      ( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ ( drop_Sum_sum_a_nat @ N @ Xs ) ) )
     => ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ).

% in_set_dropD
thf(fact_1235_append__eq__conv__conj,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
      ( ( ( append_Sum_sum_a_nat @ Xs @ Ys )
        = Zs2 )
      = ( ( Xs
          = ( take_Sum_sum_a_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ Zs2 ) )
        & ( Ys
          = ( drop_Sum_sum_a_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ Zs2 ) ) ) ) ).

% append_eq_conv_conj
thf(fact_1236_append__eq__append__conv__if,axiom,
    ! [Xs_1: list_Sum_sum_a_nat,Xs_2: list_Sum_sum_a_nat,Ys_1: list_Sum_sum_a_nat,Ys_2: list_Sum_sum_a_nat] :
      ( ( ( append_Sum_sum_a_nat @ Xs_1 @ Xs_2 )
        = ( append_Sum_sum_a_nat @ Ys_1 @ Ys_2 ) )
      = ( ( ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs_1 ) @ ( size_s5283204784079214577_a_nat @ Ys_1 ) )
         => ( ( Xs_1
              = ( take_Sum_sum_a_nat @ ( size_s5283204784079214577_a_nat @ Xs_1 ) @ Ys_1 ) )
            & ( Xs_2
              = ( append_Sum_sum_a_nat @ ( drop_Sum_sum_a_nat @ ( size_s5283204784079214577_a_nat @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs_1 ) @ ( size_s5283204784079214577_a_nat @ Ys_1 ) )
         => ( ( ( take_Sum_sum_a_nat @ ( size_s5283204784079214577_a_nat @ Ys_1 ) @ Xs_1 )
              = Ys_1 )
            & ( ( append_Sum_sum_a_nat @ ( drop_Sum_sum_a_nat @ ( size_s5283204784079214577_a_nat @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
              = Ys_2 ) ) ) ) ) ).

% append_eq_append_conv_if
thf(fact_1237_id__take__nth__drop,axiom,
    ! [I2: nat,Xs: list_Sum_sum_a_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( Xs
        = ( append_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ I2 @ Xs ) @ ( cons_Sum_sum_a_nat @ ( nth_Sum_sum_a_nat @ Xs @ I2 ) @ ( drop_Sum_sum_a_nat @ ( suc @ I2 ) @ Xs ) ) ) ) ) ).

% id_take_nth_drop
thf(fact_1238_take__hd__drop,axiom,
    ! [N: nat,Xs: list_Sum_sum_a_nat] :
      ( ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
     => ( ( append_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ N @ Xs ) @ ( cons_Sum_sum_a_nat @ ( hd_Sum_sum_a_nat @ ( drop_Sum_sum_a_nat @ N @ Xs ) ) @ nil_Sum_sum_a_nat ) )
        = ( take_Sum_sum_a_nat @ ( suc @ N ) @ Xs ) ) ) ).

% take_hd_drop
thf(fact_1239_hd__append2,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( Xs != nil_Sum_sum_a_nat )
     => ( ( hd_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
        = ( hd_Sum_sum_a_nat @ Xs ) ) ) ).

% hd_append2
thf(fact_1240_hd__append,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( ( Xs = nil_Sum_sum_a_nat )
       => ( ( hd_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
          = ( hd_Sum_sum_a_nat @ Ys ) ) )
      & ( ( Xs != nil_Sum_sum_a_nat )
       => ( ( hd_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
          = ( hd_Sum_sum_a_nat @ Xs ) ) ) ) ).

% hd_append
thf(fact_1241_longest__common__prefix,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
    ? [Ps2: list_Sum_sum_a_nat,Xs5: list_Sum_sum_a_nat,Ys6: list_Sum_sum_a_nat] :
      ( ( Xs
        = ( append_Sum_sum_a_nat @ Ps2 @ Xs5 ) )
      & ( Ys
        = ( append_Sum_sum_a_nat @ Ps2 @ Ys6 ) )
      & ( ( Xs5 = nil_Sum_sum_a_nat )
        | ( Ys6 = nil_Sum_sum_a_nat )
        | ( ( hd_Sum_sum_a_nat @ Xs5 )
         != ( hd_Sum_sum_a_nat @ Ys6 ) ) ) ) ).

% longest_common_prefix
thf(fact_1242_hd__in__set,axiom,
    ! [Xs: list_Sum_sum_a_nat] :
      ( ( Xs != nil_Sum_sum_a_nat )
     => ( member_Sum_sum_a_nat @ ( hd_Sum_sum_a_nat @ Xs ) @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ).

% hd_in_set
thf(fact_1243_list_Oset__sel_I1_J,axiom,
    ! [A: list_Sum_sum_a_nat] :
      ( ( A != nil_Sum_sum_a_nat )
     => ( member_Sum_sum_a_nat @ ( hd_Sum_sum_a_nat @ A ) @ ( set_Sum_sum_a_nat2 @ A ) ) ) ).

% list.set_sel(1)
thf(fact_1244_hd__zip,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
      ( ( Xs != nil_Sum_sum_a_nat )
     => ( ( Ys != nil_Sum_sum_a_nat )
       => ( ( hd_Pro3965030861105696466_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) )
          = ( produc1212125651291703639_a_nat @ ( hd_Sum_sum_a_nat @ Xs ) @ ( hd_Sum_sum_a_nat @ Ys ) ) ) ) ) ).

% hd_zip
thf(fact_1245_listrel1__iff__update,axiom,
    ! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,R2: set_Pr7343886759072863943_a_nat] :
      ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Ys ) @ ( listre5822253806113410398_a_nat @ R2 ) )
      = ( ? [Y3: sum_sum_a_nat,N4: nat] :
            ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ ( nth_Sum_sum_a_nat @ Xs @ N4 ) @ Y3 ) @ R2 )
            & ( ord_less_nat @ N4 @ ( size_s5283204784079214577_a_nat @ Xs ) )
            & ( Ys
              = ( list_u9138855634547462509_a_nat @ Xs @ N4 @ Y3 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1246_not__Nil__listrel1,axiom,
    ! [Xs: list_Sum_sum_a_nat,R2: set_Pr7343886759072863943_a_nat] :
      ~ ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Xs ) @ ( listre5822253806113410398_a_nat @ R2 ) ) ).

% not_Nil_listrel1
thf(fact_1247_not__listrel1__Nil,axiom,
    ! [Xs: list_Sum_sum_a_nat,R2: set_Pr7343886759072863943_a_nat] :
      ~ ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ nil_Sum_sum_a_nat ) @ ( listre5822253806113410398_a_nat @ R2 ) ) ).

% not_listrel1_Nil
thf(fact_1248_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Y: sum_sum_a_nat,R2: set_Pr7343886759072863943_a_nat] :
      ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) @ ( append_Sum_sum_a_nat @ Ys @ ( cons_Sum_sum_a_nat @ Y @ nil_Sum_sum_a_nat ) ) ) @ ( listre5822253806113410398_a_nat @ R2 ) )
      = ( ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Ys ) @ ( listre5822253806113410398_a_nat @ R2 ) )
          & ( X = Y ) )
        | ( ( Xs = Ys )
          & ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X @ Y ) @ R2 ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_1249_Compl__set__less__aux_Opelims_I3_J,axiom,
    ! [X: option_nat,Xa: list_nat,Xb: list_nat] :
      ( ~ ( set_pr2156344338276335947ux_nat @ X @ Xa @ Xb )
     => ( ( accp_P5867667535934726061st_nat @ set_pr6598557696555394320el_nat @ ( produc6570249034887359912st_nat @ X @ ( produc2694037385005941721st_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_nat )
           => ~ ( accp_P5867667535934726061st_nat @ set_pr6598557696555394320el_nat @ ( produc6570249034887359912st_nat @ X @ ( produc2694037385005941721st_nat @ nil_nat @ Xb ) ) ) )
         => ( ( ( Xb = nil_nat )
             => ~ ( accp_P5867667535934726061st_nat @ set_pr6598557696555394320el_nat @ ( produc6570249034887359912st_nat @ X @ ( produc2694037385005941721st_nat @ Xa @ nil_nat ) ) ) )
           => ~ ! [X3: nat,Xs3: list_nat] :
                  ( ( Xa
                    = ( cons_nat @ X3 @ Xs3 ) )
                 => ! [Y4: nat,Ys2: list_nat] :
                      ( ( Xb
                        = ( cons_nat @ Y4 @ Ys2 ) )
                     => ( ( accp_P5867667535934726061st_nat @ set_pr6598557696555394320el_nat @ ( produc6570249034887359912st_nat @ X @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y4 @ Ys2 ) ) ) )
                       => ( ( ( ord_less_nat @ X3 @ Y4 )
                           => ( ~ ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ X3 ) )
                              & ( set_pr2156344338276335947ux_nat @ ( some_nat @ X3 ) @ Xs3 @ ( cons_nat @ Y4 @ Ys2 ) ) ) )
                          & ( ~ ( ord_less_nat @ X3 @ Y4 )
                           => ( ( ( ord_less_nat @ Y4 @ X3 )
                               => ( ~ ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ Y4 ) )
                                  & ( set_pr2156344338276335947ux_nat @ ( some_nat @ Y4 ) @ ( cons_nat @ X3 @ Xs3 ) @ Ys2 ) ) )
                              & ( ~ ( ord_less_nat @ Y4 @ X3 )
                               => ~ ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ Y4 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Compl_set_less_aux.pelims(3)
thf(fact_1250_Compl__set__less__aux_Opelims_I1_J,axiom,
    ! [X: option_nat,Xa: list_nat,Xb: list_nat,Y: $o] :
      ( ( ( set_pr2156344338276335947ux_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P5867667535934726061st_nat @ set_pr6598557696555394320el_nat @ ( produc6570249034887359912st_nat @ X @ ( produc2694037385005941721st_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_nat )
           => ( ~ Y
             => ~ ( accp_P5867667535934726061st_nat @ set_pr6598557696555394320el_nat @ ( produc6570249034887359912st_nat @ X @ ( produc2694037385005941721st_nat @ nil_nat @ Xb ) ) ) ) )
         => ( ( ( Xb = nil_nat )
             => ( ~ Y
               => ~ ( accp_P5867667535934726061st_nat @ set_pr6598557696555394320el_nat @ ( produc6570249034887359912st_nat @ X @ ( produc2694037385005941721st_nat @ Xa @ nil_nat ) ) ) ) )
           => ~ ! [X3: nat,Xs3: list_nat] :
                  ( ( Xa
                    = ( cons_nat @ X3 @ Xs3 ) )
                 => ! [Y4: nat,Ys2: list_nat] :
                      ( ( Xb
                        = ( cons_nat @ Y4 @ Ys2 ) )
                     => ( ( Y
                          = ( ( ( ord_less_nat @ X3 @ Y4 )
                             => ( ~ ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ X3 ) )
                                & ( set_pr2156344338276335947ux_nat @ ( some_nat @ X3 ) @ Xs3 @ ( cons_nat @ Y4 @ Ys2 ) ) ) )
                            & ( ~ ( ord_less_nat @ X3 @ Y4 )
                             => ( ( ( ord_less_nat @ Y4 @ X3 )
                                 => ( ~ ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ Y4 ) )
                                    & ( set_pr2156344338276335947ux_nat @ ( some_nat @ Y4 ) @ ( cons_nat @ X3 @ Xs3 ) @ Ys2 ) ) )
                                & ( ~ ( ord_less_nat @ Y4 @ X3 )
                                 => ~ ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ Y4 ) ) ) ) ) ) )
                       => ~ ( accp_P5867667535934726061st_nat @ set_pr6598557696555394320el_nat @ ( produc6570249034887359912st_nat @ X @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y4 @ Ys2 ) ) ) ) ) ) ) ) ) ) ) ).

% Compl_set_less_aux.pelims(1)
thf(fact_1251_Compl__set__less__aux_Oelims_I3_J,axiom,
    ! [X: option_nat,Xa: list_nat,Xb: list_nat] :
      ( ~ ( set_pr2156344338276335947ux_nat @ X @ Xa @ Xb )
     => ( ( Xa != nil_nat )
       => ( ( Xb != nil_nat )
         => ~ ! [X3: nat,Xs3: list_nat] :
                ( ( Xa
                  = ( cons_nat @ X3 @ Xs3 ) )
               => ! [Y4: nat,Ys2: list_nat] :
                    ( ( Xb
                      = ( cons_nat @ Y4 @ Ys2 ) )
                   => ( ( ( ord_less_nat @ X3 @ Y4 )
                       => ( ~ ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ X3 ) )
                          & ( set_pr2156344338276335947ux_nat @ ( some_nat @ X3 ) @ Xs3 @ ( cons_nat @ Y4 @ Ys2 ) ) ) )
                      & ( ~ ( ord_less_nat @ X3 @ Y4 )
                       => ( ( ( ord_less_nat @ Y4 @ X3 )
                           => ( ~ ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ Y4 ) )
                              & ( set_pr2156344338276335947ux_nat @ ( some_nat @ Y4 ) @ ( cons_nat @ X3 @ Xs3 ) @ Ys2 ) ) )
                          & ( ~ ( ord_less_nat @ Y4 @ X3 )
                           => ~ ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ Y4 ) ) ) ) ) ) ) ) ) ) ) ).

% Compl_set_less_aux.elims(3)
thf(fact_1252_Compl__set__less__aux_Oelims_I1_J,axiom,
    ! [X: option_nat,Xa: list_nat,Xb: list_nat,Y: $o] :
      ( ( ( set_pr2156344338276335947ux_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_nat )
         => Y )
       => ( ( ( Xb = nil_nat )
           => Y )
         => ~ ! [X3: nat,Xs3: list_nat] :
                ( ( Xa
                  = ( cons_nat @ X3 @ Xs3 ) )
               => ! [Y4: nat,Ys2: list_nat] :
                    ( ( Xb
                      = ( cons_nat @ Y4 @ Ys2 ) )
                   => ( Y
                      = ( ~ ( ( ( ord_less_nat @ X3 @ Y4 )
                             => ( ~ ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ X3 ) )
                                & ( set_pr2156344338276335947ux_nat @ ( some_nat @ X3 ) @ Xs3 @ ( cons_nat @ Y4 @ Ys2 ) ) ) )
                            & ( ~ ( ord_less_nat @ X3 @ Y4 )
                             => ( ( ( ord_less_nat @ Y4 @ X3 )
                                 => ( ~ ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ Y4 ) )
                                    & ( set_pr2156344338276335947ux_nat @ ( some_nat @ Y4 ) @ ( cons_nat @ X3 @ Xs3 ) @ Ys2 ) ) )
                                & ( ~ ( ord_less_nat @ Y4 @ X3 )
                                 => ~ ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ Y4 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Compl_set_less_aux.elims(1)
thf(fact_1253_set__less__eq__aux__Compl_Opelims_I1_J,axiom,
    ! [X: option_nat,Xa: list_nat,Xb: list_nat,Y: $o] :
      ( ( ( set_pr4650252130045998533pl_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P5867667535934726061st_nat @ set_pr9200262421968554262el_nat @ ( produc6570249034887359912st_nat @ X @ ( produc2694037385005941721st_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_nat )
           => ( Y
             => ~ ( accp_P5867667535934726061st_nat @ set_pr9200262421968554262el_nat @ ( produc6570249034887359912st_nat @ X @ ( produc2694037385005941721st_nat @ nil_nat @ Xb ) ) ) ) )
         => ( ( ( Xb = nil_nat )
             => ( Y
               => ~ ( accp_P5867667535934726061st_nat @ set_pr9200262421968554262el_nat @ ( produc6570249034887359912st_nat @ X @ ( produc2694037385005941721st_nat @ Xa @ nil_nat ) ) ) ) )
           => ~ ! [X3: nat,Xs3: list_nat] :
                  ( ( Xa
                    = ( cons_nat @ X3 @ Xs3 ) )
                 => ! [Y4: nat,Ys2: list_nat] :
                      ( ( Xb
                        = ( cons_nat @ Y4 @ Ys2 ) )
                     => ( ( Y
                          = ( ( ( ord_less_nat @ X3 @ Y4 )
                             => ( ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ X3 ) )
                                | ( set_pr4650252130045998533pl_nat @ ( some_nat @ X3 ) @ Xs3 @ ( cons_nat @ Y4 @ Ys2 ) ) ) )
                            & ( ~ ( ord_less_nat @ X3 @ Y4 )
                             => ( ( ( ord_less_nat @ Y4 @ X3 )
                                 => ( ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ Y4 ) )
                                    | ( set_pr4650252130045998533pl_nat @ ( some_nat @ Y4 ) @ ( cons_nat @ X3 @ Xs3 ) @ Ys2 ) ) )
                                & ( ~ ( ord_less_nat @ Y4 @ X3 )
                                 => ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ Y4 ) ) ) ) ) ) )
                       => ~ ( accp_P5867667535934726061st_nat @ set_pr9200262421968554262el_nat @ ( produc6570249034887359912st_nat @ X @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y4 @ Ys2 ) ) ) ) ) ) ) ) ) ) ) ).

% set_less_eq_aux_Compl.pelims(1)
thf(fact_1254_set__less__eq__aux__Compl_Opelims_I2_J,axiom,
    ! [X: option_nat,Xa: list_nat,Xb: list_nat] :
      ( ( set_pr4650252130045998533pl_nat @ X @ Xa @ Xb )
     => ( ( accp_P5867667535934726061st_nat @ set_pr9200262421968554262el_nat @ ( produc6570249034887359912st_nat @ X @ ( produc2694037385005941721st_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_nat )
           => ~ ( accp_P5867667535934726061st_nat @ set_pr9200262421968554262el_nat @ ( produc6570249034887359912st_nat @ X @ ( produc2694037385005941721st_nat @ nil_nat @ Xb ) ) ) )
         => ( ( ( Xb = nil_nat )
             => ~ ( accp_P5867667535934726061st_nat @ set_pr9200262421968554262el_nat @ ( produc6570249034887359912st_nat @ X @ ( produc2694037385005941721st_nat @ Xa @ nil_nat ) ) ) )
           => ~ ! [X3: nat,Xs3: list_nat] :
                  ( ( Xa
                    = ( cons_nat @ X3 @ Xs3 ) )
                 => ! [Y4: nat,Ys2: list_nat] :
                      ( ( Xb
                        = ( cons_nat @ Y4 @ Ys2 ) )
                     => ( ( accp_P5867667535934726061st_nat @ set_pr9200262421968554262el_nat @ ( produc6570249034887359912st_nat @ X @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y4 @ Ys2 ) ) ) )
                       => ~ ( ( ( ord_less_nat @ X3 @ Y4 )
                             => ( ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ X3 ) )
                                | ( set_pr4650252130045998533pl_nat @ ( some_nat @ X3 ) @ Xs3 @ ( cons_nat @ Y4 @ Ys2 ) ) ) )
                            & ( ~ ( ord_less_nat @ X3 @ Y4 )
                             => ( ( ( ord_less_nat @ Y4 @ X3 )
                                 => ( ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ Y4 ) )
                                    | ( set_pr4650252130045998533pl_nat @ ( some_nat @ Y4 ) @ ( cons_nat @ X3 @ Xs3 ) @ Ys2 ) ) )
                                & ( ~ ( ord_less_nat @ Y4 @ X3 )
                                 => ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ Y4 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% set_less_eq_aux_Compl.pelims(2)
thf(fact_1255_set__less__eq__aux__Compl_Oelims_I1_J,axiom,
    ! [X: option_nat,Xa: list_nat,Xb: list_nat,Y: $o] :
      ( ( ( set_pr4650252130045998533pl_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_nat )
         => ~ Y )
       => ( ( ( Xb = nil_nat )
           => ~ Y )
         => ~ ! [X3: nat,Xs3: list_nat] :
                ( ( Xa
                  = ( cons_nat @ X3 @ Xs3 ) )
               => ! [Y4: nat,Ys2: list_nat] :
                    ( ( Xb
                      = ( cons_nat @ Y4 @ Ys2 ) )
                   => ( Y
                      = ( ~ ( ( ( ord_less_nat @ X3 @ Y4 )
                             => ( ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ X3 ) )
                                | ( set_pr4650252130045998533pl_nat @ ( some_nat @ X3 ) @ Xs3 @ ( cons_nat @ Y4 @ Ys2 ) ) ) )
                            & ( ~ ( ord_less_nat @ X3 @ Y4 )
                             => ( ( ( ord_less_nat @ Y4 @ X3 )
                                 => ( ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ Y4 ) )
                                    | ( set_pr4650252130045998533pl_nat @ ( some_nat @ Y4 ) @ ( cons_nat @ X3 @ Xs3 ) @ Ys2 ) ) )
                                & ( ~ ( ord_less_nat @ Y4 @ X3 )
                                 => ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ Y4 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% set_less_eq_aux_Compl.elims(1)
thf(fact_1256_set__less__eq__aux__Compl_Oelims_I2_J,axiom,
    ! [X: option_nat,Xa: list_nat,Xb: list_nat] :
      ( ( set_pr4650252130045998533pl_nat @ X @ Xa @ Xb )
     => ( ( Xa != nil_nat )
       => ( ( Xb != nil_nat )
         => ~ ! [X3: nat,Xs3: list_nat] :
                ( ( Xa
                  = ( cons_nat @ X3 @ Xs3 ) )
               => ! [Y4: nat,Ys2: list_nat] :
                    ( ( Xb
                      = ( cons_nat @ Y4 @ Ys2 ) )
                   => ~ ( ( ( ord_less_nat @ X3 @ Y4 )
                         => ( ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ X3 ) )
                            | ( set_pr4650252130045998533pl_nat @ ( some_nat @ X3 ) @ Xs3 @ ( cons_nat @ Y4 @ Ys2 ) ) ) )
                        & ( ~ ( ord_less_nat @ X3 @ Y4 )
                         => ( ( ( ord_less_nat @ Y4 @ X3 )
                             => ( ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ Y4 ) )
                                | ( set_pr4650252130045998533pl_nat @ ( some_nat @ Y4 ) @ ( cons_nat @ X3 @ Xs3 ) @ Ys2 ) ) )
                            & ( ~ ( ord_less_nat @ Y4 @ X3 )
                             => ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ Y4 ) ) ) ) ) ) ) ) ) ) ) ).

% set_less_eq_aux_Compl.elims(2)
thf(fact_1257_set__less__aux__Compl_Oelims_I1_J,axiom,
    ! [X: option_nat,Xa: list_nat,Xb: list_nat,Y: $o] :
      ( ( ( set_pr3298563537802266553pl_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_nat )
         => ( ( Xb = nil_nat )
           => ( Y
              = ( ~ ( set_pr3848668467136637714al_nat @ X @ none_nat ) ) ) ) )
       => ( ( ( Xa = nil_nat )
           => ! [Y4: nat,Ys2: list_nat] :
                ( ( Xb
                  = ( cons_nat @ Y4 @ Ys2 ) )
               => ( Y
                  = ( ~ ( ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ Y4 ) )
                        | ( set_pr3298563537802266553pl_nat @ ( some_nat @ Y4 ) @ nil_nat @ Ys2 ) ) ) ) ) )
         => ( ! [X3: nat,Xs3: list_nat] :
                ( ( Xa
                  = ( cons_nat @ X3 @ Xs3 ) )
               => ( ( Xb = nil_nat )
                 => ( Y
                    = ( ~ ( ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ X3 ) )
                          | ( set_pr3298563537802266553pl_nat @ ( some_nat @ X3 ) @ Xs3 @ nil_nat ) ) ) ) ) )
           => ~ ! [X3: nat,Xs3: list_nat] :
                  ( ( Xa
                    = ( cons_nat @ X3 @ Xs3 ) )
                 => ! [Y4: nat,Ys2: list_nat] :
                      ( ( Xb
                        = ( cons_nat @ Y4 @ Ys2 ) )
                     => ( Y
                        = ( ~ ( ( ( ord_less_nat @ X3 @ Y4 )
                               => ( ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ X3 ) )
                                  | ( set_pr3298563537802266553pl_nat @ ( some_nat @ X3 ) @ Xs3 @ ( cons_nat @ Y4 @ Ys2 ) ) ) )
                              & ( ~ ( ord_less_nat @ X3 @ Y4 )
                               => ( ( ( ord_less_nat @ Y4 @ X3 )
                                   => ( ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ Y4 ) )
                                      | ( set_pr3298563537802266553pl_nat @ ( some_nat @ Y4 ) @ ( cons_nat @ X3 @ Xs3 ) @ Ys2 ) ) )
                                  & ( ~ ( ord_less_nat @ Y4 @ X3 )
                                   => ( set_pr3848668467136637714al_nat @ X @ ( some_nat @ Y4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% set_less_aux_Compl.elims(1)

% Helper facts (7)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
      ( ( if_set_Sum_sum_a_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
      ( ( if_set_Sum_sum_a_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
      ( ( if_lis4685338526944683083_a_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
      ( ( if_lis4685338526944683083_a_nat @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    fo_nmlzd_a @ ad @ ( take_Sum_sum_a_nat @ i @ xs ) ).

%------------------------------------------------------------------------------