TPTP Problem File: SLH0240^1.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    : Prefix_Free_Code_Combinators/0000_Prefix_Free_Code_Combinators/prob_00571_019235__12023618_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1966 ( 577 unt; 683 typ;   0 def)
%            Number of atoms       : 3791 (1347 equ;   0 cnn)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives : 12150 ( 470   ~;  99   |; 217   &;9664   @)
%                                         (   0 <=>;1700  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   7 avg)
%            Number of types       :  121 ( 120 usr)
%            Number of type conns  : 2246 (2246   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  566 ( 563 usr;  40 con; 0-3 aty)
%            Number of variables   : 4141 ( 463   ^;3556   !; 122   ?;4141   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 10:02:55.762
%------------------------------------------------------------------------------
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__List__Olist_Itf__b_J_J_J,type,
    set_Pr4340016442279239812list_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__Real__Oreal_J_J,type,
    set_Pr873410624239679766b_real: $tType ).

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

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

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

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

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__Nat__Onat_J_J,type,
    set_Pr3469313283233365050_b_nat: $tType ).

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

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    set_Pr6218003697084177305l_real: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    list_P6834414599653733731al_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    set_Pr3510011417693777981al_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    set_Pr320017278500174781t_real: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    produc1540777390238407569at_nat: $tType ).

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

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

thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__List__Olist_Itf__b_J_J,type,
    produc959683350210080206list_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__Real__Oreal_J,type,
    produc4645575661736406880b_real: $tType ).

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

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__Nat__Onat_J,type,
    produc6020613750378442500_b_nat: $tType ).

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

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

thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    produc2422161461964618553l_real: $tType ).

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

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

thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
    produc3741383161447143261al_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    produc7716430852924023517t_real: $tType ).

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

thf(ty_n_t__List__Olist_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    list_nat_nat: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Real__Oreal_J_J,type,
    set_list_real: $tType ).

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

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

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

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

thf(ty_n_t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
    option_list_o: $tType ).

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

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

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

thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
    list_real: $tType ).

thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
    set_real: $tType ).

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

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

thf(ty_n_t__List__Olist_Itf__b_J,type,
    list_b: $tType ).

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

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

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

thf(ty_n_t__Real__Oreal,type,
    real: $tType ).

thf(ty_n_t__Num__Onum,type,
    num: $tType ).

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

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

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

% Explicit typings (563)
thf(sy_c_Countable_Oto__nat_001t__Nat__Onat,type,
    to_nat_nat: nat > nat ).

thf(sy_c_Countable_Oto__nat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    to_nat7320730079965953815at_nat: product_prod_nat_nat > nat ).

thf(sy_c_Countable_Oto__nat_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    to_nat5607129738308160893at_nat: produc859450856879609959at_nat > nat ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__List__Olist_Itf__b_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J_J,type,
    inj_on5686930240773211284list_b: ( list_list_b > list_P5454506040216281530list_b ) > set_list_list_b > $o ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    inj_on3049792774292151987st_nat: ( list_nat > list_nat ) > set_list_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J_J,type,
    inj_on23012040918581517list_b: ( list_nat > list_P5454506040216281530list_b ) > set_list_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    inj_on3024915551208126196at_nat: ( list_nat > list_P6011104703257516679at_nat ) > set_list_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J_J,type,
    inj_on1287217757190318181al_nat: ( list_nat > list_P189569681052899986al_nat ) > set_list_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    inj_on3011817175317237783at_nat: ( list_P6011104703257516679at_nat > list_P8469869581646625389at_nat ) > set_li5450038453877631591at_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Real__Oreal_J_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    inj_on8276318967698619124al_nat: ( list_real > list_P6834414599653733731al_nat ) > set_list_real > $o ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__b_J_001t__List__Olist_Itf__b_J,type,
    inj_on_list_b_list_b: ( list_b > list_b ) > set_list_b > $o ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__b_J_001t__Nat__Onat,type,
    inj_on_list_b_nat: ( list_b > nat ) > set_list_b > $o ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__b_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J,type,
    inj_on8733544287791916926list_b: ( list_b > produc7811952446676219690list_b ) > set_list_b > $o ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__b_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    inj_on8730813633938913709at_nat: ( list_b > product_prod_nat_nat ) > set_list_b > $o ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__b_J_001tf__a,type,
    inj_on_list_b_a: ( list_b > a ) > set_list_b > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__List__Olist_Itf__b_J,type,
    inj_on_nat_list_b: ( nat > list_b ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
    inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J,type,
    inj_on8555073082499328493list_b: ( nat > produc7811952446676219690list_b ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    inj_on5538052773655684606at_nat: ( nat > product_prod_nat_nat ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    inj_on7152781252210839493al_nat: ( nat > produc9155354469731034754al_nat ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001tf__a,type,
    inj_on_nat_a: ( nat > a ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    inj_on3847583997030603456st_nat: ( produc1540777390238407569at_nat > list_nat ) > set_Pr7549493975000385137at_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__Nat__Onat_J,type,
    inj_on6310514192432462843_b_nat: ( produc6020613750378442500_b_nat > produc6020613750378442500_b_nat ) > set_Pr3469313283233365050_b_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J_J,type,
    inj_on3137949769578664533list_b: ( produc6020613750378442500_b_nat > produc8319926803371359198list_b ) > set_Pr3469313283233365050_b_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    inj_on9121707551203860396at_nat: ( produc6020613750378442500_b_nat > produc3137399281773485411at_nat ) > set_Pr3469313283233365050_b_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J,type,
    inj_on8101852888730240033list_b: ( produc6020613750378442500_b_nat > produc7811952446676219690list_b ) > set_Pr3469313283233365050_b_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J_Mt__Nat__Onat_J,type,
    inj_on5740833130252885284_b_nat: ( produc6020613750378442500_b_nat > produc8979896897677286107_b_nat ) > set_Pr3469313283233365050_b_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__Real__Oreal_J_001t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    inj_on5023972497546319276al_nat: ( produc4645575661736406880b_real > produc1586279338094070335al_nat ) > set_Pr873410624239679766b_real > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J,type,
    inj_on4901578499414790983list_b: ( produc7811952446676219690list_b > produc7811952446676219690list_b ) > set_Pr1349601357184307552list_b > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J_J,type,
    inj_on8157835898299261438list_b: ( produc7811952446676219690list_b > produc2515358133646292571list_b ) > set_Pr1349601357184307552list_b > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J_Mt__List__Olist_Itf__b_J_J,type,
    inj_on4371743984487328109list_b: ( produc7811952446676219690list_b > produc149682169844524240list_b ) > set_Pr1349601357184307552list_b > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__List__Olist_Itf__b_J_J,type,
    inj_on6772791774875031566list_b: ( produc7811952446676219690list_b > produc1130314010222062699list_b ) > set_Pr1349601357184307552list_b > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__List__Olist_Itf__b_J,type,
    inj_on7966946029222161077list_b: ( product_prod_nat_nat > list_b ) > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    inj_on2178005380612969504at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    inj_on8969904277767023793at_nat: ( product_prod_nat_nat > product_prod_nat_nat ) > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J_J,type,
    inj_on4534532566085093899list_b: ( product_prod_nat_nat > produc2515358133646292571list_b ) > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    inj_on5416022199913362294at_nat: ( product_prod_nat_nat > produc7248412053542808358at_nat ) > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J_J,type,
    inj_on935854358587236963al_nat: ( product_prod_nat_nat > produc5963376359100343987al_nat ) > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J_Mt__Nat__Onat_J,type,
    inj_on1775699293261311627_b_nat: ( product_prod_nat_nat > produc8979896897677286107_b_nat ) > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    inj_on6541509183880663376at_nat: ( product_prod_nat_nat > produc8373899037510109440at_nat ) > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    inj_on3865918002419084311at_nat: ( product_prod_nat_nat > produc859450856879609959at_nat ) > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
    inj_on2445742330725020003at_nat: ( product_prod_nat_nat > produc7473264331238127027at_nat ) > set_Pr1261947904930325089at_nat > $o ).

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thf(sy_c_Fun_Oinj__on_001tf__a_001t__Nat__Onat,type,
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thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
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thf(sy_c_Fun__Def_Opair__leq,type,
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thf(sy_c_Fun__Def_Opair__less,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Oappend_001t__Nat__Onat,type,
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thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
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thf(sy_c_List_Odrop_001t__Nat__Onat,type,
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thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
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thf(sy_c_List_Olenlex_001t__List__Olist_Itf__b_J,type,
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thf(sy_c_List_Olexn_001tf__b,type,
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thf(sy_c_List_Olexord_001t__List__Olist_Itf__b_J,type,
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    listrel_nat_list_b: set_Pr1349601357184307552list_b > set_Pr6009741699831149654list_b ).

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

thf(sy_c_List_Olistrel_001t__Nat__Onat_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
    listre1017866332378306906al_nat: set_Pr836851361167288120al_nat > set_Pr3318565038780223662al_nat ).

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

thf(sy_c_List_Olistrel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    listre818007680106770737at_nat: set_Pr8693737435421807431at_nat > set_Pr1542805901266377927at_nat ).

thf(sy_c_List_Olistrel_001t__Real__Oreal_001t__Nat__Onat,type,
    listrel_real_nat: set_Pr3510011417693777981al_nat > set_Pr6676954283489005405st_nat ).

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

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

thf(sy_c_List_Olistrelp_001t__Nat__Onat_001t__List__Olist_Itf__b_J,type,
    listrelp_nat_list_b: ( nat > list_b > $o ) > list_nat > list_list_b > $o ).

thf(sy_c_List_Olistrelp_001t__Nat__Onat_001t__Nat__Onat,type,
    listrelp_nat_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).

thf(sy_c_List_Olistrelp_001t__Nat__Onat_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
    listre8863158135660013452al_nat: ( nat > produc3741383161447143261al_nat > $o ) > list_nat > list_P6834414599653733731al_nat > $o ).

thf(sy_c_List_Olistrelp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    listre5588581128925307775at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o ).

thf(sy_c_List_Olistrelp_001t__Real__Oreal_001t__Nat__Onat,type,
    listrelp_real_nat: ( real > nat > $o ) > list_real > list_nat > $o ).

thf(sy_c_List_Olists_001t__List__Olist_Itf__b_J,type,
    lists_list_b: set_list_b > set_list_list_b ).

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

thf(sy_c_List_Olists_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    lists_3066315782749251317at_nat: set_Pr1261947904930325089at_nat > set_li5450038453877631591at_nat ).

thf(sy_c_List_Olists_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    lists_8807101532170170459at_nat: set_Pr8693737435421807431at_nat > set_li3197816953174176717at_nat ).

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

thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001t__Nat__Onat,type,
    map_tailrec_nat_nat: ( nat > nat ) > list_nat > list_nat ).

thf(sy_c_List_Omeasures_001t__Nat__Onat,type,
    measures_nat: list_nat_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_List_Omeasures_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    measur2679027848233739777at_nat: list_P9162950289778280392at_nat > set_Pr8693737435421807431at_nat ).

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

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

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
    nth_Pr3792769368220640996al_nat: list_P6834414599653733731al_nat > nat > produc3741383161447143261al_nat ).

thf(sy_c_List_Onth_001t__Real__Oreal,type,
    nth_real: list_real > nat > real ).

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

thf(sy_c_List_Oord__class_Olexordp_001t__Nat__Onat,type,
    ord_lexordp_nat: list_nat > list_nat > $o ).

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

thf(sy_c_List_OremoveAll_001t__List__Olist_Itf__b_J,type,
    removeAll_list_b: list_b > list_list_b > list_list_b ).

thf(sy_c_List_OremoveAll_001t__Nat__Onat,type,
    removeAll_nat: nat > list_nat > list_nat ).

thf(sy_c_List_OremoveAll_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J,type,
    remove5849340258981760030list_b: produc7811952446676219690list_b > list_P5454506040216281530list_b > list_P5454506040216281530list_b ).

thf(sy_c_List_OremoveAll_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    remove3673390508374433037at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_OremoveAll_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    remove6711459154290680182al_nat: produc9155354469731034754al_nat > list_P189569681052899986al_nat > list_P189569681052899986al_nat ).

thf(sy_c_List_OremoveAll_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    remove2181804207701385843at_nat: produc859450856879609959at_nat > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat ).

thf(sy_c_List_OremoveAll_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
    remove4535377733419490665al_nat: produc3741383161447143261al_nat > list_P6834414599653733731al_nat > list_P6834414599653733731al_nat ).

thf(sy_c_List_OremoveAll_001t__Real__Oreal,type,
    removeAll_real: real > list_real > list_real ).

thf(sy_c_List_OremoveAll_001tf__b,type,
    removeAll_b: b > list_b > list_b ).

thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
    rotate1_nat: list_nat > list_nat ).

thf(sy_c_List_Orotate1_001tf__b,type,
    rotate1_b: list_b > list_b ).

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

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

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

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

thf(sy_c_List_Oupt,type,
    upt: nat > nat > list_nat ).

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

thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
    case_nat_o: $o > ( nat > $o ) > nat > $o ).

thf(sy_c_Nat_Onat_Ocase__nat_001t__List__Olist_It__Nat__Onat_J,type,
    case_nat_list_nat: list_nat > ( nat > list_nat ) > nat > list_nat ).

thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
    case_nat_nat: nat > ( nat > nat ) > nat > nat ).

thf(sy_c_Nat_Onat_Opred,type,
    pred: nat > nat ).

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    size_s1877336372972134351al_nat: list_P6834414599653733731al_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
    size_size_list_real: list_real > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__b_J,type,
    size_size_list_b: list_b > nat ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: num > nat ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__List__Olist_Itf__b_J_M_Eo_J,type,
    ord_less_list_b_o: ( list_b > $o ) > ( list_b > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
    ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    ord_le549003669493604880_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
    ord_le7432393201274343594_nat_o: ( produc859450856879609959at_nat > $o ) > ( produc859450856879609959at_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__List__Olist_It__Nat__Onat_J,type,
    ord_less_list_nat: list_nat > list_nat > $o ).

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__Real__Oreal,type,
    ord_less_real: real > real > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_Itf__b_J_J,type,
    ord_less_set_list_b: set_list_b > set_list_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le7866589430770878221at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    ord_le6428140832669894131at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_Itf__b_J_M_Eo_J,type,
    ord_less_eq_list_b_o: ( list_b > $o ) > ( list_b > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__List__Olist_Itf__b_J_M_Eo_J_J,type,
    ord_le5501773618496797319st_b_o: ( nat > list_b > $o ) > ( nat > list_b > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    ord_le2646555220125990790_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_M_Eo_J_J,type,
    ord_le1366284440460845743_nat_o: ( nat > produc3741383161447143261al_nat > $o ) > ( nat > produc3741383161447143261al_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
    ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
    ord_le5604493270027003598_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    ord_le704812498762024988_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
    ord_le7858099551454983350_nat_o: ( produc859450856879609959at_nat > $o ) > ( produc859450856879609959at_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_062_It__Nat__Onat_M_Eo_J_J,type,
    ord_le1557071757414830890_nat_o: ( real > nat > $o ) > ( real > nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__List__Olist_It__Nat__Onat_J,type,
    ord_less_eq_list_nat: list_nat > list_nat > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
    ord_less_eq_real: real > real > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J_J,type,
    ord_le972014486225453504list_b: set_Pr1349601357184307552list_b > set_Pr1349601357184307552list_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J_J,type,
    ord_le4407014575791007640al_nat: set_Pr836851361167288120al_nat > set_Pr836851361167288120al_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    ord_le3000389064537975527at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    ord_le8992644243367634909al_nat: set_Pr3510011417693777981al_nat > set_Pr3510011417693777981al_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_less_eq_set_real: set_real > set_real > $o ).

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

thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
    ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).

thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__List__Olist_Itf__b_J_J,type,
    orderi2506971327722225558list_b: ( set_list_b > set_list_b > $o ) > ( set_list_b > set_list_b > $o ) > set_list_b > $o ).

thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Nat__Onat_J,type,
    ordering_top_set_nat: ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > set_nat > $o ).

thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    orderi2172309028950807442at_nat: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    orderi92352388023179000at_nat: ( set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o ) > ( set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o ) > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__List__Olist_Itf__b_J_M_Eo_J,type,
    top_top_list_b_o: list_b > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_062_It__List__Olist_Itf__b_J_M_Eo_J_J,type,
    top_top_nat_list_b_o: nat > list_b > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    top_top_nat_nat_o: nat > nat > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_062_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_M_Eo_J_J,type,
    top_to4025840840469758207_nat_o: nat > produc3741383161447143261al_nat > $o ).

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

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
    top_to2659251575160290430_nat_o: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
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    same_fst_nat_list_b: ( nat > $o ) > ( nat > set_Pr7665282455119567943list_b ) > set_Pr3474451521773494677list_b ).

thf(sy_c_Wfrec_Osame__fst_001t__Nat__Onat_001t__Nat__Onat,type,
    same_fst_nat_nat: ( nat > $o ) > ( nat > set_Pr1261947904930325089at_nat ) > set_Pr8693737435421807431at_nat ).

thf(sy_c_Wfrec_Osame__fst_001t__Nat__Onat_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
    same_f8570761488067147039al_nat: ( nat > $o ) > ( nat > set_Pr9214095960617939271al_nat ) > set_Pr6740064708546314949al_nat ).

thf(sy_c_Wfrec_Osame__fst_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    same_f4956014544515070124at_nat: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > set_Pr8693737435421807431at_nat ) > set_Pr553994874890374343at_nat ).

thf(sy_c_Wfrec_Osame__fst_001t__Real__Oreal_001t__Nat__Onat,type,
    same_fst_real_nat: ( real > $o ) > ( real > set_Pr1261947904930325089at_nat ) > set_Pr9214095960617939271al_nat ).

thf(sy_c_fChoice_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    fChoice_nat_nat: ( ( nat > nat ) > $o ) > nat > nat ).

thf(sy_c_fChoice_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    fChoic638070965443183254at_nat: ( ( product_prod_nat_nat > nat ) > $o ) > product_prod_nat_nat > nat ).

thf(sy_c_fChoice_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
    fChoic8220946065393586812at_nat: ( ( produc859450856879609959at_nat > nat ) > $o ) > produc859450856879609959at_nat > nat ).

thf(sy_c_fChoice_001t__List__Olist_Itf__b_J,type,
    fChoice_list_b: ( list_b > $o ) > list_b ).

thf(sy_c_fChoice_001t__Nat__Onat,type,
    fChoice_nat: ( nat > $o ) > nat ).

thf(sy_c_fChoice_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J,type,
    fChoic403860114826281174list_b: ( produc7811952446676219690list_b > $o ) > produc7811952446676219690list_b ).

thf(sy_c_fChoice_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    fChoic6978938873391328853at_nat: ( product_prod_nat_nat > $o ) > product_prod_nat_nat ).

thf(sy_c_fChoice_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    fChoic5522844001492335150al_nat: ( produc9155354469731034754al_nat > $o ) > produc9155354469731034754al_nat ).

thf(sy_c_fChoice_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fChoic6673916560228149179at_nat: ( produc859450856879609959at_nat > $o ) > produc859450856879609959at_nat ).

thf(sy_c_fChoice_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
    fChoic8332448170627812017al_nat: ( produc3741383161447143261al_nat > $o ) > produc3741383161447143261al_nat ).

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

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

thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member3067507820990806192at_nat: list_P6011104703257516679at_nat > set_li5450038453877631591at_nat > $o ).

thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member3799944675974059798at_nat: list_P8469869581646625389at_nat > set_li3197816953174176717at_nat > $o ).

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

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

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

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J_J,type,
    member1290970816896746895al_nat: produc9220417599568932344al_nat > set_Pr3318565038780223662al_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_Itf__b_J_J,type,
    member6003184433318419249list_b: produc1584444791027398810list_b > set_Pr4901976648650938704list_b > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member4574794575480667280at_nat: produc289266582803401575at_nat > set_Pr4087777274317423175at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Real__Oreal_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member7421688103010841766st_nat: produc8740824290856567677st_nat > set_Pr6676954283489005405st_nat > $o ).

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__Nat__Onat_J,type,
    member6469666724224207131_b_nat: produc6020613750378442500_b_nat > set_Pr3469313283233365050_b_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member1908957324972862092at_nat: produc3137399281773485411at_nat > set_Pr2262908016442131011at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J,type,
    member8261005420521984321list_b: produc7811952446676219690list_b > set_Pr1349601357184307552list_b > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member2223272150424702269at_nat: produc7248412053542808358at_nat > set_Pr7717912310451564380at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    member7408976046919118105al_nat: produc9155354469731034754al_nat > set_Pr836851361167288120al_nat > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J_J,type,
    member5069333211937813726list_b: produc2226693793037607221list_b > set_Pr3474451521773494677list_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__List__Olist_Itf__b_J_J,type,
    member9125244090276215188list_b: produc1130314010222062699list_b > set_Pr2272649439365958987list_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    member3348759134392003351at_nat: produc8373899037510109440at_nat > set_Pr2539167527615954998at_nat > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member3048279782668807382at_nat: produc6277219514840344877at_nat > set_Pr575275573428919693at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J_J,type,
    member8283612037765495310al_nat: produc2404420509685808997al_nat > set_Pr6740064708546314949al_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    member1995966531042493578at_nat: produc5224906263214031073at_nat > set_Pr7116486347545156417at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member8062223511168850704at_nat: produc349518998152878311at_nat > set_Pr553994874890374343at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    member7366861965109168016al_nat: produc1222878302676487015al_nat > set_Pr9214095960617939271al_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
    member5805532792777349510al_nat: produc3741383161447143261al_nat > set_Pr3510011417693777981al_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    member7849222048561428706l_real: produc2422161461964618553l_real > set_Pr6218003697084177305l_real > $o ).

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

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

thf(sy_c_member_001t__Real__Oreal,type,
    member_real: real > set_real > $o ).

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

thf(sy_v_e,type,
    e: a > option_list_o ).

% Relevant facts (1269)
thf(fact_0_assms,axiom,
    prefix7485107378405021920ding_a @ e ).

% assms
thf(fact_1_inj__on__convol__ident,axiom,
    ! [F: nat > list_b,X: set_nat] :
      ( inj_on8555073082499328493list_b
      @ ^ [X2: nat] : ( produc7903367361620597084list_b @ X2 @ ( F @ X2 ) )
      @ X ) ).

% inj_on_convol_ident
thf(fact_2_inj__on__convol__ident,axiom,
    ! [F: nat > nat,X: set_nat] :
      ( inj_on5538052773655684606at_nat
      @ ^ [X2: nat] : ( product_Pair_nat_nat @ X2 @ ( F @ X2 ) )
      @ X ) ).

% inj_on_convol_ident
thf(fact_3_inj__on__convol__ident,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat,X: set_Pr1261947904930325089at_nat] :
      ( inj_on3865918002419084311at_nat
      @ ^ [X2: product_prod_nat_nat] : ( produc6161850002892822231at_nat @ X2 @ ( F @ X2 ) )
      @ X ) ).

% inj_on_convol_ident
thf(fact_4_inj__on__convol__ident,axiom,
    ! [F: nat > produc3741383161447143261al_nat,X: set_nat] :
      ( inj_on7152781252210839493al_nat
      @ ^ [X2: nat] : ( produc4490577844368043572al_nat @ X2 @ ( F @ X2 ) )
      @ X ) ).

% inj_on_convol_ident
thf(fact_5_inj__on__convol__ident,axiom,
    ! [F: real > nat,X: set_real] :
      ( inj_on8027545645978435326al_nat
      @ ^ [X2: real] : ( produc3181502643871035669al_nat @ X2 @ ( F @ X2 ) )
      @ X ) ).

% inj_on_convol_ident
thf(fact_6_UNIV__I,axiom,
    ! [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ top_to4669805908274784177at_nat ) ).

% UNIV_I
thf(fact_7_UNIV__I,axiom,
    ! [X3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X3 @ top_to8454300755403694231at_nat ) ).

% UNIV_I
thf(fact_8_UNIV__I,axiom,
    ! [X3: list_b] : ( member_list_b @ X3 @ top_top_set_list_b ) ).

% UNIV_I
thf(fact_9_UNIV__I,axiom,
    ! [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).

% UNIV_I
thf(fact_10_iso__tuple__UNIV__I,axiom,
    ! [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ top_to4669805908274784177at_nat ) ).

% iso_tuple_UNIV_I
thf(fact_11_iso__tuple__UNIV__I,axiom,
    ! [X3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X3 @ top_to8454300755403694231at_nat ) ).

% iso_tuple_UNIV_I
thf(fact_12_iso__tuple__UNIV__I,axiom,
    ! [X3: list_b] : ( member_list_b @ X3 @ top_top_set_list_b ) ).

% iso_tuple_UNIV_I
thf(fact_13_iso__tuple__UNIV__I,axiom,
    ! [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).

% iso_tuple_UNIV_I
thf(fact_14_prod_Oinject,axiom,
    ! [X1: nat,X22: list_b,Y1: nat,Y2: list_b] :
      ( ( ( produc7903367361620597084list_b @ X1 @ X22 )
        = ( produc7903367361620597084list_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_15_prod_Oinject,axiom,
    ! [X1: nat,X22: nat,Y1: nat,Y2: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X22 )
        = ( product_Pair_nat_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_16_prod_Oinject,axiom,
    ! [X1: product_prod_nat_nat,X22: product_prod_nat_nat,Y1: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ X1 @ X22 )
        = ( produc6161850002892822231at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_17_prod_Oinject,axiom,
    ! [X1: nat,X22: produc3741383161447143261al_nat,Y1: nat,Y2: produc3741383161447143261al_nat] :
      ( ( ( produc4490577844368043572al_nat @ X1 @ X22 )
        = ( produc4490577844368043572al_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_18_prod_Oinject,axiom,
    ! [X1: real,X22: nat,Y1: real,Y2: nat] :
      ( ( ( produc3181502643871035669al_nat @ X1 @ X22 )
        = ( produc3181502643871035669al_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_19_old_Oprod_Oinject,axiom,
    ! [A: nat,B: list_b,A2: nat,B2: list_b] :
      ( ( ( produc7903367361620597084list_b @ A @ B )
        = ( produc7903367361620597084list_b @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_20_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_21_old_Oprod_Oinject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A @ B )
        = ( produc6161850002892822231at_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_22_old_Oprod_Oinject,axiom,
    ! [A: nat,B: produc3741383161447143261al_nat,A2: nat,B2: produc3741383161447143261al_nat] :
      ( ( ( produc4490577844368043572al_nat @ A @ B )
        = ( produc4490577844368043572al_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_23_old_Oprod_Oinject,axiom,
    ! [A: real,B: nat,A2: real,B2: nat] :
      ( ( ( produc3181502643871035669al_nat @ A @ B )
        = ( produc3181502643871035669al_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_24_sorted__list__of__set_Oinj__on,axiom,
    ( inj_on_nat_nat
    @ ^ [X2: nat] : X2
    @ top_top_set_nat ) ).

% sorted_list_of_set.inj_on
thf(fact_25_injD,axiom,
    ! [F: real > produc3741383161447143261al_nat,X3: real,Y: real] :
      ( ( inj_on8027545645978435326al_nat @ F @ top_top_set_real )
     => ( ( ( F @ X3 )
          = ( F @ Y ) )
       => ( X3 = Y ) ) ) ).

% injD
thf(fact_26_injD,axiom,
    ! [F: nat > produc9155354469731034754al_nat,X3: nat,Y: nat] :
      ( ( inj_on7152781252210839493al_nat @ F @ top_top_set_nat )
     => ( ( ( F @ X3 )
          = ( F @ Y ) )
       => ( X3 = Y ) ) ) ).

% injD
thf(fact_27_injD,axiom,
    ! [F: nat > product_prod_nat_nat,X3: nat,Y: nat] :
      ( ( inj_on5538052773655684606at_nat @ F @ top_top_set_nat )
     => ( ( ( F @ X3 )
          = ( F @ Y ) )
       => ( X3 = Y ) ) ) ).

% injD
thf(fact_28_injD,axiom,
    ! [F: nat > produc7811952446676219690list_b,X3: nat,Y: nat] :
      ( ( inj_on8555073082499328493list_b @ F @ top_top_set_nat )
     => ( ( ( F @ X3 )
          = ( F @ Y ) )
       => ( X3 = Y ) ) ) ).

% injD
thf(fact_29_injD,axiom,
    ! [F: product_prod_nat_nat > produc859450856879609959at_nat,X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( inj_on3865918002419084311at_nat @ F @ top_to4669805908274784177at_nat )
     => ( ( ( F @ X3 )
          = ( F @ Y ) )
       => ( X3 = Y ) ) ) ).

% injD
thf(fact_30_injD,axiom,
    ! [F: list_b > produc7811952446676219690list_b,X3: list_b,Y: list_b] :
      ( ( inj_on8733544287791916926list_b @ F @ top_top_set_list_b )
     => ( ( ( F @ X3 )
          = ( F @ Y ) )
       => ( X3 = Y ) ) ) ).

% injD
thf(fact_31_injD,axiom,
    ! [F: nat > nat,X3: nat,Y: nat] :
      ( ( inj_on_nat_nat @ F @ top_top_set_nat )
     => ( ( ( F @ X3 )
          = ( F @ Y ) )
       => ( X3 = Y ) ) ) ).

% injD
thf(fact_32_injI,axiom,
    ! [F: real > produc3741383161447143261al_nat] :
      ( ! [X4: real,Y3: real] :
          ( ( ( F @ X4 )
            = ( F @ Y3 ) )
         => ( X4 = Y3 ) )
     => ( inj_on8027545645978435326al_nat @ F @ top_top_set_real ) ) ).

% injI
thf(fact_33_injI,axiom,
    ! [F: list_b > produc7811952446676219690list_b] :
      ( ! [X4: list_b,Y3: list_b] :
          ( ( ( F @ X4 )
            = ( F @ Y3 ) )
         => ( X4 = Y3 ) )
     => ( inj_on8733544287791916926list_b @ F @ top_top_set_list_b ) ) ).

% injI
thf(fact_34_injI,axiom,
    ! [F: nat > nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( ( F @ X4 )
            = ( F @ Y3 ) )
         => ( X4 = Y3 ) )
     => ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).

% injI
thf(fact_35_injI,axiom,
    ! [F: nat > produc9155354469731034754al_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( ( F @ X4 )
            = ( F @ Y3 ) )
         => ( X4 = Y3 ) )
     => ( inj_on7152781252210839493al_nat @ F @ top_top_set_nat ) ) ).

% injI
thf(fact_36_injI,axiom,
    ! [F: nat > product_prod_nat_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( ( F @ X4 )
            = ( F @ Y3 ) )
         => ( X4 = Y3 ) )
     => ( inj_on5538052773655684606at_nat @ F @ top_top_set_nat ) ) ).

% injI
thf(fact_37_injI,axiom,
    ! [F: nat > produc7811952446676219690list_b] :
      ( ! [X4: nat,Y3: nat] :
          ( ( ( F @ X4 )
            = ( F @ Y3 ) )
         => ( X4 = Y3 ) )
     => ( inj_on8555073082499328493list_b @ F @ top_top_set_nat ) ) ).

% injI
thf(fact_38_injI,axiom,
    ! [F: product_prod_nat_nat > produc859450856879609959at_nat] :
      ( ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( ( F @ X4 )
            = ( F @ Y3 ) )
         => ( X4 = Y3 ) )
     => ( inj_on3865918002419084311at_nat @ F @ top_to4669805908274784177at_nat ) ) ).

% injI
thf(fact_39_inj__eq,axiom,
    ! [F: real > produc3741383161447143261al_nat,X3: real,Y: real] :
      ( ( inj_on8027545645978435326al_nat @ F @ top_top_set_real )
     => ( ( ( F @ X3 )
          = ( F @ Y ) )
        = ( X3 = Y ) ) ) ).

% inj_eq
thf(fact_40_inj__eq,axiom,
    ! [F: list_b > produc7811952446676219690list_b,X3: list_b,Y: list_b] :
      ( ( inj_on8733544287791916926list_b @ F @ top_top_set_list_b )
     => ( ( ( F @ X3 )
          = ( F @ Y ) )
        = ( X3 = Y ) ) ) ).

% inj_eq
thf(fact_41_inj__eq,axiom,
    ! [F: nat > nat,X3: nat,Y: nat] :
      ( ( inj_on_nat_nat @ F @ top_top_set_nat )
     => ( ( ( F @ X3 )
          = ( F @ Y ) )
        = ( X3 = Y ) ) ) ).

% inj_eq
thf(fact_42_inj__eq,axiom,
    ! [F: nat > produc9155354469731034754al_nat,X3: nat,Y: nat] :
      ( ( inj_on7152781252210839493al_nat @ F @ top_top_set_nat )
     => ( ( ( F @ X3 )
          = ( F @ Y ) )
        = ( X3 = Y ) ) ) ).

% inj_eq
thf(fact_43_inj__eq,axiom,
    ! [F: nat > product_prod_nat_nat,X3: nat,Y: nat] :
      ( ( inj_on5538052773655684606at_nat @ F @ top_top_set_nat )
     => ( ( ( F @ X3 )
          = ( F @ Y ) )
        = ( X3 = Y ) ) ) ).

% inj_eq
thf(fact_44_inj__eq,axiom,
    ! [F: nat > produc7811952446676219690list_b,X3: nat,Y: nat] :
      ( ( inj_on8555073082499328493list_b @ F @ top_top_set_nat )
     => ( ( ( F @ X3 )
          = ( F @ Y ) )
        = ( X3 = Y ) ) ) ).

% inj_eq
thf(fact_45_inj__eq,axiom,
    ! [F: product_prod_nat_nat > produc859450856879609959at_nat,X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( inj_on3865918002419084311at_nat @ F @ top_to4669805908274784177at_nat )
     => ( ( ( F @ X3 )
          = ( F @ Y ) )
        = ( X3 = Y ) ) ) ).

% inj_eq
thf(fact_46_inj__def,axiom,
    ! [F: real > produc3741383161447143261al_nat] :
      ( ( inj_on8027545645978435326al_nat @ F @ top_top_set_real )
      = ( ! [X2: real,Y4: real] :
            ( ( ( F @ X2 )
              = ( F @ Y4 ) )
           => ( X2 = Y4 ) ) ) ) ).

% inj_def
thf(fact_47_inj__def,axiom,
    ! [F: list_b > produc7811952446676219690list_b] :
      ( ( inj_on8733544287791916926list_b @ F @ top_top_set_list_b )
      = ( ! [X2: list_b,Y4: list_b] :
            ( ( ( F @ X2 )
              = ( F @ Y4 ) )
           => ( X2 = Y4 ) ) ) ) ).

% inj_def
thf(fact_48_inj__def,axiom,
    ! [F: nat > nat] :
      ( ( inj_on_nat_nat @ F @ top_top_set_nat )
      = ( ! [X2: nat,Y4: nat] :
            ( ( ( F @ X2 )
              = ( F @ Y4 ) )
           => ( X2 = Y4 ) ) ) ) ).

% inj_def
thf(fact_49_inj__def,axiom,
    ! [F: nat > produc9155354469731034754al_nat] :
      ( ( inj_on7152781252210839493al_nat @ F @ top_top_set_nat )
      = ( ! [X2: nat,Y4: nat] :
            ( ( ( F @ X2 )
              = ( F @ Y4 ) )
           => ( X2 = Y4 ) ) ) ) ).

% inj_def
thf(fact_50_inj__def,axiom,
    ! [F: nat > product_prod_nat_nat] :
      ( ( inj_on5538052773655684606at_nat @ F @ top_top_set_nat )
      = ( ! [X2: nat,Y4: nat] :
            ( ( ( F @ X2 )
              = ( F @ Y4 ) )
           => ( X2 = Y4 ) ) ) ) ).

% inj_def
thf(fact_51_inj__def,axiom,
    ! [F: nat > produc7811952446676219690list_b] :
      ( ( inj_on8555073082499328493list_b @ F @ top_top_set_nat )
      = ( ! [X2: nat,Y4: nat] :
            ( ( ( F @ X2 )
              = ( F @ Y4 ) )
           => ( X2 = Y4 ) ) ) ) ).

% inj_def
thf(fact_52_inj__def,axiom,
    ! [F: product_prod_nat_nat > produc859450856879609959at_nat] :
      ( ( inj_on3865918002419084311at_nat @ F @ top_to4669805908274784177at_nat )
      = ( ! [X2: product_prod_nat_nat,Y4: product_prod_nat_nat] :
            ( ( ( F @ X2 )
              = ( F @ Y4 ) )
           => ( X2 = Y4 ) ) ) ) ).

% inj_def
thf(fact_53_top__set__def,axiom,
    ( top_top_set_list_b
    = ( collect_list_b @ top_top_list_b_o ) ) ).

% top_set_def
thf(fact_54_top__set__def,axiom,
    ( top_top_set_nat
    = ( collect_nat @ top_top_nat_o ) ) ).

% top_set_def
thf(fact_55_top__set__def,axiom,
    ( top_to8454300755403694231at_nat
    = ( collec7088162979684241874at_nat @ top_to1313005431924226822_nat_o ) ) ).

% top_set_def
thf(fact_56_top__set__def,axiom,
    ( top_to4669805908274784177at_nat
    = ( collec3392354462482085612at_nat @ top_to3137496036531963500_nat_o ) ) ).

% top_set_def
thf(fact_57_prod__induct3,axiom,
    ! [P: produc859450856879609959at_nat > $o,X3: produc859450856879609959at_nat] :
      ( ! [A3: product_prod_nat_nat,B3: nat,C: nat] : ( P @ ( produc6161850002892822231at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ C ) ) )
     => ( P @ X3 ) ) ).

% prod_induct3
thf(fact_58_prod__induct3,axiom,
    ! [P: produc9155354469731034754al_nat > $o,X3: produc9155354469731034754al_nat] :
      ( ! [A3: nat,B3: real,C: nat] : ( P @ ( produc4490577844368043572al_nat @ A3 @ ( produc3181502643871035669al_nat @ B3 @ C ) ) )
     => ( P @ X3 ) ) ).

% prod_induct3
thf(fact_59_prod__cases3,axiom,
    ! [Y: produc859450856879609959at_nat] :
      ~ ! [A3: product_prod_nat_nat,B3: nat,C: nat] :
          ( Y
         != ( produc6161850002892822231at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_60_prod__cases3,axiom,
    ! [Y: produc9155354469731034754al_nat] :
      ~ ! [A3: nat,B3: real,C: nat] :
          ( Y
         != ( produc4490577844368043572al_nat @ A3 @ ( produc3181502643871035669al_nat @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_61_Pair__inject,axiom,
    ! [A: nat,B: list_b,A2: nat,B2: list_b] :
      ( ( ( produc7903367361620597084list_b @ A @ B )
        = ( produc7903367361620597084list_b @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_62_Pair__inject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_63_Pair__inject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A @ B )
        = ( produc6161850002892822231at_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_64_Pair__inject,axiom,
    ! [A: nat,B: produc3741383161447143261al_nat,A2: nat,B2: produc3741383161447143261al_nat] :
      ( ( ( produc4490577844368043572al_nat @ A @ B )
        = ( produc4490577844368043572al_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_65_Pair__inject,axiom,
    ! [A: real,B: nat,A2: real,B2: nat] :
      ( ( ( produc3181502643871035669al_nat @ A @ B )
        = ( produc3181502643871035669al_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_66_prod__cases,axiom,
    ! [P: produc7811952446676219690list_b > $o,P2: produc7811952446676219690list_b] :
      ( ! [A3: nat,B3: list_b] : ( P @ ( produc7903367361620597084list_b @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_67_prod__cases,axiom,
    ! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_68_prod__cases,axiom,
    ! [P: produc859450856879609959at_nat > $o,P2: produc859450856879609959at_nat] :
      ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] : ( P @ ( produc6161850002892822231at_nat @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_69_prod__cases,axiom,
    ! [P: produc9155354469731034754al_nat > $o,P2: produc9155354469731034754al_nat] :
      ( ! [A3: nat,B3: produc3741383161447143261al_nat] : ( P @ ( produc4490577844368043572al_nat @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_70_prod__cases,axiom,
    ! [P: produc3741383161447143261al_nat > $o,P2: produc3741383161447143261al_nat] :
      ( ! [A3: real,B3: nat] : ( P @ ( produc3181502643871035669al_nat @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_71_surj__pair,axiom,
    ! [P2: produc7811952446676219690list_b] :
    ? [X4: nat,Y3: list_b] :
      ( P2
      = ( produc7903367361620597084list_b @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_72_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X4: nat,Y3: nat] :
      ( P2
      = ( product_Pair_nat_nat @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_73_surj__pair,axiom,
    ! [P2: produc859450856879609959at_nat] :
    ? [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( P2
      = ( produc6161850002892822231at_nat @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_74_surj__pair,axiom,
    ! [P2: produc9155354469731034754al_nat] :
    ? [X4: nat,Y3: produc3741383161447143261al_nat] :
      ( P2
      = ( produc4490577844368043572al_nat @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_75_surj__pair,axiom,
    ! [P2: produc3741383161447143261al_nat] :
    ? [X4: real,Y3: nat] :
      ( P2
      = ( produc3181502643871035669al_nat @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_76_old_Oprod_Oexhaust,axiom,
    ! [Y: produc7811952446676219690list_b] :
      ~ ! [A3: nat,B3: list_b] :
          ( Y
         != ( produc7903367361620597084list_b @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_77_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A3: nat,B3: nat] :
          ( Y
         != ( product_Pair_nat_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_78_old_Oprod_Oexhaust,axiom,
    ! [Y: produc859450856879609959at_nat] :
      ~ ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
          ( Y
         != ( produc6161850002892822231at_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_79_old_Oprod_Oexhaust,axiom,
    ! [Y: produc9155354469731034754al_nat] :
      ~ ! [A3: nat,B3: produc3741383161447143261al_nat] :
          ( Y
         != ( produc4490577844368043572al_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_80_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3741383161447143261al_nat] :
      ~ ! [A3: real,B3: nat] :
          ( Y
         != ( produc3181502643871035669al_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_81_UNIV__witness,axiom,
    ? [X4: list_b] : ( member_list_b @ X4 @ top_top_set_list_b ) ).

% UNIV_witness
thf(fact_82_UNIV__witness,axiom,
    ? [X4: nat] : ( member_nat @ X4 @ top_top_set_nat ) ).

% UNIV_witness
thf(fact_83_UNIV__witness,axiom,
    ? [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ top_to8454300755403694231at_nat ) ).

% UNIV_witness
thf(fact_84_UNIV__witness,axiom,
    ? [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ top_to4669805908274784177at_nat ) ).

% UNIV_witness
thf(fact_85_UNIV__eq__I,axiom,
    ! [A4: set_list_b] :
      ( ! [X4: list_b] : ( member_list_b @ X4 @ A4 )
     => ( top_top_set_list_b = A4 ) ) ).

% UNIV_eq_I
thf(fact_86_UNIV__eq__I,axiom,
    ! [A4: set_nat] :
      ( ! [X4: nat] : ( member_nat @ X4 @ A4 )
     => ( top_top_set_nat = A4 ) ) ).

% UNIV_eq_I
thf(fact_87_UNIV__eq__I,axiom,
    ! [A4: set_Pr8693737435421807431at_nat] :
      ( ! [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ A4 )
     => ( top_to8454300755403694231at_nat = A4 ) ) ).

% UNIV_eq_I
thf(fact_88_UNIV__eq__I,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ! [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A4 )
     => ( top_to4669805908274784177at_nat = A4 ) ) ).

% UNIV_eq_I
thf(fact_89_neq__if__length__neq,axiom,
    ! [Xs: list_b,Ys: list_b] :
      ( ( ( size_size_list_b @ Xs )
       != ( size_size_list_b @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_90_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_b] :
      ( ( size_size_list_b @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_91_inj__on__inverseI,axiom,
    ! [A4: set_real,G: produc3741383161447143261al_nat > real,F: real > produc3741383161447143261al_nat] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A4 )
         => ( ( G @ ( F @ X4 ) )
            = X4 ) )
     => ( inj_on8027545645978435326al_nat @ F @ A4 ) ) ).

% inj_on_inverseI
thf(fact_92_inj__on__inverseI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,G: produc859450856879609959at_nat > product_prod_nat_nat,F: product_prod_nat_nat > produc859450856879609959at_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ A4 )
         => ( ( G @ ( F @ X4 ) )
            = X4 ) )
     => ( inj_on3865918002419084311at_nat @ F @ A4 ) ) ).

% inj_on_inverseI
thf(fact_93_inj__on__inverseI,axiom,
    ! [A4: set_nat,G: nat > nat,F: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A4 )
         => ( ( G @ ( F @ X4 ) )
            = X4 ) )
     => ( inj_on_nat_nat @ F @ A4 ) ) ).

% inj_on_inverseI
thf(fact_94_inj__on__inverseI,axiom,
    ! [A4: set_nat,G: produc9155354469731034754al_nat > nat,F: nat > produc9155354469731034754al_nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A4 )
         => ( ( G @ ( F @ X4 ) )
            = X4 ) )
     => ( inj_on7152781252210839493al_nat @ F @ A4 ) ) ).

% inj_on_inverseI
thf(fact_95_inj__on__inverseI,axiom,
    ! [A4: set_nat,G: product_prod_nat_nat > nat,F: nat > product_prod_nat_nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A4 )
         => ( ( G @ ( F @ X4 ) )
            = X4 ) )
     => ( inj_on5538052773655684606at_nat @ F @ A4 ) ) ).

% inj_on_inverseI
thf(fact_96_inj__on__inverseI,axiom,
    ! [A4: set_nat,G: produc7811952446676219690list_b > nat,F: nat > produc7811952446676219690list_b] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A4 )
         => ( ( G @ ( F @ X4 ) )
            = X4 ) )
     => ( inj_on8555073082499328493list_b @ F @ A4 ) ) ).

% inj_on_inverseI
thf(fact_97_inj__on__inverseI,axiom,
    ! [A4: set_list_b,G: produc7811952446676219690list_b > list_b,F: list_b > produc7811952446676219690list_b] :
      ( ! [X4: list_b] :
          ( ( member_list_b @ X4 @ A4 )
         => ( ( G @ ( F @ X4 ) )
            = X4 ) )
     => ( inj_on8733544287791916926list_b @ F @ A4 ) ) ).

% inj_on_inverseI
thf(fact_98_inj__on__contraD,axiom,
    ! [F: real > produc3741383161447143261al_nat,A4: set_real,X3: real,Y: real] :
      ( ( inj_on8027545645978435326al_nat @ F @ A4 )
     => ( ( X3 != Y )
       => ( ( member_real @ X3 @ A4 )
         => ( ( member_real @ Y @ A4 )
           => ( ( F @ X3 )
             != ( F @ Y ) ) ) ) ) ) ).

% inj_on_contraD
thf(fact_99_inj__on__contraD,axiom,
    ! [F: product_prod_nat_nat > produc859450856879609959at_nat,A4: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( inj_on3865918002419084311at_nat @ F @ A4 )
     => ( ( X3 != Y )
       => ( ( member8440522571783428010at_nat @ X3 @ A4 )
         => ( ( member8440522571783428010at_nat @ Y @ A4 )
           => ( ( F @ X3 )
             != ( F @ Y ) ) ) ) ) ) ).

% inj_on_contraD
thf(fact_100_inj__on__contraD,axiom,
    ! [F: nat > nat,A4: set_nat,X3: nat,Y: nat] :
      ( ( inj_on_nat_nat @ F @ A4 )
     => ( ( X3 != Y )
       => ( ( member_nat @ X3 @ A4 )
         => ( ( member_nat @ Y @ A4 )
           => ( ( F @ X3 )
             != ( F @ Y ) ) ) ) ) ) ).

% inj_on_contraD
thf(fact_101_inj__on__contraD,axiom,
    ! [F: nat > produc9155354469731034754al_nat,A4: set_nat,X3: nat,Y: nat] :
      ( ( inj_on7152781252210839493al_nat @ F @ A4 )
     => ( ( X3 != Y )
       => ( ( member_nat @ X3 @ A4 )
         => ( ( member_nat @ Y @ A4 )
           => ( ( F @ X3 )
             != ( F @ Y ) ) ) ) ) ) ).

% inj_on_contraD
thf(fact_102_inj__on__contraD,axiom,
    ! [F: nat > product_prod_nat_nat,A4: set_nat,X3: nat,Y: nat] :
      ( ( inj_on5538052773655684606at_nat @ F @ A4 )
     => ( ( X3 != Y )
       => ( ( member_nat @ X3 @ A4 )
         => ( ( member_nat @ Y @ A4 )
           => ( ( F @ X3 )
             != ( F @ Y ) ) ) ) ) ) ).

% inj_on_contraD
thf(fact_103_inj__on__contraD,axiom,
    ! [F: nat > produc7811952446676219690list_b,A4: set_nat,X3: nat,Y: nat] :
      ( ( inj_on8555073082499328493list_b @ F @ A4 )
     => ( ( X3 != Y )
       => ( ( member_nat @ X3 @ A4 )
         => ( ( member_nat @ Y @ A4 )
           => ( ( F @ X3 )
             != ( F @ Y ) ) ) ) ) ) ).

% inj_on_contraD
thf(fact_104_inj__on__contraD,axiom,
    ! [F: list_b > produc7811952446676219690list_b,A4: set_list_b,X3: list_b,Y: list_b] :
      ( ( inj_on8733544287791916926list_b @ F @ A4 )
     => ( ( X3 != Y )
       => ( ( member_list_b @ X3 @ A4 )
         => ( ( member_list_b @ Y @ A4 )
           => ( ( F @ X3 )
             != ( F @ Y ) ) ) ) ) ) ).

% inj_on_contraD
thf(fact_105_inj__on__eq__iff,axiom,
    ! [F: real > produc3741383161447143261al_nat,A4: set_real,X3: real,Y: real] :
      ( ( inj_on8027545645978435326al_nat @ F @ A4 )
     => ( ( member_real @ X3 @ A4 )
       => ( ( member_real @ Y @ A4 )
         => ( ( ( F @ X3 )
              = ( F @ Y ) )
            = ( X3 = Y ) ) ) ) ) ).

% inj_on_eq_iff
thf(fact_106_inj__on__eq__iff,axiom,
    ! [F: product_prod_nat_nat > produc859450856879609959at_nat,A4: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( inj_on3865918002419084311at_nat @ F @ A4 )
     => ( ( member8440522571783428010at_nat @ X3 @ A4 )
       => ( ( member8440522571783428010at_nat @ Y @ A4 )
         => ( ( ( F @ X3 )
              = ( F @ Y ) )
            = ( X3 = Y ) ) ) ) ) ).

% inj_on_eq_iff
thf(fact_107_inj__on__eq__iff,axiom,
    ! [F: nat > nat,A4: set_nat,X3: nat,Y: nat] :
      ( ( inj_on_nat_nat @ F @ A4 )
     => ( ( member_nat @ X3 @ A4 )
       => ( ( member_nat @ Y @ A4 )
         => ( ( ( F @ X3 )
              = ( F @ Y ) )
            = ( X3 = Y ) ) ) ) ) ).

% inj_on_eq_iff
thf(fact_108_inj__on__eq__iff,axiom,
    ! [F: nat > produc9155354469731034754al_nat,A4: set_nat,X3: nat,Y: nat] :
      ( ( inj_on7152781252210839493al_nat @ F @ A4 )
     => ( ( member_nat @ X3 @ A4 )
       => ( ( member_nat @ Y @ A4 )
         => ( ( ( F @ X3 )
              = ( F @ Y ) )
            = ( X3 = Y ) ) ) ) ) ).

% inj_on_eq_iff
thf(fact_109_inj__on__eq__iff,axiom,
    ! [F: nat > product_prod_nat_nat,A4: set_nat,X3: nat,Y: nat] :
      ( ( inj_on5538052773655684606at_nat @ F @ A4 )
     => ( ( member_nat @ X3 @ A4 )
       => ( ( member_nat @ Y @ A4 )
         => ( ( ( F @ X3 )
              = ( F @ Y ) )
            = ( X3 = Y ) ) ) ) ) ).

% inj_on_eq_iff
thf(fact_110_inj__on__eq__iff,axiom,
    ! [F: nat > produc7811952446676219690list_b,A4: set_nat,X3: nat,Y: nat] :
      ( ( inj_on8555073082499328493list_b @ F @ A4 )
     => ( ( member_nat @ X3 @ A4 )
       => ( ( member_nat @ Y @ A4 )
         => ( ( ( F @ X3 )
              = ( F @ Y ) )
            = ( X3 = Y ) ) ) ) ) ).

% inj_on_eq_iff
thf(fact_111_inj__on__eq__iff,axiom,
    ! [F: list_b > produc7811952446676219690list_b,A4: set_list_b,X3: list_b,Y: list_b] :
      ( ( inj_on8733544287791916926list_b @ F @ A4 )
     => ( ( member_list_b @ X3 @ A4 )
       => ( ( member_list_b @ Y @ A4 )
         => ( ( ( F @ X3 )
              = ( F @ Y ) )
            = ( X3 = Y ) ) ) ) ) ).

% inj_on_eq_iff
thf(fact_112_inj__on__cong,axiom,
    ! [A4: set_real,F: real > produc3741383161447143261al_nat,G: real > produc3741383161447143261al_nat] :
      ( ! [A3: real] :
          ( ( member_real @ A3 @ A4 )
         => ( ( F @ A3 )
            = ( G @ A3 ) ) )
     => ( ( inj_on8027545645978435326al_nat @ F @ A4 )
        = ( inj_on8027545645978435326al_nat @ G @ A4 ) ) ) ).

% inj_on_cong
thf(fact_113_inj__on__cong,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > produc859450856879609959at_nat,G: product_prod_nat_nat > produc859450856879609959at_nat] :
      ( ! [A3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ A3 @ A4 )
         => ( ( F @ A3 )
            = ( G @ A3 ) ) )
     => ( ( inj_on3865918002419084311at_nat @ F @ A4 )
        = ( inj_on3865918002419084311at_nat @ G @ A4 ) ) ) ).

% inj_on_cong
thf(fact_114_inj__on__cong,axiom,
    ! [A4: set_nat,F: nat > nat,G: nat > nat] :
      ( ! [A3: nat] :
          ( ( member_nat @ A3 @ A4 )
         => ( ( F @ A3 )
            = ( G @ A3 ) ) )
     => ( ( inj_on_nat_nat @ F @ A4 )
        = ( inj_on_nat_nat @ G @ A4 ) ) ) ).

% inj_on_cong
thf(fact_115_inj__on__cong,axiom,
    ! [A4: set_nat,F: nat > produc9155354469731034754al_nat,G: nat > produc9155354469731034754al_nat] :
      ( ! [A3: nat] :
          ( ( member_nat @ A3 @ A4 )
         => ( ( F @ A3 )
            = ( G @ A3 ) ) )
     => ( ( inj_on7152781252210839493al_nat @ F @ A4 )
        = ( inj_on7152781252210839493al_nat @ G @ A4 ) ) ) ).

% inj_on_cong
thf(fact_116_inj__on__cong,axiom,
    ! [A4: set_nat,F: nat > product_prod_nat_nat,G: nat > product_prod_nat_nat] :
      ( ! [A3: nat] :
          ( ( member_nat @ A3 @ A4 )
         => ( ( F @ A3 )
            = ( G @ A3 ) ) )
     => ( ( inj_on5538052773655684606at_nat @ F @ A4 )
        = ( inj_on5538052773655684606at_nat @ G @ A4 ) ) ) ).

% inj_on_cong
thf(fact_117_inj__on__cong,axiom,
    ! [A4: set_nat,F: nat > produc7811952446676219690list_b,G: nat > produc7811952446676219690list_b] :
      ( ! [A3: nat] :
          ( ( member_nat @ A3 @ A4 )
         => ( ( F @ A3 )
            = ( G @ A3 ) ) )
     => ( ( inj_on8555073082499328493list_b @ F @ A4 )
        = ( inj_on8555073082499328493list_b @ G @ A4 ) ) ) ).

% inj_on_cong
thf(fact_118_inj__on__cong,axiom,
    ! [A4: set_list_b,F: list_b > produc7811952446676219690list_b,G: list_b > produc7811952446676219690list_b] :
      ( ! [A3: list_b] :
          ( ( member_list_b @ A3 @ A4 )
         => ( ( F @ A3 )
            = ( G @ A3 ) ) )
     => ( ( inj_on8733544287791916926list_b @ F @ A4 )
        = ( inj_on8733544287791916926list_b @ G @ A4 ) ) ) ).

% inj_on_cong
thf(fact_119_inj__on__def,axiom,
    ( inj_on8733544287791916926list_b
    = ( ^ [F2: list_b > produc7811952446676219690list_b,A5: set_list_b] :
        ! [X2: list_b] :
          ( ( member_list_b @ X2 @ A5 )
         => ! [Y4: list_b] :
              ( ( member_list_b @ Y4 @ A5 )
             => ( ( ( F2 @ X2 )
                  = ( F2 @ Y4 ) )
               => ( X2 = Y4 ) ) ) ) ) ) ).

% inj_on_def
thf(fact_120_inj__on__def,axiom,
    ( inj_on_nat_nat
    = ( ^ [F2: nat > nat,A5: set_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A5 )
         => ! [Y4: nat] :
              ( ( member_nat @ Y4 @ A5 )
             => ( ( ( F2 @ X2 )
                  = ( F2 @ Y4 ) )
               => ( X2 = Y4 ) ) ) ) ) ) ).

% inj_on_def
thf(fact_121_inj__on__def,axiom,
    ( inj_on8027545645978435326al_nat
    = ( ^ [F2: real > produc3741383161447143261al_nat,A5: set_real] :
        ! [X2: real] :
          ( ( member_real @ X2 @ A5 )
         => ! [Y4: real] :
              ( ( member_real @ Y4 @ A5 )
             => ( ( ( F2 @ X2 )
                  = ( F2 @ Y4 ) )
               => ( X2 = Y4 ) ) ) ) ) ) ).

% inj_on_def
thf(fact_122_inj__on__def,axiom,
    ( inj_on7152781252210839493al_nat
    = ( ^ [F2: nat > produc9155354469731034754al_nat,A5: set_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A5 )
         => ! [Y4: nat] :
              ( ( member_nat @ Y4 @ A5 )
             => ( ( ( F2 @ X2 )
                  = ( F2 @ Y4 ) )
               => ( X2 = Y4 ) ) ) ) ) ) ).

% inj_on_def
thf(fact_123_inj__on__def,axiom,
    ( inj_on3865918002419084311at_nat
    = ( ^ [F2: product_prod_nat_nat > produc859450856879609959at_nat,A5: set_Pr1261947904930325089at_nat] :
        ! [X2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X2 @ A5 )
         => ! [Y4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Y4 @ A5 )
             => ( ( ( F2 @ X2 )
                  = ( F2 @ Y4 ) )
               => ( X2 = Y4 ) ) ) ) ) ) ).

% inj_on_def
thf(fact_124_inj__on__def,axiom,
    ( inj_on5538052773655684606at_nat
    = ( ^ [F2: nat > product_prod_nat_nat,A5: set_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A5 )
         => ! [Y4: nat] :
              ( ( member_nat @ Y4 @ A5 )
             => ( ( ( F2 @ X2 )
                  = ( F2 @ Y4 ) )
               => ( X2 = Y4 ) ) ) ) ) ) ).

% inj_on_def
thf(fact_125_inj__on__def,axiom,
    ( inj_on8555073082499328493list_b
    = ( ^ [F2: nat > produc7811952446676219690list_b,A5: set_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A5 )
         => ! [Y4: nat] :
              ( ( member_nat @ Y4 @ A5 )
             => ( ( ( F2 @ X2 )
                  = ( F2 @ Y4 ) )
               => ( X2 = Y4 ) ) ) ) ) ) ).

% inj_on_def
thf(fact_126_inj__onI,axiom,
    ! [A4: set_real,F: real > produc3741383161447143261al_nat] :
      ( ! [X4: real,Y3: real] :
          ( ( member_real @ X4 @ A4 )
         => ( ( member_real @ Y3 @ A4 )
           => ( ( ( F @ X4 )
                = ( F @ Y3 ) )
             => ( X4 = Y3 ) ) ) )
     => ( inj_on8027545645978435326al_nat @ F @ A4 ) ) ).

% inj_onI
thf(fact_127_inj__onI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > produc859450856879609959at_nat] :
      ( ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ A4 )
         => ( ( member8440522571783428010at_nat @ Y3 @ A4 )
           => ( ( ( F @ X4 )
                = ( F @ Y3 ) )
             => ( X4 = Y3 ) ) ) )
     => ( inj_on3865918002419084311at_nat @ F @ A4 ) ) ).

% inj_onI
thf(fact_128_inj__onI,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( member_nat @ X4 @ A4 )
         => ( ( member_nat @ Y3 @ A4 )
           => ( ( ( F @ X4 )
                = ( F @ Y3 ) )
             => ( X4 = Y3 ) ) ) )
     => ( inj_on_nat_nat @ F @ A4 ) ) ).

% inj_onI
thf(fact_129_inj__onI,axiom,
    ! [A4: set_nat,F: nat > produc9155354469731034754al_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( member_nat @ X4 @ A4 )
         => ( ( member_nat @ Y3 @ A4 )
           => ( ( ( F @ X4 )
                = ( F @ Y3 ) )
             => ( X4 = Y3 ) ) ) )
     => ( inj_on7152781252210839493al_nat @ F @ A4 ) ) ).

% inj_onI
thf(fact_130_inj__onI,axiom,
    ! [A4: set_nat,F: nat > product_prod_nat_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( member_nat @ X4 @ A4 )
         => ( ( member_nat @ Y3 @ A4 )
           => ( ( ( F @ X4 )
                = ( F @ Y3 ) )
             => ( X4 = Y3 ) ) ) )
     => ( inj_on5538052773655684606at_nat @ F @ A4 ) ) ).

% inj_onI
thf(fact_131_inj__onI,axiom,
    ! [A4: set_nat,F: nat > produc7811952446676219690list_b] :
      ( ! [X4: nat,Y3: nat] :
          ( ( member_nat @ X4 @ A4 )
         => ( ( member_nat @ Y3 @ A4 )
           => ( ( ( F @ X4 )
                = ( F @ Y3 ) )
             => ( X4 = Y3 ) ) ) )
     => ( inj_on8555073082499328493list_b @ F @ A4 ) ) ).

% inj_onI
thf(fact_132_inj__onI,axiom,
    ! [A4: set_list_b,F: list_b > produc7811952446676219690list_b] :
      ( ! [X4: list_b,Y3: list_b] :
          ( ( member_list_b @ X4 @ A4 )
         => ( ( member_list_b @ Y3 @ A4 )
           => ( ( ( F @ X4 )
                = ( F @ Y3 ) )
             => ( X4 = Y3 ) ) ) )
     => ( inj_on8733544287791916926list_b @ F @ A4 ) ) ).

% inj_onI
thf(fact_133_inj__onD,axiom,
    ! [F: real > produc3741383161447143261al_nat,A4: set_real,X3: real,Y: real] :
      ( ( inj_on8027545645978435326al_nat @ F @ A4 )
     => ( ( ( F @ X3 )
          = ( F @ Y ) )
       => ( ( member_real @ X3 @ A4 )
         => ( ( member_real @ Y @ A4 )
           => ( X3 = Y ) ) ) ) ) ).

% inj_onD
thf(fact_134_inj__onD,axiom,
    ! [F: product_prod_nat_nat > produc859450856879609959at_nat,A4: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( inj_on3865918002419084311at_nat @ F @ A4 )
     => ( ( ( F @ X3 )
          = ( F @ Y ) )
       => ( ( member8440522571783428010at_nat @ X3 @ A4 )
         => ( ( member8440522571783428010at_nat @ Y @ A4 )
           => ( X3 = Y ) ) ) ) ) ).

% inj_onD
thf(fact_135_inj__onD,axiom,
    ! [F: nat > nat,A4: set_nat,X3: nat,Y: nat] :
      ( ( inj_on_nat_nat @ F @ A4 )
     => ( ( ( F @ X3 )
          = ( F @ Y ) )
       => ( ( member_nat @ X3 @ A4 )
         => ( ( member_nat @ Y @ A4 )
           => ( X3 = Y ) ) ) ) ) ).

% inj_onD
thf(fact_136_inj__onD,axiom,
    ! [F: nat > produc9155354469731034754al_nat,A4: set_nat,X3: nat,Y: nat] :
      ( ( inj_on7152781252210839493al_nat @ F @ A4 )
     => ( ( ( F @ X3 )
          = ( F @ Y ) )
       => ( ( member_nat @ X3 @ A4 )
         => ( ( member_nat @ Y @ A4 )
           => ( X3 = Y ) ) ) ) ) ).

% inj_onD
thf(fact_137_inj__onD,axiom,
    ! [F: nat > product_prod_nat_nat,A4: set_nat,X3: nat,Y: nat] :
      ( ( inj_on5538052773655684606at_nat @ F @ A4 )
     => ( ( ( F @ X3 )
          = ( F @ Y ) )
       => ( ( member_nat @ X3 @ A4 )
         => ( ( member_nat @ Y @ A4 )
           => ( X3 = Y ) ) ) ) ) ).

% inj_onD
thf(fact_138_inj__onD,axiom,
    ! [F: nat > produc7811952446676219690list_b,A4: set_nat,X3: nat,Y: nat] :
      ( ( inj_on8555073082499328493list_b @ F @ A4 )
     => ( ( ( F @ X3 )
          = ( F @ Y ) )
       => ( ( member_nat @ X3 @ A4 )
         => ( ( member_nat @ Y @ A4 )
           => ( X3 = Y ) ) ) ) ) ).

% inj_onD
thf(fact_139_inj__onD,axiom,
    ! [F: list_b > produc7811952446676219690list_b,A4: set_list_b,X3: list_b,Y: list_b] :
      ( ( inj_on8733544287791916926list_b @ F @ A4 )
     => ( ( ( F @ X3 )
          = ( F @ Y ) )
       => ( ( member_list_b @ X3 @ A4 )
         => ( ( member_list_b @ Y @ A4 )
           => ( X3 = Y ) ) ) ) ) ).

% inj_onD
thf(fact_140_encoding__compose__2,axiom,
    ! [F: a > option_list_o,G: a > a] :
      ( ( prefix7485107378405021920ding_a @ F )
     => ( ( inj_on_a_a @ G @ top_top_set_a )
       => ( prefix7485107378405021920ding_a
          @ ^ [X2: a] : ( F @ ( G @ X2 ) ) ) ) ) ).

% encoding_compose_2
thf(fact_141_encoding__compose__2,axiom,
    ! [F: a > option_list_o,G: nat > a] :
      ( ( prefix7485107378405021920ding_a @ F )
     => ( ( inj_on_nat_a @ G @ top_top_set_nat )
       => ( prefix3558185134189398382ng_nat
          @ ^ [X2: nat] : ( F @ ( G @ X2 ) ) ) ) ) ).

% encoding_compose_2
thf(fact_142_encoding__compose__2,axiom,
    ! [F: nat > option_list_o,G: a > nat] :
      ( ( prefix3558185134189398382ng_nat @ F )
     => ( ( inj_on_a_nat @ G @ top_top_set_a )
       => ( prefix7485107378405021920ding_a
          @ ^ [X2: a] : ( F @ ( G @ X2 ) ) ) ) ) ).

% encoding_compose_2
thf(fact_143_encoding__compose__2,axiom,
    ! [F: nat > option_list_o,G: nat > nat] :
      ( ( prefix3558185134189398382ng_nat @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( prefix3558185134189398382ng_nat
          @ ^ [X2: nat] : ( F @ ( G @ X2 ) ) ) ) ) ).

% encoding_compose_2
thf(fact_144_encoding__compose__2,axiom,
    ! [F: a > option_list_o,G: list_b > a] :
      ( ( prefix7485107378405021920ding_a @ F )
     => ( ( inj_on_list_b_a @ G @ top_top_set_list_b )
       => ( prefix5220018971054140391list_b
          @ ^ [X2: list_b] : ( F @ ( G @ X2 ) ) ) ) ) ).

% encoding_compose_2
thf(fact_145_encoding__compose__2,axiom,
    ! [F: nat > option_list_o,G: list_b > nat] :
      ( ( prefix3558185134189398382ng_nat @ F )
     => ( ( inj_on_list_b_nat @ G @ top_top_set_list_b )
       => ( prefix5220018971054140391list_b
          @ ^ [X2: list_b] : ( F @ ( G @ X2 ) ) ) ) ) ).

% encoding_compose_2
thf(fact_146_encoding__compose__2,axiom,
    ! [F: produc3741383161447143261al_nat > option_list_o,G: real > produc3741383161447143261al_nat] :
      ( ( prefix6066307901512586047al_nat @ F )
     => ( ( inj_on8027545645978435326al_nat @ G @ top_top_set_real )
       => ( prefix6272311651955457354g_real
          @ ^ [X2: real] : ( F @ ( G @ X2 ) ) ) ) ) ).

% encoding_compose_2
thf(fact_147_encoding__compose__2,axiom,
    ! [F: product_prod_nat_nat > option_list_o,G: nat > product_prod_nat_nat] :
      ( ( prefix5049516368958873059at_nat @ F )
     => ( ( inj_on5538052773655684606at_nat @ G @ top_top_set_nat )
       => ( prefix3558185134189398382ng_nat
          @ ^ [X2: nat] : ( F @ ( G @ X2 ) ) ) ) ) ).

% encoding_compose_2
thf(fact_148_encoding__compose__2,axiom,
    ! [F: a > option_list_o,G: product_prod_nat_nat > a] :
      ( ( prefix7485107378405021920ding_a @ F )
     => ( ( inj_on1068318659649230510_nat_a @ G @ top_to4669805908274784177at_nat )
       => ( prefix5049516368958873059at_nat
          @ ^ [X2: product_prod_nat_nat] : ( F @ ( G @ X2 ) ) ) ) ) ).

% encoding_compose_2
thf(fact_149_encoding__compose__2,axiom,
    ! [F: nat > option_list_o,G: product_prod_nat_nat > nat] :
      ( ( prefix3558185134189398382ng_nat @ F )
     => ( ( inj_on2178005380612969504at_nat @ G @ top_to4669805908274784177at_nat )
       => ( prefix5049516368958873059at_nat
          @ ^ [X2: product_prod_nat_nat] : ( F @ ( G @ X2 ) ) ) ) ) ).

% encoding_compose_2
thf(fact_150_UNIV__def,axiom,
    ( top_top_set_list_b
    = ( collect_list_b
      @ ^ [X2: list_b] : $true ) ) ).

% UNIV_def
thf(fact_151_UNIV__def,axiom,
    ( top_top_set_nat
    = ( collect_nat
      @ ^ [X2: nat] : $true ) ) ).

% UNIV_def
thf(fact_152_UNIV__def,axiom,
    ( top_to8454300755403694231at_nat
    = ( collec7088162979684241874at_nat
      @ ^ [X2: produc859450856879609959at_nat] : $true ) ) ).

% UNIV_def
thf(fact_153_UNIV__def,axiom,
    ( top_to4669805908274784177at_nat
    = ( collec3392354462482085612at_nat
      @ ^ [X2: product_prod_nat_nat] : $true ) ) ).

% UNIV_def
thf(fact_154_inj__on__id2,axiom,
    ! [A4: set_nat] :
      ( inj_on_nat_nat
      @ ^ [X2: nat] : X2
      @ A4 ) ).

% inj_on_id2
thf(fact_155_top__empty__eq2,axiom,
    ( top_top_nat_list_b_o
    = ( ^ [X2: nat,Y4: list_b] : ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ X2 @ Y4 ) @ top_to4428277177966209040list_b ) ) ) ).

% top_empty_eq2
thf(fact_156_top__empty__eq2,axiom,
    ( top_to4025840840469758207_nat_o
    = ( ^ [X2: nat,Y4: produc3741383161447143261al_nat] : ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X2 @ Y4 ) @ top_to5352222812181489128al_nat ) ) ) ).

% top_empty_eq2
thf(fact_157_top__empty__eq2,axiom,
    ( top_top_real_nat_o
    = ( ^ [X2: real,Y4: nat] : ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X2 @ Y4 ) @ top_to8045530339851357581al_nat ) ) ) ).

% top_empty_eq2
thf(fact_158_top__empty__eq2,axiom,
    ( top_to2659251575160290430_nat_o
    = ( ^ [X2: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y4 ) @ top_to8454300755403694231at_nat ) ) ) ).

% top_empty_eq2
thf(fact_159_top__empty__eq2,axiom,
    ( top_top_nat_nat_o
    = ( ^ [X2: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ top_to4669805908274784177at_nat ) ) ) ).

% top_empty_eq2
thf(fact_160_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_161_mem__Collect__eq,axiom,
    ! [A: produc859450856879609959at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ A @ ( collec7088162979684241874at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_162_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_163_mem__Collect__eq,axiom,
    ! [A: list_b,P: list_b > $o] :
      ( ( member_list_b @ A @ ( collect_list_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_164_Collect__mem__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_165_Collect__mem__eq,axiom,
    ! [A4: set_Pr8693737435421807431at_nat] :
      ( ( collec7088162979684241874at_nat
        @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_166_Collect__mem__eq,axiom,
    ! [A4: set_nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : ( member_nat @ X2 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_167_Collect__mem__eq,axiom,
    ! [A4: set_list_b] :
      ( ( collect_list_b
        @ ^ [X2: list_b] : ( member_list_b @ X2 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_168_pred__equals__eq2,axiom,
    ! [R: set_Pr1349601357184307552list_b,S: set_Pr1349601357184307552list_b] :
      ( ( ( ^ [X2: nat,Y4: list_b] : ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ X2 @ Y4 ) @ R ) )
        = ( ^ [X2: nat,Y4: list_b] : ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ X2 @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_169_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X2: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ R ) )
        = ( ^ [X2: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_170_pred__equals__eq2,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( ( ^ [X2: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y4 ) @ R ) )
        = ( ^ [X2: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_171_pred__equals__eq2,axiom,
    ! [R: set_Pr836851361167288120al_nat,S: set_Pr836851361167288120al_nat] :
      ( ( ( ^ [X2: nat,Y4: produc3741383161447143261al_nat] : ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X2 @ Y4 ) @ R ) )
        = ( ^ [X2: nat,Y4: produc3741383161447143261al_nat] : ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X2 @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_172_pred__equals__eq2,axiom,
    ! [R: set_Pr3510011417693777981al_nat,S: set_Pr3510011417693777981al_nat] :
      ( ( ( ^ [X2: real,Y4: nat] : ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X2 @ Y4 ) @ R ) )
        = ( ^ [X2: real,Y4: nat] : ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X2 @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_173_fixed__list__encoding,axiom,
    ! [E: a > option_list_o,N: nat] :
      ( ( prefix7485107378405021920ding_a @ E )
     => ( prefix5220018966750911590list_a @ ( prefix4097710381326367690Lf_e_a @ E @ N ) ) ) ).

% fixed_list_encoding
thf(fact_174_fixed__list__encoding,axiom,
    ! [E: nat > option_list_o,N: nat] :
      ( ( prefix3558185134189398382ng_nat @ E )
     => ( prefix3565147084827311486st_nat @ ( prefix4948863086554537988_e_nat @ E @ N ) ) ) ).

% fixed_list_encoding
thf(fact_175_folding__insort__key__axioms__def,axiom,
    ( foldin7842429202329594855list_b
    = ( ^ [S2: set_list_b,F2: list_b > produc7811952446676219690list_b] : ( inj_on8733544287791916926list_b @ F2 @ S2 ) ) ) ).

% folding_insort_key_axioms_def
thf(fact_176_folding__insort__key__axioms__def,axiom,
    ( foldin1360219024038166634at_nat
    = ( ^ [S2: set_nat,F2: nat > nat] : ( inj_on_nat_nat @ F2 @ S2 ) ) ) ).

% folding_insort_key_axioms_def
thf(fact_177_folding__insort__key__axioms__def,axiom,
    ( foldin7878508968674724839al_nat
    = ( ^ [S2: set_real,F2: real > produc3741383161447143261al_nat] : ( inj_on8027545645978435326al_nat @ F2 @ S2 ) ) ) ).

% folding_insort_key_axioms_def
thf(fact_178_folding__insort__key__axioms__def,axiom,
    ( foldin4846302559880288028al_nat
    = ( ^ [S2: set_nat,F2: nat > produc9155354469731034754al_nat] : ( inj_on7152781252210839493al_nat @ F2 @ S2 ) ) ) ).

% folding_insort_key_axioms_def
thf(fact_179_folding__insort__key__axioms__def,axiom,
    ( foldin8193298255865475182at_nat
    = ( ^ [S2: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > produc859450856879609959at_nat] : ( inj_on3865918002419084311at_nat @ F2 @ S2 ) ) ) ).

% folding_insort_key_axioms_def
thf(fact_180_folding__insort__key__axioms__def,axiom,
    ( foldin7992136063462046695at_nat
    = ( ^ [S2: set_nat,F2: nat > product_prod_nat_nat] : ( inj_on5538052773655684606at_nat @ F2 @ S2 ) ) ) ).

% folding_insort_key_axioms_def
thf(fact_181_folding__insort__key__axioms__def,axiom,
    ( foldin3648121289066013892list_b
    = ( ^ [S2: set_nat,F2: nat > produc7811952446676219690list_b] : ( inj_on8555073082499328493list_b @ F2 @ S2 ) ) ) ).

% folding_insort_key_axioms_def
thf(fact_182_folding__insort__key__axioms_Ointro,axiom,
    ! [F: list_b > produc7811952446676219690list_b,S: set_list_b] :
      ( ( inj_on8733544287791916926list_b @ F @ S )
     => ( foldin7842429202329594855list_b @ S @ F ) ) ).

% folding_insort_key_axioms.intro
thf(fact_183_folding__insort__key__axioms_Ointro,axiom,
    ! [F: nat > nat,S: set_nat] :
      ( ( inj_on_nat_nat @ F @ S )
     => ( foldin1360219024038166634at_nat @ S @ F ) ) ).

% folding_insort_key_axioms.intro
thf(fact_184_folding__insort__key__axioms_Ointro,axiom,
    ! [F: real > produc3741383161447143261al_nat,S: set_real] :
      ( ( inj_on8027545645978435326al_nat @ F @ S )
     => ( foldin7878508968674724839al_nat @ S @ F ) ) ).

% folding_insort_key_axioms.intro
thf(fact_185_folding__insort__key__axioms_Ointro,axiom,
    ! [F: nat > produc9155354469731034754al_nat,S: set_nat] :
      ( ( inj_on7152781252210839493al_nat @ F @ S )
     => ( foldin4846302559880288028al_nat @ S @ F ) ) ).

% folding_insort_key_axioms.intro
thf(fact_186_folding__insort__key__axioms_Ointro,axiom,
    ! [F: product_prod_nat_nat > produc859450856879609959at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( inj_on3865918002419084311at_nat @ F @ S )
     => ( foldin8193298255865475182at_nat @ S @ F ) ) ).

% folding_insort_key_axioms.intro
thf(fact_187_folding__insort__key__axioms_Ointro,axiom,
    ! [F: nat > product_prod_nat_nat,S: set_nat] :
      ( ( inj_on5538052773655684606at_nat @ F @ S )
     => ( foldin7992136063462046695at_nat @ S @ F ) ) ).

% folding_insort_key_axioms.intro
thf(fact_188_folding__insort__key__axioms_Ointro,axiom,
    ! [F: nat > produc7811952446676219690list_b,S: set_nat] :
      ( ( inj_on8555073082499328493list_b @ F @ S )
     => ( foldin3648121289066013892list_b @ S @ F ) ) ).

% folding_insort_key_axioms.intro
thf(fact_189_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X3: produc970035679548117564list_b] :
      ~ ! [F3: nat > list_b > list_b,A3: nat,B3: nat,Acc: list_b] :
          ( X3
         != ( produc6588396999937523566list_b @ F3 @ ( produc7988756899370845203list_b @ A3 @ ( produc7903367361620597084list_b @ B3 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_190_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X3: produc4471711990508489141at_nat] :
      ~ ! [F3: nat > nat > nat,A3: nat,B3: nat,Acc: nat] :
          ( X3
         != ( produc3209952032786966637at_nat @ F3 @ ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_191_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X3: produc1019696013813152916al_nat] :
      ~ ! [F3: nat > produc3741383161447143261al_nat > produc3741383161447143261al_nat,A3: nat,B3: nat,Acc: produc3741383161447143261al_nat] :
          ( X3
         != ( produc5211167162158505542al_nat @ F3 @ ( produc2879132834507146987al_nat @ A3 @ ( produc4490577844368043572al_nat @ B3 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_192_inj__apsnd,axiom,
    ! [F: product_prod_nat_nat > produc859450856879609959at_nat] :
      ( ( inj_on2018228931673164823at_nat @ ( produc3304601163070410845at_nat @ F ) @ top_to8454300755403694231at_nat )
      = ( inj_on3865918002419084311at_nat @ F @ top_to4669805908274784177at_nat ) ) ).

% inj_apsnd
thf(fact_193_inj__apsnd,axiom,
    ! [F: nat > nat] :
      ( ( inj_on8969904277767023793at_nat @ ( produc3094765310956027504at_nat @ F ) @ top_to4669805908274784177at_nat )
      = ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).

% inj_apsnd
thf(fact_194_inj__apsnd,axiom,
    ! [F: nat > produc9155354469731034754al_nat] :
      ( ( inj_on935854358587236963al_nat @ ( produc7042579502741520162at_nat @ F ) @ top_to4669805908274784177at_nat )
      = ( inj_on7152781252210839493al_nat @ F @ top_top_set_nat ) ) ).

% inj_apsnd
thf(fact_195_inj__apsnd,axiom,
    ! [F: nat > product_prod_nat_nat] :
      ( ( inj_on5416022199913362294at_nat @ ( produc1784090006485240963at_nat @ F ) @ top_to4669805908274784177at_nat )
      = ( inj_on5538052773655684606at_nat @ F @ top_top_set_nat ) ) ).

% inj_apsnd
thf(fact_196_inj__apsnd,axiom,
    ! [F: nat > produc7811952446676219690list_b] :
      ( ( inj_on4534532566085093899list_b @ ( produc7599226473598062410_b_nat @ F ) @ top_to4669805908274784177at_nat )
      = ( inj_on8555073082499328493list_b @ F @ top_top_set_nat ) ) ).

% inj_apsnd
thf(fact_197_inj__apfst,axiom,
    ! [F: product_prod_nat_nat > produc859450856879609959at_nat] :
      ( ( inj_on9127415789052672971at_nat @ ( produc5721968133327750431at_nat @ F ) @ top_to8454300755403694231at_nat )
      = ( inj_on3865918002419084311at_nat @ F @ top_to4669805908274784177at_nat ) ) ).

% inj_apfst
thf(fact_198_inj__apfst,axiom,
    ! [F: nat > nat] :
      ( ( inj_on8969904277767023793at_nat @ ( produc986720760941809198at_nat @ F ) @ top_to4669805908274784177at_nat )
      = ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).

% inj_apfst
thf(fact_199_inj__apfst,axiom,
    ! [F: nat > produc9155354469731034754al_nat] :
      ( ( inj_on2445742330725020003at_nat @ ( produc3511468602978368224at_nat @ F ) @ top_to4669805908274784177at_nat )
      = ( inj_on7152781252210839493al_nat @ F @ top_top_set_nat ) ) ).

% inj_apfst
thf(fact_200_inj__apfst,axiom,
    ! [F: nat > product_prod_nat_nat] :
      ( ( inj_on6541509183880663376at_nat @ ( produc3001624868188084037at_nat @ F ) @ top_to4669805908274784177at_nat )
      = ( inj_on5538052773655684606at_nat @ F @ top_top_set_nat ) ) ).

% inj_apfst
thf(fact_201_inj__apfst,axiom,
    ! [F: nat > produc7811952446676219690list_b] :
      ( ( inj_on1775699293261311627_b_nat @ ( produc4853665626768243208_b_nat @ F ) @ top_to4669805908274784177at_nat )
      = ( inj_on8555073082499328493list_b @ F @ top_top_set_nat ) ) ).

% inj_apfst
thf(fact_202_inj__mapI,axiom,
    ! [F: real > produc3741383161447143261al_nat] :
      ( ( inj_on8027545645978435326al_nat @ F @ top_top_set_real )
     => ( inj_on8276318967698619124al_nat @ ( map_re3328248900745367531al_nat @ F ) @ top_to7938183770042617506t_real ) ) ).

% inj_mapI
thf(fact_203_inj__mapI,axiom,
    ! [F: list_b > produc7811952446676219690list_b] :
      ( ( inj_on8733544287791916926list_b @ F @ top_top_set_list_b )
     => ( inj_on5686930240773211284list_b @ ( map_li8759865046734642667list_b @ F ) @ top_to4060329871436644795list_b ) ) ).

% inj_mapI
thf(fact_204_inj__mapI,axiom,
    ! [F: nat > produc9155354469731034754al_nat] :
      ( ( inj_on7152781252210839493al_nat @ F @ top_top_set_nat )
     => ( inj_on1287217757190318181al_nat @ ( map_na466430255352921112al_nat @ F ) @ top_top_set_list_nat ) ) ).

% inj_mapI
thf(fact_205_inj__mapI,axiom,
    ! [F: nat > product_prod_nat_nat] :
      ( ( inj_on5538052773655684606at_nat @ F @ top_top_set_nat )
     => ( inj_on3024915551208126196at_nat @ ( map_na7298421622053143531at_nat @ F ) @ top_top_set_list_nat ) ) ).

% inj_mapI
thf(fact_206_inj__mapI,axiom,
    ! [F: nat > produc7811952446676219690list_b] :
      ( ( inj_on8555073082499328493list_b @ F @ top_top_set_nat )
     => ( inj_on23012040918581517list_b @ ( map_na1272791293171509184list_b @ F ) @ top_top_set_list_nat ) ) ).

% inj_mapI
thf(fact_207_inj__mapI,axiom,
    ! [F: nat > nat] :
      ( ( inj_on_nat_nat @ F @ top_top_set_nat )
     => ( inj_on3049792774292151987st_nat @ ( map_nat_nat @ F ) @ top_top_set_list_nat ) ) ).

% inj_mapI
thf(fact_208_inj__mapI,axiom,
    ! [F: product_prod_nat_nat > produc859450856879609959at_nat] :
      ( ( inj_on3865918002419084311at_nat @ F @ top_to4669805908274784177at_nat )
     => ( inj_on3011817175317237783at_nat @ ( map_Pr9105590726435236202at_nat @ F ) @ top_to70995312096774071at_nat ) ) ).

% inj_mapI
thf(fact_209_inj__map,axiom,
    ! [F: real > produc3741383161447143261al_nat] :
      ( ( inj_on8276318967698619124al_nat @ ( map_re3328248900745367531al_nat @ F ) @ top_to7938183770042617506t_real )
      = ( inj_on8027545645978435326al_nat @ F @ top_top_set_real ) ) ).

% inj_map
thf(fact_210_inj__map,axiom,
    ! [F: list_b > produc7811952446676219690list_b] :
      ( ( inj_on5686930240773211284list_b @ ( map_li8759865046734642667list_b @ F ) @ top_to4060329871436644795list_b )
      = ( inj_on8733544287791916926list_b @ F @ top_top_set_list_b ) ) ).

% inj_map
thf(fact_211_inj__map,axiom,
    ! [F: nat > produc9155354469731034754al_nat] :
      ( ( inj_on1287217757190318181al_nat @ ( map_na466430255352921112al_nat @ F ) @ top_top_set_list_nat )
      = ( inj_on7152781252210839493al_nat @ F @ top_top_set_nat ) ) ).

% inj_map
thf(fact_212_inj__map,axiom,
    ! [F: nat > product_prod_nat_nat] :
      ( ( inj_on3024915551208126196at_nat @ ( map_na7298421622053143531at_nat @ F ) @ top_top_set_list_nat )
      = ( inj_on5538052773655684606at_nat @ F @ top_top_set_nat ) ) ).

% inj_map
thf(fact_213_inj__map,axiom,
    ! [F: nat > produc7811952446676219690list_b] :
      ( ( inj_on23012040918581517list_b @ ( map_na1272791293171509184list_b @ F ) @ top_top_set_list_nat )
      = ( inj_on8555073082499328493list_b @ F @ top_top_set_nat ) ) ).

% inj_map
thf(fact_214_inj__map,axiom,
    ! [F: nat > nat] :
      ( ( inj_on3049792774292151987st_nat @ ( map_nat_nat @ F ) @ top_top_set_list_nat )
      = ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).

% inj_map
thf(fact_215_inj__map,axiom,
    ! [F: product_prod_nat_nat > produc859450856879609959at_nat] :
      ( ( inj_on3011817175317237783at_nat @ ( map_Pr9105590726435236202at_nat @ F ) @ top_to70995312096774071at_nat )
      = ( inj_on3865918002419084311at_nat @ F @ top_to4669805908274784177at_nat ) ) ).

% inj_map
thf(fact_216_map__ident,axiom,
    ( ( map_nat_nat
      @ ^ [X2: nat] : X2 )
    = ( ^ [Xs3: list_nat] : Xs3 ) ) ).

% map_ident
thf(fact_217_length__map,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_218_length__map,axiom,
    ! [F: b > b,Xs: list_b] :
      ( ( size_size_list_b @ ( map_b_b @ F @ Xs ) )
      = ( size_size_list_b @ Xs ) ) ).

% length_map
thf(fact_219_apfst__conv,axiom,
    ! [F: nat > nat,X3: nat,Y: list_b] :
      ( ( produc8895340734784419623list_b @ F @ ( produc7903367361620597084list_b @ X3 @ Y ) )
      = ( produc7903367361620597084list_b @ ( F @ X3 ) @ Y ) ) ).

% apfst_conv
thf(fact_220_apfst__conv,axiom,
    ! [F: nat > nat,X3: nat,Y: nat] :
      ( ( produc986720760941809198at_nat @ F @ ( product_Pair_nat_nat @ X3 @ Y ) )
      = ( product_Pair_nat_nat @ ( F @ X3 ) @ Y ) ) ).

% apfst_conv
thf(fact_221_apfst__conv,axiom,
    ! [F: nat > real,X3: nat,Y: nat] :
      ( ( produc7323138200241035914al_nat @ F @ ( product_Pair_nat_nat @ X3 @ Y ) )
      = ( produc3181502643871035669al_nat @ ( F @ X3 ) @ Y ) ) ).

% apfst_conv
thf(fact_222_apfst__conv,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat,X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( produc3057951030792887557at_nat @ F @ ( produc6161850002892822231at_nat @ X3 @ Y ) )
      = ( produc6161850002892822231at_nat @ ( F @ X3 ) @ Y ) ) ).

% apfst_conv
thf(fact_223_apfst__conv,axiom,
    ! [F: nat > nat,X3: nat,Y: produc3741383161447143261al_nat] :
      ( ( produc2060316929304411775al_nat @ F @ ( produc4490577844368043572al_nat @ X3 @ Y ) )
      = ( produc4490577844368043572al_nat @ ( F @ X3 ) @ Y ) ) ).

% apfst_conv
thf(fact_224_apfst__conv,axiom,
    ! [F: real > nat,X3: real,Y: nat] :
      ( ( produc4676469564790641418at_nat @ F @ ( produc3181502643871035669al_nat @ X3 @ Y ) )
      = ( product_Pair_nat_nat @ ( F @ X3 ) @ Y ) ) ).

% apfst_conv
thf(fact_225_apfst__conv,axiom,
    ! [F: real > real,X3: real,Y: nat] :
      ( ( produc8414705825198518630al_nat @ F @ ( produc3181502643871035669al_nat @ X3 @ Y ) )
      = ( produc3181502643871035669al_nat @ ( F @ X3 ) @ Y ) ) ).

% apfst_conv
thf(fact_226_apsnd__conv,axiom,
    ! [F: nat > nat,X3: nat,Y: nat] :
      ( ( produc3094765310956027504at_nat @ F @ ( product_Pair_nat_nat @ X3 @ Y ) )
      = ( product_Pair_nat_nat @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_227_apsnd__conv,axiom,
    ! [F: nat > nat,X3: real,Y: nat] :
      ( ( produc4873413197848629580t_real @ F @ ( produc3181502643871035669al_nat @ X3 @ Y ) )
      = ( produc3181502643871035669al_nat @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_228_apsnd__conv,axiom,
    ! [F: list_b > nat,X3: nat,Y: list_b] :
      ( ( produc1645906838343065769at_nat @ F @ ( produc7903367361620597084list_b @ X3 @ Y ) )
      = ( product_Pair_nat_nat @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_229_apsnd__conv,axiom,
    ! [F: nat > list_b,X3: nat,Y: nat] :
      ( ( produc2041552131524264583_b_nat @ F @ ( product_Pair_nat_nat @ X3 @ Y ) )
      = ( produc7903367361620597084list_b @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_230_apsnd__conv,axiom,
    ! [F: list_b > list_b,X3: nat,Y: list_b] :
      ( ( produc6661256811431873806_b_nat @ F @ ( produc7903367361620597084list_b @ X3 @ Y ) )
      = ( produc7903367361620597084list_b @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_231_apsnd__conv,axiom,
    ! [F: nat > produc3741383161447143261al_nat,X3: nat,Y: nat] :
      ( ( produc6464852473767144031at_nat @ F @ ( product_Pair_nat_nat @ X3 @ Y ) )
      = ( produc4490577844368043572al_nat @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_232_apsnd__conv,axiom,
    ! [F: produc3741383161447143261al_nat > nat,X3: nat,Y: produc3741383161447143261al_nat] :
      ( ( produc4857132256083941377at_nat @ F @ ( produc4490577844368043572al_nat @ X3 @ Y ) )
      = ( product_Pair_nat_nat @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_233_apsnd__conv,axiom,
    ! [F: list_b > produc3741383161447143261al_nat,X3: nat,Y: list_b] :
      ( ( produc5057693577170282854at_nat @ F @ ( produc7903367361620597084list_b @ X3 @ Y ) )
      = ( produc4490577844368043572al_nat @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_234_apsnd__conv,axiom,
    ! [F: produc3741383161447143261al_nat > list_b,X3: nat,Y: produc3741383161447143261al_nat] :
      ( ( produc2579153942975317942_b_nat @ F @ ( produc4490577844368043572al_nat @ X3 @ Y ) )
      = ( produc7903367361620597084list_b @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_235_apsnd__conv,axiom,
    ! [F: produc3741383161447143261al_nat > produc3741383161447143261al_nat,X3: nat,Y: produc3741383161447143261al_nat] :
      ( ( produc4858856733485666318at_nat @ F @ ( produc4490577844368043572al_nat @ X3 @ Y ) )
      = ( produc4490577844368043572al_nat @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_236_inj__map__eq__map,axiom,
    ! [F: real > produc3741383161447143261al_nat,Xs: list_real,Ys: list_real] :
      ( ( inj_on8027545645978435326al_nat @ F @ top_top_set_real )
     => ( ( ( map_re3328248900745367531al_nat @ F @ Xs )
          = ( map_re3328248900745367531al_nat @ F @ Ys ) )
        = ( Xs = Ys ) ) ) ).

% inj_map_eq_map
thf(fact_237_inj__map__eq__map,axiom,
    ! [F: list_b > produc7811952446676219690list_b,Xs: list_list_b,Ys: list_list_b] :
      ( ( inj_on8733544287791916926list_b @ F @ top_top_set_list_b )
     => ( ( ( map_li8759865046734642667list_b @ F @ Xs )
          = ( map_li8759865046734642667list_b @ F @ Ys ) )
        = ( Xs = Ys ) ) ) ).

% inj_map_eq_map
thf(fact_238_inj__map__eq__map,axiom,
    ! [F: nat > produc9155354469731034754al_nat,Xs: list_nat,Ys: list_nat] :
      ( ( inj_on7152781252210839493al_nat @ F @ top_top_set_nat )
     => ( ( ( map_na466430255352921112al_nat @ F @ Xs )
          = ( map_na466430255352921112al_nat @ F @ Ys ) )
        = ( Xs = Ys ) ) ) ).

% inj_map_eq_map
thf(fact_239_inj__map__eq__map,axiom,
    ! [F: nat > product_prod_nat_nat,Xs: list_nat,Ys: list_nat] :
      ( ( inj_on5538052773655684606at_nat @ F @ top_top_set_nat )
     => ( ( ( map_na7298421622053143531at_nat @ F @ Xs )
          = ( map_na7298421622053143531at_nat @ F @ Ys ) )
        = ( Xs = Ys ) ) ) ).

% inj_map_eq_map
thf(fact_240_inj__map__eq__map,axiom,
    ! [F: nat > produc7811952446676219690list_b,Xs: list_nat,Ys: list_nat] :
      ( ( inj_on8555073082499328493list_b @ F @ top_top_set_nat )
     => ( ( ( map_na1272791293171509184list_b @ F @ Xs )
          = ( map_na1272791293171509184list_b @ F @ Ys ) )
        = ( Xs = Ys ) ) ) ).

% inj_map_eq_map
thf(fact_241_inj__map__eq__map,axiom,
    ! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
      ( ( inj_on_nat_nat @ F @ top_top_set_nat )
     => ( ( ( map_nat_nat @ F @ Xs )
          = ( map_nat_nat @ F @ Ys ) )
        = ( Xs = Ys ) ) ) ).

% inj_map_eq_map
thf(fact_242_inj__map__eq__map,axiom,
    ! [F: product_prod_nat_nat > produc859450856879609959at_nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( inj_on3865918002419084311at_nat @ F @ top_to4669805908274784177at_nat )
     => ( ( ( map_Pr9105590726435236202at_nat @ F @ Xs )
          = ( map_Pr9105590726435236202at_nat @ F @ Ys ) )
        = ( Xs = Ys ) ) ) ).

% inj_map_eq_map
thf(fact_243_list_Omap__ident,axiom,
    ! [T: list_nat] :
      ( ( map_nat_nat
        @ ^ [X2: nat] : X2
        @ T )
      = T ) ).

% list.map_ident
thf(fact_244_map__eq__imp__length__eq,axiom,
    ! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_245_map__eq__imp__length__eq,axiom,
    ! [F: nat > nat,Xs: list_nat,G: b > nat,Ys: list_b] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_b_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_b @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_246_map__eq__imp__length__eq,axiom,
    ! [F: b > nat,Xs: list_b,G: nat > nat,Ys: list_nat] :
      ( ( ( map_b_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_size_list_b @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_247_map__injective,axiom,
    ! [F: real > produc3741383161447143261al_nat,Xs: list_real,Ys: list_real] :
      ( ( ( map_re3328248900745367531al_nat @ F @ Xs )
        = ( map_re3328248900745367531al_nat @ F @ Ys ) )
     => ( ( inj_on8027545645978435326al_nat @ F @ top_top_set_real )
       => ( Xs = Ys ) ) ) ).

% map_injective
thf(fact_248_map__injective,axiom,
    ! [F: list_b > produc7811952446676219690list_b,Xs: list_list_b,Ys: list_list_b] :
      ( ( ( map_li8759865046734642667list_b @ F @ Xs )
        = ( map_li8759865046734642667list_b @ F @ Ys ) )
     => ( ( inj_on8733544287791916926list_b @ F @ top_top_set_list_b )
       => ( Xs = Ys ) ) ) ).

% map_injective
thf(fact_249_map__injective,axiom,
    ! [F: nat > produc9155354469731034754al_nat,Xs: list_nat,Ys: list_nat] :
      ( ( ( map_na466430255352921112al_nat @ F @ Xs )
        = ( map_na466430255352921112al_nat @ F @ Ys ) )
     => ( ( inj_on7152781252210839493al_nat @ F @ top_top_set_nat )
       => ( Xs = Ys ) ) ) ).

% map_injective
thf(fact_250_map__injective,axiom,
    ! [F: nat > product_prod_nat_nat,Xs: list_nat,Ys: list_nat] :
      ( ( ( map_na7298421622053143531at_nat @ F @ Xs )
        = ( map_na7298421622053143531at_nat @ F @ Ys ) )
     => ( ( inj_on5538052773655684606at_nat @ F @ top_top_set_nat )
       => ( Xs = Ys ) ) ) ).

% map_injective
thf(fact_251_map__injective,axiom,
    ! [F: nat > produc7811952446676219690list_b,Xs: list_nat,Ys: list_nat] :
      ( ( ( map_na1272791293171509184list_b @ F @ Xs )
        = ( map_na1272791293171509184list_b @ F @ Ys ) )
     => ( ( inj_on8555073082499328493list_b @ F @ top_top_set_nat )
       => ( Xs = Ys ) ) ) ).

% map_injective
thf(fact_252_map__injective,axiom,
    ! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ F @ Ys ) )
     => ( ( inj_on_nat_nat @ F @ top_top_set_nat )
       => ( Xs = Ys ) ) ) ).

% map_injective
thf(fact_253_map__injective,axiom,
    ! [F: product_prod_nat_nat > produc859450856879609959at_nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( ( map_Pr9105590726435236202at_nat @ F @ Xs )
        = ( map_Pr9105590726435236202at_nat @ F @ Ys ) )
     => ( ( inj_on3865918002419084311at_nat @ F @ top_to4669805908274784177at_nat )
       => ( Xs = Ys ) ) ) ).

% map_injective
thf(fact_254_inj__mapD,axiom,
    ! [F: real > produc3741383161447143261al_nat] :
      ( ( inj_on8276318967698619124al_nat @ ( map_re3328248900745367531al_nat @ F ) @ top_to7938183770042617506t_real )
     => ( inj_on8027545645978435326al_nat @ F @ top_top_set_real ) ) ).

% inj_mapD
thf(fact_255_inj__mapD,axiom,
    ! [F: list_b > produc7811952446676219690list_b] :
      ( ( inj_on5686930240773211284list_b @ ( map_li8759865046734642667list_b @ F ) @ top_to4060329871436644795list_b )
     => ( inj_on8733544287791916926list_b @ F @ top_top_set_list_b ) ) ).

% inj_mapD
thf(fact_256_inj__mapD,axiom,
    ! [F: nat > produc9155354469731034754al_nat] :
      ( ( inj_on1287217757190318181al_nat @ ( map_na466430255352921112al_nat @ F ) @ top_top_set_list_nat )
     => ( inj_on7152781252210839493al_nat @ F @ top_top_set_nat ) ) ).

% inj_mapD
thf(fact_257_inj__mapD,axiom,
    ! [F: nat > product_prod_nat_nat] :
      ( ( inj_on3024915551208126196at_nat @ ( map_na7298421622053143531at_nat @ F ) @ top_top_set_list_nat )
     => ( inj_on5538052773655684606at_nat @ F @ top_top_set_nat ) ) ).

% inj_mapD
thf(fact_258_inj__mapD,axiom,
    ! [F: nat > produc7811952446676219690list_b] :
      ( ( inj_on23012040918581517list_b @ ( map_na1272791293171509184list_b @ F ) @ top_top_set_list_nat )
     => ( inj_on8555073082499328493list_b @ F @ top_top_set_nat ) ) ).

% inj_mapD
thf(fact_259_inj__mapD,axiom,
    ! [F: nat > nat] :
      ( ( inj_on3049792774292151987st_nat @ ( map_nat_nat @ F ) @ top_top_set_list_nat )
     => ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).

% inj_mapD
thf(fact_260_inj__mapD,axiom,
    ! [F: product_prod_nat_nat > produc859450856879609959at_nat] :
      ( ( inj_on3011817175317237783at_nat @ ( map_Pr9105590726435236202at_nat @ F ) @ top_to70995312096774071at_nat )
     => ( inj_on3865918002419084311at_nat @ F @ top_to4669805908274784177at_nat ) ) ).

% inj_mapD
thf(fact_261_top__empty__eq,axiom,
    ( top_top_list_b_o
    = ( ^ [X2: list_b] : ( member_list_b @ X2 @ top_top_set_list_b ) ) ) ).

% top_empty_eq
thf(fact_262_top__empty__eq,axiom,
    ( top_top_nat_o
    = ( ^ [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ) ) ).

% top_empty_eq
thf(fact_263_top__empty__eq,axiom,
    ( top_to1313005431924226822_nat_o
    = ( ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ top_to8454300755403694231at_nat ) ) ) ).

% top_empty_eq
thf(fact_264_top__empty__eq,axiom,
    ( top_to3137496036531963500_nat_o
    = ( ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ top_to4669805908274784177at_nat ) ) ) ).

% top_empty_eq
thf(fact_265_apfst__convE,axiom,
    ! [Q: produc7811952446676219690list_b,F: nat > nat,P2: produc7811952446676219690list_b] :
      ( ( Q
        = ( produc8895340734784419623list_b @ F @ P2 ) )
     => ~ ! [X4: nat,Y3: list_b] :
            ( ( P2
              = ( produc7903367361620597084list_b @ X4 @ Y3 ) )
           => ( Q
             != ( produc7903367361620597084list_b @ ( F @ X4 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_266_apfst__convE,axiom,
    ! [Q: product_prod_nat_nat,F: nat > nat,P2: product_prod_nat_nat] :
      ( ( Q
        = ( produc986720760941809198at_nat @ F @ P2 ) )
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X4 @ Y3 ) )
           => ( Q
             != ( product_Pair_nat_nat @ ( F @ X4 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_267_apfst__convE,axiom,
    ! [Q: produc3741383161447143261al_nat,F: nat > real,P2: product_prod_nat_nat] :
      ( ( Q
        = ( produc7323138200241035914al_nat @ F @ P2 ) )
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X4 @ Y3 ) )
           => ( Q
             != ( produc3181502643871035669al_nat @ ( F @ X4 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_268_apfst__convE,axiom,
    ! [Q: produc859450856879609959at_nat,F: product_prod_nat_nat > product_prod_nat_nat,P2: produc859450856879609959at_nat] :
      ( ( Q
        = ( produc3057951030792887557at_nat @ F @ P2 ) )
     => ~ ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( P2
              = ( produc6161850002892822231at_nat @ X4 @ Y3 ) )
           => ( Q
             != ( produc6161850002892822231at_nat @ ( F @ X4 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_269_apfst__convE,axiom,
    ! [Q: produc9155354469731034754al_nat,F: nat > nat,P2: produc9155354469731034754al_nat] :
      ( ( Q
        = ( produc2060316929304411775al_nat @ F @ P2 ) )
     => ~ ! [X4: nat,Y3: produc3741383161447143261al_nat] :
            ( ( P2
              = ( produc4490577844368043572al_nat @ X4 @ Y3 ) )
           => ( Q
             != ( produc4490577844368043572al_nat @ ( F @ X4 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_270_apfst__convE,axiom,
    ! [Q: product_prod_nat_nat,F: real > nat,P2: produc3741383161447143261al_nat] :
      ( ( Q
        = ( produc4676469564790641418at_nat @ F @ P2 ) )
     => ~ ! [X4: real,Y3: nat] :
            ( ( P2
              = ( produc3181502643871035669al_nat @ X4 @ Y3 ) )
           => ( Q
             != ( product_Pair_nat_nat @ ( F @ X4 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_271_apfst__convE,axiom,
    ! [Q: produc3741383161447143261al_nat,F: real > real,P2: produc3741383161447143261al_nat] :
      ( ( Q
        = ( produc8414705825198518630al_nat @ F @ P2 ) )
     => ~ ! [X4: real,Y3: nat] :
            ( ( P2
              = ( produc3181502643871035669al_nat @ X4 @ Y3 ) )
           => ( Q
             != ( produc3181502643871035669al_nat @ ( F @ X4 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_272_inj__on__apsnd,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( inj_on8969904277767023793at_nat @ ( produc3094765310956027504at_nat @ F )
        @ ( produc457027306803732586at_nat @ top_top_set_nat
          @ ^ [Uu: nat] : A4 ) )
      = ( inj_on_nat_nat @ F @ A4 ) ) ).

% inj_on_apsnd
thf(fact_273_inj__on__apsnd,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( inj_on6310514192432462843_b_nat @ ( produc338270834811952229list_b @ F )
        @ ( produc1835408586239750157_b_nat @ top_top_set_list_b
          @ ^ [Uu: list_b] : A4 ) )
      = ( inj_on_nat_nat @ F @ A4 ) ) ).

% inj_on_apsnd
thf(fact_274_inj__on__apsnd,axiom,
    ! [F: real > produc3741383161447143261al_nat,A4: set_real] :
      ( ( inj_on3400758723552037294al_nat @ ( produc7698354669090771331at_nat @ F )
        @ ( produc5860367176999294534t_real @ top_top_set_nat
          @ ^ [Uu: nat] : A4 ) )
      = ( inj_on8027545645978435326al_nat @ F @ A4 ) ) ).

% inj_on_apsnd
thf(fact_275_inj__on__apsnd,axiom,
    ! [F: nat > product_prod_nat_nat,A4: set_nat] :
      ( ( inj_on5416022199913362294at_nat @ ( produc1784090006485240963at_nat @ F )
        @ ( produc457027306803732586at_nat @ top_top_set_nat
          @ ^ [Uu: nat] : A4 ) )
      = ( inj_on5538052773655684606at_nat @ F @ A4 ) ) ).

% inj_on_apsnd
thf(fact_276_inj__on__apsnd,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( inj_on6274059163302071795at_nat @ ( produc5144137399527956065at_nat @ F )
        @ ( produc7672662199629908489at_nat @ top_to4669805908274784177at_nat
          @ ^ [Uu: product_prod_nat_nat] : A4 ) )
      = ( inj_on_nat_nat @ F @ A4 ) ) ).

% inj_on_apsnd
thf(fact_277_inj__on__apsnd,axiom,
    ! [F: real > produc3741383161447143261al_nat,A4: set_real] :
      ( ( inj_on5023972497546319276al_nat @ ( produc7060922397183159698list_b @ F )
        @ ( produc2378038581037384809b_real @ top_top_set_list_b
          @ ^ [Uu: list_b] : A4 ) )
      = ( inj_on8027545645978435326al_nat @ F @ A4 ) ) ).

% inj_on_apsnd
thf(fact_278_inj__on__apsnd,axiom,
    ! [F: nat > product_prod_nat_nat,A4: set_nat] :
      ( ( inj_on9121707551203860396at_nat @ ( produc5478395629987525266list_b @ F )
        @ ( produc1835408586239750157_b_nat @ top_top_set_list_b
          @ ^ [Uu: list_b] : A4 ) )
      = ( inj_on5538052773655684606at_nat @ F @ A4 ) ) ).

% inj_on_apsnd
thf(fact_279_inj__on__apsnd,axiom,
    ! [F: nat > produc7811952446676219690list_b,A4: set_nat] :
      ( ( inj_on4534532566085093899list_b @ ( produc7599226473598062410_b_nat @ F )
        @ ( produc457027306803732586at_nat @ top_top_set_nat
          @ ^ [Uu: nat] : A4 ) )
      = ( inj_on8555073082499328493list_b @ F @ A4 ) ) ).

% inj_on_apsnd
thf(fact_280_inj__on__apsnd,axiom,
    ! [F: nat > produc7811952446676219690list_b,A4: set_nat] :
      ( ( inj_on3137949769578664533list_b @ ( produc1442933947714241291list_b @ F )
        @ ( produc1835408586239750157_b_nat @ top_top_set_list_b
          @ ^ [Uu: list_b] : A4 ) )
      = ( inj_on8555073082499328493list_b @ F @ A4 ) ) ).

% inj_on_apsnd
thf(fact_281_inj__on__apsnd,axiom,
    ! [F: list_b > produc7811952446676219690list_b,A4: set_list_b] :
      ( ( inj_on8157835898299261438list_b @ ( produc5171987171093733379_b_nat @ F )
        @ ( produc132127289527437803list_b @ top_top_set_nat
          @ ^ [Uu: nat] : A4 ) )
      = ( inj_on8733544287791916926list_b @ F @ A4 ) ) ).

% inj_on_apsnd
thf(fact_282_inj__on__apfst,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( inj_on8969904277767023793at_nat @ ( produc986720760941809198at_nat @ F )
        @ ( produc457027306803732586at_nat @ A4
          @ ^ [Uu: nat] : top_top_set_nat ) )
      = ( inj_on_nat_nat @ F @ A4 ) ) ).

% inj_on_apfst
thf(fact_283_inj__on__apfst,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( inj_on4901578499414790983list_b @ ( produc8895340734784419623list_b @ F )
        @ ( produc132127289527437803list_b @ A4
          @ ^ [Uu: nat] : top_top_set_list_b ) )
      = ( inj_on_nat_nat @ F @ A4 ) ) ).

% inj_on_apfst
thf(fact_284_inj__on__apfst,axiom,
    ! [F: real > produc3741383161447143261al_nat,A4: set_real] :
      ( ( inj_on7077800877260587912at_nat @ ( produc2212142120895633989at_nat @ F )
        @ ( produc1204303713273417414al_nat @ A4
          @ ^ [Uu: real] : top_top_set_nat ) )
      = ( inj_on8027545645978435326al_nat @ F @ A4 ) ) ).

% inj_on_apfst
thf(fact_285_inj__on__apfst,axiom,
    ! [F: nat > product_prod_nat_nat,A4: set_nat] :
      ( ( inj_on6541509183880663376at_nat @ ( produc3001624868188084037at_nat @ F )
        @ ( produc457027306803732586at_nat @ A4
          @ ^ [Uu: nat] : top_top_set_nat ) )
      = ( inj_on5538052773655684606at_nat @ F @ A4 ) ) ).

% inj_on_apfst
thf(fact_286_inj__on__apfst,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( inj_on1022140648125475135at_nat @ ( produc6361672261230799139at_nat @ F )
        @ ( produc1809337555817847783at_nat @ A4
          @ ^ [Uu: nat] : top_to4669805908274784177at_nat ) )
      = ( inj_on_nat_nat @ F @ A4 ) ) ).

% inj_on_apfst
thf(fact_287_inj__on__apfst,axiom,
    ! [F: real > produc3741383161447143261al_nat,A4: set_real] :
      ( ( inj_on6823834738547617294list_b @ ( produc241710712860514896list_b @ F )
        @ ( produc3739882541106747535list_b @ A4
          @ ^ [Uu: real] : top_top_set_list_b ) )
      = ( inj_on8027545645978435326al_nat @ F @ A4 ) ) ).

% inj_on_apfst
thf(fact_288_inj__on__apfst,axiom,
    ! [F: nat > product_prod_nat_nat,A4: set_nat] :
      ( ( inj_on6772791774875031566list_b @ ( produc2732834783157706064list_b @ F )
        @ ( produc132127289527437803list_b @ A4
          @ ^ [Uu: nat] : top_top_set_list_b ) )
      = ( inj_on5538052773655684606at_nat @ F @ A4 ) ) ).

% inj_on_apfst
thf(fact_289_inj__on__apfst,axiom,
    ! [F: nat > produc7811952446676219690list_b,A4: set_nat] :
      ( ( inj_on1775699293261311627_b_nat @ ( produc4853665626768243208_b_nat @ F )
        @ ( produc457027306803732586at_nat @ A4
          @ ^ [Uu: nat] : top_top_set_nat ) )
      = ( inj_on8555073082499328493list_b @ F @ A4 ) ) ).

% inj_on_apfst
thf(fact_290_inj__on__apfst,axiom,
    ! [F: nat > produc7811952446676219690list_b,A4: set_nat] :
      ( ( inj_on4371743984487328109list_b @ ( produc8686098695948174029list_b @ F )
        @ ( produc132127289527437803list_b @ A4
          @ ^ [Uu: nat] : top_top_set_list_b ) )
      = ( inj_on8555073082499328493list_b @ F @ A4 ) ) ).

% inj_on_apfst
thf(fact_291_inj__on__apfst,axiom,
    ! [F: list_b > produc7811952446676219690list_b,A4: set_list_b] :
      ( ( inj_on5740833130252885284_b_nat @ ( produc3191779882472890309_b_nat @ F )
        @ ( produc1835408586239750157_b_nat @ A4
          @ ^ [Uu: list_b] : top_top_set_nat ) )
      = ( inj_on8733544287791916926list_b @ F @ A4 ) ) ).

% inj_on_apfst
thf(fact_292_L_092_060_094sub_062e__def,axiom,
    ( prefix9075205949435278297_L_e_b
    = ( ^ [E2: b > option_list_o,Xs3: list_b] : ( prefix8792501375849716147list_b @ prefix8864127203703499552e_Nu_e @ ( prefix4097710381326367691Lf_e_b @ E2 ) @ ( produc7903367361620597084list_b @ ( size_size_list_b @ Xs3 ) @ Xs3 ) ) ) ) ).

% L\<^sub>e_def
thf(fact_293_map__eq__map__tailrec,axiom,
    map_nat_nat = map_tailrec_nat_nat ).

% map_eq_map_tailrec
thf(fact_294_map__removeAll__inj,axiom,
    ! [F: real > produc3741383161447143261al_nat,X3: real,Xs: list_real] :
      ( ( inj_on8027545645978435326al_nat @ F @ top_top_set_real )
     => ( ( map_re3328248900745367531al_nat @ F @ ( removeAll_real @ X3 @ Xs ) )
        = ( remove4535377733419490665al_nat @ ( F @ X3 ) @ ( map_re3328248900745367531al_nat @ F @ Xs ) ) ) ) ).

% map_removeAll_inj
thf(fact_295_map__removeAll__inj,axiom,
    ! [F: list_b > produc7811952446676219690list_b,X3: list_b,Xs: list_list_b] :
      ( ( inj_on8733544287791916926list_b @ F @ top_top_set_list_b )
     => ( ( map_li8759865046734642667list_b @ F @ ( removeAll_list_b @ X3 @ Xs ) )
        = ( remove5849340258981760030list_b @ ( F @ X3 ) @ ( map_li8759865046734642667list_b @ F @ Xs ) ) ) ) ).

% map_removeAll_inj
thf(fact_296_map__removeAll__inj,axiom,
    ! [F: nat > produc9155354469731034754al_nat,X3: nat,Xs: list_nat] :
      ( ( inj_on7152781252210839493al_nat @ F @ top_top_set_nat )
     => ( ( map_na466430255352921112al_nat @ F @ ( removeAll_nat @ X3 @ Xs ) )
        = ( remove6711459154290680182al_nat @ ( F @ X3 ) @ ( map_na466430255352921112al_nat @ F @ Xs ) ) ) ) ).

% map_removeAll_inj
thf(fact_297_map__removeAll__inj,axiom,
    ! [F: nat > product_prod_nat_nat,X3: nat,Xs: list_nat] :
      ( ( inj_on5538052773655684606at_nat @ F @ top_top_set_nat )
     => ( ( map_na7298421622053143531at_nat @ F @ ( removeAll_nat @ X3 @ Xs ) )
        = ( remove3673390508374433037at_nat @ ( F @ X3 ) @ ( map_na7298421622053143531at_nat @ F @ Xs ) ) ) ) ).

% map_removeAll_inj
thf(fact_298_map__removeAll__inj,axiom,
    ! [F: nat > produc7811952446676219690list_b,X3: nat,Xs: list_nat] :
      ( ( inj_on8555073082499328493list_b @ F @ top_top_set_nat )
     => ( ( map_na1272791293171509184list_b @ F @ ( removeAll_nat @ X3 @ Xs ) )
        = ( remove5849340258981760030list_b @ ( F @ X3 ) @ ( map_na1272791293171509184list_b @ F @ Xs ) ) ) ) ).

% map_removeAll_inj
thf(fact_299_map__removeAll__inj,axiom,
    ! [F: nat > nat,X3: nat,Xs: list_nat] :
      ( ( inj_on_nat_nat @ F @ top_top_set_nat )
     => ( ( map_nat_nat @ F @ ( removeAll_nat @ X3 @ Xs ) )
        = ( removeAll_nat @ ( F @ X3 ) @ ( map_nat_nat @ F @ Xs ) ) ) ) ).

% map_removeAll_inj
thf(fact_300_map__removeAll__inj,axiom,
    ! [F: product_prod_nat_nat > produc859450856879609959at_nat,X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( inj_on3865918002419084311at_nat @ F @ top_to4669805908274784177at_nat )
     => ( ( map_Pr9105590726435236202at_nat @ F @ ( remove3673390508374433037at_nat @ X3 @ Xs ) )
        = ( remove2181804207701385843at_nat @ ( F @ X3 ) @ ( map_Pr9105590726435236202at_nat @ F @ Xs ) ) ) ) ).

% map_removeAll_inj
thf(fact_301_inj__rotate1,axiom,
    inj_on_list_b_list_b @ rotate1_b @ top_top_set_list_b ).

% inj_rotate1
thf(fact_302_asymI,axiom,
    ! [R2: set_Pr7665282455119567943list_b] :
      ( ! [X4: list_b,Y3: list_b] :
          ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ X4 @ Y3 ) @ R2 )
         => ~ ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Y3 @ X4 ) @ R2 ) )
     => ( asym_on_list_b @ top_top_set_list_b @ R2 ) ) ).

% asymI
thf(fact_303_asymI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R2 )
         => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X4 ) @ R2 ) )
     => ( asym_on_nat @ top_top_set_nat @ R2 ) ) ).

% asymI
thf(fact_304_asymI,axiom,
    ! [R2: set_Pr553994874890374343at_nat] :
      ( ! [X4: produc859450856879609959at_nat,Y3: produc859450856879609959at_nat] :
          ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X4 @ Y3 ) @ R2 )
         => ~ ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y3 @ X4 ) @ R2 ) )
     => ( asym_o9200482492015546265at_nat @ top_to8454300755403694231at_nat @ R2 ) ) ).

% asymI
thf(fact_305_asymI,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y3 ) @ R2 )
         => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ X4 ) @ R2 ) )
     => ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R2 ) ) ).

% asymI
thf(fact_306_size__neq__size__imp__neq,axiom,
    ! [X3: list_b,Y: list_b] :
      ( ( ( size_size_list_b @ X3 )
       != ( size_size_list_b @ Y ) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_307_prod_Oinj__map,axiom,
    ! [F1: nat > nat,F22: nat > nat] :
      ( ( inj_on_nat_nat @ F1 @ top_top_set_nat )
     => ( ( inj_on_nat_nat @ F22 @ top_top_set_nat )
       => ( inj_on8969904277767023793at_nat @ ( produc6977886695330630970at_nat @ F1 @ F22 ) @ top_to4669805908274784177at_nat ) ) ) ).

% prod.inj_map
thf(fact_308_prod_Oinj__map,axiom,
    ! [F1: real > produc3741383161447143261al_nat,F22: nat > nat] :
      ( ( inj_on8027545645978435326al_nat @ F1 @ top_top_set_real )
     => ( ( inj_on_nat_nat @ F22 @ top_top_set_nat )
       => ( inj_on7077800877260587912at_nat @ ( produc94305847159717211at_nat @ F1 @ F22 ) @ top_to8045530339851357581al_nat ) ) ) ).

% prod.inj_map
thf(fact_309_prod_Oinj__map,axiom,
    ! [F1: nat > nat,F22: real > produc3741383161447143261al_nat] :
      ( ( inj_on_nat_nat @ F1 @ top_top_set_nat )
     => ( ( inj_on8027545645978435326al_nat @ F22 @ top_top_set_real )
       => ( inj_on3400758723552037294al_nat @ ( produc7218282358209923863al_nat @ F1 @ F22 ) @ top_to4855536200657754381t_real ) ) ) ).

% prod.inj_map
thf(fact_310_prod_Oinj__map,axiom,
    ! [F1: nat > nat,F22: nat > product_prod_nat_nat] :
      ( ( inj_on_nat_nat @ F1 @ top_top_set_nat )
     => ( ( inj_on5538052773655684606at_nat @ F22 @ top_top_set_nat )
       => ( inj_on5416022199913362294at_nat @ ( produc9062365837733128983at_nat @ F1 @ F22 ) @ top_to4669805908274784177at_nat ) ) ) ).

% prod.inj_map
thf(fact_311_prod_Oinj__map,axiom,
    ! [F1: nat > product_prod_nat_nat,F22: nat > nat] :
      ( ( inj_on5538052773655684606at_nat @ F1 @ top_top_set_nat )
     => ( ( inj_on_nat_nat @ F22 @ top_top_set_nat )
       => ( inj_on6541509183880663376at_nat @ ( produc8306735474602065755at_nat @ F1 @ F22 ) @ top_to4669805908274784177at_nat ) ) ) ).

% prod.inj_map
thf(fact_312_prod_Oinj__map,axiom,
    ! [F1: nat > nat,F22: nat > produc7811952446676219690list_b] :
      ( ( inj_on_nat_nat @ F1 @ top_top_set_nat )
     => ( ( inj_on8555073082499328493list_b @ F22 @ top_top_set_nat )
       => ( inj_on4534532566085093899list_b @ ( produc5470999172165745044list_b @ F1 @ F22 ) @ top_to4669805908274784177at_nat ) ) ) ).

% prod.inj_map
thf(fact_313_prod_Oinj__map,axiom,
    ! [F1: nat > produc7811952446676219690list_b,F22: nat > nat] :
      ( ( inj_on8555073082499328493list_b @ F1 @ top_top_set_nat )
     => ( ( inj_on_nat_nat @ F22 @ top_top_set_nat )
       => ( inj_on1775699293261311627_b_nat @ ( produc2270369883413640340at_nat @ F1 @ F22 ) @ top_to4669805908274784177at_nat ) ) ) ).

% prod.inj_map
thf(fact_314_prod_Oinj__map,axiom,
    ! [F1: real > produc3741383161447143261al_nat,F22: real > produc3741383161447143261al_nat] :
      ( ( inj_on8027545645978435326al_nat @ F1 @ top_top_set_real )
     => ( ( inj_on8027545645978435326al_nat @ F22 @ top_top_set_real )
       => ( inj_on986035600877344991al_nat @ ( produc3923160613318430390al_nat @ F1 @ F22 ) @ top_to1175844992842649833l_real ) ) ) ).

% prod.inj_map
thf(fact_315_prod_Oinj__map,axiom,
    ! [F1: real > produc3741383161447143261al_nat,F22: nat > product_prod_nat_nat] :
      ( ( inj_on8027545645978435326al_nat @ F1 @ top_top_set_real )
     => ( ( inj_on5538052773655684606at_nat @ F22 @ top_top_set_nat )
       => ( inj_on394277883117196255at_nat @ ( produc6091599326012681654at_nat @ F1 @ F22 ) @ top_to8045530339851357581al_nat ) ) ) ).

% prod.inj_map
thf(fact_316_prod_Oinj__map,axiom,
    ! [F1: list_b > produc7811952446676219690list_b,F22: nat > nat] :
      ( ( inj_on8733544287791916926list_b @ F1 @ top_top_set_list_b )
     => ( ( inj_on_nat_nat @ F22 @ top_top_set_nat )
       => ( inj_on5740833130252885284_b_nat @ ( produc8169669994075221083at_nat @ F1 @ F22 ) @ top_to6547989104015266538_b_nat ) ) ) ).

% prod.inj_map
thf(fact_317_ex__inj,axiom,
    ? [To_nat: nat > nat] : ( inj_on_nat_nat @ To_nat @ top_top_set_nat ) ).

% ex_inj
thf(fact_318_ex__inj,axiom,
    ? [To_nat: produc859450856879609959at_nat > nat] : ( inj_on490725545064285830at_nat @ To_nat @ top_to8454300755403694231at_nat ) ).

% ex_inj
thf(fact_319_ex__inj,axiom,
    ? [To_nat: product_prod_nat_nat > nat] : ( inj_on2178005380612969504at_nat @ To_nat @ top_to4669805908274784177at_nat ) ).

% ex_inj
thf(fact_320_mem__Sigma__iff,axiom,
    ! [A: nat,B: nat,A4: set_nat,B4: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A4 @ B4 ) )
      = ( ( member_nat @ A @ A4 )
        & ( member_nat @ B @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_321_mem__Sigma__iff,axiom,
    ! [A: real,B: nat,A4: set_real,B4: real > set_nat] :
      ( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ A @ B ) @ ( produc1204303713273417414al_nat @ A4 @ B4 ) )
      = ( ( member_real @ A @ A4 )
        & ( member_nat @ B @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_322_mem__Sigma__iff,axiom,
    ! [A: list_b,B: nat,A4: set_list_b,B4: list_b > set_nat] :
      ( ( member6469666724224207131_b_nat @ ( produc383276621478133630_b_nat @ A @ B ) @ ( produc1835408586239750157_b_nat @ A4 @ B4 ) )
      = ( ( member_list_b @ A @ A4 )
        & ( member_nat @ B @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_323_mem__Sigma__iff,axiom,
    ! [A: nat,B: list_b,A4: set_nat,B4: nat > set_list_b] :
      ( ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ A @ B ) @ ( produc132127289527437803list_b @ A4 @ B4 ) )
      = ( ( member_nat @ A @ A4 )
        & ( member_list_b @ B @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_324_mem__Sigma__iff,axiom,
    ! [A: product_prod_nat_nat,B: nat,A4: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_nat] :
      ( ( member3348759134392003351at_nat @ ( produc6350711070570205562at_nat @ A @ B ) @ ( produc7672662199629908489at_nat @ A4 @ B4 ) )
      = ( ( member8440522571783428010at_nat @ A @ A4 )
        & ( member_nat @ B @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_325_mem__Sigma__iff,axiom,
    ! [A: nat,B: product_prod_nat_nat,A4: set_nat,B4: nat > set_Pr1261947904930325089at_nat] :
      ( ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ A @ B ) @ ( produc1809337555817847783at_nat @ A4 @ B4 ) )
      = ( ( member_nat @ A @ A4 )
        & ( member8440522571783428010at_nat @ B @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_326_mem__Sigma__iff,axiom,
    ! [A: list_b,B: list_b,A4: set_list_b,B4: list_b > set_list_b] :
      ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ A @ B ) @ ( produc7475029858066475656list_b @ A4 @ B4 ) )
      = ( ( member_list_b @ A @ A4 )
        & ( member_list_b @ B @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_327_mem__Sigma__iff,axiom,
    ! [A: nat,B: produc3741383161447143261al_nat,A4: set_nat,B4: nat > set_Pr3510011417693777981al_nat] :
      ( ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ A @ B ) @ ( produc3292543272716750659al_nat @ A4 @ B4 ) )
      = ( ( member_nat @ A @ A4 )
        & ( member5805532792777349510al_nat @ B @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_328_mem__Sigma__iff,axiom,
    ! [A: product_prod_nat_nat,B: list_b,A4: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_list_b] :
      ( ( member9125244090276215188list_b @ ( produc7400629846093677787list_b @ A @ B ) @ ( produc4636629702015616396list_b @ A4 @ B4 ) )
      = ( ( member8440522571783428010at_nat @ A @ A4 )
        & ( member_list_b @ B @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_329_mem__Sigma__iff,axiom,
    ! [A: list_b,B: product_prod_nat_nat,A4: set_list_b,B4: list_b > set_Pr1261947904930325089at_nat] :
      ( ( member1908957324972862092at_nat @ ( produc8164497450810430419at_nat @ A @ B ) @ ( produc5400497306732369028at_nat @ A4 @ B4 ) )
      = ( ( member_list_b @ A @ A4 )
        & ( member8440522571783428010at_nat @ B @ ( B4 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_330_SigmaI,axiom,
    ! [A: nat,A4: set_nat,B: nat,B4: nat > set_nat] :
      ( ( member_nat @ A @ A4 )
     => ( ( member_nat @ B @ ( B4 @ A ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A4 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_331_SigmaI,axiom,
    ! [A: real,A4: set_real,B: nat,B4: real > set_nat] :
      ( ( member_real @ A @ A4 )
     => ( ( member_nat @ B @ ( B4 @ A ) )
       => ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ A @ B ) @ ( produc1204303713273417414al_nat @ A4 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_332_SigmaI,axiom,
    ! [A: list_b,A4: set_list_b,B: nat,B4: list_b > set_nat] :
      ( ( member_list_b @ A @ A4 )
     => ( ( member_nat @ B @ ( B4 @ A ) )
       => ( member6469666724224207131_b_nat @ ( produc383276621478133630_b_nat @ A @ B ) @ ( produc1835408586239750157_b_nat @ A4 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_333_SigmaI,axiom,
    ! [A: nat,A4: set_nat,B: list_b,B4: nat > set_list_b] :
      ( ( member_nat @ A @ A4 )
     => ( ( member_list_b @ B @ ( B4 @ A ) )
       => ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ A @ B ) @ ( produc132127289527437803list_b @ A4 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_334_SigmaI,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: nat,B4: product_prod_nat_nat > set_nat] :
      ( ( member8440522571783428010at_nat @ A @ A4 )
     => ( ( member_nat @ B @ ( B4 @ A ) )
       => ( member3348759134392003351at_nat @ ( produc6350711070570205562at_nat @ A @ B ) @ ( produc7672662199629908489at_nat @ A4 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_335_SigmaI,axiom,
    ! [A: nat,A4: set_nat,B: product_prod_nat_nat,B4: nat > set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ A4 )
     => ( ( member8440522571783428010at_nat @ B @ ( B4 @ A ) )
       => ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ A @ B ) @ ( produc1809337555817847783at_nat @ A4 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_336_SigmaI,axiom,
    ! [A: list_b,A4: set_list_b,B: list_b,B4: list_b > set_list_b] :
      ( ( member_list_b @ A @ A4 )
     => ( ( member_list_b @ B @ ( B4 @ A ) )
       => ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ A @ B ) @ ( produc7475029858066475656list_b @ A4 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_337_SigmaI,axiom,
    ! [A: nat,A4: set_nat,B: produc3741383161447143261al_nat,B4: nat > set_Pr3510011417693777981al_nat] :
      ( ( member_nat @ A @ A4 )
     => ( ( member5805532792777349510al_nat @ B @ ( B4 @ A ) )
       => ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ A @ B ) @ ( produc3292543272716750659al_nat @ A4 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_338_SigmaI,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: list_b,B4: product_prod_nat_nat > set_list_b] :
      ( ( member8440522571783428010at_nat @ A @ A4 )
     => ( ( member_list_b @ B @ ( B4 @ A ) )
       => ( member9125244090276215188list_b @ ( produc7400629846093677787list_b @ A @ B ) @ ( produc4636629702015616396list_b @ A4 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_339_SigmaI,axiom,
    ! [A: list_b,A4: set_list_b,B: product_prod_nat_nat,B4: list_b > set_Pr1261947904930325089at_nat] :
      ( ( member_list_b @ A @ A4 )
     => ( ( member8440522571783428010at_nat @ B @ ( B4 @ A ) )
       => ( member1908957324972862092at_nat @ ( produc8164497450810430419at_nat @ A @ B ) @ ( produc5400497306732369028at_nat @ A4 @ B4 ) ) ) ) ).

% SigmaI
thf(fact_340_map__prod__simp,axiom,
    ! [F: nat > nat,G: nat > nat,A: nat,B: nat] :
      ( ( produc6977886695330630970at_nat @ F @ G @ ( product_Pair_nat_nat @ A @ B ) )
      = ( product_Pair_nat_nat @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_341_map__prod__simp,axiom,
    ! [F: nat > real,G: nat > nat,A: nat,B: nat] :
      ( ( produc1775294054601952278at_nat @ F @ G @ ( product_Pair_nat_nat @ A @ B ) )
      = ( produc3181502643871035669al_nat @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_342_map__prod__simp,axiom,
    ! [F: real > nat,G: nat > nat,A: real,B: nat] :
      ( ( produc2087023291286373526at_nat @ F @ G @ ( produc3181502643871035669al_nat @ A @ B ) )
      = ( product_Pair_nat_nat @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_343_map__prod__simp,axiom,
    ! [F: real > real,G: nat > nat,A: real,B: nat] :
      ( ( produc7190006219768567154at_nat @ F @ G @ ( produc3181502643871035669al_nat @ A @ B ) )
      = ( produc3181502643871035669al_nat @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_344_map__prod__simp,axiom,
    ! [F: nat > nat,G: list_b > nat,A: nat,B: list_b] :
      ( ( produc2348337553441767229_b_nat @ F @ G @ ( produc7903367361620597084list_b @ A @ B ) )
      = ( product_Pair_nat_nat @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_345_map__prod__simp,axiom,
    ! [F: nat > real,G: list_b > nat,A: nat,B: list_b] :
      ( ( produc5514980190314929505_b_nat @ F @ G @ ( produc7903367361620597084list_b @ A @ B ) )
      = ( produc3181502643871035669al_nat @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_346_map__prod__simp,axiom,
    ! [F: nat > nat,G: nat > list_b,A: nat,B: nat] :
      ( ( produc645056256729454875list_b @ F @ G @ ( product_Pair_nat_nat @ A @ B ) )
      = ( produc7903367361620597084list_b @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_347_map__prod__simp,axiom,
    ! [F: real > nat,G: nat > list_b,A: real,B: nat] :
      ( ( produc5957278046674525887list_b @ F @ G @ ( produc3181502643871035669al_nat @ A @ B ) )
      = ( produc7903367361620597084list_b @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_348_map__prod__simp,axiom,
    ! [F: nat > nat,G: list_b > list_b,A: nat,B: list_b] :
      ( ( produc5356872882700738392list_b @ F @ G @ ( produc7903367361620597084list_b @ A @ B ) )
      = ( produc7903367361620597084list_b @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_349_map__prod__simp,axiom,
    ! [F: nat > nat,G: nat > produc3741383161447143261al_nat,A: nat,B: nat] :
      ( ( produc1642924643756656243al_nat @ F @ G @ ( product_Pair_nat_nat @ A @ B ) )
      = ( produc4490577844368043572al_nat @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_350_asym__onI,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat] :
      ( ! [X4: produc859450856879609959at_nat,Y3: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X4 @ A4 )
         => ( ( member8206827879206165904at_nat @ Y3 @ A4 )
           => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X4 @ Y3 ) @ R2 )
             => ~ ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y3 @ X4 ) @ R2 ) ) ) )
     => ( asym_o9200482492015546265at_nat @ A4 @ R2 ) ) ).

% asym_onI
thf(fact_351_asym__onI,axiom,
    ! [A4: set_list_b,R2: set_Pr7665282455119567943list_b] :
      ( ! [X4: list_b,Y3: list_b] :
          ( ( member_list_b @ X4 @ A4 )
         => ( ( member_list_b @ Y3 @ A4 )
           => ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ X4 @ Y3 ) @ R2 )
             => ~ ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Y3 @ X4 ) @ R2 ) ) ) )
     => ( asym_on_list_b @ A4 @ R2 ) ) ).

% asym_onI
thf(fact_352_asym__onI,axiom,
    ! [A4: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( member_nat @ X4 @ A4 )
         => ( ( member_nat @ Y3 @ A4 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R2 )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X4 ) @ R2 ) ) ) )
     => ( asym_on_nat @ A4 @ R2 ) ) ).

% asym_onI
thf(fact_353_asym__onI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ A4 )
         => ( ( member8440522571783428010at_nat @ Y3 @ A4 )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y3 ) @ R2 )
             => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ X4 ) @ R2 ) ) ) )
     => ( asym_o8546584678760569651at_nat @ A4 @ R2 ) ) ).

% asym_onI
thf(fact_354_length__rotate1,axiom,
    ! [Xs: list_b] :
      ( ( size_size_list_b @ ( rotate1_b @ Xs ) )
      = ( size_size_list_b @ Xs ) ) ).

% length_rotate1
thf(fact_355_UNIV__Times__UNIV,axiom,
    ( ( produc457027306803732586at_nat @ top_top_set_nat
      @ ^ [Uu: nat] : top_top_set_nat )
    = top_to4669805908274784177at_nat ) ).

% UNIV_Times_UNIV
thf(fact_356_UNIV__Times__UNIV,axiom,
    ( ( produc1835408586239750157_b_nat @ top_top_set_list_b
      @ ^ [Uu: list_b] : top_top_set_nat )
    = top_to6547989104015266538_b_nat ) ).

% UNIV_Times_UNIV
thf(fact_357_UNIV__Times__UNIV,axiom,
    ( ( produc132127289527437803list_b @ top_top_set_nat
      @ ^ [Uu: nat] : top_top_set_list_b )
    = top_to4428277177966209040list_b ) ).

% UNIV_Times_UNIV
thf(fact_358_UNIV__Times__UNIV,axiom,
    ( ( produc7475029858066475656list_b @ top_top_set_list_b
      @ ^ [Uu: list_b] : top_top_set_list_b )
    = top_to6408289459866364311list_b ) ).

% UNIV_Times_UNIV
thf(fact_359_UNIV__Times__UNIV,axiom,
    ( ( produc1809337555817847783at_nat @ top_top_set_nat
      @ ^ [Uu: nat] : top_to4669805908274784177at_nat )
    = top_to7745748144976627212at_nat ) ).

% UNIV_Times_UNIV
thf(fact_360_UNIV__Times__UNIV,axiom,
    ( ( produc7672662199629908489at_nat @ top_to4669805908274784177at_nat
      @ ^ [Uu: product_prod_nat_nat] : top_top_set_nat )
    = top_to2567003362141017830at_nat ) ).

% UNIV_Times_UNIV
thf(fact_361_UNIV__Times__UNIV,axiom,
    ( ( produc5400497306732369028at_nat @ top_top_set_list_b
      @ ^ [Uu: list_b] : top_to4669805908274784177at_nat )
    = top_to538300756415451027at_nat ) ).

% UNIV_Times_UNIV
thf(fact_362_UNIV__Times__UNIV,axiom,
    ( ( produc4636629702015616396list_b @ top_to4669805908274784177at_nat
      @ ^ [Uu: product_prod_nat_nat] : top_top_set_list_b )
    = top_to548042179339279003list_b ) ).

% UNIV_Times_UNIV
thf(fact_363_UNIV__Times__UNIV,axiom,
    ( ( produc2761391749766926216at_nat @ top_to4669805908274784177at_nat
      @ ^ [Uu: product_prod_nat_nat] : top_to4669805908274784177at_nat )
    = top_to8454300755403694231at_nat ) ).

% UNIV_Times_UNIV
thf(fact_364_UNIV__Times__UNIV,axiom,
    ( ( produc5822243379040957005at_nat @ top_top_set_nat
      @ ^ [Uu: nat] : top_to8454300755403694231at_nat )
    = top_to1012081421760046066at_nat ) ).

% UNIV_Times_UNIV
thf(fact_365_unary__encoding,axiom,
    prefix3558185134189398382ng_nat @ prefix8864127203703499552e_Nu_e ).

% unary_encoding
thf(fact_366_SigmaE2,axiom,
    ! [A: nat,B: nat,A4: set_nat,B4: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A4 @ B4 ) )
     => ~ ( ( member_nat @ A @ A4 )
         => ~ ( member_nat @ B @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_367_SigmaE2,axiom,
    ! [A: real,B: nat,A4: set_real,B4: real > set_nat] :
      ( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ A @ B ) @ ( produc1204303713273417414al_nat @ A4 @ B4 ) )
     => ~ ( ( member_real @ A @ A4 )
         => ~ ( member_nat @ B @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_368_SigmaE2,axiom,
    ! [A: list_b,B: nat,A4: set_list_b,B4: list_b > set_nat] :
      ( ( member6469666724224207131_b_nat @ ( produc383276621478133630_b_nat @ A @ B ) @ ( produc1835408586239750157_b_nat @ A4 @ B4 ) )
     => ~ ( ( member_list_b @ A @ A4 )
         => ~ ( member_nat @ B @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_369_SigmaE2,axiom,
    ! [A: nat,B: list_b,A4: set_nat,B4: nat > set_list_b] :
      ( ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ A @ B ) @ ( produc132127289527437803list_b @ A4 @ B4 ) )
     => ~ ( ( member_nat @ A @ A4 )
         => ~ ( member_list_b @ B @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_370_SigmaE2,axiom,
    ! [A: product_prod_nat_nat,B: nat,A4: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_nat] :
      ( ( member3348759134392003351at_nat @ ( produc6350711070570205562at_nat @ A @ B ) @ ( produc7672662199629908489at_nat @ A4 @ B4 ) )
     => ~ ( ( member8440522571783428010at_nat @ A @ A4 )
         => ~ ( member_nat @ B @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_371_SigmaE2,axiom,
    ! [A: nat,B: product_prod_nat_nat,A4: set_nat,B4: nat > set_Pr1261947904930325089at_nat] :
      ( ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ A @ B ) @ ( produc1809337555817847783at_nat @ A4 @ B4 ) )
     => ~ ( ( member_nat @ A @ A4 )
         => ~ ( member8440522571783428010at_nat @ B @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_372_SigmaE2,axiom,
    ! [A: list_b,B: list_b,A4: set_list_b,B4: list_b > set_list_b] :
      ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ A @ B ) @ ( produc7475029858066475656list_b @ A4 @ B4 ) )
     => ~ ( ( member_list_b @ A @ A4 )
         => ~ ( member_list_b @ B @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_373_SigmaE2,axiom,
    ! [A: nat,B: produc3741383161447143261al_nat,A4: set_nat,B4: nat > set_Pr3510011417693777981al_nat] :
      ( ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ A @ B ) @ ( produc3292543272716750659al_nat @ A4 @ B4 ) )
     => ~ ( ( member_nat @ A @ A4 )
         => ~ ( member5805532792777349510al_nat @ B @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_374_SigmaE2,axiom,
    ! [A: product_prod_nat_nat,B: list_b,A4: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_list_b] :
      ( ( member9125244090276215188list_b @ ( produc7400629846093677787list_b @ A @ B ) @ ( produc4636629702015616396list_b @ A4 @ B4 ) )
     => ~ ( ( member8440522571783428010at_nat @ A @ A4 )
         => ~ ( member_list_b @ B @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_375_SigmaE2,axiom,
    ! [A: list_b,B: product_prod_nat_nat,A4: set_list_b,B4: list_b > set_Pr1261947904930325089at_nat] :
      ( ( member1908957324972862092at_nat @ ( produc8164497450810430419at_nat @ A @ B ) @ ( produc5400497306732369028at_nat @ A4 @ B4 ) )
     => ~ ( ( member_list_b @ A @ A4 )
         => ~ ( member8440522571783428010at_nat @ B @ ( B4 @ A ) ) ) ) ).

% SigmaE2
thf(fact_376_SigmaD2,axiom,
    ! [A: nat,B: list_b,A4: set_nat,B4: nat > set_list_b] :
      ( ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ A @ B ) @ ( produc132127289527437803list_b @ A4 @ B4 ) )
     => ( member_list_b @ B @ ( B4 @ A ) ) ) ).

% SigmaD2
thf(fact_377_SigmaD2,axiom,
    ! [A: nat,B: nat,A4: set_nat,B4: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A4 @ B4 ) )
     => ( member_nat @ B @ ( B4 @ A ) ) ) ).

% SigmaD2
thf(fact_378_SigmaD2,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc2761391749766926216at_nat @ A4 @ B4 ) )
     => ( member8440522571783428010at_nat @ B @ ( B4 @ A ) ) ) ).

% SigmaD2
thf(fact_379_SigmaD2,axiom,
    ! [A: nat,B: produc3741383161447143261al_nat,A4: set_nat,B4: nat > set_Pr3510011417693777981al_nat] :
      ( ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ A @ B ) @ ( produc3292543272716750659al_nat @ A4 @ B4 ) )
     => ( member5805532792777349510al_nat @ B @ ( B4 @ A ) ) ) ).

% SigmaD2
thf(fact_380_SigmaD2,axiom,
    ! [A: real,B: nat,A4: set_real,B4: real > set_nat] :
      ( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ A @ B ) @ ( produc1204303713273417414al_nat @ A4 @ B4 ) )
     => ( member_nat @ B @ ( B4 @ A ) ) ) ).

% SigmaD2
thf(fact_381_SigmaD1,axiom,
    ! [A: nat,B: list_b,A4: set_nat,B4: nat > set_list_b] :
      ( ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ A @ B ) @ ( produc132127289527437803list_b @ A4 @ B4 ) )
     => ( member_nat @ A @ A4 ) ) ).

% SigmaD1
thf(fact_382_SigmaD1,axiom,
    ! [A: nat,B: nat,A4: set_nat,B4: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A4 @ B4 ) )
     => ( member_nat @ A @ A4 ) ) ).

% SigmaD1
thf(fact_383_SigmaD1,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc2761391749766926216at_nat @ A4 @ B4 ) )
     => ( member8440522571783428010at_nat @ A @ A4 ) ) ).

% SigmaD1
thf(fact_384_SigmaD1,axiom,
    ! [A: nat,B: produc3741383161447143261al_nat,A4: set_nat,B4: nat > set_Pr3510011417693777981al_nat] :
      ( ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ A @ B ) @ ( produc3292543272716750659al_nat @ A4 @ B4 ) )
     => ( member_nat @ A @ A4 ) ) ).

% SigmaD1
thf(fact_385_SigmaD1,axiom,
    ! [A: real,B: nat,A4: set_real,B4: real > set_nat] :
      ( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ A @ B ) @ ( produc1204303713273417414al_nat @ A4 @ B4 ) )
     => ( member_real @ A @ A4 ) ) ).

% SigmaD1
thf(fact_386_SigmaE,axiom,
    ! [C2: product_prod_nat_nat,A4: set_nat,B4: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( produc457027306803732586at_nat @ A4 @ B4 ) )
     => ~ ! [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( B4 @ X4 ) )
               => ( C2
                 != ( product_Pair_nat_nat @ X4 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_387_SigmaE,axiom,
    ! [C2: produc3741383161447143261al_nat,A4: set_real,B4: real > set_nat] :
      ( ( member5805532792777349510al_nat @ C2 @ ( produc1204303713273417414al_nat @ A4 @ B4 ) )
     => ~ ! [X4: real] :
            ( ( member_real @ X4 @ A4 )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( B4 @ X4 ) )
               => ( C2
                 != ( produc3181502643871035669al_nat @ X4 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_388_SigmaE,axiom,
    ! [C2: produc6020613750378442500_b_nat,A4: set_list_b,B4: list_b > set_nat] :
      ( ( member6469666724224207131_b_nat @ C2 @ ( produc1835408586239750157_b_nat @ A4 @ B4 ) )
     => ~ ! [X4: list_b] :
            ( ( member_list_b @ X4 @ A4 )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( B4 @ X4 ) )
               => ( C2
                 != ( produc383276621478133630_b_nat @ X4 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_389_SigmaE,axiom,
    ! [C2: produc7811952446676219690list_b,A4: set_nat,B4: nat > set_list_b] :
      ( ( member8261005420521984321list_b @ C2 @ ( produc132127289527437803list_b @ A4 @ B4 ) )
     => ~ ! [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
           => ! [Y3: list_b] :
                ( ( member_list_b @ Y3 @ ( B4 @ X4 ) )
               => ( C2
                 != ( produc7903367361620597084list_b @ X4 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_390_SigmaE,axiom,
    ! [C2: produc8373899037510109440at_nat,A4: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_nat] :
      ( ( member3348759134392003351at_nat @ C2 @ ( produc7672662199629908489at_nat @ A4 @ B4 ) )
     => ~ ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A4 )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( B4 @ X4 ) )
               => ( C2
                 != ( produc6350711070570205562at_nat @ X4 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_391_SigmaE,axiom,
    ! [C2: produc7248412053542808358at_nat,A4: set_nat,B4: nat > set_Pr1261947904930325089at_nat] :
      ( ( member2223272150424702269at_nat @ C2 @ ( produc1809337555817847783at_nat @ A4 @ B4 ) )
     => ~ ! [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
           => ! [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ ( B4 @ X4 ) )
               => ( C2
                 != ( produc487386426758144856at_nat @ X4 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_392_SigmaE,axiom,
    ! [C2: produc3963297410138542439list_b,A4: set_list_b,B4: list_b > set_list_b] :
      ( ( member2990321877988238992list_b @ C2 @ ( produc7475029858066475656list_b @ A4 @ B4 ) )
     => ~ ! [X4: list_b] :
            ( ( member_list_b @ X4 @ A4 )
           => ! [Y3: list_b] :
                ( ( member_list_b @ Y3 @ ( B4 @ X4 ) )
               => ( C2
                 != ( produc1564554178308465111list_b @ X4 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_393_SigmaE,axiom,
    ! [C2: produc9155354469731034754al_nat,A4: set_nat,B4: nat > set_Pr3510011417693777981al_nat] :
      ( ( member7408976046919118105al_nat @ C2 @ ( produc3292543272716750659al_nat @ A4 @ B4 ) )
     => ~ ! [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
           => ! [Y3: produc3741383161447143261al_nat] :
                ( ( member5805532792777349510al_nat @ Y3 @ ( B4 @ X4 ) )
               => ( C2
                 != ( produc4490577844368043572al_nat @ X4 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_394_SigmaE,axiom,
    ! [C2: produc1130314010222062699list_b,A4: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_list_b] :
      ( ( member9125244090276215188list_b @ C2 @ ( produc4636629702015616396list_b @ A4 @ B4 ) )
     => ~ ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A4 )
           => ! [Y3: list_b] :
                ( ( member_list_b @ Y3 @ ( B4 @ X4 ) )
               => ( C2
                 != ( produc7400629846093677787list_b @ X4 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_395_SigmaE,axiom,
    ! [C2: produc3137399281773485411at_nat,A4: set_list_b,B4: list_b > set_Pr1261947904930325089at_nat] :
      ( ( member1908957324972862092at_nat @ C2 @ ( produc5400497306732369028at_nat @ A4 @ B4 ) )
     => ~ ! [X4: list_b] :
            ( ( member_list_b @ X4 @ A4 )
           => ! [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ ( B4 @ X4 ) )
               => ( C2
                 != ( produc8164497450810430419at_nat @ X4 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_396_map__prod__inj__on,axiom,
    ! [F: nat > nat,A4: set_nat,G: nat > nat,B4: set_nat] :
      ( ( inj_on_nat_nat @ F @ A4 )
     => ( ( inj_on_nat_nat @ G @ B4 )
       => ( inj_on8969904277767023793at_nat @ ( produc6977886695330630970at_nat @ F @ G )
          @ ( produc457027306803732586at_nat @ A4
            @ ^ [Uu: nat] : B4 ) ) ) ) ).

% map_prod_inj_on
thf(fact_397_map__prod__inj__on,axiom,
    ! [F: nat > nat,A4: set_nat,G: real > produc3741383161447143261al_nat,B4: set_real] :
      ( ( inj_on_nat_nat @ F @ A4 )
     => ( ( inj_on8027545645978435326al_nat @ G @ B4 )
       => ( inj_on3400758723552037294al_nat @ ( produc7218282358209923863al_nat @ F @ G )
          @ ( produc5860367176999294534t_real @ A4
            @ ^ [Uu: nat] : B4 ) ) ) ) ).

% map_prod_inj_on
thf(fact_398_map__prod__inj__on,axiom,
    ! [F: nat > nat,A4: set_nat,G: nat > product_prod_nat_nat,B4: set_nat] :
      ( ( inj_on_nat_nat @ F @ A4 )
     => ( ( inj_on5538052773655684606at_nat @ G @ B4 )
       => ( inj_on5416022199913362294at_nat @ ( produc9062365837733128983at_nat @ F @ G )
          @ ( produc457027306803732586at_nat @ A4
            @ ^ [Uu: nat] : B4 ) ) ) ) ).

% map_prod_inj_on
thf(fact_399_map__prod__inj__on,axiom,
    ! [F: real > produc3741383161447143261al_nat,A4: set_real,G: nat > nat,B4: set_nat] :
      ( ( inj_on8027545645978435326al_nat @ F @ A4 )
     => ( ( inj_on_nat_nat @ G @ B4 )
       => ( inj_on7077800877260587912at_nat @ ( produc94305847159717211at_nat @ F @ G )
          @ ( produc1204303713273417414al_nat @ A4
            @ ^ [Uu: real] : B4 ) ) ) ) ).

% map_prod_inj_on
thf(fact_400_map__prod__inj__on,axiom,
    ! [F: nat > product_prod_nat_nat,A4: set_nat,G: nat > nat,B4: set_nat] :
      ( ( inj_on5538052773655684606at_nat @ F @ A4 )
     => ( ( inj_on_nat_nat @ G @ B4 )
       => ( inj_on6541509183880663376at_nat @ ( produc8306735474602065755at_nat @ F @ G )
          @ ( produc457027306803732586at_nat @ A4
            @ ^ [Uu: nat] : B4 ) ) ) ) ).

% map_prod_inj_on
thf(fact_401_map__prod__inj__on,axiom,
    ! [F: nat > nat,A4: set_nat,G: nat > produc7811952446676219690list_b,B4: set_nat] :
      ( ( inj_on_nat_nat @ F @ A4 )
     => ( ( inj_on8555073082499328493list_b @ G @ B4 )
       => ( inj_on4534532566085093899list_b @ ( produc5470999172165745044list_b @ F @ G )
          @ ( produc457027306803732586at_nat @ A4
            @ ^ [Uu: nat] : B4 ) ) ) ) ).

% map_prod_inj_on
thf(fact_402_map__prod__inj__on,axiom,
    ! [F: nat > produc7811952446676219690list_b,A4: set_nat,G: nat > nat,B4: set_nat] :
      ( ( inj_on8555073082499328493list_b @ F @ A4 )
     => ( ( inj_on_nat_nat @ G @ B4 )
       => ( inj_on1775699293261311627_b_nat @ ( produc2270369883413640340at_nat @ F @ G )
          @ ( produc457027306803732586at_nat @ A4
            @ ^ [Uu: nat] : B4 ) ) ) ) ).

% map_prod_inj_on
thf(fact_403_map__prod__inj__on,axiom,
    ! [F: list_b > produc7811952446676219690list_b,A4: set_list_b,G: nat > nat,B4: set_nat] :
      ( ( inj_on8733544287791916926list_b @ F @ A4 )
     => ( ( inj_on_nat_nat @ G @ B4 )
       => ( inj_on5740833130252885284_b_nat @ ( produc8169669994075221083at_nat @ F @ G )
          @ ( produc1835408586239750157_b_nat @ A4
            @ ^ [Uu: list_b] : B4 ) ) ) ) ).

% map_prod_inj_on
thf(fact_404_map__prod__inj__on,axiom,
    ! [F: nat > nat,A4: set_nat,G: list_b > produc7811952446676219690list_b,B4: set_list_b] :
      ( ( inj_on_nat_nat @ F @ A4 )
     => ( ( inj_on8733544287791916926list_b @ G @ B4 )
       => ( inj_on8157835898299261438list_b @ ( produc3770422184237617943list_b @ F @ G )
          @ ( produc132127289527437803list_b @ A4
            @ ^ [Uu: nat] : B4 ) ) ) ) ).

% map_prod_inj_on
thf(fact_405_map__prod__inj__on,axiom,
    ! [F: nat > nat,A4: set_nat,G: nat > produc9155354469731034754al_nat,B4: set_nat] :
      ( ( inj_on_nat_nat @ F @ A4 )
     => ( ( inj_on7152781252210839493al_nat @ G @ B4 )
       => ( inj_on935854358587236963al_nat @ ( produc7322023129190663148al_nat @ F @ G )
          @ ( produc457027306803732586at_nat @ A4
            @ ^ [Uu: nat] : B4 ) ) ) ) ).

% map_prod_inj_on
thf(fact_406_asym__on__def,axiom,
    ( asym_on_nat
    = ( ^ [A5: set_nat,R3: set_Pr1261947904930325089at_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A5 )
         => ! [Y4: nat] :
              ( ( member_nat @ Y4 @ A5 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ R3 )
               => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X2 ) @ R3 ) ) ) ) ) ) ).

% asym_on_def
thf(fact_407_asym__on__def,axiom,
    ( asym_o8546584678760569651at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat] :
        ! [X2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X2 @ A5 )
         => ! [Y4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Y4 @ A5 )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y4 ) @ R3 )
               => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ X2 ) @ R3 ) ) ) ) ) ) ).

% asym_on_def
thf(fact_408_asym__onD,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat,X3: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
      ( ( asym_o9200482492015546265at_nat @ A4 @ R2 )
     => ( ( member8206827879206165904at_nat @ X3 @ A4 )
       => ( ( member8206827879206165904at_nat @ Y @ A4 )
         => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X3 @ Y ) @ R2 )
           => ~ ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y @ X3 ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_409_asym__onD,axiom,
    ! [A4: set_list_b,R2: set_Pr7665282455119567943list_b,X3: list_b,Y: list_b] :
      ( ( asym_on_list_b @ A4 @ R2 )
     => ( ( member_list_b @ X3 @ A4 )
       => ( ( member_list_b @ Y @ A4 )
         => ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ X3 @ Y ) @ R2 )
           => ~ ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Y @ X3 ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_410_asym__onD,axiom,
    ! [A4: set_nat,R2: set_Pr1261947904930325089at_nat,X3: nat,Y: nat] :
      ( ( asym_on_nat @ A4 @ R2 )
     => ( ( member_nat @ X3 @ A4 )
       => ( ( member_nat @ Y @ A4 )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R2 )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X3 ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_411_asym__onD,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( asym_o8546584678760569651at_nat @ A4 @ R2 )
     => ( ( member8440522571783428010at_nat @ X3 @ A4 )
       => ( ( member8440522571783428010at_nat @ Y @ A4 )
         => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R2 )
           => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ X3 ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_412_rotate1__map,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( rotate1_nat @ ( map_nat_nat @ F @ Xs ) )
      = ( map_nat_nat @ F @ ( rotate1_nat @ Xs ) ) ) ).

% rotate1_map
thf(fact_413_asym__iff,axiom,
    ! [R2: set_Pr7665282455119567943list_b] :
      ( ( asym_on_list_b @ top_top_set_list_b @ R2 )
      = ( ! [X2: list_b,Y4: list_b] :
            ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ X2 @ Y4 ) @ R2 )
           => ~ ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Y4 @ X2 ) @ R2 ) ) ) ) ).

% asym_iff
thf(fact_414_asym__iff,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R2 )
      = ( ! [X2: nat,Y4: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ R2 )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X2 ) @ R2 ) ) ) ) ).

% asym_iff
thf(fact_415_asym__iff,axiom,
    ! [R2: set_Pr553994874890374343at_nat] :
      ( ( asym_o9200482492015546265at_nat @ top_to8454300755403694231at_nat @ R2 )
      = ( ! [X2: produc859450856879609959at_nat,Y4: produc859450856879609959at_nat] :
            ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X2 @ Y4 ) @ R2 )
           => ~ ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y4 @ X2 ) @ R2 ) ) ) ) ).

% asym_iff
thf(fact_416_asym__iff,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R2 )
      = ( ! [X2: product_prod_nat_nat,Y4: product_prod_nat_nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y4 ) @ R2 )
           => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ X2 ) @ R2 ) ) ) ) ).

% asym_iff
thf(fact_417_asymD,axiom,
    ! [R2: set_Pr7665282455119567943list_b,X3: list_b,Y: list_b] :
      ( ( asym_on_list_b @ top_top_set_list_b @ R2 )
     => ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ X3 @ Y ) @ R2 )
       => ~ ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Y @ X3 ) @ R2 ) ) ) ).

% asymD
thf(fact_418_asymD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X3: nat,Y: nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R2 )
       => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X3 ) @ R2 ) ) ) ).

% asymD
thf(fact_419_asymD,axiom,
    ! [R2: set_Pr553994874890374343at_nat,X3: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
      ( ( asym_o9200482492015546265at_nat @ top_to8454300755403694231at_nat @ R2 )
     => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X3 @ Y ) @ R2 )
       => ~ ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y @ X3 ) @ R2 ) ) ) ).

% asymD
thf(fact_420_asymD,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R2 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R2 )
       => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ X3 ) @ R2 ) ) ) ).

% asymD
thf(fact_421_member__product,axiom,
    ! [X3: product_prod_nat_nat,A4: set_nat,B4: set_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( produc929334515565554804at_nat @ A4 @ B4 ) )
      = ( member8440522571783428010at_nat @ X3
        @ ( produc457027306803732586at_nat @ A4
          @ ^ [Uu: nat] : B4 ) ) ) ).

% member_product
thf(fact_422_member__product,axiom,
    ! [X3: produc859450856879609959at_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( produc6773009516019950738at_nat @ A4 @ B4 ) )
      = ( member8206827879206165904at_nat @ X3
        @ ( produc2761391749766926216at_nat @ A4
          @ ^ [Uu: product_prod_nat_nat] : B4 ) ) ) ).

% member_product
thf(fact_423_asymp__asym__eq,axiom,
    ! [R2: set_Pr7665282455119567943list_b] :
      ( ( asymp_on_list_b @ top_top_set_list_b
        @ ^ [X2: list_b,Y4: list_b] : ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ X2 @ Y4 ) @ R2 ) )
      = ( asym_on_list_b @ top_top_set_list_b @ R2 ) ) ).

% asymp_asym_eq
thf(fact_424_asymp__asym__eq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( asymp_on_nat @ top_top_set_nat
        @ ^ [X2: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ R2 ) )
      = ( asym_on_nat @ top_top_set_nat @ R2 ) ) ).

% asymp_asym_eq
thf(fact_425_asymp__asym__eq,axiom,
    ! [R2: set_Pr553994874890374343at_nat] :
      ( ( asymp_3317030538120956165at_nat @ top_to8454300755403694231at_nat
        @ ^ [X2: produc859450856879609959at_nat,Y4: produc859450856879609959at_nat] : ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X2 @ Y4 ) @ R2 ) )
      = ( asym_o9200482492015546265at_nat @ top_to8454300755403694231at_nat @ R2 ) ) ).

% asymp_asym_eq
thf(fact_426_asymp__asym__eq,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ( asymp_3201933403969103519at_nat @ top_to4669805908274784177at_nat
        @ ^ [X2: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y4 ) @ R2 ) )
      = ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R2 ) ) ).

% asymp_asym_eq
thf(fact_427_inj__to__nat,axiom,
    inj_on_nat_nat @ to_nat_nat @ top_top_set_nat ).

% inj_to_nat
thf(fact_428_inj__to__nat,axiom,
    inj_on490725545064285830at_nat @ to_nat5607129738308160893at_nat @ top_to8454300755403694231at_nat ).

% inj_to_nat
thf(fact_429_inj__to__nat,axiom,
    inj_on2178005380612969504at_nat @ to_nat7320730079965953815at_nat @ top_to4669805908274784177at_nat ).

% inj_to_nat
thf(fact_430_asym__lenlex,axiom,
    ! [R: set_Product_prod_b_b] :
      ( ( asym_on_b @ top_top_set_b @ R )
     => ( asym_on_list_b @ top_top_set_list_b @ ( lenlex_b @ R ) ) ) ).

% asym_lenlex
thf(fact_431_asym__lenlex,axiom,
    ! [R: set_Pr7665282455119567943list_b] :
      ( ( asym_on_list_b @ top_top_set_list_b @ R )
     => ( asym_on_list_list_b @ top_to4060329871436644795list_b @ ( lenlex_list_b @ R ) ) ) ).

% asym_lenlex
thf(fact_432_asym__lenlex,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R )
     => ( asym_on_list_nat @ top_top_set_list_nat @ ( lenlex_nat @ R ) ) ) ).

% asym_lenlex
thf(fact_433_asym__lenlex,axiom,
    ! [R: set_Pr553994874890374343at_nat] :
      ( ( asym_o9200482492015546265at_nat @ top_to8454300755403694231at_nat @ R )
     => ( asym_o6078720637282775199at_nat @ top_to2355589028646918429at_nat @ ( lenlex4912532420890773362at_nat @ R ) ) ) ).

% asym_lenlex
thf(fact_434_asym__lenlex,axiom,
    ! [R: set_Pr8693737435421807431at_nat] :
      ( ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R )
     => ( asym_o3606958092715058105at_nat @ top_to70995312096774071at_nat @ ( lenlex325483962726685836at_nat @ R ) ) ) ).

% asym_lenlex
thf(fact_435_inj__on__to__nat,axiom,
    ! [S: set_nat] : ( inj_on_nat_nat @ to_nat_nat @ S ) ).

% inj_on_to_nat
thf(fact_436_asym__lex,axiom,
    ! [R: set_Product_prod_b_b] :
      ( ( asym_on_b @ top_top_set_b @ R )
     => ( asym_on_list_b @ top_top_set_list_b @ ( lex_b @ R ) ) ) ).

% asym_lex
thf(fact_437_asym__lex,axiom,
    ! [R: set_Pr7665282455119567943list_b] :
      ( ( asym_on_list_b @ top_top_set_list_b @ R )
     => ( asym_on_list_list_b @ top_to4060329871436644795list_b @ ( lex_list_b @ R ) ) ) ).

% asym_lex
thf(fact_438_asym__lex,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R )
     => ( asym_on_list_nat @ top_top_set_list_nat @ ( lex_nat @ R ) ) ) ).

% asym_lex
thf(fact_439_asym__lex,axiom,
    ! [R: set_Pr553994874890374343at_nat] :
      ( ( asym_o9200482492015546265at_nat @ top_to8454300755403694231at_nat @ R )
     => ( asym_o6078720637282775199at_nat @ top_to2355589028646918429at_nat @ ( lex_Pr9100905268475511505at_nat @ R ) ) ) ).

% asym_lex
thf(fact_440_asym__lex,axiom,
    ! [R: set_Pr8693737435421807431at_nat] :
      ( ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R )
     => ( asym_o3606958092715058105at_nat @ top_to70995312096774071at_nat @ ( lex_Pr8571645452597969515at_nat @ R ) ) ) ).

% asym_lex
thf(fact_441_lexord__asym,axiom,
    ! [R: set_Product_prod_b_b] :
      ( ( asym_on_b @ top_top_set_b @ R )
     => ( asym_on_list_b @ top_top_set_list_b @ ( lexord_b @ R ) ) ) ).

% lexord_asym
thf(fact_442_lexord__asym,axiom,
    ! [R: set_Pr7665282455119567943list_b] :
      ( ( asym_on_list_b @ top_top_set_list_b @ R )
     => ( asym_on_list_list_b @ top_to4060329871436644795list_b @ ( lexord_list_b @ R ) ) ) ).

% lexord_asym
thf(fact_443_lexord__asym,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R )
     => ( asym_on_list_nat @ top_top_set_list_nat @ ( lexord_nat @ R ) ) ) ).

% lexord_asym
thf(fact_444_lexord__asym,axiom,
    ! [R: set_Pr553994874890374343at_nat] :
      ( ( asym_o9200482492015546265at_nat @ top_to8454300755403694231at_nat @ R )
     => ( asym_o6078720637282775199at_nat @ top_to2355589028646918429at_nat @ ( lexord5831005462426227802at_nat @ R ) ) ) ).

% lexord_asym
thf(fact_445_lexord__asym,axiom,
    ! [R: set_Pr8693737435421807431at_nat] :
      ( ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R )
     => ( asym_o3606958092715058105at_nat @ top_to70995312096774071at_nat @ ( lexord2841853652668343668at_nat @ R ) ) ) ).

% lexord_asym
thf(fact_446_asym__inv__image,axiom,
    ! [R: set_Pr1261947904930325089at_nat,F: nat > nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R )
     => ( asym_on_nat @ top_top_set_nat @ ( inv_image_nat_nat @ R @ F ) ) ) ).

% asym_inv_image
thf(fact_447_asym__inv__image,axiom,
    ! [R: set_Pr7665282455119567943list_b,F: nat > list_b] :
      ( ( asym_on_list_b @ top_top_set_list_b @ R )
     => ( asym_on_nat @ top_top_set_nat @ ( inv_image_list_b_nat @ R @ F ) ) ) ).

% asym_inv_image
thf(fact_448_asym__inv__image,axiom,
    ! [R: set_Pr1261947904930325089at_nat,F: list_b > nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R )
     => ( asym_on_list_b @ top_top_set_list_b @ ( inv_image_nat_list_b @ R @ F ) ) ) ).

% asym_inv_image
thf(fact_449_asym__inv__image,axiom,
    ! [R: set_Pr7665282455119567943list_b,F: list_b > list_b] :
      ( ( asym_on_list_b @ top_top_set_list_b @ R )
     => ( asym_on_list_b @ top_top_set_list_b @ ( inv_im183700145293098386list_b @ R @ F ) ) ) ).

% asym_inv_image
thf(fact_450_asym__inv__image,axiom,
    ! [R: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R )
     => ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ ( inv_im2857804880773004637at_nat @ R @ F ) ) ) ).

% asym_inv_image
thf(fact_451_asym__inv__image,axiom,
    ! [R: set_Pr8693737435421807431at_nat,F: nat > product_prod_nat_nat] :
      ( ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R )
     => ( asym_on_nat @ top_top_set_nat @ ( inv_im8721129524585065343at_nat @ R @ F ) ) ) ).

% asym_inv_image
thf(fact_452_asym__inv__image,axiom,
    ! [R: set_Pr7665282455119567943list_b,F: product_prod_nat_nat > list_b] :
      ( ( asym_on_list_b @ top_top_set_list_b @ R )
     => ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ ( inv_im5978152666696423310at_nat @ R @ F ) ) ) ).

% asym_inv_image
thf(fact_453_asym__inv__image,axiom,
    ! [R: set_Pr8693737435421807431at_nat,F: list_b > product_prod_nat_nat] :
      ( ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R )
     => ( asym_on_list_b @ top_top_set_list_b @ ( inv_im5214285061979670678list_b @ R @ F ) ) ) ).

% asym_inv_image
thf(fact_454_asym__inv__image,axiom,
    ! [R: set_Pr8693737435421807431at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R )
     => ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ ( inv_im5896302226278515346at_nat @ R @ F ) ) ) ).

% asym_inv_image
thf(fact_455_asym__inv__image,axiom,
    ! [R: set_Pr1261947904930325089at_nat,F: produc859450856879609959at_nat > nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R )
     => ( asym_o9200482492015546265at_nat @ top_to8454300755403694231at_nat @ ( inv_im87553201807674051at_nat @ R @ F ) ) ) ).

% asym_inv_image
thf(fact_456_asymp__on__asym__on__eq,axiom,
    ! [A4: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( asymp_on_nat @ A4
        @ ^ [X2: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ R2 ) )
      = ( asym_on_nat @ A4 @ R2 ) ) ).

% asymp_on_asym_on_eq
thf(fact_457_asymp__on__asym__on__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( asymp_3201933403969103519at_nat @ A4
        @ ^ [X2: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y4 ) @ R2 ) )
      = ( asym_o8546584678760569651at_nat @ A4 @ R2 ) ) ).

% asymp_on_asym_on_eq
thf(fact_458_asymp__onI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ A4 )
         => ( ( member8440522571783428010at_nat @ Y3 @ A4 )
           => ( ( R @ X4 @ Y3 )
             => ~ ( R @ Y3 @ X4 ) ) ) )
     => ( asymp_3201933403969103519at_nat @ A4 @ R ) ) ).

% asymp_onI
thf(fact_459_asymp__onI,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,R: produc859450856879609959at_nat > produc859450856879609959at_nat > $o] :
      ( ! [X4: produc859450856879609959at_nat,Y3: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X4 @ A4 )
         => ( ( member8206827879206165904at_nat @ Y3 @ A4 )
           => ( ( R @ X4 @ Y3 )
             => ~ ( R @ Y3 @ X4 ) ) ) )
     => ( asymp_3317030538120956165at_nat @ A4 @ R ) ) ).

% asymp_onI
thf(fact_460_asymp__onI,axiom,
    ! [A4: set_nat,R: nat > nat > $o] :
      ( ! [X4: nat,Y3: nat] :
          ( ( member_nat @ X4 @ A4 )
         => ( ( member_nat @ Y3 @ A4 )
           => ( ( R @ X4 @ Y3 )
             => ~ ( R @ Y3 @ X4 ) ) ) )
     => ( asymp_on_nat @ A4 @ R ) ) ).

% asymp_onI
thf(fact_461_asymp__onI,axiom,
    ! [A4: set_list_b,R: list_b > list_b > $o] :
      ( ! [X4: list_b,Y3: list_b] :
          ( ( member_list_b @ X4 @ A4 )
         => ( ( member_list_b @ Y3 @ A4 )
           => ( ( R @ X4 @ Y3 )
             => ~ ( R @ Y3 @ X4 ) ) ) )
     => ( asymp_on_list_b @ A4 @ R ) ) ).

% asymp_onI
thf(fact_462_asympI,axiom,
    ! [R: list_b > list_b > $o] :
      ( ! [X4: list_b,Y3: list_b] :
          ( ( R @ X4 @ Y3 )
         => ~ ( R @ Y3 @ X4 ) )
     => ( asymp_on_list_b @ top_top_set_list_b @ R ) ) ).

% asympI
thf(fact_463_asympI,axiom,
    ! [R: nat > nat > $o] :
      ( ! [X4: nat,Y3: nat] :
          ( ( R @ X4 @ Y3 )
         => ~ ( R @ Y3 @ X4 ) )
     => ( asymp_on_nat @ top_top_set_nat @ R ) ) ).

% asympI
thf(fact_464_asympI,axiom,
    ! [R: produc859450856879609959at_nat > produc859450856879609959at_nat > $o] :
      ( ! [X4: produc859450856879609959at_nat,Y3: produc859450856879609959at_nat] :
          ( ( R @ X4 @ Y3 )
         => ~ ( R @ Y3 @ X4 ) )
     => ( asymp_3317030538120956165at_nat @ top_to8454300755403694231at_nat @ R ) ) ).

% asympI
thf(fact_465_asympI,axiom,
    ! [R: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( R @ X4 @ Y3 )
         => ~ ( R @ Y3 @ X4 ) )
     => ( asymp_3201933403969103519at_nat @ top_to4669805908274784177at_nat @ R ) ) ).

% asympI
thf(fact_466_in__inv__image,axiom,
    ! [X3: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( inv_image_nat_nat @ R2 @ F ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X3 ) @ ( F @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_467_in__inv__image,axiom,
    ! [X3: nat,Y: nat,R2: set_Pr8693737435421807431at_nat,F: nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( inv_im8721129524585065343at_nat @ R2 @ F ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F @ X3 ) @ ( F @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_468_in__inv__image,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( inv_im2857804880773004637at_nat @ R2 @ F ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X3 ) @ ( F @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_469_in__inv__image,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( inv_im5896302226278515346at_nat @ R2 @ F ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F @ X3 ) @ ( F @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_470_asymp__onD,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( asymp_3201933403969103519at_nat @ A4 @ R )
     => ( ( member8440522571783428010at_nat @ X3 @ A4 )
       => ( ( member8440522571783428010at_nat @ Y @ A4 )
         => ( ( R @ X3 @ Y )
           => ~ ( R @ Y @ X3 ) ) ) ) ) ).

% asymp_onD
thf(fact_471_asymp__onD,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,R: produc859450856879609959at_nat > produc859450856879609959at_nat > $o,X3: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
      ( ( asymp_3317030538120956165at_nat @ A4 @ R )
     => ( ( member8206827879206165904at_nat @ X3 @ A4 )
       => ( ( member8206827879206165904at_nat @ Y @ A4 )
         => ( ( R @ X3 @ Y )
           => ~ ( R @ Y @ X3 ) ) ) ) ) ).

% asymp_onD
thf(fact_472_asymp__onD,axiom,
    ! [A4: set_nat,R: nat > nat > $o,X3: nat,Y: nat] :
      ( ( asymp_on_nat @ A4 @ R )
     => ( ( member_nat @ X3 @ A4 )
       => ( ( member_nat @ Y @ A4 )
         => ( ( R @ X3 @ Y )
           => ~ ( R @ Y @ X3 ) ) ) ) ) ).

% asymp_onD
thf(fact_473_asymp__onD,axiom,
    ! [A4: set_list_b,R: list_b > list_b > $o,X3: list_b,Y: list_b] :
      ( ( asymp_on_list_b @ A4 @ R )
     => ( ( member_list_b @ X3 @ A4 )
       => ( ( member_list_b @ Y @ A4 )
         => ( ( R @ X3 @ Y )
           => ~ ( R @ Y @ X3 ) ) ) ) ) ).

% asymp_onD
thf(fact_474_lexord__lex,axiom,
    ! [X3: list_b,Y: list_b,R2: set_Product_prod_b_b] :
      ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ X3 @ Y ) @ ( lex_b @ R2 ) )
      = ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ X3 @ Y ) @ ( lexord_b @ R2 ) )
        & ( ( size_size_list_b @ X3 )
          = ( size_size_list_b @ Y ) ) ) ) ).

% lexord_lex
thf(fact_475_lexord__linear,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X3: list_nat,Y: list_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ R2 )
          | ( A3 = B3 )
          | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A3 ) @ R2 ) )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y ) @ ( lexord_nat @ R2 ) )
        | ( X3 = Y )
        | ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y @ X3 ) @ ( lexord_nat @ R2 ) ) ) ) ).

% lexord_linear
thf(fact_476_lexord__linear,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,X3: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A3 @ B3 ) @ R2 )
          | ( A3 = B3 )
          | ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B3 @ A3 ) @ R2 ) )
     => ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ X3 @ Y ) @ ( lexord2841853652668343668at_nat @ R2 ) )
        | ( X3 = Y )
        | ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Y @ X3 ) @ ( lexord2841853652668343668at_nat @ R2 ) ) ) ) ).

% lexord_linear
thf(fact_477_lexord__irreflexive,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ! [X4: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R2 )
     => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lexord_nat @ R2 ) ) ) ).

% lexord_irreflexive
thf(fact_478_lexord__irreflexive,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ X4 ) @ R2 )
     => ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Xs ) @ ( lexord2841853652668343668at_nat @ R2 ) ) ) ).

% lexord_irreflexive
thf(fact_479_asympD,axiom,
    ! [R: list_b > list_b > $o,X3: list_b,Y: list_b] :
      ( ( asymp_on_list_b @ top_top_set_list_b @ R )
     => ( ( R @ X3 @ Y )
       => ~ ( R @ Y @ X3 ) ) ) ).

% asympD
thf(fact_480_asympD,axiom,
    ! [R: nat > nat > $o,X3: nat,Y: nat] :
      ( ( asymp_on_nat @ top_top_set_nat @ R )
     => ( ( R @ X3 @ Y )
       => ~ ( R @ Y @ X3 ) ) ) ).

% asympD
thf(fact_481_asympD,axiom,
    ! [R: produc859450856879609959at_nat > produc859450856879609959at_nat > $o,X3: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
      ( ( asymp_3317030538120956165at_nat @ top_to8454300755403694231at_nat @ R )
     => ( ( R @ X3 @ Y )
       => ~ ( R @ Y @ X3 ) ) ) ).

% asympD
thf(fact_482_asympD,axiom,
    ! [R: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( asymp_3201933403969103519at_nat @ top_to4669805908274784177at_nat @ R )
     => ( ( R @ X3 @ Y )
       => ~ ( R @ Y @ X3 ) ) ) ).

% asympD
thf(fact_483_lenlex__irreflexive,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ! [X4: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R2 )
     => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lenlex_nat @ R2 ) ) ) ).

% lenlex_irreflexive
thf(fact_484_lenlex__irreflexive,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ X4 ) @ R2 )
     => ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Xs ) @ ( lenlex325483962726685836at_nat @ R2 ) ) ) ).

% lenlex_irreflexive
thf(fact_485_lexord__asymmetric,axiom,
    ! [R: set_Pr7665282455119567943list_b,A: list_list_b,B: list_list_b] :
      ( ( asym_on_list_b @ top_top_set_list_b @ R )
     => ( ( member7435197128963948048list_b @ ( produc8181704880241203287list_b @ A @ B ) @ ( lexord_list_b @ R ) )
       => ~ ( member7435197128963948048list_b @ ( produc8181704880241203287list_b @ B @ A ) @ ( lexord_list_b @ R ) ) ) ) ).

% lexord_asymmetric
thf(fact_486_lexord__asymmetric,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: list_nat,B: list_nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( lexord_nat @ R ) )
       => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ A ) @ ( lexord_nat @ R ) ) ) ) ).

% lexord_asymmetric
thf(fact_487_lexord__asymmetric,axiom,
    ! [R: set_Pr553994874890374343at_nat,A: list_P8469869581646625389at_nat,B: list_P8469869581646625389at_nat] :
      ( ( asym_o9200482492015546265at_nat @ top_to8454300755403694231at_nat @ R )
     => ( ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ A @ B ) @ ( lexord5831005462426227802at_nat @ R ) )
       => ~ ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ B @ A ) @ ( lexord5831005462426227802at_nat @ R ) ) ) ) ).

% lexord_asymmetric
thf(fact_488_lexord__asymmetric,axiom,
    ! [R: set_Pr8693737435421807431at_nat,A: list_P6011104703257516679at_nat,B: list_P6011104703257516679at_nat] :
      ( ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R )
     => ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ A @ B ) @ ( lexord2841853652668343668at_nat @ R ) )
       => ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ B @ A ) @ ( lexord2841853652668343668at_nat @ R ) ) ) ) ).

% lexord_asymmetric
thf(fact_489_elias__gamma__encoding,axiom,
    prefix3558185134189398382ng_nat @ prefix1649127329469935890e_Ng_e ).

% elias_gamma_encoding
thf(fact_490_lenlex__def,axiom,
    ( lenlex_b
    = ( ^ [R3: set_Product_prod_b_b] :
          ( inv_im2558614453762373639list_b @ ( lex_prod_nat_list_b @ less_than @ ( lex_b @ R3 ) )
          @ ^ [Xs3: list_b] : ( produc7903367361620597084list_b @ ( size_size_list_b @ Xs3 ) @ Xs3 ) ) ) ) ).

% lenlex_def
thf(fact_491_bounded__nat__encoding,axiom,
    ! [L: nat] : ( prefix3558185134189398382ng_nat @ ( prefix6319276831915272717e_Nb_e @ L ) ) ).

% bounded_nat_encoding
thf(fact_492_exp__golomb__encoding,axiom,
    prefix3558185134189398382ng_nat @ prefix_Free_Code_N_e ).

% exp_golomb_encoding
thf(fact_493_Cons__in__lex,axiom,
    ! [X3: nat,Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( lex_nat @ R2 ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R2 )
          & ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) ) )
        | ( ( X3 = Y )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_494_Cons__in__lex,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys ) ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) )
      = ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R2 )
          & ( ( size_s5460976970255530739at_nat @ Xs )
            = ( size_s5460976970255530739at_nat @ Ys ) ) )
        | ( ( X3 = Y )
          & ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_495_Cons__in__lex,axiom,
    ! [X3: b,Xs: list_b,Y: b,Ys: list_b,R2: set_Product_prod_b_b] :
      ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ ( cons_b @ X3 @ Xs ) @ ( cons_b @ Y @ Ys ) ) @ ( lex_b @ R2 ) )
      = ( ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X3 @ Y ) @ R2 )
          & ( ( size_size_list_b @ Xs )
            = ( size_size_list_b @ Ys ) ) )
        | ( ( X3 = Y )
          & ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Xs @ Ys ) @ ( lex_b @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_496_total__inv__image,axiom,
    ! [F: nat > nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( inj_on_nat_nat @ F @ top_top_set_nat )
     => ( ( total_on_nat @ top_top_set_nat @ R2 )
       => ( total_on_nat @ top_top_set_nat @ ( inv_image_nat_nat @ R2 @ F ) ) ) ) ).

% total_inv_image
thf(fact_497_total__inv__image,axiom,
    ! [F: list_b > nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( inj_on_list_b_nat @ F @ top_top_set_list_b )
     => ( ( total_on_nat @ top_top_set_nat @ R2 )
       => ( total_on_list_b @ top_top_set_list_b @ ( inv_image_nat_list_b @ R2 @ F ) ) ) ) ).

% total_inv_image
thf(fact_498_total__inv__image,axiom,
    ! [F: nat > list_b,R2: set_Pr7665282455119567943list_b] :
      ( ( inj_on_nat_list_b @ F @ top_top_set_nat )
     => ( ( total_on_list_b @ top_top_set_list_b @ R2 )
       => ( total_on_nat @ top_top_set_nat @ ( inv_image_list_b_nat @ R2 @ F ) ) ) ) ).

% total_inv_image
thf(fact_499_total__inv__image,axiom,
    ! [F: real > produc3741383161447143261al_nat,R2: set_Pr9214095960617939271al_nat] :
      ( ( inj_on8027545645978435326al_nat @ F @ top_top_set_real )
     => ( ( total_8412025274382043873al_nat @ top_to8045530339851357581al_nat @ R2 )
       => ( total_on_real @ top_top_set_real @ ( inv_im9037750907728042167t_real @ R2 @ F ) ) ) ) ).

% total_inv_image
thf(fact_500_total__inv__image,axiom,
    ! [F: list_b > list_b,R2: set_Pr7665282455119567943list_b] :
      ( ( inj_on_list_b_list_b @ F @ top_top_set_list_b )
     => ( ( total_on_list_b @ top_top_set_list_b @ R2 )
       => ( total_on_list_b @ top_top_set_list_b @ ( inv_im183700145293098386list_b @ R2 @ F ) ) ) ) ).

% total_inv_image
thf(fact_501_total__inv__image,axiom,
    ! [F: nat > product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( inj_on5538052773655684606at_nat @ F @ top_top_set_nat )
     => ( ( total_3592101749530773125at_nat @ top_to4669805908274784177at_nat @ R2 )
       => ( total_on_nat @ top_top_set_nat @ ( inv_im8721129524585065343at_nat @ R2 @ F ) ) ) ) ).

% total_inv_image
thf(fact_502_total__inv__image,axiom,
    ! [F: product_prod_nat_nat > nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( inj_on2178005380612969504at_nat @ F @ top_to4669805908274784177at_nat )
     => ( ( total_on_nat @ top_top_set_nat @ R2 )
       => ( total_3592101749530773125at_nat @ top_to4669805908274784177at_nat @ ( inv_im2857804880773004637at_nat @ R2 @ F ) ) ) ) ).

% total_inv_image
thf(fact_503_total__inv__image,axiom,
    ! [F: list_b > product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( inj_on8730813633938913709at_nat @ F @ top_top_set_list_b )
     => ( ( total_3592101749530773125at_nat @ top_to4669805908274784177at_nat @ R2 )
       => ( total_on_list_b @ top_top_set_list_b @ ( inv_im5214285061979670678list_b @ R2 @ F ) ) ) ) ).

% total_inv_image
thf(fact_504_total__inv__image,axiom,
    ! [F: nat > produc7811952446676219690list_b,R2: set_Pr3474451521773494677list_b] :
      ( ( inj_on8555073082499328493list_b @ F @ top_top_set_nat )
     => ( ( total_2330929641541938342list_b @ top_to4428277177966209040list_b @ R2 )
       => ( total_on_nat @ top_top_set_nat @ ( inv_im7335115905590207822_b_nat @ R2 @ F ) ) ) ) ).

% total_inv_image
thf(fact_505_total__inv__image,axiom,
    ! [F: product_prod_nat_nat > list_b,R2: set_Pr7665282455119567943list_b] :
      ( ( inj_on7966946029222161077list_b @ F @ top_to4669805908274784177at_nat )
     => ( ( total_on_list_b @ top_top_set_list_b @ R2 )
       => ( total_3592101749530773125at_nat @ top_to4669805908274784177at_nat @ ( inv_im5978152666696423310at_nat @ R2 @ F ) ) ) ) ).

% total_inv_image
thf(fact_506_to__nat__def,axiom,
    ( to_nat_nat
    = ( fChoice_nat_nat
      @ ^ [F2: nat > nat] : ( inj_on_nat_nat @ F2 @ top_top_set_nat ) ) ) ).

% to_nat_def
thf(fact_507_to__nat__def,axiom,
    ( to_nat5607129738308160893at_nat
    = ( fChoic8220946065393586812at_nat
      @ ^ [F2: produc859450856879609959at_nat > nat] : ( inj_on490725545064285830at_nat @ F2 @ top_to8454300755403694231at_nat ) ) ) ).

% to_nat_def
thf(fact_508_to__nat__def,axiom,
    ( to_nat7320730079965953815at_nat
    = ( fChoic638070965443183254at_nat
      @ ^ [F2: product_prod_nat_nat > nat] : ( inj_on2178005380612969504at_nat @ F2 @ top_to4669805908274784177at_nat ) ) ) ).

% to_nat_def
thf(fact_509_asym__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R_A )
     => ( ( asym_on_nat @ top_top_set_nat @ R_B )
       => ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).

% asym_lex_prod
thf(fact_510_asym__lex__prod,axiom,
    ! [R_A: set_Pr7665282455119567943list_b,R_B: set_Pr1261947904930325089at_nat] :
      ( ( asym_on_list_b @ top_top_set_list_b @ R_A )
     => ( ( asym_on_nat @ top_top_set_nat @ R_B )
       => ( asym_o2166927944166264402_b_nat @ top_to6547989104015266538_b_nat @ ( lex_prod_list_b_nat @ R_A @ R_B ) ) ) ) ).

% asym_lex_prod
thf(fact_511_asym__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr7665282455119567943list_b] :
      ( ( asym_on_nat @ top_top_set_nat @ R_A )
     => ( ( asym_on_list_b @ top_top_set_list_b @ R_B )
       => ( asym_o3958266640464041592list_b @ top_to4428277177966209040list_b @ ( lex_prod_nat_list_b @ R_A @ R_B ) ) ) ) ).

% asym_lex_prod
thf(fact_512_asym__lex__prod,axiom,
    ! [R_A: set_Pr7665282455119567943list_b,R_B: set_Pr7665282455119567943list_b] :
      ( ( asym_on_list_b @ top_top_set_list_b @ R_A )
     => ( ( asym_on_list_b @ top_top_set_list_b @ R_B )
       => ( asym_o4805641579978188953list_b @ top_to6408289459866364311list_b @ ( lex_pr3125412782666178031list_b @ R_A @ R_B ) ) ) ) ).

% asym_lex_prod
thf(fact_513_asym__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr8693737435421807431at_nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R_A )
     => ( ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R_B )
       => ( asym_o3144787024281459316at_nat @ top_to7745748144976627212at_nat @ ( lex_pr8029265285556086080at_nat @ R_A @ R_B ) ) ) ) ).

% asym_lex_prod
thf(fact_514_asym__lex__prod,axiom,
    ! [R_A: set_Pr8693737435421807431at_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R_A )
     => ( ( asym_on_nat @ top_top_set_nat @ R_B )
       => ( asym_o4270274008248760398at_nat @ top_to2567003362141017830at_nat @ ( lex_pr4669217892513370978at_nat @ R_A @ R_B ) ) ) ) ).

% asym_lex_prod
thf(fact_515_asym__lex__prod,axiom,
    ! [R_A: set_Pr7665282455119567943list_b,R_B: set_Pr8693737435421807431at_nat] :
      ( ( asym_on_list_b @ top_top_set_list_b @ R_A )
     => ( ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R_B )
       => ( asym_o796278632387110037at_nat @ top_to538300756415451027at_nat @ ( lex_pr8139799934699638763at_nat @ R_A @ R_B ) ) ) ) ).

% asym_lex_prod
thf(fact_516_asym__lex__prod,axiom,
    ! [R_A: set_Pr8693737435421807431at_nat,R_B: set_Pr7665282455119567943list_b] :
      ( ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R_A )
     => ( ( asym_on_list_b @ top_top_set_list_b @ R_B )
       => ( asym_o8012565397690463133list_b @ top_to548042179339279003list_b @ ( lex_pr7375932329982886131list_b @ R_A @ R_B ) ) ) ) ).

% asym_lex_prod
thf(fact_517_asym__lex__prod,axiom,
    ! [R_A: set_Pr8693737435421807431at_nat,R_B: set_Pr8693737435421807431at_nat] :
      ( ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R_A )
     => ( ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R_B )
       => ( asym_o9200482492015546265at_nat @ top_to8454300755403694231at_nat @ ( lex_pr8801849515957261039at_nat @ R_A @ R_B ) ) ) ) ).

% asym_lex_prod
thf(fact_518_asym__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr553994874890374343at_nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R_A )
     => ( ( asym_o9200482492015546265at_nat @ top_to8454300755403694231at_nat @ R_B )
       => ( asym_o1289167481111577306at_nat @ top_to1012081421760046066at_nat @ ( lex_pr5131707101520687654at_nat @ R_A @ R_B ) ) ) ) ).

% asym_lex_prod
thf(fact_519_list_Oinject,axiom,
    ! [X21: nat,X222: list_nat,Y21: nat,Y22: list_nat] :
      ( ( ( cons_nat @ X21 @ X222 )
        = ( cons_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% list.inject
thf(fact_520_in__lex__prod,axiom,
    ! [A: nat,B: product_prod_nat_nat,A2: nat,B2: product_prod_nat_nat,R2: set_Pr1261947904930325089at_nat,S3: set_Pr8693737435421807431at_nat] :
      ( ( member3048279782668807382at_nat @ ( produc2653010282673554021at_nat @ ( produc487386426758144856at_nat @ A @ B ) @ ( produc487386426758144856at_nat @ A2 @ B2 ) ) @ ( lex_pr8029265285556086080at_nat @ R2 @ S3 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A2 ) @ R2 )
        | ( ( A = A2 )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ B2 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_521_in__lex__prod,axiom,
    ! [A: product_prod_nat_nat,B: nat,A2: product_prod_nat_nat,B2: nat,R2: set_Pr8693737435421807431at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( member1995966531042493578at_nat @ ( produc7904928797850150681at_nat @ ( produc6350711070570205562at_nat @ A @ B ) @ ( produc6350711070570205562at_nat @ A2 @ B2 ) ) @ ( lex_pr4669217892513370978at_nat @ R2 @ S3 ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A2 ) @ R2 )
        | ( ( A = A2 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_522_in__lex__prod,axiom,
    ! [A: nat,B: list_b,A2: nat,B2: list_b,R2: set_Pr1261947904930325089at_nat,S3: set_Pr7665282455119567943list_b] :
      ( ( member5069333211937813726list_b @ ( produc8430590441843559533list_b @ ( produc7903367361620597084list_b @ A @ B ) @ ( produc7903367361620597084list_b @ A2 @ B2 ) ) @ ( lex_prod_nat_list_b @ R2 @ S3 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A2 ) @ R2 )
        | ( ( A = A2 )
          & ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ B @ B2 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_523_in__lex__prod,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc6161850002892822231at_nat @ A2 @ B2 ) ) @ ( lex_pr8801849515957261039at_nat @ R2 @ S3 ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A2 ) @ R2 )
        | ( ( A = A2 )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ B2 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_524_in__lex__prod,axiom,
    ! [A: nat,B: produc3741383161447143261al_nat,A2: nat,B2: produc3741383161447143261al_nat,R2: set_Pr1261947904930325089at_nat,S3: set_Pr9214095960617939271al_nat] :
      ( ( member8283612037765495310al_nat @ ( produc8240651324928177565al_nat @ ( produc4490577844368043572al_nat @ A @ B ) @ ( produc4490577844368043572al_nat @ A2 @ B2 ) ) @ ( lex_pr3067795221024046108al_nat @ R2 @ S3 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A2 ) @ R2 )
        | ( ( A = A2 )
          & ( member7366861965109168016al_nat @ ( produc2826530874179193559al_nat @ B @ B2 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_525_in__lex__prod,axiom,
    ! [A: real,B: nat,A2: real,B2: nat,R2: set_Pr6218003697084177305l_real,S3: set_Pr1261947904930325089at_nat] :
      ( ( member7366861965109168016al_nat @ ( produc2826530874179193559al_nat @ ( produc3181502643871035669al_nat @ A @ B ) @ ( produc3181502643871035669al_nat @ A2 @ B2 ) ) @ ( lex_prod_real_nat @ R2 @ S3 ) )
      = ( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ A @ A2 ) @ R2 )
        | ( ( A = A2 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_526_in__lex__prod,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat,R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ A2 @ B2 ) ) @ ( lex_prod_nat_nat @ R2 @ S3 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A2 ) @ R2 )
        | ( ( A = A2 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_527_lexord__cons__cons,axiom,
    ! [A: nat,X3: list_nat,B: nat,Y: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ A @ X3 ) @ ( cons_nat @ B @ Y ) ) @ ( lexord_nat @ R2 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
        | ( ( A = B )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y ) @ ( lexord_nat @ R2 ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_528_lexord__cons__cons,axiom,
    ! [A: product_prod_nat_nat,X3: list_P6011104703257516679at_nat,B: product_prod_nat_nat,Y: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ A @ X3 ) @ ( cons_P6512896166579812791at_nat @ B @ Y ) ) @ ( lexord2841853652668343668at_nat @ R2 ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 )
        | ( ( A = B )
          & ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ X3 @ Y ) @ ( lexord2841853652668343668at_nat @ R2 ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_529_asym__less__than,axiom,
    asym_on_nat @ top_top_set_nat @ less_than ).

% asym_less_than
thf(fact_530_total__less__than,axiom,
    total_on_nat @ top_top_set_nat @ less_than ).

% total_less_than
thf(fact_531_total__on__less__than,axiom,
    ! [A4: set_nat] : ( total_on_nat @ A4 @ less_than ) ).

% total_on_less_than
thf(fact_532_not__Cons__self2,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( cons_nat @ X3 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_533_total__on__lex__prod,axiom,
    ! [A4: set_nat,R_A: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,R_B: set_Pr8693737435421807431at_nat] :
      ( ( total_on_nat @ A4 @ R_A )
     => ( ( total_3592101749530773125at_nat @ B4 @ R_B )
       => ( total_2766879334212059298at_nat
          @ ( produc1809337555817847783at_nat @ A4
            @ ^ [Uu: nat] : B4 )
          @ ( lex_pr8029265285556086080at_nat @ R_A @ R_B ) ) ) ) ).

% total_on_lex_prod
thf(fact_534_total__on__lex__prod,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,R_A: set_Pr8693737435421807431at_nat,B4: set_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( total_3592101749530773125at_nat @ A4 @ R_A )
     => ( ( total_on_nat @ B4 @ R_B )
       => ( total_3892366318179360380at_nat
          @ ( produc7672662199629908489at_nat @ A4
            @ ^ [Uu: product_prod_nat_nat] : B4 )
          @ ( lex_pr4669217892513370978at_nat @ R_A @ R_B ) ) ) ) ).

% total_on_lex_prod
thf(fact_535_total__on__lex__prod,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,R_A: set_Pr8693737435421807431at_nat,B4: set_Pr1261947904930325089at_nat,R_B: set_Pr8693737435421807431at_nat] :
      ( ( total_3592101749530773125at_nat @ A4 @ R_A )
     => ( ( total_3592101749530773125at_nat @ B4 @ R_B )
       => ( total_3247520894076811243at_nat
          @ ( produc2761391749766926216at_nat @ A4
            @ ^ [Uu: product_prod_nat_nat] : B4 )
          @ ( lex_pr8801849515957261039at_nat @ R_A @ R_B ) ) ) ) ).

% total_on_lex_prod
thf(fact_536_total__on__lex__prod,axiom,
    ! [A4: set_nat,R_A: set_Pr1261947904930325089at_nat,B4: set_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( total_on_nat @ A4 @ R_A )
     => ( ( total_on_nat @ B4 @ R_B )
       => ( total_3592101749530773125at_nat
          @ ( produc457027306803732586at_nat @ A4
            @ ^ [Uu: nat] : B4 )
          @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).

% total_on_lex_prod
thf(fact_537_total__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( total_on_nat @ top_top_set_nat @ R_A )
     => ( ( total_on_nat @ top_top_set_nat @ R_B )
       => ( total_3592101749530773125at_nat @ top_to4669805908274784177at_nat @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).

% total_lex_prod
thf(fact_538_total__lex__prod,axiom,
    ! [R_A: set_Pr7665282455119567943list_b,R_B: set_Pr1261947904930325089at_nat] :
      ( ( total_on_list_b @ top_top_set_list_b @ R_A )
     => ( ( total_on_nat @ top_top_set_nat @ R_B )
       => ( total_539590945244161152_b_nat @ top_to6547989104015266538_b_nat @ ( lex_prod_list_b_nat @ R_A @ R_B ) ) ) ) ).

% total_lex_prod
thf(fact_539_total__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr7665282455119567943list_b] :
      ( ( total_on_nat @ top_top_set_nat @ R_A )
     => ( ( total_on_list_b @ top_top_set_list_b @ R_B )
       => ( total_2330929641541938342list_b @ top_to4428277177966209040list_b @ ( lex_prod_nat_list_b @ R_A @ R_B ) ) ) ) ).

% total_lex_prod
thf(fact_540_total__lex__prod,axiom,
    ! [R_A: set_Pr7665282455119567943list_b,R_B: set_Pr7665282455119567943list_b] :
      ( ( total_on_list_b @ top_top_set_list_b @ R_A )
     => ( ( total_on_list_b @ top_top_set_list_b @ R_B )
       => ( total_4973729943858802923list_b @ top_to6408289459866364311list_b @ ( lex_pr3125412782666178031list_b @ R_A @ R_B ) ) ) ) ).

% total_lex_prod
thf(fact_541_total__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr8693737435421807431at_nat] :
      ( ( total_on_nat @ top_top_set_nat @ R_A )
     => ( ( total_3592101749530773125at_nat @ top_to4669805908274784177at_nat @ R_B )
       => ( total_2766879334212059298at_nat @ top_to7745748144976627212at_nat @ ( lex_pr8029265285556086080at_nat @ R_A @ R_B ) ) ) ) ).

% total_lex_prod
thf(fact_542_total__lex__prod,axiom,
    ! [R_A: set_Pr8693737435421807431at_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( total_3592101749530773125at_nat @ top_to4669805908274784177at_nat @ R_A )
     => ( ( total_on_nat @ top_top_set_nat @ R_B )
       => ( total_3892366318179360380at_nat @ top_to2567003362141017830at_nat @ ( lex_pr4669217892513370978at_nat @ R_A @ R_B ) ) ) ) ).

% total_lex_prod
thf(fact_543_total__lex__prod,axiom,
    ! [R_A: set_Pr7665282455119567943list_b,R_B: set_Pr8693737435421807431at_nat] :
      ( ( total_on_list_b @ top_top_set_list_b @ R_A )
     => ( ( total_3592101749530773125at_nat @ top_to4669805908274784177at_nat @ R_B )
       => ( total_3959100607063743719at_nat @ top_to538300756415451027at_nat @ ( lex_pr8139799934699638763at_nat @ R_A @ R_B ) ) ) ) ).

% total_lex_prod
thf(fact_544_total__lex__prod,axiom,
    ! [R_A: set_Pr8693737435421807431at_nat,R_B: set_Pr7665282455119567943list_b] :
      ( ( total_3592101749530773125at_nat @ top_to4669805908274784177at_nat @ R_A )
     => ( ( total_on_list_b @ top_top_set_list_b @ R_B )
       => ( total_1952015335512321007list_b @ top_to548042179339279003list_b @ ( lex_pr7375932329982886131list_b @ R_A @ R_B ) ) ) ) ).

% total_lex_prod
thf(fact_545_total__lex__prod,axiom,
    ! [R_A: set_Pr8693737435421807431at_nat,R_B: set_Pr8693737435421807431at_nat] :
      ( ( total_3592101749530773125at_nat @ top_to4669805908274784177at_nat @ R_A )
     => ( ( total_3592101749530773125at_nat @ top_to4669805908274784177at_nat @ R_B )
       => ( total_3247520894076811243at_nat @ top_to8454300755403694231at_nat @ ( lex_pr8801849515957261039at_nat @ R_A @ R_B ) ) ) ) ).

% total_lex_prod
thf(fact_546_total__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr553994874890374343at_nat] :
      ( ( total_on_nat @ top_top_set_nat @ R_A )
     => ( ( total_3247520894076811243at_nat @ top_to8454300755403694231at_nat @ R_B )
       => ( total_6604280431924587528at_nat @ top_to1012081421760046066at_nat @ ( lex_pr5131707101520687654at_nat @ R_A @ R_B ) ) ) ) ).

% total_lex_prod
thf(fact_547_map__eq__Cons__conv,axiom,
    ! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( cons_nat @ Y @ Ys ) )
      = ( ? [Z: nat,Zs: list_nat] :
            ( ( Xs
              = ( cons_nat @ Z @ Zs ) )
            & ( ( F @ Z )
              = Y )
            & ( ( map_nat_nat @ F @ Zs )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_548_Cons__eq__map__conv,axiom,
    ! [X3: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
      ( ( ( cons_nat @ X3 @ Xs )
        = ( map_nat_nat @ F @ Ys ) )
      = ( ? [Z: nat,Zs: list_nat] :
            ( ( Ys
              = ( cons_nat @ Z @ Zs ) )
            & ( X3
              = ( F @ Z ) )
            & ( Xs
              = ( map_nat_nat @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_549_map__eq__Cons__D,axiom,
    ! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( cons_nat @ Y @ Ys ) )
     => ? [Z2: nat,Zs2: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z2 @ Zs2 ) )
          & ( ( F @ Z2 )
            = Y )
          & ( ( map_nat_nat @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_550_Cons__eq__map__D,axiom,
    ! [X3: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
      ( ( ( cons_nat @ X3 @ Xs )
        = ( map_nat_nat @ F @ Ys ) )
     => ? [Z2: nat,Zs2: list_nat] :
          ( ( Ys
            = ( cons_nat @ Z2 @ Zs2 ) )
          & ( X3
            = ( F @ Z2 ) )
          & ( Xs
            = ( map_nat_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_551_list_Osimps_I9_J,axiom,
    ! [F: nat > nat,X21: nat,X222: list_nat] :
      ( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X222 ) )
      = ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_552_total__on__def,axiom,
    ( total_on_nat
    = ( ^ [A5: set_nat,R3: set_Pr1261947904930325089at_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A5 )
         => ! [Y4: nat] :
              ( ( member_nat @ Y4 @ A5 )
             => ( ( X2 != Y4 )
               => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ R3 )
                  | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X2 ) @ R3 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_553_total__on__def,axiom,
    ( total_3592101749530773125at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat] :
        ! [X2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X2 @ A5 )
         => ! [Y4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Y4 @ A5 )
             => ( ( X2 != Y4 )
               => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y4 ) @ R3 )
                  | ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ X2 ) @ R3 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_554_total__onI,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat] :
      ( ! [X4: produc859450856879609959at_nat,Y3: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X4 @ A4 )
         => ( ( member8206827879206165904at_nat @ Y3 @ A4 )
           => ( ( X4 != Y3 )
             => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X4 @ Y3 ) @ R2 )
                | ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y3 @ X4 ) @ R2 ) ) ) ) )
     => ( total_3247520894076811243at_nat @ A4 @ R2 ) ) ).

% total_onI
thf(fact_555_total__onI,axiom,
    ! [A4: set_list_b,R2: set_Pr7665282455119567943list_b] :
      ( ! [X4: list_b,Y3: list_b] :
          ( ( member_list_b @ X4 @ A4 )
         => ( ( member_list_b @ Y3 @ A4 )
           => ( ( X4 != Y3 )
             => ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ X4 @ Y3 ) @ R2 )
                | ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Y3 @ X4 ) @ R2 ) ) ) ) )
     => ( total_on_list_b @ A4 @ R2 ) ) ).

% total_onI
thf(fact_556_total__onI,axiom,
    ! [A4: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( member_nat @ X4 @ A4 )
         => ( ( member_nat @ Y3 @ A4 )
           => ( ( X4 != Y3 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R2 )
                | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X4 ) @ R2 ) ) ) ) )
     => ( total_on_nat @ A4 @ R2 ) ) ).

% total_onI
thf(fact_557_total__onI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ A4 )
         => ( ( member8440522571783428010at_nat @ Y3 @ A4 )
           => ( ( X4 != Y3 )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y3 ) @ R2 )
                | ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ X4 ) @ R2 ) ) ) ) )
     => ( total_3592101749530773125at_nat @ A4 @ R2 ) ) ).

% total_onI
thf(fact_558_removeAll_Osimps_I2_J,axiom,
    ! [X3: nat,Y: nat,Xs: list_nat] :
      ( ( ( X3 = Y )
       => ( ( removeAll_nat @ X3 @ ( cons_nat @ Y @ Xs ) )
          = ( removeAll_nat @ X3 @ Xs ) ) )
      & ( ( X3 != Y )
       => ( ( removeAll_nat @ X3 @ ( cons_nat @ Y @ Xs ) )
          = ( cons_nat @ Y @ ( removeAll_nat @ X3 @ Xs ) ) ) ) ) ).

% removeAll.simps(2)
thf(fact_559_inj__on__Cons1,axiom,
    ! [X3: nat,A4: set_list_nat] : ( inj_on3049792774292151987st_nat @ ( cons_nat @ X3 ) @ A4 ) ).

% inj_on_Cons1
thf(fact_560_total__lexord,axiom,
    ! [R2: set_Product_prod_b_b] :
      ( ( total_on_b @ top_top_set_b @ R2 )
     => ( total_on_list_b @ top_top_set_list_b @ ( lexord_b @ R2 ) ) ) ).

% total_lexord
thf(fact_561_total__lexord,axiom,
    ! [R2: set_Pr7665282455119567943list_b] :
      ( ( total_on_list_b @ top_top_set_list_b @ R2 )
     => ( total_on_list_list_b @ top_to4060329871436644795list_b @ ( lexord_list_b @ R2 ) ) ) ).

% total_lexord
thf(fact_562_total__lexord,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( total_on_nat @ top_top_set_nat @ R2 )
     => ( total_on_list_nat @ top_top_set_list_nat @ ( lexord_nat @ R2 ) ) ) ).

% total_lexord
thf(fact_563_total__lexord,axiom,
    ! [R2: set_Pr553994874890374343at_nat] :
      ( ( total_3247520894076811243at_nat @ top_to8454300755403694231at_nat @ R2 )
     => ( total_1246546807855253489at_nat @ top_to2355589028646918429at_nat @ ( lexord5831005462426227802at_nat @ R2 ) ) ) ).

% total_lexord
thf(fact_564_total__lexord,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ( total_3592101749530773125at_nat @ top_to4669805908274784177at_nat @ R2 )
     => ( total_5612938278750584331at_nat @ top_to70995312096774071at_nat @ ( lexord2841853652668343668at_nat @ R2 ) ) ) ).

% total_lexord
thf(fact_565_total__lenlex,axiom,
    ! [R2: set_Product_prod_b_b] :
      ( ( total_on_b @ top_top_set_b @ R2 )
     => ( total_on_list_b @ top_top_set_list_b @ ( lenlex_b @ R2 ) ) ) ).

% total_lenlex
thf(fact_566_total__lenlex,axiom,
    ! [R2: set_Pr7665282455119567943list_b] :
      ( ( total_on_list_b @ top_top_set_list_b @ R2 )
     => ( total_on_list_list_b @ top_to4060329871436644795list_b @ ( lenlex_list_b @ R2 ) ) ) ).

% total_lenlex
thf(fact_567_total__lenlex,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( total_on_nat @ top_top_set_nat @ R2 )
     => ( total_on_list_nat @ top_top_set_list_nat @ ( lenlex_nat @ R2 ) ) ) ).

% total_lenlex
thf(fact_568_total__lenlex,axiom,
    ! [R2: set_Pr553994874890374343at_nat] :
      ( ( total_3247520894076811243at_nat @ top_to8454300755403694231at_nat @ R2 )
     => ( total_1246546807855253489at_nat @ top_to2355589028646918429at_nat @ ( lenlex4912532420890773362at_nat @ R2 ) ) ) ).

% total_lenlex
thf(fact_569_total__lenlex,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ( total_3592101749530773125at_nat @ top_to4669805908274784177at_nat @ R2 )
     => ( total_5612938278750584331at_nat @ top_to70995312096774071at_nat @ ( lenlex325483962726685836at_nat @ R2 ) ) ) ).

% total_lenlex
thf(fact_570_totalI,axiom,
    ! [R2: set_Pr7665282455119567943list_b] :
      ( ! [X4: list_b,Y3: list_b] :
          ( ( X4 != Y3 )
         => ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ X4 @ Y3 ) @ R2 )
            | ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Y3 @ X4 ) @ R2 ) ) )
     => ( total_on_list_b @ top_top_set_list_b @ R2 ) ) ).

% totalI
thf(fact_571_totalI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( X4 != Y3 )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R2 )
            | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X4 ) @ R2 ) ) )
     => ( total_on_nat @ top_top_set_nat @ R2 ) ) ).

% totalI
thf(fact_572_totalI,axiom,
    ! [R2: set_Pr553994874890374343at_nat] :
      ( ! [X4: produc859450856879609959at_nat,Y3: produc859450856879609959at_nat] :
          ( ( X4 != Y3 )
         => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X4 @ Y3 ) @ R2 )
            | ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y3 @ X4 ) @ R2 ) ) )
     => ( total_3247520894076811243at_nat @ top_to8454300755403694231at_nat @ R2 ) ) ).

% totalI
thf(fact_573_totalI,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( X4 != Y3 )
         => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y3 ) @ R2 )
            | ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ X4 ) @ R2 ) ) )
     => ( total_3592101749530773125at_nat @ top_to4669805908274784177at_nat @ R2 ) ) ).

% totalI
thf(fact_574_asym__on__lex__prod,axiom,
    ! [A4: set_nat,R_A: set_Pr1261947904930325089at_nat,B4: set_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( asym_on_nat @ A4 @ R_A )
     => ( ( asym_on_nat @ B4 @ R_B )
       => ( asym_o8546584678760569651at_nat
          @ ( produc457027306803732586at_nat @ A4
            @ ^ [Uu: nat] : B4 )
          @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).

% asym_on_lex_prod
thf(fact_575_mlex__prod__def,axiom,
    ( mlex_prod_list_b
    = ( ^ [F2: list_b > nat,R4: set_Pr7665282455119567943list_b] :
          ( inv_im2558614453762373639list_b @ ( lex_prod_nat_list_b @ less_than @ R4 )
          @ ^ [X2: list_b] : ( produc7903367361620597084list_b @ ( F2 @ X2 ) @ X2 ) ) ) ) ).

% mlex_prod_def
thf(fact_576_mlex__prod__def,axiom,
    ( mlex_p7481444158671011052al_nat
    = ( ^ [F2: produc3741383161447143261al_nat > nat,R4: set_Pr9214095960617939271al_nat] :
          ( inv_im2751506247701626759al_nat @ ( lex_pr3067795221024046108al_nat @ less_than @ R4 )
          @ ^ [X2: produc3741383161447143261al_nat] : ( produc4490577844368043572al_nat @ ( F2 @ X2 ) @ X2 ) ) ) ) ).

% mlex_prod_def
thf(fact_577_mlex__prod__def,axiom,
    ( mlex_prod_nat
    = ( ^ [F2: nat > nat,R4: set_Pr1261947904930325089at_nat] :
          ( inv_im8721129524585065343at_nat @ ( lex_prod_nat_nat @ less_than @ R4 )
          @ ^ [X2: nat] : ( product_Pair_nat_nat @ ( F2 @ X2 ) @ X2 ) ) ) ) ).

% mlex_prod_def
thf(fact_578_Cons__lenlex__iff,axiom,
    ! [M: nat,Ms: list_nat,N: nat,Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R2 ) )
      = ( ( ord_less_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) )
        | ( ( ( size_size_list_nat @ Ms )
            = ( size_size_list_nat @ Ns ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ R2 ) )
        | ( ( M = N )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R2 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_579_Cons__lenlex__iff,axiom,
    ! [M: product_prod_nat_nat,Ms: list_P6011104703257516679at_nat,N: product_prod_nat_nat,Ns: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ M @ Ms ) @ ( cons_P6512896166579812791at_nat @ N @ Ns ) ) @ ( lenlex325483962726685836at_nat @ R2 ) )
      = ( ( ord_less_nat @ ( size_s5460976970255530739at_nat @ Ms ) @ ( size_s5460976970255530739at_nat @ Ns ) )
        | ( ( ( size_s5460976970255530739at_nat @ Ms )
            = ( size_s5460976970255530739at_nat @ Ns ) )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ M @ N ) @ R2 ) )
        | ( ( M = N )
          & ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ms @ Ns ) @ ( lenlex325483962726685836at_nat @ R2 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_580_Cons__lenlex__iff,axiom,
    ! [M: b,Ms: list_b,N: b,Ns: list_b,R2: set_Product_prod_b_b] :
      ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ ( cons_b @ M @ Ms ) @ ( cons_b @ N @ Ns ) ) @ ( lenlex_b @ R2 ) )
      = ( ( ord_less_nat @ ( size_size_list_b @ Ms ) @ ( size_size_list_b @ Ns ) )
        | ( ( ( size_size_list_b @ Ms )
            = ( size_size_list_b @ Ns ) )
          & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ M @ N ) @ R2 ) )
        | ( ( M = N )
          & ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Ms @ Ns ) @ ( lenlex_b @ R2 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_581_less__than__iff,axiom,
    ! [X3: nat,Y: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ less_than )
      = ( ord_less_nat @ X3 @ Y ) ) ).

% less_than_iff
thf(fact_582_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
      = ( ( ord_less_nat @ M @ N )
        | ( ord_less_nat @ N @ M ) ) ) ).

% nat_neq_iff
thf(fact_583_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_584_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_585_less__not__refl3,axiom,
    ! [S3: nat,T: nat] :
      ( ( ord_less_nat @ S3 @ T )
     => ( S3 != T ) ) ).

% less_not_refl3
thf(fact_586_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_587_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N2 )
             => ( P @ M2 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_588_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ~ ( P @ N2 )
         => ? [M2: nat] :
              ( ( ord_less_nat @ M2 @ N2 )
              & ~ ( P @ M2 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_589_linorder__neqE__nat,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
     => ( ~ ( ord_less_nat @ X3 @ Y )
       => ( ord_less_nat @ Y @ X3 ) ) ) ).

% linorder_neqE_nat
thf(fact_590_gt__ex,axiom,
    ! [X3: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).

% gt_ex
thf(fact_591_less__imp__neq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( X3 != Y ) ) ).

% less_imp_neq
thf(fact_592_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_593_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_594_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_595_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X4: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X4 )
             => ( P @ Y5 ) )
         => ( P @ X4 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_596_antisym__conv3,axiom,
    ! [Y: nat,X3: nat] :
      ( ~ ( ord_less_nat @ Y @ X3 )
     => ( ( ~ ( ord_less_nat @ X3 @ Y ) )
        = ( X3 = Y ) ) ) ).

% antisym_conv3
thf(fact_597_linorder__cases,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y )
     => ( ( X3 != Y )
       => ( ord_less_nat @ Y @ X3 ) ) ) ).

% linorder_cases
thf(fact_598_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_599_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_600_exists__least__iff,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X5: nat] : ( P3 @ X5 ) )
    = ( ^ [P4: nat > $o] :
        ? [N3: nat] :
          ( ( P4 @ N3 )
          & ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N3 )
             => ~ ( P4 @ M3 ) ) ) ) ) ).

% exists_least_iff
thf(fact_601_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_nat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: nat] : ( P @ A3 @ A3 )
       => ( ! [A3: nat,B3: nat] :
              ( ( P @ B3 @ A3 )
             => ( P @ A3 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_602_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_603_not__less__iff__gr__or__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y ) )
      = ( ( ord_less_nat @ Y @ X3 )
        | ( X3 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_604_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_605_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_606_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_607_linorder__neqE,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
     => ( ~ ( ord_less_nat @ X3 @ Y )
       => ( ord_less_nat @ Y @ X3 ) ) ) ).

% linorder_neqE
thf(fact_608_order__less__asym,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ~ ( ord_less_nat @ Y @ X3 ) ) ).

% order_less_asym
thf(fact_609_linorder__neq__iff,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
      = ( ( ord_less_nat @ X3 @ Y )
        | ( ord_less_nat @ Y @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_610_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_611_order__less__trans,axiom,
    ! [X3: nat,Y: nat,Z3: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( ( ord_less_nat @ Y @ Z3 )
       => ( ord_less_nat @ X3 @ Z3 ) ) ) ).

% order_less_trans
thf(fact_612_ord__eq__less__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_613_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_614_order__less__irrefl,axiom,
    ! [X3: nat] :
      ~ ( ord_less_nat @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_615_order__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_616_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_617_order__less__not__sym,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ~ ( ord_less_nat @ Y @ X3 ) ) ).

% order_less_not_sym
thf(fact_618_order__less__imp__triv,axiom,
    ! [X3: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( ( ord_less_nat @ Y @ X3 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_619_linorder__less__linear,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
      | ( X3 = Y )
      | ( ord_less_nat @ Y @ X3 ) ) ).

% linorder_less_linear
thf(fact_620_order__less__imp__not__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( X3 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_621_order__less__imp__not__eq2,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( Y != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_622_order__less__imp__not__less,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ~ ( ord_less_nat @ Y @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_623_mlex__iff,axiom,
    ! [X3: nat,Y: nat,F: nat > nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( mlex_prod_nat @ F @ R ) )
      = ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) )
        | ( ( ( F @ X3 )
            = ( F @ Y ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R ) ) ) ) ).

% mlex_iff
thf(fact_624_mlex__iff,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,F: product_prod_nat_nat > nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( mlex_p6366001652026297872at_nat @ F @ R ) )
      = ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) )
        | ( ( ( F @ X3 )
            = ( F @ Y ) )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R ) ) ) ) ).

% mlex_iff
thf(fact_625_mlex__less,axiom,
    ! [F: nat > nat,X3: nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( mlex_prod_nat @ F @ R ) ) ) ).

% mlex_less
thf(fact_626_mlex__less,axiom,
    ! [F: product_prod_nat_nat > nat,X3: product_prod_nat_nat,Y: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( mlex_p6366001652026297872at_nat @ F @ R ) ) ) ).

% mlex_less
thf(fact_627_top_Oextremum__strict,axiom,
    ! [A: set_list_b] :
      ~ ( ord_less_set_list_b @ top_top_set_list_b @ A ) ).

% top.extremum_strict
thf(fact_628_top_Oextremum__strict,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ top_top_set_nat @ A ) ).

% top.extremum_strict
thf(fact_629_top_Oextremum__strict,axiom,
    ! [A: set_Pr8693737435421807431at_nat] :
      ~ ( ord_le6428140832669894131at_nat @ top_to8454300755403694231at_nat @ A ) ).

% top.extremum_strict
thf(fact_630_top_Oextremum__strict,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ~ ( ord_le7866589430770878221at_nat @ top_to4669805908274784177at_nat @ A ) ).

% top.extremum_strict
thf(fact_631_top_Onot__eq__extremum,axiom,
    ! [A: set_list_b] :
      ( ( A != top_top_set_list_b )
      = ( ord_less_set_list_b @ A @ top_top_set_list_b ) ) ).

% top.not_eq_extremum
thf(fact_632_top_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != top_top_set_nat )
      = ( ord_less_set_nat @ A @ top_top_set_nat ) ) ).

% top.not_eq_extremum
thf(fact_633_top_Onot__eq__extremum,axiom,
    ! [A: set_Pr8693737435421807431at_nat] :
      ( ( A != top_to8454300755403694231at_nat )
      = ( ord_le6428140832669894131at_nat @ A @ top_to8454300755403694231at_nat ) ) ).

% top.not_eq_extremum
thf(fact_634_top_Onot__eq__extremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( A != top_to4669805908274784177at_nat )
      = ( ord_le7866589430770878221at_nat @ A @ top_to4669805908274784177at_nat ) ) ).

% top.not_eq_extremum
thf(fact_635_linorder__inj__onI_H,axiom,
    ! [A4: set_real,F: real > produc3741383161447143261al_nat] :
      ( ! [I: real,J: real] :
          ( ( member_real @ I @ A4 )
         => ( ( member_real @ J @ A4 )
           => ( ( ord_less_real @ I @ J )
             => ( ( F @ I )
               != ( F @ J ) ) ) ) )
     => ( inj_on8027545645978435326al_nat @ F @ A4 ) ) ).

% linorder_inj_onI'
thf(fact_636_linorder__inj__onI_H,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ! [I: nat,J: nat] :
          ( ( member_nat @ I @ A4 )
         => ( ( member_nat @ J @ A4 )
           => ( ( ord_less_nat @ I @ J )
             => ( ( F @ I )
               != ( F @ J ) ) ) ) )
     => ( inj_on_nat_nat @ F @ A4 ) ) ).

% linorder_inj_onI'
thf(fact_637_linorder__inj__onI_H,axiom,
    ! [A4: set_nat,F: nat > produc9155354469731034754al_nat] :
      ( ! [I: nat,J: nat] :
          ( ( member_nat @ I @ A4 )
         => ( ( member_nat @ J @ A4 )
           => ( ( ord_less_nat @ I @ J )
             => ( ( F @ I )
               != ( F @ J ) ) ) ) )
     => ( inj_on7152781252210839493al_nat @ F @ A4 ) ) ).

% linorder_inj_onI'
thf(fact_638_linorder__inj__onI_H,axiom,
    ! [A4: set_nat,F: nat > product_prod_nat_nat] :
      ( ! [I: nat,J: nat] :
          ( ( member_nat @ I @ A4 )
         => ( ( member_nat @ J @ A4 )
           => ( ( ord_less_nat @ I @ J )
             => ( ( F @ I )
               != ( F @ J ) ) ) ) )
     => ( inj_on5538052773655684606at_nat @ F @ A4 ) ) ).

% linorder_inj_onI'
thf(fact_639_linorder__inj__onI_H,axiom,
    ! [A4: set_nat,F: nat > produc7811952446676219690list_b] :
      ( ! [I: nat,J: nat] :
          ( ( member_nat @ I @ A4 )
         => ( ( member_nat @ J @ A4 )
           => ( ( ord_less_nat @ I @ J )
             => ( ( F @ I )
               != ( F @ J ) ) ) ) )
     => ( inj_on8555073082499328493list_b @ F @ A4 ) ) ).

% linorder_inj_onI'
thf(fact_640_length__induct,axiom,
    ! [P: list_b > $o,Xs: list_b] :
      ( ! [Xs2: list_b] :
          ( ! [Ys2: list_b] :
              ( ( ord_less_nat @ ( size_size_list_b @ Ys2 ) @ ( size_size_list_b @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_641_asymp__on__less,axiom,
    ! [A4: set_nat] : ( asymp_on_nat @ A4 @ ord_less_nat ) ).

% asymp_on_less
thf(fact_642_asymp__on__greater,axiom,
    ! [A4: set_nat] :
      ( asymp_on_nat @ A4
      @ ^ [X2: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X2 ) ) ).

% asymp_on_greater
thf(fact_643_linorder__injI,axiom,
    ! [F: real > produc3741383161447143261al_nat] :
      ( ! [X4: real,Y3: real] :
          ( ( ord_less_real @ X4 @ Y3 )
         => ( ( F @ X4 )
           != ( F @ Y3 ) ) )
     => ( inj_on8027545645978435326al_nat @ F @ top_top_set_real ) ) ).

% linorder_injI
thf(fact_644_linorder__injI,axiom,
    ! [F: nat > nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( ord_less_nat @ X4 @ Y3 )
         => ( ( F @ X4 )
           != ( F @ Y3 ) ) )
     => ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).

% linorder_injI
thf(fact_645_linorder__injI,axiom,
    ! [F: nat > produc9155354469731034754al_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( ord_less_nat @ X4 @ Y3 )
         => ( ( F @ X4 )
           != ( F @ Y3 ) ) )
     => ( inj_on7152781252210839493al_nat @ F @ top_top_set_nat ) ) ).

% linorder_injI
thf(fact_646_linorder__injI,axiom,
    ! [F: nat > product_prod_nat_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( ord_less_nat @ X4 @ Y3 )
         => ( ( F @ X4 )
           != ( F @ Y3 ) ) )
     => ( inj_on5538052773655684606at_nat @ F @ top_top_set_nat ) ) ).

% linorder_injI
thf(fact_647_linorder__injI,axiom,
    ! [F: nat > produc7811952446676219690list_b] :
      ( ! [X4: nat,Y3: nat] :
          ( ( ord_less_nat @ X4 @ Y3 )
         => ( ( F @ X4 )
           != ( F @ Y3 ) ) )
     => ( inj_on8555073082499328493list_b @ F @ top_top_set_nat ) ) ).

% linorder_injI
thf(fact_648_someI2__bex,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o,Q2: product_prod_nat_nat > $o] :
      ( ? [X6: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X6 @ A4 )
          & ( P @ X6 ) )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( ( member8440522571783428010at_nat @ X4 @ A4 )
              & ( P @ X4 ) )
           => ( Q2 @ X4 ) )
       => ( Q2
          @ ( fChoic6978938873391328853at_nat
            @ ^ [X2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X2 @ A4 )
                & ( P @ X2 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_649_someI2__bex,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o,Q2: produc859450856879609959at_nat > $o] :
      ( ? [X6: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X6 @ A4 )
          & ( P @ X6 ) )
     => ( ! [X4: produc859450856879609959at_nat] :
            ( ( ( member8206827879206165904at_nat @ X4 @ A4 )
              & ( P @ X4 ) )
           => ( Q2 @ X4 ) )
       => ( Q2
          @ ( fChoic6673916560228149179at_nat
            @ ^ [X2: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X2 @ A4 )
                & ( P @ X2 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_650_someI2__bex,axiom,
    ! [A4: set_nat,P: nat > $o,Q2: nat > $o] :
      ( ? [X6: nat] :
          ( ( member_nat @ X6 @ A4 )
          & ( P @ X6 ) )
     => ( ! [X4: nat] :
            ( ( ( member_nat @ X4 @ A4 )
              & ( P @ X4 ) )
           => ( Q2 @ X4 ) )
       => ( Q2
          @ ( fChoice_nat
            @ ^ [X2: nat] :
                ( ( member_nat @ X2 @ A4 )
                & ( P @ X2 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_651_someI2__bex,axiom,
    ! [A4: set_list_b,P: list_b > $o,Q2: list_b > $o] :
      ( ? [X6: list_b] :
          ( ( member_list_b @ X6 @ A4 )
          & ( P @ X6 ) )
     => ( ! [X4: list_b] :
            ( ( ( member_list_b @ X4 @ A4 )
              & ( P @ X4 ) )
           => ( Q2 @ X4 ) )
       => ( Q2
          @ ( fChoice_list_b
            @ ^ [X2: list_b] :
                ( ( member_list_b @ X2 @ A4 )
                & ( P @ X2 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_652_Cons__less__Cons,axiom,
    ! [A: nat,X3: list_nat,B: nat,Y: list_nat] :
      ( ( ord_less_list_nat @ ( cons_nat @ A @ X3 ) @ ( cons_nat @ B @ Y ) )
      = ( ( ord_less_nat @ A @ B )
        | ( ( A = B )
          & ( ord_less_list_nat @ X3 @ Y ) ) ) ) ).

% Cons_less_Cons
thf(fact_653_in__measure,axiom,
    ! [X3: nat,Y: nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( measure_nat @ F ) )
      = ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) ) ).

% in_measure
thf(fact_654_in__measure,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,F: product_prod_nat_nat > nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( measur8038558561449204169at_nat @ F ) )
      = ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) ) ).

% in_measure
thf(fact_655_less__list__code_I3_J,axiom,
    ! [X3: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( ord_less_list_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) )
      = ( ( ord_less_nat @ X3 @ Y )
        | ( ( X3 = Y )
          & ( ord_less_list_nat @ Xs @ Ys ) ) ) ) ).

% less_list_code(3)
thf(fact_656_in__measures_I2_J,axiom,
    ! [X3: nat,Y: nat,F: nat > nat,Fs: list_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) )
      = ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) )
        | ( ( ( F @ X3 )
            = ( F @ Y ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( measures_nat @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_657_in__measures_I2_J,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,F: product_prod_nat_nat > nat,Fs: list_P9162950289778280392at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( measur2679027848233739777at_nat @ ( cons_P4861729644591583992at_nat @ F @ Fs ) ) )
      = ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) )
        | ( ( ( F @ X3 )
            = ( F @ Y ) )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( measur2679027848233739777at_nat @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_658_lenlex__conv,axiom,
    ( lenlex_b
    = ( ^ [R3: set_Product_prod_b_b] :
          ( collec4964980819554056914list_b
          @ ( produc6051822690389492396st_b_o
            @ ^ [Xs3: list_b,Ys3: list_b] :
                ( ( ord_less_nat @ ( size_size_list_b @ Xs3 ) @ ( size_size_list_b @ Ys3 ) )
                | ( ( ( size_size_list_b @ Xs3 )
                    = ( size_size_list_b @ Ys3 ) )
                  & ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Xs3 @ Ys3 ) @ ( lex_b @ R3 ) ) ) ) ) ) ) ) ).

% lenlex_conv
thf(fact_659_same__fstI,axiom,
    ! [P: nat > $o,X3: nat,Y6: list_b,Y: list_b,R: nat > set_Pr7665282455119567943list_b] :
      ( ( P @ X3 )
     => ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Y6 @ Y ) @ ( R @ X3 ) )
       => ( member5069333211937813726list_b @ ( produc8430590441843559533list_b @ ( produc7903367361620597084list_b @ X3 @ Y6 ) @ ( produc7903367361620597084list_b @ X3 @ Y ) ) @ ( same_fst_nat_list_b @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_660_same__fstI,axiom,
    ! [P: nat > $o,X3: nat,Y6: produc3741383161447143261al_nat,Y: produc3741383161447143261al_nat,R: nat > set_Pr9214095960617939271al_nat] :
      ( ( P @ X3 )
     => ( ( member7366861965109168016al_nat @ ( produc2826530874179193559al_nat @ Y6 @ Y ) @ ( R @ X3 ) )
       => ( member8283612037765495310al_nat @ ( produc8240651324928177565al_nat @ ( produc4490577844368043572al_nat @ X3 @ Y6 ) @ ( produc4490577844368043572al_nat @ X3 @ Y ) ) @ ( same_f8570761488067147039al_nat @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_661_same__fstI,axiom,
    ! [P: real > $o,X3: real,Y6: nat,Y: nat,R: real > set_Pr1261947904930325089at_nat] :
      ( ( P @ X3 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y6 @ Y ) @ ( R @ X3 ) )
       => ( member7366861965109168016al_nat @ ( produc2826530874179193559al_nat @ ( produc3181502643871035669al_nat @ X3 @ Y6 ) @ ( produc3181502643871035669al_nat @ X3 @ Y ) ) @ ( same_fst_real_nat @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_662_same__fstI,axiom,
    ! [P: nat > $o,X3: nat,Y6: nat,Y: nat,R: nat > set_Pr1261947904930325089at_nat] :
      ( ( P @ X3 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y6 @ Y ) @ ( R @ X3 ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X3 @ Y6 ) @ ( product_Pair_nat_nat @ X3 @ Y ) ) @ ( same_fst_nat_nat @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_663_same__fstI,axiom,
    ! [P: product_prod_nat_nat > $o,X3: product_prod_nat_nat,Y6: product_prod_nat_nat,Y: product_prod_nat_nat,R: product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
      ( ( P @ X3 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y6 @ Y ) @ ( R @ X3 ) )
       => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y6 ) @ ( produc6161850002892822231at_nat @ X3 @ Y ) ) @ ( same_f4956014544515070124at_nat @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_664_lexn__length,axiom,
    ! [Xs: list_b,Ys: list_b,R2: set_Product_prod_b_b,N: nat] :
      ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Xs @ Ys ) @ ( lexn_b @ R2 @ N ) )
     => ( ( ( size_size_list_b @ Xs )
          = N )
        & ( ( size_size_list_b @ Ys )
          = N ) ) ) ).

% lexn_length
thf(fact_665_case__prodI2,axiom,
    ! [P2: produc7811952446676219690list_b,C2: nat > list_b > $o] :
      ( ! [A3: nat,B3: list_b] :
          ( ( P2
            = ( produc7903367361620597084list_b @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc4544114555777700815st_b_o @ C2 @ P2 ) ) ).

% case_prodI2
thf(fact_666_case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,C2: nat > nat > $o] :
      ( ! [A3: nat,B3: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc6081775807080527818_nat_o @ C2 @ P2 ) ) ).

% case_prodI2
thf(fact_667_case__prodI2,axiom,
    ! [P2: produc859450856879609959at_nat,C2: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
          ( ( P2
            = ( produc6161850002892822231at_nat @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc6590410687421337004_nat_o @ C2 @ P2 ) ) ).

% case_prodI2
thf(fact_668_case__prodI2,axiom,
    ! [P2: produc9155354469731034754al_nat,C2: nat > produc3741383161447143261al_nat > $o] :
      ( ! [A3: nat,B3: produc3741383161447143261al_nat] :
          ( ( P2
            = ( produc4490577844368043572al_nat @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc8534146015018131191_nat_o @ C2 @ P2 ) ) ).

% case_prodI2
thf(fact_669_case__prodI2,axiom,
    ! [P2: produc3741383161447143261al_nat,C2: real > nat > $o] :
      ( ! [A3: real,B3: nat] :
          ( ( P2
            = ( produc3181502643871035669al_nat @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc5286614342184123374_nat_o @ C2 @ P2 ) ) ).

% case_prodI2
thf(fact_670_case__prodI,axiom,
    ! [F: nat > list_b > $o,A: nat,B: list_b] :
      ( ( F @ A @ B )
     => ( produc4544114555777700815st_b_o @ F @ ( produc7903367361620597084list_b @ A @ B ) ) ) ).

% case_prodI
thf(fact_671_case__prodI,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( F @ A @ B )
     => ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_672_case__prodI,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( F @ A @ B )
     => ( produc6590410687421337004_nat_o @ F @ ( produc6161850002892822231at_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_673_case__prodI,axiom,
    ! [F: nat > produc3741383161447143261al_nat > $o,A: nat,B: produc3741383161447143261al_nat] :
      ( ( F @ A @ B )
     => ( produc8534146015018131191_nat_o @ F @ ( produc4490577844368043572al_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_674_case__prodI,axiom,
    ! [F: real > nat > $o,A: real,B: nat] :
      ( ( F @ A @ B )
     => ( produc5286614342184123374_nat_o @ F @ ( produc3181502643871035669al_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_675_Eps__case__prod__eq,axiom,
    ! [X3: nat,Y: list_b] :
      ( ( fChoic403860114826281174list_b
        @ ( produc4544114555777700815st_b_o
          @ ^ [X7: nat,Y7: list_b] :
              ( ( X3 = X7 )
              & ( Y = Y7 ) ) ) )
      = ( produc7903367361620597084list_b @ X3 @ Y ) ) ).

% Eps_case_prod_eq
thf(fact_676_Eps__case__prod__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( fChoic6978938873391328853at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X7: nat,Y7: nat] :
              ( ( X3 = X7 )
              & ( Y = Y7 ) ) ) )
      = ( product_Pair_nat_nat @ X3 @ Y ) ) ).

% Eps_case_prod_eq
thf(fact_677_Eps__case__prod__eq,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( fChoic6673916560228149179at_nat
        @ ( produc6590410687421337004_nat_o
          @ ^ [X7: product_prod_nat_nat,Y7: product_prod_nat_nat] :
              ( ( X3 = X7 )
              & ( Y = Y7 ) ) ) )
      = ( produc6161850002892822231at_nat @ X3 @ Y ) ) ).

% Eps_case_prod_eq
thf(fact_678_Eps__case__prod__eq,axiom,
    ! [X3: nat,Y: produc3741383161447143261al_nat] :
      ( ( fChoic5522844001492335150al_nat
        @ ( produc8534146015018131191_nat_o
          @ ^ [X7: nat,Y7: produc3741383161447143261al_nat] :
              ( ( X3 = X7 )
              & ( Y = Y7 ) ) ) )
      = ( produc4490577844368043572al_nat @ X3 @ Y ) ) ).

% Eps_case_prod_eq
thf(fact_679_Eps__case__prod__eq,axiom,
    ! [X3: real,Y: nat] :
      ( ( fChoic8332448170627812017al_nat
        @ ( produc5286614342184123374_nat_o
          @ ^ [X7: real,Y7: nat] :
              ( ( X3 = X7 )
              & ( Y = Y7 ) ) ) )
      = ( produc3181502643871035669al_nat @ X3 @ Y ) ) ).

% Eps_case_prod_eq
thf(fact_680_case__prodE,axiom,
    ! [C2: nat > list_b > $o,P2: produc7811952446676219690list_b] :
      ( ( produc4544114555777700815st_b_o @ C2 @ P2 )
     => ~ ! [X4: nat,Y3: list_b] :
            ( ( P2
              = ( produc7903367361620597084list_b @ X4 @ Y3 ) )
           => ~ ( C2 @ X4 @ Y3 ) ) ) ).

% case_prodE
thf(fact_681_case__prodE,axiom,
    ! [C2: nat > nat > $o,P2: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ C2 @ P2 )
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X4 @ Y3 ) )
           => ~ ( C2 @ X4 @ Y3 ) ) ) ).

% case_prodE
thf(fact_682_case__prodE,axiom,
    ! [C2: product_prod_nat_nat > product_prod_nat_nat > $o,P2: produc859450856879609959at_nat] :
      ( ( produc6590410687421337004_nat_o @ C2 @ P2 )
     => ~ ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( P2
              = ( produc6161850002892822231at_nat @ X4 @ Y3 ) )
           => ~ ( C2 @ X4 @ Y3 ) ) ) ).

% case_prodE
thf(fact_683_case__prodE,axiom,
    ! [C2: nat > produc3741383161447143261al_nat > $o,P2: produc9155354469731034754al_nat] :
      ( ( produc8534146015018131191_nat_o @ C2 @ P2 )
     => ~ ! [X4: nat,Y3: produc3741383161447143261al_nat] :
            ( ( P2
              = ( produc4490577844368043572al_nat @ X4 @ Y3 ) )
           => ~ ( C2 @ X4 @ Y3 ) ) ) ).

% case_prodE
thf(fact_684_case__prodE,axiom,
    ! [C2: real > nat > $o,P2: produc3741383161447143261al_nat] :
      ( ( produc5286614342184123374_nat_o @ C2 @ P2 )
     => ~ ! [X4: real,Y3: nat] :
            ( ( P2
              = ( produc3181502643871035669al_nat @ X4 @ Y3 ) )
           => ~ ( C2 @ X4 @ Y3 ) ) ) ).

% case_prodE
thf(fact_685_case__prodD,axiom,
    ! [F: nat > list_b > $o,A: nat,B: list_b] :
      ( ( produc4544114555777700815st_b_o @ F @ ( produc7903367361620597084list_b @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_686_case__prodD,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_687_case__prodD,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( produc6590410687421337004_nat_o @ F @ ( produc6161850002892822231at_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_688_case__prodD,axiom,
    ! [F: nat > produc3741383161447143261al_nat > $o,A: nat,B: produc3741383161447143261al_nat] :
      ( ( produc8534146015018131191_nat_o @ F @ ( produc4490577844368043572al_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_689_case__prodD,axiom,
    ! [F: real > nat > $o,A: real,B: nat] :
      ( ( produc5286614342184123374_nat_o @ F @ ( produc3181502643871035669al_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_690_split__paired__Eps,axiom,
    ( fChoic403860114826281174list_b
    = ( ^ [P4: produc7811952446676219690list_b > $o] :
          ( fChoic403860114826281174list_b
          @ ( produc4544114555777700815st_b_o
            @ ^ [A6: nat,B5: list_b] : ( P4 @ ( produc7903367361620597084list_b @ A6 @ B5 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_691_split__paired__Eps,axiom,
    ( fChoic6978938873391328853at_nat
    = ( ^ [P4: product_prod_nat_nat > $o] :
          ( fChoic6978938873391328853at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [A6: nat,B5: nat] : ( P4 @ ( product_Pair_nat_nat @ A6 @ B5 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_692_split__paired__Eps,axiom,
    ( fChoic6673916560228149179at_nat
    = ( ^ [P4: produc859450856879609959at_nat > $o] :
          ( fChoic6673916560228149179at_nat
          @ ( produc6590410687421337004_nat_o
            @ ^ [A6: product_prod_nat_nat,B5: product_prod_nat_nat] : ( P4 @ ( produc6161850002892822231at_nat @ A6 @ B5 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_693_split__paired__Eps,axiom,
    ( fChoic5522844001492335150al_nat
    = ( ^ [P4: produc9155354469731034754al_nat > $o] :
          ( fChoic5522844001492335150al_nat
          @ ( produc8534146015018131191_nat_o
            @ ^ [A6: nat,B5: produc3741383161447143261al_nat] : ( P4 @ ( produc4490577844368043572al_nat @ A6 @ B5 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_694_split__paired__Eps,axiom,
    ( fChoic8332448170627812017al_nat
    = ( ^ [P4: produc3741383161447143261al_nat > $o] :
          ( fChoic8332448170627812017al_nat
          @ ( produc5286614342184123374_nat_o
            @ ^ [A6: real,B5: nat] : ( P4 @ ( produc3181502643871035669al_nat @ A6 @ B5 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_695_list__less__def,axiom,
    ( ord_less_list_nat
    = ( ^ [Xs3: list_nat,Ys3: list_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys3 ) @ ( lexord_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ) ) ) ).

% list_less_def
thf(fact_696_measures__less,axiom,
    ! [F: nat > nat,X3: nat,Y: nat,Fs: list_nat_nat] :
      ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) ) ) ).

% measures_less
thf(fact_697_measures__less,axiom,
    ! [F: product_prod_nat_nat > nat,X3: product_prod_nat_nat,Y: product_prod_nat_nat,Fs: list_P9162950289778280392at_nat] :
      ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( measur2679027848233739777at_nat @ ( cons_P4861729644591583992at_nat @ F @ Fs ) ) ) ) ).

% measures_less
thf(fact_698_lexordp__conv__lexord,axiom,
    ( ord_lexordp_nat
    = ( ^ [Xs3: list_nat,Ys3: list_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys3 ) @ ( lexord_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ) ) ) ).

% lexordp_conv_lexord
thf(fact_699_measures__lesseq,axiom,
    ! [F: nat > nat,X3: nat,Y: nat,Fs: list_nat_nat] :
      ( ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( measures_nat @ Fs ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) ) ) ) ).

% measures_lesseq
thf(fact_700_measures__lesseq,axiom,
    ! [F: product_prod_nat_nat > nat,X3: product_prod_nat_nat,Y: product_prod_nat_nat,Fs: list_P9162950289778280392at_nat] :
      ( ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( measur2679027848233739777at_nat @ Fs ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( measur2679027848233739777at_nat @ ( cons_P4861729644591583992at_nat @ F @ Fs ) ) ) ) ) ).

% measures_lesseq
thf(fact_701_lexord__Nil__left,axiom,
    ! [Y: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Y ) @ ( lexord_nat @ R2 ) )
      = ( ? [A6: nat,X2: list_nat] :
            ( Y
            = ( cons_nat @ A6 @ X2 ) ) ) ) ).

% lexord_Nil_left
thf(fact_702_order__refl,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).

% order_refl
thf(fact_703_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_704_mem__case__prodI,axiom,
    ! [Z3: nat,C2: nat > nat > set_nat,A: nat,B: nat] :
      ( ( member_nat @ Z3 @ ( C2 @ A @ B ) )
     => ( member_nat @ Z3 @ ( produc6189476227299908564et_nat @ C2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_705_mem__case__prodI,axiom,
    ! [Z3: nat,C2: real > nat > set_nat,A: real,B: nat] :
      ( ( member_nat @ Z3 @ ( C2 @ A @ B ) )
     => ( member_nat @ Z3 @ ( produc3765401888635566256et_nat @ C2 @ ( produc3181502643871035669al_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_706_mem__case__prodI,axiom,
    ! [Z3: nat,C2: nat > list_b > set_nat,A: nat,B: list_b] :
      ( ( member_nat @ Z3 @ ( C2 @ A @ B ) )
     => ( member_nat @ Z3 @ ( produc3913824297223167759et_nat @ C2 @ ( produc7903367361620597084list_b @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_707_mem__case__prodI,axiom,
    ! [Z3: list_b,C2: nat > nat > set_list_b,A: nat,B: nat] :
      ( ( member_list_b @ Z3 @ ( C2 @ A @ B ) )
     => ( member_list_b @ Z3 @ ( produc944277461895274647list_b @ C2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_708_mem__case__prodI,axiom,
    ! [Z3: list_b,C2: real > nat > set_list_b,A: real,B: nat] :
      ( ( member_list_b @ Z3 @ ( C2 @ A @ B ) )
     => ( member_list_b @ Z3 @ ( produc1885509172603543739list_b @ C2 @ ( produc3181502643871035669al_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_709_mem__case__prodI,axiom,
    ! [Z3: list_b,C2: nat > list_b > set_list_b,A: nat,B: list_b] :
      ( ( member_list_b @ Z3 @ ( C2 @ A @ B ) )
     => ( member_list_b @ Z3 @ ( produc6393485962720571292list_b @ C2 @ ( produc7903367361620597084list_b @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_710_mem__case__prodI,axiom,
    ! [Z3: product_prod_nat_nat,C2: nat > nat > set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( member8440522571783428010at_nat @ Z3 @ ( C2 @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z3 @ ( produc8197505143624133779at_nat @ C2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_711_mem__case__prodI,axiom,
    ! [Z3: nat,C2: nat > produc3741383161447143261al_nat > set_nat,A: nat,B: produc3741383161447143261al_nat] :
      ( ( member_nat @ Z3 @ ( C2 @ A @ B ) )
     => ( member_nat @ Z3 @ ( produc5025032356412427751et_nat @ C2 @ ( produc4490577844368043572al_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_712_mem__case__prodI,axiom,
    ! [Z3: product_prod_nat_nat,C2: real > nat > set_Pr1261947904930325089at_nat,A: real,B: nat] :
      ( ( member8440522571783428010at_nat @ Z3 @ ( C2 @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z3 @ ( produc3648867147378838711at_nat @ C2 @ ( produc3181502643871035669al_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_713_mem__case__prodI,axiom,
    ! [Z3: product_prod_nat_nat,C2: nat > list_b > set_Pr1261947904930325089at_nat,A: nat,B: list_b] :
      ( ( member8440522571783428010at_nat @ Z3 @ ( C2 @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z3 @ ( produc3725398101618676632at_nat @ C2 @ ( produc7903367361620597084list_b @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_714_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z3: nat,C2: nat > nat > set_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A3 @ B3 ) )
         => ( member_nat @ Z3 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_nat @ Z3 @ ( produc6189476227299908564et_nat @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_715_mem__case__prodI2,axiom,
    ! [P2: produc3741383161447143261al_nat,Z3: nat,C2: real > nat > set_nat] :
      ( ! [A3: real,B3: nat] :
          ( ( P2
            = ( produc3181502643871035669al_nat @ A3 @ B3 ) )
         => ( member_nat @ Z3 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_nat @ Z3 @ ( produc3765401888635566256et_nat @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_716_mem__case__prodI2,axiom,
    ! [P2: produc7811952446676219690list_b,Z3: nat,C2: nat > list_b > set_nat] :
      ( ! [A3: nat,B3: list_b] :
          ( ( P2
            = ( produc7903367361620597084list_b @ A3 @ B3 ) )
         => ( member_nat @ Z3 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_nat @ Z3 @ ( produc3913824297223167759et_nat @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_717_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z3: list_b,C2: nat > nat > set_list_b] :
      ( ! [A3: nat,B3: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A3 @ B3 ) )
         => ( member_list_b @ Z3 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_list_b @ Z3 @ ( produc944277461895274647list_b @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_718_mem__case__prodI2,axiom,
    ! [P2: produc3741383161447143261al_nat,Z3: list_b,C2: real > nat > set_list_b] :
      ( ! [A3: real,B3: nat] :
          ( ( P2
            = ( produc3181502643871035669al_nat @ A3 @ B3 ) )
         => ( member_list_b @ Z3 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_list_b @ Z3 @ ( produc1885509172603543739list_b @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_719_mem__case__prodI2,axiom,
    ! [P2: produc7811952446676219690list_b,Z3: list_b,C2: nat > list_b > set_list_b] :
      ( ! [A3: nat,B3: list_b] :
          ( ( P2
            = ( produc7903367361620597084list_b @ A3 @ B3 ) )
         => ( member_list_b @ Z3 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_list_b @ Z3 @ ( produc6393485962720571292list_b @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_720_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z3: product_prod_nat_nat,C2: nat > nat > set_Pr1261947904930325089at_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A3 @ B3 ) )
         => ( member8440522571783428010at_nat @ Z3 @ ( C2 @ A3 @ B3 ) ) )
     => ( member8440522571783428010at_nat @ Z3 @ ( produc8197505143624133779at_nat @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_721_mem__case__prodI2,axiom,
    ! [P2: produc9155354469731034754al_nat,Z3: nat,C2: nat > produc3741383161447143261al_nat > set_nat] :
      ( ! [A3: nat,B3: produc3741383161447143261al_nat] :
          ( ( P2
            = ( produc4490577844368043572al_nat @ A3 @ B3 ) )
         => ( member_nat @ Z3 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_nat @ Z3 @ ( produc5025032356412427751et_nat @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_722_mem__case__prodI2,axiom,
    ! [P2: produc3741383161447143261al_nat,Z3: product_prod_nat_nat,C2: real > nat > set_Pr1261947904930325089at_nat] :
      ( ! [A3: real,B3: nat] :
          ( ( P2
            = ( produc3181502643871035669al_nat @ A3 @ B3 ) )
         => ( member8440522571783428010at_nat @ Z3 @ ( C2 @ A3 @ B3 ) ) )
     => ( member8440522571783428010at_nat @ Z3 @ ( produc3648867147378838711at_nat @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_723_mem__case__prodI2,axiom,
    ! [P2: produc7811952446676219690list_b,Z3: product_prod_nat_nat,C2: nat > list_b > set_Pr1261947904930325089at_nat] :
      ( ! [A3: nat,B3: list_b] :
          ( ( P2
            = ( produc7903367361620597084list_b @ A3 @ B3 ) )
         => ( member8440522571783428010at_nat @ Z3 @ ( C2 @ A3 @ B3 ) ) )
     => ( member8440522571783428010at_nat @ Z3 @ ( produc3725398101618676632at_nat @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_724_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_725_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_726_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_727_lexordp__simps_I2_J,axiom,
    ! [Xs: list_nat] :
      ~ ( ord_lexordp_nat @ Xs @ nil_nat ) ).

% lexordp_simps(2)
thf(fact_728_lexordp__simps_I1_J,axiom,
    ! [Ys: list_nat] :
      ( ( ord_lexordp_nat @ nil_nat @ Ys )
      = ( Ys != nil_nat ) ) ).

% lexordp_simps(1)
thf(fact_729_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ( rotate1_nat @ Xs )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% rotate1_is_Nil_conv
thf(fact_730_in__measures_I1_J,axiom,
    ! [X3: nat,Y: nat] :
      ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( measures_nat @ nil_nat_nat ) ) ).

% in_measures(1)
thf(fact_731_in__measures_I1_J,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( measur2679027848233739777at_nat @ nil_Pr2865493887535707976at_nat ) ) ).

% in_measures(1)
thf(fact_732_lexordp__simps_I3_J,axiom,
    ! [X3: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( ord_lexordp_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) )
      = ( ( ord_less_nat @ X3 @ Y )
        | ( ~ ( ord_less_nat @ Y @ X3 )
          & ( ord_lexordp_nat @ Xs @ Ys ) ) ) ) ).

% lexordp_simps(3)
thf(fact_733_Nil__lenlex__iff1,axiom,
    ! [Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ns ) @ ( lenlex_nat @ R2 ) )
      = ( Ns != nil_nat ) ) ).

% Nil_lenlex_iff1
thf(fact_734_transpose_Ocases,axiom,
    ! [X3: list_list_nat] :
      ( ( X3 != nil_list_nat )
     => ( ! [Xss: list_list_nat] :
            ( X3
           != ( cons_list_nat @ nil_nat @ Xss ) )
       => ~ ! [X4: nat,Xs2: list_nat,Xss: list_list_nat] :
              ( X3
             != ( cons_list_nat @ ( cons_nat @ X4 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_735_lexordp_ONil,axiom,
    ! [Y: nat,Ys: list_nat] : ( ord_lexordp_nat @ nil_nat @ ( cons_nat @ Y @ Ys ) ) ).

% lexordp.Nil
thf(fact_736_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_737_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_738_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_739_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_740_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_741_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ? [X4: nat] :
            ( ( P @ X4 )
            & ! [Y5: nat] :
                ( ( P @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_742_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_743_le__cases3,axiom,
    ! [X3: nat,Y: nat,Z3: nat] :
      ( ( ( ord_less_eq_nat @ X3 @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z3 ) )
     => ( ( ( ord_less_eq_nat @ Y @ X3 )
         => ~ ( ord_less_eq_nat @ X3 @ Z3 ) )
       => ( ( ( ord_less_eq_nat @ X3 @ Z3 )
           => ~ ( ord_less_eq_nat @ Z3 @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z3 @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X3 ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z3 )
               => ~ ( ord_less_eq_nat @ Z3 @ X3 ) )
             => ~ ( ( ord_less_eq_nat @ Z3 @ X3 )
                 => ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_744_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y8: nat,Z4: nat] : ( Y8 = Z4 ) )
    = ( ^ [X2: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y4 )
          & ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_745_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_746_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_747_order__antisym,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ Y @ X3 )
       => ( X3 = Y ) ) ) ).

% order_antisym
thf(fact_748_order_Otrans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% order.trans
thf(fact_749_order__trans,axiom,
    ! [X3: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z3 )
       => ( ord_less_eq_nat @ X3 @ Z3 ) ) ) ).

% order_trans
thf(fact_750_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: nat,B3: nat] :
            ( ( P @ B3 @ A3 )
           => ( P @ A3 @ B3 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_751_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y8: nat,Z4: nat] : ( Y8 = Z4 ) )
    = ( ^ [A6: nat,B5: nat] :
          ( ( ord_less_eq_nat @ B5 @ A6 )
          & ( ord_less_eq_nat @ A6 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_752_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_753_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_754_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_755_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y8: nat,Z4: nat] : ( Y8 = Z4 ) )
    = ( ^ [A6: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A6 @ B5 )
          & ( ord_less_eq_nat @ B5 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_756_order__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_757_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_758_order__eq__refl,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 = Y )
     => ( ord_less_eq_nat @ X3 @ Y ) ) ).

% order_eq_refl
thf(fact_759_linorder__linear,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
      | ( ord_less_eq_nat @ Y @ X3 ) ) ).

% linorder_linear
thf(fact_760_ord__eq__le__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_761_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_762_linorder__le__cases,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X3 @ Y )
     => ( ord_less_eq_nat @ Y @ X3 ) ) ).

% linorder_le_cases
thf(fact_763_order__antisym__conv,axiom,
    ! [Y: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y @ X3 )
     => ( ( ord_less_eq_nat @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% order_antisym_conv
thf(fact_764_bounded__Max__nat,axiom,
    ! [P: nat > $o,X3: nat,M4: nat] :
      ( ( P @ X3 )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( ord_less_eq_nat @ X4 @ M4 ) )
       => ~ ! [M5: nat] :
              ( ( P @ M5 )
             => ~ ! [X6: nat] :
                    ( ( P @ X6 )
                   => ( ord_less_eq_nat @ X6 @ M5 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_765_lexordp__induct,axiom,
    ! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
      ( ( ord_lexordp_nat @ Xs @ Ys )
     => ( ! [Y3: nat,Ys4: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y3 @ Ys4 ) )
       => ( ! [X4: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
         => ( ! [X4: nat,Xs2: list_nat,Ys4: list_nat] :
                ( ( ord_lexordp_nat @ Xs2 @ Ys4 )
               => ( ( P @ Xs2 @ Ys4 )
                 => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ X4 @ Ys4 ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% lexordp_induct
thf(fact_766_lexordp__cases,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ord_lexordp_nat @ Xs @ Ys )
     => ( ( ( Xs = nil_nat )
         => ! [Y3: nat,Ys5: list_nat] :
              ( Ys
             != ( cons_nat @ Y3 @ Ys5 ) ) )
       => ( ! [X4: nat] :
              ( ? [Xs4: list_nat] :
                  ( Xs
                  = ( cons_nat @ X4 @ Xs4 ) )
             => ! [Y3: nat] :
                  ( ? [Ys5: list_nat] :
                      ( Ys
                      = ( cons_nat @ Y3 @ Ys5 ) )
                 => ~ ( ord_less_nat @ X4 @ Y3 ) ) )
         => ~ ! [X4: nat,Xs4: list_nat] :
                ( ( Xs
                  = ( cons_nat @ X4 @ Xs4 ) )
               => ! [Ys5: list_nat] :
                    ( ( Ys
                      = ( cons_nat @ X4 @ Ys5 ) )
                   => ~ ( ord_lexordp_nat @ Xs4 @ Ys5 ) ) ) ) ) ) ).

% lexordp_cases
thf(fact_767_lexordp_Osimps,axiom,
    ( ord_lexordp_nat
    = ( ^ [A1: list_nat,A22: list_nat] :
          ( ? [Y4: nat,Ys3: list_nat] :
              ( ( A1 = nil_nat )
              & ( A22
                = ( cons_nat @ Y4 @ Ys3 ) ) )
          | ? [X2: nat,Y4: nat,Xs3: list_nat,Ys3: list_nat] :
              ( ( A1
                = ( cons_nat @ X2 @ Xs3 ) )
              & ( A22
                = ( cons_nat @ Y4 @ Ys3 ) )
              & ( ord_less_nat @ X2 @ Y4 ) )
          | ? [X2: nat,Y4: nat,Xs3: list_nat,Ys3: list_nat] :
              ( ( A1
                = ( cons_nat @ X2 @ Xs3 ) )
              & ( A22
                = ( cons_nat @ Y4 @ Ys3 ) )
              & ~ ( ord_less_nat @ X2 @ Y4 )
              & ~ ( ord_less_nat @ Y4 @ X2 )
              & ( ord_lexordp_nat @ Xs3 @ Ys3 ) ) ) ) ) ).

% lexordp.simps
thf(fact_768_lexordp_Ocases,axiom,
    ! [A12: list_nat,A23: list_nat] :
      ( ( ord_lexordp_nat @ A12 @ A23 )
     => ( ( ( A12 = nil_nat )
         => ! [Y3: nat,Ys4: list_nat] :
              ( A23
             != ( cons_nat @ Y3 @ Ys4 ) ) )
       => ( ! [X4: nat] :
              ( ? [Xs2: list_nat] :
                  ( A12
                  = ( cons_nat @ X4 @ Xs2 ) )
             => ! [Y3: nat] :
                  ( ? [Ys4: list_nat] :
                      ( A23
                      = ( cons_nat @ Y3 @ Ys4 ) )
                 => ~ ( ord_less_nat @ X4 @ Y3 ) ) )
         => ~ ! [X4: nat,Y3: nat,Xs2: list_nat] :
                ( ( A12
                  = ( cons_nat @ X4 @ Xs2 ) )
               => ! [Ys4: list_nat] :
                    ( ( A23
                      = ( cons_nat @ Y3 @ Ys4 ) )
                   => ( ~ ( ord_less_nat @ X4 @ Y3 )
                     => ( ~ ( ord_less_nat @ Y3 @ X4 )
                       => ~ ( ord_lexordp_nat @ Xs2 @ Ys4 ) ) ) ) ) ) ) ) ).

% lexordp.cases
thf(fact_769_mem__case__prodE,axiom,
    ! [Z3: nat,C2: nat > nat > set_nat,P2: product_prod_nat_nat] :
      ( ( member_nat @ Z3 @ ( produc6189476227299908564et_nat @ C2 @ P2 ) )
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X4 @ Y3 ) )
           => ~ ( member_nat @ Z3 @ ( C2 @ X4 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_770_mem__case__prodE,axiom,
    ! [Z3: nat,C2: real > nat > set_nat,P2: produc3741383161447143261al_nat] :
      ( ( member_nat @ Z3 @ ( produc3765401888635566256et_nat @ C2 @ P2 ) )
     => ~ ! [X4: real,Y3: nat] :
            ( ( P2
              = ( produc3181502643871035669al_nat @ X4 @ Y3 ) )
           => ~ ( member_nat @ Z3 @ ( C2 @ X4 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_771_mem__case__prodE,axiom,
    ! [Z3: nat,C2: nat > list_b > set_nat,P2: produc7811952446676219690list_b] :
      ( ( member_nat @ Z3 @ ( produc3913824297223167759et_nat @ C2 @ P2 ) )
     => ~ ! [X4: nat,Y3: list_b] :
            ( ( P2
              = ( produc7903367361620597084list_b @ X4 @ Y3 ) )
           => ~ ( member_nat @ Z3 @ ( C2 @ X4 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_772_mem__case__prodE,axiom,
    ! [Z3: list_b,C2: nat > nat > set_list_b,P2: product_prod_nat_nat] :
      ( ( member_list_b @ Z3 @ ( produc944277461895274647list_b @ C2 @ P2 ) )
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X4 @ Y3 ) )
           => ~ ( member_list_b @ Z3 @ ( C2 @ X4 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_773_mem__case__prodE,axiom,
    ! [Z3: list_b,C2: real > nat > set_list_b,P2: produc3741383161447143261al_nat] :
      ( ( member_list_b @ Z3 @ ( produc1885509172603543739list_b @ C2 @ P2 ) )
     => ~ ! [X4: real,Y3: nat] :
            ( ( P2
              = ( produc3181502643871035669al_nat @ X4 @ Y3 ) )
           => ~ ( member_list_b @ Z3 @ ( C2 @ X4 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_774_mem__case__prodE,axiom,
    ! [Z3: list_b,C2: nat > list_b > set_list_b,P2: produc7811952446676219690list_b] :
      ( ( member_list_b @ Z3 @ ( produc6393485962720571292list_b @ C2 @ P2 ) )
     => ~ ! [X4: nat,Y3: list_b] :
            ( ( P2
              = ( produc7903367361620597084list_b @ X4 @ Y3 ) )
           => ~ ( member_list_b @ Z3 @ ( C2 @ X4 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_775_mem__case__prodE,axiom,
    ! [Z3: product_prod_nat_nat,C2: nat > nat > set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ Z3 @ ( produc8197505143624133779at_nat @ C2 @ P2 ) )
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X4 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z3 @ ( C2 @ X4 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_776_mem__case__prodE,axiom,
    ! [Z3: nat,C2: nat > produc3741383161447143261al_nat > set_nat,P2: produc9155354469731034754al_nat] :
      ( ( member_nat @ Z3 @ ( produc5025032356412427751et_nat @ C2 @ P2 ) )
     => ~ ! [X4: nat,Y3: produc3741383161447143261al_nat] :
            ( ( P2
              = ( produc4490577844368043572al_nat @ X4 @ Y3 ) )
           => ~ ( member_nat @ Z3 @ ( C2 @ X4 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_777_mem__case__prodE,axiom,
    ! [Z3: product_prod_nat_nat,C2: real > nat > set_Pr1261947904930325089at_nat,P2: produc3741383161447143261al_nat] :
      ( ( member8440522571783428010at_nat @ Z3 @ ( produc3648867147378838711at_nat @ C2 @ P2 ) )
     => ~ ! [X4: real,Y3: nat] :
            ( ( P2
              = ( produc3181502643871035669al_nat @ X4 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z3 @ ( C2 @ X4 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_778_mem__case__prodE,axiom,
    ! [Z3: product_prod_nat_nat,C2: nat > list_b > set_Pr1261947904930325089at_nat,P2: produc7811952446676219690list_b] :
      ( ( member8440522571783428010at_nat @ Z3 @ ( produc3725398101618676632at_nat @ C2 @ P2 ) )
     => ~ ! [X4: nat,Y3: list_b] :
            ( ( P2
              = ( produc7903367361620597084list_b @ X4 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z3 @ ( C2 @ X4 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_779_leD,axiom,
    ! [Y: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y @ X3 )
     => ~ ( ord_less_nat @ X3 @ Y ) ) ).

% leD
thf(fact_780_leI,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y )
     => ( ord_less_eq_nat @ Y @ X3 ) ) ).

% leI
thf(fact_781_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_782_antisym__conv1,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% antisym_conv1
thf(fact_783_antisym__conv2,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ~ ( ord_less_nat @ X3 @ Y ) )
        = ( X3 = Y ) ) ) ).

% antisym_conv2
thf(fact_784_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X2: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y4 )
          & ~ ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_785_not__le__imp__less,axiom,
    ! [Y: nat,X3: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X3 )
     => ( ord_less_nat @ X3 @ Y ) ) ).

% not_le_imp_less
thf(fact_786_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A6: nat,B5: nat] :
          ( ( ord_less_nat @ A6 @ B5 )
          | ( A6 = B5 ) ) ) ) ).

% order.order_iff_strict
thf(fact_787_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A6: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A6 @ B5 )
          & ( A6 != B5 ) ) ) ) ).

% order.strict_iff_order
thf(fact_788_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_789_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_790_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A6: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A6 @ B5 )
          & ~ ( ord_less_eq_nat @ B5 @ A6 ) ) ) ) ).

% order.strict_iff_not
thf(fact_791_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B5: nat,A6: nat] :
          ( ( ord_less_nat @ B5 @ A6 )
          | ( A6 = B5 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_792_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B5: nat,A6: nat] :
          ( ( ord_less_eq_nat @ B5 @ A6 )
          & ( A6 != B5 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_793_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_794_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_795_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B5: nat,A6: nat] :
          ( ( ord_less_eq_nat @ B5 @ A6 )
          & ~ ( ord_less_eq_nat @ A6 @ B5 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_796_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_797_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_798_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y4: nat] :
          ( ( ord_less_nat @ X2 @ Y4 )
          | ( X2 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_799_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X2: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y4 )
          & ( X2 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_800_linorder__not__le,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X3 @ Y ) )
      = ( ord_less_nat @ Y @ X3 ) ) ).

% linorder_not_le
thf(fact_801_linorder__not__less,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y ) )
      = ( ord_less_eq_nat @ Y @ X3 ) ) ).

% linorder_not_less
thf(fact_802_order__less__imp__le,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( ord_less_eq_nat @ X3 @ Y ) ) ).

% order_less_imp_le
thf(fact_803_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_804_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_805_order__le__less__trans,axiom,
    ! [X3: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_nat @ Y @ Z3 )
       => ( ord_less_nat @ X3 @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_806_order__less__le__trans,axiom,
    ! [X3: nat,Y: nat,Z3: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z3 )
       => ( ord_less_nat @ X3 @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_807_order__le__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_808_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_809_order__less__le__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_810_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_811_linorder__le__less__linear,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
      | ( ord_less_nat @ Y @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_812_order__le__imp__less__or__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_nat @ X3 @ Y )
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_813_top__greatest,axiom,
    ! [A: set_list_b] : ( ord_le8932221534207217157list_b @ A @ top_top_set_list_b ) ).

% top_greatest
thf(fact_814_top__greatest,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).

% top_greatest
thf(fact_815_top__greatest,axiom,
    ! [A: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ A @ top_to8454300755403694231at_nat ) ).

% top_greatest
thf(fact_816_top__greatest,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ A @ top_to4669805908274784177at_nat ) ).

% top_greatest
thf(fact_817_top_Oextremum__unique,axiom,
    ! [A: set_list_b] :
      ( ( ord_le8932221534207217157list_b @ top_top_set_list_b @ A )
      = ( A = top_top_set_list_b ) ) ).

% top.extremum_unique
thf(fact_818_top_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
      = ( A = top_top_set_nat ) ) ).

% top.extremum_unique
thf(fact_819_top_Oextremum__unique,axiom,
    ! [A: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ top_to8454300755403694231at_nat @ A )
      = ( A = top_to8454300755403694231at_nat ) ) ).

% top.extremum_unique
thf(fact_820_top_Oextremum__unique,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ top_to4669805908274784177at_nat @ A )
      = ( A = top_to4669805908274784177at_nat ) ) ).

% top.extremum_unique
thf(fact_821_top_Oextremum__uniqueI,axiom,
    ! [A: set_list_b] :
      ( ( ord_le8932221534207217157list_b @ top_top_set_list_b @ A )
     => ( A = top_top_set_list_b ) ) ).

% top.extremum_uniqueI
thf(fact_822_top_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
     => ( A = top_top_set_nat ) ) ).

% top.extremum_uniqueI
thf(fact_823_top_Oextremum__uniqueI,axiom,
    ! [A: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ top_to8454300755403694231at_nat @ A )
     => ( A = top_to8454300755403694231at_nat ) ) ).

% top.extremum_uniqueI
thf(fact_824_top_Oextremum__uniqueI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ top_to4669805908274784177at_nat @ A )
     => ( A = top_to4669805908274784177at_nat ) ) ).

% top.extremum_uniqueI
thf(fact_825_sorted__wrt_Ocases,axiom,
    ! [X3: produc254973753779126261st_nat] :
      ( ! [P5: nat > nat > $o] :
          ( X3
         != ( produc4727192421694094319st_nat @ P5 @ nil_nat ) )
     => ~ ! [P5: nat > nat > $o,X4: nat,Ys4: list_nat] :
            ( X3
           != ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X4 @ Ys4 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_826_successively_Ocases,axiom,
    ! [X3: produc254973753779126261st_nat] :
      ( ! [P5: nat > nat > $o] :
          ( X3
         != ( produc4727192421694094319st_nat @ P5 @ nil_nat ) )
     => ( ! [P5: nat > nat > $o,X4: nat] :
            ( X3
           != ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X4 @ nil_nat ) ) )
       => ~ ! [P5: nat > nat > $o,X4: nat,Y3: nat,Xs2: list_nat] :
              ( X3
             != ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X4 @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_827_list_Odistinct_I1_J,axiom,
    ! [X21: nat,X222: list_nat] :
      ( nil_nat
     != ( cons_nat @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_828_list_OdiscI,axiom,
    ! [List: list_nat,X21: nat,X222: list_nat] :
      ( ( List
        = ( cons_nat @ X21 @ X222 ) )
     => ( List != nil_nat ) ) ).

% list.discI
thf(fact_829_list_Oexhaust,axiom,
    ! [Y: list_nat] :
      ( ( Y != nil_nat )
     => ~ ! [X212: nat,X223: list_nat] :
            ( Y
           != ( cons_nat @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_830_min__list_Ocases,axiom,
    ! [X3: list_nat] :
      ( ! [X4: nat,Xs2: list_nat] :
          ( X3
         != ( cons_nat @ X4 @ Xs2 ) )
     => ( X3 = nil_nat ) ) ).

% min_list.cases
thf(fact_831_remdups__adj_Ocases,axiom,
    ! [X3: list_nat] :
      ( ( X3 != nil_nat )
     => ( ! [X4: nat] :
            ( X3
           != ( cons_nat @ X4 @ nil_nat ) )
       => ~ ! [X4: nat,Y3: nat,Xs2: list_nat] :
              ( X3
             != ( cons_nat @ X4 @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_832_neq__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
      = ( ? [Y4: nat,Ys3: list_nat] :
            ( Xs
            = ( cons_nat @ Y4 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_833_list__induct2_H,axiom,
    ! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
      ( ( P @ nil_nat @ nil_nat )
     => ( ! [X4: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X4 @ Xs2 ) @ nil_nat )
       => ( ! [Y3: nat,Ys4: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y3 @ Ys4 ) )
         => ( ! [X4: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat] :
                ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_834_list__nonempty__induct,axiom,
    ! [Xs: list_nat,P: list_nat > $o] :
      ( ( Xs != nil_nat )
     => ( ! [X4: nat] : ( P @ ( cons_nat @ X4 @ nil_nat ) )
       => ( ! [X4: nat,Xs2: list_nat] :
              ( ( Xs2 != nil_nat )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_nat @ X4 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_835_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
          & ( M3 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_836_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_837_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N3: nat] :
          ( ( ord_less_nat @ M3 @ N3 )
          | ( M3 = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_838_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_839_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_840_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_841_list_Osimps_I8_J,axiom,
    ! [F: nat > nat] :
      ( ( map_nat_nat @ F @ nil_nat )
      = nil_nat ) ).

% list.simps(8)
thf(fact_842_lexordp__irreflexive,axiom,
    ! [Xs: list_nat] :
      ( ! [X4: nat] :
          ~ ( ord_less_nat @ X4 @ X4 )
     => ~ ( ord_lexordp_nat @ Xs @ Xs ) ) ).

% lexordp_irreflexive
thf(fact_843_case__prod__Pair__iden,axiom,
    ! [P2: produc7811952446676219690list_b] :
      ( ( produc1929794673946095923list_b @ produc7903367361620597084list_b @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_844_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat @ product_Pair_nat_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_845_case__prod__Pair__iden,axiom,
    ! [P2: produc859450856879609959at_nat] :
      ( ( produc6744312701629110395at_nat @ produc6161850002892822231at_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_846_case__prod__Pair__iden,axiom,
    ! [P2: produc9155354469731034754al_nat] :
      ( ( produc8416291181653828323al_nat @ produc4490577844368043572al_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_847_case__prod__Pair__iden,axiom,
    ! [P2: produc3741383161447143261al_nat] :
      ( ( produc8907022329947360947al_nat @ produc3181502643871035669al_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_848_removeAll_Osimps_I1_J,axiom,
    ! [X3: nat] :
      ( ( removeAll_nat @ X3 @ nil_nat )
      = nil_nat ) ).

% removeAll.simps(1)
thf(fact_849_rotate1_Osimps_I1_J,axiom,
    ( ( rotate1_nat @ nil_nat )
    = nil_nat ) ).

% rotate1.simps(1)
thf(fact_850_inj__split__Cons,axiom,
    ! [X: set_Pr7549493975000385137at_nat] :
      ( inj_on3847583997030603456st_nat
      @ ( produc1629833771042763070st_nat
        @ ^ [Xs3: list_nat,N3: nat] : ( cons_nat @ N3 @ Xs3 ) )
      @ X ) ).

% inj_split_Cons
thf(fact_851_linorder__inj__onI,axiom,
    ! [A4: set_real,F: real > produc3741383161447143261al_nat] :
      ( ! [X4: real,Y3: real] :
          ( ( ord_less_real @ X4 @ Y3 )
         => ( ( member_real @ X4 @ A4 )
           => ( ( member_real @ Y3 @ A4 )
             => ( ( F @ X4 )
               != ( F @ Y3 ) ) ) ) )
     => ( ! [X4: real,Y3: real] :
            ( ( member_real @ X4 @ A4 )
           => ( ( member_real @ Y3 @ A4 )
             => ( ( ord_less_eq_real @ X4 @ Y3 )
                | ( ord_less_eq_real @ Y3 @ X4 ) ) ) )
       => ( inj_on8027545645978435326al_nat @ F @ A4 ) ) ) ).

% linorder_inj_onI
thf(fact_852_linorder__inj__onI,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( ord_less_nat @ X4 @ Y3 )
         => ( ( member_nat @ X4 @ A4 )
           => ( ( member_nat @ Y3 @ A4 )
             => ( ( F @ X4 )
               != ( F @ Y3 ) ) ) ) )
     => ( ! [X4: nat,Y3: nat] :
            ( ( member_nat @ X4 @ A4 )
           => ( ( member_nat @ Y3 @ A4 )
             => ( ( ord_less_eq_nat @ X4 @ Y3 )
                | ( ord_less_eq_nat @ Y3 @ X4 ) ) ) )
       => ( inj_on_nat_nat @ F @ A4 ) ) ) ).

% linorder_inj_onI
thf(fact_853_linorder__inj__onI,axiom,
    ! [A4: set_nat,F: nat > produc9155354469731034754al_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( ord_less_nat @ X4 @ Y3 )
         => ( ( member_nat @ X4 @ A4 )
           => ( ( member_nat @ Y3 @ A4 )
             => ( ( F @ X4 )
               != ( F @ Y3 ) ) ) ) )
     => ( ! [X4: nat,Y3: nat] :
            ( ( member_nat @ X4 @ A4 )
           => ( ( member_nat @ Y3 @ A4 )
             => ( ( ord_less_eq_nat @ X4 @ Y3 )
                | ( ord_less_eq_nat @ Y3 @ X4 ) ) ) )
       => ( inj_on7152781252210839493al_nat @ F @ A4 ) ) ) ).

% linorder_inj_onI
thf(fact_854_linorder__inj__onI,axiom,
    ! [A4: set_nat,F: nat > product_prod_nat_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( ord_less_nat @ X4 @ Y3 )
         => ( ( member_nat @ X4 @ A4 )
           => ( ( member_nat @ Y3 @ A4 )
             => ( ( F @ X4 )
               != ( F @ Y3 ) ) ) ) )
     => ( ! [X4: nat,Y3: nat] :
            ( ( member_nat @ X4 @ A4 )
           => ( ( member_nat @ Y3 @ A4 )
             => ( ( ord_less_eq_nat @ X4 @ Y3 )
                | ( ord_less_eq_nat @ Y3 @ X4 ) ) ) )
       => ( inj_on5538052773655684606at_nat @ F @ A4 ) ) ) ).

% linorder_inj_onI
thf(fact_855_linorder__inj__onI,axiom,
    ! [A4: set_nat,F: nat > produc7811952446676219690list_b] :
      ( ! [X4: nat,Y3: nat] :
          ( ( ord_less_nat @ X4 @ Y3 )
         => ( ( member_nat @ X4 @ A4 )
           => ( ( member_nat @ Y3 @ A4 )
             => ( ( F @ X4 )
               != ( F @ Y3 ) ) ) ) )
     => ( ! [X4: nat,Y3: nat] :
            ( ( member_nat @ X4 @ A4 )
           => ( ( member_nat @ Y3 @ A4 )
             => ( ( ord_less_eq_nat @ X4 @ Y3 )
                | ( ord_less_eq_nat @ Y3 @ X4 ) ) ) )
       => ( inj_on8555073082499328493list_b @ F @ A4 ) ) ) ).

% linorder_inj_onI
thf(fact_856_impossible__Cons,axiom,
    ! [Xs: list_nat,Ys: list_nat,X3: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
     => ( Xs
       != ( cons_nat @ X3 @ Ys ) ) ) ).

% impossible_Cons
thf(fact_857_impossible__Cons,axiom,
    ! [Xs: list_b,Ys: list_b,X3: b] :
      ( ( ord_less_eq_nat @ ( size_size_list_b @ Xs ) @ ( size_size_list_b @ Ys ) )
     => ( Xs
       != ( cons_b @ X3 @ Ys ) ) ) ).

% impossible_Cons
thf(fact_858_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_nat @ nil_nat )
       => ( ! [X4: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat] :
              ( ( ( size_size_list_nat @ Xs2 )
                = ( size_size_list_nat @ Ys4 ) )
             => ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_859_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_b,P: list_nat > list_b > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( P @ nil_nat @ nil_b )
       => ( ! [X4: nat,Xs2: list_nat,Y3: b,Ys4: list_b] :
              ( ( ( size_size_list_nat @ Xs2 )
                = ( size_size_list_b @ Ys4 ) )
             => ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_b @ Y3 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_860_list__induct2,axiom,
    ! [Xs: list_b,Ys: list_nat,P: list_b > list_nat > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_b @ nil_nat )
       => ( ! [X4: b,Xs2: list_b,Y3: nat,Ys4: list_nat] :
              ( ( ( size_size_list_b @ Xs2 )
                = ( size_size_list_nat @ Ys4 ) )
             => ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_b @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_861_list__induct2,axiom,
    ! [Xs: list_b,Ys: list_b,P: list_b > list_b > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( P @ nil_b @ nil_b )
       => ( ! [X4: b,Xs2: list_b,Y3: b,Ys4: list_b] :
              ( ( ( size_size_list_b @ Xs2 )
                = ( size_size_list_b @ Ys4 ) )
             => ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_b @ X4 @ Xs2 ) @ ( cons_b @ Y3 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_862_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs3: list_nat,P: list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs3 ) )
       => ( ( P @ nil_nat @ nil_nat @ nil_nat )
         => ( ! [X4: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat,Z2: nat,Zs2: list_nat] :
                ( ( ( size_size_list_nat @ Xs2 )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ( ( size_size_list_nat @ Ys4 )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_863_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs3: list_b,P: list_nat > list_nat > list_b > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_b @ Zs3 ) )
       => ( ( P @ nil_nat @ nil_nat @ nil_b )
         => ( ! [X4: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat,Z2: b,Zs2: list_b] :
                ( ( ( size_size_list_nat @ Xs2 )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ( ( size_size_list_nat @ Ys4 )
                    = ( size_size_list_b @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_b @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_864_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_b,Zs3: list_nat,P: list_nat > list_b > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( size_size_list_b @ Ys )
          = ( size_size_list_nat @ Zs3 ) )
       => ( ( P @ nil_nat @ nil_b @ nil_nat )
         => ( ! [X4: nat,Xs2: list_nat,Y3: b,Ys4: list_b,Z2: nat,Zs2: list_nat] :
                ( ( ( size_size_list_nat @ Xs2 )
                  = ( size_size_list_b @ Ys4 ) )
               => ( ( ( size_size_list_b @ Ys4 )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_b @ Y3 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_865_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_b,Zs3: list_b,P: list_nat > list_b > list_b > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( size_size_list_b @ Ys )
          = ( size_size_list_b @ Zs3 ) )
       => ( ( P @ nil_nat @ nil_b @ nil_b )
         => ( ! [X4: nat,Xs2: list_nat,Y3: b,Ys4: list_b,Z2: b,Zs2: list_b] :
                ( ( ( size_size_list_nat @ Xs2 )
                  = ( size_size_list_b @ Ys4 ) )
               => ( ( ( size_size_list_b @ Ys4 )
                    = ( size_size_list_b @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_b @ Y3 @ Ys4 ) @ ( cons_b @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_866_list__induct3,axiom,
    ! [Xs: list_b,Ys: list_nat,Zs3: list_nat,P: list_b > list_nat > list_nat > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs3 ) )
       => ( ( P @ nil_b @ nil_nat @ nil_nat )
         => ( ! [X4: b,Xs2: list_b,Y3: nat,Ys4: list_nat,Z2: nat,Zs2: list_nat] :
                ( ( ( size_size_list_b @ Xs2 )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ( ( size_size_list_nat @ Ys4 )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_b @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_867_list__induct3,axiom,
    ! [Xs: list_b,Ys: list_nat,Zs3: list_b,P: list_b > list_nat > list_b > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_b @ Zs3 ) )
       => ( ( P @ nil_b @ nil_nat @ nil_b )
         => ( ! [X4: b,Xs2: list_b,Y3: nat,Ys4: list_nat,Z2: b,Zs2: list_b] :
                ( ( ( size_size_list_b @ Xs2 )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ( ( size_size_list_nat @ Ys4 )
                    = ( size_size_list_b @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_b @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_b @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_868_list__induct3,axiom,
    ! [Xs: list_b,Ys: list_b,Zs3: list_nat,P: list_b > list_b > list_nat > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( size_size_list_b @ Ys )
          = ( size_size_list_nat @ Zs3 ) )
       => ( ( P @ nil_b @ nil_b @ nil_nat )
         => ( ! [X4: b,Xs2: list_b,Y3: b,Ys4: list_b,Z2: nat,Zs2: list_nat] :
                ( ( ( size_size_list_b @ Xs2 )
                  = ( size_size_list_b @ Ys4 ) )
               => ( ( ( size_size_list_b @ Ys4 )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_b @ X4 @ Xs2 ) @ ( cons_b @ Y3 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_869_list__induct3,axiom,
    ! [Xs: list_b,Ys: list_b,Zs3: list_b,P: list_b > list_b > list_b > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( size_size_list_b @ Ys )
          = ( size_size_list_b @ Zs3 ) )
       => ( ( P @ nil_b @ nil_b @ nil_b )
         => ( ! [X4: b,Xs2: list_b,Y3: b,Ys4: list_b,Z2: b,Zs2: list_b] :
                ( ( ( size_size_list_b @ Xs2 )
                  = ( size_size_list_b @ Ys4 ) )
               => ( ( ( size_size_list_b @ Ys4 )
                    = ( size_size_list_b @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_b @ X4 @ Xs2 ) @ ( cons_b @ Y3 @ Ys4 ) @ ( cons_b @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_870_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs3: list_nat,Ws: list_nat,P: list_nat > list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs3 ) )
       => ( ( ( size_size_list_nat @ Zs3 )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
           => ( ! [X4: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat,Z2: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_871_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs3: list_nat,Ws: list_b,P: list_nat > list_nat > list_nat > list_b > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs3 ) )
       => ( ( ( size_size_list_nat @ Zs3 )
            = ( size_size_list_b @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_b )
           => ( ! [X4: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat,Z2: nat,Zs2: list_nat,W: b,Ws2: list_b] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_b @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_b @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_872_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs3: list_b,Ws: list_nat,P: list_nat > list_nat > list_b > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_b @ Zs3 ) )
       => ( ( ( size_size_list_b @ Zs3 )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_b @ nil_nat )
           => ( ! [X4: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat,Z2: b,Zs2: list_b,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_size_list_b @ Zs2 ) )
                   => ( ( ( size_size_list_b @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_b @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_873_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs3: list_b,Ws: list_b,P: list_nat > list_nat > list_b > list_b > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_b @ Zs3 ) )
       => ( ( ( size_size_list_b @ Zs3 )
            = ( size_size_list_b @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_b @ nil_b )
           => ( ! [X4: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat,Z2: b,Zs2: list_b,W: b,Ws2: list_b] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_size_list_b @ Zs2 ) )
                   => ( ( ( size_size_list_b @ Zs2 )
                        = ( size_size_list_b @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_b @ Z2 @ Zs2 ) @ ( cons_b @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_874_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_b,Zs3: list_nat,Ws: list_nat,P: list_nat > list_b > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( size_size_list_b @ Ys )
          = ( size_size_list_nat @ Zs3 ) )
       => ( ( ( size_size_list_nat @ Zs3 )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_b @ nil_nat @ nil_nat )
           => ( ! [X4: nat,Xs2: list_nat,Y3: b,Ys4: list_b,Z2: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_b @ Ys4 ) )
                 => ( ( ( size_size_list_b @ Ys4 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_b @ Y3 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_875_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_b,Zs3: list_nat,Ws: list_b,P: list_nat > list_b > list_nat > list_b > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( size_size_list_b @ Ys )
          = ( size_size_list_nat @ Zs3 ) )
       => ( ( ( size_size_list_nat @ Zs3 )
            = ( size_size_list_b @ Ws ) )
         => ( ( P @ nil_nat @ nil_b @ nil_nat @ nil_b )
           => ( ! [X4: nat,Xs2: list_nat,Y3: b,Ys4: list_b,Z2: nat,Zs2: list_nat,W: b,Ws2: list_b] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_b @ Ys4 ) )
                 => ( ( ( size_size_list_b @ Ys4 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_b @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_b @ Y3 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_b @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_876_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_b,Zs3: list_b,Ws: list_nat,P: list_nat > list_b > list_b > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( size_size_list_b @ Ys )
          = ( size_size_list_b @ Zs3 ) )
       => ( ( ( size_size_list_b @ Zs3 )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_b @ nil_b @ nil_nat )
           => ( ! [X4: nat,Xs2: list_nat,Y3: b,Ys4: list_b,Z2: b,Zs2: list_b,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_b @ Ys4 ) )
                 => ( ( ( size_size_list_b @ Ys4 )
                      = ( size_size_list_b @ Zs2 ) )
                   => ( ( ( size_size_list_b @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_b @ Y3 @ Ys4 ) @ ( cons_b @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_877_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_b,Zs3: list_b,Ws: list_b,P: list_nat > list_b > list_b > list_b > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( size_size_list_b @ Ys )
          = ( size_size_list_b @ Zs3 ) )
       => ( ( ( size_size_list_b @ Zs3 )
            = ( size_size_list_b @ Ws ) )
         => ( ( P @ nil_nat @ nil_b @ nil_b @ nil_b )
           => ( ! [X4: nat,Xs2: list_nat,Y3: b,Ys4: list_b,Z2: b,Zs2: list_b,W: b,Ws2: list_b] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_b @ Ys4 ) )
                 => ( ( ( size_size_list_b @ Ys4 )
                      = ( size_size_list_b @ Zs2 ) )
                   => ( ( ( size_size_list_b @ Zs2 )
                        = ( size_size_list_b @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_b @ Y3 @ Ys4 ) @ ( cons_b @ Z2 @ Zs2 ) @ ( cons_b @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_878_list__induct4,axiom,
    ! [Xs: list_b,Ys: list_nat,Zs3: list_nat,Ws: list_nat,P: list_b > list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs3 ) )
       => ( ( ( size_size_list_nat @ Zs3 )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_b @ nil_nat @ nil_nat @ nil_nat )
           => ( ! [X4: b,Xs2: list_b,Y3: nat,Ys4: list_nat,Z2: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_b @ Xs2 )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_b @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_879_list__induct4,axiom,
    ! [Xs: list_b,Ys: list_nat,Zs3: list_nat,Ws: list_b,P: list_b > list_nat > list_nat > list_b > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs3 ) )
       => ( ( ( size_size_list_nat @ Zs3 )
            = ( size_size_list_b @ Ws ) )
         => ( ( P @ nil_b @ nil_nat @ nil_nat @ nil_b )
           => ( ! [X4: b,Xs2: list_b,Y3: nat,Ys4: list_nat,Z2: nat,Zs2: list_nat,W: b,Ws2: list_b] :
                  ( ( ( size_size_list_b @ Xs2 )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_b @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_b @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_b @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_880_splice_Ocases,axiom,
    ! [X3: produc1828647624359046049st_nat] :
      ( ! [Ys4: list_nat] :
          ( X3
         != ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
     => ~ ! [X4: nat,Xs2: list_nat,Ys4: list_nat] :
            ( X3
           != ( produc2694037385005941721st_nat @ ( cons_nat @ X4 @ Xs2 ) @ Ys4 ) ) ) ).

% splice.cases
thf(fact_881_shuffles_Ocases,axiom,
    ! [X3: produc1828647624359046049st_nat] :
      ( ! [Ys4: list_nat] :
          ( X3
         != ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
     => ( ! [Xs2: list_nat] :
            ( X3
           != ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) )
       => ~ ! [X4: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat] :
              ( X3
             != ( produc2694037385005941721st_nat @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) ) ) ).

% shuffles.cases
thf(fact_882_lexordp_OCons__eq,axiom,
    ! [X3: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ~ ( ord_less_nat @ X3 @ Y )
     => ( ~ ( ord_less_nat @ Y @ X3 )
       => ( ( ord_lexordp_nat @ Xs @ Ys )
         => ( ord_lexordp_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ) ) ).

% lexordp.Cons_eq
thf(fact_883_lexordp_OCons,axiom,
    ! [X3: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( ord_lexordp_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ).

% lexordp.Cons
thf(fact_884_length__removeAll__less__eq,axiom,
    ! [X3: b,Xs: list_b] : ( ord_less_eq_nat @ ( size_size_list_b @ ( removeAll_b @ X3 @ Xs ) ) @ ( size_size_list_b @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_885_lexord__Nil__right,axiom,
    ! [X3: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ nil_nat ) @ ( lexord_nat @ R2 ) ) ).

% lexord_Nil_right
thf(fact_886_Nil__notin__lex,axiom,
    ! [Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys ) @ ( lex_nat @ R2 ) ) ).

% Nil_notin_lex
thf(fact_887_Nil2__notin__lex,axiom,
    ! [Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( lex_nat @ R2 ) ) ).

% Nil2_notin_lex
thf(fact_888_Nil__lenlex__iff2,axiom,
    ! [Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ns @ nil_nat ) @ ( lenlex_nat @ R2 ) ) ).

% Nil_lenlex_iff2
thf(fact_889_Nil__less__Cons,axiom,
    ! [A: nat,X3: list_nat] : ( ord_less_list_nat @ nil_nat @ ( cons_nat @ A @ X3 ) ) ).

% Nil_less_Cons
thf(fact_890_less__list__code_I2_J,axiom,
    ! [X3: nat,Xs: list_nat] : ( ord_less_list_nat @ nil_nat @ ( cons_nat @ X3 @ Xs ) ) ).

% less_list_code(2)
thf(fact_891_swap__inj__on,axiom,
    ! [A4: set_Pr3469313283233365050_b_nat] :
      ( inj_on8101852888730240033list_b
      @ ( produc5976188580467221589list_b
        @ ^ [I3: list_b,J3: nat] : ( produc7903367361620597084list_b @ J3 @ I3 ) )
      @ A4 ) ).

% swap_inj_on
thf(fact_892_swap__inj__on,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( inj_on8969904277767023793at_nat
      @ ( produc2626176000494625587at_nat
        @ ^ [I3: nat,J3: nat] : ( product_Pair_nat_nat @ J3 @ I3 ) )
      @ A4 ) ).

% swap_inj_on
thf(fact_893_swap__inj__on,axiom,
    ! [A4: set_Pr8693737435421807431at_nat] :
      ( inj_on7797493862353638193at_nat
      @ ( produc6744312701629110395at_nat
        @ ^ [I3: product_prod_nat_nat,J3: product_prod_nat_nat] : ( produc6161850002892822231at_nat @ J3 @ I3 ) )
      @ A4 ) ).

% swap_inj_on
thf(fact_894_swap__inj__on,axiom,
    ! [A4: set_Pr7780494974915207058at_nat] :
      ( inj_on2333172051303441361al_nat
      @ ( produc3507466112089236613al_nat
        @ ^ [I3: produc3741383161447143261al_nat,J3: nat] : ( produc4490577844368043572al_nat @ J3 @ I3 ) )
      @ A4 ) ).

% swap_inj_on
thf(fact_895_swap__inj__on,axiom,
    ! [A4: set_Pr320017278500174781t_real] :
      ( inj_on7965676736892427057al_nat
      @ ( produc4028160848036898611al_nat
        @ ^ [I3: nat,J3: real] : ( produc3181502643871035669al_nat @ J3 @ I3 ) )
      @ A4 ) ).

% swap_inj_on
thf(fact_896_mlex__leq,axiom,
    ! [F: nat > nat,X3: nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( mlex_prod_nat @ F @ R ) ) ) ) ).

% mlex_leq
thf(fact_897_mlex__leq,axiom,
    ! [F: product_prod_nat_nat > nat,X3: product_prod_nat_nat,Y: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( mlex_p6366001652026297872at_nat @ F @ R ) ) ) ) ).

% mlex_leq
thf(fact_898_lenlex__length,axiom,
    ! [Ms: list_b,Ns: list_b,R2: set_Product_prod_b_b] :
      ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Ms @ Ns ) @ ( lenlex_b @ R2 ) )
     => ( ord_less_eq_nat @ ( size_size_list_b @ Ms ) @ ( size_size_list_b @ Ns ) ) ) ).

% lenlex_length
thf(fact_899_lex__prod__def,axiom,
    ( lex_pr8029265285556086080at_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr8693737435421807431at_nat] :
          ( collec3253044014973990424at_nat
          @ ( produc2869316064735525022_nat_o
            @ ( produc7480869177107369060_nat_o
              @ ^ [A6: nat,B5: product_prod_nat_nat] :
                  ( produc5864757623865647827_nat_o
                  @ ^ [A7: nat,B6: product_prod_nat_nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ A7 ) @ Ra )
                      | ( ( A6 = A7 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B5 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_900_lex__prod__def,axiom,
    ( lex_pr4669217892513370978at_nat
    = ( ^ [Ra: set_Pr8693737435421807431at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec2200730763347676620at_nat
          @ ( produc3600351907215576810_nat_o
            @ ( produc6198819664464039784_nat_o
              @ ^ [A6: product_prod_nat_nat,B5: nat] :
                  ( produc8758269395934548017_nat_o
                  @ ^ [A7: product_prod_nat_nat,B6: nat] :
                      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A6 @ A7 ) @ Ra )
                      | ( ( A6 = A7 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B5 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_901_lex__prod__def,axiom,
    ( lex_pr8801849515957261039at_nat
    = ( ^ [Ra: set_Pr8693737435421807431at_nat,Rb: set_Pr8693737435421807431at_nat] :
          ( collec7334067512558549330at_nat
          @ ( produc3603361296207914284_nat_o
            @ ( produc5077327844045112642_nat_o
              @ ^ [A6: product_prod_nat_nat,B5: product_prod_nat_nat] :
                  ( produc6590410687421337004_nat_o
                  @ ^ [A7: product_prod_nat_nat,B6: product_prod_nat_nat] :
                      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A6 @ A7 ) @ Ra )
                      | ( ( A6 = A7 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B5 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_902_lex__prod__def,axiom,
    ( lex_prod_nat_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [A6: nat,B5: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [A7: nat,B6: nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ A7 ) @ Ra )
                      | ( ( A6 = A7 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B5 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_903_mlex__eq,axiom,
    ( mlex_prod_nat
    = ( ^ [F2: nat > nat,R4: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X2: nat,Y4: nat] :
                ( ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y4 ) )
                | ( ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y4 ) )
                  & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ R4 ) ) ) ) ) ) ) ).

% mlex_eq
thf(fact_904_mlex__eq,axiom,
    ( mlex_p6366001652026297872at_nat
    = ( ^ [F2: product_prod_nat_nat > nat,R4: set_Pr8693737435421807431at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ^ [X2: product_prod_nat_nat,Y4: product_prod_nat_nat] :
                ( ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y4 ) )
                | ( ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y4 ) )
                  & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y4 ) @ R4 ) ) ) ) ) ) ) ).

% mlex_eq
thf(fact_905_longest__common__prefix_Ocases,axiom,
    ! [X3: produc1828647624359046049st_nat] :
      ( ! [X4: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat] :
          ( X3
         != ( produc2694037385005941721st_nat @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
     => ( ! [Uv: list_nat] :
            ( X3
           != ( produc2694037385005941721st_nat @ nil_nat @ Uv ) )
       => ~ ! [Uu2: list_nat] :
              ( X3
             != ( produc2694037385005941721st_nat @ Uu2 @ nil_nat ) ) ) ) ).

% longest_common_prefix.cases
thf(fact_906_map__rec,axiom,
    ( map_nat_nat
    = ( ^ [F2: nat > nat] :
          ( rec_li7516600145284979816at_nat @ nil_nat
          @ ^ [X2: nat,Uu: list_nat] : ( cons_nat @ ( F2 @ X2 ) ) ) ) ) ).

% map_rec
thf(fact_907_pair__lessI2,axiom,
    ! [A: nat,B: nat,S3: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ S3 @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_908_pair__less__iff1,axiom,
    ! [X3: nat,Y: nat,Z3: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( product_Pair_nat_nat @ X3 @ Z3 ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y @ Z3 ) ) ).

% pair_less_iff1
thf(fact_909_total__pair__less,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : ( total_3592101749530773125at_nat @ A4 @ fun_pair_less ) ).

% total_pair_less
thf(fact_910_Cons__le__Cons,axiom,
    ! [A: nat,X3: list_nat,B: nat,Y: list_nat] :
      ( ( ord_less_eq_list_nat @ ( cons_nat @ A @ X3 ) @ ( cons_nat @ B @ Y ) )
      = ( ( ord_less_nat @ A @ B )
        | ( ( A = B )
          & ( ord_less_eq_list_nat @ X3 @ Y ) ) ) ) ).

% Cons_le_Cons
thf(fact_911_pred__subset__eq2,axiom,
    ! [R: set_Pr1349601357184307552list_b,S: set_Pr1349601357184307552list_b] :
      ( ( ord_le5501773618496797319st_b_o
        @ ^ [X2: nat,Y4: list_b] : ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ X2 @ Y4 ) @ R )
        @ ^ [X2: nat,Y4: list_b] : ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ X2 @ Y4 ) @ S ) )
      = ( ord_le972014486225453504list_b @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_912_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X2: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ R )
        @ ^ [X2: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ S ) )
      = ( ord_le3146513528884898305at_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_913_pred__subset__eq2,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( ord_le5604493270027003598_nat_o
        @ ^ [X2: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y4 ) @ R )
        @ ^ [X2: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y4 ) @ S ) )
      = ( ord_le3000389064537975527at_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_914_pred__subset__eq2,axiom,
    ! [R: set_Pr836851361167288120al_nat,S: set_Pr836851361167288120al_nat] :
      ( ( ord_le1366284440460845743_nat_o
        @ ^ [X2: nat,Y4: produc3741383161447143261al_nat] : ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X2 @ Y4 ) @ R )
        @ ^ [X2: nat,Y4: produc3741383161447143261al_nat] : ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X2 @ Y4 ) @ S ) )
      = ( ord_le4407014575791007640al_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_915_pred__subset__eq2,axiom,
    ! [R: set_Pr3510011417693777981al_nat,S: set_Pr3510011417693777981al_nat] :
      ( ( ord_le1557071757414830890_nat_o
        @ ^ [X2: real,Y4: nat] : ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X2 @ Y4 ) @ R )
        @ ^ [X2: real,Y4: nat] : ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X2 @ Y4 ) @ S ) )
      = ( ord_le8992644243367634909al_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_916_subrelI,axiom,
    ! [R2: set_Pr1349601357184307552list_b,S3: set_Pr1349601357184307552list_b] :
      ( ! [X4: nat,Y3: list_b] :
          ( ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ X4 @ Y3 ) @ R2 )
         => ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ X4 @ Y3 ) @ S3 ) )
     => ( ord_le972014486225453504list_b @ R2 @ S3 ) ) ).

% subrelI
thf(fact_917_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ S3 ) )
     => ( ord_le3146513528884898305at_nat @ R2 @ S3 ) ) ).

% subrelI
thf(fact_918_subrelI,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat] :
      ( ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y3 ) @ R2 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y3 ) @ S3 ) )
     => ( ord_le3000389064537975527at_nat @ R2 @ S3 ) ) ).

% subrelI
thf(fact_919_subrelI,axiom,
    ! [R2: set_Pr836851361167288120al_nat,S3: set_Pr836851361167288120al_nat] :
      ( ! [X4: nat,Y3: produc3741383161447143261al_nat] :
          ( ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X4 @ Y3 ) @ R2 )
         => ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X4 @ Y3 ) @ S3 ) )
     => ( ord_le4407014575791007640al_nat @ R2 @ S3 ) ) ).

% subrelI
thf(fact_920_subrelI,axiom,
    ! [R2: set_Pr3510011417693777981al_nat,S3: set_Pr3510011417693777981al_nat] :
      ( ! [X4: real,Y3: nat] :
          ( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X4 @ Y3 ) @ R2 )
         => ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X4 @ Y3 ) @ S3 ) )
     => ( ord_le8992644243367634909al_nat @ R2 @ S3 ) ) ).

% subrelI
thf(fact_921_subset__UNIV,axiom,
    ! [A4: set_list_b] : ( ord_le8932221534207217157list_b @ A4 @ top_top_set_list_b ) ).

% subset_UNIV
thf(fact_922_subset__UNIV,axiom,
    ! [A4: set_nat] : ( ord_less_eq_set_nat @ A4 @ top_top_set_nat ) ).

% subset_UNIV
thf(fact_923_subset__UNIV,axiom,
    ! [A4: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ A4 @ top_to8454300755403694231at_nat ) ).

% subset_UNIV
thf(fact_924_subset__UNIV,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ A4 @ top_to4669805908274784177at_nat ) ).

% subset_UNIV
thf(fact_925_subset__inj__on,axiom,
    ! [F: list_b > produc7811952446676219690list_b,B4: set_list_b,A4: set_list_b] :
      ( ( inj_on8733544287791916926list_b @ F @ B4 )
     => ( ( ord_le8932221534207217157list_b @ A4 @ B4 )
       => ( inj_on8733544287791916926list_b @ F @ A4 ) ) ) ).

% subset_inj_on
thf(fact_926_subset__inj__on,axiom,
    ! [F: nat > nat,B4: set_nat,A4: set_nat] :
      ( ( inj_on_nat_nat @ F @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( inj_on_nat_nat @ F @ A4 ) ) ) ).

% subset_inj_on
thf(fact_927_subset__inj__on,axiom,
    ! [F: real > produc3741383161447143261al_nat,B4: set_real,A4: set_real] :
      ( ( inj_on8027545645978435326al_nat @ F @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( inj_on8027545645978435326al_nat @ F @ A4 ) ) ) ).

% subset_inj_on
thf(fact_928_subset__inj__on,axiom,
    ! [F: nat > produc9155354469731034754al_nat,B4: set_nat,A4: set_nat] :
      ( ( inj_on7152781252210839493al_nat @ F @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( inj_on7152781252210839493al_nat @ F @ A4 ) ) ) ).

% subset_inj_on
thf(fact_929_subset__inj__on,axiom,
    ! [F: product_prod_nat_nat > produc859450856879609959at_nat,B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( inj_on3865918002419084311at_nat @ F @ B4 )
     => ( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
       => ( inj_on3865918002419084311at_nat @ F @ A4 ) ) ) ).

% subset_inj_on
thf(fact_930_subset__inj__on,axiom,
    ! [F: nat > product_prod_nat_nat,B4: set_nat,A4: set_nat] :
      ( ( inj_on5538052773655684606at_nat @ F @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( inj_on5538052773655684606at_nat @ F @ A4 ) ) ) ).

% subset_inj_on
thf(fact_931_subset__inj__on,axiom,
    ! [F: nat > produc7811952446676219690list_b,B4: set_nat,A4: set_nat] :
      ( ( inj_on8555073082499328493list_b @ F @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( inj_on8555073082499328493list_b @ F @ A4 ) ) ) ).

% subset_inj_on
thf(fact_932_inj__on__subset,axiom,
    ! [F: list_b > produc7811952446676219690list_b,A4: set_list_b,B4: set_list_b] :
      ( ( inj_on8733544287791916926list_b @ F @ A4 )
     => ( ( ord_le8932221534207217157list_b @ B4 @ A4 )
       => ( inj_on8733544287791916926list_b @ F @ B4 ) ) ) ).

% inj_on_subset
thf(fact_933_inj__on__subset,axiom,
    ! [F: nat > nat,A4: set_nat,B4: set_nat] :
      ( ( inj_on_nat_nat @ F @ A4 )
     => ( ( ord_less_eq_set_nat @ B4 @ A4 )
       => ( inj_on_nat_nat @ F @ B4 ) ) ) ).

% inj_on_subset
thf(fact_934_inj__on__subset,axiom,
    ! [F: real > produc3741383161447143261al_nat,A4: set_real,B4: set_real] :
      ( ( inj_on8027545645978435326al_nat @ F @ A4 )
     => ( ( ord_less_eq_set_real @ B4 @ A4 )
       => ( inj_on8027545645978435326al_nat @ F @ B4 ) ) ) ).

% inj_on_subset
thf(fact_935_inj__on__subset,axiom,
    ! [F: nat > produc9155354469731034754al_nat,A4: set_nat,B4: set_nat] :
      ( ( inj_on7152781252210839493al_nat @ F @ A4 )
     => ( ( ord_less_eq_set_nat @ B4 @ A4 )
       => ( inj_on7152781252210839493al_nat @ F @ B4 ) ) ) ).

% inj_on_subset
thf(fact_936_inj__on__subset,axiom,
    ! [F: product_prod_nat_nat > produc859450856879609959at_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( inj_on3865918002419084311at_nat @ F @ A4 )
     => ( ( ord_le3146513528884898305at_nat @ B4 @ A4 )
       => ( inj_on3865918002419084311at_nat @ F @ B4 ) ) ) ).

% inj_on_subset
thf(fact_937_inj__on__subset,axiom,
    ! [F: nat > product_prod_nat_nat,A4: set_nat,B4: set_nat] :
      ( ( inj_on5538052773655684606at_nat @ F @ A4 )
     => ( ( ord_less_eq_set_nat @ B4 @ A4 )
       => ( inj_on5538052773655684606at_nat @ F @ B4 ) ) ) ).

% inj_on_subset
thf(fact_938_inj__on__subset,axiom,
    ! [F: nat > produc7811952446676219690list_b,A4: set_nat,B4: set_nat] :
      ( ( inj_on8555073082499328493list_b @ F @ A4 )
     => ( ( ord_less_eq_set_nat @ B4 @ A4 )
       => ( inj_on8555073082499328493list_b @ F @ B4 ) ) ) ).

% inj_on_subset
thf(fact_939_total__on__subset,axiom,
    ! [A4: set_nat,R2: set_Pr1261947904930325089at_nat,B4: set_nat] :
      ( ( total_on_nat @ A4 @ R2 )
     => ( ( ord_less_eq_set_nat @ B4 @ A4 )
       => ( total_on_nat @ B4 @ R2 ) ) ) ).

% total_on_subset
thf(fact_940_total__on__subset,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( total_3592101749530773125at_nat @ A4 @ R2 )
     => ( ( ord_le3146513528884898305at_nat @ B4 @ A4 )
       => ( total_3592101749530773125at_nat @ B4 @ R2 ) ) ) ).

% total_on_subset
thf(fact_941_asym__on__subset,axiom,
    ! [A4: set_nat,R2: set_Pr1261947904930325089at_nat,B4: set_nat] :
      ( ( asym_on_nat @ A4 @ R2 )
     => ( ( ord_less_eq_set_nat @ B4 @ A4 )
       => ( asym_on_nat @ B4 @ R2 ) ) ) ).

% asym_on_subset
thf(fact_942_less__eq__list__code_I3_J,axiom,
    ! [X3: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( ord_less_eq_list_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) )
      = ( ( ord_less_nat @ X3 @ Y )
        | ( ( X3 = Y )
          & ( ord_less_eq_list_nat @ Xs @ Ys ) ) ) ) ).

% less_eq_list_code(3)
thf(fact_943_less__eq__list__code_I1_J,axiom,
    ! [X3: nat,Xs: list_nat] :
      ~ ( ord_less_eq_list_nat @ ( cons_nat @ X3 @ Xs ) @ nil_nat ) ).

% less_eq_list_code(1)
thf(fact_944_pair__less__def,axiom,
    ( fun_pair_less
    = ( lex_prod_nat_nat @ less_than @ less_than ) ) ).

% pair_less_def
thf(fact_945_pair__lessI1,axiom,
    ! [A: nat,B: nat,S3: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_946_pair__leqI2,axiom,
    ! [A: nat,B: nat,S3: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ S3 @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_947_pair__leqI1,axiom,
    ! [A: nat,B: nat,S3: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_948_sublists_Osimps_I1_J,axiom,
    ( ( sublists_nat @ nil_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% sublists.simps(1)
thf(fact_949_subsetI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ A4 )
         => ( member8440522571783428010at_nat @ X4 @ B4 ) )
     => ( ord_le3146513528884898305at_nat @ A4 @ B4 ) ) ).

% subsetI
thf(fact_950_subsetI,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
      ( ! [X4: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X4 @ A4 )
         => ( member8206827879206165904at_nat @ X4 @ B4 ) )
     => ( ord_le3000389064537975527at_nat @ A4 @ B4 ) ) ).

% subsetI
thf(fact_951_subsetI,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A4 )
         => ( member_nat @ X4 @ B4 ) )
     => ( ord_less_eq_set_nat @ A4 @ B4 ) ) ).

% subsetI
thf(fact_952_subsetI,axiom,
    ! [A4: set_list_b,B4: set_list_b] :
      ( ! [X4: list_b] :
          ( ( member_list_b @ X4 @ A4 )
         => ( member_list_b @ X4 @ B4 ) )
     => ( ord_le8932221534207217157list_b @ A4 @ B4 ) ) ).

% subsetI
thf(fact_953_pred__subset__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_le704812498762024988_nat_o
        @ ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ R )
        @ ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ S ) )
      = ( ord_le3146513528884898305at_nat @ R @ S ) ) ).

% pred_subset_eq
thf(fact_954_pred__subset__eq,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( ord_le7858099551454983350_nat_o
        @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ R )
        @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ S ) )
      = ( ord_le3000389064537975527at_nat @ R @ S ) ) ).

% pred_subset_eq
thf(fact_955_pred__subset__eq,axiom,
    ! [R: set_nat,S: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X2: nat] : ( member_nat @ X2 @ R )
        @ ^ [X2: nat] : ( member_nat @ X2 @ S ) )
      = ( ord_less_eq_set_nat @ R @ S ) ) ).

% pred_subset_eq
thf(fact_956_pred__subset__eq,axiom,
    ! [R: set_list_b,S: set_list_b] :
      ( ( ord_less_eq_list_b_o
        @ ^ [X2: list_b] : ( member_list_b @ X2 @ R )
        @ ^ [X2: list_b] : ( member_list_b @ X2 @ S ) )
      = ( ord_le8932221534207217157list_b @ R @ S ) ) ).

% pred_subset_eq
thf(fact_957_subset__CollectI,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,Q2: product_prod_nat_nat > $o,P: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ B4 @ A4 )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ B4 )
           => ( ( Q2 @ X4 )
             => ( P @ X4 ) ) )
       => ( ord_le3146513528884898305at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X2 @ B4 )
                & ( Q2 @ X2 ) ) )
          @ ( collec3392354462482085612at_nat
            @ ^ [X2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X2 @ A4 )
                & ( P @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_958_subset__CollectI,axiom,
    ! [B4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat,Q2: produc859450856879609959at_nat > $o,P: produc859450856879609959at_nat > $o] :
      ( ( ord_le3000389064537975527at_nat @ B4 @ A4 )
     => ( ! [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ B4 )
           => ( ( Q2 @ X4 )
             => ( P @ X4 ) ) )
       => ( ord_le3000389064537975527at_nat
          @ ( collec7088162979684241874at_nat
            @ ^ [X2: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X2 @ B4 )
                & ( Q2 @ X2 ) ) )
          @ ( collec7088162979684241874at_nat
            @ ^ [X2: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X2 @ A4 )
                & ( P @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_959_subset__CollectI,axiom,
    ! [B4: set_nat,A4: set_nat,Q2: nat > $o,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ B4 )
           => ( ( Q2 @ X4 )
             => ( P @ X4 ) ) )
       => ( ord_less_eq_set_nat
          @ ( collect_nat
            @ ^ [X2: nat] :
                ( ( member_nat @ X2 @ B4 )
                & ( Q2 @ X2 ) ) )
          @ ( collect_nat
            @ ^ [X2: nat] :
                ( ( member_nat @ X2 @ A4 )
                & ( P @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_960_subset__CollectI,axiom,
    ! [B4: set_list_b,A4: set_list_b,Q2: list_b > $o,P: list_b > $o] :
      ( ( ord_le8932221534207217157list_b @ B4 @ A4 )
     => ( ! [X4: list_b] :
            ( ( member_list_b @ X4 @ B4 )
           => ( ( Q2 @ X4 )
             => ( P @ X4 ) ) )
       => ( ord_le8932221534207217157list_b
          @ ( collect_list_b
            @ ^ [X2: list_b] :
                ( ( member_list_b @ X2 @ B4 )
                & ( Q2 @ X2 ) ) )
          @ ( collect_list_b
            @ ^ [X2: list_b] :
                ( ( member_list_b @ X2 @ A4 )
                & ( P @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_961_subset__Collect__iff,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ B4 @ A4 )
     => ( ( ord_le3146513528884898305at_nat @ B4
          @ ( collec3392354462482085612at_nat
            @ ^ [X2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X2 @ A4 )
                & ( P @ X2 ) ) ) )
        = ( ! [X2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X2 @ B4 )
             => ( P @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_962_subset__Collect__iff,axiom,
    ! [B4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( ord_le3000389064537975527at_nat @ B4 @ A4 )
     => ( ( ord_le3000389064537975527at_nat @ B4
          @ ( collec7088162979684241874at_nat
            @ ^ [X2: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X2 @ A4 )
                & ( P @ X2 ) ) ) )
        = ( ! [X2: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X2 @ B4 )
             => ( P @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_963_subset__Collect__iff,axiom,
    ! [B4: set_nat,A4: set_nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( ord_less_eq_set_nat @ B4
          @ ( collect_nat
            @ ^ [X2: nat] :
                ( ( member_nat @ X2 @ A4 )
                & ( P @ X2 ) ) ) )
        = ( ! [X2: nat] :
              ( ( member_nat @ X2 @ B4 )
             => ( P @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_964_subset__Collect__iff,axiom,
    ! [B4: set_list_b,A4: set_list_b,P: list_b > $o] :
      ( ( ord_le8932221534207217157list_b @ B4 @ A4 )
     => ( ( ord_le8932221534207217157list_b @ B4
          @ ( collect_list_b
            @ ^ [X2: list_b] :
                ( ( member_list_b @ X2 @ A4 )
                & ( P @ X2 ) ) ) )
        = ( ! [X2: list_b] :
              ( ( member_list_b @ X2 @ B4 )
             => ( P @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_965_less__eq__set__def,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
          ( ord_le704812498762024988_nat_o
          @ ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ A5 )
          @ ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ B7 ) ) ) ) ).

% less_eq_set_def
thf(fact_966_less__eq__set__def,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A5: set_Pr8693737435421807431at_nat,B7: set_Pr8693737435421807431at_nat] :
          ( ord_le7858099551454983350_nat_o
          @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ A5 )
          @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ B7 ) ) ) ) ).

% less_eq_set_def
thf(fact_967_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B7: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X2: nat] : ( member_nat @ X2 @ A5 )
          @ ^ [X2: nat] : ( member_nat @ X2 @ B7 ) ) ) ) ).

% less_eq_set_def
thf(fact_968_less__eq__set__def,axiom,
    ( ord_le8932221534207217157list_b
    = ( ^ [A5: set_list_b,B7: set_list_b] :
          ( ord_less_eq_list_b_o
          @ ^ [X2: list_b] : ( member_list_b @ X2 @ A5 )
          @ ^ [X2: list_b] : ( member_list_b @ X2 @ B7 ) ) ) ) ).

% less_eq_set_def
thf(fact_969_in__mono,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
     => ( ( member8440522571783428010at_nat @ X3 @ A4 )
       => ( member8440522571783428010at_nat @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_970_in__mono,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat,X3: produc859450856879609959at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A4 @ B4 )
     => ( ( member8206827879206165904at_nat @ X3 @ A4 )
       => ( member8206827879206165904at_nat @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_971_in__mono,axiom,
    ! [A4: set_nat,B4: set_nat,X3: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( member_nat @ X3 @ A4 )
       => ( member_nat @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_972_in__mono,axiom,
    ! [A4: set_list_b,B4: set_list_b,X3: list_b] :
      ( ( ord_le8932221534207217157list_b @ A4 @ B4 )
     => ( ( member_list_b @ X3 @ A4 )
       => ( member_list_b @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_973_subsetD,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C2: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
     => ( ( member8440522571783428010at_nat @ C2 @ A4 )
       => ( member8440522571783428010at_nat @ C2 @ B4 ) ) ) ).

% subsetD
thf(fact_974_subsetD,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat,C2: produc859450856879609959at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A4 @ B4 )
     => ( ( member8206827879206165904at_nat @ C2 @ A4 )
       => ( member8206827879206165904at_nat @ C2 @ B4 ) ) ) ).

% subsetD
thf(fact_975_subsetD,axiom,
    ! [A4: set_nat,B4: set_nat,C2: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( member_nat @ C2 @ A4 )
       => ( member_nat @ C2 @ B4 ) ) ) ).

% subsetD
thf(fact_976_subsetD,axiom,
    ! [A4: set_list_b,B4: set_list_b,C2: list_b] :
      ( ( ord_le8932221534207217157list_b @ A4 @ B4 )
     => ( ( member_list_b @ C2 @ A4 )
       => ( member_list_b @ C2 @ B4 ) ) ) ).

% subsetD
thf(fact_977_subset__eq,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
        ! [X2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X2 @ A5 )
         => ( member8440522571783428010at_nat @ X2 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_978_subset__eq,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A5: set_Pr8693737435421807431at_nat,B7: set_Pr8693737435421807431at_nat] :
        ! [X2: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X2 @ A5 )
         => ( member8206827879206165904at_nat @ X2 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_979_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B7: set_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A5 )
         => ( member_nat @ X2 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_980_subset__eq,axiom,
    ( ord_le8932221534207217157list_b
    = ( ^ [A5: set_list_b,B7: set_list_b] :
        ! [X2: list_b] :
          ( ( member_list_b @ X2 @ A5 )
         => ( member_list_b @ X2 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_981_subset__iff,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
        ! [T2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ T2 @ A5 )
         => ( member8440522571783428010at_nat @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_982_subset__iff,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A5: set_Pr8693737435421807431at_nat,B7: set_Pr8693737435421807431at_nat] :
        ! [T2: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ T2 @ A5 )
         => ( member8206827879206165904at_nat @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_983_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B7: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A5 )
         => ( member_nat @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_984_subset__iff,axiom,
    ( ord_le8932221534207217157list_b
    = ( ^ [A5: set_list_b,B7: set_list_b] :
        ! [T2: list_b] :
          ( ( member_list_b @ T2 @ A5 )
         => ( member_list_b @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_985_Collect__subset,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ord_le3146513528884898305at_nat
      @ ( collec3392354462482085612at_nat
        @ ^ [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ A4 )
            & ( P @ X2 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_986_Collect__subset,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ord_le3000389064537975527at_nat
      @ ( collec7088162979684241874at_nat
        @ ^ [X2: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X2 @ A4 )
            & ( P @ X2 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_987_Collect__subset,axiom,
    ! [A4: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X2: nat] :
            ( ( member_nat @ X2 @ A4 )
            & ( P @ X2 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_988_Collect__subset,axiom,
    ! [A4: set_list_b,P: list_b > $o] :
      ( ord_le8932221534207217157list_b
      @ ( collect_list_b
        @ ^ [X2: list_b] :
            ( ( member_list_b @ X2 @ A4 )
            & ( P @ X2 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_989_product__lists_Osimps_I1_J,axiom,
    ( ( product_lists_nat @ nil_list_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% product_lists.simps(1)
thf(fact_990_subseqs_Osimps_I1_J,axiom,
    ( ( subseqs_nat @ nil_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% subseqs.simps(1)
thf(fact_991_less__set__def,axiom,
    ( ord_le7866589430770878221at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
          ( ord_le549003669493604880_nat_o
          @ ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ A5 )
          @ ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ B7 ) ) ) ) ).

% less_set_def
thf(fact_992_less__set__def,axiom,
    ( ord_le6428140832669894131at_nat
    = ( ^ [A5: set_Pr8693737435421807431at_nat,B7: set_Pr8693737435421807431at_nat] :
          ( ord_le7432393201274343594_nat_o
          @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ A5 )
          @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ B7 ) ) ) ) ).

% less_set_def
thf(fact_993_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A5: set_nat,B7: set_nat] :
          ( ord_less_nat_o
          @ ^ [X2: nat] : ( member_nat @ X2 @ A5 )
          @ ^ [X2: nat] : ( member_nat @ X2 @ B7 ) ) ) ) ).

% less_set_def
thf(fact_994_less__set__def,axiom,
    ( ord_less_set_list_b
    = ( ^ [A5: set_list_b,B7: set_list_b] :
          ( ord_less_list_b_o
          @ ^ [X2: list_b] : ( member_list_b @ X2 @ A5 )
          @ ^ [X2: list_b] : ( member_list_b @ X2 @ B7 ) ) ) ) ).

% less_set_def
thf(fact_995_psubsetD,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C2: product_prod_nat_nat] :
      ( ( ord_le7866589430770878221at_nat @ A4 @ B4 )
     => ( ( member8440522571783428010at_nat @ C2 @ A4 )
       => ( member8440522571783428010at_nat @ C2 @ B4 ) ) ) ).

% psubsetD
thf(fact_996_psubsetD,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat,C2: produc859450856879609959at_nat] :
      ( ( ord_le6428140832669894131at_nat @ A4 @ B4 )
     => ( ( member8206827879206165904at_nat @ C2 @ A4 )
       => ( member8206827879206165904at_nat @ C2 @ B4 ) ) ) ).

% psubsetD
thf(fact_997_psubsetD,axiom,
    ! [A4: set_nat,B4: set_nat,C2: nat] :
      ( ( ord_less_set_nat @ A4 @ B4 )
     => ( ( member_nat @ C2 @ A4 )
       => ( member_nat @ C2 @ B4 ) ) ) ).

% psubsetD
thf(fact_998_psubsetD,axiom,
    ! [A4: set_list_b,B4: set_list_b,C2: list_b] :
      ( ( ord_less_set_list_b @ A4 @ B4 )
     => ( ( member_list_b @ C2 @ A4 )
       => ( member_list_b @ C2 @ B4 ) ) ) ).

% psubsetD
thf(fact_999_insert__Nil,axiom,
    ! [X3: nat] :
      ( ( insert_nat @ X3 @ nil_nat )
      = ( cons_nat @ X3 @ nil_nat ) ) ).

% insert_Nil
thf(fact_1000_top_Oordering__top__axioms,axiom,
    orderi2506971327722225558list_b @ ord_le8932221534207217157list_b @ ord_less_set_list_b @ top_top_set_list_b ).

% top.ordering_top_axioms
thf(fact_1001_top_Oordering__top__axioms,axiom,
    ordering_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat @ top_top_set_nat ).

% top.ordering_top_axioms
thf(fact_1002_top_Oordering__top__axioms,axiom,
    orderi92352388023179000at_nat @ ord_le3000389064537975527at_nat @ ord_le6428140832669894131at_nat @ top_to8454300755403694231at_nat ).

% top.ordering_top_axioms
thf(fact_1003_top_Oordering__top__axioms,axiom,
    orderi2172309028950807442at_nat @ ord_le3146513528884898305at_nat @ ord_le7866589430770878221at_nat @ top_to4669805908274784177at_nat ).

% top.ordering_top_axioms
thf(fact_1004_listrel_Osimps,axiom,
    ! [A12: list_nat,A23: list_list_b,R2: set_Pr1349601357184307552list_b] :
      ( ( member4916451157736907575list_b @ ( produc2084500703101687378list_b @ A12 @ A23 ) @ ( listrel_nat_list_b @ R2 ) )
      = ( ( ( A12 = nil_nat )
          & ( A23 = nil_list_b ) )
        | ? [X2: nat,Y4: list_b,Xs3: list_nat,Ys3: list_list_b] :
            ( ( A12
              = ( cons_nat @ X2 @ Xs3 ) )
            & ( A23
              = ( cons_list_b @ Y4 @ Ys3 ) )
            & ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ X2 @ Y4 ) @ R2 )
            & ( member4916451157736907575list_b @ ( produc2084500703101687378list_b @ Xs3 @ Ys3 ) @ ( listrel_nat_list_b @ R2 ) ) ) ) ) ).

% listrel.simps
thf(fact_1005_listrel_Osimps,axiom,
    ! [A12: list_nat,A23: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A12 @ A23 ) @ ( listrel_nat_nat @ R2 ) )
      = ( ( ( A12 = nil_nat )
          & ( A23 = nil_nat ) )
        | ? [X2: nat,Y4: nat,Xs3: list_nat,Ys3: list_nat] :
            ( ( A12
              = ( cons_nat @ X2 @ Xs3 ) )
            & ( A23
              = ( cons_nat @ Y4 @ Ys3 ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ R2 )
            & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys3 ) @ ( listrel_nat_nat @ R2 ) ) ) ) ) ).

% listrel.simps
thf(fact_1006_listrel_Osimps,axiom,
    ! [A12: list_P6011104703257516679at_nat,A23: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ A12 @ A23 ) @ ( listre818007680106770737at_nat @ R2 ) )
      = ( ( ( A12 = nil_Pr5478986624290739719at_nat )
          & ( A23 = nil_Pr5478986624290739719at_nat ) )
        | ? [X2: product_prod_nat_nat,Y4: product_prod_nat_nat,Xs3: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat] :
            ( ( A12
              = ( cons_P6512896166579812791at_nat @ X2 @ Xs3 ) )
            & ( A23
              = ( cons_P6512896166579812791at_nat @ Y4 @ Ys3 ) )
            & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y4 ) @ R2 )
            & ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs3 @ Ys3 ) @ ( listre818007680106770737at_nat @ R2 ) ) ) ) ) ).

% listrel.simps
thf(fact_1007_listrel_Osimps,axiom,
    ! [A12: list_nat,A23: list_P6834414599653733731al_nat,R2: set_Pr836851361167288120al_nat] :
      ( ( member1290970816896746895al_nat @ ( produc7399767566154494378al_nat @ A12 @ A23 ) @ ( listre1017866332378306906al_nat @ R2 ) )
      = ( ( ( A12 = nil_nat )
          & ( A23 = nil_Pr1917482104270272867al_nat ) )
        | ? [X2: nat,Y4: produc3741383161447143261al_nat,Xs3: list_nat,Ys3: list_P6834414599653733731al_nat] :
            ( ( A12
              = ( cons_nat @ X2 @ Xs3 ) )
            & ( A23
              = ( cons_P500833500243608851al_nat @ Y4 @ Ys3 ) )
            & ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X2 @ Y4 ) @ R2 )
            & ( member1290970816896746895al_nat @ ( produc7399767566154494378al_nat @ Xs3 @ Ys3 ) @ ( listre1017866332378306906al_nat @ R2 ) ) ) ) ) ).

% listrel.simps
thf(fact_1008_listrel_Osimps,axiom,
    ! [A12: list_real,A23: list_nat,R2: set_Pr3510011417693777981al_nat] :
      ( ( member7421688103010841766st_nat @ ( produc4048774278581505077st_nat @ A12 @ A23 ) @ ( listrel_real_nat @ R2 ) )
      = ( ( ( A12 = nil_real )
          & ( A23 = nil_nat ) )
        | ? [X2: real,Y4: nat,Xs3: list_real,Ys3: list_nat] :
            ( ( A12
              = ( cons_real @ X2 @ Xs3 ) )
            & ( A23
              = ( cons_nat @ Y4 @ Ys3 ) )
            & ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X2 @ Y4 ) @ R2 )
            & ( member7421688103010841766st_nat @ ( produc4048774278581505077st_nat @ Xs3 @ Ys3 ) @ ( listrel_real_nat @ R2 ) ) ) ) ) ).

% listrel.simps
thf(fact_1009_ordering__top_Oextremum,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ( Less_eq @ A @ Top ) ) ).

% ordering_top.extremum
thf(fact_1010_ordering__top_Oextremum__strict,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ~ ( Less @ Top @ A ) ) ).

% ordering_top.extremum_strict
thf(fact_1011_ordering__top_Oextremum__unique,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ( ( Less_eq @ Top @ A )
        = ( A = Top ) ) ) ).

% ordering_top.extremum_unique
thf(fact_1012_ordering__top_Onot__eq__extremum,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ( ( A != Top )
        = ( Less @ A @ Top ) ) ) ).

% ordering_top.not_eq_extremum
thf(fact_1013_ordering__top_Oextremum__uniqueI,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ( ( Less_eq @ Top @ A )
       => ( A = Top ) ) ) ).

% ordering_top.extremum_uniqueI
thf(fact_1014_listrel__Nil2,axiom,
    ! [Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel_nat_nat @ R2 ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil2
thf(fact_1015_listrel__Nil1,axiom,
    ! [Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel_nat_nat @ R2 ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil1
thf(fact_1016_listrel_ONil,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) @ ( listrel_nat_nat @ R2 ) ) ).

% listrel.Nil
thf(fact_1017_listrel__eq__len,axiom,
    ! [Xs: list_b,Ys: list_b,R2: set_Product_prod_b_b] :
      ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Xs @ Ys ) @ ( listrel_b_b @ R2 ) )
     => ( ( size_size_list_b @ Xs )
        = ( size_size_list_b @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_1018_listrel_OCons,axiom,
    ! [X3: nat,Y: list_b,R2: set_Pr1349601357184307552list_b,Xs: list_nat,Ys: list_list_b] :
      ( ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ X3 @ Y ) @ R2 )
     => ( ( member4916451157736907575list_b @ ( produc2084500703101687378list_b @ Xs @ Ys ) @ ( listrel_nat_list_b @ R2 ) )
       => ( member4916451157736907575list_b @ ( produc2084500703101687378list_b @ ( cons_nat @ X3 @ Xs ) @ ( cons_list_b @ Y @ Ys ) ) @ ( listrel_nat_list_b @ R2 ) ) ) ) ).

% listrel.Cons
thf(fact_1019_listrel_OCons,axiom,
    ! [X3: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R2 )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R2 ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat @ R2 ) ) ) ) ).

% listrel.Cons
thf(fact_1020_listrel_OCons,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R2 )
     => ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( listre818007680106770737at_nat @ R2 ) )
       => ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys ) ) @ ( listre818007680106770737at_nat @ R2 ) ) ) ) ).

% listrel.Cons
thf(fact_1021_listrel_OCons,axiom,
    ! [X3: nat,Y: produc3741383161447143261al_nat,R2: set_Pr836851361167288120al_nat,Xs: list_nat,Ys: list_P6834414599653733731al_nat] :
      ( ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X3 @ Y ) @ R2 )
     => ( ( member1290970816896746895al_nat @ ( produc7399767566154494378al_nat @ Xs @ Ys ) @ ( listre1017866332378306906al_nat @ R2 ) )
       => ( member1290970816896746895al_nat @ ( produc7399767566154494378al_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_P500833500243608851al_nat @ Y @ Ys ) ) @ ( listre1017866332378306906al_nat @ R2 ) ) ) ) ).

% listrel.Cons
thf(fact_1022_listrel_OCons,axiom,
    ! [X3: real,Y: nat,R2: set_Pr3510011417693777981al_nat,Xs: list_real,Ys: list_nat] :
      ( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X3 @ Y ) @ R2 )
     => ( ( member7421688103010841766st_nat @ ( produc4048774278581505077st_nat @ Xs @ Ys ) @ ( listrel_real_nat @ R2 ) )
       => ( member7421688103010841766st_nat @ ( produc4048774278581505077st_nat @ ( cons_real @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_real_nat @ R2 ) ) ) ) ).

% listrel.Cons
thf(fact_1023_listrel__Cons1,axiom,
    ! [Y: nat,Ys: list_nat,Xs: list_list_b,R2: set_Pr1349601357184307552list_b] :
      ( ( member4916451157736907575list_b @ ( produc2084500703101687378list_b @ ( cons_nat @ Y @ Ys ) @ Xs ) @ ( listrel_nat_list_b @ R2 ) )
     => ~ ! [Y3: list_b,Ys4: list_list_b] :
            ( ( Xs
              = ( cons_list_b @ Y3 @ Ys4 ) )
           => ( ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ Y @ Y3 ) @ R2 )
             => ~ ( member4916451157736907575list_b @ ( produc2084500703101687378list_b @ Ys @ Ys4 ) @ ( listrel_nat_list_b @ R2 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_1024_listrel__Cons1,axiom,
    ! [Y: nat,Ys: list_nat,Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y @ Ys ) @ Xs ) @ ( listrel_nat_nat @ R2 ) )
     => ~ ! [Y3: nat,Ys4: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y3 @ Ys4 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Y3 ) @ R2 )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Ys4 ) @ ( listrel_nat_nat @ R2 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_1025_listrel__Cons1,axiom,
    ! [Y: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Xs: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ Y @ Ys ) @ Xs ) @ ( listre818007680106770737at_nat @ R2 ) )
     => ~ ! [Y3: product_prod_nat_nat,Ys4: list_P6011104703257516679at_nat] :
            ( ( Xs
              = ( cons_P6512896166579812791at_nat @ Y3 @ Ys4 ) )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Y3 ) @ R2 )
             => ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ys @ Ys4 ) @ ( listre818007680106770737at_nat @ R2 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_1026_listrel__Cons1,axiom,
    ! [Y: nat,Ys: list_nat,Xs: list_P6834414599653733731al_nat,R2: set_Pr836851361167288120al_nat] :
      ( ( member1290970816896746895al_nat @ ( produc7399767566154494378al_nat @ ( cons_nat @ Y @ Ys ) @ Xs ) @ ( listre1017866332378306906al_nat @ R2 ) )
     => ~ ! [Y3: produc3741383161447143261al_nat,Ys4: list_P6834414599653733731al_nat] :
            ( ( Xs
              = ( cons_P500833500243608851al_nat @ Y3 @ Ys4 ) )
           => ( ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ Y @ Y3 ) @ R2 )
             => ~ ( member1290970816896746895al_nat @ ( produc7399767566154494378al_nat @ Ys @ Ys4 ) @ ( listre1017866332378306906al_nat @ R2 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_1027_listrel__Cons1,axiom,
    ! [Y: real,Ys: list_real,Xs: list_nat,R2: set_Pr3510011417693777981al_nat] :
      ( ( member7421688103010841766st_nat @ ( produc4048774278581505077st_nat @ ( cons_real @ Y @ Ys ) @ Xs ) @ ( listrel_real_nat @ R2 ) )
     => ~ ! [Y3: nat,Ys4: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y3 @ Ys4 ) )
           => ( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ Y @ Y3 ) @ R2 )
             => ~ ( member7421688103010841766st_nat @ ( produc4048774278581505077st_nat @ Ys @ Ys4 ) @ ( listrel_real_nat @ R2 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_1028_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y: list_b,Ys: list_list_b,R2: set_Pr1349601357184307552list_b] :
      ( ( member4916451157736907575list_b @ ( produc2084500703101687378list_b @ Xs @ ( cons_list_b @ Y @ Ys ) ) @ ( listrel_nat_list_b @ R2 ) )
     => ~ ! [X4: nat,Xs2: list_nat] :
            ( ( Xs
              = ( cons_nat @ X4 @ Xs2 ) )
           => ( ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ X4 @ Y ) @ R2 )
             => ~ ( member4916451157736907575list_b @ ( produc2084500703101687378list_b @ Xs2 @ Ys ) @ ( listrel_nat_list_b @ R2 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_1029_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat @ R2 ) )
     => ~ ! [X4: nat,Xs2: list_nat] :
            ( ( Xs
              = ( cons_nat @ X4 @ Xs2 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y ) @ R2 )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( listrel_nat_nat @ R2 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_1030_listrel__Cons2,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ Y @ Ys ) ) @ ( listre818007680106770737at_nat @ R2 ) )
     => ~ ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
            ( ( Xs
              = ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y ) @ R2 )
             => ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Ys ) @ ( listre818007680106770737at_nat @ R2 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_1031_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y: produc3741383161447143261al_nat,Ys: list_P6834414599653733731al_nat,R2: set_Pr836851361167288120al_nat] :
      ( ( member1290970816896746895al_nat @ ( produc7399767566154494378al_nat @ Xs @ ( cons_P500833500243608851al_nat @ Y @ Ys ) ) @ ( listre1017866332378306906al_nat @ R2 ) )
     => ~ ! [X4: nat,Xs2: list_nat] :
            ( ( Xs
              = ( cons_nat @ X4 @ Xs2 ) )
           => ( ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X4 @ Y ) @ R2 )
             => ~ ( member1290970816896746895al_nat @ ( produc7399767566154494378al_nat @ Xs2 @ Ys ) @ ( listre1017866332378306906al_nat @ R2 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_1032_listrel__Cons2,axiom,
    ! [Xs: list_real,Y: nat,Ys: list_nat,R2: set_Pr3510011417693777981al_nat] :
      ( ( member7421688103010841766st_nat @ ( produc4048774278581505077st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_real_nat @ R2 ) )
     => ~ ! [X4: real,Xs2: list_real] :
            ( ( Xs
              = ( cons_real @ X4 @ Xs2 ) )
           => ( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X4 @ Y ) @ R2 )
             => ~ ( member7421688103010841766st_nat @ ( produc4048774278581505077st_nat @ Xs2 @ Ys ) @ ( listrel_real_nat @ R2 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_1033_listrel_Ocases,axiom,
    ! [A12: list_nat,A23: list_list_b,R2: set_Pr1349601357184307552list_b] :
      ( ( member4916451157736907575list_b @ ( produc2084500703101687378list_b @ A12 @ A23 ) @ ( listrel_nat_list_b @ R2 ) )
     => ( ( ( A12 = nil_nat )
         => ( A23 != nil_list_b ) )
       => ~ ! [X4: nat,Y3: list_b,Xs2: list_nat] :
              ( ( A12
                = ( cons_nat @ X4 @ Xs2 ) )
             => ! [Ys4: list_list_b] :
                  ( ( A23
                    = ( cons_list_b @ Y3 @ Ys4 ) )
                 => ( ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ X4 @ Y3 ) @ R2 )
                   => ~ ( member4916451157736907575list_b @ ( produc2084500703101687378list_b @ Xs2 @ Ys4 ) @ ( listrel_nat_list_b @ R2 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_1034_listrel_Ocases,axiom,
    ! [A12: list_nat,A23: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A12 @ A23 ) @ ( listrel_nat_nat @ R2 ) )
     => ( ( ( A12 = nil_nat )
         => ( A23 != nil_nat ) )
       => ~ ! [X4: nat,Y3: nat,Xs2: list_nat] :
              ( ( A12
                = ( cons_nat @ X4 @ Xs2 ) )
             => ! [Ys4: list_nat] :
                  ( ( A23
                    = ( cons_nat @ Y3 @ Ys4 ) )
                 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R2 )
                   => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys4 ) @ ( listrel_nat_nat @ R2 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_1035_listrel_Ocases,axiom,
    ! [A12: list_P6011104703257516679at_nat,A23: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ A12 @ A23 ) @ ( listre818007680106770737at_nat @ R2 ) )
     => ( ( ( A12 = nil_Pr5478986624290739719at_nat )
         => ( A23 != nil_Pr5478986624290739719at_nat ) )
       => ~ ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
              ( ( A12
                = ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) )
             => ! [Ys4: list_P6011104703257516679at_nat] :
                  ( ( A23
                    = ( cons_P6512896166579812791at_nat @ Y3 @ Ys4 ) )
                 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y3 ) @ R2 )
                   => ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Ys4 ) @ ( listre818007680106770737at_nat @ R2 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_1036_listrel_Ocases,axiom,
    ! [A12: list_nat,A23: list_P6834414599653733731al_nat,R2: set_Pr836851361167288120al_nat] :
      ( ( member1290970816896746895al_nat @ ( produc7399767566154494378al_nat @ A12 @ A23 ) @ ( listre1017866332378306906al_nat @ R2 ) )
     => ( ( ( A12 = nil_nat )
         => ( A23 != nil_Pr1917482104270272867al_nat ) )
       => ~ ! [X4: nat,Y3: produc3741383161447143261al_nat,Xs2: list_nat] :
              ( ( A12
                = ( cons_nat @ X4 @ Xs2 ) )
             => ! [Ys4: list_P6834414599653733731al_nat] :
                  ( ( A23
                    = ( cons_P500833500243608851al_nat @ Y3 @ Ys4 ) )
                 => ( ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X4 @ Y3 ) @ R2 )
                   => ~ ( member1290970816896746895al_nat @ ( produc7399767566154494378al_nat @ Xs2 @ Ys4 ) @ ( listre1017866332378306906al_nat @ R2 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_1037_listrel_Ocases,axiom,
    ! [A12: list_real,A23: list_nat,R2: set_Pr3510011417693777981al_nat] :
      ( ( member7421688103010841766st_nat @ ( produc4048774278581505077st_nat @ A12 @ A23 ) @ ( listrel_real_nat @ R2 ) )
     => ( ( ( A12 = nil_real )
         => ( A23 != nil_nat ) )
       => ~ ! [X4: real,Y3: nat,Xs2: list_real] :
              ( ( A12
                = ( cons_real @ X4 @ Xs2 ) )
             => ! [Ys4: list_nat] :
                  ( ( A23
                    = ( cons_nat @ Y3 @ Ys4 ) )
                 => ( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X4 @ Y3 ) @ R2 )
                   => ~ ( member7421688103010841766st_nat @ ( produc4048774278581505077st_nat @ Xs2 @ Ys4 ) @ ( listrel_real_nat @ R2 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_1038_listrel__def,axiom,
    ( listrel_nat_list_b
    = ( ^ [R3: set_Pr1349601357184307552list_b] :
          ( collec6045138553467186037list_b
          @ ( produc2506431402601369945st_b_o
            @ ( listrelp_nat_list_b
              @ ^ [X2: nat,Y4: list_b] : ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ X2 @ Y4 ) @ R3 ) ) ) ) ) ) ).

% listrel_def
thf(fact_1039_listrel__def,axiom,
    ( listrel_nat_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat] :
          ( collec1570431334306492044st_nat
          @ ( produc6382697403170970154_nat_o
            @ ( listrelp_nat_nat
              @ ^ [X2: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ R3 ) ) ) ) ) ) ).

% listrel_def
thf(fact_1040_listrel__def,axiom,
    ( listre818007680106770737at_nat
    = ( ^ [R3: set_Pr8693737435421807431at_nat] :
          ( collec2094671920101799250at_nat
          @ ( produc2929669097083596076_nat_o
            @ ( listre5588581128925307775at_nat
              @ ^ [X2: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y4 ) @ R3 ) ) ) ) ) ) ).

% listrel_def
thf(fact_1041_listrel__def,axiom,
    ( listre1017866332378306906al_nat
    = ( ^ [R3: set_Pr836851361167288120al_nat] :
          ( collec885236078990679757al_nat
          @ ( produc8525291796600174081_nat_o
            @ ( listre8863158135660013452al_nat
              @ ^ [X2: nat,Y4: produc3741383161447143261al_nat] : ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X2 @ Y4 ) @ R3 ) ) ) ) ) ) ).

% listrel_def
thf(fact_1042_listrel__def,axiom,
    ( listrel_real_nat
    = ( ^ [R3: set_Pr3510011417693777981al_nat] :
          ( collec8567191420325495016st_nat
          @ ( produc4974167048199089358_nat_o
            @ ( listrelp_real_nat
              @ ^ [X2: real,Y4: nat] : ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X2 @ Y4 ) @ R3 ) ) ) ) ) ) ).

% listrel_def
thf(fact_1043_listrelp__listrel__eq,axiom,
    ! [R2: set_Pr1349601357184307552list_b] :
      ( ( listrelp_nat_list_b
        @ ^ [X2: nat,Y4: list_b] : ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ X2 @ Y4 ) @ R2 ) )
      = ( ^ [X2: list_nat,Y4: list_list_b] : ( member4916451157736907575list_b @ ( produc2084500703101687378list_b @ X2 @ Y4 ) @ ( listrel_nat_list_b @ R2 ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_1044_listrelp__listrel__eq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( listrelp_nat_nat
        @ ^ [X2: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ R2 ) )
      = ( ^ [X2: list_nat,Y4: list_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X2 @ Y4 ) @ ( listrel_nat_nat @ R2 ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_1045_listrelp__listrel__eq,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ( listre5588581128925307775at_nat
        @ ^ [X2: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y4 ) @ R2 ) )
      = ( ^ [X2: list_P6011104703257516679at_nat,Y4: list_P6011104703257516679at_nat] : ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ X2 @ Y4 ) @ ( listre818007680106770737at_nat @ R2 ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_1046_listrelp__listrel__eq,axiom,
    ! [R2: set_Pr836851361167288120al_nat] :
      ( ( listre8863158135660013452al_nat
        @ ^ [X2: nat,Y4: produc3741383161447143261al_nat] : ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ X2 @ Y4 ) @ R2 ) )
      = ( ^ [X2: list_nat,Y4: list_P6834414599653733731al_nat] : ( member1290970816896746895al_nat @ ( produc7399767566154494378al_nat @ X2 @ Y4 ) @ ( listre1017866332378306906al_nat @ R2 ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_1047_listrelp__listrel__eq,axiom,
    ! [R2: set_Pr3510011417693777981al_nat] :
      ( ( listrelp_real_nat
        @ ^ [X2: real,Y4: nat] : ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X2 @ Y4 ) @ R2 ) )
      = ( ^ [X2: list_real,Y4: list_nat] : ( member7421688103010841766st_nat @ ( produc4048774278581505077st_nat @ X2 @ Y4 ) @ ( listrel_real_nat @ R2 ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_1048_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_list_b,R2: set_Pr1349601357184307552list_b] :
      ( ( member4916451157736907575list_b @ ( produc2084500703101687378list_b @ Xs @ Ys ) @ ( listrel_nat_list_b @ R2 ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_s420531428170919671list_b @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs ) )
           => ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ ( nth_nat @ Xs @ N3 ) @ ( nth_list_b @ Ys @ N3 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1049_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R2 ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N3 ) @ ( nth_nat @ Ys @ N3 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1050_listrel__iff__nth,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( listre818007680106770737at_nat @ R2 ) )
      = ( ( ( size_s5460976970255530739at_nat @ Xs )
          = ( size_s5460976970255530739at_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s5460976970255530739at_nat @ Xs ) )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N3 ) @ ( nth_Pr7617993195940197384at_nat @ Ys @ N3 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1051_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_P6834414599653733731al_nat,R2: set_Pr836851361167288120al_nat] :
      ( ( member1290970816896746895al_nat @ ( produc7399767566154494378al_nat @ Xs @ Ys ) @ ( listre1017866332378306906al_nat @ R2 ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_s1877336372972134351al_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs ) )
           => ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ ( nth_nat @ Xs @ N3 ) @ ( nth_Pr3792769368220640996al_nat @ Ys @ N3 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1052_listrel__iff__nth,axiom,
    ! [Xs: list_real,Ys: list_nat,R2: set_Pr3510011417693777981al_nat] :
      ( ( member7421688103010841766st_nat @ ( produc4048774278581505077st_nat @ Xs @ Ys ) @ ( listrel_real_nat @ R2 ) )
      = ( ( ( size_size_list_real @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_real @ Xs ) )
           => ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ ( nth_real @ Xs @ N3 ) @ ( nth_nat @ Ys @ N3 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1053_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_b,R2: set_Pr4264375888882495962_nat_b] :
      ( ( member6003184433318419249list_b @ ( produc7723716014355252812list_b @ Xs @ Ys ) @ ( listrel_nat_b @ R2 ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_b @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs ) )
           => ( member8962352056413324475_nat_b @ ( product_Pair_nat_b @ ( nth_nat @ Xs @ N3 ) @ ( nth_b @ Ys @ N3 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1054_listrel__iff__nth,axiom,
    ! [Xs: list_b,Ys: list_nat,R2: set_Pr1307281990691478580_b_nat] :
      ( ( member5518038250373942315st_nat @ ( produc5459849259740801934st_nat @ Xs @ Ys ) @ ( listrel_b_nat @ R2 ) )
      = ( ( ( size_size_list_b @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_b @ Xs ) )
           => ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ ( nth_b @ Xs @ N3 ) @ ( nth_nat @ Ys @ N3 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1055_listrel__iff__nth,axiom,
    ! [Xs: list_b,Ys: list_b,R2: set_Product_prod_b_b] :
      ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Xs @ Ys ) @ ( listrel_b_b @ R2 ) )
      = ( ( ( size_size_list_b @ Xs )
          = ( size_size_list_b @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_b @ Xs ) )
           => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( nth_b @ Xs @ N3 ) @ ( nth_b @ Ys @ N3 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1056_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F: nat > nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( map_nat_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_1057_nth__map,axiom,
    ! [N: nat,Xs: list_b,F: b > nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_b @ Xs ) )
     => ( ( nth_nat @ ( map_b_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_b @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_1058_listrelp_OCons,axiom,
    ! [R2: nat > nat > $o,X3: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( R2 @ X3 @ Y )
     => ( ( listrelp_nat_nat @ R2 @ Xs @ Ys )
       => ( listrelp_nat_nat @ R2 @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ) ).

% listrelp.Cons
thf(fact_1059_listrelp_ONil,axiom,
    ! [R2: nat > nat > $o] : ( listrelp_nat_nat @ R2 @ nil_nat @ nil_nat ) ).

% listrelp.Nil
thf(fact_1060_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y8: list_nat,Z4: list_nat] : ( Y8 = Z4 ) )
    = ( ^ [Xs3: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs3 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
             => ( ( nth_nat @ Xs3 @ I3 )
                = ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1061_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y8: list_b,Z4: list_b] : ( Y8 = Z4 ) )
    = ( ^ [Xs3: list_b,Ys3: list_b] :
          ( ( ( size_size_list_b @ Xs3 )
            = ( size_size_list_b @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Xs3 ) )
             => ( ( nth_b @ Xs3 @ I3 )
                = ( nth_b @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1062_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X8: nat] : ( P @ I3 @ X8 ) ) )
      = ( ? [Xs3: list_nat] :
            ( ( ( size_size_list_nat @ Xs3 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1063_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > b > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X8: b] : ( P @ I3 @ X8 ) ) )
      = ( ? [Xs3: list_b] :
            ( ( ( size_size_list_b @ Xs3 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_b @ Xs3 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1064_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I )
              = ( nth_nat @ Ys @ I ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_1065_nth__equalityI,axiom,
    ! [Xs: list_b,Ys: list_b] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
           => ( ( nth_b @ Xs @ I )
              = ( nth_b @ Ys @ I ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_1066_map__equality__iff,axiom,
    ! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I3 ) )
              = ( G @ ( nth_nat @ Ys @ I3 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_1067_map__equality__iff,axiom,
    ! [F: nat > nat,Xs: list_nat,G: b > nat,Ys: list_b] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_b_nat @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_b @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_b @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I3 ) )
              = ( G @ ( nth_b @ Ys @ I3 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_1068_map__equality__iff,axiom,
    ! [F: b > nat,Xs: list_b,G: nat > nat,Ys: list_nat] :
      ( ( ( map_b_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
      = ( ( ( size_size_list_b @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_b @ Xs @ I3 ) )
              = ( G @ ( nth_nat @ Ys @ I3 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_1069_listrelp_Ocases,axiom,
    ! [R2: nat > nat > $o,A12: list_nat,A23: list_nat] :
      ( ( listrelp_nat_nat @ R2 @ A12 @ A23 )
     => ( ( ( A12 = nil_nat )
         => ( A23 != nil_nat ) )
       => ~ ! [X4: nat,Y3: nat,Xs2: list_nat] :
              ( ( A12
                = ( cons_nat @ X4 @ Xs2 ) )
             => ! [Ys4: list_nat] :
                  ( ( A23
                    = ( cons_nat @ Y3 @ Ys4 ) )
                 => ( ( R2 @ X4 @ Y3 )
                   => ~ ( listrelp_nat_nat @ R2 @ Xs2 @ Ys4 ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_1070_listrelp_Osimps,axiom,
    ( listrelp_nat_nat
    = ( ^ [R3: nat > nat > $o,A1: list_nat,A22: list_nat] :
          ( ( ( A1 = nil_nat )
            & ( A22 = nil_nat ) )
          | ? [X2: nat,Y4: nat,Xs3: list_nat,Ys3: list_nat] :
              ( ( A1
                = ( cons_nat @ X2 @ Xs3 ) )
              & ( A22
                = ( cons_nat @ Y4 @ Ys3 ) )
              & ( R3 @ X2 @ Y4 )
              & ( listrelp_nat_nat @ R3 @ Xs3 @ Ys3 ) ) ) ) ) ).

% listrelp.simps
thf(fact_1071_lex__take__index,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R2 ) )
     => ~ ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
           => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
             => ( ( ( take_nat @ I @ Xs )
                  = ( take_nat @ I @ Ys ) )
               => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) @ R2 ) ) ) ) ) ).

% lex_take_index
thf(fact_1072_lex__take__index,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) )
     => ~ ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Xs ) )
           => ( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Ys ) )
             => ( ( ( take_P2173866234530122223at_nat @ I @ Xs )
                  = ( take_P2173866234530122223at_nat @ I @ Ys ) )
               => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I ) @ ( nth_Pr7617993195940197384at_nat @ Ys @ I ) ) @ R2 ) ) ) ) ) ).

% lex_take_index
thf(fact_1073_lex__take__index,axiom,
    ! [Xs: list_b,Ys: list_b,R2: set_Product_prod_b_b] :
      ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Xs @ Ys ) @ ( lex_b @ R2 ) )
     => ~ ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
           => ( ( ord_less_nat @ I @ ( size_size_list_b @ Ys ) )
             => ( ( ( take_b @ I @ Xs )
                  = ( take_b @ I @ Ys ) )
               => ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( nth_b @ Xs @ I ) @ ( nth_b @ Ys @ I ) ) @ R2 ) ) ) ) ) ).

% lex_take_index
thf(fact_1074_list__ex__length,axiom,
    ( list_ex_nat
    = ( ^ [P4: nat > $o,Xs3: list_nat] :
        ? [N3: nat] :
          ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs3 ) )
          & ( P4 @ ( nth_nat @ Xs3 @ N3 ) ) ) ) ) ).

% list_ex_length
thf(fact_1075_list__ex__length,axiom,
    ( list_ex_b
    = ( ^ [P4: b > $o,Xs3: list_b] :
        ? [N3: nat] :
          ( ( ord_less_nat @ N3 @ ( size_size_list_b @ Xs3 ) )
          & ( P4 @ ( nth_b @ Xs3 @ N3 ) ) ) ) ) ).

% list_ex_length
thf(fact_1076_Collect__restrict,axiom,
    ! [X: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ord_le3146513528884898305at_nat
      @ ( collec3392354462482085612at_nat
        @ ^ [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ X )
            & ( P @ X2 ) ) )
      @ X ) ).

% Collect_restrict
thf(fact_1077_Collect__restrict,axiom,
    ! [X: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ord_le3000389064537975527at_nat
      @ ( collec7088162979684241874at_nat
        @ ^ [X2: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X2 @ X )
            & ( P @ X2 ) ) )
      @ X ) ).

% Collect_restrict
thf(fact_1078_Collect__restrict,axiom,
    ! [X: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X2: nat] :
            ( ( member_nat @ X2 @ X )
            & ( P @ X2 ) ) )
      @ X ) ).

% Collect_restrict
thf(fact_1079_Collect__restrict,axiom,
    ! [X: set_list_b,P: list_b > $o] :
      ( ord_le8932221534207217157list_b
      @ ( collect_list_b
        @ ^ [X2: list_b] :
            ( ( member_list_b @ X2 @ X )
            & ( P @ X2 ) ) )
      @ X ) ).

% Collect_restrict
thf(fact_1080_Cons__in__lists__iff,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member3067507820990806192at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( lists_3066315782749251317at_nat @ A4 ) )
      = ( ( member8440522571783428010at_nat @ X3 @ A4 )
        & ( member3067507820990806192at_nat @ Xs @ ( lists_3066315782749251317at_nat @ A4 ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_1081_Cons__in__lists__iff,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( member3799944675974059798at_nat @ ( cons_P8732206157123786781at_nat @ X3 @ Xs ) @ ( lists_8807101532170170459at_nat @ A4 ) )
      = ( ( member8206827879206165904at_nat @ X3 @ A4 )
        & ( member3799944675974059798at_nat @ Xs @ ( lists_8807101532170170459at_nat @ A4 ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_1082_Cons__in__lists__iff,axiom,
    ! [X3: list_b,Xs: list_list_b,A4: set_list_b] :
      ( ( member_list_list_b @ ( cons_list_b @ X3 @ Xs ) @ ( lists_list_b @ A4 ) )
      = ( ( member_list_b @ X3 @ A4 )
        & ( member_list_list_b @ Xs @ ( lists_list_b @ A4 ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_1083_Cons__in__lists__iff,axiom,
    ! [X3: b,Xs: list_b,A4: set_b] :
      ( ( member_list_b @ ( cons_b @ X3 @ Xs ) @ ( lists_b @ A4 ) )
      = ( ( member_b @ X3 @ A4 )
        & ( member_list_b @ Xs @ ( lists_b @ A4 ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_1084_Cons__in__lists__iff,axiom,
    ! [X3: nat,Xs: list_nat,A4: set_nat] :
      ( ( member_list_nat @ ( cons_nat @ X3 @ Xs ) @ ( lists_nat @ A4 ) )
      = ( ( member_nat @ X3 @ A4 )
        & ( member_list_nat @ Xs @ ( lists_nat @ A4 ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_1085_list__ex__simps_I1_J,axiom,
    ! [P: nat > $o,X3: nat,Xs: list_nat] :
      ( ( list_ex_nat @ P @ ( cons_nat @ X3 @ Xs ) )
      = ( ( P @ X3 )
        | ( list_ex_nat @ P @ Xs ) ) ) ).

% list_ex_simps(1)
thf(fact_1086_list__ex__simps_I2_J,axiom,
    ! [P: nat > $o] :
      ~ ( list_ex_nat @ P @ nil_nat ) ).

% list_ex_simps(2)
thf(fact_1087_take__all,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
     => ( ( take_nat @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_1088_take__all,axiom,
    ! [Xs: list_b,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_b @ Xs ) @ N )
     => ( ( take_b @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_1089_take__all__iff,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ( take_nat @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_1090_take__all__iff,axiom,
    ! [N: nat,Xs: list_b] :
      ( ( ( take_b @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_size_list_b @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_1091_nth__take,axiom,
    ! [I2: nat,N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ I2 @ N )
     => ( ( nth_nat @ ( take_nat @ N @ Xs ) @ I2 )
        = ( nth_nat @ Xs @ I2 ) ) ) ).

% nth_take
thf(fact_1092_lists__UNIV,axiom,
    ( ( lists_b @ top_top_set_b )
    = top_top_set_list_b ) ).

% lists_UNIV
thf(fact_1093_lists__UNIV,axiom,
    ( ( lists_list_b @ top_top_set_list_b )
    = top_to4060329871436644795list_b ) ).

% lists_UNIV
thf(fact_1094_lists__UNIV,axiom,
    ( ( lists_nat @ top_top_set_nat )
    = top_top_set_list_nat ) ).

% lists_UNIV
thf(fact_1095_lists__UNIV,axiom,
    ( ( lists_8807101532170170459at_nat @ top_to8454300755403694231at_nat )
    = top_to2355589028646918429at_nat ) ).

% lists_UNIV
thf(fact_1096_lists__UNIV,axiom,
    ( ( lists_3066315782749251317at_nat @ top_to4669805908274784177at_nat )
    = top_to70995312096774071at_nat ) ).

% lists_UNIV
thf(fact_1097_take__Nil,axiom,
    ! [N: nat] :
      ( ( take_nat @ N @ nil_nat )
      = nil_nat ) ).

% take_Nil
thf(fact_1098_take__map,axiom,
    ! [N: nat,F: nat > nat,Xs: list_nat] :
      ( ( take_nat @ N @ ( map_nat_nat @ F @ Xs ) )
      = ( map_nat_nat @ F @ ( take_nat @ N @ Xs ) ) ) ).

% take_map
thf(fact_1099_listsE,axiom,
    ! [X3: product_prod_nat_nat,L: list_P6011104703257516679at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member3067507820990806192at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ L ) @ ( lists_3066315782749251317at_nat @ A4 ) )
     => ~ ( ( member8440522571783428010at_nat @ X3 @ A4 )
         => ~ ( member3067507820990806192at_nat @ L @ ( lists_3066315782749251317at_nat @ A4 ) ) ) ) ).

% listsE
thf(fact_1100_listsE,axiom,
    ! [X3: produc859450856879609959at_nat,L: list_P8469869581646625389at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( member3799944675974059798at_nat @ ( cons_P8732206157123786781at_nat @ X3 @ L ) @ ( lists_8807101532170170459at_nat @ A4 ) )
     => ~ ( ( member8206827879206165904at_nat @ X3 @ A4 )
         => ~ ( member3799944675974059798at_nat @ L @ ( lists_8807101532170170459at_nat @ A4 ) ) ) ) ).

% listsE
thf(fact_1101_listsE,axiom,
    ! [X3: list_b,L: list_list_b,A4: set_list_b] :
      ( ( member_list_list_b @ ( cons_list_b @ X3 @ L ) @ ( lists_list_b @ A4 ) )
     => ~ ( ( member_list_b @ X3 @ A4 )
         => ~ ( member_list_list_b @ L @ ( lists_list_b @ A4 ) ) ) ) ).

% listsE
thf(fact_1102_listsE,axiom,
    ! [X3: b,L: list_b,A4: set_b] :
      ( ( member_list_b @ ( cons_b @ X3 @ L ) @ ( lists_b @ A4 ) )
     => ~ ( ( member_b @ X3 @ A4 )
         => ~ ( member_list_b @ L @ ( lists_b @ A4 ) ) ) ) ).

% listsE
thf(fact_1103_listsE,axiom,
    ! [X3: nat,L: list_nat,A4: set_nat] :
      ( ( member_list_nat @ ( cons_nat @ X3 @ L ) @ ( lists_nat @ A4 ) )
     => ~ ( ( member_nat @ X3 @ A4 )
         => ~ ( member_list_nat @ L @ ( lists_nat @ A4 ) ) ) ) ).

% listsE
thf(fact_1104_lists_OCons,axiom,
    ! [A: b,A4: set_b,L: list_b] :
      ( ( member_b @ A @ A4 )
     => ( ( member_list_b @ L @ ( lists_b @ A4 ) )
       => ( member_list_b @ ( cons_b @ A @ L ) @ ( lists_b @ A4 ) ) ) ) ).

% lists.Cons
thf(fact_1105_lists_OCons,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,L: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A4 )
     => ( ( member3067507820990806192at_nat @ L @ ( lists_3066315782749251317at_nat @ A4 ) )
       => ( member3067507820990806192at_nat @ ( cons_P6512896166579812791at_nat @ A @ L ) @ ( lists_3066315782749251317at_nat @ A4 ) ) ) ) ).

% lists.Cons
thf(fact_1106_lists_OCons,axiom,
    ! [A: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,L: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ A @ A4 )
     => ( ( member3799944675974059798at_nat @ L @ ( lists_8807101532170170459at_nat @ A4 ) )
       => ( member3799944675974059798at_nat @ ( cons_P8732206157123786781at_nat @ A @ L ) @ ( lists_8807101532170170459at_nat @ A4 ) ) ) ) ).

% lists.Cons
thf(fact_1107_lists_OCons,axiom,
    ! [A: list_b,A4: set_list_b,L: list_list_b] :
      ( ( member_list_b @ A @ A4 )
     => ( ( member_list_list_b @ L @ ( lists_list_b @ A4 ) )
       => ( member_list_list_b @ ( cons_list_b @ A @ L ) @ ( lists_list_b @ A4 ) ) ) ) ).

% lists.Cons
thf(fact_1108_lists_OCons,axiom,
    ! [A: nat,A4: set_nat,L: list_nat] :
      ( ( member_nat @ A @ A4 )
     => ( ( member_list_nat @ L @ ( lists_nat @ A4 ) )
       => ( member_list_nat @ ( cons_nat @ A @ L ) @ ( lists_nat @ A4 ) ) ) ) ).

% lists.Cons
thf(fact_1109_lists_ONil,axiom,
    ! [A4: set_b] : ( member_list_b @ nil_b @ ( lists_b @ A4 ) ) ).

% lists.Nil
thf(fact_1110_lists_ONil,axiom,
    ! [A4: set_nat] : ( member_list_nat @ nil_nat @ ( lists_nat @ A4 ) ) ).

% lists.Nil
thf(fact_1111_take__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ! [I: nat] :
          ( ( take_nat @ I @ Xs )
          = ( take_nat @ I @ Ys ) )
     => ( Xs = Ys ) ) ).

% take_equalityI
thf(fact_1112_lists_Ocases,axiom,
    ! [A: list_P6011104703257516679at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member3067507820990806192at_nat @ A @ ( lists_3066315782749251317at_nat @ A4 ) )
     => ( ( A != nil_Pr5478986624290739719at_nat )
       => ~ ! [A3: product_prod_nat_nat,L2: list_P6011104703257516679at_nat] :
              ( ( A
                = ( cons_P6512896166579812791at_nat @ A3 @ L2 ) )
             => ( ( member8440522571783428010at_nat @ A3 @ A4 )
               => ~ ( member3067507820990806192at_nat @ L2 @ ( lists_3066315782749251317at_nat @ A4 ) ) ) ) ) ) ).

% lists.cases
thf(fact_1113_lists_Ocases,axiom,
    ! [A: list_P8469869581646625389at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( member3799944675974059798at_nat @ A @ ( lists_8807101532170170459at_nat @ A4 ) )
     => ( ( A != nil_Pr2582115297535392877at_nat )
       => ~ ! [A3: produc859450856879609959at_nat,L2: list_P8469869581646625389at_nat] :
              ( ( A
                = ( cons_P8732206157123786781at_nat @ A3 @ L2 ) )
             => ( ( member8206827879206165904at_nat @ A3 @ A4 )
               => ~ ( member3799944675974059798at_nat @ L2 @ ( lists_8807101532170170459at_nat @ A4 ) ) ) ) ) ) ).

% lists.cases
thf(fact_1114_lists_Ocases,axiom,
    ! [A: list_list_b,A4: set_list_b] :
      ( ( member_list_list_b @ A @ ( lists_list_b @ A4 ) )
     => ( ( A != nil_list_b )
       => ~ ! [A3: list_b,L2: list_list_b] :
              ( ( A
                = ( cons_list_b @ A3 @ L2 ) )
             => ( ( member_list_b @ A3 @ A4 )
               => ~ ( member_list_list_b @ L2 @ ( lists_list_b @ A4 ) ) ) ) ) ) ).

% lists.cases
thf(fact_1115_lists_Ocases,axiom,
    ! [A: list_b,A4: set_b] :
      ( ( member_list_b @ A @ ( lists_b @ A4 ) )
     => ( ( A != nil_b )
       => ~ ! [A3: b,L2: list_b] :
              ( ( A
                = ( cons_b @ A3 @ L2 ) )
             => ( ( member_b @ A3 @ A4 )
               => ~ ( member_list_b @ L2 @ ( lists_b @ A4 ) ) ) ) ) ) ).

% lists.cases
thf(fact_1116_lists_Ocases,axiom,
    ! [A: list_nat,A4: set_nat] :
      ( ( member_list_nat @ A @ ( lists_nat @ A4 ) )
     => ( ( A != nil_nat )
       => ~ ! [A3: nat,L2: list_nat] :
              ( ( A
                = ( cons_nat @ A3 @ L2 ) )
             => ( ( member_nat @ A3 @ A4 )
               => ~ ( member_list_nat @ L2 @ ( lists_nat @ A4 ) ) ) ) ) ) ).

% lists.cases
thf(fact_1117_lists_Osimps,axiom,
    ! [A: list_P6011104703257516679at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member3067507820990806192at_nat @ A @ ( lists_3066315782749251317at_nat @ A4 ) )
      = ( ( A = nil_Pr5478986624290739719at_nat )
        | ? [A6: product_prod_nat_nat,L3: list_P6011104703257516679at_nat] :
            ( ( A
              = ( cons_P6512896166579812791at_nat @ A6 @ L3 ) )
            & ( member8440522571783428010at_nat @ A6 @ A4 )
            & ( member3067507820990806192at_nat @ L3 @ ( lists_3066315782749251317at_nat @ A4 ) ) ) ) ) ).

% lists.simps
thf(fact_1118_lists_Osimps,axiom,
    ! [A: list_P8469869581646625389at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( member3799944675974059798at_nat @ A @ ( lists_8807101532170170459at_nat @ A4 ) )
      = ( ( A = nil_Pr2582115297535392877at_nat )
        | ? [A6: produc859450856879609959at_nat,L3: list_P8469869581646625389at_nat] :
            ( ( A
              = ( cons_P8732206157123786781at_nat @ A6 @ L3 ) )
            & ( member8206827879206165904at_nat @ A6 @ A4 )
            & ( member3799944675974059798at_nat @ L3 @ ( lists_8807101532170170459at_nat @ A4 ) ) ) ) ) ).

% lists.simps
thf(fact_1119_lists_Osimps,axiom,
    ! [A: list_list_b,A4: set_list_b] :
      ( ( member_list_list_b @ A @ ( lists_list_b @ A4 ) )
      = ( ( A = nil_list_b )
        | ? [A6: list_b,L3: list_list_b] :
            ( ( A
              = ( cons_list_b @ A6 @ L3 ) )
            & ( member_list_b @ A6 @ A4 )
            & ( member_list_list_b @ L3 @ ( lists_list_b @ A4 ) ) ) ) ) ).

% lists.simps
thf(fact_1120_lists_Osimps,axiom,
    ! [A: list_b,A4: set_b] :
      ( ( member_list_b @ A @ ( lists_b @ A4 ) )
      = ( ( A = nil_b )
        | ? [A6: b,L3: list_b] :
            ( ( A
              = ( cons_b @ A6 @ L3 ) )
            & ( member_b @ A6 @ A4 )
            & ( member_list_b @ L3 @ ( lists_b @ A4 ) ) ) ) ) ).

% lists.simps
thf(fact_1121_lists_Osimps,axiom,
    ! [A: list_nat,A4: set_nat] :
      ( ( member_list_nat @ A @ ( lists_nat @ A4 ) )
      = ( ( A = nil_nat )
        | ? [A6: nat,L3: list_nat] :
            ( ( A
              = ( cons_nat @ A6 @ L3 ) )
            & ( member_nat @ A6 @ A4 )
            & ( member_list_nat @ L3 @ ( lists_nat @ A4 ) ) ) ) ) ).

% lists.simps
thf(fact_1122_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S3: list_b,R: set_Pr1349601357184307552list_b,S4: list_b] :
      ( ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ R2 @ S3 ) @ R )
     => ( ( S4 = S3 )
       => ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1123_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S3: nat,R: set_Pr1261947904930325089at_nat,S4: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S3 ) @ R )
     => ( ( S4 = S3 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1124_ssubst__Pair__rhs,axiom,
    ! [R2: product_prod_nat_nat,S3: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,S4: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R2 @ S3 ) @ R )
     => ( ( S4 = S3 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1125_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S3: produc3741383161447143261al_nat,R: set_Pr836851361167288120al_nat,S4: produc3741383161447143261al_nat] :
      ( ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ R2 @ S3 ) @ R )
     => ( ( S4 = S3 )
       => ( member7408976046919118105al_nat @ ( produc4490577844368043572al_nat @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1126_ssubst__Pair__rhs,axiom,
    ! [R2: real,S3: nat,R: set_Pr3510011417693777981al_nat,S4: nat] :
      ( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ R2 @ S3 ) @ R )
     => ( ( S4 = S3 )
       => ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1127_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Ys ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ K )
             => ( ( nth_nat @ Xs @ I )
                = ( nth_nat @ Ys @ I ) ) )
         => ( ( take_nat @ K @ Xs )
            = ( take_nat @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_1128_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_b,Ys: list_b] :
      ( ( ord_less_eq_nat @ K @ ( size_size_list_b @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_size_list_b @ Ys ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ K )
             => ( ( nth_b @ Xs @ I )
                = ( nth_b @ Ys @ I ) ) )
         => ( ( take_b @ K @ Xs )
            = ( take_b @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_1129_prop__restrict,axiom,
    ! [X3: product_prod_nat_nat,Z5: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ X3 @ Z5 )
     => ( ( ord_le3146513528884898305at_nat @ Z5
          @ ( collec3392354462482085612at_nat
            @ ^ [X2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X2 @ X )
                & ( P @ X2 ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_1130_prop__restrict,axiom,
    ! [X3: produc859450856879609959at_nat,Z5: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ X3 @ Z5 )
     => ( ( ord_le3000389064537975527at_nat @ Z5
          @ ( collec7088162979684241874at_nat
            @ ^ [X2: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X2 @ X )
                & ( P @ X2 ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_1131_prop__restrict,axiom,
    ! [X3: nat,Z5: set_nat,X: set_nat,P: nat > $o] :
      ( ( member_nat @ X3 @ Z5 )
     => ( ( ord_less_eq_set_nat @ Z5
          @ ( collect_nat
            @ ^ [X2: nat] :
                ( ( member_nat @ X2 @ X )
                & ( P @ X2 ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_1132_prop__restrict,axiom,
    ! [X3: list_b,Z5: set_list_b,X: set_list_b,P: list_b > $o] :
      ( ( member_list_b @ X3 @ Z5 )
     => ( ( ord_le8932221534207217157list_b @ Z5
          @ ( collect_list_b
            @ ^ [X2: list_b] :
                ( ( member_list_b @ X2 @ X )
                & ( P @ X2 ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_1133_lexord__take__index__conv,axiom,
    ! [X3: list_nat,Y: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y ) @ ( lexord_nat @ R2 ) )
      = ( ( ( ord_less_nat @ ( size_size_list_nat @ X3 ) @ ( size_size_list_nat @ Y ) )
          & ( ( take_nat @ ( size_size_list_nat @ X3 ) @ Y )
            = X3 ) )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( ord_min_nat @ ( size_size_list_nat @ X3 ) @ ( size_size_list_nat @ Y ) ) )
            & ( ( take_nat @ I3 @ X3 )
              = ( take_nat @ I3 @ Y ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ X3 @ I3 ) @ ( nth_nat @ Y @ I3 ) ) @ R2 ) ) ) ) ).

% lexord_take_index_conv
thf(fact_1134_lexord__take__index__conv,axiom,
    ! [X3: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ X3 @ Y ) @ ( lexord2841853652668343668at_nat @ R2 ) )
      = ( ( ( ord_less_nat @ ( size_s5460976970255530739at_nat @ X3 ) @ ( size_s5460976970255530739at_nat @ Y ) )
          & ( ( take_P2173866234530122223at_nat @ ( size_s5460976970255530739at_nat @ X3 ) @ Y )
            = X3 ) )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( ord_min_nat @ ( size_s5460976970255530739at_nat @ X3 ) @ ( size_s5460976970255530739at_nat @ Y ) ) )
            & ( ( take_P2173866234530122223at_nat @ I3 @ X3 )
              = ( take_P2173866234530122223at_nat @ I3 @ Y ) )
            & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ X3 @ I3 ) @ ( nth_Pr7617993195940197384at_nat @ Y @ I3 ) ) @ R2 ) ) ) ) ).

% lexord_take_index_conv
thf(fact_1135_lexord__take__index__conv,axiom,
    ! [X3: list_b,Y: list_b,R2: set_Product_prod_b_b] :
      ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ X3 @ Y ) @ ( lexord_b @ R2 ) )
      = ( ( ( ord_less_nat @ ( size_size_list_b @ X3 ) @ ( size_size_list_b @ Y ) )
          & ( ( take_b @ ( size_size_list_b @ X3 ) @ Y )
            = X3 ) )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( ord_min_nat @ ( size_size_list_b @ X3 ) @ ( size_size_list_b @ Y ) ) )
            & ( ( take_b @ I3 @ X3 )
              = ( take_b @ I3 @ Y ) )
            & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ ( nth_b @ X3 @ I3 ) @ ( nth_b @ Y @ I3 ) ) @ R2 ) ) ) ) ).

% lexord_take_index_conv
thf(fact_1136_take__Cons,axiom,
    ! [N: nat,X3: nat,Xs: list_nat] :
      ( ( take_nat @ N @ ( cons_nat @ X3 @ Xs ) )
      = ( case_nat_list_nat @ nil_nat
        @ ^ [M3: nat] : ( cons_nat @ X3 @ ( take_nat @ M3 @ Xs ) )
        @ N ) ) ).

% take_Cons
thf(fact_1137_concat__map__singleton,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( concat_nat
        @ ( map_nat_list_nat
          @ ^ [X2: nat] : ( cons_nat @ ( F @ X2 ) @ nil_nat )
          @ Xs ) )
      = ( map_nat_nat @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_1138_nat_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% nat.inject
thf(fact_1139_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_1140_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_less_eq
thf(fact_1141_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_1142_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_1143_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_1144_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_1145_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X4: nat] : ( R @ X4 @ X4 )
       => ( ! [X4: nat,Y3: nat,Z2: nat] :
              ( ( R @ X4 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X4 @ Z2 ) ) )
         => ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
           => ( R @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_1146_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_1147_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
             => ( P @ M2 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_1148_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_1149_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_1150_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_1151_Suc__le__D,axiom,
    ! [N: nat,M6: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
     => ? [M5: nat] :
          ( M6
          = ( suc @ M5 ) ) ) ).

% Suc_le_D
thf(fact_1152_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_1153_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_1154_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_1155_inj__Suc,axiom,
    ! [N4: set_nat] : ( inj_on_nat_nat @ suc @ N4 ) ).

% inj_Suc
thf(fact_1156_Suc__inject,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ( suc @ X3 )
        = ( suc @ Y ) )
     => ( X3 = Y ) ) ).

% Suc_inject
thf(fact_1157_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_1158_Nat_OlessE,axiom,
    ! [I2: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ K )
     => ( ( K
         != ( suc @ I2 ) )
       => ~ ! [J: nat] :
              ( ( ord_less_nat @ I2 @ J )
             => ( K
               != ( suc @ J ) ) ) ) ) ).

% Nat.lessE
thf(fact_1159_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_1160_Suc__lessE,axiom,
    ! [I2: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I2 ) @ K )
     => ~ ! [J: nat] :
            ( ( ord_less_nat @ I2 @ J )
           => ( K
             != ( suc @ J ) ) ) ) ).

% Suc_lessE
thf(fact_1161_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ( suc @ M )
         != N )
       => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).

% Suc_lessI
thf(fact_1162_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_1163_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_1164_Ex__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
            & ( P @ I3 ) ) )
      = ( ( P @ N )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
            & ( P @ I3 ) ) ) ) ).

% Ex_less_Suc
thf(fact_1165_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( M = N ) ) ) ).

% less_Suc_eq
thf(fact_1166_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_1167_All__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
           => ( P @ I3 ) ) )
      = ( ( P @ N )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
           => ( P @ I3 ) ) ) ) ).

% All_less_Suc
thf(fact_1168_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M )
      = ( ? [M7: nat] :
            ( ( M
              = ( suc @ M7 ) )
            & ( ord_less_nat @ N @ M7 ) ) ) ) ).

% Suc_less_eq2
thf(fact_1169_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_1170_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_less_SucD
thf(fact_1171_less__trans__Suc,axiom,
    ! [I2: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( ord_less_nat @ J2 @ K )
       => ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_1172_less__Suc__induct,axiom,
    ! [I2: nat,J2: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ! [I: nat] : ( P @ I @ ( suc @ I ) )
       => ( ! [I: nat,J: nat,K2: nat] :
              ( ( ord_less_nat @ I @ J )
             => ( ( ord_less_nat @ J @ K2 )
               => ( ( P @ I @ J )
                 => ( ( P @ J @ K2 )
                   => ( P @ I @ K2 ) ) ) ) )
         => ( P @ I2 @ J2 ) ) ) ) ).

% less_Suc_induct
thf(fact_1173_strict__inc__induct,axiom,
    ! [I2: nat,J2: nat,P: nat > $o] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ! [I: nat] :
            ( ( J2
              = ( suc @ I ) )
           => ( P @ I ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ J2 )
             => ( ( P @ ( suc @ I ) )
               => ( P @ I ) ) )
         => ( P @ I2 ) ) ) ) ).

% strict_inc_induct
thf(fact_1174_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_1175_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).

% less_eq_nat.simps(2)
thf(fact_1176_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_1177_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_1178_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_1179_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_1180_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_1181_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_1182_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_1183_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1184_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_1185_irrefl__less__than,axiom,
    irrefl_on_nat @ top_top_set_nat @ less_than ).

% irrefl_less_than
thf(fact_1186_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_1187_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_1188_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_1189_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_1190_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_1191_min__0R,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_1192_min__0L,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% min_0L
thf(fact_1193_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_1194_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_1195_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
      = ( case_nat_o @ $false
        @ ^ [Uu: nat] : $true
        @ Nat ) ) ).

% nat.disc_eq_case(2)
thf(fact_1196_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
      = ( case_nat_o @ $true
        @ ^ [Uu: nat] : $false
        @ Nat ) ) ).

% nat.disc_eq_case(1)
thf(fact_1197_Ex__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
            & ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
            & ( P @ ( suc @ I3 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_1198_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M3: nat] :
            ( N
            = ( suc @ M3 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_1199_All__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
           => ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
           => ( P @ ( suc @ I3 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_1200_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M5: nat] :
          ( N
          = ( suc @ M5 ) ) ) ).

% gr0_implies_Suc
thf(fact_1201_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_1202_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K2 )
               => ~ ( P @ I4 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_1203_encode__bounded__nat_Ocases,axiom,
    ! [X3: product_prod_nat_nat] :
      ( ! [L2: nat,N2: nat] :
          ( X3
         != ( product_Pair_nat_nat @ ( suc @ L2 ) @ N2 ) )
     => ~ ! [Uu2: nat] :
            ( X3
           != ( product_Pair_nat_nat @ zero_zero_nat @ Uu2 ) ) ) ).

% encode_bounded_nat.cases
thf(fact_1204_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_1205_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_1206_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_1207_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_1208_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ~ ( P @ N2 )
             => ? [M2: nat] :
                  ( ( ord_less_nat @ M2 @ N2 )
                  & ~ ( P @ M2 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_1209_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_1210_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_1211_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_1212_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_1213_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_1214_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_1215_encode__unary__nat_Ocases,axiom,
    ! [X3: nat] :
      ( ! [L2: nat] :
          ( X3
         != ( suc @ L2 ) )
     => ( X3 = zero_zero_nat ) ) ).

% encode_unary_nat.cases
thf(fact_1216_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M5: nat] :
          ( N
          = ( suc @ M5 ) ) ) ).

% not0_implies_Suc
thf(fact_1217_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_1218_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_1219_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_1220_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_1221_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
     => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
       => ( ! [X4: nat,Y3: nat] :
              ( ( P @ X4 @ Y3 )
             => ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
         => ( P @ M @ N ) ) ) ) ).

% diff_induct
thf(fact_1222_nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P @ N2 )
           => ( P @ ( suc @ N2 ) ) )
       => ( P @ N ) ) ) ).

% nat_induct
thf(fact_1223_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_1224_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_1225_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_1226_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_1227_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_1228_bot__nat__0_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X2: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X2 )
    @ ^ [X2: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X2 )
    @ zero_zero_nat ) ).

% bot_nat_0.ordering_top_axioms
thf(fact_1229_min__Suc2,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_min_nat @ M @ ( suc @ N ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M7: nat] : ( suc @ ( ord_min_nat @ M7 @ N ) )
        @ M ) ) ).

% min_Suc2
thf(fact_1230_min__Suc1,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_min_nat @ ( suc @ N ) @ M )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M7: nat] : ( suc @ ( ord_min_nat @ N @ M7 ) )
        @ M ) ) ).

% min_Suc1
thf(fact_1231_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K2 )
               => ~ ( P @ I4 ) )
            & ( P @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_1232_tl__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( tl_nat @ ( upt @ M @ N ) )
      = ( upt @ ( suc @ M ) @ N ) ) ).

% tl_upt
thf(fact_1233_hd__upt,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( hd_nat @ ( upt @ I2 @ J2 ) )
        = I2 ) ) ).

% hd_upt
thf(fact_1234_upt__conv__Nil,axiom,
    ! [J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ J2 @ I2 )
     => ( ( upt @ I2 @ J2 )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_1235_upt__eq__Nil__conv,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ( upt @ I2 @ J2 )
        = nil_nat )
      = ( ( J2 = zero_zero_nat )
        | ( ord_less_eq_nat @ J2 @ I2 ) ) ) ).

% upt_eq_Nil_conv
thf(fact_1236_upt__0,axiom,
    ! [I2: nat] :
      ( ( upt @ I2 @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_1237_upt__Suc,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J2 )
       => ( ( upt @ I2 @ ( suc @ J2 ) )
          = ( append_nat @ ( upt @ I2 @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I2 @ J2 )
       => ( ( upt @ I2 @ ( suc @ J2 ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_1238_upt__Suc__append,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( upt @ I2 @ ( suc @ J2 ) )
        = ( append_nat @ ( upt @ I2 @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_1239_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N: nat,Ns: list_nat,Q: nat] :
      ( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
        = ( upt @ M @ Q ) )
      = ( ( cons_nat @ N @ Ns )
        = ( upt @ ( suc @ M ) @ Q ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_1240_upt__conv__Cons,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( upt @ I2 @ J2 )
        = ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J2 ) ) ) ) ).

% upt_conv_Cons
thf(fact_1241_upt__rec,axiom,
    ( upt
    = ( ^ [I3: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I3 @ J3 ) @ ( cons_nat @ I3 @ ( upt @ ( suc @ I3 ) @ J3 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_1242_map__Suc__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
      = ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% map_Suc_upt
thf(fact_1243_pred__def,axiom,
    ( pred
    = ( case_nat_nat @ zero_zero_nat
      @ ^ [X23: nat] : X23 ) ) ).

% pred_def
thf(fact_1244_list__encode_Ocases,axiom,
    ! [X3: list_nat] :
      ( ( X3 != nil_nat )
     => ~ ! [X4: nat,Xs2: list_nat] :
            ( X3
           != ( cons_nat @ X4 @ Xs2 ) ) ) ).

% list_encode.cases
thf(fact_1245_prod__decode__aux_Ocases,axiom,
    ! [X3: product_prod_nat_nat] :
      ~ ! [K2: nat,M5: nat] :
          ( X3
         != ( product_Pair_nat_nat @ K2 @ M5 ) ) ).

% prod_decode_aux.cases
thf(fact_1246_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_1247_power__down_Ocases,axiom,
    ! [X3: produc9155354469731034754al_nat] :
      ( ! [P6: nat,X4: real] :
          ( X3
         != ( produc4490577844368043572al_nat @ P6 @ ( produc3181502643871035669al_nat @ X4 @ zero_zero_nat ) ) )
     => ~ ! [P6: nat,X4: real,N2: nat] :
            ( X3
           != ( produc4490577844368043572al_nat @ P6 @ ( produc3181502643871035669al_nat @ X4 @ ( suc @ N2 ) ) ) ) ) ).

% power_down.cases
thf(fact_1248_add__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc_right
thf(fact_1249_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_1250_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_1251_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_1252_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_1253_drop__upt,axiom,
    ! [M: nat,I2: nat,J2: nat] :
      ( ( drop_nat @ M @ ( upt @ I2 @ J2 ) )
      = ( upt @ ( plus_plus_nat @ I2 @ M ) @ J2 ) ) ).

% drop_upt
thf(fact_1254_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_1255_nth__upt,axiom,
    ! [I2: nat,K: nat,J2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 )
     => ( ( nth_nat @ ( upt @ I2 @ J2 ) @ K )
        = ( plus_plus_nat @ I2 @ K ) ) ) ).

% nth_upt
thf(fact_1256_take__upt,axiom,
    ! [I2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ M ) @ N )
     => ( ( take_nat @ M @ ( upt @ I2 @ N ) )
        = ( upt @ I2 @ ( plus_plus_nat @ I2 @ M ) ) ) ) ).

% take_upt
thf(fact_1257_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_1258_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_1259_less__imp__add__positive,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I2 @ K2 )
            = J2 ) ) ) ).

% less_imp_add_positive
thf(fact_1260_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_1261_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_1262_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N3: nat] :
        ? [K3: nat] :
          ( N3
          = ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_1263_trans__le__add2,axiom,
    ! [I2: nat,J2: nat,M: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J2 ) ) ) ).

% trans_le_add2
thf(fact_1264_trans__le__add1,axiom,
    ! [I2: nat,J2: nat,M: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J2 @ M ) ) ) ).

% trans_le_add1
thf(fact_1265_add__le__mono1,axiom,
    ! [I2: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).

% add_le_mono1
thf(fact_1266_add__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 @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).

% add_le_mono
thf(fact_1267_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N2: nat] :
          ( L
          = ( plus_plus_nat @ K @ N2 ) ) ) ).

% le_Suc_ex
thf(fact_1268_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ K @ N ) ) ).

% add_leD2

% Helper facts (13)
thf(help_fChoice_1_1_fChoice_001t__Nat__Onat_T,axiom,
    ! [P: nat > $o] :
      ( ( P @ ( fChoice_nat @ P ) )
      = ( ? [X8: nat] : ( P @ X8 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__List__Olist_Itf__b_J_T,axiom,
    ! [P: list_b > $o] :
      ( ( P @ ( fChoice_list_b @ P ) )
      = ( ? [X8: list_b] : ( P @ X8 ) ) ) ).

thf(help_If_3_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X3: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X3: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_fChoice_1_1_fChoice_001_062_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [P: ( nat > nat ) > $o] :
      ( ( P @ ( fChoice_nat_nat @ P ) )
      = ( ? [X8: nat > nat] : ( P @ X8 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( P @ ( fChoic6978938873391328853at_nat @ P ) )
      = ( ? [X8: product_prod_nat_nat] : ( P @ X8 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_T,axiom,
    ! [P: produc3741383161447143261al_nat > $o] :
      ( ( P @ ( fChoic8332448170627812017al_nat @ P ) )
      = ( ? [X8: produc3741383161447143261al_nat] : ( P @ X8 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J_T,axiom,
    ! [P: produc7811952446676219690list_b > $o] :
      ( ( P @ ( fChoic403860114826281174list_b @ P ) )
      = ( ? [X8: produc7811952446676219690list_b] : ( P @ X8 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_T,axiom,
    ! [P: ( product_prod_nat_nat > nat ) > $o] :
      ( ( P @ ( fChoic638070965443183254at_nat @ P ) )
      = ( ? [X8: product_prod_nat_nat > nat] : ( P @ X8 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J_T,axiom,
    ! [P: produc9155354469731034754al_nat > $o] :
      ( ( P @ ( fChoic5522844001492335150al_nat @ P ) )
      = ( ? [X8: produc9155354469731034754al_nat] : ( P @ X8 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [P: produc859450856879609959at_nat > $o] :
      ( ( P @ ( fChoic6673916560228149179at_nat @ P ) )
      = ( ? [X8: produc859450856879609959at_nat] : ( P @ X8 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_T,axiom,
    ! [P: ( produc859450856879609959at_nat > nat ) > $o] :
      ( ( P @ ( fChoic8220946065393586812at_nat @ P ) )
      = ( ? [X8: produc859450856879609959at_nat > nat] : ( P @ X8 ) ) ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( inj_on8733544287791916926list_b
    @ ^ [Xs3: list_b] : ( produc7903367361620597084list_b @ ( size_size_list_b @ Xs3 ) @ Xs3 )
    @ top_top_set_list_b ) ).

%------------------------------------------------------------------------------