TPTP Problem File: SLH0849^1.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    : Query_Optimization/0012_IKKBZ/prob_02958_129714__15629462_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1804 ( 446 unt; 533 typ;   0 def)
%            Number of atoms       : 4107 (1266 equ;   0 cnn)
%            Maximal formula atoms :   27 (   3 avg)
%            Number of connectives : 14754 ( 549   ~;  39   |; 245   &;11855   @)
%                                         (   0 <=>;2066  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   26 (   8 avg)
%            Number of types       :  105 ( 104 usr)
%            Number of type conns  : 1365 (1365   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  431 ( 429 usr;  32 con; 0-4 aty)
%            Number of variables   : 4683 ( 370   ^;4092   !; 221   ?;4683   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-18 16:07:34.559
%------------------------------------------------------------------------------
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_J,type,
    list_P2281458131032066856st_a_b: $tType ).

thf(ty_n_t__Dtree__Odtree_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J,type,
    dtree_2614965411716861998_a_b_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_J,type,
    set_Pr9033091269622839554st_a_b: $tType ).

thf(ty_n_t__List__Olist_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J,type,
    list_dtree_list_a_b: $tType ).

thf(ty_n_t__FSet__Ofset_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J,type,
    fset_dtree_list_a_b: $tType ).

thf(ty_n_t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
    dtree_list_a_list_a: $tType ).

thf(ty_n_t__Set__Oset_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J,type,
    set_dtree_list_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J,type,
    produc2579390649552321826st_a_b: $tType ).

thf(ty_n_t__Dtree__Odtree_It__Real__Oreal_Mt__List__Olist_Itf__a_J_J,type,
    dtree_real_list_a: $tType ).

thf(ty_n_t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mt__Real__Oreal_J,type,
    dtree_list_a_real: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    set_set_list_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__List__Olist_Itf__a_J_M_Eo_J_J,type,
    set_list_a_o: $tType ).

thf(ty_n_t__Dtree__Odtree_It__Real__Oreal_Mt__Real__Oreal_J,type,
    dtree_real_real: $tType ).

thf(ty_n_t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
    dtree_list_a_b: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
    set_set_real: $tType ).

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

thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_M_Eo_J_J,type,
    set_real_o: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    list_list_a: $tType ).

thf(ty_n_t__FSet__Ofset_It__List__Olist_Itf__a_J_J,type,
    fset_list_a: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    set_list_a: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    set_set_a: $tType ).

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

thf(ty_n_t__FSet__Ofset_It__Real__Oreal_J,type,
    fset_real: $tType ).

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

thf(ty_n_t__FSet__Ofset_It__Nat__Onat_J,type,
    fset_nat: $tType ).

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

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

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

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

thf(ty_n_t__Set__Oset_I_Eo_J,type,
    set_o: $tType ).

thf(ty_n_t__Real__Oreal,type,
    real: $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 (429)
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
    bNF_Ca4824630504345799472_a_nat: set_Pr4048851178543822343list_a > ( list_a > nat ) > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__List__Olist_Itf__a_J_001t__Real__Oreal,type,
    bNF_Ca1029839828519009804a_real: set_Pr4048851178543822343list_a > ( list_a > real ) > $o ).

thf(sy_c_Basic__BNFs_Ofsts_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_001tf__b,type,
    basic_5528648086025252546_a_b_b: produc6499617310964463488_a_b_b > set_dtree_list_a_b ).

thf(sy_c_Basic__BNFs_Osnds_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_001tf__b,type,
    basic_3193785740840883972_a_b_b: produc6499617310964463488_a_b_b > set_b ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_M_062_Itf__b_M_Eo_J_J,type,
    comple5639742380414582692_b_b_o: set_dt4649734747845939591_b_b_o > dtree_list_a_b > b > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_M_Eo_J,type,
    comple8325939764076033224_a_b_o: set_dtree_list_a_b_o > dtree_list_a_b > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__List__Olist_Itf__a_J_M_062_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_M_Eo_J_J,type,
    comple396325037688036751_a_b_o: set_li4074879222405492210_a_b_o > list_a > dtree_list_a_b > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__List__Olist_Itf__a_J_M_062_It__List__Olist_Itf__a_J_M_Eo_J_J,type,
    comple3554826314717667675st_a_o: set_list_a_list_a_o > list_a > list_a > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__List__Olist_Itf__a_J_M_062_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J_M_Eo_J_J,type,
    comple8439212109263825820_a_b_o: set_li7652136325503127935_a_b_o > list_a > produc111314985273491367st_a_b > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__List__Olist_Itf__a_J_M_Eo_J,type,
    comple7218726029605528012st_a_o: set_list_a_o > list_a > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J_M_Eo_J,type,
    comple1499699326375664188_b_b_o: set_Pr5782749397791665583_b_b_o > produc6499617310964463488_a_b_b > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Real__Oreal_M_Eo_J,type,
    comple3015195443809154064real_o: set_real_o > real > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
    complete_Sup_Sup_o: set_o > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
    complete_Sup_Sup_nat: set_nat > nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
    comple1385675409528146559p_real: set_real > real ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J,type,
    comple3787021748553583613st_a_b: set_se4773973851007363424st_a_b > set_dtree_list_a_b ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    comple6928918032620976721list_a: set_set_list_a > set_list_a ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
    comple7399068483239264473et_nat: set_set_nat > set_nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J_J,type,
    comple5737441310262272393_a_b_b: set_se486103598301167852_a_b_b > set_Pr3443975907877334966_a_b_b ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J_J,type,
    comple8655260255396921136st_a_b: set_se3485313402130513683st_a_b > set_Pr5727314569254441693st_a_b ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
    comple1497117618345130356list_a: set_se4091964020838898919list_a > set_Pr4048851178543822343list_a ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J_J_J,type,
    comple3151476520729156835st_a_b: set_se3213122578561875782st_a_b > set_Pr8451570673126195472st_a_b ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Real__Oreal_J,type,
    comple3096694443085538997t_real: set_set_real > set_real ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
    comple2307003609928055243_set_a: set_set_a > set_a ).

thf(sy_c_Dtree_Odhead_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_001t__List__Olist_Itf__a_J,type,
    dhead_2549497505893332128list_a: dtree_8284992029673318451list_a > ( list_a > dtree_list_a_b ) > list_a > dtree_list_a_b ).

thf(sy_c_Dtree_Odhead_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_001t__Real__Oreal,type,
    dhead_5461920678355059344b_real: dtree_8345463500551380051b_real > ( real > dtree_list_a_b ) > real > dtree_list_a_b ).

thf(sy_c_Dtree_Odhead_001t__List__Olist_Itf__a_J_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
    dhead_7954756475827843210st_a_b: dtree_4875458026974816597st_a_b > ( dtree_list_a_b > list_a ) > dtree_list_a_b > list_a ).

thf(sy_c_Dtree_Odhead_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    dhead_list_a_list_a: dtree_list_a_list_a > ( list_a > list_a ) > list_a > list_a ).

thf(sy_c_Dtree_Odhead_001t__List__Olist_Itf__a_J_001t__Real__Oreal,type,
    dhead_list_a_real: dtree_list_a_real > ( real > list_a ) > real > list_a ).

thf(sy_c_Dtree_Odhead_001t__List__Olist_Itf__a_J_001tf__b,type,
    dhead_list_a_b: dtree_list_a_b > ( b > list_a ) > b > list_a ).

thf(sy_c_Dtree_Odhead_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J_001t__Real__Oreal,type,
    dhead_8830158393902075676b_real: dtree_310814809473010911b_real > ( real > produc6499617310964463488_a_b_b ) > real > produc6499617310964463488_a_b_b ).

thf(sy_c_Dtree_Odhead_001t__Real__Oreal_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
    dhead_5523335947888865680st_a_b: dtree_3696687526968142675st_a_b > ( dtree_list_a_b > real ) > dtree_list_a_b > real ).

thf(sy_c_Dtree_Odhead_001t__Real__Oreal_001t__List__Olist_Itf__a_J,type,
    dhead_real_list_a: dtree_real_list_a > ( list_a > real ) > list_a > real ).

thf(sy_c_Dtree_Odhead_001t__Real__Oreal_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J,type,
    dhead_8734047501402136348_a_b_b: dtree_7302148650479942367_a_b_b > ( produc6499617310964463488_a_b_b > real ) > produc6499617310964463488_a_b_b > real ).

thf(sy_c_Dtree_Odhead_001t__Real__Oreal_001t__Real__Oreal,type,
    dhead_real_real: dtree_real_real > ( real > real ) > real > real ).

thf(sy_c_Dtree_Odtail_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_001t__List__Olist_Itf__a_J,type,
    dtail_7497147172532560816list_a: dtree_8284992029673318451list_a > ( list_a > dtree_list_a_b ) > list_a > dtree_list_a_b ).

thf(sy_c_Dtree_Odtail_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_001t__Real__Oreal,type,
    dtail_8649134905595490176b_real: dtree_8345463500551380051b_real > ( real > dtree_list_a_b ) > real > dtree_list_a_b ).

thf(sy_c_Dtree_Odtail_001t__List__Olist_Itf__a_J_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
    dtail_3679034105612296090st_a_b: dtree_4875458026974816597st_a_b > ( dtree_list_a_b > list_a ) > dtree_list_a_b > list_a ).

thf(sy_c_Dtree_Odtail_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    dtail_list_a_list_a: dtree_list_a_list_a > ( list_a > list_a ) > list_a > list_a ).

thf(sy_c_Dtree_Odtail_001t__List__Olist_Itf__a_J_001t__Real__Oreal,type,
    dtail_list_a_real: dtree_list_a_real > ( real > list_a ) > real > list_a ).

thf(sy_c_Dtree_Odtail_001t__List__Olist_Itf__a_J_001tf__b,type,
    dtail_list_a_b: dtree_list_a_b > ( b > list_a ) > b > list_a ).

thf(sy_c_Dtree_Odtail_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J_001t__Real__Oreal,type,
    dtail_6110999216585821196b_real: dtree_310814809473010911b_real > ( real > produc6499617310964463488_a_b_b ) > real > produc6499617310964463488_a_b_b ).

thf(sy_c_Dtree_Odtail_001t__Real__Oreal_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
    dtail_8710550175129296512st_a_b: dtree_3696687526968142675st_a_b > ( dtree_list_a_b > real ) > dtree_list_a_b > real ).

thf(sy_c_Dtree_Odtail_001t__Real__Oreal_001t__List__Olist_Itf__a_J,type,
    dtail_real_list_a: dtree_real_list_a > ( list_a > real ) > list_a > real ).

thf(sy_c_Dtree_Odtail_001t__Real__Oreal_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J,type,
    dtail_6014888324085881868_a_b_b: dtree_7302148650479942367_a_b_b > ( produc6499617310964463488_a_b_b > real ) > produc6499617310964463488_a_b_b > real ).

thf(sy_c_Dtree_Odtail_001t__Real__Oreal_001t__Real__Oreal,type,
    dtail_real_real: dtree_real_real > ( real > real ) > real > real ).

thf(sy_c_Dtree_Odtree_ONode_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_001t__List__Olist_Itf__a_J,type,
    node_d2950405300658840400list_a: dtree_list_a_b > fset_P5658613431985659038list_a > dtree_8284992029673318451list_a ).

thf(sy_c_Dtree_Odtree_ONode_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_001t__Real__Oreal,type,
    node_d335693152011667424b_real: dtree_list_a_b > fset_P9074655572916523974l_real > dtree_8345463500551380051b_real ).

thf(sy_c_Dtree_Odtree_ONode_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_001tf__b,type,
    node_d3522988236360801483_a_b_b: dtree_list_a_b > fset_P8033862543129508702_b_b_b > dtree_2614965411716861998_a_b_b ).

thf(sy_c_Dtree_Odtree_ONode_001t__List__Olist_Itf__a_J_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
    node_l8355664270593351482st_a_b: list_a > fset_P6340920821931359400st_a_b > dtree_4875458026974816597st_a_b ).

thf(sy_c_Dtree_Odtree_ONode_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    node_list_a_list_a: list_a > fset_P6656299774166858224list_a > dtree_list_a_list_a ).

thf(sy_c_Dtree_Odtree_ONode_001t__List__Olist_Itf__a_J_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J,type,
    node_l6315829471324094445st_a_b: list_a > fset_P1196402277819410702st_a_b > dtree_5894418989983070344st_a_b ).

thf(sy_c_Dtree_Odtree_ONode_001t__List__Olist_Itf__a_J_001t__Real__Oreal,type,
    node_list_a_real: list_a > fset_P6497461246188700184l_real > dtree_list_a_real ).

thf(sy_c_Dtree_Odtree_ONode_001t__List__Olist_Itf__a_J_001tf__b,type,
    node_list_a_b: list_a > fset_P2153231429829016240_a_b_b > dtree_list_a_b ).

thf(sy_c_Dtree_Odtree_ONode_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J_001t__Real__Oreal,type,
    node_P8407631645849338988b_real: produc6499617310964463488_a_b_b > fset_P1597832662759650386l_real > dtree_310814809473010911b_real ).

thf(sy_c_Dtree_Odtree_ONode_001t__Real__Oreal_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
    node_r397108421545473760st_a_b: real > fset_P5746387328717196814st_a_b > dtree_3696687526968142675st_a_b ).

thf(sy_c_Dtree_Odtree_ONode_001t__Real__Oreal_001t__List__Olist_Itf__a_J,type,
    node_real_list_a: real > fset_P1040399934802234454list_a > dtree_real_list_a ).

thf(sy_c_Dtree_Odtree_ONode_001t__Real__Oreal_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J,type,
    node_r8311520753349399660_a_b_b: real > fset_P2088967965053398822_a_b_b > dtree_7302148650479942367_a_b_b ).

thf(sy_c_Dtree_Odtree_ONode_001t__Real__Oreal_001t__Real__Oreal,type,
    node_real_real: real > fset_P7499862231470980990l_real > dtree_real_real ).

thf(sy_c_Dtree_Odtree_Odarcs_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_001t__List__Olist_Itf__a_J,type,
    darcs_4898656622333939429list_a: dtree_8284992029673318451list_a > set_list_a ).

thf(sy_c_Dtree_Odtree_Odarcs_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_001t__Real__Oreal,type,
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thf(sy_c_IKKBZ_Odenormalize_001tf__a_001tf__b,type,
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thf(sy_c_IKKBZ_Oold_001tf__a_001tf__b,type,
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thf(sy_c_IKKBZ_Oranked__dtree_001tf__a_001tf__b,type,
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thf(sy_c_IKKBZ_Oranked__dtree_Omerge_001tf__a_001tf__b,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_M_062_Itf__b_M_Eo_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_M_Eo_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__List__Olist_Itf__a_J_M_062_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_M_Eo_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__List__Olist_Itf__a_J_M_Eo_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Real__Oreal,type,
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thf(sy_c_List_Oappend_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
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thf(sy_c_List_Oappend_001tf__a,type,
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    image_real_list_a: ( real > list_a ) > set_real > set_list_a ).

thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Nat__Onat,type,
    image_real_nat: ( real > nat ) > set_real > set_nat ).

thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J,type,
    image_3400966383075225011_a_b_b: ( real > produc6499617310964463488_a_b_b ) > set_real > set_Pr3443975907877334966_a_b_b ).

thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
    image_real_real: ( real > real ) > set_real > set_real ).

thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Set__Oset_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J,type,
    image_5051693651339568221st_a_b: ( real > set_dtree_list_a_b ) > set_real > set_se4773973851007363424st_a_b ).

thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    image_3983288985320528369list_a: ( real > set_list_a ) > set_real > set_set_list_a ).

thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Set__Oset_It__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J_J,type,
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thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Set__Oset_It__Real__Oreal_J,type,
    image_real_set_real: ( real > set_real ) > set_real > set_set_real ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J_001_062_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_M_Eo_J,type,
    image_430045828685645930_a_b_o: ( set_dtree_list_a_b > dtree_list_a_b > $o ) > set_se4773973851007363424st_a_b > set_dtree_list_a_b_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J_001_Eo,type,
    image_6096855620777653543_a_b_o: ( set_dtree_list_a_b > $o ) > set_se4773973851007363424st_a_b > set_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001_062_It__List__Olist_Itf__a_J_M_Eo_J,type,
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thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001_Eo,type,
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thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J_J_001_062_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_M_062_Itf__b_M_Eo_J_J,type,
    image_7324209900139855886_b_b_o: ( set_Pr3443975907877334966_a_b_b > dtree_list_a_b > b > $o ) > set_se486103598301167852_a_b_b > set_dt4649734747845939591_b_b_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J_J_001_062_It__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J_M_Eo_J,type,
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thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J_J_001_Eo,type,
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thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J_J_001_062_It__List__Olist_Itf__a_J_M_062_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_M_Eo_J_J,type,
    image_3304428138862881312_a_b_o: ( set_Pr5727314569254441693st_a_b > list_a > dtree_list_a_b > $o ) > set_se3485313402130513683st_a_b > set_li4074879222405492210_a_b_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_001_062_It__List__Olist_Itf__a_J_M_062_It__List__Olist_Itf__a_J_M_Eo_J_J,type,
    image_8564175721147914668st_a_o: ( set_Pr4048851178543822343list_a > list_a > list_a > $o ) > set_se4091964020838898919list_a > set_list_a_list_a_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J_J_J_001_062_It__List__Olist_Itf__a_J_M_062_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J_M_Eo_J_J,type,
    image_1933873314476848224_a_b_o: ( set_Pr8451570673126195472st_a_b > list_a > produc111314985273491367st_a_b > $o ) > set_se3213122578561875782st_a_b > set_li7652136325503127935_a_b_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001_062_It__Real__Oreal_M_Eo_J,type,
    image_5650221686686655994real_o: ( set_real > real > $o ) > set_set_real > set_real_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001_Eo,type,
    image_set_real_o: ( set_real > $o ) > set_set_real > set_o ).

thf(sy_c_Set_Oinsert_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
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thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
    insert_list_a: list_a > set_list_a > set_list_a ).

thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
    insert_nat: nat > set_nat > set_nat ).

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J,type,
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thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
    insert_real: real > set_real > set_real ).

thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
    is_empty_nat: set_nat > $o ).

thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J,type,
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thf(sy_c_Wellfounded_Oaccp_001t__Dtree__Odtree_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J,type,
    accp_d827264278867218999_a_b_b: ( dtree_2614965411716861998_a_b_b > dtree_2614965411716861998_a_b_b > $o ) > dtree_2614965411716861998_a_b_b > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mt__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J,type,
    accp_d9081407519697605342st_a_b: ( dtree_4875458026974816597st_a_b > dtree_4875458026974816597st_a_b > $o ) > dtree_4875458026974816597st_a_b > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
    accp_d2674248120870298800list_a: ( dtree_list_a_list_a > dtree_list_a_list_a > $o ) > dtree_list_a_list_a > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J_J,type,
    accp_d766546008032286865st_a_b: ( dtree_5894418989983070344st_a_b > dtree_5894418989983070344st_a_b > $o ) > dtree_5894418989983070344st_a_b > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
    accp_dtree_list_a_b: ( dtree_list_a_b > dtree_list_a_b > $o ) > dtree_list_a_b > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J,type,
    accp_P3553483059299338846st_a_b: ( produc111314985273491367st_a_b > produc111314985273491367st_a_b > $o ) > produc111314985273491367st_a_b > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J_J,type,
    accp_P2814348900907978769st_a_b: ( produc7147531718898801626st_a_b > produc7147531718898801626st_a_b > $o ) > produc7147531718898801626st_a_b > $o ).

thf(sy_c_Wellfounded_Omeasure_001t__List__Olist_Itf__a_J,type,
    measure_list_a: ( list_a > nat ) > set_Pr4048851178543822343list_a ).

thf(sy_c_member_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
    member551035911493665803st_a_b: dtree_list_a_b > set_dtree_list_a_b > $o ).

thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
    member_list_a: list_a > set_list_a > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mt__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
    member2153700486347808993list_a: produc6498716942866525624list_a > set_Pr7758080164884194840list_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mt__Real__Oreal_J_Mt__Real__Oreal_J,type,
    member8458687784735308553l_real: produc7375242215863181152l_real > set_Pr2219270211170116160l_real > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mt__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J_Mt__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J,type,
    member8722885590174353295st_a_b: produc7192846479455950328st_a_b > set_Pr8515797714057792174st_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
    member535470554364689367list_a: produc7575571382841205696list_a > set_Pr6765974203441051894list_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mt__Real__Oreal_J_Mt__Real__Oreal_J,type,
    member4849742732303649023l_real: produc7420957250391652712l_real > set_Pr6334390857388424222l_real > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J,type,
    member4695696432722591383_a_b_b: produc6499617310964463488_a_b_b > set_Pr3443975907877334966_a_b_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_It__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J_Mt__Real__Oreal_J_Mt__Real__Oreal_J,type,
    member4680567731829293973l_real: produc4416935485724539116l_real > set_Pr555168240263297228l_real > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_It__Real__Oreal_Mt__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J_Mt__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J,type,
    member692244352973388625st_a_b: produc428612106868633768st_a_b > set_Pr4703722906220843656st_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_It__Real__Oreal_Mt__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
    member5451813156226304665list_a: produc2619705089329645168list_a > set_Pr1354202033690271696list_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_It__Real__Oreal_Mt__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J_J,type,
    member1217632944401329257_a_b_b: produc7654046354941100992_a_b_b > set_Pr5289323960836120480_a_b_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_It__Real__Oreal_Mt__Real__Oreal_J_Mt__Real__Oreal_J,type,
    member2617108185285288129l_real: produc5734500334197453848l_real > set_Pr4673054925704665080l_real > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J,type,
    member6763472504040263486st_a_b: produc111314985273491367st_a_b > set_Pr5727314569254441693st_a_b > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J_J,type,
    member4895573663636194545st_a_b: produc7147531718898801626st_a_b > set_Pr8451570673126195472st_a_b > $o ).

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

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

thf(sy_c_member_001tf__a,type,
    member_a: a > set_a > $o ).

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

thf(sy_v_pa____,type,
    pa: list_a ).

thf(sy_v_r____,type,
    r: list_a ).

thf(sy_v_rank,type,
    rank: list_a > real ).

thf(sy_v_v,type,
    v: list_a ).

thf(sy_v_x,type,
    x: list_a ).

thf(sy_v_xs____,type,
    xs: fset_P2153231429829016240_a_b_b ).

thf(sy_v_y,type,
    y: list_a ).

% Relevant facts (1270)
thf(fact_0_local_Omerge_Ocases,axiom,
    ! [X: dtree_list_a_b] :
      ~ ! [R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
          ( X
         != ( node_list_a_b @ R @ Xs ) ) ).

% local.merge.cases
thf(fact_1_False,axiom,
    ~ ( ord_less_eq_real @ ( rank @ ( rev_a @ v ) ) @ ( rank @ ( rev_a @ r ) ) ) ).

% False
thf(fact_2_find__pos__aux__rel_Ocong,axiom,
    find_pos_aux_rel_a_b = find_pos_aux_rel_a_b ).

% find_pos_aux_rel.cong
thf(fact_3_dtree_Oinject,axiom,
    ! [X1: list_a,X2: fset_P2153231429829016240_a_b_b,Y1: list_a,Y2: fset_P2153231429829016240_a_b_b] :
      ( ( ( node_list_a_b @ X1 @ X2 )
        = ( node_list_a_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% dtree.inject
thf(fact_4_dtree_Oset__intros_I1_J,axiom,
    ! [X1: list_a,X2: fset_P2153231429829016240_a_b_b] : ( member_list_a @ X1 @ ( dverts_list_a_b @ ( node_list_a_b @ X1 @ X2 ) ) ) ).

% dtree.set_intros(1)
thf(fact_5_Node_Oprems,axiom,
    ( ( produc6837034575241423639list_a @ x @ y )
    = ( find_pos_aux_a_b @ rank @ v @ pa @ ( node_list_a_b @ r @ xs ) ) ) ).

% Node.prems
thf(fact_6_wf__dlverts_Ocases,axiom,
    ! [X: dtree_list_a_b] :
      ~ ! [R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
          ( X
         != ( node_list_a_b @ R @ Xs ) ) ).

% wf_dlverts.cases
thf(fact_7_Node_OIH,axiom,
    ! [X2aa: produc6499617310964463488_a_b_b,X2aaa: dtree_list_a_b,P: list_a] :
      ( ( member4695696432722591383_a_b_b @ X2aa @ ( fset_P9138963618725001425_a_b_b @ xs ) )
     => ( ( member551035911493665803st_a_b @ X2aaa @ ( basic_5528648086025252546_a_b_b @ X2aa ) )
       => ( ( ( produc6837034575241423639list_a @ x @ y )
            = ( find_pos_aux_a_b @ rank @ v @ P @ X2aaa ) )
         => ( member_list_a @ y @ ( dverts_list_a_b @ X2aaa ) ) ) ) ) ).

% Node.IH
thf(fact_8_dtree_Oexhaust,axiom,
    ! [Y: dtree_list_a_b] :
      ~ ! [X12: list_a,X22: fset_P2153231429829016240_a_b_b] :
          ( Y
         != ( node_list_a_b @ X12 @ X22 ) ) ).

% dtree.exhaust
thf(fact_9_dverts__mset_Ocases,axiom,
    ! [X: dtree_list_a_b] :
      ~ ! [R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
          ( X
         != ( node_list_a_b @ R @ Xs ) ) ).

% dverts_mset.cases
thf(fact_10_dtree_Oset__cases_I1_J,axiom,
    ! [E: list_a,A: dtree_list_a_b] :
      ( ( member_list_a @ E @ ( dverts_list_a_b @ A ) )
     => ( ! [Z2: fset_P2153231429829016240_a_b_b] :
            ( A
           != ( node_list_a_b @ E @ Z2 ) )
       => ~ ! [Z1: list_a,Z2: fset_P2153231429829016240_a_b_b] :
              ( ( A
                = ( node_list_a_b @ Z1 @ Z2 ) )
             => ! [X3: produc6499617310964463488_a_b_b] :
                  ( ( member4695696432722591383_a_b_b @ X3 @ ( fset_P9138963618725001425_a_b_b @ Z2 ) )
                 => ! [Xa: dtree_list_a_b] :
                      ( ( member551035911493665803st_a_b @ Xa @ ( basic_5528648086025252546_a_b_b @ X3 ) )
                     => ~ ( member_list_a @ E @ ( dverts_list_a_b @ Xa ) ) ) ) ) ) ) ).

% dtree.set_cases(1)
thf(fact_11_dtree_Oset__intros_I2_J,axiom,
    ! [Y: produc6499617310964463488_a_b_b,X2: fset_P2153231429829016240_a_b_b,Ya: dtree_list_a_b,Yb: list_a,X1: list_a] :
      ( ( member4695696432722591383_a_b_b @ Y @ ( fset_P9138963618725001425_a_b_b @ X2 ) )
     => ( ( member551035911493665803st_a_b @ Ya @ ( basic_5528648086025252546_a_b_b @ Y ) )
       => ( ( member_list_a @ Yb @ ( dverts_list_a_b @ Ya ) )
         => ( member_list_a @ Yb @ ( dverts_list_a_b @ ( node_list_a_b @ X1 @ X2 ) ) ) ) ) ) ).

% dtree.set_intros(2)
thf(fact_12_old_Ofind__pos__aux_Ocong,axiom,
    find_pos_aux_a_b = find_pos_aux_a_b ).

% old.find_pos_aux.cong
thf(fact_13_find__pos__rank__less,axiom,
    ! [V: list_a,R2: list_a,P: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
      ( ( ord_less_eq_real @ ( rank @ ( rev_a @ V ) ) @ ( rank @ ( rev_a @ R2 ) ) )
     => ( ( find_pos_aux_a_b @ rank @ V @ P @ ( node_list_a_b @ R2 @ Xs2 ) )
        = ( produc6837034575241423639list_a @ P @ R2 ) ) ) ).

% find_pos_rank_less
thf(fact_14_rev__rev__ident,axiom,
    ! [Xs2: list_a] :
      ( ( rev_a @ ( rev_a @ Xs2 ) )
      = Xs2 ) ).

% rev_rev_ident
thf(fact_15_rev__is__rev__conv,axiom,
    ! [Xs2: list_a,Ys: list_a] :
      ( ( ( rev_a @ Xs2 )
        = ( rev_a @ Ys ) )
      = ( Xs2 = Ys ) ) ).

% rev_is_rev_conv
thf(fact_16_prod_Oinject,axiom,
    ! [X1: list_a,X2: list_a,Y1: list_a,Y2: list_a] :
      ( ( ( produc6837034575241423639list_a @ X1 @ X2 )
        = ( produc6837034575241423639list_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_17_prod_Oinject,axiom,
    ! [X1: list_a,X2: dtree_list_a_b,Y1: list_a,Y2: dtree_list_a_b] :
      ( ( ( produc148520996349637281st_a_b @ X1 @ X2 )
        = ( produc148520996349637281st_a_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_18_prod_Oinject,axiom,
    ! [X1: dtree_list_a_b,X2: b,Y1: dtree_list_a_b,Y2: b] :
      ( ( ( produc7704165765595008946_a_b_b @ X1 @ X2 )
        = ( produc7704165765595008946_a_b_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_19_prod_Oinject,axiom,
    ! [X1: list_a,X2: produc111314985273491367st_a_b,Y1: list_a,Y2: produc111314985273491367st_a_b] :
      ( ( ( produc673257793671328980st_a_b @ X1 @ X2 )
        = ( produc673257793671328980st_a_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_20_old_Oprod_Oinject,axiom,
    ! [A: list_a,B: list_a,A2: list_a,B2: list_a] :
      ( ( ( produc6837034575241423639list_a @ A @ B )
        = ( produc6837034575241423639list_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_21_old_Oprod_Oinject,axiom,
    ! [A: list_a,B: dtree_list_a_b,A2: list_a,B2: dtree_list_a_b] :
      ( ( ( produc148520996349637281st_a_b @ A @ B )
        = ( produc148520996349637281st_a_b @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_22_old_Oprod_Oinject,axiom,
    ! [A: dtree_list_a_b,B: b,A2: dtree_list_a_b,B2: b] :
      ( ( ( produc7704165765595008946_a_b_b @ A @ B )
        = ( produc7704165765595008946_a_b_b @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_23_old_Oprod_Oinject,axiom,
    ! [A: list_a,B: produc111314985273491367st_a_b,A2: list_a,B2: produc111314985273491367st_a_b] :
      ( ( ( produc673257793671328980st_a_b @ A @ B )
        = ( produc673257793671328980st_a_b @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_24_order__refl,axiom,
    ! [X: real] : ( ord_less_eq_real @ X @ X ) ).

% order_refl
thf(fact_25_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_26_dual__order_Orefl,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ A ) ).

% dual_order.refl
thf(fact_27_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_28_old_Ofind__pos__rank__less,axiom,
    ! [T: dtree_list_a_b,Rank: list_a > real,V: list_a,R2: list_a,P: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
      ( ( old_a_b @ T )
     => ( ( ord_less_eq_real @ ( Rank @ ( rev_a @ V ) ) @ ( Rank @ ( rev_a @ R2 ) ) )
       => ( ( find_pos_aux_a_b @ Rank @ V @ P @ ( node_list_a_b @ R2 @ Xs2 ) )
          = ( produc6837034575241423639list_a @ P @ R2 ) ) ) ) ).

% old.find_pos_rank_less
thf(fact_29_find__pos__aux_Osimps_I2_J,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,V: list_a,R2: list_a,P: list_a] :
      ( ! [X3: produc6499617310964463488_a_b_b] :
          ( Xs2
         != ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
     => ( ( ( ord_less_eq_real @ ( rank @ ( rev_a @ V ) ) @ ( rank @ ( rev_a @ R2 ) ) )
         => ( ( find_pos_aux_a_b @ rank @ V @ P @ ( node_list_a_b @ R2 @ Xs2 ) )
            = ( produc6837034575241423639list_a @ P @ R2 ) ) )
        & ( ~ ( ord_less_eq_real @ ( rank @ ( rev_a @ V ) ) @ ( rank @ ( rev_a @ R2 ) ) )
         => ( ( find_pos_aux_a_b @ rank @ V @ P @ ( node_list_a_b @ R2 @ Xs2 ) )
            = ( produc6837034575241423639list_a @ R2 @ R2 ) ) ) ) ) ).

% find_pos_aux.simps(2)
thf(fact_30_pred__equals__eq2,axiom,
    ! [R3: set_Pr4048851178543822343list_a,S: set_Pr4048851178543822343list_a] :
      ( ( ( ^ [X4: list_a,Y3: list_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Y3 ) @ R3 ) )
        = ( ^ [X4: list_a,Y3: list_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Y3 ) @ S ) ) )
      = ( R3 = S ) ) ).

% pred_equals_eq2
thf(fact_31_pred__equals__eq2,axiom,
    ! [R3: set_Pr5727314569254441693st_a_b,S: set_Pr5727314569254441693st_a_b] :
      ( ( ( ^ [X4: list_a,Y3: dtree_list_a_b] : ( member6763472504040263486st_a_b @ ( produc148520996349637281st_a_b @ X4 @ Y3 ) @ R3 ) )
        = ( ^ [X4: list_a,Y3: dtree_list_a_b] : ( member6763472504040263486st_a_b @ ( produc148520996349637281st_a_b @ X4 @ Y3 ) @ S ) ) )
      = ( R3 = S ) ) ).

% pred_equals_eq2
thf(fact_32_pred__equals__eq2,axiom,
    ! [R3: set_Pr3443975907877334966_a_b_b,S: set_Pr3443975907877334966_a_b_b] :
      ( ( ( ^ [X4: dtree_list_a_b,Y3: b] : ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X4 @ Y3 ) @ R3 ) )
        = ( ^ [X4: dtree_list_a_b,Y3: b] : ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X4 @ Y3 ) @ S ) ) )
      = ( R3 = S ) ) ).

% pred_equals_eq2
thf(fact_33_pred__equals__eq2,axiom,
    ! [R3: set_Pr8451570673126195472st_a_b,S: set_Pr8451570673126195472st_a_b] :
      ( ( ( ^ [X4: list_a,Y3: produc111314985273491367st_a_b] : ( member4895573663636194545st_a_b @ ( produc673257793671328980st_a_b @ X4 @ Y3 ) @ R3 ) )
        = ( ^ [X4: list_a,Y3: produc111314985273491367st_a_b] : ( member4895573663636194545st_a_b @ ( produc673257793671328980st_a_b @ X4 @ Y3 ) @ S ) ) )
      = ( R3 = S ) ) ).

% pred_equals_eq2
thf(fact_34_relChain__def,axiom,
    ( bNF_Ca1029839828519009804a_real
    = ( ^ [R4: set_Pr4048851178543822343list_a,As: list_a > real] :
        ! [I: list_a,J: list_a] :
          ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ I @ J ) @ R4 )
         => ( ord_less_eq_real @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_35_relChain__def,axiom,
    ( bNF_Ca4824630504345799472_a_nat
    = ( ^ [R4: set_Pr4048851178543822343list_a,As: list_a > nat] :
        ! [I: list_a,J: list_a] :
          ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ I @ J ) @ R4 )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_36_rev__swap,axiom,
    ! [Xs2: list_a,Ys: list_a] :
      ( ( ( rev_a @ Xs2 )
        = Ys )
      = ( Xs2
        = ( rev_a @ Ys ) ) ) ).

% rev_swap
thf(fact_37_dtree_Oset__sel_I2_J,axiom,
    ! [X: produc6499617310964463488_a_b_b,A: dtree_list_a_b,Xa2: dtree_list_a_b,Xb: list_a] :
      ( ( member4695696432722591383_a_b_b @ X @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ A ) ) )
     => ( ( member551035911493665803st_a_b @ Xa2 @ ( basic_5528648086025252546_a_b_b @ X ) )
       => ( ( member_list_a @ Xb @ ( dverts_list_a_b @ Xa2 ) )
         => ( member_list_a @ Xb @ ( dverts_list_a_b @ A ) ) ) ) ) ).

% dtree.set_sel(2)
thf(fact_38_find__pos_Ocases,axiom,
    ! [X: produc111314985273491367st_a_b] :
      ( ! [V2: list_a,R: list_a,T1: dtree_list_a_b,Uu: b] :
          ( X
         != ( produc148520996349637281st_a_b @ V2 @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
     => ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
            ( ! [X5: produc6499617310964463488_a_b_b] :
                ( Xs
               != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
           => ! [V2: list_a,R: list_a] :
                ( X
               != ( produc148520996349637281st_a_b @ V2 @ ( node_list_a_b @ R @ Xs ) ) ) ) ) ).

% find_pos.cases
thf(fact_39_find__pos__aux_Ocases,axiom,
    ! [X: produc7147531718898801626st_a_b] :
      ( ! [V2: list_a,P2: list_a,R: list_a,T1: dtree_list_a_b,Uu: b] :
          ( X
         != ( produc673257793671328980st_a_b @ V2 @ ( produc148520996349637281st_a_b @ P2 @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) )
     => ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
            ( ! [X5: produc6499617310964463488_a_b_b] :
                ( Xs
               != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
           => ! [V2: list_a,P2: list_a,R: list_a] :
                ( X
               != ( produc673257793671328980st_a_b @ V2 @ ( produc148520996349637281st_a_b @ P2 @ ( node_list_a_b @ R @ Xs ) ) ) ) ) ) ).

% find_pos_aux.cases
thf(fact_40_find__pos__aux_Oelims,axiom,
    ! [X: list_a,Xa2: list_a,Xb: dtree_list_a_b,Y: produc9164743771328383783list_a] :
      ( ( ( find_pos_aux_a_b @ rank @ X @ Xa2 @ Xb )
        = Y )
     => ( ! [R: list_a,T1: dtree_list_a_b] :
            ( ? [Uu: b] :
                ( Xb
                = ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) )
           => ~ ( ( ( ord_less_eq_real @ ( rank @ ( rev_a @ X ) ) @ ( rank @ ( rev_a @ R ) ) )
                 => ( Y
                    = ( produc6837034575241423639list_a @ Xa2 @ R ) ) )
                & ( ~ ( ord_less_eq_real @ ( rank @ ( rev_a @ X ) ) @ ( rank @ ( rev_a @ R ) ) )
                 => ( Y
                    = ( find_pos_aux_a_b @ rank @ X @ R @ T1 ) ) ) ) )
       => ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
              ( ! [X5: produc6499617310964463488_a_b_b] :
                  ( Xs
                 != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
             => ! [R: list_a] :
                  ( ( Xb
                    = ( node_list_a_b @ R @ Xs ) )
                 => ~ ( ( ( ord_less_eq_real @ ( rank @ ( rev_a @ X ) ) @ ( rank @ ( rev_a @ R ) ) )
                       => ( Y
                          = ( produc6837034575241423639list_a @ Xa2 @ R ) ) )
                      & ( ~ ( ord_less_eq_real @ ( rank @ ( rev_a @ X ) ) @ ( rank @ ( rev_a @ R ) ) )
                       => ( Y
                          = ( produc6837034575241423639list_a @ R @ R ) ) ) ) ) ) ) ) ).

% find_pos_aux.elims
thf(fact_41_find__pos__aux_Osimps_I1_J,axiom,
    ! [V: list_a,R2: list_a,P: list_a,T12: dtree_list_a_b,Uu2: b] :
      ( ( ( ord_less_eq_real @ ( rank @ ( rev_a @ V ) ) @ ( rank @ ( rev_a @ R2 ) ) )
       => ( ( find_pos_aux_a_b @ rank @ V @ P @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ Uu2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
          = ( produc6837034575241423639list_a @ P @ R2 ) ) )
      & ( ~ ( ord_less_eq_real @ ( rank @ ( rev_a @ V ) ) @ ( rank @ ( rev_a @ R2 ) ) )
       => ( ( find_pos_aux_a_b @ rank @ V @ P @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ Uu2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
          = ( find_pos_aux_a_b @ rank @ V @ R2 @ T12 ) ) ) ) ).

% find_pos_aux.simps(1)
thf(fact_42_denormalize_Ocases,axiom,
    ! [X: dtree_list_a_b] :
      ( ! [R: list_a,T2: dtree_list_a_b,E2: b] :
          ( X
         != ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
     => ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
            ( ! [X5: produc6499617310964463488_a_b_b] :
                ( Xs
               != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
           => ! [R: list_a] :
                ( X
               != ( node_list_a_b @ R @ Xs ) ) ) ) ).

% denormalize.cases
thf(fact_43_old_Ofind__pos_Ocases,axiom,
    ! [T: dtree_list_a_b,X: produc111314985273491367st_a_b] :
      ( ( old_a_b @ T )
     => ( ! [V2: list_a,R: list_a,T1: dtree_list_a_b,Uu: b] :
            ( X
           != ( produc148520996349637281st_a_b @ V2 @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
       => ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
              ( ! [X5: produc6499617310964463488_a_b_b] :
                  ( Xs
                 != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
             => ! [V2: list_a,R: list_a] :
                  ( X
                 != ( produc148520996349637281st_a_b @ V2 @ ( node_list_a_b @ R @ Xs ) ) ) ) ) ) ).

% old.find_pos.cases
thf(fact_44_dtree__to__list_Ocases,axiom,
    ! [X: dtree_list_a_b] :
      ( ! [R: list_a,T2: dtree_list_a_b,E2: b] :
          ( X
         != ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
     => ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
            ( ! [X5: produc6499617310964463488_a_b_b] :
                ( Xs
               != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
           => ! [R: list_a] :
                ( X
               != ( node_list_a_b @ R @ Xs ) ) ) ) ).

% dtree_to_list.cases
thf(fact_45_insert__between_Ocases,axiom,
    ! [X: produc9008341577332299707st_a_b] :
      ~ ! [V2: list_a,E2: b,X3: list_a,Y4: list_a,R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
          ( X
         != ( produc1848684973559390389st_a_b @ V2 @ ( produc5064203622704112514st_a_b @ E2 @ ( produc673257793671328980st_a_b @ X3 @ ( produc148520996349637281st_a_b @ Y4 @ ( node_list_a_b @ R @ Xs ) ) ) ) ) ) ).

% insert_between.cases
thf(fact_46_mem__Collect__eq,axiom,
    ! [A: list_a,P3: list_a > $o] :
      ( ( member_list_a @ A @ ( collect_list_a @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
    ! [A: produc6499617310964463488_a_b_b,P3: produc6499617310964463488_a_b_b > $o] :
      ( ( member4695696432722591383_a_b_b @ A @ ( collec3241703154345475797_a_b_b @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
    ! [A: dtree_list_a_b,P3: dtree_list_a_b > $o] :
      ( ( member551035911493665803st_a_b @ A @ ( collec2944820760411501129st_a_b @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_49_mem__Collect__eq,axiom,
    ! [A: real,P3: real > $o] :
      ( ( member_real @ A @ ( collect_real @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_50_Collect__mem__eq,axiom,
    ! [A3: set_list_a] :
      ( ( collect_list_a
        @ ^ [X4: list_a] : ( member_list_a @ X4 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
    ! [A3: set_Pr3443975907877334966_a_b_b] :
      ( ( collec3241703154345475797_a_b_b
        @ ^ [X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_52_Collect__mem__eq,axiom,
    ! [A3: set_dtree_list_a_b] :
      ( ( collec2944820760411501129st_a_b
        @ ^ [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_53_Collect__mem__eq,axiom,
    ! [A3: set_real] :
      ( ( collect_real
        @ ^ [X4: real] : ( member_real @ X4 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_54_old_Ofind__pos__aux_Ocases,axiom,
    ! [T: dtree_list_a_b,X: produc7147531718898801626st_a_b] :
      ( ( old_a_b @ T )
     => ( ! [V2: list_a,P2: list_a,R: list_a,T1: dtree_list_a_b,Uu: b] :
            ( X
           != ( produc673257793671328980st_a_b @ V2 @ ( produc148520996349637281st_a_b @ P2 @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) )
       => ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
              ( ! [X5: produc6499617310964463488_a_b_b] :
                  ( Xs
                 != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
             => ! [V2: list_a,P2: list_a,R: list_a] :
                  ( X
                 != ( produc673257793671328980st_a_b @ V2 @ ( produc148520996349637281st_a_b @ P2 @ ( node_list_a_b @ R @ Xs ) ) ) ) ) ) ) ).

% old.find_pos_aux.cases
thf(fact_55_singleton__uneq,axiom,
    ! [R2: list_a,T: dtree_list_a_b,E: b] :
      ( ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) )
     != T ) ).

% singleton_uneq
thf(fact_56_singleton__uneq_H,axiom,
    ! [R2: list_a,T: dtree_list_a_b,E: b,V: list_a] :
      ( ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) )
     != ( node_list_a_b @ V @ ( sucs_list_a_b @ T ) ) ) ).

% singleton_uneq'
thf(fact_57_pred__subset__eq2,axiom,
    ! [R3: set_Pr4048851178543822343list_a,S: set_Pr4048851178543822343list_a] :
      ( ( ord_le5542992221119063950st_a_o
        @ ^ [X4: list_a,Y3: list_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Y3 ) @ R3 )
        @ ^ [X4: list_a,Y3: list_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Y3 ) @ S ) )
      = ( ord_le7857023143581076903list_a @ R3 @ S ) ) ).

% pred_subset_eq2
thf(fact_58_pred__subset__eq2,axiom,
    ! [R3: set_Pr5727314569254441693st_a_b,S: set_Pr5727314569254441693st_a_b] :
      ( ( ord_le8921296852080140188_a_b_o
        @ ^ [X4: list_a,Y3: dtree_list_a_b] : ( member6763472504040263486st_a_b @ ( produc148520996349637281st_a_b @ X4 @ Y3 ) @ R3 )
        @ ^ [X4: list_a,Y3: dtree_list_a_b] : ( member6763472504040263486st_a_b @ ( produc148520996349637281st_a_b @ X4 @ Y3 ) @ S ) )
      = ( ord_le6820904450577167165st_a_b @ R3 @ S ) ) ).

% pred_subset_eq2
thf(fact_59_pred__subset__eq2,axiom,
    ! [R3: set_Pr3443975907877334966_a_b_b,S: set_Pr3443975907877334966_a_b_b] :
      ( ( ord_le4372168669586844721_b_b_o
        @ ^ [X4: dtree_list_a_b,Y3: b] : ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X4 @ Y3 ) @ R3 )
        @ ^ [X4: dtree_list_a_b,Y3: b] : ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X4 @ Y3 ) @ S ) )
      = ( ord_le1619362961161175062_a_b_b @ R3 @ S ) ) ).

% pred_subset_eq2
thf(fact_60_pred__subset__eq2,axiom,
    ! [R3: set_Pr8451570673126195472st_a_b,S: set_Pr8451570673126195472st_a_b] :
      ( ( ord_le6349127272349855273_a_b_o
        @ ^ [X4: list_a,Y3: produc111314985273491367st_a_b] : ( member4895573663636194545st_a_b @ ( produc673257793671328980st_a_b @ X4 @ Y3 ) @ R3 )
        @ ^ [X4: list_a,Y3: produc111314985273491367st_a_b] : ( member4895573663636194545st_a_b @ ( produc673257793671328980st_a_b @ X4 @ Y3 ) @ S ) )
      = ( ord_le3570456948295111024st_a_b @ R3 @ S ) ) ).

% pred_subset_eq2
thf(fact_61_combine_Ocases,axiom,
    ! [X: produc7147531718898801626st_a_b] :
      ~ ! [X3: list_a,Y4: list_a,R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
          ( X
         != ( produc673257793671328980st_a_b @ X3 @ ( produc148520996349637281st_a_b @ Y4 @ ( node_list_a_b @ R @ Xs ) ) ) ) ).

% combine.cases
thf(fact_62_bot_Oextremum__uniqueI,axiom,
    ! [A: fset_P2153231429829016240_a_b_b] :
      ( ( ord_le8870638447146015504_a_b_b @ A @ bot_bo2248824169281960260_a_b_b )
     => ( A = bot_bo2248824169281960260_a_b_b ) ) ).

% bot.extremum_uniqueI
thf(fact_63_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_64_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_65_bot_Oextremum__unique,axiom,
    ! [A: fset_P2153231429829016240_a_b_b] :
      ( ( ord_le8870638447146015504_a_b_b @ A @ bot_bo2248824169281960260_a_b_b )
      = ( A = bot_bo2248824169281960260_a_b_b ) ) ).

% bot.extremum_unique
thf(fact_66_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
thf(fact_67_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_68_bot_Oextremum,axiom,
    ! [A: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ bot_bo2248824169281960260_a_b_b @ A ) ).

% bot.extremum
thf(fact_69_bot_Oextremum,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% bot.extremum
thf(fact_70_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_71_subrelI,axiom,
    ! [R2: set_Pr4048851178543822343list_a,S2: set_Pr4048851178543822343list_a] :
      ( ! [X3: list_a,Y4: list_a] :
          ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X3 @ Y4 ) @ R2 )
         => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X3 @ Y4 ) @ S2 ) )
     => ( ord_le7857023143581076903list_a @ R2 @ S2 ) ) ).

% subrelI
thf(fact_72_subrelI,axiom,
    ! [R2: set_Pr5727314569254441693st_a_b,S2: set_Pr5727314569254441693st_a_b] :
      ( ! [X3: list_a,Y4: dtree_list_a_b] :
          ( ( member6763472504040263486st_a_b @ ( produc148520996349637281st_a_b @ X3 @ Y4 ) @ R2 )
         => ( member6763472504040263486st_a_b @ ( produc148520996349637281st_a_b @ X3 @ Y4 ) @ S2 ) )
     => ( ord_le6820904450577167165st_a_b @ R2 @ S2 ) ) ).

% subrelI
thf(fact_73_subrelI,axiom,
    ! [R2: set_Pr3443975907877334966_a_b_b,S2: set_Pr3443975907877334966_a_b_b] :
      ( ! [X3: dtree_list_a_b,Y4: b] :
          ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X3 @ Y4 ) @ R2 )
         => ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X3 @ Y4 ) @ S2 ) )
     => ( ord_le1619362961161175062_a_b_b @ R2 @ S2 ) ) ).

% subrelI
thf(fact_74_subrelI,axiom,
    ! [R2: set_Pr8451570673126195472st_a_b,S2: set_Pr8451570673126195472st_a_b] :
      ( ! [X3: list_a,Y4: produc111314985273491367st_a_b] :
          ( ( member4895573663636194545st_a_b @ ( produc673257793671328980st_a_b @ X3 @ Y4 ) @ R2 )
         => ( member4895573663636194545st_a_b @ ( produc673257793671328980st_a_b @ X3 @ Y4 ) @ S2 ) )
     => ( ord_le3570456948295111024st_a_b @ R2 @ S2 ) ) ).

% subrelI
thf(fact_75_dtail_Ocases,axiom,
    ! [X: produc1920479565126685823list_a] :
      ~ ! [R: list_a,Xs: fset_P2153231429829016240_a_b_b,Def: b > list_a] :
          ( X
         != ( produc2621617146629198007list_a @ ( node_list_a_b @ R @ Xs ) @ Def ) ) ).

% dtail.cases
thf(fact_76_dtree_Osel_I2_J,axiom,
    ! [X1: list_a,X2: fset_P2153231429829016240_a_b_b] :
      ( ( sucs_list_a_b @ ( node_list_a_b @ X1 @ X2 ) )
      = X2 ) ).

% dtree.sel(2)
thf(fact_77_is__subtree_Ocases,axiom,
    ! [X: produc1510363273921914569st_a_b] :
      ~ ! [X3: dtree_list_a_b,R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
          ( X
         != ( produc783528831147138817st_a_b @ X3 @ ( node_list_a_b @ R @ Xs ) ) ) ).

% is_subtree.cases
thf(fact_78_old_Ofind__pos__aux_Oelims,axiom,
    ! [T: dtree_list_a_b,Rank: list_a > real,X: list_a,Xa2: list_a,Xb: dtree_list_a_b,Y: produc9164743771328383783list_a] :
      ( ( old_a_b @ T )
     => ( ( ( find_pos_aux_a_b @ Rank @ X @ Xa2 @ Xb )
          = Y )
       => ( ! [R: list_a,T1: dtree_list_a_b] :
              ( ? [Uu: b] :
                  ( Xb
                  = ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) )
             => ~ ( ( ( ord_less_eq_real @ ( Rank @ ( rev_a @ X ) ) @ ( Rank @ ( rev_a @ R ) ) )
                   => ( Y
                      = ( produc6837034575241423639list_a @ Xa2 @ R ) ) )
                  & ( ~ ( ord_less_eq_real @ ( Rank @ ( rev_a @ X ) ) @ ( Rank @ ( rev_a @ R ) ) )
                   => ( Y
                      = ( find_pos_aux_a_b @ Rank @ X @ R @ T1 ) ) ) ) )
         => ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
                ( ! [X5: produc6499617310964463488_a_b_b] :
                    ( Xs
                   != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
               => ! [R: list_a] :
                    ( ( Xb
                      = ( node_list_a_b @ R @ Xs ) )
                   => ~ ( ( ( ord_less_eq_real @ ( Rank @ ( rev_a @ X ) ) @ ( Rank @ ( rev_a @ R ) ) )
                         => ( Y
                            = ( produc6837034575241423639list_a @ Xa2 @ R ) ) )
                        & ( ~ ( ord_less_eq_real @ ( Rank @ ( rev_a @ X ) ) @ ( Rank @ ( rev_a @ R ) ) )
                         => ( Y
                            = ( produc6837034575241423639list_a @ R @ R ) ) ) ) ) ) ) ) ) ).

% old.find_pos_aux.elims
thf(fact_79_old_Ofind__pos__aux_Osimps_I1_J,axiom,
    ! [T: dtree_list_a_b,Rank: list_a > real,V: list_a,R2: list_a,P: list_a,T12: dtree_list_a_b,Uu2: b] :
      ( ( old_a_b @ T )
     => ( ( ( ord_less_eq_real @ ( Rank @ ( rev_a @ V ) ) @ ( Rank @ ( rev_a @ R2 ) ) )
         => ( ( find_pos_aux_a_b @ Rank @ V @ P @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ Uu2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
            = ( produc6837034575241423639list_a @ P @ R2 ) ) )
        & ( ~ ( ord_less_eq_real @ ( Rank @ ( rev_a @ V ) ) @ ( Rank @ ( rev_a @ R2 ) ) )
         => ( ( find_pos_aux_a_b @ Rank @ V @ P @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ Uu2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
            = ( find_pos_aux_a_b @ Rank @ V @ R2 @ T12 ) ) ) ) ) ).

% old.find_pos_aux.simps(1)
thf(fact_80_old_Omerge_Ocases,axiom,
    ! [T: dtree_list_a_b,X: dtree_list_a_b] :
      ( ( old_a_b @ T )
     => ~ ! [R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
            ( X
           != ( node_list_a_b @ R @ Xs ) ) ) ).

% old.merge.cases
thf(fact_81_old_Ofind__pos__aux_Osimps_I2_J,axiom,
    ! [T: dtree_list_a_b,Xs2: fset_P2153231429829016240_a_b_b,Rank: list_a > real,V: list_a,R2: list_a,P: list_a] :
      ( ( old_a_b @ T )
     => ( ! [X3: produc6499617310964463488_a_b_b] :
            ( Xs2
           != ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
       => ( ( ( ord_less_eq_real @ ( Rank @ ( rev_a @ V ) ) @ ( Rank @ ( rev_a @ R2 ) ) )
           => ( ( find_pos_aux_a_b @ Rank @ V @ P @ ( node_list_a_b @ R2 @ Xs2 ) )
              = ( produc6837034575241423639list_a @ P @ R2 ) ) )
          & ( ~ ( ord_less_eq_real @ ( Rank @ ( rev_a @ V ) ) @ ( Rank @ ( rev_a @ R2 ) ) )
           => ( ( find_pos_aux_a_b @ Rank @ V @ P @ ( node_list_a_b @ R2 @ Xs2 ) )
              = ( produc6837034575241423639list_a @ R2 @ R2 ) ) ) ) ) ) ).

% old.find_pos_aux.simps(2)
thf(fact_82_order__antisym__conv,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ( ( ord_less_eq_real @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_83_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_84_linorder__le__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_real @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_85_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_86_ord__le__eq__subst,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_eq_real @ X3 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_87_ord__le__eq__subst,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_eq_real @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_88_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > real,C: real] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_89_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_90_ord__eq__le__subst,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_eq_real @ X3 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_91_ord__eq__le__subst,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_eq_real @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_92_ord__eq__le__subst,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_93_ord__eq__le__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_94_linorder__linear,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
      | ( ord_less_eq_real @ Y @ X ) ) ).

% linorder_linear
thf(fact_95_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_96_order__eq__refl,axiom,
    ! [X: real,Y: real] :
      ( ( X = Y )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% order_eq_refl
thf(fact_97_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_98_order__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_eq_real @ X3 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_99_order__subst2,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_eq_real @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_100_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > real,C: real] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_101_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_102_order__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_eq_real @ X3 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_103_order__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_104_order__subst1,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_eq_real @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_105_order__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_106_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: real,Z: real] : ( Y5 = Z ) )
    = ( ^ [A4: real,B3: real] :
          ( ( ord_less_eq_real @ A4 @ B3 )
          & ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_107_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
          & ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_108_antisym,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_109_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_110_dual__order_Otrans,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_eq_real @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_111_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_112_dual__order_Oantisym,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_113_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_114_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: real,Z: real] : ( Y5 = Z ) )
    = ( ^ [A4: real,B3: real] :
          ( ( ord_less_eq_real @ B3 @ A4 )
          & ( ord_less_eq_real @ A4 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_115_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A4 )
          & ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_116_linorder__wlog,axiom,
    ! [P3: real > real > $o,A: real,B: real] :
      ( ! [A5: real,B4: real] :
          ( ( ord_less_eq_real @ A5 @ B4 )
         => ( P3 @ A5 @ B4 ) )
     => ( ! [A5: real,B4: real] :
            ( ( P3 @ B4 @ A5 )
           => ( P3 @ A5 @ B4 ) )
       => ( P3 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_117_linorder__wlog,axiom,
    ! [P3: nat > nat > $o,A: nat,B: nat] :
      ( ! [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A5 @ B4 )
         => ( P3 @ A5 @ B4 ) )
     => ( ! [A5: nat,B4: nat] :
            ( ( P3 @ B4 @ A5 )
           => ( P3 @ A5 @ B4 ) )
       => ( P3 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_118_order__trans,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_eq_real @ Y @ Z3 )
       => ( ord_less_eq_real @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_119_order__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z3 )
       => ( ord_less_eq_nat @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_120_order_Otrans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_eq_real @ A @ C ) ) ) ).

% order.trans
thf(fact_121_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_122_order__antisym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_123_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_124_ord__le__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_real @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_125_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_126_ord__eq__le__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_eq_real @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_127_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_128_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: real,Z: real] : ( Y5 = Z ) )
    = ( ^ [X4: real,Y3: real] :
          ( ( ord_less_eq_real @ X4 @ Y3 )
          & ( ord_less_eq_real @ Y3 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_129_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
    = ( ^ [X4: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y3 )
          & ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_130_le__cases3,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( ( ord_less_eq_real @ X @ Y )
       => ~ ( ord_less_eq_real @ Y @ Z3 ) )
     => ( ( ( ord_less_eq_real @ Y @ X )
         => ~ ( ord_less_eq_real @ X @ Z3 ) )
       => ( ( ( ord_less_eq_real @ X @ Z3 )
           => ~ ( ord_less_eq_real @ Z3 @ Y ) )
         => ( ( ( ord_less_eq_real @ Z3 @ Y )
             => ~ ( ord_less_eq_real @ Y @ X ) )
           => ( ( ( ord_less_eq_real @ Y @ Z3 )
               => ~ ( ord_less_eq_real @ Z3 @ X ) )
             => ~ ( ( ord_less_eq_real @ Z3 @ X )
                 => ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_131_le__cases3,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z3 ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z3 ) )
       => ( ( ( ord_less_eq_nat @ X @ Z3 )
           => ~ ( ord_less_eq_nat @ Z3 @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z3 @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z3 )
               => ~ ( ord_less_eq_nat @ Z3 @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z3 @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_132_nle__le,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( ord_less_eq_real @ A @ B ) )
      = ( ( ord_less_eq_real @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_133_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_134_prod__induct3,axiom,
    ! [P3: produc7147531718898801626st_a_b > $o,X: produc7147531718898801626st_a_b] :
      ( ! [A5: list_a,B4: list_a,C2: dtree_list_a_b] : ( P3 @ ( produc673257793671328980st_a_b @ A5 @ ( produc148520996349637281st_a_b @ B4 @ C2 ) ) )
     => ( P3 @ X ) ) ).

% prod_induct3
thf(fact_135_prod__cases3,axiom,
    ! [Y: produc7147531718898801626st_a_b] :
      ~ ! [A5: list_a,B4: list_a,C2: dtree_list_a_b] :
          ( Y
         != ( produc673257793671328980st_a_b @ A5 @ ( produc148520996349637281st_a_b @ B4 @ C2 ) ) ) ).

% prod_cases3
thf(fact_136_Pair__inject,axiom,
    ! [A: list_a,B: list_a,A2: list_a,B2: list_a] :
      ( ( ( produc6837034575241423639list_a @ A @ B )
        = ( produc6837034575241423639list_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_137_Pair__inject,axiom,
    ! [A: list_a,B: dtree_list_a_b,A2: list_a,B2: dtree_list_a_b] :
      ( ( ( produc148520996349637281st_a_b @ A @ B )
        = ( produc148520996349637281st_a_b @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_138_Pair__inject,axiom,
    ! [A: dtree_list_a_b,B: b,A2: dtree_list_a_b,B2: b] :
      ( ( ( produc7704165765595008946_a_b_b @ A @ B )
        = ( produc7704165765595008946_a_b_b @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_139_Pair__inject,axiom,
    ! [A: list_a,B: produc111314985273491367st_a_b,A2: list_a,B2: produc111314985273491367st_a_b] :
      ( ( ( produc673257793671328980st_a_b @ A @ B )
        = ( produc673257793671328980st_a_b @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_140_prod__cases,axiom,
    ! [P3: produc9164743771328383783list_a > $o,P: produc9164743771328383783list_a] :
      ( ! [A5: list_a,B4: list_a] : ( P3 @ ( produc6837034575241423639list_a @ A5 @ B4 ) )
     => ( P3 @ P ) ) ).

% prod_cases
thf(fact_141_prod__cases,axiom,
    ! [P3: produc111314985273491367st_a_b > $o,P: produc111314985273491367st_a_b] :
      ( ! [A5: list_a,B4: dtree_list_a_b] : ( P3 @ ( produc148520996349637281st_a_b @ A5 @ B4 ) )
     => ( P3 @ P ) ) ).

% prod_cases
thf(fact_142_prod__cases,axiom,
    ! [P3: produc6499617310964463488_a_b_b > $o,P: produc6499617310964463488_a_b_b] :
      ( ! [A5: dtree_list_a_b,B4: b] : ( P3 @ ( produc7704165765595008946_a_b_b @ A5 @ B4 ) )
     => ( P3 @ P ) ) ).

% prod_cases
thf(fact_143_prod__cases,axiom,
    ! [P3: produc7147531718898801626st_a_b > $o,P: produc7147531718898801626st_a_b] :
      ( ! [A5: list_a,B4: produc111314985273491367st_a_b] : ( P3 @ ( produc673257793671328980st_a_b @ A5 @ B4 ) )
     => ( P3 @ P ) ) ).

% prod_cases
thf(fact_144_surj__pair,axiom,
    ! [P: produc9164743771328383783list_a] :
    ? [X3: list_a,Y4: list_a] :
      ( P
      = ( produc6837034575241423639list_a @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_145_surj__pair,axiom,
    ! [P: produc111314985273491367st_a_b] :
    ? [X3: list_a,Y4: dtree_list_a_b] :
      ( P
      = ( produc148520996349637281st_a_b @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_146_surj__pair,axiom,
    ! [P: produc6499617310964463488_a_b_b] :
    ? [X3: dtree_list_a_b,Y4: b] :
      ( P
      = ( produc7704165765595008946_a_b_b @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_147_surj__pair,axiom,
    ! [P: produc7147531718898801626st_a_b] :
    ? [X3: list_a,Y4: produc111314985273491367st_a_b] :
      ( P
      = ( produc673257793671328980st_a_b @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_148_old_Oprod_Oexhaust,axiom,
    ! [Y: produc9164743771328383783list_a] :
      ~ ! [A5: list_a,B4: list_a] :
          ( Y
         != ( produc6837034575241423639list_a @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_149_old_Oprod_Oexhaust,axiom,
    ! [Y: produc111314985273491367st_a_b] :
      ~ ! [A5: list_a,B4: dtree_list_a_b] :
          ( Y
         != ( produc148520996349637281st_a_b @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_150_old_Oprod_Oexhaust,axiom,
    ! [Y: produc6499617310964463488_a_b_b] :
      ~ ! [A5: dtree_list_a_b,B4: b] :
          ( Y
         != ( produc7704165765595008946_a_b_b @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_151_old_Oprod_Oexhaust,axiom,
    ! [Y: produc7147531718898801626st_a_b] :
      ~ ! [A5: list_a,B4: produc111314985273491367st_a_b] :
          ( Y
         != ( produc673257793671328980st_a_b @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_152_find__pos_Oelims,axiom,
    ! [X: list_a,Xa2: dtree_list_a_b,Y: produc9164743771328383783list_a] :
      ( ( ( find_pos_a_b @ rank @ X @ Xa2 )
        = Y )
     => ( ! [R: list_a,T1: dtree_list_a_b] :
            ( ? [Uu: b] :
                ( Xa2
                = ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) )
           => ( Y
             != ( find_pos_aux_a_b @ rank @ X @ R @ T1 ) ) )
       => ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
              ( ! [X5: produc6499617310964463488_a_b_b] :
                  ( Xs
                 != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
             => ! [R: list_a] :
                  ( ( Xa2
                    = ( node_list_a_b @ R @ Xs ) )
                 => ( Y
                   != ( produc6837034575241423639list_a @ R @ R ) ) ) ) ) ) ).

% find_pos.elims
thf(fact_153_find__pos_Osimps_I1_J,axiom,
    ! [V: list_a,R2: list_a,T12: dtree_list_a_b,Uu2: b] :
      ( ( find_pos_a_b @ rank @ V @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ Uu2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
      = ( find_pos_aux_a_b @ rank @ V @ R2 @ T12 ) ) ).

% find_pos.simps(1)
thf(fact_154_find__pos_Osimps_I2_J,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,V: list_a,R2: list_a] :
      ( ! [X3: produc6499617310964463488_a_b_b] :
          ( Xs2
         != ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
     => ( ( find_pos_a_b @ rank @ V @ ( node_list_a_b @ R2 @ Xs2 ) )
        = ( produc6837034575241423639list_a @ R2 @ R2 ) ) ) ).

% find_pos.simps(2)
thf(fact_155_find__pos__aux_Opelims,axiom,
    ! [X: list_a,Xa2: list_a,Xb: dtree_list_a_b,Y: produc9164743771328383783list_a] :
      ( ( ( find_pos_aux_a_b @ rank @ X @ Xa2 @ Xb )
        = Y )
     => ( ( accp_P2814348900907978769st_a_b @ ( find_pos_aux_rel_a_b @ rank ) @ ( produc673257793671328980st_a_b @ X @ ( produc148520996349637281st_a_b @ Xa2 @ Xb ) ) )
       => ( ! [R: list_a,T1: dtree_list_a_b,Uu: b] :
              ( ( Xb
                = ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) )
             => ( ( ( ( ord_less_eq_real @ ( rank @ ( rev_a @ X ) ) @ ( rank @ ( rev_a @ R ) ) )
                   => ( Y
                      = ( produc6837034575241423639list_a @ Xa2 @ R ) ) )
                  & ( ~ ( ord_less_eq_real @ ( rank @ ( rev_a @ X ) ) @ ( rank @ ( rev_a @ R ) ) )
                   => ( Y
                      = ( find_pos_aux_a_b @ rank @ X @ R @ T1 ) ) ) )
               => ~ ( accp_P2814348900907978769st_a_b @ ( find_pos_aux_rel_a_b @ rank ) @ ( produc673257793671328980st_a_b @ X @ ( produc148520996349637281st_a_b @ Xa2 @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ) )
         => ( ! [Xs: fset_P2153231429829016240_a_b_b,V2: list_a,P2: list_a,R: list_a] :
                ( ( accp_P2814348900907978769st_a_b @ ( find_pos_aux_rel_a_b @ rank ) @ ( produc673257793671328980st_a_b @ V2 @ ( produc148520996349637281st_a_b @ P2 @ ( node_list_a_b @ R @ Xs ) ) ) )
               => ( ! [X5: produc6499617310964463488_a_b_b] :
                      ( Xs
                     != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
                 => ( ( X = V2 )
                   => ( ( Xa2 = P2 )
                     => ( ( Xb
                          = ( node_list_a_b @ R @ Xs ) )
                       => ( accp_P2814348900907978769st_a_b @ ( find_pos_aux_rel_a_b @ rank ) @ ( produc673257793671328980st_a_b @ V2 @ ( produc148520996349637281st_a_b @ P2 @ ( node_list_a_b @ R @ Xs ) ) ) ) ) ) ) ) )
           => ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
                  ( ! [X5: produc6499617310964463488_a_b_b] :
                      ( Xs
                     != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
                 => ! [R: list_a] :
                      ( ( Xb
                        = ( node_list_a_b @ R @ Xs ) )
                     => ( ( ( ( ord_less_eq_real @ ( rank @ ( rev_a @ X ) ) @ ( rank @ ( rev_a @ R ) ) )
                           => ( Y
                              = ( produc6837034575241423639list_a @ Xa2 @ R ) ) )
                          & ( ~ ( ord_less_eq_real @ ( rank @ ( rev_a @ X ) ) @ ( rank @ ( rev_a @ R ) ) )
                           => ( Y
                              = ( produc6837034575241423639list_a @ R @ R ) ) ) )
                       => ~ ( accp_P2814348900907978769st_a_b @ ( find_pos_aux_rel_a_b @ rank ) @ ( produc673257793671328980st_a_b @ X @ ( produc148520996349637281st_a_b @ Xa2 @ ( node_list_a_b @ R @ Xs ) ) ) ) ) ) ) ) ) ) ) ).

% find_pos_aux.pelims
thf(fact_156_find__pos__aux_Opsimps_I1_J,axiom,
    ! [V: list_a,P: list_a,R2: list_a,T12: dtree_list_a_b,Uu2: b] :
      ( ( accp_P2814348900907978769st_a_b @ ( find_pos_aux_rel_a_b @ rank ) @ ( produc673257793671328980st_a_b @ V @ ( produc148520996349637281st_a_b @ P @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ Uu2 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) )
     => ( ( ( ord_less_eq_real @ ( rank @ ( rev_a @ V ) ) @ ( rank @ ( rev_a @ R2 ) ) )
         => ( ( find_pos_aux_a_b @ rank @ V @ P @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ Uu2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
            = ( produc6837034575241423639list_a @ P @ R2 ) ) )
        & ( ~ ( ord_less_eq_real @ ( rank @ ( rev_a @ V ) ) @ ( rank @ ( rev_a @ R2 ) ) )
         => ( ( find_pos_aux_a_b @ rank @ V @ P @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ Uu2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
            = ( find_pos_aux_a_b @ rank @ V @ R2 @ T12 ) ) ) ) ) ).

% find_pos_aux.psimps(1)
thf(fact_157_find__pos__aux_Opsimps_I2_J,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,V: list_a,P: list_a,R2: list_a] :
      ( ! [X3: produc6499617310964463488_a_b_b] :
          ( Xs2
         != ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
     => ( ( accp_P2814348900907978769st_a_b @ ( find_pos_aux_rel_a_b @ rank ) @ ( produc673257793671328980st_a_b @ V @ ( produc148520996349637281st_a_b @ P @ ( node_list_a_b @ R2 @ Xs2 ) ) ) )
       => ( ( ( ord_less_eq_real @ ( rank @ ( rev_a @ V ) ) @ ( rank @ ( rev_a @ R2 ) ) )
           => ( ( find_pos_aux_a_b @ rank @ V @ P @ ( node_list_a_b @ R2 @ Xs2 ) )
              = ( produc6837034575241423639list_a @ P @ R2 ) ) )
          & ( ~ ( ord_less_eq_real @ ( rank @ ( rev_a @ V ) ) @ ( rank @ ( rev_a @ R2 ) ) )
           => ( ( find_pos_aux_a_b @ rank @ V @ P @ ( node_list_a_b @ R2 @ Xs2 ) )
              = ( produc6837034575241423639list_a @ R2 @ R2 ) ) ) ) ) ) ).

% find_pos_aux.psimps(2)
thf(fact_158_find__pos__aux_Opinduct,axiom,
    ! [A0: list_a,A1: list_a,A22: dtree_list_a_b,P3: list_a > list_a > dtree_list_a_b > $o] :
      ( ( accp_P2814348900907978769st_a_b @ ( find_pos_aux_rel_a_b @ rank ) @ ( produc673257793671328980st_a_b @ A0 @ ( produc148520996349637281st_a_b @ A1 @ A22 ) ) )
     => ( ! [V2: list_a,P2: list_a,R: list_a,T1: dtree_list_a_b,Uu: b] :
            ( ( accp_P2814348900907978769st_a_b @ ( find_pos_aux_rel_a_b @ rank ) @ ( produc673257793671328980st_a_b @ V2 @ ( produc148520996349637281st_a_b @ P2 @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) )
           => ( ( ~ ( ord_less_eq_real @ ( rank @ ( rev_a @ V2 ) ) @ ( rank @ ( rev_a @ R ) ) )
               => ( P3 @ V2 @ R @ T1 ) )
             => ( P3 @ V2 @ P2 @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) )
       => ( ! [Xs: fset_P2153231429829016240_a_b_b,V2: list_a,P2: list_a,R: list_a] :
              ( ( accp_P2814348900907978769st_a_b @ ( find_pos_aux_rel_a_b @ rank ) @ ( produc673257793671328980st_a_b @ V2 @ ( produc148520996349637281st_a_b @ P2 @ ( node_list_a_b @ R @ Xs ) ) ) )
             => ( ! [X5: produc6499617310964463488_a_b_b] :
                    ( Xs
                   != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
               => ( P3 @ V2 @ P2 @ ( node_list_a_b @ R @ Xs ) ) ) )
         => ( P3 @ A0 @ A1 @ A22 ) ) ) ) ).

% find_pos_aux.pinduct
thf(fact_159_old_Ofind__pos_Oelims,axiom,
    ! [T: dtree_list_a_b,Rank: list_a > real,X: list_a,Xa2: dtree_list_a_b,Y: produc9164743771328383783list_a] :
      ( ( old_a_b @ T )
     => ( ( ( find_pos_a_b @ Rank @ X @ Xa2 )
          = Y )
       => ( ! [R: list_a,T1: dtree_list_a_b] :
              ( ? [Uu: b] :
                  ( Xa2
                  = ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) )
             => ( Y
               != ( find_pos_aux_a_b @ Rank @ X @ R @ T1 ) ) )
         => ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
                ( ! [X5: produc6499617310964463488_a_b_b] :
                    ( Xs
                   != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
               => ! [R: list_a] :
                    ( ( Xa2
                      = ( node_list_a_b @ R @ Xs ) )
                   => ( Y
                     != ( produc6837034575241423639list_a @ R @ R ) ) ) ) ) ) ) ).

% old.find_pos.elims
thf(fact_160_old_Ofind__pos__aux_Opelims,axiom,
    ! [T: dtree_list_a_b,Rank: list_a > real,X: list_a,Xa2: list_a,Xb: dtree_list_a_b,Y: produc9164743771328383783list_a] :
      ( ( old_a_b @ T )
     => ( ( ( find_pos_aux_a_b @ Rank @ X @ Xa2 @ Xb )
          = Y )
       => ( ( accp_P2814348900907978769st_a_b @ ( find_pos_aux_rel_a_b @ Rank ) @ ( produc673257793671328980st_a_b @ X @ ( produc148520996349637281st_a_b @ Xa2 @ Xb ) ) )
         => ( ! [R: list_a,T1: dtree_list_a_b,Uu: b] :
                ( ( Xb
                  = ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) )
               => ( ( ( ( ord_less_eq_real @ ( Rank @ ( rev_a @ X ) ) @ ( Rank @ ( rev_a @ R ) ) )
                     => ( Y
                        = ( produc6837034575241423639list_a @ Xa2 @ R ) ) )
                    & ( ~ ( ord_less_eq_real @ ( Rank @ ( rev_a @ X ) ) @ ( Rank @ ( rev_a @ R ) ) )
                     => ( Y
                        = ( find_pos_aux_a_b @ Rank @ X @ R @ T1 ) ) ) )
                 => ~ ( accp_P2814348900907978769st_a_b @ ( find_pos_aux_rel_a_b @ Rank ) @ ( produc673257793671328980st_a_b @ X @ ( produc148520996349637281st_a_b @ Xa2 @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ) )
           => ( ! [Xs: fset_P2153231429829016240_a_b_b,V2: list_a,P2: list_a,R: list_a] :
                  ( ( accp_P2814348900907978769st_a_b @ ( find_pos_aux_rel_a_b @ Rank ) @ ( produc673257793671328980st_a_b @ V2 @ ( produc148520996349637281st_a_b @ P2 @ ( node_list_a_b @ R @ Xs ) ) ) )
                 => ( ! [X5: produc6499617310964463488_a_b_b] :
                        ( Xs
                       != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
                   => ( ( X = V2 )
                     => ( ( Xa2 = P2 )
                       => ( ( Xb
                            = ( node_list_a_b @ R @ Xs ) )
                         => ( accp_P2814348900907978769st_a_b @ ( find_pos_aux_rel_a_b @ Rank ) @ ( produc673257793671328980st_a_b @ V2 @ ( produc148520996349637281st_a_b @ P2 @ ( node_list_a_b @ R @ Xs ) ) ) ) ) ) ) ) )
             => ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
                    ( ! [X5: produc6499617310964463488_a_b_b] :
                        ( Xs
                       != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
                   => ! [R: list_a] :
                        ( ( Xb
                          = ( node_list_a_b @ R @ Xs ) )
                       => ( ( ( ( ord_less_eq_real @ ( Rank @ ( rev_a @ X ) ) @ ( Rank @ ( rev_a @ R ) ) )
                             => ( Y
                                = ( produc6837034575241423639list_a @ Xa2 @ R ) ) )
                            & ( ~ ( ord_less_eq_real @ ( Rank @ ( rev_a @ X ) ) @ ( Rank @ ( rev_a @ R ) ) )
                             => ( Y
                                = ( produc6837034575241423639list_a @ R @ R ) ) ) )
                         => ~ ( accp_P2814348900907978769st_a_b @ ( find_pos_aux_rel_a_b @ Rank ) @ ( produc673257793671328980st_a_b @ X @ ( produc148520996349637281st_a_b @ Xa2 @ ( node_list_a_b @ R @ Xs ) ) ) ) ) ) ) ) ) ) ) ) ).

% old.find_pos_aux.pelims
thf(fact_161_old_Ofind__pos__aux_Opsimps_I1_J,axiom,
    ! [T: dtree_list_a_b,Rank: list_a > real,V: list_a,P: list_a,R2: list_a,T12: dtree_list_a_b,Uu2: b] :
      ( ( old_a_b @ T )
     => ( ( accp_P2814348900907978769st_a_b @ ( find_pos_aux_rel_a_b @ Rank ) @ ( produc673257793671328980st_a_b @ V @ ( produc148520996349637281st_a_b @ P @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ Uu2 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) )
       => ( ( ( ord_less_eq_real @ ( Rank @ ( rev_a @ V ) ) @ ( Rank @ ( rev_a @ R2 ) ) )
           => ( ( find_pos_aux_a_b @ Rank @ V @ P @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ Uu2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
              = ( produc6837034575241423639list_a @ P @ R2 ) ) )
          & ( ~ ( ord_less_eq_real @ ( Rank @ ( rev_a @ V ) ) @ ( Rank @ ( rev_a @ R2 ) ) )
           => ( ( find_pos_aux_a_b @ Rank @ V @ P @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ Uu2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
              = ( find_pos_aux_a_b @ Rank @ V @ R2 @ T12 ) ) ) ) ) ) ).

% old.find_pos_aux.psimps(1)
thf(fact_162_pred__subset__eq,axiom,
    ! [R3: set_list_a,S: set_list_a] :
      ( ( ord_less_eq_list_a_o
        @ ^ [X4: list_a] : ( member_list_a @ X4 @ R3 )
        @ ^ [X4: list_a] : ( member_list_a @ X4 @ S ) )
      = ( ord_le8861187494160871172list_a @ R3 @ S ) ) ).

% pred_subset_eq
thf(fact_163_pred__subset__eq,axiom,
    ! [R3: set_Pr3443975907877334966_a_b_b,S: set_Pr3443975907877334966_a_b_b] :
      ( ( ord_le5434074902312169967_b_b_o
        @ ^ [X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ R3 )
        @ ^ [X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ S ) )
      = ( ord_le1619362961161175062_a_b_b @ R3 @ S ) ) ).

% pred_subset_eq
thf(fact_164_pred__subset__eq,axiom,
    ! [R3: set_dtree_list_a_b,S: set_dtree_list_a_b] :
      ( ( ord_le739112815111206779_a_b_o
        @ ^ [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ R3 )
        @ ^ [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ S ) )
      = ( ord_le7599451563663638410st_a_b @ R3 @ S ) ) ).

% pred_subset_eq
thf(fact_165_pred__subset__eq,axiom,
    ! [R3: set_real,S: set_real] :
      ( ( ord_less_eq_real_o
        @ ^ [X4: real] : ( member_real @ X4 @ R3 )
        @ ^ [X4: real] : ( member_real @ X4 @ S ) )
      = ( ord_less_eq_set_real @ R3 @ S ) ) ).

% pred_subset_eq
thf(fact_166_old_Ofind__pos_Ocong,axiom,
    find_pos_a_b = find_pos_a_b ).

% old.find_pos.cong
thf(fact_167_bot__empty__eq2,axiom,
    ( bot_bo1672404152767791706st_a_o
    = ( ^ [X4: list_a,Y3: list_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Y3 ) @ bot_bo2955605580254355571list_a ) ) ) ).

% bot_empty_eq2
thf(fact_168_bot__empty__eq2,axiom,
    ( bot_bo6194466349879338448_a_b_o
    = ( ^ [X4: list_a,Y3: dtree_list_a_b] : ( member6763472504040263486st_a_b @ ( produc148520996349637281st_a_b @ X4 @ Y3 ) @ bot_bo8571636681367830897st_a_b ) ) ) ).

% bot_empty_eq2
thf(fact_169_bot__empty__eq2,axiom,
    ( bot_bo771552775820516453_b_b_o
    = ( ^ [X4: dtree_list_a_b,Y3: b] : ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X4 @ Y3 ) @ bot_bo4897374000430069834_a_b_b ) ) ) ).

% bot_empty_eq2
thf(fact_170_bot__empty__eq2,axiom,
    ( bot_bo5096458512222287453_a_b_o
    = ( ^ [X4: list_a,Y3: produc111314985273491367st_a_b] : ( member4895573663636194545st_a_b @ ( produc673257793671328980st_a_b @ X4 @ Y3 ) @ bot_bo2137710973035897252st_a_b ) ) ) ).

% bot_empty_eq2
thf(fact_171_dverts__nempty,axiom,
    ! [T: dtree_list_a_b] :
      ( ( dverts_list_a_b @ T )
     != bot_bot_set_list_a ) ).

% dverts_nempty
thf(fact_172_old_Ofind__pos__aux_Opinduct,axiom,
    ! [T: dtree_list_a_b,Rank: list_a > real,A0: list_a,A1: list_a,A22: dtree_list_a_b,P3: list_a > list_a > dtree_list_a_b > $o] :
      ( ( old_a_b @ T )
     => ( ( accp_P2814348900907978769st_a_b @ ( find_pos_aux_rel_a_b @ Rank ) @ ( produc673257793671328980st_a_b @ A0 @ ( produc148520996349637281st_a_b @ A1 @ A22 ) ) )
       => ( ! [V2: list_a,P2: list_a,R: list_a,T1: dtree_list_a_b,Uu: b] :
              ( ( accp_P2814348900907978769st_a_b @ ( find_pos_aux_rel_a_b @ Rank ) @ ( produc673257793671328980st_a_b @ V2 @ ( produc148520996349637281st_a_b @ P2 @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) )
             => ( ( ~ ( ord_less_eq_real @ ( Rank @ ( rev_a @ V2 ) ) @ ( Rank @ ( rev_a @ R ) ) )
                 => ( P3 @ V2 @ R @ T1 ) )
               => ( P3 @ V2 @ P2 @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) )
         => ( ! [Xs: fset_P2153231429829016240_a_b_b,V2: list_a,P2: list_a,R: list_a] :
                ( ( accp_P2814348900907978769st_a_b @ ( find_pos_aux_rel_a_b @ Rank ) @ ( produc673257793671328980st_a_b @ V2 @ ( produc148520996349637281st_a_b @ P2 @ ( node_list_a_b @ R @ Xs ) ) ) )
               => ( ! [X5: produc6499617310964463488_a_b_b] :
                      ( Xs
                     != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
                 => ( P3 @ V2 @ P2 @ ( node_list_a_b @ R @ Xs ) ) ) )
           => ( P3 @ A0 @ A1 @ A22 ) ) ) ) ) ).

% old.find_pos_aux.pinduct
thf(fact_173_old_Ofind__pos_Osimps_I2_J,axiom,
    ! [T: dtree_list_a_b,Xs2: fset_P2153231429829016240_a_b_b,Rank: list_a > real,V: list_a,R2: list_a] :
      ( ( old_a_b @ T )
     => ( ! [X3: produc6499617310964463488_a_b_b] :
            ( Xs2
           != ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
       => ( ( find_pos_a_b @ Rank @ V @ ( node_list_a_b @ R2 @ Xs2 ) )
          = ( produc6837034575241423639list_a @ R2 @ R2 ) ) ) ) ).

% old.find_pos.simps(2)
thf(fact_174_old_Ofind__pos_Osimps_I1_J,axiom,
    ! [T: dtree_list_a_b,Rank: list_a > real,V: list_a,R2: list_a,T12: dtree_list_a_b,Uu2: b] :
      ( ( old_a_b @ T )
     => ( ( find_pos_a_b @ Rank @ V @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ Uu2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
        = ( find_pos_aux_a_b @ Rank @ V @ R2 @ T12 ) ) ) ).

% old.find_pos.simps(1)
thf(fact_175_old_Ofind__pos__aux_Opsimps_I2_J,axiom,
    ! [T: dtree_list_a_b,Xs2: fset_P2153231429829016240_a_b_b,Rank: list_a > real,V: list_a,P: list_a,R2: list_a] :
      ( ( old_a_b @ T )
     => ( ! [X3: produc6499617310964463488_a_b_b] :
            ( Xs2
           != ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
       => ( ( accp_P2814348900907978769st_a_b @ ( find_pos_aux_rel_a_b @ Rank ) @ ( produc673257793671328980st_a_b @ V @ ( produc148520996349637281st_a_b @ P @ ( node_list_a_b @ R2 @ Xs2 ) ) ) )
         => ( ( ( ord_less_eq_real @ ( Rank @ ( rev_a @ V ) ) @ ( Rank @ ( rev_a @ R2 ) ) )
             => ( ( find_pos_aux_a_b @ Rank @ V @ P @ ( node_list_a_b @ R2 @ Xs2 ) )
                = ( produc6837034575241423639list_a @ P @ R2 ) ) )
            & ( ~ ( ord_less_eq_real @ ( Rank @ ( rev_a @ V ) ) @ ( Rank @ ( rev_a @ R2 ) ) )
             => ( ( find_pos_aux_a_b @ Rank @ V @ P @ ( node_list_a_b @ R2 @ Xs2 ) )
                = ( produc6837034575241423639list_a @ R2 @ R2 ) ) ) ) ) ) ) ).

% old.find_pos_aux.psimps(2)
thf(fact_176_bot__fset_Orep__eq,axiom,
    ( ( fset_nat2 @ bot_bot_fset_nat )
    = bot_bot_set_nat ) ).

% bot_fset.rep_eq
thf(fact_177_bot__fset_Orep__eq,axiom,
    ( ( fset_P9138963618725001425_a_b_b @ bot_bo2248824169281960260_a_b_b )
    = bot_bo4897374000430069834_a_b_b ) ).

% bot_fset.rep_eq
thf(fact_178_find__pos_Opelims,axiom,
    ! [X: list_a,Xa2: dtree_list_a_b,Y: produc9164743771328383783list_a] :
      ( ( ( find_pos_a_b @ rank @ X @ Xa2 )
        = Y )
     => ( ( accp_P3553483059299338846st_a_b @ find_pos_rel_a_b @ ( produc148520996349637281st_a_b @ X @ Xa2 ) )
       => ( ! [R: list_a,T1: dtree_list_a_b,Uu: b] :
              ( ( Xa2
                = ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) )
             => ( ( Y
                  = ( find_pos_aux_a_b @ rank @ X @ R @ T1 ) )
               => ~ ( accp_P3553483059299338846st_a_b @ find_pos_rel_a_b @ ( produc148520996349637281st_a_b @ X @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) )
         => ( ! [Xs: fset_P2153231429829016240_a_b_b,V2: list_a,R: list_a] :
                ( ( accp_P3553483059299338846st_a_b @ find_pos_rel_a_b @ ( produc148520996349637281st_a_b @ V2 @ ( node_list_a_b @ R @ Xs ) ) )
               => ( ! [X5: produc6499617310964463488_a_b_b] :
                      ( Xs
                     != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
                 => ( ( X = V2 )
                   => ( ( Xa2
                        = ( node_list_a_b @ R @ Xs ) )
                     => ( accp_P3553483059299338846st_a_b @ find_pos_rel_a_b @ ( produc148520996349637281st_a_b @ V2 @ ( node_list_a_b @ R @ Xs ) ) ) ) ) ) )
           => ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
                  ( ! [X5: produc6499617310964463488_a_b_b] :
                      ( Xs
                     != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
                 => ! [R: list_a] :
                      ( ( Xa2
                        = ( node_list_a_b @ R @ Xs ) )
                     => ( ( Y
                          = ( produc6837034575241423639list_a @ R @ R ) )
                       => ~ ( accp_P3553483059299338846st_a_b @ find_pos_rel_a_b @ ( produc148520996349637281st_a_b @ X @ ( node_list_a_b @ R @ Xs ) ) ) ) ) ) ) ) ) ) ).

% find_pos.pelims
thf(fact_179_find__pos_Opsimps_I1_J,axiom,
    ! [V: list_a,R2: list_a,T12: dtree_list_a_b,Uu2: b] :
      ( ( accp_P3553483059299338846st_a_b @ find_pos_rel_a_b @ ( produc148520996349637281st_a_b @ V @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ Uu2 ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
     => ( ( find_pos_a_b @ rank @ V @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ Uu2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
        = ( find_pos_aux_a_b @ rank @ V @ R2 @ T12 ) ) ) ).

% find_pos.psimps(1)
thf(fact_180_finsert__absorb2,axiom,
    ! [X: produc6499617310964463488_a_b_b,A3: fset_P2153231429829016240_a_b_b] :
      ( ( finser2303212525150181944_a_b_b @ X @ ( finser2303212525150181944_a_b_b @ X @ A3 ) )
      = ( finser2303212525150181944_a_b_b @ X @ A3 ) ) ).

% finsert_absorb2
thf(fact_181_fempty__fsubsetI,axiom,
    ! [X: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ bot_bo2248824169281960260_a_b_b @ X ) ).

% fempty_fsubsetI
thf(fact_182_fsubset__fempty,axiom,
    ! [A3: fset_P2153231429829016240_a_b_b] :
      ( ( ord_le8870638447146015504_a_b_b @ A3 @ bot_bo2248824169281960260_a_b_b )
      = ( A3 = bot_bo2248824169281960260_a_b_b ) ) ).

% fsubset_fempty
thf(fact_183_subsetI,axiom,
    ! [A3: set_list_a,B5: set_list_a] :
      ( ! [X3: list_a] :
          ( ( member_list_a @ X3 @ A3 )
         => ( member_list_a @ X3 @ B5 ) )
     => ( ord_le8861187494160871172list_a @ A3 @ B5 ) ) ).

% subsetI
thf(fact_184_subsetI,axiom,
    ! [A3: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
      ( ! [X3: produc6499617310964463488_a_b_b] :
          ( ( member4695696432722591383_a_b_b @ X3 @ A3 )
         => ( member4695696432722591383_a_b_b @ X3 @ B5 ) )
     => ( ord_le1619362961161175062_a_b_b @ A3 @ B5 ) ) ).

% subsetI
thf(fact_185_subsetI,axiom,
    ! [A3: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
      ( ! [X3: dtree_list_a_b] :
          ( ( member551035911493665803st_a_b @ X3 @ A3 )
         => ( member551035911493665803st_a_b @ X3 @ B5 ) )
     => ( ord_le7599451563663638410st_a_b @ A3 @ B5 ) ) ).

% subsetI
thf(fact_186_subsetI,axiom,
    ! [A3: set_real,B5: set_real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A3 )
         => ( member_real @ X3 @ B5 ) )
     => ( ord_less_eq_set_real @ A3 @ B5 ) ) ).

% subsetI
thf(fact_187_subset__empty,axiom,
    ! [A3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat )
      = ( A3 = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_188_empty__subsetI,axiom,
    ! [A3: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A3 ) ).

% empty_subsetI
thf(fact_189_empty__Collect__eq,axiom,
    ! [P3: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P3 ) )
      = ( ! [X4: nat] :
            ~ ( P3 @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_190_Collect__empty__eq,axiom,
    ! [P3: nat > $o] :
      ( ( ( collect_nat @ P3 )
        = bot_bot_set_nat )
      = ( ! [X4: nat] :
            ~ ( P3 @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_191_all__not__in__conv,axiom,
    ! [A3: set_list_a] :
      ( ( ! [X4: list_a] :
            ~ ( member_list_a @ X4 @ A3 ) )
      = ( A3 = bot_bot_set_list_a ) ) ).

% all_not_in_conv
thf(fact_192_all__not__in__conv,axiom,
    ! [A3: set_Pr3443975907877334966_a_b_b] :
      ( ( ! [X4: produc6499617310964463488_a_b_b] :
            ~ ( member4695696432722591383_a_b_b @ X4 @ A3 ) )
      = ( A3 = bot_bo4897374000430069834_a_b_b ) ) ).

% all_not_in_conv
thf(fact_193_all__not__in__conv,axiom,
    ! [A3: set_dtree_list_a_b] :
      ( ( ! [X4: dtree_list_a_b] :
            ~ ( member551035911493665803st_a_b @ X4 @ A3 ) )
      = ( A3 = bot_bo798015271861357502st_a_b ) ) ).

% all_not_in_conv
thf(fact_194_all__not__in__conv,axiom,
    ! [A3: set_real] :
      ( ( ! [X4: real] :
            ~ ( member_real @ X4 @ A3 ) )
      = ( A3 = bot_bot_set_real ) ) ).

% all_not_in_conv
thf(fact_195_all__not__in__conv,axiom,
    ! [A3: set_nat] :
      ( ( ! [X4: nat] :
            ~ ( member_nat @ X4 @ A3 ) )
      = ( A3 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_196_empty__iff,axiom,
    ! [C: list_a] :
      ~ ( member_list_a @ C @ bot_bot_set_list_a ) ).

% empty_iff
thf(fact_197_empty__iff,axiom,
    ! [C: produc6499617310964463488_a_b_b] :
      ~ ( member4695696432722591383_a_b_b @ C @ bot_bo4897374000430069834_a_b_b ) ).

% empty_iff
thf(fact_198_empty__iff,axiom,
    ! [C: dtree_list_a_b] :
      ~ ( member551035911493665803st_a_b @ C @ bot_bo798015271861357502st_a_b ) ).

% empty_iff
thf(fact_199_empty__iff,axiom,
    ! [C: real] :
      ~ ( member_real @ C @ bot_bot_set_real ) ).

% empty_iff
thf(fact_200_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_201_find__pos_Opinduct,axiom,
    ! [A0: list_a,A1: dtree_list_a_b,P3: list_a > dtree_list_a_b > $o] :
      ( ( accp_P3553483059299338846st_a_b @ find_pos_rel_a_b @ ( produc148520996349637281st_a_b @ A0 @ A1 ) )
     => ( ! [V2: list_a,R: list_a,T1: dtree_list_a_b,Uu: b] :
            ( ( accp_P3553483059299338846st_a_b @ find_pos_rel_a_b @ ( produc148520996349637281st_a_b @ V2 @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
           => ( P3 @ V2 @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
       => ( ! [Xs: fset_P2153231429829016240_a_b_b,V2: list_a,R: list_a] :
              ( ( accp_P3553483059299338846st_a_b @ find_pos_rel_a_b @ ( produc148520996349637281st_a_b @ V2 @ ( node_list_a_b @ R @ Xs ) ) )
             => ( ! [X5: produc6499617310964463488_a_b_b] :
                    ( Xs
                   != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
               => ( P3 @ V2 @ ( node_list_a_b @ R @ Xs ) ) ) )
         => ( P3 @ A0 @ A1 ) ) ) ) ).

% find_pos.pinduct
thf(fact_202_find__pos_Opsimps_I2_J,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,V: list_a,R2: list_a] :
      ( ! [X3: produc6499617310964463488_a_b_b] :
          ( Xs2
         != ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
     => ( ( accp_P3553483059299338846st_a_b @ find_pos_rel_a_b @ ( produc148520996349637281st_a_b @ V @ ( node_list_a_b @ R2 @ Xs2 ) ) )
       => ( ( find_pos_a_b @ rank @ V @ ( node_list_a_b @ R2 @ Xs2 ) )
          = ( produc6837034575241423639list_a @ R2 @ R2 ) ) ) ) ).

% find_pos.psimps(2)
thf(fact_203_empty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X4: nat] : $false ) ) ).

% empty_def
thf(fact_204_bot__empty__eq,axiom,
    ( bot_bot_list_a_o
    = ( ^ [X4: list_a] : ( member_list_a @ X4 @ bot_bot_set_list_a ) ) ) ).

% bot_empty_eq
thf(fact_205_bot__empty__eq,axiom,
    ( bot_bo9167104602940019387_b_b_o
    = ( ^ [X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ bot_bo4897374000430069834_a_b_b ) ) ) ).

% bot_empty_eq
thf(fact_206_bot__empty__eq,axiom,
    ( bot_bo84193016852448327_a_b_o
    = ( ^ [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ bot_bo798015271861357502st_a_b ) ) ) ).

% bot_empty_eq
thf(fact_207_bot__empty__eq,axiom,
    ( bot_bot_real_o
    = ( ^ [X4: real] : ( member_real @ X4 @ bot_bot_set_real ) ) ) ).

% bot_empty_eq
thf(fact_208_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X4: nat] : ( member_nat @ X4 @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_209_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_210_ex__in__conv,axiom,
    ! [A3: set_list_a] :
      ( ( ? [X4: list_a] : ( member_list_a @ X4 @ A3 ) )
      = ( A3 != bot_bot_set_list_a ) ) ).

% ex_in_conv
thf(fact_211_ex__in__conv,axiom,
    ! [A3: set_Pr3443975907877334966_a_b_b] :
      ( ( ? [X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ A3 ) )
      = ( A3 != bot_bo4897374000430069834_a_b_b ) ) ).

% ex_in_conv
thf(fact_212_ex__in__conv,axiom,
    ! [A3: set_dtree_list_a_b] :
      ( ( ? [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ A3 ) )
      = ( A3 != bot_bo798015271861357502st_a_b ) ) ).

% ex_in_conv
thf(fact_213_ex__in__conv,axiom,
    ! [A3: set_real] :
      ( ( ? [X4: real] : ( member_real @ X4 @ A3 ) )
      = ( A3 != bot_bot_set_real ) ) ).

% ex_in_conv
thf(fact_214_ex__in__conv,axiom,
    ! [A3: set_nat] :
      ( ( ? [X4: nat] : ( member_nat @ X4 @ A3 ) )
      = ( A3 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_215_equals0I,axiom,
    ! [A3: set_list_a] :
      ( ! [Y4: list_a] :
          ~ ( member_list_a @ Y4 @ A3 )
     => ( A3 = bot_bot_set_list_a ) ) ).

% equals0I
thf(fact_216_equals0I,axiom,
    ! [A3: set_Pr3443975907877334966_a_b_b] :
      ( ! [Y4: produc6499617310964463488_a_b_b] :
          ~ ( member4695696432722591383_a_b_b @ Y4 @ A3 )
     => ( A3 = bot_bo4897374000430069834_a_b_b ) ) ).

% equals0I
thf(fact_217_equals0I,axiom,
    ! [A3: set_dtree_list_a_b] :
      ( ! [Y4: dtree_list_a_b] :
          ~ ( member551035911493665803st_a_b @ Y4 @ A3 )
     => ( A3 = bot_bo798015271861357502st_a_b ) ) ).

% equals0I
thf(fact_218_equals0I,axiom,
    ! [A3: set_real] :
      ( ! [Y4: real] :
          ~ ( member_real @ Y4 @ A3 )
     => ( A3 = bot_bot_set_real ) ) ).

% equals0I
thf(fact_219_equals0I,axiom,
    ! [A3: set_nat] :
      ( ! [Y4: nat] :
          ~ ( member_nat @ Y4 @ A3 )
     => ( A3 = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_220_equals0D,axiom,
    ! [A3: set_list_a,A: list_a] :
      ( ( A3 = bot_bot_set_list_a )
     => ~ ( member_list_a @ A @ A3 ) ) ).

% equals0D
thf(fact_221_equals0D,axiom,
    ! [A3: set_Pr3443975907877334966_a_b_b,A: produc6499617310964463488_a_b_b] :
      ( ( A3 = bot_bo4897374000430069834_a_b_b )
     => ~ ( member4695696432722591383_a_b_b @ A @ A3 ) ) ).

% equals0D
thf(fact_222_equals0D,axiom,
    ! [A3: set_dtree_list_a_b,A: dtree_list_a_b] :
      ( ( A3 = bot_bo798015271861357502st_a_b )
     => ~ ( member551035911493665803st_a_b @ A @ A3 ) ) ).

% equals0D
thf(fact_223_equals0D,axiom,
    ! [A3: set_real,A: real] :
      ( ( A3 = bot_bot_set_real )
     => ~ ( member_real @ A @ A3 ) ) ).

% equals0D
thf(fact_224_equals0D,axiom,
    ! [A3: set_nat,A: nat] :
      ( ( A3 = bot_bot_set_nat )
     => ~ ( member_nat @ A @ A3 ) ) ).

% equals0D
thf(fact_225_emptyE,axiom,
    ! [A: list_a] :
      ~ ( member_list_a @ A @ bot_bot_set_list_a ) ).

% emptyE
thf(fact_226_emptyE,axiom,
    ! [A: produc6499617310964463488_a_b_b] :
      ~ ( member4695696432722591383_a_b_b @ A @ bot_bo4897374000430069834_a_b_b ) ).

% emptyE
thf(fact_227_emptyE,axiom,
    ! [A: dtree_list_a_b] :
      ~ ( member551035911493665803st_a_b @ A @ bot_bo798015271861357502st_a_b ) ).

% emptyE
thf(fact_228_emptyE,axiom,
    ! [A: real] :
      ~ ( member_real @ A @ bot_bot_set_real ) ).

% emptyE
thf(fact_229_emptyE,axiom,
    ! [A: nat] :
      ~ ( member_nat @ A @ bot_bot_set_nat ) ).

% emptyE
thf(fact_230_subset__iff,axiom,
    ( ord_le8861187494160871172list_a
    = ( ^ [A6: set_list_a,B6: set_list_a] :
        ! [T3: list_a] :
          ( ( member_list_a @ T3 @ A6 )
         => ( member_list_a @ T3 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_231_subset__iff,axiom,
    ( ord_le1619362961161175062_a_b_b
    = ( ^ [A6: set_Pr3443975907877334966_a_b_b,B6: set_Pr3443975907877334966_a_b_b] :
        ! [T3: produc6499617310964463488_a_b_b] :
          ( ( member4695696432722591383_a_b_b @ T3 @ A6 )
         => ( member4695696432722591383_a_b_b @ T3 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_232_subset__iff,axiom,
    ( ord_le7599451563663638410st_a_b
    = ( ^ [A6: set_dtree_list_a_b,B6: set_dtree_list_a_b] :
        ! [T3: dtree_list_a_b] :
          ( ( member551035911493665803st_a_b @ T3 @ A6 )
         => ( member551035911493665803st_a_b @ T3 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_233_subset__iff,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
        ! [T3: real] :
          ( ( member_real @ T3 @ A6 )
         => ( member_real @ T3 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_234_subset__eq,axiom,
    ( ord_le8861187494160871172list_a
    = ( ^ [A6: set_list_a,B6: set_list_a] :
        ! [X4: list_a] :
          ( ( member_list_a @ X4 @ A6 )
         => ( member_list_a @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_235_subset__eq,axiom,
    ( ord_le1619362961161175062_a_b_b
    = ( ^ [A6: set_Pr3443975907877334966_a_b_b,B6: set_Pr3443975907877334966_a_b_b] :
        ! [X4: produc6499617310964463488_a_b_b] :
          ( ( member4695696432722591383_a_b_b @ X4 @ A6 )
         => ( member4695696432722591383_a_b_b @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_236_subset__eq,axiom,
    ( ord_le7599451563663638410st_a_b
    = ( ^ [A6: set_dtree_list_a_b,B6: set_dtree_list_a_b] :
        ! [X4: dtree_list_a_b] :
          ( ( member551035911493665803st_a_b @ X4 @ A6 )
         => ( member551035911493665803st_a_b @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_237_subset__eq,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
        ! [X4: real] :
          ( ( member_real @ X4 @ A6 )
         => ( member_real @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_238_subsetD,axiom,
    ! [A3: set_list_a,B5: set_list_a,C: list_a] :
      ( ( ord_le8861187494160871172list_a @ A3 @ B5 )
     => ( ( member_list_a @ C @ A3 )
       => ( member_list_a @ C @ B5 ) ) ) ).

% subsetD
thf(fact_239_subsetD,axiom,
    ! [A3: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b,C: produc6499617310964463488_a_b_b] :
      ( ( ord_le1619362961161175062_a_b_b @ A3 @ B5 )
     => ( ( member4695696432722591383_a_b_b @ C @ A3 )
       => ( member4695696432722591383_a_b_b @ C @ B5 ) ) ) ).

% subsetD
thf(fact_240_subsetD,axiom,
    ! [A3: set_dtree_list_a_b,B5: set_dtree_list_a_b,C: dtree_list_a_b] :
      ( ( ord_le7599451563663638410st_a_b @ A3 @ B5 )
     => ( ( member551035911493665803st_a_b @ C @ A3 )
       => ( member551035911493665803st_a_b @ C @ B5 ) ) ) ).

% subsetD
thf(fact_241_subsetD,axiom,
    ! [A3: set_real,B5: set_real,C: real] :
      ( ( ord_less_eq_set_real @ A3 @ B5 )
     => ( ( member_real @ C @ A3 )
       => ( member_real @ C @ B5 ) ) ) ).

% subsetD
thf(fact_242_in__mono,axiom,
    ! [A3: set_list_a,B5: set_list_a,X: list_a] :
      ( ( ord_le8861187494160871172list_a @ A3 @ B5 )
     => ( ( member_list_a @ X @ A3 )
       => ( member_list_a @ X @ B5 ) ) ) ).

% in_mono
thf(fact_243_in__mono,axiom,
    ! [A3: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b,X: produc6499617310964463488_a_b_b] :
      ( ( ord_le1619362961161175062_a_b_b @ A3 @ B5 )
     => ( ( member4695696432722591383_a_b_b @ X @ A3 )
       => ( member4695696432722591383_a_b_b @ X @ B5 ) ) ) ).

% in_mono
thf(fact_244_in__mono,axiom,
    ! [A3: set_dtree_list_a_b,B5: set_dtree_list_a_b,X: dtree_list_a_b] :
      ( ( ord_le7599451563663638410st_a_b @ A3 @ B5 )
     => ( ( member551035911493665803st_a_b @ X @ A3 )
       => ( member551035911493665803st_a_b @ X @ B5 ) ) ) ).

% in_mono
thf(fact_245_in__mono,axiom,
    ! [A3: set_real,B5: set_real,X: real] :
      ( ( ord_less_eq_set_real @ A3 @ B5 )
     => ( ( member_real @ X @ A3 )
       => ( member_real @ X @ B5 ) ) ) ).

% in_mono
thf(fact_246_fset__cong,axiom,
    ! [X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b] :
      ( ( ( fset_P9138963618725001425_a_b_b @ X )
        = ( fset_P9138963618725001425_a_b_b @ Y ) )
      = ( X = Y ) ) ).

% fset_cong
thf(fact_247_finsert__mono,axiom,
    ! [C3: fset_P2153231429829016240_a_b_b,D: fset_P2153231429829016240_a_b_b,A: produc6499617310964463488_a_b_b] :
      ( ( ord_le8870638447146015504_a_b_b @ C3 @ D )
     => ( ord_le8870638447146015504_a_b_b @ ( finser2303212525150181944_a_b_b @ A @ C3 ) @ ( finser2303212525150181944_a_b_b @ A @ D ) ) ) ).

% finsert_mono
thf(fact_248_fsubset__finsertI,axiom,
    ! [B5: fset_P2153231429829016240_a_b_b,A: produc6499617310964463488_a_b_b] : ( ord_le8870638447146015504_a_b_b @ B5 @ ( finser2303212525150181944_a_b_b @ A @ B5 ) ) ).

% fsubset_finsertI
thf(fact_249_fsubset__finsertI2,axiom,
    ! [A3: fset_P2153231429829016240_a_b_b,B5: fset_P2153231429829016240_a_b_b,B: produc6499617310964463488_a_b_b] :
      ( ( ord_le8870638447146015504_a_b_b @ A3 @ B5 )
     => ( ord_le8870638447146015504_a_b_b @ A3 @ ( finser2303212525150181944_a_b_b @ B @ B5 ) ) ) ).

% fsubset_finsertI2
thf(fact_250_finsert__commute,axiom,
    ! [X: produc6499617310964463488_a_b_b,Y: produc6499617310964463488_a_b_b,A3: fset_P2153231429829016240_a_b_b] :
      ( ( finser2303212525150181944_a_b_b @ X @ ( finser2303212525150181944_a_b_b @ Y @ A3 ) )
      = ( finser2303212525150181944_a_b_b @ Y @ ( finser2303212525150181944_a_b_b @ X @ A3 ) ) ) ).

% finsert_commute
thf(fact_251_old_Ofind__pos_Opinduct,axiom,
    ! [T: dtree_list_a_b,A0: list_a,A1: dtree_list_a_b,P3: list_a > dtree_list_a_b > $o] :
      ( ( old_a_b @ T )
     => ( ( accp_P3553483059299338846st_a_b @ find_pos_rel_a_b @ ( produc148520996349637281st_a_b @ A0 @ A1 ) )
       => ( ! [V2: list_a,R: list_a,T1: dtree_list_a_b,Uu: b] :
              ( ( accp_P3553483059299338846st_a_b @ find_pos_rel_a_b @ ( produc148520996349637281st_a_b @ V2 @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
             => ( P3 @ V2 @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
         => ( ! [Xs: fset_P2153231429829016240_a_b_b,V2: list_a,R: list_a] :
                ( ( accp_P3553483059299338846st_a_b @ find_pos_rel_a_b @ ( produc148520996349637281st_a_b @ V2 @ ( node_list_a_b @ R @ Xs ) ) )
               => ( ! [X5: produc6499617310964463488_a_b_b] :
                      ( Xs
                     != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
                 => ( P3 @ V2 @ ( node_list_a_b @ R @ Xs ) ) ) )
           => ( P3 @ A0 @ A1 ) ) ) ) ) ).

% old.find_pos.pinduct
thf(fact_252_less__eq__set__def,axiom,
    ( ord_le8861187494160871172list_a
    = ( ^ [A6: set_list_a,B6: set_list_a] :
          ( ord_less_eq_list_a_o
          @ ^ [X4: list_a] : ( member_list_a @ X4 @ A6 )
          @ ^ [X4: list_a] : ( member_list_a @ X4 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_253_less__eq__set__def,axiom,
    ( ord_le1619362961161175062_a_b_b
    = ( ^ [A6: set_Pr3443975907877334966_a_b_b,B6: set_Pr3443975907877334966_a_b_b] :
          ( ord_le5434074902312169967_b_b_o
          @ ^ [X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ A6 )
          @ ^ [X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_254_less__eq__set__def,axiom,
    ( ord_le7599451563663638410st_a_b
    = ( ^ [A6: set_dtree_list_a_b,B6: set_dtree_list_a_b] :
          ( ord_le739112815111206779_a_b_o
          @ ^ [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ A6 )
          @ ^ [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_255_less__eq__set__def,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( ord_less_eq_real_o
          @ ^ [X4: real] : ( member_real @ X4 @ A6 )
          @ ^ [X4: real] : ( member_real @ X4 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_256_Collect__subset,axiom,
    ! [A3: set_list_a,P3: list_a > $o] :
      ( ord_le8861187494160871172list_a
      @ ( collect_list_a
        @ ^ [X4: list_a] :
            ( ( member_list_a @ X4 @ A3 )
            & ( P3 @ X4 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_257_Collect__subset,axiom,
    ! [A3: set_Pr3443975907877334966_a_b_b,P3: produc6499617310964463488_a_b_b > $o] :
      ( ord_le1619362961161175062_a_b_b
      @ ( collec3241703154345475797_a_b_b
        @ ^ [X4: produc6499617310964463488_a_b_b] :
            ( ( member4695696432722591383_a_b_b @ X4 @ A3 )
            & ( P3 @ X4 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_258_Collect__subset,axiom,
    ! [A3: set_dtree_list_a_b,P3: dtree_list_a_b > $o] :
      ( ord_le7599451563663638410st_a_b
      @ ( collec2944820760411501129st_a_b
        @ ^ [X4: dtree_list_a_b] :
            ( ( member551035911493665803st_a_b @ X4 @ A3 )
            & ( P3 @ X4 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_259_Collect__subset,axiom,
    ! [A3: set_real,P3: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X4: real] :
            ( ( member_real @ X4 @ A3 )
            & ( P3 @ X4 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_260_old_Ofind__pos_Opsimps_I2_J,axiom,
    ! [T: dtree_list_a_b,Xs2: fset_P2153231429829016240_a_b_b,V: list_a,R2: list_a,Rank: list_a > real] :
      ( ( old_a_b @ T )
     => ( ! [X3: produc6499617310964463488_a_b_b] :
            ( Xs2
           != ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
       => ( ( accp_P3553483059299338846st_a_b @ find_pos_rel_a_b @ ( produc148520996349637281st_a_b @ V @ ( node_list_a_b @ R2 @ Xs2 ) ) )
         => ( ( find_pos_a_b @ Rank @ V @ ( node_list_a_b @ R2 @ Xs2 ) )
            = ( produc6837034575241423639list_a @ R2 @ R2 ) ) ) ) ) ).

% old.find_pos.psimps(2)
thf(fact_261_old_Ofind__pos_Opsimps_I1_J,axiom,
    ! [T: dtree_list_a_b,V: list_a,R2: list_a,T12: dtree_list_a_b,Uu2: b,Rank: list_a > real] :
      ( ( old_a_b @ T )
     => ( ( accp_P3553483059299338846st_a_b @ find_pos_rel_a_b @ ( produc148520996349637281st_a_b @ V @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ Uu2 ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
       => ( ( find_pos_a_b @ Rank @ V @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ Uu2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
          = ( find_pos_aux_a_b @ Rank @ V @ R2 @ T12 ) ) ) ) ).

% old.find_pos.psimps(1)
thf(fact_262_old_Ofind__pos_Opelims,axiom,
    ! [T: dtree_list_a_b,Rank: list_a > real,X: list_a,Xa2: dtree_list_a_b,Y: produc9164743771328383783list_a] :
      ( ( old_a_b @ T )
     => ( ( ( find_pos_a_b @ Rank @ X @ Xa2 )
          = Y )
       => ( ( accp_P3553483059299338846st_a_b @ find_pos_rel_a_b @ ( produc148520996349637281st_a_b @ X @ Xa2 ) )
         => ( ! [R: list_a,T1: dtree_list_a_b,Uu: b] :
                ( ( Xa2
                  = ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) )
               => ( ( Y
                    = ( find_pos_aux_a_b @ Rank @ X @ R @ T1 ) )
                 => ~ ( accp_P3553483059299338846st_a_b @ find_pos_rel_a_b @ ( produc148520996349637281st_a_b @ X @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) )
           => ( ! [Xs: fset_P2153231429829016240_a_b_b,V2: list_a,R: list_a] :
                  ( ( accp_P3553483059299338846st_a_b @ find_pos_rel_a_b @ ( produc148520996349637281st_a_b @ V2 @ ( node_list_a_b @ R @ Xs ) ) )
                 => ( ! [X5: produc6499617310964463488_a_b_b] :
                        ( Xs
                       != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
                   => ( ( X = V2 )
                     => ( ( Xa2
                          = ( node_list_a_b @ R @ Xs ) )
                       => ( accp_P3553483059299338846st_a_b @ find_pos_rel_a_b @ ( produc148520996349637281st_a_b @ V2 @ ( node_list_a_b @ R @ Xs ) ) ) ) ) ) )
             => ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
                    ( ! [X5: produc6499617310964463488_a_b_b] :
                        ( Xs
                       != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
                   => ! [R: list_a] :
                        ( ( Xa2
                          = ( node_list_a_b @ R @ Xs ) )
                       => ( ( Y
                            = ( produc6837034575241423639list_a @ R @ R ) )
                         => ~ ( accp_P3553483059299338846st_a_b @ find_pos_rel_a_b @ ( produc148520996349637281st_a_b @ X @ ( node_list_a_b @ R @ Xs ) ) ) ) ) ) ) ) ) ) ) ).

% old.find_pos.pelims
thf(fact_263_less__eq__fset_Orep__eq,axiom,
    ( ord_le8870638447146015504_a_b_b
    = ( ^ [X4: fset_P2153231429829016240_a_b_b,Xa3: fset_P2153231429829016240_a_b_b] : ( ord_le1619362961161175062_a_b_b @ ( fset_P9138963618725001425_a_b_b @ X4 ) @ ( fset_P9138963618725001425_a_b_b @ Xa3 ) ) ) ) ).

% less_eq_fset.rep_eq
thf(fact_264_fsubset__fsingletonD,axiom,
    ! [A3: fset_P2153231429829016240_a_b_b,X: produc6499617310964463488_a_b_b] :
      ( ( ord_le8870638447146015504_a_b_b @ A3 @ ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) )
     => ( ( A3 = bot_bo2248824169281960260_a_b_b )
        | ( A3
          = ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).

% fsubset_fsingletonD
thf(fact_265_finsert__not__fempty,axiom,
    ! [A: produc6499617310964463488_a_b_b,A3: fset_P2153231429829016240_a_b_b] :
      ( ( finser2303212525150181944_a_b_b @ A @ A3 )
     != bot_bo2248824169281960260_a_b_b ) ).

% finsert_not_fempty
thf(fact_266_fsingleton__inject,axiom,
    ! [A: produc6499617310964463488_a_b_b,B: produc6499617310964463488_a_b_b] :
      ( ( ( finser2303212525150181944_a_b_b @ A @ bot_bo2248824169281960260_a_b_b )
        = ( finser2303212525150181944_a_b_b @ B @ bot_bo2248824169281960260_a_b_b ) )
     => ( A = B ) ) ).

% fsingleton_inject
thf(fact_267_fdoubleton__eq__iff,axiom,
    ! [A: produc6499617310964463488_a_b_b,B: produc6499617310964463488_a_b_b,C: produc6499617310964463488_a_b_b,D2: produc6499617310964463488_a_b_b] :
      ( ( ( finser2303212525150181944_a_b_b @ A @ ( finser2303212525150181944_a_b_b @ B @ bot_bo2248824169281960260_a_b_b ) )
        = ( finser2303212525150181944_a_b_b @ C @ ( finser2303212525150181944_a_b_b @ D2 @ bot_bo2248824169281960260_a_b_b ) ) )
      = ( ( ( A = C )
          & ( B = D2 ) )
        | ( ( A = D2 )
          & ( B = C ) ) ) ) ).

% fdoubleton_eq_iff
thf(fact_268_fset__exhaust,axiom,
    ! [S: fset_P2153231429829016240_a_b_b] :
      ( ( S != bot_bo2248824169281960260_a_b_b )
     => ~ ! [X3: produc6499617310964463488_a_b_b,S3: fset_P2153231429829016240_a_b_b] :
            ( S
           != ( finser2303212525150181944_a_b_b @ X3 @ S3 ) ) ) ).

% fset_exhaust
thf(fact_269_FSet_Ofset__induct,axiom,
    ! [P3: fset_P2153231429829016240_a_b_b > $o,S: fset_P2153231429829016240_a_b_b] :
      ( ( P3 @ bot_bo2248824169281960260_a_b_b )
     => ( ! [X3: produc6499617310964463488_a_b_b,S4: fset_P2153231429829016240_a_b_b] :
            ( ( P3 @ S4 )
           => ( P3 @ ( finser2303212525150181944_a_b_b @ X3 @ S4 ) ) )
       => ( P3 @ S ) ) ) ).

% FSet.fset_induct
thf(fact_270_accp__subset,axiom,
    ! [R1: produc7147531718898801626st_a_b > produc7147531718898801626st_a_b > $o,R22: produc7147531718898801626st_a_b > produc7147531718898801626st_a_b > $o] :
      ( ( ord_le5270784759268031602_a_b_o @ R1 @ R22 )
     => ( ord_le4286200605579700501_a_b_o @ ( accp_P2814348900907978769st_a_b @ R22 ) @ ( accp_P2814348900907978769st_a_b @ R1 ) ) ) ).

% accp_subset
thf(fact_271_accp__subset,axiom,
    ! [R1: produc111314985273491367st_a_b > produc111314985273491367st_a_b > $o,R22: produc111314985273491367st_a_b > produc111314985273491367st_a_b > $o] :
      ( ( ord_le3835420409775173976_a_b_o @ R1 @ R22 )
     => ( ord_le2928166020984067464_a_b_o @ ( accp_P3553483059299338846st_a_b @ R22 ) @ ( accp_P3553483059299338846st_a_b @ R1 ) ) ) ).

% accp_subset
thf(fact_272_fthe__felem__eq,axiom,
    ! [X: produc6499617310964463488_a_b_b] :
      ( ( fthe_e8731401527248658499_a_b_b @ ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) )
      = X ) ).

% fthe_felem_eq
thf(fact_273_accp__subset__induct,axiom,
    ! [D: produc7147531718898801626st_a_b > $o,R3: produc7147531718898801626st_a_b > produc7147531718898801626st_a_b > $o,X: produc7147531718898801626st_a_b,P3: produc7147531718898801626st_a_b > $o] :
      ( ( ord_le4286200605579700501_a_b_o @ D @ ( accp_P2814348900907978769st_a_b @ R3 ) )
     => ( ! [X3: produc7147531718898801626st_a_b,Z4: produc7147531718898801626st_a_b] :
            ( ( D @ X3 )
           => ( ( R3 @ Z4 @ X3 )
             => ( D @ Z4 ) ) )
       => ( ( D @ X )
         => ( ! [X3: produc7147531718898801626st_a_b] :
                ( ( D @ X3 )
               => ( ! [Z5: produc7147531718898801626st_a_b] :
                      ( ( R3 @ Z5 @ X3 )
                     => ( P3 @ Z5 ) )
                 => ( P3 @ X3 ) ) )
           => ( P3 @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_274_accp__subset__induct,axiom,
    ! [D: produc111314985273491367st_a_b > $o,R3: produc111314985273491367st_a_b > produc111314985273491367st_a_b > $o,X: produc111314985273491367st_a_b,P3: produc111314985273491367st_a_b > $o] :
      ( ( ord_le2928166020984067464_a_b_o @ D @ ( accp_P3553483059299338846st_a_b @ R3 ) )
     => ( ! [X3: produc111314985273491367st_a_b,Z4: produc111314985273491367st_a_b] :
            ( ( D @ X3 )
           => ( ( R3 @ Z4 @ X3 )
             => ( D @ Z4 ) ) )
       => ( ( D @ X )
         => ( ! [X3: produc111314985273491367st_a_b] :
                ( ( D @ X3 )
               => ( ! [Z5: produc111314985273491367st_a_b] :
                      ( ( R3 @ Z5 @ X3 )
                     => ( P3 @ Z5 ) )
                 => ( P3 @ X3 ) ) )
           => ( P3 @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_275_subset__emptyI,axiom,
    ! [A3: set_list_a] :
      ( ! [X3: list_a] :
          ~ ( member_list_a @ X3 @ A3 )
     => ( ord_le8861187494160871172list_a @ A3 @ bot_bot_set_list_a ) ) ).

% subset_emptyI
thf(fact_276_subset__emptyI,axiom,
    ! [A3: set_Pr3443975907877334966_a_b_b] :
      ( ! [X3: produc6499617310964463488_a_b_b] :
          ~ ( member4695696432722591383_a_b_b @ X3 @ A3 )
     => ( ord_le1619362961161175062_a_b_b @ A3 @ bot_bo4897374000430069834_a_b_b ) ) ).

% subset_emptyI
thf(fact_277_subset__emptyI,axiom,
    ! [A3: set_dtree_list_a_b] :
      ( ! [X3: dtree_list_a_b] :
          ~ ( member551035911493665803st_a_b @ X3 @ A3 )
     => ( ord_le7599451563663638410st_a_b @ A3 @ bot_bo798015271861357502st_a_b ) ) ).

% subset_emptyI
thf(fact_278_subset__emptyI,axiom,
    ! [A3: set_real] :
      ( ! [X3: real] :
          ~ ( member_real @ X3 @ A3 )
     => ( ord_less_eq_set_real @ A3 @ bot_bot_set_real ) ) ).

% subset_emptyI
thf(fact_279_subset__emptyI,axiom,
    ! [A3: set_nat] :
      ( ! [X3: nat] :
          ~ ( member_nat @ X3 @ A3 )
     => ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_280_num__leaves__singleton,axiom,
    ! [R2: list_a,T: dtree_list_a_b,E: b] :
      ( ( num_leaves_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
      = ( num_leaves_list_a_b @ T ) ) ).

% num_leaves_singleton
thf(fact_281_darcs__sub__if__children__sub,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,Ys: fset_P2153231429829016240_a_b_b,R2: list_a,V: list_a] :
      ( ( ord_le8870638447146015504_a_b_b @ Xs2 @ Ys )
     => ( ord_less_eq_set_b @ ( darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) @ ( darcs_list_a_b @ ( node_list_a_b @ V @ Ys ) ) ) ) ).

% darcs_sub_if_children_sub
thf(fact_282_Set_Ois__empty__def,axiom,
    ( is_empty_nat
    = ( ^ [A6: set_nat] : ( A6 = bot_bot_set_nat ) ) ) ).

% Set.is_empty_def
thf(fact_283_num__leaves__root,axiom,
    ! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,R5: list_a] :
      ( ( num_leaves_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
      = ( num_leaves_list_a_b @ ( node_list_a_b @ R5 @ Xs2 ) ) ) ).

% num_leaves_root
thf(fact_284_ssubst__Pair__rhs,axiom,
    ! [R2: list_a,S2: list_a,R3: set_Pr4048851178543822343list_a,S5: list_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ R2 @ S2 ) @ R3 )
     => ( ( S5 = S2 )
       => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ R2 @ S5 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_285_ssubst__Pair__rhs,axiom,
    ! [R2: list_a,S2: dtree_list_a_b,R3: set_Pr5727314569254441693st_a_b,S5: dtree_list_a_b] :
      ( ( member6763472504040263486st_a_b @ ( produc148520996349637281st_a_b @ R2 @ S2 ) @ R3 )
     => ( ( S5 = S2 )
       => ( member6763472504040263486st_a_b @ ( produc148520996349637281st_a_b @ R2 @ S5 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_286_ssubst__Pair__rhs,axiom,
    ! [R2: dtree_list_a_b,S2: b,R3: set_Pr3443975907877334966_a_b_b,S5: b] :
      ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ R2 @ S2 ) @ R3 )
     => ( ( S5 = S2 )
       => ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ R2 @ S5 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_287_ssubst__Pair__rhs,axiom,
    ! [R2: list_a,S2: produc111314985273491367st_a_b,R3: set_Pr8451570673126195472st_a_b,S5: produc111314985273491367st_a_b] :
      ( ( member4895573663636194545st_a_b @ ( produc673257793671328980st_a_b @ R2 @ S2 ) @ R3 )
     => ( ( S5 = S2 )
       => ( member4895573663636194545st_a_b @ ( produc673257793671328980st_a_b @ R2 @ S5 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_288_dtree_Oset__intros_I3_J,axiom,
    ! [Ye: produc6499617310964463488_a_b_b,X2: fset_P2153231429829016240_a_b_b,Yf: dtree_list_a_b,Yh: b,X1: list_a] :
      ( ( member4695696432722591383_a_b_b @ Ye @ ( fset_P9138963618725001425_a_b_b @ X2 ) )
     => ( ( member551035911493665803st_a_b @ Yf @ ( basic_5528648086025252546_a_b_b @ Ye ) )
       => ( ( member_b @ Yh @ ( darcs_list_a_b @ Yf ) )
         => ( member_b @ Yh @ ( darcs_list_a_b @ ( node_list_a_b @ X1 @ X2 ) ) ) ) ) ) ).

% dtree.set_intros(3)
thf(fact_289_dtree_Oset__sel_I3_J,axiom,
    ! [Xe: produc6499617310964463488_a_b_b,A: dtree_list_a_b,Xf: dtree_list_a_b,Xh: b] :
      ( ( member4695696432722591383_a_b_b @ Xe @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ A ) ) )
     => ( ( member551035911493665803st_a_b @ Xf @ ( basic_5528648086025252546_a_b_b @ Xe ) )
       => ( ( member_b @ Xh @ ( darcs_list_a_b @ Xf ) )
         => ( member_b @ Xh @ ( darcs_list_a_b @ A ) ) ) ) ) ).

% dtree.set_sel(3)
thf(fact_290_accp_Ocases,axiom,
    ! [R2: produc7147531718898801626st_a_b > produc7147531718898801626st_a_b > $o,A: produc7147531718898801626st_a_b] :
      ( ( accp_P2814348900907978769st_a_b @ R2 @ A )
     => ! [Y6: produc7147531718898801626st_a_b] :
          ( ( R2 @ Y6 @ A )
         => ( accp_P2814348900907978769st_a_b @ R2 @ Y6 ) ) ) ).

% accp.cases
thf(fact_291_accp_Ocases,axiom,
    ! [R2: produc111314985273491367st_a_b > produc111314985273491367st_a_b > $o,A: produc111314985273491367st_a_b] :
      ( ( accp_P3553483059299338846st_a_b @ R2 @ A )
     => ! [Y6: produc111314985273491367st_a_b] :
          ( ( R2 @ Y6 @ A )
         => ( accp_P3553483059299338846st_a_b @ R2 @ Y6 ) ) ) ).

% accp.cases
thf(fact_292_accp_Osimps,axiom,
    ( accp_P2814348900907978769st_a_b
    = ( ^ [R4: produc7147531718898801626st_a_b > produc7147531718898801626st_a_b > $o,A4: produc7147531718898801626st_a_b] :
        ? [X4: produc7147531718898801626st_a_b] :
          ( ( A4 = X4 )
          & ! [Y3: produc7147531718898801626st_a_b] :
              ( ( R4 @ Y3 @ X4 )
             => ( accp_P2814348900907978769st_a_b @ R4 @ Y3 ) ) ) ) ) ).

% accp.simps
thf(fact_293_accp_Osimps,axiom,
    ( accp_P3553483059299338846st_a_b
    = ( ^ [R4: produc111314985273491367st_a_b > produc111314985273491367st_a_b > $o,A4: produc111314985273491367st_a_b] :
        ? [X4: produc111314985273491367st_a_b] :
          ( ( A4 = X4 )
          & ! [Y3: produc111314985273491367st_a_b] :
              ( ( R4 @ Y3 @ X4 )
             => ( accp_P3553483059299338846st_a_b @ R4 @ Y3 ) ) ) ) ) ).

% accp.simps
thf(fact_294_accpI,axiom,
    ! [R2: produc7147531718898801626st_a_b > produc7147531718898801626st_a_b > $o,X: produc7147531718898801626st_a_b] :
      ( ! [Y4: produc7147531718898801626st_a_b] :
          ( ( R2 @ Y4 @ X )
         => ( accp_P2814348900907978769st_a_b @ R2 @ Y4 ) )
     => ( accp_P2814348900907978769st_a_b @ R2 @ X ) ) ).

% accpI
thf(fact_295_accpI,axiom,
    ! [R2: produc111314985273491367st_a_b > produc111314985273491367st_a_b > $o,X: produc111314985273491367st_a_b] :
      ( ! [Y4: produc111314985273491367st_a_b] :
          ( ( R2 @ Y4 @ X )
         => ( accp_P3553483059299338846st_a_b @ R2 @ Y4 ) )
     => ( accp_P3553483059299338846st_a_b @ R2 @ X ) ) ).

% accpI
thf(fact_296_accp__induct,axiom,
    ! [R2: produc7147531718898801626st_a_b > produc7147531718898801626st_a_b > $o,A: produc7147531718898801626st_a_b,P3: produc7147531718898801626st_a_b > $o] :
      ( ( accp_P2814348900907978769st_a_b @ R2 @ A )
     => ( ! [X3: produc7147531718898801626st_a_b] :
            ( ( accp_P2814348900907978769st_a_b @ R2 @ X3 )
           => ( ! [Y6: produc7147531718898801626st_a_b] :
                  ( ( R2 @ Y6 @ X3 )
                 => ( P3 @ Y6 ) )
             => ( P3 @ X3 ) ) )
       => ( P3 @ A ) ) ) ).

% accp_induct
thf(fact_297_accp__induct,axiom,
    ! [R2: produc111314985273491367st_a_b > produc111314985273491367st_a_b > $o,A: produc111314985273491367st_a_b,P3: produc111314985273491367st_a_b > $o] :
      ( ( accp_P3553483059299338846st_a_b @ R2 @ A )
     => ( ! [X3: produc111314985273491367st_a_b] :
            ( ( accp_P3553483059299338846st_a_b @ R2 @ X3 )
           => ( ! [Y6: produc111314985273491367st_a_b] :
                  ( ( R2 @ Y6 @ X3 )
                 => ( P3 @ Y6 ) )
             => ( P3 @ X3 ) ) )
       => ( P3 @ A ) ) ) ).

% accp_induct
thf(fact_298_accp__downward,axiom,
    ! [R2: produc7147531718898801626st_a_b > produc7147531718898801626st_a_b > $o,B: produc7147531718898801626st_a_b,A: produc7147531718898801626st_a_b] :
      ( ( accp_P2814348900907978769st_a_b @ R2 @ B )
     => ( ( R2 @ A @ B )
       => ( accp_P2814348900907978769st_a_b @ R2 @ A ) ) ) ).

% accp_downward
thf(fact_299_accp__downward,axiom,
    ! [R2: produc111314985273491367st_a_b > produc111314985273491367st_a_b > $o,B: produc111314985273491367st_a_b,A: produc111314985273491367st_a_b] :
      ( ( accp_P3553483059299338846st_a_b @ R2 @ B )
     => ( ( R2 @ A @ B )
       => ( accp_P3553483059299338846st_a_b @ R2 @ A ) ) ) ).

% accp_downward
thf(fact_300_not__accp__down,axiom,
    ! [R3: produc7147531718898801626st_a_b > produc7147531718898801626st_a_b > $o,X: produc7147531718898801626st_a_b] :
      ( ~ ( accp_P2814348900907978769st_a_b @ R3 @ X )
     => ~ ! [Z4: produc7147531718898801626st_a_b] :
            ( ( R3 @ Z4 @ X )
           => ( accp_P2814348900907978769st_a_b @ R3 @ Z4 ) ) ) ).

% not_accp_down
thf(fact_301_not__accp__down,axiom,
    ! [R3: produc111314985273491367st_a_b > produc111314985273491367st_a_b > $o,X: produc111314985273491367st_a_b] :
      ( ~ ( accp_P3553483059299338846st_a_b @ R3 @ X )
     => ~ ! [Z4: produc111314985273491367st_a_b] :
            ( ( R3 @ Z4 @ X )
           => ( accp_P3553483059299338846st_a_b @ R3 @ Z4 ) ) ) ).

% not_accp_down
thf(fact_302_accp__induct__rule,axiom,
    ! [R2: produc7147531718898801626st_a_b > produc7147531718898801626st_a_b > $o,A: produc7147531718898801626st_a_b,P3: produc7147531718898801626st_a_b > $o] :
      ( ( accp_P2814348900907978769st_a_b @ R2 @ A )
     => ( ! [X3: produc7147531718898801626st_a_b] :
            ( ( accp_P2814348900907978769st_a_b @ R2 @ X3 )
           => ( ! [Y6: produc7147531718898801626st_a_b] :
                  ( ( R2 @ Y6 @ X3 )
                 => ( P3 @ Y6 ) )
             => ( P3 @ X3 ) ) )
       => ( P3 @ A ) ) ) ).

% accp_induct_rule
thf(fact_303_accp__induct__rule,axiom,
    ! [R2: produc111314985273491367st_a_b > produc111314985273491367st_a_b > $o,A: produc111314985273491367st_a_b,P3: produc111314985273491367st_a_b > $o] :
      ( ( accp_P3553483059299338846st_a_b @ R2 @ A )
     => ( ! [X3: produc111314985273491367st_a_b] :
            ( ( accp_P3553483059299338846st_a_b @ R2 @ X3 )
           => ( ! [Y6: produc111314985273491367st_a_b] :
                  ( ( R2 @ Y6 @ X3 )
                 => ( P3 @ Y6 ) )
             => ( P3 @ X3 ) ) )
       => ( P3 @ A ) ) ) ).

% accp_induct_rule
thf(fact_304_prop__restrict,axiom,
    ! [X: list_a,Z6: set_list_a,X6: set_list_a,P3: list_a > $o] :
      ( ( member_list_a @ X @ Z6 )
     => ( ( ord_le8861187494160871172list_a @ Z6
          @ ( collect_list_a
            @ ^ [X4: list_a] :
                ( ( member_list_a @ X4 @ X6 )
                & ( P3 @ X4 ) ) ) )
       => ( P3 @ X ) ) ) ).

% prop_restrict
thf(fact_305_prop__restrict,axiom,
    ! [X: produc6499617310964463488_a_b_b,Z6: set_Pr3443975907877334966_a_b_b,X6: set_Pr3443975907877334966_a_b_b,P3: produc6499617310964463488_a_b_b > $o] :
      ( ( member4695696432722591383_a_b_b @ X @ Z6 )
     => ( ( ord_le1619362961161175062_a_b_b @ Z6
          @ ( collec3241703154345475797_a_b_b
            @ ^ [X4: produc6499617310964463488_a_b_b] :
                ( ( member4695696432722591383_a_b_b @ X4 @ X6 )
                & ( P3 @ X4 ) ) ) )
       => ( P3 @ X ) ) ) ).

% prop_restrict
thf(fact_306_prop__restrict,axiom,
    ! [X: dtree_list_a_b,Z6: set_dtree_list_a_b,X6: set_dtree_list_a_b,P3: dtree_list_a_b > $o] :
      ( ( member551035911493665803st_a_b @ X @ Z6 )
     => ( ( ord_le7599451563663638410st_a_b @ Z6
          @ ( collec2944820760411501129st_a_b
            @ ^ [X4: dtree_list_a_b] :
                ( ( member551035911493665803st_a_b @ X4 @ X6 )
                & ( P3 @ X4 ) ) ) )
       => ( P3 @ X ) ) ) ).

% prop_restrict
thf(fact_307_prop__restrict,axiom,
    ! [X: real,Z6: set_real,X6: set_real,P3: real > $o] :
      ( ( member_real @ X @ Z6 )
     => ( ( ord_less_eq_set_real @ Z6
          @ ( collect_real
            @ ^ [X4: real] :
                ( ( member_real @ X4 @ X6 )
                & ( P3 @ X4 ) ) ) )
       => ( P3 @ X ) ) ) ).

% prop_restrict
thf(fact_308_Collect__restrict,axiom,
    ! [X6: set_list_a,P3: list_a > $o] :
      ( ord_le8861187494160871172list_a
      @ ( collect_list_a
        @ ^ [X4: list_a] :
            ( ( member_list_a @ X4 @ X6 )
            & ( P3 @ X4 ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_309_Collect__restrict,axiom,
    ! [X6: set_Pr3443975907877334966_a_b_b,P3: produc6499617310964463488_a_b_b > $o] :
      ( ord_le1619362961161175062_a_b_b
      @ ( collec3241703154345475797_a_b_b
        @ ^ [X4: produc6499617310964463488_a_b_b] :
            ( ( member4695696432722591383_a_b_b @ X4 @ X6 )
            & ( P3 @ X4 ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_310_Collect__restrict,axiom,
    ! [X6: set_dtree_list_a_b,P3: dtree_list_a_b > $o] :
      ( ord_le7599451563663638410st_a_b
      @ ( collec2944820760411501129st_a_b
        @ ^ [X4: dtree_list_a_b] :
            ( ( member551035911493665803st_a_b @ X4 @ X6 )
            & ( P3 @ X4 ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_311_Collect__restrict,axiom,
    ! [X6: set_real,P3: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X4: real] :
            ( ( member_real @ X4 @ X6 )
            & ( P3 @ X4 ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_312_Collect__empty__eq__bot,axiom,
    ! [P3: nat > $o] :
      ( ( ( collect_nat @ P3 )
        = bot_bot_set_nat )
      = ( P3 = bot_bot_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_313_denormalize_Opinduct,axiom,
    ! [A0: dtree_list_a_b,P3: dtree_list_a_b > $o] :
      ( ( accp_dtree_list_a_b @ denormalize_rel_a_b @ A0 )
     => ( ! [R: list_a,T2: dtree_list_a_b,E2: b] :
            ( ( accp_dtree_list_a_b @ denormalize_rel_a_b @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
           => ( ( P3 @ T2 )
             => ( P3 @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) )
       => ( ! [Xs: fset_P2153231429829016240_a_b_b,R: list_a] :
              ( ( accp_dtree_list_a_b @ denormalize_rel_a_b @ ( node_list_a_b @ R @ Xs ) )
             => ( ! [X5: produc6499617310964463488_a_b_b] :
                    ( Xs
                   != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
               => ( P3 @ ( node_list_a_b @ R @ Xs ) ) ) )
         => ( P3 @ A0 ) ) ) ) ).

% denormalize.pinduct
thf(fact_314_dtree__to__list_Opinduct,axiom,
    ! [A0: dtree_list_a_b,P3: dtree_list_a_b > $o] :
      ( ( accp_dtree_list_a_b @ dtree_452016980859282703st_a_b @ A0 )
     => ( ! [R: list_a,T2: dtree_list_a_b,E2: b] :
            ( ( accp_dtree_list_a_b @ dtree_452016980859282703st_a_b @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
           => ( ( P3 @ T2 )
             => ( P3 @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) )
       => ( ! [Xs: fset_P2153231429829016240_a_b_b,R: list_a] :
              ( ( accp_dtree_list_a_b @ dtree_452016980859282703st_a_b @ ( node_list_a_b @ R @ Xs ) )
             => ( ! [X5: produc6499617310964463488_a_b_b] :
                    ( Xs
                   != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
               => ( P3 @ ( node_list_a_b @ R @ Xs ) ) ) )
         => ( P3 @ A0 ) ) ) ) ).

% dtree_to_list.pinduct
thf(fact_315_dtree_Oset__cases_I2_J,axiom,
    ! [E: b,A: dtree_list_a_b] :
      ( ( member_b @ E @ ( darcs_list_a_b @ A ) )
     => ( ! [Z1: list_a,Z2: fset_P2153231429829016240_a_b_b] :
            ( ( A
              = ( node_list_a_b @ Z1 @ Z2 ) )
           => ! [X3: produc6499617310964463488_a_b_b] :
                ( ( member4695696432722591383_a_b_b @ X3 @ ( fset_P9138963618725001425_a_b_b @ Z2 ) )
               => ! [Xa: dtree_list_a_b] :
                    ( ( member551035911493665803st_a_b @ Xa @ ( basic_5528648086025252546_a_b_b @ X3 ) )
                   => ~ ( member_b @ E @ ( darcs_list_a_b @ Xa ) ) ) ) )
       => ~ ! [Z1: list_a,Z2: fset_P2153231429829016240_a_b_b] :
              ( ( A
                = ( node_list_a_b @ Z1 @ Z2 ) )
             => ! [X3: produc6499617310964463488_a_b_b] :
                  ( ( member4695696432722591383_a_b_b @ X3 @ ( fset_P9138963618725001425_a_b_b @ Z2 ) )
                 => ~ ( member_b @ E @ ( basic_3193785740840883972_a_b_b @ X3 ) ) ) ) ) ) ).

% dtree.set_cases(2)
thf(fact_316_wf__darcs__sucs,axiom,
    ! [T: dtree_list_a_b,X: produc6499617310964463488_a_b_b,R2: list_a] :
      ( ( wf_darcs_list_a_b @ T )
     => ( ( member4695696432722591383_a_b_b @ X @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) )
       => ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ).

% wf_darcs_sucs
thf(fact_317_single__subtree__root__dverts,axiom,
    ! [V22: list_a,T22: dtree_list_a_b,E22: b,T12: dtree_list_a_b] :
      ( ( is_subtree_list_a_b @ ( node_list_a_b @ V22 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T12 )
     => ( member_list_a @ V22 @ ( dverts_list_a_b @ T12 ) ) ) ).

% single_subtree_root_dverts
thf(fact_318_fthe__elem_Orep__eq,axiom,
    ( fthe_e8731401527248658499_a_b_b
    = ( ^ [X4: fset_P2153231429829016240_a_b_b] : ( the_el1530768284204824267_a_b_b @ ( fset_P9138963618725001425_a_b_b @ X4 ) ) ) ) ).

% fthe_elem.rep_eq
thf(fact_319_dverts__subtree__subset,axiom,
    ! [X: dtree_list_a_b,Y: dtree_list_a_b] :
      ( ( is_subtree_list_a_b @ X @ Y )
     => ( ord_le8861187494160871172list_a @ ( dverts_list_a_b @ X ) @ ( dverts_list_a_b @ Y ) ) ) ).

% dverts_subtree_subset
thf(fact_320_subtree__root__if__dverts,axiom,
    ! [X: list_a,T: dtree_list_a_b] :
      ( ( member_list_a @ X @ ( dverts_list_a_b @ T ) )
     => ? [Xs: fset_P2153231429829016240_a_b_b] : ( is_subtree_list_a_b @ ( node_list_a_b @ X @ Xs ) @ T ) ) ).

% subtree_root_if_dverts
thf(fact_321_wf__darcs__sub,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,Ys: fset_P2153231429829016240_a_b_b,R5: list_a,R2: list_a] :
      ( ( ord_le8870638447146015504_a_b_b @ Xs2 @ Ys )
     => ( ( wf_darcs_list_a_b @ ( node_list_a_b @ R5 @ Ys ) )
       => ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ).

% wf_darcs_sub
thf(fact_322_disjoint__darcs__if__wf__aux4,axiom,
    ! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T12: dtree_list_a_b,E1: b,T22: dtree_list_a_b,E22: b] :
      ( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
     => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
       => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
         => ( ( ( produc7704165765595008946_a_b_b @ T12 @ E1 )
             != ( produc7704165765595008946_a_b_b @ T22 @ E22 ) )
           => ( E1 != E22 ) ) ) ) ) ).

% disjoint_darcs_if_wf_aux4
thf(fact_323_dtree_Oset__intros_I4_J,axiom,
    ! [Ye: produc6499617310964463488_a_b_b,X2: fset_P2153231429829016240_a_b_b,Yi: b,X1: list_a] :
      ( ( member4695696432722591383_a_b_b @ Ye @ ( fset_P9138963618725001425_a_b_b @ X2 ) )
     => ( ( member_b @ Yi @ ( basic_3193785740840883972_a_b_b @ Ye ) )
       => ( member_b @ Yi @ ( darcs_list_a_b @ ( node_list_a_b @ X1 @ X2 ) ) ) ) ) ).

% dtree.set_intros(4)
thf(fact_324_disjoint__darcs__if__wf__aux3,axiom,
    ! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T12: dtree_list_a_b,E1: b,T22: dtree_list_a_b,E22: b] :
      ( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
     => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
       => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
         => ~ ( member_b @ E22 @ ( darcs_list_a_b @ T12 ) ) ) ) ) ).

% disjoint_darcs_if_wf_aux3
thf(fact_325_disjoint__darcs__if__wf__aux1,axiom,
    ! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T12: dtree_list_a_b,E1: b] :
      ( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
     => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
       => ~ ( member_b @ E1 @ ( darcs_list_a_b @ T12 ) ) ) ) ).

% disjoint_darcs_if_wf_aux1
thf(fact_326_dtree_Oset__sel_I4_J,axiom,
    ! [Xe: produc6499617310964463488_a_b_b,A: dtree_list_a_b,Xi: b] :
      ( ( member4695696432722591383_a_b_b @ Xe @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ A ) ) )
     => ( ( member_b @ Xi @ ( basic_3193785740840883972_a_b_b @ Xe ) )
       => ( member_b @ Xi @ ( darcs_list_a_b @ A ) ) ) ) ).

% dtree.set_sel(4)
thf(fact_327_denormalize_Opsimps_I2_J,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
      ( ! [X3: produc6499617310964463488_a_b_b] :
          ( Xs2
         != ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
     => ( ( accp_dtree_list_a_b @ denormalize_rel_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
       => ( ( denormalize_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
          = R2 ) ) ) ).

% denormalize.psimps(2)
thf(fact_328_disjoint__darcs__if__wf__aux2,axiom,
    ! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T12: dtree_list_a_b,E1: b,T22: dtree_list_a_b,E22: b] :
      ( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
     => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
       => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
         => ( ( ( produc7704165765595008946_a_b_b @ T12 @ E1 )
             != ( produc7704165765595008946_a_b_b @ T22 @ E22 ) )
           => ( ( inf_inf_set_b @ ( darcs_list_a_b @ T12 ) @ ( darcs_list_a_b @ T22 ) )
              = bot_bot_set_b ) ) ) ) ) ).

% disjoint_darcs_if_wf_aux2
thf(fact_329_single__subtree__child__root__dverts,axiom,
    ! [V22: list_a,T22: dtree_list_a_b,E22: b,T12: dtree_list_a_b] :
      ( ( is_subtree_list_a_b @ ( node_list_a_b @ V22 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T12 )
     => ( member_list_a @ ( root_list_a_b @ T22 ) @ ( dverts_list_a_b @ T12 ) ) ) ).

% single_subtree_child_root_dverts
thf(fact_330_dtail__in__child__eq__child,axiom,
    ! [T: dtree_list_a_b,E1: b,Xs2: fset_P2153231429829016240_a_b_b,E: b,R2: list_a,Def2: b > list_a] :
      ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
     => ( ( member_b @ E @ ( darcs_list_a_b @ T ) )
       => ( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
         => ( ( dtail_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ Def2 @ E )
            = ( dtail_list_a_b @ T @ Def2 @ E ) ) ) ) ) ).

% dtail_in_child_eq_child
thf(fact_331_dhead__in__child__eq__child,axiom,
    ! [T: dtree_list_a_b,E1: b,Xs2: fset_P2153231429829016240_a_b_b,E: b,R2: list_a,Def2: b > list_a] :
      ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
     => ( ( member_b @ E @ ( darcs_list_a_b @ T ) )
       => ( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
         => ( ( dhead_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ Def2 @ E )
            = ( dhead_list_a_b @ T @ Def2 @ E ) ) ) ) ) ).

% dhead_in_child_eq_child
thf(fact_332_mdeg__child__sucs__le,axiom,
    ! [V: list_a,T: dtree_list_a_b,R2: list_a,E: b] : ( ord_less_eq_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ V @ ( sucs_list_a_b @ T ) ) ) @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).

% mdeg_child_sucs_le
thf(fact_333_subset__Collect__iff,axiom,
    ! [B5: set_list_a,A3: set_list_a,P3: list_a > $o] :
      ( ( ord_le8861187494160871172list_a @ B5 @ A3 )
     => ( ( ord_le8861187494160871172list_a @ B5
          @ ( collect_list_a
            @ ^ [X4: list_a] :
                ( ( member_list_a @ X4 @ A3 )
                & ( P3 @ X4 ) ) ) )
        = ( ! [X4: list_a] :
              ( ( member_list_a @ X4 @ B5 )
             => ( P3 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_334_subset__Collect__iff,axiom,
    ! [B5: set_Pr3443975907877334966_a_b_b,A3: set_Pr3443975907877334966_a_b_b,P3: produc6499617310964463488_a_b_b > $o] :
      ( ( ord_le1619362961161175062_a_b_b @ B5 @ A3 )
     => ( ( ord_le1619362961161175062_a_b_b @ B5
          @ ( collec3241703154345475797_a_b_b
            @ ^ [X4: produc6499617310964463488_a_b_b] :
                ( ( member4695696432722591383_a_b_b @ X4 @ A3 )
                & ( P3 @ X4 ) ) ) )
        = ( ! [X4: produc6499617310964463488_a_b_b] :
              ( ( member4695696432722591383_a_b_b @ X4 @ B5 )
             => ( P3 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_335_subset__Collect__iff,axiom,
    ! [B5: set_dtree_list_a_b,A3: set_dtree_list_a_b,P3: dtree_list_a_b > $o] :
      ( ( ord_le7599451563663638410st_a_b @ B5 @ A3 )
     => ( ( ord_le7599451563663638410st_a_b @ B5
          @ ( collec2944820760411501129st_a_b
            @ ^ [X4: dtree_list_a_b] :
                ( ( member551035911493665803st_a_b @ X4 @ A3 )
                & ( P3 @ X4 ) ) ) )
        = ( ! [X4: dtree_list_a_b] :
              ( ( member551035911493665803st_a_b @ X4 @ B5 )
             => ( P3 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_336_subset__Collect__iff,axiom,
    ! [B5: set_real,A3: set_real,P3: real > $o] :
      ( ( ord_less_eq_set_real @ B5 @ A3 )
     => ( ( ord_less_eq_set_real @ B5
          @ ( collect_real
            @ ^ [X4: real] :
                ( ( member_real @ X4 @ A3 )
                & ( P3 @ X4 ) ) ) )
        = ( ! [X4: real] :
              ( ( member_real @ X4 @ B5 )
             => ( P3 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_337_Int__iff,axiom,
    ! [C: list_a,A3: set_list_a,B5: set_list_a] :
      ( ( member_list_a @ C @ ( inf_inf_set_list_a @ A3 @ B5 ) )
      = ( ( member_list_a @ C @ A3 )
        & ( member_list_a @ C @ B5 ) ) ) ).

% Int_iff
thf(fact_338_Int__iff,axiom,
    ! [C: produc6499617310964463488_a_b_b,A3: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
      ( ( member4695696432722591383_a_b_b @ C @ ( inf_in7962574851115124040_a_b_b @ A3 @ B5 ) )
      = ( ( member4695696432722591383_a_b_b @ C @ A3 )
        & ( member4695696432722591383_a_b_b @ C @ B5 ) ) ) ).

% Int_iff
thf(fact_339_Int__iff,axiom,
    ! [C: dtree_list_a_b,A3: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
      ( ( member551035911493665803st_a_b @ C @ ( inf_in3355993651213403836st_a_b @ A3 @ B5 ) )
      = ( ( member551035911493665803st_a_b @ C @ A3 )
        & ( member551035911493665803st_a_b @ C @ B5 ) ) ) ).

% Int_iff
thf(fact_340_Int__iff,axiom,
    ! [C: real,A3: set_real,B5: set_real] :
      ( ( member_real @ C @ ( inf_inf_set_real @ A3 @ B5 ) )
      = ( ( member_real @ C @ A3 )
        & ( member_real @ C @ B5 ) ) ) ).

% Int_iff
thf(fact_341_IntI,axiom,
    ! [C: list_a,A3: set_list_a,B5: set_list_a] :
      ( ( member_list_a @ C @ A3 )
     => ( ( member_list_a @ C @ B5 )
       => ( member_list_a @ C @ ( inf_inf_set_list_a @ A3 @ B5 ) ) ) ) ).

% IntI
thf(fact_342_IntI,axiom,
    ! [C: produc6499617310964463488_a_b_b,A3: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
      ( ( member4695696432722591383_a_b_b @ C @ A3 )
     => ( ( member4695696432722591383_a_b_b @ C @ B5 )
       => ( member4695696432722591383_a_b_b @ C @ ( inf_in7962574851115124040_a_b_b @ A3 @ B5 ) ) ) ) ).

% IntI
thf(fact_343_IntI,axiom,
    ! [C: dtree_list_a_b,A3: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
      ( ( member551035911493665803st_a_b @ C @ A3 )
     => ( ( member551035911493665803st_a_b @ C @ B5 )
       => ( member551035911493665803st_a_b @ C @ ( inf_in3355993651213403836st_a_b @ A3 @ B5 ) ) ) ) ).

% IntI
thf(fact_344_IntI,axiom,
    ! [C: real,A3: set_real,B5: set_real] :
      ( ( member_real @ C @ A3 )
     => ( ( member_real @ C @ B5 )
       => ( member_real @ C @ ( inf_inf_set_real @ A3 @ B5 ) ) ) ) ).

% IntI
thf(fact_345_dtree_Ocollapse,axiom,
    ! [Dtree: dtree_list_a_b] :
      ( ( node_list_a_b @ ( root_list_a_b @ Dtree ) @ ( sucs_list_a_b @ Dtree ) )
      = Dtree ) ).

% dtree.collapse
thf(fact_346_Int__def,axiom,
    ( inf_inf_set_list_a
    = ( ^ [A6: set_list_a,B6: set_list_a] :
          ( collect_list_a
          @ ^ [X4: list_a] :
              ( ( member_list_a @ X4 @ A6 )
              & ( member_list_a @ X4 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_347_Int__def,axiom,
    ( inf_in7962574851115124040_a_b_b
    = ( ^ [A6: set_Pr3443975907877334966_a_b_b,B6: set_Pr3443975907877334966_a_b_b] :
          ( collec3241703154345475797_a_b_b
          @ ^ [X4: produc6499617310964463488_a_b_b] :
              ( ( member4695696432722591383_a_b_b @ X4 @ A6 )
              & ( member4695696432722591383_a_b_b @ X4 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_348_Int__def,axiom,
    ( inf_in3355993651213403836st_a_b
    = ( ^ [A6: set_dtree_list_a_b,B6: set_dtree_list_a_b] :
          ( collec2944820760411501129st_a_b
          @ ^ [X4: dtree_list_a_b] :
              ( ( member551035911493665803st_a_b @ X4 @ A6 )
              & ( member551035911493665803st_a_b @ X4 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_349_Int__def,axiom,
    ( inf_inf_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( collect_real
          @ ^ [X4: real] :
              ( ( member_real @ X4 @ A6 )
              & ( member_real @ X4 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_350_Int__Collect,axiom,
    ! [X: list_a,A3: set_list_a,P3: list_a > $o] :
      ( ( member_list_a @ X @ ( inf_inf_set_list_a @ A3 @ ( collect_list_a @ P3 ) ) )
      = ( ( member_list_a @ X @ A3 )
        & ( P3 @ X ) ) ) ).

% Int_Collect
thf(fact_351_Int__Collect,axiom,
    ! [X: produc6499617310964463488_a_b_b,A3: set_Pr3443975907877334966_a_b_b,P3: produc6499617310964463488_a_b_b > $o] :
      ( ( member4695696432722591383_a_b_b @ X @ ( inf_in7962574851115124040_a_b_b @ A3 @ ( collec3241703154345475797_a_b_b @ P3 ) ) )
      = ( ( member4695696432722591383_a_b_b @ X @ A3 )
        & ( P3 @ X ) ) ) ).

% Int_Collect
thf(fact_352_Int__Collect,axiom,
    ! [X: dtree_list_a_b,A3: set_dtree_list_a_b,P3: dtree_list_a_b > $o] :
      ( ( member551035911493665803st_a_b @ X @ ( inf_in3355993651213403836st_a_b @ A3 @ ( collec2944820760411501129st_a_b @ P3 ) ) )
      = ( ( member551035911493665803st_a_b @ X @ A3 )
        & ( P3 @ X ) ) ) ).

% Int_Collect
thf(fact_353_Int__Collect,axiom,
    ! [X: real,A3: set_real,P3: real > $o] :
      ( ( member_real @ X @ ( inf_inf_set_real @ A3 @ ( collect_real @ P3 ) ) )
      = ( ( member_real @ X @ A3 )
        & ( P3 @ X ) ) ) ).

% Int_Collect
thf(fact_354_IntD2,axiom,
    ! [C: list_a,A3: set_list_a,B5: set_list_a] :
      ( ( member_list_a @ C @ ( inf_inf_set_list_a @ A3 @ B5 ) )
     => ( member_list_a @ C @ B5 ) ) ).

% IntD2
thf(fact_355_IntD2,axiom,
    ! [C: produc6499617310964463488_a_b_b,A3: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
      ( ( member4695696432722591383_a_b_b @ C @ ( inf_in7962574851115124040_a_b_b @ A3 @ B5 ) )
     => ( member4695696432722591383_a_b_b @ C @ B5 ) ) ).

% IntD2
thf(fact_356_IntD2,axiom,
    ! [C: dtree_list_a_b,A3: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
      ( ( member551035911493665803st_a_b @ C @ ( inf_in3355993651213403836st_a_b @ A3 @ B5 ) )
     => ( member551035911493665803st_a_b @ C @ B5 ) ) ).

% IntD2
thf(fact_357_IntD2,axiom,
    ! [C: real,A3: set_real,B5: set_real] :
      ( ( member_real @ C @ ( inf_inf_set_real @ A3 @ B5 ) )
     => ( member_real @ C @ B5 ) ) ).

% IntD2
thf(fact_358_IntD1,axiom,
    ! [C: list_a,A3: set_list_a,B5: set_list_a] :
      ( ( member_list_a @ C @ ( inf_inf_set_list_a @ A3 @ B5 ) )
     => ( member_list_a @ C @ A3 ) ) ).

% IntD1
thf(fact_359_IntD1,axiom,
    ! [C: produc6499617310964463488_a_b_b,A3: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
      ( ( member4695696432722591383_a_b_b @ C @ ( inf_in7962574851115124040_a_b_b @ A3 @ B5 ) )
     => ( member4695696432722591383_a_b_b @ C @ A3 ) ) ).

% IntD1
thf(fact_360_IntD1,axiom,
    ! [C: dtree_list_a_b,A3: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
      ( ( member551035911493665803st_a_b @ C @ ( inf_in3355993651213403836st_a_b @ A3 @ B5 ) )
     => ( member551035911493665803st_a_b @ C @ A3 ) ) ).

% IntD1
thf(fact_361_IntD1,axiom,
    ! [C: real,A3: set_real,B5: set_real] :
      ( ( member_real @ C @ ( inf_inf_set_real @ A3 @ B5 ) )
     => ( member_real @ C @ A3 ) ) ).

% IntD1
thf(fact_362_IntE,axiom,
    ! [C: list_a,A3: set_list_a,B5: set_list_a] :
      ( ( member_list_a @ C @ ( inf_inf_set_list_a @ A3 @ B5 ) )
     => ~ ( ( member_list_a @ C @ A3 )
         => ~ ( member_list_a @ C @ B5 ) ) ) ).

% IntE
thf(fact_363_IntE,axiom,
    ! [C: produc6499617310964463488_a_b_b,A3: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
      ( ( member4695696432722591383_a_b_b @ C @ ( inf_in7962574851115124040_a_b_b @ A3 @ B5 ) )
     => ~ ( ( member4695696432722591383_a_b_b @ C @ A3 )
         => ~ ( member4695696432722591383_a_b_b @ C @ B5 ) ) ) ).

% IntE
thf(fact_364_IntE,axiom,
    ! [C: dtree_list_a_b,A3: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
      ( ( member551035911493665803st_a_b @ C @ ( inf_in3355993651213403836st_a_b @ A3 @ B5 ) )
     => ~ ( ( member551035911493665803st_a_b @ C @ A3 )
         => ~ ( member551035911493665803st_a_b @ C @ B5 ) ) ) ).

% IntE
thf(fact_365_IntE,axiom,
    ! [C: real,A3: set_real,B5: set_real] :
      ( ( member_real @ C @ ( inf_inf_set_real @ A3 @ B5 ) )
     => ~ ( ( member_real @ C @ A3 )
         => ~ ( member_real @ C @ B5 ) ) ) ).

% IntE
thf(fact_366_mdeg__root,axiom,
    ! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,V: list_a] :
      ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
      = ( max_deg_list_a_b @ ( node_list_a_b @ V @ Xs2 ) ) ) ).

% mdeg_root
thf(fact_367_dtree_Osel_I1_J,axiom,
    ! [X1: list_a,X2: fset_P2153231429829016240_a_b_b] :
      ( ( root_list_a_b @ ( node_list_a_b @ X1 @ X2 ) )
      = X1 ) ).

% dtree.sel(1)
thf(fact_368_dtree_Oset__sel_I1_J,axiom,
    ! [A: dtree_list_a_b] : ( member_list_a @ ( root_list_a_b @ A ) @ ( dverts_list_a_b @ A ) ) ).

% dtree.set_sel(1)
thf(fact_369_Int__emptyI,axiom,
    ! [A3: set_list_a,B5: set_list_a] :
      ( ! [X3: list_a] :
          ( ( member_list_a @ X3 @ A3 )
         => ~ ( member_list_a @ X3 @ B5 ) )
     => ( ( inf_inf_set_list_a @ A3 @ B5 )
        = bot_bot_set_list_a ) ) ).

% Int_emptyI
thf(fact_370_Int__emptyI,axiom,
    ! [A3: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
      ( ! [X3: produc6499617310964463488_a_b_b] :
          ( ( member4695696432722591383_a_b_b @ X3 @ A3 )
         => ~ ( member4695696432722591383_a_b_b @ X3 @ B5 ) )
     => ( ( inf_in7962574851115124040_a_b_b @ A3 @ B5 )
        = bot_bo4897374000430069834_a_b_b ) ) ).

% Int_emptyI
thf(fact_371_Int__emptyI,axiom,
    ! [A3: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
      ( ! [X3: dtree_list_a_b] :
          ( ( member551035911493665803st_a_b @ X3 @ A3 )
         => ~ ( member551035911493665803st_a_b @ X3 @ B5 ) )
     => ( ( inf_in3355993651213403836st_a_b @ A3 @ B5 )
        = bot_bo798015271861357502st_a_b ) ) ).

% Int_emptyI
thf(fact_372_Int__emptyI,axiom,
    ! [A3: set_real,B5: set_real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A3 )
         => ~ ( member_real @ X3 @ B5 ) )
     => ( ( inf_inf_set_real @ A3 @ B5 )
        = bot_bot_set_real ) ) ).

% Int_emptyI
thf(fact_373_Int__emptyI,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ~ ( member_nat @ X3 @ B5 ) )
     => ( ( inf_inf_set_nat @ A3 @ B5 )
        = bot_bot_set_nat ) ) ).

% Int_emptyI
thf(fact_374_disjoint__iff,axiom,
    ! [A3: set_list_a,B5: set_list_a] :
      ( ( ( inf_inf_set_list_a @ A3 @ B5 )
        = bot_bot_set_list_a )
      = ( ! [X4: list_a] :
            ( ( member_list_a @ X4 @ A3 )
           => ~ ( member_list_a @ X4 @ B5 ) ) ) ) ).

% disjoint_iff
thf(fact_375_disjoint__iff,axiom,
    ! [A3: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
      ( ( ( inf_in7962574851115124040_a_b_b @ A3 @ B5 )
        = bot_bo4897374000430069834_a_b_b )
      = ( ! [X4: produc6499617310964463488_a_b_b] :
            ( ( member4695696432722591383_a_b_b @ X4 @ A3 )
           => ~ ( member4695696432722591383_a_b_b @ X4 @ B5 ) ) ) ) ).

% disjoint_iff
thf(fact_376_disjoint__iff,axiom,
    ! [A3: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
      ( ( ( inf_in3355993651213403836st_a_b @ A3 @ B5 )
        = bot_bo798015271861357502st_a_b )
      = ( ! [X4: dtree_list_a_b] :
            ( ( member551035911493665803st_a_b @ X4 @ A3 )
           => ~ ( member551035911493665803st_a_b @ X4 @ B5 ) ) ) ) ).

% disjoint_iff
thf(fact_377_disjoint__iff,axiom,
    ! [A3: set_real,B5: set_real] :
      ( ( ( inf_inf_set_real @ A3 @ B5 )
        = bot_bot_set_real )
      = ( ! [X4: real] :
            ( ( member_real @ X4 @ A3 )
           => ~ ( member_real @ X4 @ B5 ) ) ) ) ).

% disjoint_iff
thf(fact_378_disjoint__iff,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( ( ( inf_inf_set_nat @ A3 @ B5 )
        = bot_bot_set_nat )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ~ ( member_nat @ X4 @ B5 ) ) ) ) ).

% disjoint_iff
thf(fact_379_Int__empty__left,axiom,
    ! [B5: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ B5 )
      = bot_bot_set_nat ) ).

% Int_empty_left
thf(fact_380_Int__empty__right,axiom,
    ! [A3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% Int_empty_right
thf(fact_381_disjoint__iff__not__equal,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( ( ( inf_inf_set_nat @ A3 @ B5 )
        = bot_bot_set_nat )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ B5 )
               => ( X4 != Y3 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_382_Int__Collect__mono,axiom,
    ! [A3: set_list_a,B5: set_list_a,P3: list_a > $o,Q: list_a > $o] :
      ( ( ord_le8861187494160871172list_a @ A3 @ B5 )
     => ( ! [X3: list_a] :
            ( ( member_list_a @ X3 @ A3 )
           => ( ( P3 @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A3 @ ( collect_list_a @ P3 ) ) @ ( inf_inf_set_list_a @ B5 @ ( collect_list_a @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_383_Int__Collect__mono,axiom,
    ! [A3: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b,P3: produc6499617310964463488_a_b_b > $o,Q: produc6499617310964463488_a_b_b > $o] :
      ( ( ord_le1619362961161175062_a_b_b @ A3 @ B5 )
     => ( ! [X3: produc6499617310964463488_a_b_b] :
            ( ( member4695696432722591383_a_b_b @ X3 @ A3 )
           => ( ( P3 @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_le1619362961161175062_a_b_b @ ( inf_in7962574851115124040_a_b_b @ A3 @ ( collec3241703154345475797_a_b_b @ P3 ) ) @ ( inf_in7962574851115124040_a_b_b @ B5 @ ( collec3241703154345475797_a_b_b @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_384_Int__Collect__mono,axiom,
    ! [A3: set_dtree_list_a_b,B5: set_dtree_list_a_b,P3: dtree_list_a_b > $o,Q: dtree_list_a_b > $o] :
      ( ( ord_le7599451563663638410st_a_b @ A3 @ B5 )
     => ( ! [X3: dtree_list_a_b] :
            ( ( member551035911493665803st_a_b @ X3 @ A3 )
           => ( ( P3 @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_le7599451563663638410st_a_b @ ( inf_in3355993651213403836st_a_b @ A3 @ ( collec2944820760411501129st_a_b @ P3 ) ) @ ( inf_in3355993651213403836st_a_b @ B5 @ ( collec2944820760411501129st_a_b @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_385_Int__Collect__mono,axiom,
    ! [A3: set_real,B5: set_real,P3: real > $o,Q: real > $o] :
      ( ( ord_less_eq_set_real @ A3 @ B5 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A3 )
           => ( ( P3 @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_less_eq_set_real @ ( inf_inf_set_real @ A3 @ ( collect_real @ P3 ) ) @ ( inf_inf_set_real @ B5 @ ( collect_real @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_386_dhead__in__set__eq__root,axiom,
    ! [T: dtree_list_a_b,E: b,Xs2: fset_P2153231429829016240_a_b_b,R2: list_a,Def2: b > list_a] :
      ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
     => ( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
       => ( ( dhead_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ Def2 @ E )
          = ( root_list_a_b @ T ) ) ) ) ).

% dhead_in_set_eq_root
thf(fact_387_dtree_Oexhaust__sel,axiom,
    ! [Dtree: dtree_list_a_b] :
      ( Dtree
      = ( node_list_a_b @ ( root_list_a_b @ Dtree ) @ ( sucs_list_a_b @ Dtree ) ) ) ).

% dtree.exhaust_sel
thf(fact_388_mdeg__ge__child,axiom,
    ! [T12: dtree_list_a_b,E1: b,Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
      ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
     => ( ord_less_eq_nat @ ( max_deg_list_a_b @ T12 ) @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ).

% mdeg_ge_child
thf(fact_389_denormalize_Osimps_I2_J,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
      ( ! [X3: produc6499617310964463488_a_b_b] :
          ( Xs2
         != ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
     => ( ( denormalize_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
        = R2 ) ) ).

% denormalize.simps(2)
thf(fact_390_subset__CollectI,axiom,
    ! [B5: set_list_a,A3: set_list_a,Q: list_a > $o,P3: list_a > $o] :
      ( ( ord_le8861187494160871172list_a @ B5 @ A3 )
     => ( ! [X3: list_a] :
            ( ( member_list_a @ X3 @ B5 )
           => ( ( Q @ X3 )
             => ( P3 @ X3 ) ) )
       => ( ord_le8861187494160871172list_a
          @ ( collect_list_a
            @ ^ [X4: list_a] :
                ( ( member_list_a @ X4 @ B5 )
                & ( Q @ X4 ) ) )
          @ ( collect_list_a
            @ ^ [X4: list_a] :
                ( ( member_list_a @ X4 @ A3 )
                & ( P3 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_391_subset__CollectI,axiom,
    ! [B5: set_Pr3443975907877334966_a_b_b,A3: set_Pr3443975907877334966_a_b_b,Q: produc6499617310964463488_a_b_b > $o,P3: produc6499617310964463488_a_b_b > $o] :
      ( ( ord_le1619362961161175062_a_b_b @ B5 @ A3 )
     => ( ! [X3: produc6499617310964463488_a_b_b] :
            ( ( member4695696432722591383_a_b_b @ X3 @ B5 )
           => ( ( Q @ X3 )
             => ( P3 @ X3 ) ) )
       => ( ord_le1619362961161175062_a_b_b
          @ ( collec3241703154345475797_a_b_b
            @ ^ [X4: produc6499617310964463488_a_b_b] :
                ( ( member4695696432722591383_a_b_b @ X4 @ B5 )
                & ( Q @ X4 ) ) )
          @ ( collec3241703154345475797_a_b_b
            @ ^ [X4: produc6499617310964463488_a_b_b] :
                ( ( member4695696432722591383_a_b_b @ X4 @ A3 )
                & ( P3 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_392_subset__CollectI,axiom,
    ! [B5: set_dtree_list_a_b,A3: set_dtree_list_a_b,Q: dtree_list_a_b > $o,P3: dtree_list_a_b > $o] :
      ( ( ord_le7599451563663638410st_a_b @ B5 @ A3 )
     => ( ! [X3: dtree_list_a_b] :
            ( ( member551035911493665803st_a_b @ X3 @ B5 )
           => ( ( Q @ X3 )
             => ( P3 @ X3 ) ) )
       => ( ord_le7599451563663638410st_a_b
          @ ( collec2944820760411501129st_a_b
            @ ^ [X4: dtree_list_a_b] :
                ( ( member551035911493665803st_a_b @ X4 @ B5 )
                & ( Q @ X4 ) ) )
          @ ( collec2944820760411501129st_a_b
            @ ^ [X4: dtree_list_a_b] :
                ( ( member551035911493665803st_a_b @ X4 @ A3 )
                & ( P3 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_393_subset__CollectI,axiom,
    ! [B5: set_real,A3: set_real,Q: real > $o,P3: real > $o] :
      ( ( ord_less_eq_set_real @ B5 @ A3 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ B5 )
           => ( ( Q @ X3 )
             => ( P3 @ X3 ) ) )
       => ( ord_less_eq_set_real
          @ ( collect_real
            @ ^ [X4: real] :
                ( ( member_real @ X4 @ B5 )
                & ( Q @ X4 ) ) )
          @ ( collect_real
            @ ^ [X4: real] :
                ( ( member_real @ X4 @ A3 )
                & ( P3 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_394_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_zero_right
thf(fact_395_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_zero_left
thf(fact_396_inf__bot__right,axiom,
    ! [X: fset_P2153231429829016240_a_b_b] :
      ( ( inf_in525273840840435522_a_b_b @ X @ bot_bo2248824169281960260_a_b_b )
      = bot_bo2248824169281960260_a_b_b ) ).

% inf_bot_right
thf(fact_397_inf__bot__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% inf_bot_right
thf(fact_398_inf__bot__left,axiom,
    ! [X: fset_P2153231429829016240_a_b_b] :
      ( ( inf_in525273840840435522_a_b_b @ bot_bo2248824169281960260_a_b_b @ X )
      = bot_bo2248824169281960260_a_b_b ) ).

% inf_bot_left
thf(fact_399_inf__bot__left,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
      = bot_bot_set_nat ) ).

% inf_bot_left
thf(fact_400_inf_Obounded__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( inf_inf_real @ B @ C ) )
      = ( ( ord_less_eq_real @ A @ B )
        & ( ord_less_eq_real @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_401_inf_Obounded__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
      = ( ( ord_less_eq_nat @ A @ B )
        & ( ord_less_eq_nat @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_402_le__inf__iff,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z3 ) )
      = ( ( ord_less_eq_real @ X @ Y )
        & ( ord_less_eq_real @ X @ Z3 ) ) ) ).

% le_inf_iff
thf(fact_403_le__inf__iff,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z3 ) )
      = ( ( ord_less_eq_nat @ X @ Y )
        & ( ord_less_eq_nat @ X @ Z3 ) ) ) ).

% le_inf_iff
thf(fact_404_subtree__child__if__dvert__notroot__mdeg__le1,axiom,
    ! [T: dtree_list_a_b,V: list_a] :
      ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T ) @ one_one_nat )
     => ( ( V
         != ( root_list_a_b @ T ) )
       => ( ( member_list_a @ V @ ( dverts_list_a_b @ T ) )
         => ? [R6: list_a,E2: b,Zs: fset_P2153231429829016240_a_b_b] : ( is_subtree_list_a_b @ ( node_list_a_b @ R6 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ Zs ) @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T ) ) ) ) ).

% subtree_child_if_dvert_notroot_mdeg_le1
thf(fact_405_finsert__inter__finsert,axiom,
    ! [A: produc6499617310964463488_a_b_b,A3: fset_P2153231429829016240_a_b_b,B5: fset_P2153231429829016240_a_b_b] :
      ( ( inf_in525273840840435522_a_b_b @ ( finser2303212525150181944_a_b_b @ A @ A3 ) @ ( finser2303212525150181944_a_b_b @ A @ B5 ) )
      = ( finser2303212525150181944_a_b_b @ A @ ( inf_in525273840840435522_a_b_b @ A3 @ B5 ) ) ) ).

% finsert_inter_finsert
thf(fact_406_inter__fset,axiom,
    ! [X: fset_P2153231429829016240_a_b_b,Xa2: fset_P2153231429829016240_a_b_b] :
      ( ( fset_P9138963618725001425_a_b_b @ ( inf_in525273840840435522_a_b_b @ X @ Xa2 ) )
      = ( inf_in7962574851115124040_a_b_b @ ( fset_P9138963618725001425_a_b_b @ X ) @ ( fset_P9138963618725001425_a_b_b @ Xa2 ) ) ) ).

% inter_fset
thf(fact_407_finter__fempty__left,axiom,
    ! [B5: fset_P2153231429829016240_a_b_b] :
      ( ( inf_in525273840840435522_a_b_b @ bot_bo2248824169281960260_a_b_b @ B5 )
      = bot_bo2248824169281960260_a_b_b ) ).

% finter_fempty_left
thf(fact_408_finter__fempty__right,axiom,
    ! [A3: fset_P2153231429829016240_a_b_b] :
      ( ( inf_in525273840840435522_a_b_b @ A3 @ bot_bo2248824169281960260_a_b_b )
      = bot_bo2248824169281960260_a_b_b ) ).

% finter_fempty_right
thf(fact_409_inf__Int__eq2,axiom,
    ! [R3: set_Pr4048851178543822343list_a,S: set_Pr4048851178543822343list_a] :
      ( ( inf_in2412930180582663644st_a_o
        @ ^ [X4: list_a,Y3: list_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Y3 ) @ R3 )
        @ ^ [X4: list_a,Y3: list_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Y3 ) @ S ) )
      = ( ^ [X4: list_a,Y3: list_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Y3 ) @ ( inf_in5086278457404209141list_a @ R3 @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_410_inf__Int__eq2,axiom,
    ! [R3: set_Pr5727314569254441693st_a_b,S: set_Pr5727314569254441693st_a_b] :
      ( ( inf_in3567057492039880654_a_b_o
        @ ^ [X4: list_a,Y3: dtree_list_a_b] : ( member6763472504040263486st_a_b @ ( produc148520996349637281st_a_b @ X4 @ Y3 ) @ R3 )
        @ ^ [X4: list_a,Y3: dtree_list_a_b] : ( member6763472504040263486st_a_b @ ( produc148520996349637281st_a_b @ X4 @ Y3 ) @ S ) )
      = ( ^ [X4: list_a,Y3: dtree_list_a_b] : ( member6763472504040263486st_a_b @ ( produc148520996349637281st_a_b @ X4 @ Y3 ) @ ( inf_in3489975765262956399st_a_b @ R3 @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_411_inf__Int__eq2,axiom,
    ! [R3: set_Pr3443975907877334966_a_b_b,S: set_Pr3443975907877334966_a_b_b] :
      ( ( inf_in6661427951089147235_b_b_o
        @ ^ [X4: dtree_list_a_b,Y3: b] : ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X4 @ Y3 ) @ R3 )
        @ ^ [X4: dtree_list_a_b,Y3: b] : ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X4 @ Y3 ) @ S ) )
      = ( ^ [X4: dtree_list_a_b,Y3: b] : ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X4 @ Y3 ) @ ( inf_in7962574851115124040_a_b_b @ R3 @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_412_inf__Int__eq2,axiom,
    ! [R3: set_Pr8451570673126195472st_a_b,S: set_Pr8451570673126195472st_a_b] :
      ( ( inf_in7119230168153742171_a_b_o
        @ ^ [X4: list_a,Y3: produc111314985273491367st_a_b] : ( member4895573663636194545st_a_b @ ( produc673257793671328980st_a_b @ X4 @ Y3 ) @ R3 )
        @ ^ [X4: list_a,Y3: produc111314985273491367st_a_b] : ( member4895573663636194545st_a_b @ ( produc673257793671328980st_a_b @ X4 @ Y3 ) @ S ) )
      = ( ^ [X4: list_a,Y3: produc111314985273491367st_a_b] : ( member4895573663636194545st_a_b @ ( produc673257793671328980st_a_b @ X4 @ Y3 ) @ ( inf_in3501900006459691682st_a_b @ R3 @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_413_inf__Int__eq,axiom,
    ! [R3: set_list_a,S: set_list_a] :
      ( ( inf_inf_list_a_o
        @ ^ [X4: list_a] : ( member_list_a @ X4 @ R3 )
        @ ^ [X4: list_a] : ( member_list_a @ X4 @ S ) )
      = ( ^ [X4: list_a] : ( member_list_a @ X4 @ ( inf_inf_set_list_a @ R3 @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_414_inf__Int__eq,axiom,
    ! [R3: set_Pr3443975907877334966_a_b_b,S: set_Pr3443975907877334966_a_b_b] :
      ( ( inf_in9196368829404727101_b_b_o
        @ ^ [X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ R3 )
        @ ^ [X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ S ) )
      = ( ^ [X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ ( inf_in7962574851115124040_a_b_b @ R3 @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_415_inf__Int__eq,axiom,
    ! [R3: set_dtree_list_a_b,S: set_dtree_list_a_b] :
      ( ( inf_in579736399448705737_a_b_o
        @ ^ [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ R3 )
        @ ^ [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ S ) )
      = ( ^ [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ ( inf_in3355993651213403836st_a_b @ R3 @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_416_inf__Int__eq,axiom,
    ! [R3: set_real,S: set_real] :
      ( ( inf_inf_real_o
        @ ^ [X4: real] : ( member_real @ X4 @ R3 )
        @ ^ [X4: real] : ( member_real @ X4 @ S ) )
      = ( ^ [X4: real] : ( member_real @ X4 @ ( inf_inf_set_real @ R3 @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_417_inf__set__def,axiom,
    ( inf_inf_set_list_a
    = ( ^ [A6: set_list_a,B6: set_list_a] :
          ( collect_list_a
          @ ( inf_inf_list_a_o
            @ ^ [X4: list_a] : ( member_list_a @ X4 @ A6 )
            @ ^ [X4: list_a] : ( member_list_a @ X4 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_418_inf__set__def,axiom,
    ( inf_in7962574851115124040_a_b_b
    = ( ^ [A6: set_Pr3443975907877334966_a_b_b,B6: set_Pr3443975907877334966_a_b_b] :
          ( collec3241703154345475797_a_b_b
          @ ( inf_in9196368829404727101_b_b_o
            @ ^ [X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ A6 )
            @ ^ [X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_419_inf__set__def,axiom,
    ( inf_in3355993651213403836st_a_b
    = ( ^ [A6: set_dtree_list_a_b,B6: set_dtree_list_a_b] :
          ( collec2944820760411501129st_a_b
          @ ( inf_in579736399448705737_a_b_o
            @ ^ [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ A6 )
            @ ^ [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_420_inf__set__def,axiom,
    ( inf_inf_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( collect_real
          @ ( inf_inf_real_o
            @ ^ [X4: real] : ( member_real @ X4 @ A6 )
            @ ^ [X4: real] : ( member_real @ X4 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_421_mdeg__1__singleton,axiom,
    ! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
      ( ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
        = one_one_nat )
     => ? [X3: produc6499617310964463488_a_b_b] :
          ( Xs2
          = ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) ) ) ).

% mdeg_1_singleton
thf(fact_422_empty__fset__if__mdeg__le1__not__single,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
      ( ! [X3: produc6499617310964463488_a_b_b] :
          ( Xs2
         != ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
     => ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) @ one_one_nat )
       => ( Xs2 = bot_bo2248824169281960260_a_b_b ) ) ) ).

% empty_fset_if_mdeg_le1_not_single
thf(fact_423_empty__fset__if__mdeg__le1__not__single__suc,axiom,
    ! [T: dtree_list_a_b] :
      ( ! [X3: produc6499617310964463488_a_b_b] :
          ( ( sucs_list_a_b @ T )
         != ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
     => ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T ) @ one_one_nat )
       => ( ( sucs_list_a_b @ T )
          = bot_bo2248824169281960260_a_b_b ) ) ) ).

% empty_fset_if_mdeg_le1_not_single_suc
thf(fact_424_singleton__if__mdeg__le1__elem,axiom,
    ! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: produc6499617310964463488_a_b_b] :
      ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) @ one_one_nat )
     => ( ( member4695696432722591383_a_b_b @ X @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
       => ( Xs2
          = ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).

% singleton_if_mdeg_le1_elem
thf(fact_425_inf__sup__ord_I2_J,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_426_inf__sup__ord_I2_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_427_inf__sup__ord_I1_J,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_428_inf__sup__ord_I1_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_429_inf__le1,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_430_inf__le1,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_431_inf__le2,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_432_inf__le2,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_433_le__infE,axiom,
    ! [X: real,A: real,B: real] :
      ( ( ord_less_eq_real @ X @ ( inf_inf_real @ A @ B ) )
     => ~ ( ( ord_less_eq_real @ X @ A )
         => ~ ( ord_less_eq_real @ X @ B ) ) ) ).

% le_infE
thf(fact_434_le__infE,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) )
     => ~ ( ( ord_less_eq_nat @ X @ A )
         => ~ ( ord_less_eq_nat @ X @ B ) ) ) ).

% le_infE
thf(fact_435_le__infI,axiom,
    ! [X: real,A: real,B: real] :
      ( ( ord_less_eq_real @ X @ A )
     => ( ( ord_less_eq_real @ X @ B )
       => ( ord_less_eq_real @ X @ ( inf_inf_real @ A @ B ) ) ) ) ).

% le_infI
thf(fact_436_le__infI,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X @ A )
     => ( ( ord_less_eq_nat @ X @ B )
       => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_437_inf__mono,axiom,
    ! [A: real,C: real,B: real,D2: real] :
      ( ( ord_less_eq_real @ A @ C )
     => ( ( ord_less_eq_real @ B @ D2 )
       => ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ ( inf_inf_real @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_438_inf__mono,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ D2 )
       => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_439_le__infI1,axiom,
    ! [A: real,X: real,B: real] :
      ( ( ord_less_eq_real @ A @ X )
     => ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_440_le__infI1,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ X )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_441_le__infI2,axiom,
    ! [B: real,X: real,A: real] :
      ( ( ord_less_eq_real @ B @ X )
     => ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_442_le__infI2,axiom,
    ! [B: nat,X: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ X )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_443_inf_OorderE,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( A
        = ( inf_inf_real @ A @ B ) ) ) ).

% inf.orderE
thf(fact_444_inf_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( A
        = ( inf_inf_nat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_445_inf_OorderI,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( inf_inf_real @ A @ B ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% inf.orderI
thf(fact_446_inf_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( inf_inf_nat @ A @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% inf.orderI
thf(fact_447_inf__unique,axiom,
    ! [F: real > real > real,X: real,Y: real] :
      ( ! [X3: real,Y4: real] : ( ord_less_eq_real @ ( F @ X3 @ Y4 ) @ X3 )
     => ( ! [X3: real,Y4: real] : ( ord_less_eq_real @ ( F @ X3 @ Y4 ) @ Y4 )
       => ( ! [X3: real,Y4: real,Z4: real] :
              ( ( ord_less_eq_real @ X3 @ Y4 )
             => ( ( ord_less_eq_real @ X3 @ Z4 )
               => ( ord_less_eq_real @ X3 @ ( F @ Y4 @ Z4 ) ) ) )
         => ( ( inf_inf_real @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_448_inf__unique,axiom,
    ! [F: nat > nat > nat,X: nat,Y: nat] :
      ( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y4 ) @ X3 )
     => ( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y4 ) @ Y4 )
       => ( ! [X3: nat,Y4: nat,Z4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ( ord_less_eq_nat @ X3 @ Z4 )
               => ( ord_less_eq_nat @ X3 @ ( F @ Y4 @ Z4 ) ) ) )
         => ( ( inf_inf_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_449_le__iff__inf,axiom,
    ( ord_less_eq_real
    = ( ^ [X4: real,Y3: real] :
          ( ( inf_inf_real @ X4 @ Y3 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_450_le__iff__inf,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y3: nat] :
          ( ( inf_inf_nat @ X4 @ Y3 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_451_inf_Oabsorb1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( inf_inf_real @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_452_inf_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( inf_inf_nat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_453_inf_Oabsorb2,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( inf_inf_real @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_454_inf_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( inf_inf_nat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_455_inf__absorb1,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( inf_inf_real @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_456_inf__absorb1,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( inf_inf_nat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_457_inf__absorb2,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ( ( inf_inf_real @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_458_inf__absorb2,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( inf_inf_nat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_459_inf_OboundedE,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( inf_inf_real @ B @ C ) )
     => ~ ( ( ord_less_eq_real @ A @ B )
         => ~ ( ord_less_eq_real @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_460_inf_OboundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
     => ~ ( ( ord_less_eq_nat @ A @ B )
         => ~ ( ord_less_eq_nat @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_461_inf_OboundedI,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ A @ C )
       => ( ord_less_eq_real @ A @ ( inf_inf_real @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_462_inf_OboundedI,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ C )
       => ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_463_inf__greatest,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_eq_real @ X @ Z3 )
       => ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z3 ) ) ) ) ).

% inf_greatest
thf(fact_464_inf__greatest,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Z3 )
       => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z3 ) ) ) ) ).

% inf_greatest
thf(fact_465_inf_Oorder__iff,axiom,
    ( ord_less_eq_real
    = ( ^ [A4: real,B3: real] :
          ( A4
          = ( inf_inf_real @ A4 @ B3 ) ) ) ) ).

% inf.order_iff
thf(fact_466_inf_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B3: nat] :
          ( A4
          = ( inf_inf_nat @ A4 @ B3 ) ) ) ) ).

% inf.order_iff
thf(fact_467_inf_Ocobounded1,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_468_inf_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_469_inf_Ocobounded2,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_470_inf_Ocobounded2,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_471_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_real
    = ( ^ [A4: real,B3: real] :
          ( ( inf_inf_real @ A4 @ B3 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_472_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( inf_inf_nat @ A4 @ B3 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_473_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_real
    = ( ^ [B3: real,A4: real] :
          ( ( inf_inf_real @ A4 @ B3 )
          = B3 ) ) ) ).

% inf.absorb_iff2
thf(fact_474_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( inf_inf_nat @ A4 @ B3 )
          = B3 ) ) ) ).

% inf.absorb_iff2
thf(fact_475_inf_OcoboundedI1,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A @ C )
     => ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_476_inf_OcoboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_477_inf_OcoboundedI2,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_eq_real @ B @ C )
     => ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_478_inf_OcoboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_479_singleton__if__mdeg__le1__elem__suc,axiom,
    ! [T: dtree_list_a_b,X: produc6499617310964463488_a_b_b] :
      ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T ) @ one_one_nat )
     => ( ( member4695696432722591383_a_b_b @ X @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) )
       => ( ( sucs_list_a_b @ T )
          = ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).

% singleton_if_mdeg_le1_elem_suc
thf(fact_480_subtree__child__if__dvert__notr__mdeg__le1,axiom,
    ! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,V: list_a] :
      ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) @ one_one_nat )
     => ( ( V != R2 )
       => ( ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
         => ? [R6: list_a,E2: b,Zs: fset_P2153231429829016240_a_b_b] : ( is_subtree_list_a_b @ ( node_list_a_b @ R6 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ Zs ) @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ) ).

% subtree_child_if_dvert_notr_mdeg_le1
thf(fact_481_denormalize_Opsimps_I1_J,axiom,
    ! [R2: list_a,T: dtree_list_a_b,E: b] :
      ( ( accp_dtree_list_a_b @ denormalize_rel_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
     => ( ( denormalize_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
        = ( append_a @ R2 @ ( denormalize_a_b @ T ) ) ) ) ).

% denormalize.psimps(1)
thf(fact_482_denormalize_Opelims,axiom,
    ! [X: dtree_list_a_b,Y: list_a] :
      ( ( ( denormalize_a_b @ X )
        = Y )
     => ( ( accp_dtree_list_a_b @ denormalize_rel_a_b @ X )
       => ( ! [R: list_a,T2: dtree_list_a_b,E2: b] :
              ( ( X
                = ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
             => ( ( Y
                  = ( append_a @ R @ ( denormalize_a_b @ T2 ) ) )
               => ~ ( accp_dtree_list_a_b @ denormalize_rel_a_b @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) )
         => ( ! [Xs: fset_P2153231429829016240_a_b_b,R: list_a] :
                ( ( accp_dtree_list_a_b @ denormalize_rel_a_b @ ( node_list_a_b @ R @ Xs ) )
               => ( ! [X5: produc6499617310964463488_a_b_b] :
                      ( Xs
                     != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
                 => ( ( X
                      = ( node_list_a_b @ R @ Xs ) )
                   => ( accp_dtree_list_a_b @ denormalize_rel_a_b @ ( node_list_a_b @ R @ Xs ) ) ) ) )
           => ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
                  ( ! [X5: produc6499617310964463488_a_b_b] :
                      ( Xs
                     != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
                 => ! [R: list_a] :
                      ( ( X
                        = ( node_list_a_b @ R @ Xs ) )
                     => ( ( Y = R )
                       => ~ ( accp_dtree_list_a_b @ denormalize_rel_a_b @ ( node_list_a_b @ R @ Xs ) ) ) ) ) ) ) ) ) ).

% denormalize.pelims
thf(fact_483_mdeg__child__sucs__eq__if__gt1,axiom,
    ! [R2: list_a,T: dtree_list_a_b,E: b,V: list_a] :
      ( ( ord_less_nat @ one_one_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
     => ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
        = ( max_deg_list_a_b @ ( node_list_a_b @ V @ ( sucs_list_a_b @ T ) ) ) ) ) ).

% mdeg_child_sucs_eq_if_gt1
thf(fact_484_wf__dtree_Odhead__in__verts__if__dtail,axiom,
    ! [T: dtree_real_real,Dt: real > real,P: real,X: dtree_real_real,R2: real,Xs2: fset_P7499862231470980990l_real,E: real,Dh: real > real] :
      ( ( wf_dtree_real_real @ T )
     => ( ( member_real @ ( dtail_real_real @ T @ Dt @ P ) @ ( dverts_real_real @ X ) )
       => ( ( member_real @ P @ ( darcs_real_real @ T ) )
         => ( ( T
              = ( node_real_real @ R2 @ Xs2 ) )
           => ( ( member2617108185285288129l_real @ ( produc3949691677225608016l_real @ X @ E ) @ ( fset_P7992937044511435015l_real @ Xs2 ) )
             => ( member_real @ ( dhead_real_real @ T @ Dh @ P ) @ ( dverts_real_real @ X ) ) ) ) ) ) ) ).

% wf_dtree.dhead_in_verts_if_dtail
thf(fact_485_wf__dtree_Odhead__in__verts__if__dtail,axiom,
    ! [T: dtree_list_a_real,Dt: real > list_a,P: real,X: dtree_list_a_real,R2: list_a,Xs2: fset_P6497461246188700184l_real,E: real,Dh: real > list_a] :
      ( ( wf_dtree_list_a_real @ T )
     => ( ( member_list_a @ ( dtail_list_a_real @ T @ Dt @ P ) @ ( dverts_list_a_real @ X ) )
       => ( ( member_real @ P @ ( darcs_list_a_real @ T ) )
         => ( ( T
              = ( node_list_a_real @ R2 @ Xs2 ) )
           => ( ( member4849742732303649023l_real @ ( produc2747972018350754914l_real @ X @ E ) @ ( fset_P6949987765245255225l_real @ Xs2 ) )
             => ( member_list_a @ ( dhead_list_a_real @ T @ Dh @ P ) @ ( dverts_list_a_real @ X ) ) ) ) ) ) ) ).

% wf_dtree.dhead_in_verts_if_dtail
thf(fact_486_wf__dtree_Odhead__in__verts__if__dtail,axiom,
    ! [T: dtree_real_list_a,Dt: list_a > real,P: list_a,X: dtree_real_list_a,R2: real,Xs2: fset_P1040399934802234454list_a,E: list_a,Dh: list_a > real] :
      ( ( wf_dtree_real_list_a @ T )
     => ( ( member_real @ ( dtail_real_list_a @ T @ Dt @ P ) @ ( dverts_real_list_a @ X ) )
       => ( ( member_list_a @ P @ ( darcs_real_list_a @ T ) )
         => ( ( T
              = ( node_real_list_a @ R2 @ Xs2 ) )
           => ( ( member5451813156226304665list_a @ ( produc8292410999216705376list_a @ X @ E ) @ ( fset_P7319268309571958751list_a @ Xs2 ) )
             => ( member_real @ ( dhead_real_list_a @ T @ Dh @ P ) @ ( dverts_real_list_a @ X ) ) ) ) ) ) ) ).

% wf_dtree.dhead_in_verts_if_dtail
thf(fact_487_wf__dtree_Odhead__in__verts__if__dtail,axiom,
    ! [T: dtree_list_a_b,Dt: b > list_a,P: b,X: dtree_list_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,E: b,Dh: b > list_a] :
      ( ( wf_dtree_list_a_b @ T )
     => ( ( member_list_a @ ( dtail_list_a_b @ T @ Dt @ P ) @ ( dverts_list_a_b @ X ) )
       => ( ( member_b @ P @ ( darcs_list_a_b @ T ) )
         => ( ( T
              = ( node_list_a_b @ R2 @ Xs2 ) )
           => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
             => ( member_list_a @ ( dhead_list_a_b @ T @ Dh @ P ) @ ( dverts_list_a_b @ X ) ) ) ) ) ) ) ).

% wf_dtree.dhead_in_verts_if_dtail
thf(fact_488_wf__dtree_Odhead__in__verts__if__dtail,axiom,
    ! [T: dtree_list_a_list_a,Dt: list_a > list_a,P: list_a,X: dtree_list_a_list_a,R2: list_a,Xs2: fset_P6656299774166858224list_a,E: list_a,Dh: list_a > list_a] :
      ( ( wf_dtr2792200986692785213list_a @ T )
     => ( ( member_list_a @ ( dtail_list_a_list_a @ T @ Dt @ P ) @ ( dverts_list_a_list_a @ X ) )
       => ( ( member_list_a @ P @ ( darcs_list_a_list_a @ T ) )
         => ( ( T
              = ( node_list_a_list_a @ R2 @ Xs2 ) )
           => ( ( member535470554364689367list_a @ ( produc6276725351364364146list_a @ X @ E ) @ ( fset_P2575872134735801361list_a @ Xs2 ) )
             => ( member_list_a @ ( dhead_list_a_list_a @ T @ Dh @ P ) @ ( dverts_list_a_list_a @ X ) ) ) ) ) ) ) ).

% wf_dtree.dhead_in_verts_if_dtail
thf(fact_489_wf__dtree_Odhead__in__verts__if__dtail,axiom,
    ! [T: dtree_8345463500551380051b_real,Dt: real > dtree_list_a_b,P: real,X: dtree_8345463500551380051b_real,R2: dtree_list_a_b,Xs2: fset_P9074655572916523974l_real,E: real,Dh: real > dtree_list_a_b] :
      ( ( wf_dtr8910464690835127007b_real @ T )
     => ( ( member551035911493665803st_a_b @ ( dtail_8649134905595490176b_real @ T @ Dt @ P ) @ ( dverts4508812070415817884b_real @ X ) )
       => ( ( member_real @ P @ ( darcs_3884030719700794635b_real @ T ) )
         => ( ( T
              = ( node_d335693152011667424b_real @ R2 @ Xs2 ) )
           => ( ( member8458687784735308553l_real @ ( produc1938216529971361688l_real @ X @ E ) @ ( fset_P344769348705914191l_real @ Xs2 ) )
             => ( member551035911493665803st_a_b @ ( dhead_5461920678355059344b_real @ T @ Dh @ P ) @ ( dverts4508812070415817884b_real @ X ) ) ) ) ) ) ) ).

% wf_dtree.dhead_in_verts_if_dtail
thf(fact_490_wf__dtree_Odhead__in__verts__if__dtail,axiom,
    ! [T: dtree_3696687526968142675st_a_b,Dt: dtree_list_a_b > real,P: dtree_list_a_b,X: dtree_3696687526968142675st_a_b,R2: real,Xs2: fset_P5746387328717196814st_a_b,E: dtree_list_a_b,Dh: dtree_list_a_b > real] :
      ( ( wf_dtr8971879960368933343st_a_b @ T )
     => ( ( member_real @ ( dtail_8710550175129296512st_a_b @ T @ Dt @ P ) @ ( dverts4570227339949624220st_a_b @ X ) )
       => ( ( member551035911493665803st_a_b @ P @ ( darcs_3945445989234600971st_a_b @ T ) )
         => ( ( T
              = ( node_r397108421545473760st_a_b @ R2 @ Xs2 ) )
           => ( ( member692244352973388625st_a_b @ ( produc4763843483075138016st_a_b @ X @ E ) @ ( fset_P8515478733839210391st_a_b @ Xs2 ) )
             => ( member_real @ ( dhead_5523335947888865680st_a_b @ T @ Dh @ P ) @ ( dverts4570227339949624220st_a_b @ X ) ) ) ) ) ) ) ).

% wf_dtree.dhead_in_verts_if_dtail
thf(fact_491_wf__dtree_Odhead__in__verts__if__dtail,axiom,
    ! [T: dtree_4875458026974816597st_a_b,Dt: dtree_list_a_b > list_a,P: dtree_list_a_b,X: dtree_4875458026974816597st_a_b,R2: list_a,Xs2: fset_P6340920821931359400st_a_b,E: dtree_list_a_b,Dh: dtree_list_a_b > list_a] :
      ( ( wf_dtr4469746621066521083st_a_b @ T )
     => ( ( member_list_a @ ( dtail_3679034105612296090st_a_b @ T @ Dt @ P ) @ ( dverts3100150328211173758st_a_b @ X ) )
       => ( ( member551035911493665803st_a_b @ P @ ( darcs_1080543555413674703st_a_b @ T ) )
         => ( ( T
              = ( node_l8355664270593351482st_a_b @ R2 @ Xs2 ) )
           => ( ( member8722885590174353295st_a_b @ ( produc4942817220686395122st_a_b @ X @ E ) @ ( fset_P623098064157935817st_a_b @ Xs2 ) )
             => ( member_list_a @ ( dhead_7954756475827843210st_a_b @ T @ Dh @ P ) @ ( dverts3100150328211173758st_a_b @ X ) ) ) ) ) ) ) ).

% wf_dtree.dhead_in_verts_if_dtail
thf(fact_492_wf__dtree_Odhead__in__verts__if__dtail,axiom,
    ! [T: dtree_8284992029673318451list_a,Dt: list_a > dtree_list_a_b,P: list_a,X: dtree_8284992029673318451list_a,R2: dtree_list_a_b,Xs2: fset_P5658613431985659038list_a,E: list_a,Dh: list_a > dtree_list_a_b] :
      ( ( wf_dtr8287859687986785809list_a @ T )
     => ( ( member551035911493665803st_a_b @ ( dtail_7497147172532560816list_a @ T @ Dt @ P ) @ ( dverts6918263395131438484list_a @ X ) )
       => ( ( member_list_a @ P @ ( darcs_4898656622333939429list_a @ T ) )
         => ( ( T
              = ( node_d2950405300658840400list_a @ R2 @ Xs2 ) )
           => ( ( member2153700486347808993list_a @ ( produc3720128825147924392list_a @ X @ E ) @ ( fset_P5083850707481761831list_a @ Xs2 ) )
             => ( member551035911493665803st_a_b @ ( dhead_2549497505893332128list_a @ T @ Dh @ P ) @ ( dverts6918263395131438484list_a @ X ) ) ) ) ) ) ) ).

% wf_dtree.dhead_in_verts_if_dtail
thf(fact_493_wf__dtree_Odhead__in__verts__if__dtail,axiom,
    ! [T: dtree_310814809473010911b_real,Dt: real > produc6499617310964463488_a_b_b,P: real,X: dtree_310814809473010911b_real,R2: produc6499617310964463488_a_b_b,Xs2: fset_P1597832662759650386l_real,E: real,Dh: real > produc6499617310964463488_a_b_b] :
      ( ( wf_dtr8634579813312766059b_real @ T )
     => ( ( member4695696432722591383_a_b_b @ ( dtail_6110999216585821196b_real @ T @ Dt @ P ) @ ( dverts8626872187801812264b_real @ X ) )
       => ( ( member_real @ P @ ( darcs_3291380776206734999b_real @ T ) )
         => ( ( T
              = ( node_P8407631645849338988b_real @ R2 @ Xs2 ) )
           => ( ( member4680567731829293973l_real @ ( produc4041975618813999652l_real @ X @ E ) @ ( fset_P3280430075840339931l_real @ Xs2 ) )
             => ( member4695696432722591383_a_b_b @ ( dhead_8830158393902075676b_real @ T @ Dh @ P ) @ ( dverts8626872187801812264b_real @ X ) ) ) ) ) ) ) ).

% wf_dtree.dhead_in_verts_if_dtail
thf(fact_494_wf__dtree_Odarc__in__sub__if__dtail__in__sub,axiom,
    ! [T: dtree_real_real,Dt: real > real,E: real,V: real,X: dtree_real_real,E1: real,Xs2: fset_P7499862231470980990l_real,T12: dtree_real_real,R2: real] :
      ( ( wf_dtree_real_real @ T )
     => ( ( ( dtail_real_real @ T @ Dt @ E )
          = V )
       => ( ( member_real @ E @ ( darcs_real_real @ T ) )
         => ( ( member2617108185285288129l_real @ ( produc3949691677225608016l_real @ X @ E1 ) @ ( fset_P7992937044511435015l_real @ Xs2 ) )
           => ( ( is_subtree_real_real @ T12 @ X )
             => ( ( ( node_real_real @ R2 @ Xs2 )
                  = T )
               => ( ( member_real @ V @ ( dverts_real_real @ T12 ) )
                 => ( member_real @ E @ ( darcs_real_real @ X ) ) ) ) ) ) ) ) ) ).

% wf_dtree.darc_in_sub_if_dtail_in_sub
thf(fact_495_wf__dtree_Odarc__in__sub__if__dtail__in__sub,axiom,
    ! [T: dtree_real_list_a,Dt: list_a > real,E: list_a,V: real,X: dtree_real_list_a,E1: list_a,Xs2: fset_P1040399934802234454list_a,T12: dtree_real_list_a,R2: real] :
      ( ( wf_dtree_real_list_a @ T )
     => ( ( ( dtail_real_list_a @ T @ Dt @ E )
          = V )
       => ( ( member_list_a @ E @ ( darcs_real_list_a @ T ) )
         => ( ( member5451813156226304665list_a @ ( produc8292410999216705376list_a @ X @ E1 ) @ ( fset_P7319268309571958751list_a @ Xs2 ) )
           => ( ( is_sub4113186846137662008list_a @ T12 @ X )
             => ( ( ( node_real_list_a @ R2 @ Xs2 )
                  = T )
               => ( ( member_real @ V @ ( dverts_real_list_a @ T12 ) )
                 => ( member_list_a @ E @ ( darcs_real_list_a @ X ) ) ) ) ) ) ) ) ) ).

% wf_dtree.darc_in_sub_if_dtail_in_sub
thf(fact_496_wf__dtree_Odarc__in__sub__if__dtail__in__sub,axiom,
    ! [T: dtree_list_a_real,Dt: real > list_a,E: real,V: list_a,X: dtree_list_a_real,E1: real,Xs2: fset_P6497461246188700184l_real,T12: dtree_list_a_real,R2: list_a] :
      ( ( wf_dtree_list_a_real @ T )
     => ( ( ( dtail_list_a_real @ T @ Dt @ E )
          = V )
       => ( ( member_real @ E @ ( darcs_list_a_real @ T ) )
         => ( ( member4849742732303649023l_real @ ( produc2747972018350754914l_real @ X @ E1 ) @ ( fset_P6949987765245255225l_real @ Xs2 ) )
           => ( ( is_sub5530047198231550610a_real @ T12 @ X )
             => ( ( ( node_list_a_real @ R2 @ Xs2 )
                  = T )
               => ( ( member_list_a @ V @ ( dverts_list_a_real @ T12 ) )
                 => ( member_real @ E @ ( darcs_list_a_real @ X ) ) ) ) ) ) ) ) ) ).

% wf_dtree.darc_in_sub_if_dtail_in_sub
thf(fact_497_wf__dtree_Odarc__in__sub__if__dtail__in__sub,axiom,
    ! [T: dtree_list_a_b,Dt: b > list_a,E: b,V: list_a,X: dtree_list_a_b,E1: b,Xs2: fset_P2153231429829016240_a_b_b,T12: dtree_list_a_b,R2: list_a] :
      ( ( wf_dtree_list_a_b @ T )
     => ( ( ( dtail_list_a_b @ T @ Dt @ E )
          = V )
       => ( ( member_b @ E @ ( darcs_list_a_b @ T ) )
         => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
           => ( ( is_subtree_list_a_b @ T12 @ X )
             => ( ( ( node_list_a_b @ R2 @ Xs2 )
                  = T )
               => ( ( member_list_a @ V @ ( dverts_list_a_b @ T12 ) )
                 => ( member_b @ E @ ( darcs_list_a_b @ X ) ) ) ) ) ) ) ) ) ).

% wf_dtree.darc_in_sub_if_dtail_in_sub
thf(fact_498_wf__dtree_Odarc__in__sub__if__dtail__in__sub,axiom,
    ! [T: dtree_list_a_list_a,Dt: list_a > list_a,E: list_a,V: list_a,X: dtree_list_a_list_a,E1: list_a,Xs2: fset_P6656299774166858224list_a,T12: dtree_list_a_list_a,R2: list_a] :
      ( ( wf_dtr2792200986692785213list_a @ T )
     => ( ( ( dtail_list_a_list_a @ T @ Dt @ E )
          = V )
       => ( ( member_list_a @ E @ ( darcs_list_a_list_a @ T ) )
         => ( ( member535470554364689367list_a @ ( produc6276725351364364146list_a @ X @ E1 ) @ ( fset_P2575872134735801361list_a @ Xs2 ) )
           => ( ( is_sub7392010792700354846list_a @ T12 @ X )
             => ( ( ( node_list_a_list_a @ R2 @ Xs2 )
                  = T )
               => ( ( member_list_a @ V @ ( dverts_list_a_list_a @ T12 ) )
                 => ( member_list_a @ E @ ( darcs_list_a_list_a @ X ) ) ) ) ) ) ) ) ) ).

% wf_dtree.darc_in_sub_if_dtail_in_sub
thf(fact_499_wf__dtree_Odarc__in__sub__if__dtail__in__sub,axiom,
    ! [T: dtree_3696687526968142675st_a_b,Dt: dtree_list_a_b > real,E: dtree_list_a_b,V: real,X: dtree_3696687526968142675st_a_b,E1: dtree_list_a_b,Xs2: fset_P5746387328717196814st_a_b,T12: dtree_3696687526968142675st_a_b,R2: real] :
      ( ( wf_dtr8971879960368933343st_a_b @ T )
     => ( ( ( dtail_8710550175129296512st_a_b @ T @ Dt @ E )
          = V )
       => ( ( member551035911493665803st_a_b @ E @ ( darcs_3945445989234600971st_a_b @ T ) )
         => ( ( member692244352973388625st_a_b @ ( produc4763843483075138016st_a_b @ X @ E1 ) @ ( fset_P8515478733839210391st_a_b @ Xs2 ) )
           => ( ( is_sub776737362830463552st_a_b @ T12 @ X )
             => ( ( ( node_r397108421545473760st_a_b @ R2 @ Xs2 )
                  = T )
               => ( ( member_real @ V @ ( dverts4570227339949624220st_a_b @ T12 ) )
                 => ( member551035911493665803st_a_b @ E @ ( darcs_3945445989234600971st_a_b @ X ) ) ) ) ) ) ) ) ) ).

% wf_dtree.darc_in_sub_if_dtail_in_sub
thf(fact_500_wf__dtree_Odarc__in__sub__if__dtail__in__sub,axiom,
    ! [T: dtree_8345463500551380051b_real,Dt: real > dtree_list_a_b,E: real,V: dtree_list_a_b,X: dtree_8345463500551380051b_real,E1: real,Xs2: fset_P9074655572916523974l_real,T12: dtree_8345463500551380051b_real,R2: dtree_list_a_b] :
      ( ( wf_dtr8910464690835127007b_real @ T )
     => ( ( ( dtail_8649134905595490176b_real @ T @ Dt @ E )
          = V )
       => ( ( member_real @ E @ ( darcs_3884030719700794635b_real @ T ) )
         => ( ( member8458687784735308553l_real @ ( produc1938216529971361688l_real @ X @ E1 ) @ ( fset_P344769348705914191l_real @ Xs2 ) )
           => ( ( is_sub715322093296657216b_real @ T12 @ X )
             => ( ( ( node_d335693152011667424b_real @ R2 @ Xs2 )
                  = T )
               => ( ( member551035911493665803st_a_b @ V @ ( dverts4508812070415817884b_real @ T12 ) )
                 => ( member_real @ E @ ( darcs_3884030719700794635b_real @ X ) ) ) ) ) ) ) ) ) ).

% wf_dtree.darc_in_sub_if_dtail_in_sub
thf(fact_501_wf__dtree_Odarc__in__sub__if__dtail__in__sub,axiom,
    ! [T: dtree_8284992029673318451list_a,Dt: list_a > dtree_list_a_b,E: list_a,V: dtree_list_a_b,X: dtree_8284992029673318451list_a,E1: list_a,Xs2: fset_P5658613431985659038list_a,T12: dtree_8284992029673318451list_a,R2: dtree_list_a_b] :
      ( ( wf_dtr8287859687986785809list_a @ T )
     => ( ( ( dtail_7497147172532560816list_a @ T @ Dt @ E )
          = V )
       => ( ( member_list_a @ E @ ( darcs_4898656622333939429list_a @ T ) )
         => ( ( member2153700486347808993list_a @ ( produc3720128825147924392list_a @ X @ E1 ) @ ( fset_P5083850707481761831list_a @ Xs2 ) )
           => ( ( is_sub986895638982090736list_a @ T12 @ X )
             => ( ( ( node_d2950405300658840400list_a @ R2 @ Xs2 )
                  = T )
               => ( ( member551035911493665803st_a_b @ V @ ( dverts6918263395131438484list_a @ T12 ) )
                 => ( member_list_a @ E @ ( darcs_4898656622333939429list_a @ X ) ) ) ) ) ) ) ) ) ).

% wf_dtree.darc_in_sub_if_dtail_in_sub
thf(fact_502_wf__dtree_Odarc__in__sub__if__dtail__in__sub,axiom,
    ! [T: dtree_4875458026974816597st_a_b,Dt: dtree_list_a_b > list_a,E: dtree_list_a_b,V: list_a,X: dtree_4875458026974816597st_a_b,E1: dtree_list_a_b,Xs2: fset_P6340920821931359400st_a_b,T12: dtree_4875458026974816597st_a_b,R2: list_a] :
      ( ( wf_dtr4469746621066521083st_a_b @ T )
     => ( ( ( dtail_3679034105612296090st_a_b @ T @ Dt @ E )
          = V )
       => ( ( member551035911493665803st_a_b @ E @ ( darcs_1080543555413674703st_a_b @ T ) )
         => ( ( member8722885590174353295st_a_b @ ( produc4942817220686395122st_a_b @ X @ E1 ) @ ( fset_P623098064157935817st_a_b @ Xs2 ) )
           => ( ( is_sub6392154608916601818st_a_b @ T12 @ X )
             => ( ( ( node_l8355664270593351482st_a_b @ R2 @ Xs2 )
                  = T )
               => ( ( member_list_a @ V @ ( dverts3100150328211173758st_a_b @ T12 ) )
                 => ( member551035911493665803st_a_b @ E @ ( darcs_1080543555413674703st_a_b @ X ) ) ) ) ) ) ) ) ) ).

% wf_dtree.darc_in_sub_if_dtail_in_sub
thf(fact_503_wf__dtree_Odarc__in__sub__if__dtail__in__sub,axiom,
    ! [T: dtree_7302148650479942367_a_b_b,Dt: produc6499617310964463488_a_b_b > real,E: produc6499617310964463488_a_b_b,V: real,X: dtree_7302148650479942367_a_b_b,E1: produc6499617310964463488_a_b_b,Xs2: fset_P2088967965053398822_a_b_b,T12: dtree_7302148650479942367_a_b_b,R2: real] :
      ( ( wf_dtr8538468920812826731_a_b_b @ T )
     => ( ( ( dtail_6014888324085881868_a_b_b @ T @ Dt @ E )
          = V )
       => ( ( member4695696432722591383_a_b_b @ E @ ( darcs_3195269883706795671_a_b_b @ T ) )
         => ( ( member1217632944401329257_a_b_b @ ( produc2872623051376081144_a_b_b @ X @ E1 ) @ ( fset_P7156727148642671791_a_b_b @ Xs2 ) )
           => ( ( is_sub6349751723147071436_a_b_b @ T12 @ X )
             => ( ( ( node_r8311520753349399660_a_b_b @ R2 @ Xs2 )
                  = T )
               => ( ( member_real @ V @ ( dverts8530761295301872936_a_b_b @ T12 ) )
                 => ( member4695696432722591383_a_b_b @ E @ ( darcs_3195269883706795671_a_b_b @ X ) ) ) ) ) ) ) ) ) ).

% wf_dtree.darc_in_sub_if_dtail_in_sub
thf(fact_504_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_505_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_506_rev__append,axiom,
    ! [Xs2: list_a,Ys: list_a] :
      ( ( rev_a @ ( append_a @ Xs2 @ Ys ) )
      = ( append_a @ ( rev_a @ Ys ) @ ( rev_a @ Xs2 ) ) ) ).

% rev_append
thf(fact_507_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_508_order__less__imp__not__less,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_509_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_510_order__less__imp__not__eq2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_511_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_512_order__less__imp__not__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_513_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
      | ( X = Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_514_linorder__less__linear,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
      | ( X = Y )
      | ( ord_less_real @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_515_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P3: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P3 ) ) ).

% order_less_imp_triv
thf(fact_516_order__less__imp__triv,axiom,
    ! [X: real,Y: real,P3: $o] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_real @ Y @ X )
       => P3 ) ) ).

% order_less_imp_triv
thf(fact_517_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_518_order__less__not__sym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_519_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_520_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > real,C: real] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_521_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_522_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_523_order__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_524_order__less__subst1,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_525_order__less__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_526_order__less__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_527_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_528_order__less__irrefl,axiom,
    ! [X: real] :
      ~ ( ord_less_real @ X @ X ) ).

% order_less_irrefl
thf(fact_529_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_530_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > real,C: real] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_531_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_532_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_533_ord__eq__less__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_534_ord__eq__less__subst,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_535_ord__eq__less__subst,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_536_ord__eq__less__subst,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_537_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z3 )
       => ( ord_less_nat @ X @ Z3 ) ) ) ).

% order_less_trans
thf(fact_538_order__less__trans,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_real @ Y @ Z3 )
       => ( ord_less_real @ X @ Z3 ) ) ) ).

% order_less_trans
thf(fact_539_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_540_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order_less_asym'
thf(fact_541_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      = ( ( ord_less_nat @ X @ Y )
        | ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_542_linorder__neq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
      = ( ( ord_less_real @ X @ Y )
        | ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_543_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_544_order__less__asym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_asym
thf(fact_545_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_546_linorder__neqE,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ ( ord_less_real @ X @ Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_547_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_548_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_549_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_550_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_551_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_552_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_553_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_554_not__less__iff__gr__or__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_real @ X @ Y ) )
      = ( ( ord_less_real @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_555_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_556_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_557_linorder__less__wlog,axiom,
    ! [P3: nat > nat > $o,A: nat,B: nat] :
      ( ! [A5: nat,B4: nat] :
          ( ( ord_less_nat @ A5 @ B4 )
         => ( P3 @ A5 @ B4 ) )
     => ( ! [A5: nat] : ( P3 @ A5 @ A5 )
       => ( ! [A5: nat,B4: nat] :
              ( ( P3 @ B4 @ A5 )
             => ( P3 @ A5 @ B4 ) )
         => ( P3 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_558_linorder__less__wlog,axiom,
    ! [P3: real > real > $o,A: real,B: real] :
      ( ! [A5: real,B4: real] :
          ( ( ord_less_real @ A5 @ B4 )
         => ( P3 @ A5 @ B4 ) )
     => ( ! [A5: real] : ( P3 @ A5 @ A5 )
       => ( ! [A5: real,B4: real] :
              ( ( P3 @ B4 @ A5 )
             => ( P3 @ A5 @ B4 ) )
         => ( P3 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_559_exists__least__iff,axiom,
    ( ( ^ [P4: nat > $o] :
        ? [X7: nat] : ( P4 @ X7 ) )
    = ( ^ [P5: nat > $o] :
        ? [N: nat] :
          ( ( P5 @ N )
          & ! [M: nat] :
              ( ( ord_less_nat @ M @ N )
             => ~ ( P5 @ M ) ) ) ) ) ).

% exists_least_iff
thf(fact_560_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_561_dual__order_Oirrefl,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% dual_order.irrefl
thf(fact_562_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_563_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ~ ( ord_less_real @ A @ B ) ) ).

% dual_order.asym
thf(fact_564_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_565_linorder__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_566_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_567_antisym__conv3,axiom,
    ! [Y: real,X: real] :
      ( ~ ( ord_less_real @ Y @ X )
     => ( ( ~ ( ord_less_real @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_568_less__induct,axiom,
    ! [P3: nat > $o,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y6: nat] :
              ( ( ord_less_nat @ Y6 @ X3 )
             => ( P3 @ Y6 ) )
         => ( P3 @ X3 ) )
     => ( P3 @ A ) ) ).

% less_induct
thf(fact_569_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_570_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( B = C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_571_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_572_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_573_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_574_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order.asym
thf(fact_575_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_576_less__imp__neq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_577_dense,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ? [Z4: real] :
          ( ( ord_less_real @ X @ Z4 )
          & ( ord_less_real @ Z4 @ Y ) ) ) ).

% dense
thf(fact_578_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).

% gt_ex
thf(fact_579_gt__ex,axiom,
    ! [X: real] :
    ? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).

% gt_ex
thf(fact_580_lt__ex,axiom,
    ! [X: real] :
    ? [Y4: real] : ( ord_less_real @ Y4 @ X ) ).

% lt_ex
thf(fact_581_leD,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ~ ( ord_less_real @ X @ Y ) ) ).

% leD
thf(fact_582_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_583_leI,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ord_less_eq_real @ Y @ X ) ) ).

% leI
thf(fact_584_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_585_nless__le,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( ord_less_real @ A @ B ) )
      = ( ~ ( ord_less_eq_real @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_586_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_587_antisym__conv1,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ( ord_less_eq_real @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_588_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_589_antisym__conv2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ~ ( ord_less_real @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_590_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_591_dense__ge,axiom,
    ! [Z3: real,Y: real] :
      ( ! [X3: real] :
          ( ( ord_less_real @ Z3 @ X3 )
         => ( ord_less_eq_real @ Y @ X3 ) )
     => ( ord_less_eq_real @ Y @ Z3 ) ) ).

% dense_ge
thf(fact_592_dense__le,axiom,
    ! [Y: real,Z3: real] :
      ( ! [X3: real] :
          ( ( ord_less_real @ X3 @ Y )
         => ( ord_less_eq_real @ X3 @ Z3 ) )
     => ( ord_less_eq_real @ Y @ Z3 ) ) ).

% dense_le
thf(fact_593_less__le__not__le,axiom,
    ( ord_less_real
    = ( ^ [X4: real,Y3: real] :
          ( ( ord_less_eq_real @ X4 @ Y3 )
          & ~ ( ord_less_eq_real @ Y3 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_594_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y3 )
          & ~ ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_595_not__le__imp__less,axiom,
    ! [Y: real,X: real] :
      ( ~ ( ord_less_eq_real @ Y @ X )
     => ( ord_less_real @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_596_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X )
     => ( ord_less_nat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_597_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [A4: real,B3: real] :
          ( ( ord_less_real @ A4 @ B3 )
          | ( A4 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_598_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_nat @ A4 @ B3 )
          | ( A4 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_599_order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [A4: real,B3: real] :
          ( ( ord_less_eq_real @ A4 @ B3 )
          & ( A4 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_600_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
          & ( A4 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_601_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_602_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_603_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_604_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_605_order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [A4: real,B3: real] :
          ( ( ord_less_eq_real @ A4 @ B3 )
          & ~ ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_606_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
          & ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_607_dense__ge__bounded,axiom,
    ! [Z3: real,X: real,Y: real] :
      ( ( ord_less_real @ Z3 @ X )
     => ( ! [W: real] :
            ( ( ord_less_real @ Z3 @ W )
           => ( ( ord_less_real @ W @ X )
             => ( ord_less_eq_real @ Y @ W ) ) )
       => ( ord_less_eq_real @ Y @ Z3 ) ) ) ).

% dense_ge_bounded
thf(fact_608_dense__le__bounded,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ! [W: real] :
            ( ( ord_less_real @ X @ W )
           => ( ( ord_less_real @ W @ Y )
             => ( ord_less_eq_real @ W @ Z3 ) ) )
       => ( ord_less_eq_real @ Y @ Z3 ) ) ) ).

% dense_le_bounded
thf(fact_609_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [B3: real,A4: real] :
          ( ( ord_less_real @ B3 @ A4 )
          | ( A4 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_610_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( ord_less_nat @ B3 @ A4 )
          | ( A4 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_611_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [B3: real,A4: real] :
          ( ( ord_less_eq_real @ B3 @ A4 )
          & ( A4 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_612_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B3 @ A4 )
          & ( A4 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_613_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_614_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_615_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_616_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_617_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [B3: real,A4: real] :
          ( ( ord_less_eq_real @ B3 @ A4 )
          & ~ ( ord_less_eq_real @ A4 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_618_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B3 @ A4 )
          & ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_619_order_Ostrict__implies__order,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_eq_real @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_620_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_621_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ord_less_eq_real @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_622_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_623_order__le__less,axiom,
    ( ord_less_eq_real
    = ( ^ [X4: real,Y3: real] :
          ( ( ord_less_real @ X4 @ Y3 )
          | ( X4 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_624_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y3: nat] :
          ( ( ord_less_nat @ X4 @ Y3 )
          | ( X4 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_625_order__less__le,axiom,
    ( ord_less_real
    = ( ^ [X4: real,Y3: real] :
          ( ( ord_less_eq_real @ X4 @ Y3 )
          & ( X4 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_626_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y3 )
          & ( X4 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_627_linorder__not__le,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_eq_real @ X @ Y ) )
      = ( ord_less_real @ Y @ X ) ) ).

% linorder_not_le
thf(fact_628_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_629_linorder__not__less,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_real @ X @ Y ) )
      = ( ord_less_eq_real @ Y @ X ) ) ).

% linorder_not_less
thf(fact_630_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_631_order__less__imp__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_632_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_633_order__le__neq__trans,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( A != B )
       => ( ord_less_real @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_634_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_635_order__neq__le__trans,axiom,
    ! [A: real,B: real] :
      ( ( A != B )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_real @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_636_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_637_order__le__less__trans,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_real @ Y @ Z3 )
       => ( ord_less_real @ X @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_638_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z3 )
       => ( ord_less_nat @ X @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_639_order__less__le__trans,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_eq_real @ Y @ Z3 )
       => ( ord_less_real @ X @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_640_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z3 )
       => ( ord_less_nat @ X @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_641_order__le__less__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_642_order__le__less__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_643_order__le__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_644_order__le__less__subst1,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_645_order__le__less__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_eq_real @ X3 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_646_order__le__less__subst2,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_eq_real @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_647_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > real,C: real] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_648_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_649_order__less__le__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_eq_real @ X3 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_650_order__less__le__subst1,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_eq_real @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_651_order__less__le__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_652_order__less__le__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_653_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > real,C: real] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_654_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_655_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_656_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_657_linorder__le__less__linear,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
      | ( ord_less_real @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_658_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_659_order__le__imp__less__or__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_real @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_660_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_661_bot_Oextremum__strict,axiom,
    ! [A: fset_P2153231429829016240_a_b_b] :
      ~ ( ord_le6631730213922513156_a_b_b @ A @ bot_bo2248824169281960260_a_b_b ) ).

% bot.extremum_strict
thf(fact_662_bot_Oextremum__strict,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_663_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_664_bot_Onot__eq__extremum,axiom,
    ! [A: fset_P2153231429829016240_a_b_b] :
      ( ( A != bot_bo2248824169281960260_a_b_b )
      = ( ord_le6631730213922513156_a_b_b @ bot_bo2248824169281960260_a_b_b @ A ) ) ).

% bot.not_eq_extremum
thf(fact_665_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
      = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_666_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_667_nat__descend__induct,axiom,
    ! [N2: nat,P3: nat > $o,M2: nat] :
      ( ! [K: nat] :
          ( ( ord_less_nat @ N2 @ K )
         => ( P3 @ K ) )
     => ( ! [K: nat] :
            ( ( ord_less_eq_nat @ K @ N2 )
           => ( ! [I2: nat] :
                  ( ( ord_less_nat @ K @ I2 )
                 => ( P3 @ I2 ) )
             => ( P3 @ K ) ) )
       => ( P3 @ M2 ) ) ) ).

% nat_descend_induct
thf(fact_668_wf__dtree_Owf__dtree__rec,axiom,
    ! [T: dtree_list_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E: b] :
      ( ( wf_dtree_list_a_b @ T )
     => ( ( ( node_list_a_b @ R2 @ Xs2 )
          = T )
       => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
         => ( wf_dtree_list_a_b @ X ) ) ) ) ).

% wf_dtree.wf_dtree_rec
thf(fact_669_wf__dtree_Oarc__uneq__if__subtree__uneq,axiom,
    ! [T: dtree_list_a_b,X1: dtree_list_a_b,E1: b,Xs2: fset_P2153231429829016240_a_b_b,X2: dtree_list_a_b,E22: b,R2: list_a] :
      ( ( wf_dtree_list_a_b @ T )
     => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X1 @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
       => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X2 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
         => ( ( X1 != X2 )
           => ( ( ( node_list_a_b @ R2 @ Xs2 )
                = T )
             => ( E1 != E22 ) ) ) ) ) ) ).

% wf_dtree.arc_uneq_if_subtree_uneq
thf(fact_670_wf__dtree_Osubtree__uneq__if__arc__uneq,axiom,
    ! [T: dtree_list_a_b,X1: dtree_list_a_b,E1: b,Xs2: fset_P2153231429829016240_a_b_b,X2: dtree_list_a_b,E22: b,R2: list_a] :
      ( ( wf_dtree_list_a_b @ T )
     => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X1 @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
       => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X2 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
         => ( ( E1 != E22 )
           => ( ( ( node_list_a_b @ R2 @ Xs2 )
                = T )
             => ( X1 != X2 ) ) ) ) ) ) ).

% wf_dtree.subtree_uneq_if_arc_uneq
thf(fact_671_wf__dtree_Odhead__in__dverts,axiom,
    ! [T: dtree_real_real,E: real,Def2: real > real] :
      ( ( wf_dtree_real_real @ T )
     => ( ( member_real @ E @ ( darcs_real_real @ T ) )
       => ( member_real @ ( dhead_real_real @ T @ Def2 @ E ) @ ( dverts_real_real @ T ) ) ) ) ).

% wf_dtree.dhead_in_dverts
thf(fact_672_wf__dtree_Odhead__in__dverts,axiom,
    ! [T: dtree_real_list_a,E: list_a,Def2: list_a > real] :
      ( ( wf_dtree_real_list_a @ T )
     => ( ( member_list_a @ E @ ( darcs_real_list_a @ T ) )
       => ( member_real @ ( dhead_real_list_a @ T @ Def2 @ E ) @ ( dverts_real_list_a @ T ) ) ) ) ).

% wf_dtree.dhead_in_dverts
thf(fact_673_wf__dtree_Odhead__in__dverts,axiom,
    ! [T: dtree_list_a_real,E: real,Def2: real > list_a] :
      ( ( wf_dtree_list_a_real @ T )
     => ( ( member_real @ E @ ( darcs_list_a_real @ T ) )
       => ( member_list_a @ ( dhead_list_a_real @ T @ Def2 @ E ) @ ( dverts_list_a_real @ T ) ) ) ) ).

% wf_dtree.dhead_in_dverts
thf(fact_674_wf__dtree_Odhead__in__dverts,axiom,
    ! [T: dtree_list_a_b,E: b,Def2: b > list_a] :
      ( ( wf_dtree_list_a_b @ T )
     => ( ( member_b @ E @ ( darcs_list_a_b @ T ) )
       => ( member_list_a @ ( dhead_list_a_b @ T @ Def2 @ E ) @ ( dverts_list_a_b @ T ) ) ) ) ).

% wf_dtree.dhead_in_dverts
thf(fact_675_wf__dtree_Odhead__in__dverts,axiom,
    ! [T: dtree_list_a_list_a,E: list_a,Def2: list_a > list_a] :
      ( ( wf_dtr2792200986692785213list_a @ T )
     => ( ( member_list_a @ E @ ( darcs_list_a_list_a @ T ) )
       => ( member_list_a @ ( dhead_list_a_list_a @ T @ Def2 @ E ) @ ( dverts_list_a_list_a @ T ) ) ) ) ).

% wf_dtree.dhead_in_dverts
thf(fact_676_wf__dtree_Odhead__in__dverts,axiom,
    ! [T: dtree_3696687526968142675st_a_b,E: dtree_list_a_b,Def2: dtree_list_a_b > real] :
      ( ( wf_dtr8971879960368933343st_a_b @ T )
     => ( ( member551035911493665803st_a_b @ E @ ( darcs_3945445989234600971st_a_b @ T ) )
       => ( member_real @ ( dhead_5523335947888865680st_a_b @ T @ Def2 @ E ) @ ( dverts4570227339949624220st_a_b @ T ) ) ) ) ).

% wf_dtree.dhead_in_dverts
thf(fact_677_wf__dtree_Odhead__in__dverts,axiom,
    ! [T: dtree_8345463500551380051b_real,E: real,Def2: real > dtree_list_a_b] :
      ( ( wf_dtr8910464690835127007b_real @ T )
     => ( ( member_real @ E @ ( darcs_3884030719700794635b_real @ T ) )
       => ( member551035911493665803st_a_b @ ( dhead_5461920678355059344b_real @ T @ Def2 @ E ) @ ( dverts4508812070415817884b_real @ T ) ) ) ) ).

% wf_dtree.dhead_in_dverts
thf(fact_678_wf__dtree_Odhead__in__dverts,axiom,
    ! [T: dtree_8284992029673318451list_a,E: list_a,Def2: list_a > dtree_list_a_b] :
      ( ( wf_dtr8287859687986785809list_a @ T )
     => ( ( member_list_a @ E @ ( darcs_4898656622333939429list_a @ T ) )
       => ( member551035911493665803st_a_b @ ( dhead_2549497505893332128list_a @ T @ Def2 @ E ) @ ( dverts6918263395131438484list_a @ T ) ) ) ) ).

% wf_dtree.dhead_in_dverts
thf(fact_679_wf__dtree_Odhead__in__dverts,axiom,
    ! [T: dtree_4875458026974816597st_a_b,E: dtree_list_a_b,Def2: dtree_list_a_b > list_a] :
      ( ( wf_dtr4469746621066521083st_a_b @ T )
     => ( ( member551035911493665803st_a_b @ E @ ( darcs_1080543555413674703st_a_b @ T ) )
       => ( member_list_a @ ( dhead_7954756475827843210st_a_b @ T @ Def2 @ E ) @ ( dverts3100150328211173758st_a_b @ T ) ) ) ) ).

% wf_dtree.dhead_in_dverts
thf(fact_680_wf__dtree_Odhead__in__dverts,axiom,
    ! [T: dtree_7302148650479942367_a_b_b,E: produc6499617310964463488_a_b_b,Def2: produc6499617310964463488_a_b_b > real] :
      ( ( wf_dtr8538468920812826731_a_b_b @ T )
     => ( ( member4695696432722591383_a_b_b @ E @ ( darcs_3195269883706795671_a_b_b @ T ) )
       => ( member_real @ ( dhead_8734047501402136348_a_b_b @ T @ Def2 @ E ) @ ( dverts8530761295301872936_a_b_b @ T ) ) ) ) ).

% wf_dtree.dhead_in_dverts
thf(fact_681_wf__dtree_Odtail__in__dverts,axiom,
    ! [T: dtree_real_real,E: real,Def2: real > real] :
      ( ( wf_dtree_real_real @ T )
     => ( ( member_real @ E @ ( darcs_real_real @ T ) )
       => ( member_real @ ( dtail_real_real @ T @ Def2 @ E ) @ ( dverts_real_real @ T ) ) ) ) ).

% wf_dtree.dtail_in_dverts
thf(fact_682_wf__dtree_Odtail__in__dverts,axiom,
    ! [T: dtree_real_list_a,E: list_a,Def2: list_a > real] :
      ( ( wf_dtree_real_list_a @ T )
     => ( ( member_list_a @ E @ ( darcs_real_list_a @ T ) )
       => ( member_real @ ( dtail_real_list_a @ T @ Def2 @ E ) @ ( dverts_real_list_a @ T ) ) ) ) ).

% wf_dtree.dtail_in_dverts
thf(fact_683_wf__dtree_Odtail__in__dverts,axiom,
    ! [T: dtree_list_a_real,E: real,Def2: real > list_a] :
      ( ( wf_dtree_list_a_real @ T )
     => ( ( member_real @ E @ ( darcs_list_a_real @ T ) )
       => ( member_list_a @ ( dtail_list_a_real @ T @ Def2 @ E ) @ ( dverts_list_a_real @ T ) ) ) ) ).

% wf_dtree.dtail_in_dverts
thf(fact_684_wf__dtree_Odtail__in__dverts,axiom,
    ! [T: dtree_list_a_b,E: b,Def2: b > list_a] :
      ( ( wf_dtree_list_a_b @ T )
     => ( ( member_b @ E @ ( darcs_list_a_b @ T ) )
       => ( member_list_a @ ( dtail_list_a_b @ T @ Def2 @ E ) @ ( dverts_list_a_b @ T ) ) ) ) ).

% wf_dtree.dtail_in_dverts
thf(fact_685_wf__dtree_Odtail__in__dverts,axiom,
    ! [T: dtree_list_a_list_a,E: list_a,Def2: list_a > list_a] :
      ( ( wf_dtr2792200986692785213list_a @ T )
     => ( ( member_list_a @ E @ ( darcs_list_a_list_a @ T ) )
       => ( member_list_a @ ( dtail_list_a_list_a @ T @ Def2 @ E ) @ ( dverts_list_a_list_a @ T ) ) ) ) ).

% wf_dtree.dtail_in_dverts
thf(fact_686_wf__dtree_Odtail__in__dverts,axiom,
    ! [T: dtree_3696687526968142675st_a_b,E: dtree_list_a_b,Def2: dtree_list_a_b > real] :
      ( ( wf_dtr8971879960368933343st_a_b @ T )
     => ( ( member551035911493665803st_a_b @ E @ ( darcs_3945445989234600971st_a_b @ T ) )
       => ( member_real @ ( dtail_8710550175129296512st_a_b @ T @ Def2 @ E ) @ ( dverts4570227339949624220st_a_b @ T ) ) ) ) ).

% wf_dtree.dtail_in_dverts
thf(fact_687_wf__dtree_Odtail__in__dverts,axiom,
    ! [T: dtree_8345463500551380051b_real,E: real,Def2: real > dtree_list_a_b] :
      ( ( wf_dtr8910464690835127007b_real @ T )
     => ( ( member_real @ E @ ( darcs_3884030719700794635b_real @ T ) )
       => ( member551035911493665803st_a_b @ ( dtail_8649134905595490176b_real @ T @ Def2 @ E ) @ ( dverts4508812070415817884b_real @ T ) ) ) ) ).

% wf_dtree.dtail_in_dverts
thf(fact_688_wf__dtree_Odtail__in__dverts,axiom,
    ! [T: dtree_8284992029673318451list_a,E: list_a,Def2: list_a > dtree_list_a_b] :
      ( ( wf_dtr8287859687986785809list_a @ T )
     => ( ( member_list_a @ E @ ( darcs_4898656622333939429list_a @ T ) )
       => ( member551035911493665803st_a_b @ ( dtail_7497147172532560816list_a @ T @ Def2 @ E ) @ ( dverts6918263395131438484list_a @ T ) ) ) ) ).

% wf_dtree.dtail_in_dverts
thf(fact_689_wf__dtree_Odtail__in__dverts,axiom,
    ! [T: dtree_4875458026974816597st_a_b,E: dtree_list_a_b,Def2: dtree_list_a_b > list_a] :
      ( ( wf_dtr4469746621066521083st_a_b @ T )
     => ( ( member551035911493665803st_a_b @ E @ ( darcs_1080543555413674703st_a_b @ T ) )
       => ( member_list_a @ ( dtail_3679034105612296090st_a_b @ T @ Def2 @ E ) @ ( dverts3100150328211173758st_a_b @ T ) ) ) ) ).

% wf_dtree.dtail_in_dverts
thf(fact_690_wf__dtree_Odtail__in__dverts,axiom,
    ! [T: dtree_7302148650479942367_a_b_b,E: produc6499617310964463488_a_b_b,Def2: produc6499617310964463488_a_b_b > real] :
      ( ( wf_dtr8538468920812826731_a_b_b @ T )
     => ( ( member4695696432722591383_a_b_b @ E @ ( darcs_3195269883706795671_a_b_b @ T ) )
       => ( member_real @ ( dtail_6014888324085881868_a_b_b @ T @ Def2 @ E ) @ ( dverts8530761295301872936_a_b_b @ T ) ) ) ) ).

% wf_dtree.dtail_in_dverts
thf(fact_691_wf__dtree_Ochild__arc__not__subtree,axiom,
    ! [T: dtree_list_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E1: b] :
      ( ( wf_dtree_list_a_b @ T )
     => ( ( ( node_list_a_b @ R2 @ Xs2 )
          = T )
       => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
         => ~ ( member_b @ E1 @ ( darcs_list_a_b @ X ) ) ) ) ) ).

% wf_dtree.child_arc_not_subtree
thf(fact_692_wf__dtree_Osubtree__root__not__root,axiom,
    ! [T: dtree_list_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E: b] :
      ( ( wf_dtree_list_a_b @ T )
     => ( ( T
          = ( node_list_a_b @ R2 @ Xs2 ) )
       => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
         => ( ( root_list_a_b @ X )
           != R2 ) ) ) ) ).

% wf_dtree.subtree_root_not_root
thf(fact_693_wf__dtree_Odtail__in__subdverts,axiom,
    ! [T: dtree_real_real,E: real,X: dtree_real_real,Def2: real > real] :
      ( ( wf_dtree_real_real @ T )
     => ( ( member_real @ E @ ( darcs_real_real @ X ) )
       => ( ( is_subtree_real_real @ X @ T )
         => ( member_real @ ( dtail_real_real @ T @ Def2 @ E ) @ ( dverts_real_real @ X ) ) ) ) ) ).

% wf_dtree.dtail_in_subdverts
thf(fact_694_wf__dtree_Odtail__in__subdverts,axiom,
    ! [T: dtree_real_list_a,E: list_a,X: dtree_real_list_a,Def2: list_a > real] :
      ( ( wf_dtree_real_list_a @ T )
     => ( ( member_list_a @ E @ ( darcs_real_list_a @ X ) )
       => ( ( is_sub4113186846137662008list_a @ X @ T )
         => ( member_real @ ( dtail_real_list_a @ T @ Def2 @ E ) @ ( dverts_real_list_a @ X ) ) ) ) ) ).

% wf_dtree.dtail_in_subdverts
thf(fact_695_wf__dtree_Odtail__in__subdverts,axiom,
    ! [T: dtree_list_a_real,E: real,X: dtree_list_a_real,Def2: real > list_a] :
      ( ( wf_dtree_list_a_real @ T )
     => ( ( member_real @ E @ ( darcs_list_a_real @ X ) )
       => ( ( is_sub5530047198231550610a_real @ X @ T )
         => ( member_list_a @ ( dtail_list_a_real @ T @ Def2 @ E ) @ ( dverts_list_a_real @ X ) ) ) ) ) ).

% wf_dtree.dtail_in_subdverts
thf(fact_696_wf__dtree_Odtail__in__subdverts,axiom,
    ! [T: dtree_list_a_b,E: b,X: dtree_list_a_b,Def2: b > list_a] :
      ( ( wf_dtree_list_a_b @ T )
     => ( ( member_b @ E @ ( darcs_list_a_b @ X ) )
       => ( ( is_subtree_list_a_b @ X @ T )
         => ( member_list_a @ ( dtail_list_a_b @ T @ Def2 @ E ) @ ( dverts_list_a_b @ X ) ) ) ) ) ).

% wf_dtree.dtail_in_subdverts
thf(fact_697_wf__dtree_Odtail__in__subdverts,axiom,
    ! [T: dtree_list_a_list_a,E: list_a,X: dtree_list_a_list_a,Def2: list_a > list_a] :
      ( ( wf_dtr2792200986692785213list_a @ T )
     => ( ( member_list_a @ E @ ( darcs_list_a_list_a @ X ) )
       => ( ( is_sub7392010792700354846list_a @ X @ T )
         => ( member_list_a @ ( dtail_list_a_list_a @ T @ Def2 @ E ) @ ( dverts_list_a_list_a @ X ) ) ) ) ) ).

% wf_dtree.dtail_in_subdverts
thf(fact_698_wf__dtree_Odtail__in__subdverts,axiom,
    ! [T: dtree_3696687526968142675st_a_b,E: dtree_list_a_b,X: dtree_3696687526968142675st_a_b,Def2: dtree_list_a_b > real] :
      ( ( wf_dtr8971879960368933343st_a_b @ T )
     => ( ( member551035911493665803st_a_b @ E @ ( darcs_3945445989234600971st_a_b @ X ) )
       => ( ( is_sub776737362830463552st_a_b @ X @ T )
         => ( member_real @ ( dtail_8710550175129296512st_a_b @ T @ Def2 @ E ) @ ( dverts4570227339949624220st_a_b @ X ) ) ) ) ) ).

% wf_dtree.dtail_in_subdverts
thf(fact_699_wf__dtree_Odtail__in__subdverts,axiom,
    ! [T: dtree_8345463500551380051b_real,E: real,X: dtree_8345463500551380051b_real,Def2: real > dtree_list_a_b] :
      ( ( wf_dtr8910464690835127007b_real @ T )
     => ( ( member_real @ E @ ( darcs_3884030719700794635b_real @ X ) )
       => ( ( is_sub715322093296657216b_real @ X @ T )
         => ( member551035911493665803st_a_b @ ( dtail_8649134905595490176b_real @ T @ Def2 @ E ) @ ( dverts4508812070415817884b_real @ X ) ) ) ) ) ).

% wf_dtree.dtail_in_subdverts
thf(fact_700_wf__dtree_Odtail__in__subdverts,axiom,
    ! [T: dtree_8284992029673318451list_a,E: list_a,X: dtree_8284992029673318451list_a,Def2: list_a > dtree_list_a_b] :
      ( ( wf_dtr8287859687986785809list_a @ T )
     => ( ( member_list_a @ E @ ( darcs_4898656622333939429list_a @ X ) )
       => ( ( is_sub986895638982090736list_a @ X @ T )
         => ( member551035911493665803st_a_b @ ( dtail_7497147172532560816list_a @ T @ Def2 @ E ) @ ( dverts6918263395131438484list_a @ X ) ) ) ) ) ).

% wf_dtree.dtail_in_subdverts
thf(fact_701_wf__dtree_Odtail__in__subdverts,axiom,
    ! [T: dtree_4875458026974816597st_a_b,E: dtree_list_a_b,X: dtree_4875458026974816597st_a_b,Def2: dtree_list_a_b > list_a] :
      ( ( wf_dtr4469746621066521083st_a_b @ T )
     => ( ( member551035911493665803st_a_b @ E @ ( darcs_1080543555413674703st_a_b @ X ) )
       => ( ( is_sub6392154608916601818st_a_b @ X @ T )
         => ( member_list_a @ ( dtail_3679034105612296090st_a_b @ T @ Def2 @ E ) @ ( dverts3100150328211173758st_a_b @ X ) ) ) ) ) ).

% wf_dtree.dtail_in_subdverts
thf(fact_702_wf__dtree_Odtail__in__subdverts,axiom,
    ! [T: dtree_7302148650479942367_a_b_b,E: produc6499617310964463488_a_b_b,X: dtree_7302148650479942367_a_b_b,Def2: produc6499617310964463488_a_b_b > real] :
      ( ( wf_dtr8538468920812826731_a_b_b @ T )
     => ( ( member4695696432722591383_a_b_b @ E @ ( darcs_3195269883706795671_a_b_b @ X ) )
       => ( ( is_sub6349751723147071436_a_b_b @ X @ T )
         => ( member_real @ ( dtail_6014888324085881868_a_b_b @ T @ Def2 @ E ) @ ( dverts8530761295301872936_a_b_b @ X ) ) ) ) ) ).

% wf_dtree.dtail_in_subdverts
thf(fact_703_combine__uneq,axiom,
    ! [R2: list_a,T12: dtree_list_a_b,E1: b] :
      ( ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) )
     != ( node_list_a_b @ ( append_a @ R2 @ ( root_list_a_b @ T12 ) ) @ ( sucs_list_a_b @ T12 ) ) ) ).

% combine_uneq
thf(fact_704_denormalize_Oelims,axiom,
    ! [X: dtree_list_a_b,Y: list_a] :
      ( ( ( denormalize_a_b @ X )
        = Y )
     => ( ! [R: list_a,T2: dtree_list_a_b] :
            ( ? [E2: b] :
                ( X
                = ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
           => ( Y
             != ( append_a @ R @ ( denormalize_a_b @ T2 ) ) ) )
       => ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
              ( ! [X5: produc6499617310964463488_a_b_b] :
                  ( Xs
                 != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
             => ! [R: list_a] :
                  ( ( X
                    = ( node_list_a_b @ R @ Xs ) )
                 => ( Y != R ) ) ) ) ) ).

% denormalize.elims
thf(fact_705_denormalize_Osimps_I1_J,axiom,
    ! [R2: list_a,T: dtree_list_a_b,E: b] :
      ( ( denormalize_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
      = ( append_a @ R2 @ ( denormalize_a_b @ T ) ) ) ).

% denormalize.simps(1)
thf(fact_706_wf__dtree_Odhead__in__childverts,axiom,
    ! [T: dtree_real_real,E: real,X: dtree_real_real,E3: real,Xs2: fset_P7499862231470980990l_real,R2: real,Def2: real > real] :
      ( ( wf_dtree_real_real @ T )
     => ( ( member_real @ E @ ( darcs_real_real @ X ) )
       => ( ( member2617108185285288129l_real @ ( produc3949691677225608016l_real @ X @ E3 ) @ ( fset_P7992937044511435015l_real @ Xs2 ) )
         => ( ( ( node_real_real @ R2 @ Xs2 )
              = T )
           => ( member_real @ ( dhead_real_real @ T @ Def2 @ E ) @ ( dverts_real_real @ X ) ) ) ) ) ) ).

% wf_dtree.dhead_in_childverts
thf(fact_707_wf__dtree_Odhead__in__childverts,axiom,
    ! [T: dtree_real_list_a,E: list_a,X: dtree_real_list_a,E3: list_a,Xs2: fset_P1040399934802234454list_a,R2: real,Def2: list_a > real] :
      ( ( wf_dtree_real_list_a @ T )
     => ( ( member_list_a @ E @ ( darcs_real_list_a @ X ) )
       => ( ( member5451813156226304665list_a @ ( produc8292410999216705376list_a @ X @ E3 ) @ ( fset_P7319268309571958751list_a @ Xs2 ) )
         => ( ( ( node_real_list_a @ R2 @ Xs2 )
              = T )
           => ( member_real @ ( dhead_real_list_a @ T @ Def2 @ E ) @ ( dverts_real_list_a @ X ) ) ) ) ) ) ).

% wf_dtree.dhead_in_childverts
thf(fact_708_wf__dtree_Odhead__in__childverts,axiom,
    ! [T: dtree_list_a_real,E: real,X: dtree_list_a_real,E3: real,Xs2: fset_P6497461246188700184l_real,R2: list_a,Def2: real > list_a] :
      ( ( wf_dtree_list_a_real @ T )
     => ( ( member_real @ E @ ( darcs_list_a_real @ X ) )
       => ( ( member4849742732303649023l_real @ ( produc2747972018350754914l_real @ X @ E3 ) @ ( fset_P6949987765245255225l_real @ Xs2 ) )
         => ( ( ( node_list_a_real @ R2 @ Xs2 )
              = T )
           => ( member_list_a @ ( dhead_list_a_real @ T @ Def2 @ E ) @ ( dverts_list_a_real @ X ) ) ) ) ) ) ).

% wf_dtree.dhead_in_childverts
thf(fact_709_wf__dtree_Odhead__in__childverts,axiom,
    ! [T: dtree_list_a_b,E: b,X: dtree_list_a_b,E3: b,Xs2: fset_P2153231429829016240_a_b_b,R2: list_a,Def2: b > list_a] :
      ( ( wf_dtree_list_a_b @ T )
     => ( ( member_b @ E @ ( darcs_list_a_b @ X ) )
       => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E3 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
         => ( ( ( node_list_a_b @ R2 @ Xs2 )
              = T )
           => ( member_list_a @ ( dhead_list_a_b @ T @ Def2 @ E ) @ ( dverts_list_a_b @ X ) ) ) ) ) ) ).

% wf_dtree.dhead_in_childverts
thf(fact_710_wf__dtree_Odhead__in__childverts,axiom,
    ! [T: dtree_list_a_list_a,E: list_a,X: dtree_list_a_list_a,E3: list_a,Xs2: fset_P6656299774166858224list_a,R2: list_a,Def2: list_a > list_a] :
      ( ( wf_dtr2792200986692785213list_a @ T )
     => ( ( member_list_a @ E @ ( darcs_list_a_list_a @ X ) )
       => ( ( member535470554364689367list_a @ ( produc6276725351364364146list_a @ X @ E3 ) @ ( fset_P2575872134735801361list_a @ Xs2 ) )
         => ( ( ( node_list_a_list_a @ R2 @ Xs2 )
              = T )
           => ( member_list_a @ ( dhead_list_a_list_a @ T @ Def2 @ E ) @ ( dverts_list_a_list_a @ X ) ) ) ) ) ) ).

% wf_dtree.dhead_in_childverts
thf(fact_711_wf__dtree_Odhead__in__childverts,axiom,
    ! [T: dtree_3696687526968142675st_a_b,E: dtree_list_a_b,X: dtree_3696687526968142675st_a_b,E3: dtree_list_a_b,Xs2: fset_P5746387328717196814st_a_b,R2: real,Def2: dtree_list_a_b > real] :
      ( ( wf_dtr8971879960368933343st_a_b @ T )
     => ( ( member551035911493665803st_a_b @ E @ ( darcs_3945445989234600971st_a_b @ X ) )
       => ( ( member692244352973388625st_a_b @ ( produc4763843483075138016st_a_b @ X @ E3 ) @ ( fset_P8515478733839210391st_a_b @ Xs2 ) )
         => ( ( ( node_r397108421545473760st_a_b @ R2 @ Xs2 )
              = T )
           => ( member_real @ ( dhead_5523335947888865680st_a_b @ T @ Def2 @ E ) @ ( dverts4570227339949624220st_a_b @ X ) ) ) ) ) ) ).

% wf_dtree.dhead_in_childverts
thf(fact_712_wf__dtree_Odhead__in__childverts,axiom,
    ! [T: dtree_8345463500551380051b_real,E: real,X: dtree_8345463500551380051b_real,E3: real,Xs2: fset_P9074655572916523974l_real,R2: dtree_list_a_b,Def2: real > dtree_list_a_b] :
      ( ( wf_dtr8910464690835127007b_real @ T )
     => ( ( member_real @ E @ ( darcs_3884030719700794635b_real @ X ) )
       => ( ( member8458687784735308553l_real @ ( produc1938216529971361688l_real @ X @ E3 ) @ ( fset_P344769348705914191l_real @ Xs2 ) )
         => ( ( ( node_d335693152011667424b_real @ R2 @ Xs2 )
              = T )
           => ( member551035911493665803st_a_b @ ( dhead_5461920678355059344b_real @ T @ Def2 @ E ) @ ( dverts4508812070415817884b_real @ X ) ) ) ) ) ) ).

% wf_dtree.dhead_in_childverts
thf(fact_713_wf__dtree_Odhead__in__childverts,axiom,
    ! [T: dtree_8284992029673318451list_a,E: list_a,X: dtree_8284992029673318451list_a,E3: list_a,Xs2: fset_P5658613431985659038list_a,R2: dtree_list_a_b,Def2: list_a > dtree_list_a_b] :
      ( ( wf_dtr8287859687986785809list_a @ T )
     => ( ( member_list_a @ E @ ( darcs_4898656622333939429list_a @ X ) )
       => ( ( member2153700486347808993list_a @ ( produc3720128825147924392list_a @ X @ E3 ) @ ( fset_P5083850707481761831list_a @ Xs2 ) )
         => ( ( ( node_d2950405300658840400list_a @ R2 @ Xs2 )
              = T )
           => ( member551035911493665803st_a_b @ ( dhead_2549497505893332128list_a @ T @ Def2 @ E ) @ ( dverts6918263395131438484list_a @ X ) ) ) ) ) ) ).

% wf_dtree.dhead_in_childverts
thf(fact_714_wf__dtree_Odhead__in__childverts,axiom,
    ! [T: dtree_4875458026974816597st_a_b,E: dtree_list_a_b,X: dtree_4875458026974816597st_a_b,E3: dtree_list_a_b,Xs2: fset_P6340920821931359400st_a_b,R2: list_a,Def2: dtree_list_a_b > list_a] :
      ( ( wf_dtr4469746621066521083st_a_b @ T )
     => ( ( member551035911493665803st_a_b @ E @ ( darcs_1080543555413674703st_a_b @ X ) )
       => ( ( member8722885590174353295st_a_b @ ( produc4942817220686395122st_a_b @ X @ E3 ) @ ( fset_P623098064157935817st_a_b @ Xs2 ) )
         => ( ( ( node_l8355664270593351482st_a_b @ R2 @ Xs2 )
              = T )
           => ( member_list_a @ ( dhead_7954756475827843210st_a_b @ T @ Def2 @ E ) @ ( dverts3100150328211173758st_a_b @ X ) ) ) ) ) ) ).

% wf_dtree.dhead_in_childverts
thf(fact_715_wf__dtree_Odhead__in__childverts,axiom,
    ! [T: dtree_7302148650479942367_a_b_b,E: produc6499617310964463488_a_b_b,X: dtree_7302148650479942367_a_b_b,E3: produc6499617310964463488_a_b_b,Xs2: fset_P2088967965053398822_a_b_b,R2: real,Def2: produc6499617310964463488_a_b_b > real] :
      ( ( wf_dtr8538468920812826731_a_b_b @ T )
     => ( ( member4695696432722591383_a_b_b @ E @ ( darcs_3195269883706795671_a_b_b @ X ) )
       => ( ( member1217632944401329257_a_b_b @ ( produc2872623051376081144_a_b_b @ X @ E3 ) @ ( fset_P7156727148642671791_a_b_b @ Xs2 ) )
         => ( ( ( node_r8311520753349399660_a_b_b @ R2 @ Xs2 )
              = T )
           => ( member_real @ ( dhead_8734047501402136348_a_b_b @ T @ Def2 @ E ) @ ( dverts8530761295301872936_a_b_b @ X ) ) ) ) ) ) ).

% wf_dtree.dhead_in_childverts
thf(fact_716_wf__dtree_Odtail__in__childverts,axiom,
    ! [T: dtree_real_real,E: real,X: dtree_real_real,E3: real,Xs2: fset_P7499862231470980990l_real,R2: real,Def2: real > real] :
      ( ( wf_dtree_real_real @ T )
     => ( ( member_real @ E @ ( darcs_real_real @ X ) )
       => ( ( member2617108185285288129l_real @ ( produc3949691677225608016l_real @ X @ E3 ) @ ( fset_P7992937044511435015l_real @ Xs2 ) )
         => ( ( ( node_real_real @ R2 @ Xs2 )
              = T )
           => ( member_real @ ( dtail_real_real @ T @ Def2 @ E ) @ ( dverts_real_real @ X ) ) ) ) ) ) ).

% wf_dtree.dtail_in_childverts
thf(fact_717_wf__dtree_Odtail__in__childverts,axiom,
    ! [T: dtree_real_list_a,E: list_a,X: dtree_real_list_a,E3: list_a,Xs2: fset_P1040399934802234454list_a,R2: real,Def2: list_a > real] :
      ( ( wf_dtree_real_list_a @ T )
     => ( ( member_list_a @ E @ ( darcs_real_list_a @ X ) )
       => ( ( member5451813156226304665list_a @ ( produc8292410999216705376list_a @ X @ E3 ) @ ( fset_P7319268309571958751list_a @ Xs2 ) )
         => ( ( ( node_real_list_a @ R2 @ Xs2 )
              = T )
           => ( member_real @ ( dtail_real_list_a @ T @ Def2 @ E ) @ ( dverts_real_list_a @ X ) ) ) ) ) ) ).

% wf_dtree.dtail_in_childverts
thf(fact_718_wf__dtree_Odtail__in__childverts,axiom,
    ! [T: dtree_list_a_real,E: real,X: dtree_list_a_real,E3: real,Xs2: fset_P6497461246188700184l_real,R2: list_a,Def2: real > list_a] :
      ( ( wf_dtree_list_a_real @ T )
     => ( ( member_real @ E @ ( darcs_list_a_real @ X ) )
       => ( ( member4849742732303649023l_real @ ( produc2747972018350754914l_real @ X @ E3 ) @ ( fset_P6949987765245255225l_real @ Xs2 ) )
         => ( ( ( node_list_a_real @ R2 @ Xs2 )
              = T )
           => ( member_list_a @ ( dtail_list_a_real @ T @ Def2 @ E ) @ ( dverts_list_a_real @ X ) ) ) ) ) ) ).

% wf_dtree.dtail_in_childverts
thf(fact_719_wf__dtree_Odtail__in__childverts,axiom,
    ! [T: dtree_list_a_b,E: b,X: dtree_list_a_b,E3: b,Xs2: fset_P2153231429829016240_a_b_b,R2: list_a,Def2: b > list_a] :
      ( ( wf_dtree_list_a_b @ T )
     => ( ( member_b @ E @ ( darcs_list_a_b @ X ) )
       => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E3 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
         => ( ( ( node_list_a_b @ R2 @ Xs2 )
              = T )
           => ( member_list_a @ ( dtail_list_a_b @ T @ Def2 @ E ) @ ( dverts_list_a_b @ X ) ) ) ) ) ) ).

% wf_dtree.dtail_in_childverts
thf(fact_720_wf__dtree_Odtail__in__childverts,axiom,
    ! [T: dtree_list_a_list_a,E: list_a,X: dtree_list_a_list_a,E3: list_a,Xs2: fset_P6656299774166858224list_a,R2: list_a,Def2: list_a > list_a] :
      ( ( wf_dtr2792200986692785213list_a @ T )
     => ( ( member_list_a @ E @ ( darcs_list_a_list_a @ X ) )
       => ( ( member535470554364689367list_a @ ( produc6276725351364364146list_a @ X @ E3 ) @ ( fset_P2575872134735801361list_a @ Xs2 ) )
         => ( ( ( node_list_a_list_a @ R2 @ Xs2 )
              = T )
           => ( member_list_a @ ( dtail_list_a_list_a @ T @ Def2 @ E ) @ ( dverts_list_a_list_a @ X ) ) ) ) ) ) ).

% wf_dtree.dtail_in_childverts
thf(fact_721_wf__dtree_Odtail__in__childverts,axiom,
    ! [T: dtree_3696687526968142675st_a_b,E: dtree_list_a_b,X: dtree_3696687526968142675st_a_b,E3: dtree_list_a_b,Xs2: fset_P5746387328717196814st_a_b,R2: real,Def2: dtree_list_a_b > real] :
      ( ( wf_dtr8971879960368933343st_a_b @ T )
     => ( ( member551035911493665803st_a_b @ E @ ( darcs_3945445989234600971st_a_b @ X ) )
       => ( ( member692244352973388625st_a_b @ ( produc4763843483075138016st_a_b @ X @ E3 ) @ ( fset_P8515478733839210391st_a_b @ Xs2 ) )
         => ( ( ( node_r397108421545473760st_a_b @ R2 @ Xs2 )
              = T )
           => ( member_real @ ( dtail_8710550175129296512st_a_b @ T @ Def2 @ E ) @ ( dverts4570227339949624220st_a_b @ X ) ) ) ) ) ) ).

% wf_dtree.dtail_in_childverts
thf(fact_722_wf__dtree_Odtail__in__childverts,axiom,
    ! [T: dtree_8345463500551380051b_real,E: real,X: dtree_8345463500551380051b_real,E3: real,Xs2: fset_P9074655572916523974l_real,R2: dtree_list_a_b,Def2: real > dtree_list_a_b] :
      ( ( wf_dtr8910464690835127007b_real @ T )
     => ( ( member_real @ E @ ( darcs_3884030719700794635b_real @ X ) )
       => ( ( member8458687784735308553l_real @ ( produc1938216529971361688l_real @ X @ E3 ) @ ( fset_P344769348705914191l_real @ Xs2 ) )
         => ( ( ( node_d335693152011667424b_real @ R2 @ Xs2 )
              = T )
           => ( member551035911493665803st_a_b @ ( dtail_8649134905595490176b_real @ T @ Def2 @ E ) @ ( dverts4508812070415817884b_real @ X ) ) ) ) ) ) ).

% wf_dtree.dtail_in_childverts
thf(fact_723_wf__dtree_Odtail__in__childverts,axiom,
    ! [T: dtree_8284992029673318451list_a,E: list_a,X: dtree_8284992029673318451list_a,E3: list_a,Xs2: fset_P5658613431985659038list_a,R2: dtree_list_a_b,Def2: list_a > dtree_list_a_b] :
      ( ( wf_dtr8287859687986785809list_a @ T )
     => ( ( member_list_a @ E @ ( darcs_4898656622333939429list_a @ X ) )
       => ( ( member2153700486347808993list_a @ ( produc3720128825147924392list_a @ X @ E3 ) @ ( fset_P5083850707481761831list_a @ Xs2 ) )
         => ( ( ( node_d2950405300658840400list_a @ R2 @ Xs2 )
              = T )
           => ( member551035911493665803st_a_b @ ( dtail_7497147172532560816list_a @ T @ Def2 @ E ) @ ( dverts6918263395131438484list_a @ X ) ) ) ) ) ) ).

% wf_dtree.dtail_in_childverts
thf(fact_724_wf__dtree_Odtail__in__childverts,axiom,
    ! [T: dtree_4875458026974816597st_a_b,E: dtree_list_a_b,X: dtree_4875458026974816597st_a_b,E3: dtree_list_a_b,Xs2: fset_P6340920821931359400st_a_b,R2: list_a,Def2: dtree_list_a_b > list_a] :
      ( ( wf_dtr4469746621066521083st_a_b @ T )
     => ( ( member551035911493665803st_a_b @ E @ ( darcs_1080543555413674703st_a_b @ X ) )
       => ( ( member8722885590174353295st_a_b @ ( produc4942817220686395122st_a_b @ X @ E3 ) @ ( fset_P623098064157935817st_a_b @ Xs2 ) )
         => ( ( ( node_l8355664270593351482st_a_b @ R2 @ Xs2 )
              = T )
           => ( member_list_a @ ( dtail_3679034105612296090st_a_b @ T @ Def2 @ E ) @ ( dverts3100150328211173758st_a_b @ X ) ) ) ) ) ) ).

% wf_dtree.dtail_in_childverts
thf(fact_725_wf__dtree_Odtail__in__childverts,axiom,
    ! [T: dtree_7302148650479942367_a_b_b,E: produc6499617310964463488_a_b_b,X: dtree_7302148650479942367_a_b_b,E3: produc6499617310964463488_a_b_b,Xs2: fset_P2088967965053398822_a_b_b,R2: real,Def2: produc6499617310964463488_a_b_b > real] :
      ( ( wf_dtr8538468920812826731_a_b_b @ T )
     => ( ( member4695696432722591383_a_b_b @ E @ ( darcs_3195269883706795671_a_b_b @ X ) )
       => ( ( member1217632944401329257_a_b_b @ ( produc2872623051376081144_a_b_b @ X @ E3 ) @ ( fset_P7156727148642671791_a_b_b @ Xs2 ) )
         => ( ( ( node_r8311520753349399660_a_b_b @ R2 @ Xs2 )
              = T )
           => ( member_real @ ( dtail_6014888324085881868_a_b_b @ T @ Def2 @ E ) @ ( dverts8530761295301872936_a_b_b @ X ) ) ) ) ) ) ).

% wf_dtree.dtail_in_childverts
thf(fact_726_wf__dtree_Oarc__in__subtree__if__tail__in__subtree,axiom,
    ! [T: dtree_real_real,Dt: real > real,P: real,X: dtree_real_real,R2: real,Xs2: fset_P7499862231470980990l_real,E: real] :
      ( ( wf_dtree_real_real @ T )
     => ( ( member_real @ ( dtail_real_real @ T @ Dt @ P ) @ ( dverts_real_real @ X ) )
       => ( ( member_real @ P @ ( darcs_real_real @ T ) )
         => ( ( T
              = ( node_real_real @ R2 @ Xs2 ) )
           => ( ( member2617108185285288129l_real @ ( produc3949691677225608016l_real @ X @ E ) @ ( fset_P7992937044511435015l_real @ Xs2 ) )
             => ( member_real @ P @ ( darcs_real_real @ X ) ) ) ) ) ) ) ).

% wf_dtree.arc_in_subtree_if_tail_in_subtree
thf(fact_727_wf__dtree_Oarc__in__subtree__if__tail__in__subtree,axiom,
    ! [T: dtree_list_a_real,Dt: real > list_a,P: real,X: dtree_list_a_real,R2: list_a,Xs2: fset_P6497461246188700184l_real,E: real] :
      ( ( wf_dtree_list_a_real @ T )
     => ( ( member_list_a @ ( dtail_list_a_real @ T @ Dt @ P ) @ ( dverts_list_a_real @ X ) )
       => ( ( member_real @ P @ ( darcs_list_a_real @ T ) )
         => ( ( T
              = ( node_list_a_real @ R2 @ Xs2 ) )
           => ( ( member4849742732303649023l_real @ ( produc2747972018350754914l_real @ X @ E ) @ ( fset_P6949987765245255225l_real @ Xs2 ) )
             => ( member_real @ P @ ( darcs_list_a_real @ X ) ) ) ) ) ) ) ).

% wf_dtree.arc_in_subtree_if_tail_in_subtree
thf(fact_728_wf__dtree_Oarc__in__subtree__if__tail__in__subtree,axiom,
    ! [T: dtree_real_list_a,Dt: list_a > real,P: list_a,X: dtree_real_list_a,R2: real,Xs2: fset_P1040399934802234454list_a,E: list_a] :
      ( ( wf_dtree_real_list_a @ T )
     => ( ( member_real @ ( dtail_real_list_a @ T @ Dt @ P ) @ ( dverts_real_list_a @ X ) )
       => ( ( member_list_a @ P @ ( darcs_real_list_a @ T ) )
         => ( ( T
              = ( node_real_list_a @ R2 @ Xs2 ) )
           => ( ( member5451813156226304665list_a @ ( produc8292410999216705376list_a @ X @ E ) @ ( fset_P7319268309571958751list_a @ Xs2 ) )
             => ( member_list_a @ P @ ( darcs_real_list_a @ X ) ) ) ) ) ) ) ).

% wf_dtree.arc_in_subtree_if_tail_in_subtree
thf(fact_729_wf__dtree_Oarc__in__subtree__if__tail__in__subtree,axiom,
    ! [T: dtree_list_a_b,Dt: b > list_a,P: b,X: dtree_list_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,E: b] :
      ( ( wf_dtree_list_a_b @ T )
     => ( ( member_list_a @ ( dtail_list_a_b @ T @ Dt @ P ) @ ( dverts_list_a_b @ X ) )
       => ( ( member_b @ P @ ( darcs_list_a_b @ T ) )
         => ( ( T
              = ( node_list_a_b @ R2 @ Xs2 ) )
           => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
             => ( member_b @ P @ ( darcs_list_a_b @ X ) ) ) ) ) ) ) ).

% wf_dtree.arc_in_subtree_if_tail_in_subtree
thf(fact_730_wf__dtree_Oarc__in__subtree__if__tail__in__subtree,axiom,
    ! [T: dtree_list_a_list_a,Dt: list_a > list_a,P: list_a,X: dtree_list_a_list_a,R2: list_a,Xs2: fset_P6656299774166858224list_a,E: list_a] :
      ( ( wf_dtr2792200986692785213list_a @ T )
     => ( ( member_list_a @ ( dtail_list_a_list_a @ T @ Dt @ P ) @ ( dverts_list_a_list_a @ X ) )
       => ( ( member_list_a @ P @ ( darcs_list_a_list_a @ T ) )
         => ( ( T
              = ( node_list_a_list_a @ R2 @ Xs2 ) )
           => ( ( member535470554364689367list_a @ ( produc6276725351364364146list_a @ X @ E ) @ ( fset_P2575872134735801361list_a @ Xs2 ) )
             => ( member_list_a @ P @ ( darcs_list_a_list_a @ X ) ) ) ) ) ) ) ).

% wf_dtree.arc_in_subtree_if_tail_in_subtree
thf(fact_731_wf__dtree_Oarc__in__subtree__if__tail__in__subtree,axiom,
    ! [T: dtree_8345463500551380051b_real,Dt: real > dtree_list_a_b,P: real,X: dtree_8345463500551380051b_real,R2: dtree_list_a_b,Xs2: fset_P9074655572916523974l_real,E: real] :
      ( ( wf_dtr8910464690835127007b_real @ T )
     => ( ( member551035911493665803st_a_b @ ( dtail_8649134905595490176b_real @ T @ Dt @ P ) @ ( dverts4508812070415817884b_real @ X ) )
       => ( ( member_real @ P @ ( darcs_3884030719700794635b_real @ T ) )
         => ( ( T
              = ( node_d335693152011667424b_real @ R2 @ Xs2 ) )
           => ( ( member8458687784735308553l_real @ ( produc1938216529971361688l_real @ X @ E ) @ ( fset_P344769348705914191l_real @ Xs2 ) )
             => ( member_real @ P @ ( darcs_3884030719700794635b_real @ X ) ) ) ) ) ) ) ).

% wf_dtree.arc_in_subtree_if_tail_in_subtree
thf(fact_732_wf__dtree_Oarc__in__subtree__if__tail__in__subtree,axiom,
    ! [T: dtree_3696687526968142675st_a_b,Dt: dtree_list_a_b > real,P: dtree_list_a_b,X: dtree_3696687526968142675st_a_b,R2: real,Xs2: fset_P5746387328717196814st_a_b,E: dtree_list_a_b] :
      ( ( wf_dtr8971879960368933343st_a_b @ T )
     => ( ( member_real @ ( dtail_8710550175129296512st_a_b @ T @ Dt @ P ) @ ( dverts4570227339949624220st_a_b @ X ) )
       => ( ( member551035911493665803st_a_b @ P @ ( darcs_3945445989234600971st_a_b @ T ) )
         => ( ( T
              = ( node_r397108421545473760st_a_b @ R2 @ Xs2 ) )
           => ( ( member692244352973388625st_a_b @ ( produc4763843483075138016st_a_b @ X @ E ) @ ( fset_P8515478733839210391st_a_b @ Xs2 ) )
             => ( member551035911493665803st_a_b @ P @ ( darcs_3945445989234600971st_a_b @ X ) ) ) ) ) ) ) ).

% wf_dtree.arc_in_subtree_if_tail_in_subtree
thf(fact_733_wf__dtree_Oarc__in__subtree__if__tail__in__subtree,axiom,
    ! [T: dtree_4875458026974816597st_a_b,Dt: dtree_list_a_b > list_a,P: dtree_list_a_b,X: dtree_4875458026974816597st_a_b,R2: list_a,Xs2: fset_P6340920821931359400st_a_b,E: dtree_list_a_b] :
      ( ( wf_dtr4469746621066521083st_a_b @ T )
     => ( ( member_list_a @ ( dtail_3679034105612296090st_a_b @ T @ Dt @ P ) @ ( dverts3100150328211173758st_a_b @ X ) )
       => ( ( member551035911493665803st_a_b @ P @ ( darcs_1080543555413674703st_a_b @ T ) )
         => ( ( T
              = ( node_l8355664270593351482st_a_b @ R2 @ Xs2 ) )
           => ( ( member8722885590174353295st_a_b @ ( produc4942817220686395122st_a_b @ X @ E ) @ ( fset_P623098064157935817st_a_b @ Xs2 ) )
             => ( member551035911493665803st_a_b @ P @ ( darcs_1080543555413674703st_a_b @ X ) ) ) ) ) ) ) ).

% wf_dtree.arc_in_subtree_if_tail_in_subtree
thf(fact_734_wf__dtree_Oarc__in__subtree__if__tail__in__subtree,axiom,
    ! [T: dtree_8284992029673318451list_a,Dt: list_a > dtree_list_a_b,P: list_a,X: dtree_8284992029673318451list_a,R2: dtree_list_a_b,Xs2: fset_P5658613431985659038list_a,E: list_a] :
      ( ( wf_dtr8287859687986785809list_a @ T )
     => ( ( member551035911493665803st_a_b @ ( dtail_7497147172532560816list_a @ T @ Dt @ P ) @ ( dverts6918263395131438484list_a @ X ) )
       => ( ( member_list_a @ P @ ( darcs_4898656622333939429list_a @ T ) )
         => ( ( T
              = ( node_d2950405300658840400list_a @ R2 @ Xs2 ) )
           => ( ( member2153700486347808993list_a @ ( produc3720128825147924392list_a @ X @ E ) @ ( fset_P5083850707481761831list_a @ Xs2 ) )
             => ( member_list_a @ P @ ( darcs_4898656622333939429list_a @ X ) ) ) ) ) ) ) ).

% wf_dtree.arc_in_subtree_if_tail_in_subtree
thf(fact_735_wf__dtree_Oarc__in__subtree__if__tail__in__subtree,axiom,
    ! [T: dtree_310814809473010911b_real,Dt: real > produc6499617310964463488_a_b_b,P: real,X: dtree_310814809473010911b_real,R2: produc6499617310964463488_a_b_b,Xs2: fset_P1597832662759650386l_real,E: real] :
      ( ( wf_dtr8634579813312766059b_real @ T )
     => ( ( member4695696432722591383_a_b_b @ ( dtail_6110999216585821196b_real @ T @ Dt @ P ) @ ( dverts8626872187801812264b_real @ X ) )
       => ( ( member_real @ P @ ( darcs_3291380776206734999b_real @ T ) )
         => ( ( T
              = ( node_P8407631645849338988b_real @ R2 @ Xs2 ) )
           => ( ( member4680567731829293973l_real @ ( produc4041975618813999652l_real @ X @ E ) @ ( fset_P3280430075840339931l_real @ Xs2 ) )
             => ( member_real @ P @ ( darcs_3291380776206734999b_real @ X ) ) ) ) ) ) ) ).

% wf_dtree.arc_in_subtree_if_tail_in_subtree
thf(fact_736_mdeg__eq__child__if__singleton__gt1,axiom,
    ! [R2: list_a,T12: dtree_list_a_b,E1: b] :
      ( ( ord_less_nat @ one_one_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
     => ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) )
        = ( max_deg_list_a_b @ T12 ) ) ) ).

% mdeg_eq_child_if_singleton_gt1
thf(fact_737_in__measure,axiom,
    ! [X: list_a,Y: list_a,F: list_a > nat] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( measure_list_a @ F ) )
      = ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).

% in_measure
thf(fact_738_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M: nat,N: nat] :
          ( ( ord_less_eq_nat @ M @ N )
          & ( M != N ) ) ) ) ).

% nat_less_le
thf(fact_739_less__imp__le__nat,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% less_imp_le_nat
thf(fact_740_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M: nat,N: nat] :
          ( ( ord_less_nat @ M @ N )
          | ( M = N ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_741_less__or__eq__imp__le,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( ord_less_nat @ M2 @ N2 )
        | ( M2 = N2 ) )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% less_or_eq_imp_le
thf(fact_742_le__neq__implies__less,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( M2 != N2 )
       => ( ord_less_nat @ M2 @ N2 ) ) ) ).

% le_neq_implies_less
thf(fact_743_not__psubset__empty,axiom,
    ! [A3: set_nat] :
      ~ ( ord_less_set_nat @ A3 @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_744_not__pfsubset__fempty,axiom,
    ! [A3: fset_P2153231429829016240_a_b_b] :
      ~ ( ord_le6631730213922513156_a_b_b @ A3 @ bot_bo2248824169281960260_a_b_b ) ).

% not_pfsubset_fempty
thf(fact_745_le__refl,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).

% le_refl
thf(fact_746_le__trans,axiom,
    ! [I3: nat,J2: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I3 @ J2 )
     => ( ( ord_less_eq_nat @ J2 @ K2 )
       => ( ord_less_eq_nat @ I3 @ K2 ) ) ) ).

% le_trans
thf(fact_747_eq__imp__le,axiom,
    ! [M2: nat,N2: nat] :
      ( ( M2 = N2 )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% eq_imp_le
thf(fact_748_le__antisym,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( M2 = N2 ) ) ) ).

% le_antisym
thf(fact_749_nat__le__linear,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
      | ( ord_less_eq_nat @ N2 @ M2 ) ) ).

% nat_le_linear
thf(fact_750_Nat_Oex__has__greatest__nat,axiom,
    ! [P3: nat > $o,K2: nat,B: nat] :
      ( ( P3 @ K2 )
     => ( ! [Y4: nat] :
            ( ( P3 @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ? [X3: nat] :
            ( ( P3 @ X3 )
            & ! [Y6: nat] :
                ( ( P3 @ Y6 )
               => ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_751_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I3: nat,J2: nat] :
      ( ! [I4: nat,J3: nat] :
          ( ( ord_less_nat @ I4 @ J3 )
         => ( ord_less_nat @ ( F @ I4 ) @ ( F @ J3 ) ) )
     => ( ( ord_less_eq_nat @ I3 @ J2 )
       => ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J2 ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_752_size__combine__decr,axiom,
    ! [R2: list_a,T12: dtree_list_a_b,E1: b] : ( ord_less_nat @ ( size_s415192292648992904st_a_b @ ( node_list_a_b @ ( append_a @ R2 @ ( root_list_a_b @ T12 ) ) @ ( sucs_list_a_b @ T12 ) ) ) @ ( size_s415192292648992904st_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).

% size_combine_decr
thf(fact_753_complete__interval,axiom,
    ! [A: real,B: real,P3: real > $o] :
      ( ( ord_less_real @ A @ B )
     => ( ( P3 @ A )
       => ( ~ ( P3 @ B )
         => ? [C2: real] :
              ( ( ord_less_eq_real @ A @ C2 )
              & ( ord_less_eq_real @ C2 @ B )
              & ! [X5: real] :
                  ( ( ( ord_less_eq_real @ A @ X5 )
                    & ( ord_less_real @ X5 @ C2 ) )
                 => ( P3 @ X5 ) )
              & ! [D3: real] :
                  ( ! [X3: real] :
                      ( ( ( ord_less_eq_real @ A @ X3 )
                        & ( ord_less_real @ X3 @ D3 ) )
                     => ( P3 @ X3 ) )
                 => ( ord_less_eq_real @ D3 @ C2 ) ) ) ) ) ) ).

% complete_interval
thf(fact_754_complete__interval,axiom,
    ! [A: nat,B: nat,P3: nat > $o] :
      ( ( ord_less_nat @ A @ B )
     => ( ( P3 @ A )
       => ( ~ ( P3 @ B )
         => ? [C2: nat] :
              ( ( ord_less_eq_nat @ A @ C2 )
              & ( ord_less_eq_nat @ C2 @ B )
              & ! [X5: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X5 )
                    & ( ord_less_nat @ X5 @ C2 ) )
                 => ( P3 @ X5 ) )
              & ! [D3: nat] :
                  ( ! [X3: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X3 )
                        & ( ord_less_nat @ X3 @ D3 ) )
                     => ( P3 @ X3 ) )
                 => ( ord_less_eq_nat @ D3 @ C2 ) ) ) ) ) ) ).

% complete_interval
thf(fact_755_less__set__def,axiom,
    ( ord_less_set_list_a
    = ( ^ [A6: set_list_a,B6: set_list_a] :
          ( ord_less_list_a_o
          @ ^ [X4: list_a] : ( member_list_a @ X4 @ A6 )
          @ ^ [X4: list_a] : ( member_list_a @ X4 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_756_less__set__def,axiom,
    ( ord_le7768603707070775306_a_b_b
    = ( ^ [A6: set_Pr3443975907877334966_a_b_b,B6: set_Pr3443975907877334966_a_b_b] :
          ( ord_le5065067911184022267_b_b_o
          @ ^ [X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ A6 )
          @ ^ [X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_757_less__set__def,axiom,
    ( ord_le4535551246020252542st_a_b
    = ( ^ [A6: set_dtree_list_a_b,B6: set_dtree_list_a_b] :
          ( ord_le3461360805275315335_a_b_o
          @ ^ [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ A6 )
          @ ^ [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_758_less__set__def,axiom,
    ( ord_less_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( ord_less_real_o
          @ ^ [X4: real] : ( member_real @ X4 @ A6 )
          @ ^ [X4: real] : ( member_real @ X4 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_759_psubsetD,axiom,
    ! [A3: set_list_a,B5: set_list_a,C: list_a] :
      ( ( ord_less_set_list_a @ A3 @ B5 )
     => ( ( member_list_a @ C @ A3 )
       => ( member_list_a @ C @ B5 ) ) ) ).

% psubsetD
thf(fact_760_psubsetD,axiom,
    ! [A3: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b,C: produc6499617310964463488_a_b_b] :
      ( ( ord_le7768603707070775306_a_b_b @ A3 @ B5 )
     => ( ( member4695696432722591383_a_b_b @ C @ A3 )
       => ( member4695696432722591383_a_b_b @ C @ B5 ) ) ) ).

% psubsetD
thf(fact_761_psubsetD,axiom,
    ! [A3: set_dtree_list_a_b,B5: set_dtree_list_a_b,C: dtree_list_a_b] :
      ( ( ord_le4535551246020252542st_a_b @ A3 @ B5 )
     => ( ( member551035911493665803st_a_b @ C @ A3 )
       => ( member551035911493665803st_a_b @ C @ B5 ) ) ) ).

% psubsetD
thf(fact_762_psubsetD,axiom,
    ! [A3: set_real,B5: set_real,C: real] :
      ( ( ord_less_set_real @ A3 @ B5 )
     => ( ( member_real @ C @ A3 )
       => ( member_real @ C @ B5 ) ) ) ).

% psubsetD
thf(fact_763_ranked__dtree_Omerge1_Ocases,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,X: dtree_list_a_b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ~ ! [R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
            ( X
           != ( node_list_a_b @ R @ Xs ) ) ) ).

% ranked_dtree.merge1.cases
thf(fact_764_dtree__size__eq__root,axiom,
    ! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,R5: list_a] :
      ( ( size_s415192292648992904st_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
      = ( size_s415192292648992904st_a_b @ ( node_list_a_b @ R5 @ Xs2 ) ) ) ).

% dtree_size_eq_root
thf(fact_765_ranked__dtree_Oranked__dtree__rec,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E: b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( ( node_list_a_b @ R2 @ Xs2 )
          = T )
       => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
         => ( ranked_dtree_a_b @ X @ Cmp ) ) ) ) ).

% ranked_dtree.ranked_dtree_rec
thf(fact_766_ranked__dtree_Oranked__dtree__rec__suc,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,X: dtree_list_a_b,E: b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) )
       => ( ranked_dtree_a_b @ X @ Cmp ) ) ) ).

% ranked_dtree.ranked_dtree_rec_suc
thf(fact_767_size__le__if__child__subset,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,Ys: fset_P2153231429829016240_a_b_b,R2: list_a,V: list_a] :
      ( ( ord_le8870638447146015504_a_b_b @ Xs2 @ Ys )
     => ( ord_less_eq_nat @ ( size_s415192292648992904st_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) @ ( size_s415192292648992904st_a_b @ ( node_list_a_b @ V @ Ys ) ) ) ) ).

% size_le_if_child_subset
thf(fact_768_ranked__dtree_Onormalize1_Ocases,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,X: dtree_list_a_b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ! [R: list_a,T1: dtree_list_a_b,E2: b] :
            ( X
           != ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
       => ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
              ( ! [X5: produc6499617310964463488_a_b_b] :
                  ( Xs
                 != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
             => ! [R: list_a] :
                  ( X
                 != ( node_list_a_b @ R @ Xs ) ) ) ) ) ).

% ranked_dtree.normalize1.cases
thf(fact_769_dtree__size__decr__aux,axiom,
    ! [X: dtree_list_a_b,Y: b,Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
      ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ Y ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
     => ( ord_less_nat @ ( size_s415192292648992904st_a_b @ X ) @ ( size_s415192292648992904st_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ).

% dtree_size_decr_aux
thf(fact_770_ranked__dtree_Onormalize1_Opinduct,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Rank: list_a > real,A0: dtree_list_a_b,P3: dtree_list_a_b > $o] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( accp_dtree_list_a_b @ ( ranked37597841108724548el_a_b @ Rank ) @ A0 )
       => ( ! [R: list_a,T1: dtree_list_a_b,E2: b] :
              ( ( accp_dtree_list_a_b @ ( ranked37597841108724548el_a_b @ Rank ) @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
             => ( ( ~ ( ord_less_real @ ( Rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( Rank @ ( rev_a @ R ) ) )
                 => ( P3 @ T1 ) )
               => ( P3 @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) )
         => ( ! [Xs: fset_P2153231429829016240_a_b_b,R: list_a] :
                ( ( accp_dtree_list_a_b @ ( ranked37597841108724548el_a_b @ Rank ) @ ( node_list_a_b @ R @ Xs ) )
               => ( ! [X5: produc6499617310964463488_a_b_b] :
                      ( Xs
                     != ( finser2303212525150181944_a_b_b @ X5 @ bot_bo2248824169281960260_a_b_b ) )
                 => ( ! [Z5: produc6499617310964463488_a_b_b,X5: dtree_list_a_b,Y6: b] :
                        ( ( member4695696432722591383_a_b_b @ Z5 @ ( fset_P9138963618725001425_a_b_b @ Xs ) )
                       => ( ( ( produc7704165765595008946_a_b_b @ X5 @ Y6 )
                            = Z5 )
                         => ( P3 @ X5 ) ) )
                   => ( P3 @ ( node_list_a_b @ R @ Xs ) ) ) ) )
           => ( P3 @ A0 ) ) ) ) ) ).

% ranked_dtree.normalize1.pinduct
thf(fact_771_ranked__dtree_Onormalize__sorted__ranks,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,R2: list_a,T12: dtree_list_a_b,E1: b,Rank: list_a > real] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked_normalize_a_b @ Rank @ T ) )
       => ( ord_less_eq_real @ ( Rank @ ( rev_a @ R2 ) ) @ ( Rank @ ( rev_a @ ( root_list_a_b @ T12 ) ) ) ) ) ) ).

% ranked_dtree.normalize_sorted_ranks
thf(fact_772_ranked__dtree_Onormalize1_Osimps_I1_J,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Rank: list_a > real,T12: dtree_list_a_b,R2: list_a,E: b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( ( ord_less_real @ ( Rank @ ( rev_a @ ( root_list_a_b @ T12 ) ) ) @ ( Rank @ ( rev_a @ R2 ) ) )
         => ( ( ranked8905849569120154423e1_a_b @ Rank @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
            = ( node_list_a_b @ ( append_a @ R2 @ ( root_list_a_b @ T12 ) ) @ ( sucs_list_a_b @ T12 ) ) ) )
        & ( ~ ( ord_less_real @ ( Rank @ ( rev_a @ ( root_list_a_b @ T12 ) ) ) @ ( Rank @ ( rev_a @ R2 ) ) )
         => ( ( ranked8905849569120154423e1_a_b @ Rank @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
            = ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ( ranked8905849569120154423e1_a_b @ Rank @ T12 ) @ E ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ) ).

% ranked_dtree.normalize1.simps(1)
thf(fact_773_ranked__dtree_Onormalize1__dverts__contr__subtree,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,V: list_a,Rank: list_a > real,T12: dtree_list_a_b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( member_list_a @ V @ ( dverts_list_a_b @ ( ranked8905849569120154423e1_a_b @ Rank @ T12 ) ) )
       => ( ~ ( member_list_a @ V @ ( dverts_list_a_b @ T12 ) )
         => ? [V23: list_a,T23: dtree_list_a_b,E23: b] :
              ( ( is_subtree_list_a_b @ ( node_list_a_b @ V23 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T12 )
              & ( ( append_a @ V23 @ ( root_list_a_b @ T23 ) )
                = V )
              & ( ord_less_real @ ( Rank @ ( rev_a @ ( root_list_a_b @ T23 ) ) ) @ ( Rank @ ( rev_a @ V23 ) ) ) ) ) ) ) ).

% ranked_dtree.normalize1_dverts_contr_subtree
thf(fact_774_ranked__dtree_Onormalize1__dverts__app__contr,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,V: list_a,Rank: list_a > real,T12: dtree_list_a_b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( member_list_a @ V @ ( dverts_list_a_b @ ( ranked8905849569120154423e1_a_b @ Rank @ T12 ) ) )
       => ( ~ ( member_list_a @ V @ ( dverts_list_a_b @ T12 ) )
         => ? [X3: list_a] :
              ( ( member_list_a @ X3 @ ( dverts_list_a_b @ T12 ) )
              & ? [Xa: list_a] :
                  ( ( member_list_a @ Xa @ ( dverts_list_a_b @ T12 ) )
                  & ( ( append_a @ X3 @ Xa )
                    = V )
                  & ( ord_less_real @ ( Rank @ ( rev_a @ Xa ) ) @ ( Rank @ ( rev_a @ X3 ) ) ) ) ) ) ) ) ).

% ranked_dtree.normalize1_dverts_app_contr
thf(fact_775_ranked__dtree_Oroot__normalize1__eq2,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Xs2: fset_P2153231429829016240_a_b_b,Rank: list_a > real,R2: list_a] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ! [X3: produc6499617310964463488_a_b_b] :
            ( Xs2
           != ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
       => ( ( root_list_a_b @ ( ranked8905849569120154423e1_a_b @ Rank @ ( node_list_a_b @ R2 @ Xs2 ) ) )
          = R2 ) ) ) ).

% ranked_dtree.root_normalize1_eq2
thf(fact_776_ranked__dtree_Onormalize1_Opsimps_I1_J,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Rank: list_a > real,R2: list_a,T12: dtree_list_a_b,E: b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( accp_dtree_list_a_b @ ( ranked37597841108724548el_a_b @ Rank ) @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
       => ( ( ( ord_less_real @ ( Rank @ ( rev_a @ ( root_list_a_b @ T12 ) ) ) @ ( Rank @ ( rev_a @ R2 ) ) )
           => ( ( ranked8905849569120154423e1_a_b @ Rank @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
              = ( node_list_a_b @ ( append_a @ R2 @ ( root_list_a_b @ T12 ) ) @ ( sucs_list_a_b @ T12 ) ) ) )
          & ( ~ ( ord_less_real @ ( Rank @ ( rev_a @ ( root_list_a_b @ T12 ) ) ) @ ( Rank @ ( rev_a @ R2 ) ) )
           => ( ( ranked8905849569120154423e1_a_b @ Rank @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
              = ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ( ranked8905849569120154423e1_a_b @ Rank @ T12 ) @ E ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ) ) ).

% ranked_dtree.normalize1.psimps(1)
thf(fact_777_ranked__dtree_Onormalize1__uneq__if__contr,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,R12: list_a,T12: dtree_list_a_b,E1: b,T22: dtree_list_a_b,Rank: list_a > real] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( is_subtree_list_a_b @ ( node_list_a_b @ R12 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T22 )
       => ( ( ord_less_real @ ( Rank @ ( rev_a @ ( root_list_a_b @ T12 ) ) ) @ ( Rank @ ( rev_a @ R12 ) ) )
         => ( ( wf_darcs_list_a_b @ T22 )
           => ( T22
             != ( ranked8905849569120154423e1_a_b @ Rank @ T22 ) ) ) ) ) ) ).

% ranked_dtree.normalize1_uneq_if_contr
thf(fact_778_ranked__dtree_Ocontr__before__normalize1,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,V: list_a,T12: dtree_list_a_b,E1: b,Rank: list_a > real,T32: dtree_list_a_b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( is_subtree_list_a_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked8905849569120154423e1_a_b @ Rank @ T32 ) )
       => ( ( ord_less_real @ ( Rank @ ( rev_a @ ( root_list_a_b @ T12 ) ) ) @ ( Rank @ ( rev_a @ V ) ) )
         => ? [V3: list_a,T23: dtree_list_a_b,E23: b] :
              ( ( is_subtree_list_a_b @ ( node_list_a_b @ V3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T32 )
              & ( ord_less_real @ ( Rank @ ( rev_a @ ( root_list_a_b @ T23 ) ) ) @ ( Rank @ ( rev_a @ V3 ) ) ) ) ) ) ) ).

% ranked_dtree.contr_before_normalize1
thf(fact_779_ranked__dtree_Ocontr__if__normalize1__uneq,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Rank: list_a > real,T12: dtree_list_a_b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( ( ranked8905849569120154423e1_a_b @ Rank @ T12 )
         != T12 )
       => ? [V2: list_a,T23: dtree_list_a_b,E23: b] :
            ( ( is_subtree_list_a_b @ ( node_list_a_b @ V2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T12 )
            & ( ord_less_real @ ( Rank @ ( rev_a @ ( root_list_a_b @ T23 ) ) ) @ ( Rank @ ( rev_a @ V2 ) ) ) ) ) ) ).

% ranked_dtree.contr_if_normalize1_uneq
thf(fact_780_ranked__dtree_Oroot__normalize1__eq1,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Rank: list_a > real,T12: dtree_list_a_b,R2: list_a,E1: b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ~ ( ord_less_real @ ( Rank @ ( rev_a @ ( root_list_a_b @ T12 ) ) ) @ ( Rank @ ( rev_a @ R2 ) ) )
       => ( ( root_list_a_b @ ( ranked8905849569120154423e1_a_b @ Rank @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
          = R2 ) ) ) ).

% ranked_dtree.root_normalize1_eq1
thf(fact_781_ranked__dtree_Osorted__ranks__if__normalize1__eq,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,T22: dtree_list_a_b,R12: list_a,T12: dtree_list_a_b,E1: b,Rank: list_a > real] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( wf_darcs_list_a_b @ T22 )
       => ( ( is_subtree_list_a_b @ ( node_list_a_b @ R12 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T22 )
         => ( ( T22
              = ( ranked8905849569120154423e1_a_b @ Rank @ T22 ) )
           => ( ord_less_eq_real @ ( Rank @ ( rev_a @ R12 ) ) @ ( Rank @ ( rev_a @ ( root_list_a_b @ T12 ) ) ) ) ) ) ) ) ).

% ranked_dtree.sorted_ranks_if_normalize1_eq
thf(fact_782_ranked__dtree_Oroot__normalize1__eq1_H,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Rank: list_a > real,T12: dtree_list_a_b,R2: list_a,E1: b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ~ ( ord_less_eq_real @ ( Rank @ ( rev_a @ ( root_list_a_b @ T12 ) ) ) @ ( Rank @ ( rev_a @ R2 ) ) )
       => ( ( root_list_a_b @ ( ranked8905849569120154423e1_a_b @ Rank @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
          = R2 ) ) ) ).

% ranked_dtree.root_normalize1_eq1'
thf(fact_783_ranked__dtree_Omerge__single__root__sucs,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,T22: dtree_list_a_b,E22: b,Rank: list_a > real,T12: dtree_list_a_b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ ( ranked_merge_a_b @ Rank @ Cmp @ T12 ) ) ) )
       => ( ( ranked_merge_a_b @ Rank @ Cmp @ T12 )
          = ( node_list_a_b @ ( root_list_a_b @ T12 ) @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ).

% ranked_dtree.merge_single_root_sucs
thf(fact_784_ranked__dtree_Omerge__single__root,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,T22: dtree_list_a_b,E22: b,Rank: list_a > real,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ ( ranked_merge_a_b @ Rank @ Cmp @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) )
       => ( ( ranked_merge_a_b @ Rank @ Cmp @ ( node_list_a_b @ R2 @ Xs2 ) )
          = ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ).

% ranked_dtree.merge_single_root
thf(fact_785_fcard__finsert__le,axiom,
    ! [A3: fset_P2153231429829016240_a_b_b,X: produc6499617310964463488_a_b_b] : ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ A3 ) @ ( fcard_4742106318756258927_a_b_b @ ( finser2303212525150181944_a_b_b @ X @ A3 ) ) ) ).

% fcard_finsert_le
thf(fact_786_mdeg__ge__fcard,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] : ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ).

% mdeg_ge_fcard
thf(fact_787_mdeg__eq__fcard__if__empty,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
      ( ( Xs2 = bot_bo2248824169281960260_a_b_b )
     => ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
        = ( fcard_4742106318756258927_a_b_b @ Xs2 ) ) ) ).

% mdeg_eq_fcard_if_empty
thf(fact_788_num__leaves__ge__card,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] : ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) @ ( num_leaves_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ).

% num_leaves_ge_card
thf(fact_789_fcard1__if__le1__elem,axiom,
    ! [Xs2: fset_list_a,X: list_a] :
      ( ( ord_less_eq_nat @ ( fcard_list_a @ Xs2 ) @ one_one_nat )
     => ( ( member_list_a @ X @ ( fset_list_a2 @ Xs2 ) )
       => ( ( fcard_list_a @ Xs2 )
          = one_one_nat ) ) ) ).

% fcard1_if_le1_elem
thf(fact_790_fcard1__if__le1__elem,axiom,
    ! [Xs2: fset_dtree_list_a_b,X: dtree_list_a_b] :
      ( ( ord_less_eq_nat @ ( fcard_dtree_list_a_b @ Xs2 ) @ one_one_nat )
     => ( ( member551035911493665803st_a_b @ X @ ( fset_dtree_list_a_b2 @ Xs2 ) )
       => ( ( fcard_dtree_list_a_b @ Xs2 )
          = one_one_nat ) ) ) ).

% fcard1_if_le1_elem
thf(fact_791_fcard1__if__le1__elem,axiom,
    ! [Xs2: fset_real,X: real] :
      ( ( ord_less_eq_nat @ ( fcard_real @ Xs2 ) @ one_one_nat )
     => ( ( member_real @ X @ ( fset_real2 @ Xs2 ) )
       => ( ( fcard_real @ Xs2 )
          = one_one_nat ) ) ) ).

% fcard1_if_le1_elem
thf(fact_792_fcard1__if__le1__elem,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,X: produc6499617310964463488_a_b_b] :
      ( ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) @ one_one_nat )
     => ( ( member4695696432722591383_a_b_b @ X @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
       => ( ( fcard_4742106318756258927_a_b_b @ Xs2 )
          = one_one_nat ) ) ) ).

% fcard1_if_le1_elem
thf(fact_793_fcard__single__1__iff,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b] :
      ( ( ( fcard_4742106318756258927_a_b_b @ Xs2 )
        = one_one_nat )
      = ( ? [X4: produc6499617310964463488_a_b_b] :
            ( Xs2
            = ( finser2303212525150181944_a_b_b @ X4 @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).

% fcard_single_1_iff
thf(fact_794_fcard__single__1,axiom,
    ! [X: produc6499617310964463488_a_b_b] :
      ( ( fcard_4742106318756258927_a_b_b @ ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) )
      = one_one_nat ) ).

% fcard_single_1
thf(fact_795_ranked__dtree_Omerge__empty,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Rank: list_a > real,R2: list_a] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( ranked_merge_a_b @ Rank @ Cmp @ ( node_list_a_b @ R2 @ bot_bo2248824169281960260_a_b_b ) )
        = ( node_list_a_b @ R2 @ bot_bo2248824169281960260_a_b_b ) ) ) ).

% ranked_dtree.merge_empty
thf(fact_796_nempty__if__mdeg__gt__fcard,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
      ( ( ord_less_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
     => ( Xs2 != bot_bo2248824169281960260_a_b_b ) ) ).

% nempty_if_mdeg_gt_fcard
thf(fact_797_empty__fset__if__fcard__le1__not__singleton,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b] :
      ( ! [X3: produc6499617310964463488_a_b_b] :
          ( Xs2
         != ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
     => ( ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) @ one_one_nat )
       => ( Xs2 = bot_bo2248824169281960260_a_b_b ) ) ) ).

% empty_fset_if_fcard_le1_not_singleton
thf(fact_798_singleton__if__fcard__le1__elem,axiom,
    ! [Xs2: fset_list_a,X: list_a] :
      ( ( ord_less_eq_nat @ ( fcard_list_a @ Xs2 ) @ one_one_nat )
     => ( ( member_list_a @ X @ ( fset_list_a2 @ Xs2 ) )
       => ( Xs2
          = ( finsert_list_a @ X @ bot_bot_fset_list_a ) ) ) ) ).

% singleton_if_fcard_le1_elem
thf(fact_799_singleton__if__fcard__le1__elem,axiom,
    ! [Xs2: fset_dtree_list_a_b,X: dtree_list_a_b] :
      ( ( ord_less_eq_nat @ ( fcard_dtree_list_a_b @ Xs2 ) @ one_one_nat )
     => ( ( member551035911493665803st_a_b @ X @ ( fset_dtree_list_a_b2 @ Xs2 ) )
       => ( Xs2
          = ( finser8636396436308191404st_a_b @ X @ bot_bo4748119319284029112st_a_b ) ) ) ) ).

% singleton_if_fcard_le1_elem
thf(fact_800_singleton__if__fcard__le1__elem,axiom,
    ! [Xs2: fset_real,X: real] :
      ( ( ord_less_eq_nat @ ( fcard_real @ Xs2 ) @ one_one_nat )
     => ( ( member_real @ X @ ( fset_real2 @ Xs2 ) )
       => ( Xs2
          = ( finsert_real @ X @ bot_bot_fset_real ) ) ) ) ).

% singleton_if_fcard_le1_elem
thf(fact_801_singleton__if__fcard__le1__elem,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,X: produc6499617310964463488_a_b_b] :
      ( ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) @ one_one_nat )
     => ( ( member4695696432722591383_a_b_b @ X @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
       => ( Xs2
          = ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).

% singleton_if_fcard_le1_elem
thf(fact_802_ranked__dtree_Omerge__cases,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Rank: list_a > real,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ? [T2: dtree_list_a_b,E2: b] :
          ( ( ( ranked_merge_a_b @ Rank @ Cmp @ ( node_list_a_b @ R2 @ Xs2 ) )
            = ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
          | ( ( ranked_merge_a_b @ Rank @ Cmp @ ( node_list_a_b @ R2 @ Xs2 ) )
            = ( node_list_a_b @ R2 @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).

% ranked_dtree.merge_cases
thf(fact_803_ranked__dtree_Omerge__empty__sucs,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,T12: dtree_list_a_b,Rank: list_a > real] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( ( sucs_list_a_b @ T12 )
          = bot_bo2248824169281960260_a_b_b )
       => ( ( ranked_merge_a_b @ Rank @ Cmp @ T12 )
          = ( node_list_a_b @ ( root_list_a_b @ T12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).

% ranked_dtree.merge_empty_sucs
thf(fact_804_ranked__dtree_Omerge__root__child__eq,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Rank: list_a > real,R2: list_a,T12: dtree_list_a_b,E1: b,T22: dtree_list_a_b,E22: b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( ( ranked_merge_a_b @ Rank @ Cmp @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) )
          = ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ bot_bo2248824169281960260_a_b_b ) ) )
       => ( ( root_list_a_b @ T12 )
          = ( root_list_a_b @ T22 ) ) ) ) ).

% ranked_dtree.merge_root_child_eq
thf(fact_805_ranked__dtree_Omerge__cases__sucs,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Rank: list_a > real,T12: dtree_list_a_b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ? [T2: dtree_list_a_b,E2: b] :
          ( ( ( ranked_merge_a_b @ Rank @ Cmp @ T12 )
            = ( node_list_a_b @ ( root_list_a_b @ T12 ) @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
          | ( ( ranked_merge_a_b @ Rank @ Cmp @ T12 )
            = ( node_list_a_b @ ( root_list_a_b @ T12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).

% ranked_dtree.merge_cases_sucs
thf(fact_806_ranked__dtree_Omerge__strict__subtree__nocontr__sucs2,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,T22: dtree_list_a_b,Rank: list_a > real,R12: list_a,T12: dtree_list_a_b,E1: b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ! [R13: list_a,T1: dtree_list_a_b,E12: b] :
            ( ( is_subtree_list_a_b @ ( node_list_a_b @ R13 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T22 )
           => ( ord_less_eq_real @ ( Rank @ ( rev_a @ R13 ) ) @ ( Rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) ) )
       => ( ( strict8995144569104247066st_a_b @ ( node_list_a_b @ R12 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked_merge_a_b @ Rank @ Cmp @ T22 ) )
         => ( ord_less_eq_real @ ( Rank @ ( rev_a @ R12 ) ) @ ( Rank @ ( rev_a @ ( root_list_a_b @ T12 ) ) ) ) ) ) ) ).

% ranked_dtree.merge_strict_subtree_nocontr_sucs2
thf(fact_807_ranked__dtree_Omerge__strict__subtree__nocontr2,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,Rank: list_a > real,R12: list_a,T12: dtree_list_a_b,E1: b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ! [R13: list_a,T1: dtree_list_a_b,E12: b] :
            ( ( is_subtree_list_a_b @ ( node_list_a_b @ R13 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ R2 @ Xs2 ) )
           => ( ord_less_eq_real @ ( Rank @ ( rev_a @ R13 ) ) @ ( Rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) ) )
       => ( ( strict8995144569104247066st_a_b @ ( node_list_a_b @ R12 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked_merge_a_b @ Rank @ Cmp @ ( node_list_a_b @ R2 @ Xs2 ) ) )
         => ( ord_less_eq_real @ ( Rank @ ( rev_a @ R12 ) ) @ ( Rank @ ( rev_a @ ( root_list_a_b @ T12 ) ) ) ) ) ) ) ).

% ranked_dtree.merge_strict_subtree_nocontr2
thf(fact_808_fcard0__if__mdeg__le1__not__single__suc,axiom,
    ! [T: dtree_list_a_b] :
      ( ! [X3: produc6499617310964463488_a_b_b] :
          ( ( sucs_list_a_b @ T )
         != ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
     => ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T ) @ one_one_nat )
       => ( ( fcard_4742106318756258927_a_b_b @ ( sucs_list_a_b @ T ) )
          = zero_zero_nat ) ) ) ).

% fcard0_if_mdeg_le1_not_single_suc
thf(fact_809_fcard0__if__mdeg__le1__not__single,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
      ( ! [X3: produc6499617310964463488_a_b_b] :
          ( Xs2
         != ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
     => ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) @ one_one_nat )
       => ( ( fcard_4742106318756258927_a_b_b @ Xs2 )
          = zero_zero_nat ) ) ) ).

% fcard0_if_mdeg_le1_not_single
thf(fact_810_le0,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% le0
thf(fact_811_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_812_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_813_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_814_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_815_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% less_eq_nat.simps(1)
thf(fact_816_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_817_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_818_le__0__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_819_ex__least__nat__le,axiom,
    ! [P3: nat > $o,N2: nat] :
      ( ( P3 @ N2 )
     => ( ~ ( P3 @ zero_zero_nat )
       => ? [K: nat] :
            ( ( ord_less_eq_nat @ K @ N2 )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ~ ( P3 @ I2 ) )
            & ( P3 @ K ) ) ) ) ).

% ex_least_nat_le
thf(fact_820_fcard__not0__if__elem,axiom,
    ! [Xs2: fset_list_a] :
      ( ? [X5: list_a] : ( member_list_a @ X5 @ ( fset_list_a2 @ Xs2 ) )
     => ( ( fcard_list_a @ Xs2 )
       != zero_zero_nat ) ) ).

% fcard_not0_if_elem
thf(fact_821_fcard__not0__if__elem,axiom,
    ! [Xs2: fset_dtree_list_a_b] :
      ( ? [X5: dtree_list_a_b] : ( member551035911493665803st_a_b @ X5 @ ( fset_dtree_list_a_b2 @ Xs2 ) )
     => ( ( fcard_dtree_list_a_b @ Xs2 )
       != zero_zero_nat ) ) ).

% fcard_not0_if_elem
thf(fact_822_fcard__not0__if__elem,axiom,
    ! [Xs2: fset_real] :
      ( ? [X5: real] : ( member_real @ X5 @ ( fset_real2 @ Xs2 ) )
     => ( ( fcard_real @ Xs2 )
       != zero_zero_nat ) ) ).

% fcard_not0_if_elem
thf(fact_823_fcard__not0__if__elem,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b] :
      ( ? [X5: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X5 @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
     => ( ( fcard_4742106318756258927_a_b_b @ Xs2 )
       != zero_zero_nat ) ) ).

% fcard_not0_if_elem
thf(fact_824_fcard__fempty,axiom,
    ( ( fcard_4742106318756258927_a_b_b @ bot_bo2248824169281960260_a_b_b )
    = zero_zero_nat ) ).

% fcard_fempty
thf(fact_825_strict__subtree__subset,axiom,
    ! [T: dtree_list_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,Ys: fset_P2153231429829016240_a_b_b] :
      ( ( strict8995144569104247066st_a_b @ T @ ( node_list_a_b @ R2 @ Xs2 ) )
     => ( ( ord_le8870638447146015504_a_b_b @ Xs2 @ Ys )
       => ( strict8995144569104247066st_a_b @ T @ ( node_list_a_b @ R2 @ Ys ) ) ) ) ).

% strict_subtree_subset
thf(fact_826_empty__if__mdeg__0,axiom,
    ! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
      ( ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
        = zero_zero_nat )
     => ( Xs2 = bot_bo2248824169281960260_a_b_b ) ) ).

% empty_if_mdeg_0
thf(fact_827_empty__iff__mdeg__0,axiom,
    ! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
      ( ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
        = zero_zero_nat )
      = ( Xs2 = bot_bo2248824169281960260_a_b_b ) ) ).

% empty_iff_mdeg_0
thf(fact_828_nempty__if__mdeg__n0,axiom,
    ! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
      ( ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
       != zero_zero_nat )
     => ( Xs2 != bot_bo2248824169281960260_a_b_b ) ) ).

% nempty_if_mdeg_n0
thf(fact_829_mdeg0__iff__fcard0,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
      ( ( ( fcard_4742106318756258927_a_b_b @ Xs2 )
        = zero_zero_nat )
      = ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
        = zero_zero_nat ) ) ).

% mdeg0_iff_fcard0
thf(fact_830_mdeg0__if__fcard0,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
      ( ( ( fcard_4742106318756258927_a_b_b @ Xs2 )
        = zero_zero_nat )
     => ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
        = zero_zero_nat ) ) ).

% mdeg0_if_fcard0
thf(fact_831_mdeg__gt__0__if__nempty,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
      ( ( Xs2 != bot_bo2248824169281960260_a_b_b )
     => ( ord_less_nat @ zero_zero_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ).

% mdeg_gt_0_if_nempty
thf(fact_832_fcard0__if__le1__not__singleton,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b] :
      ( ! [X3: produc6499617310964463488_a_b_b] :
          ( Xs2
         != ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
     => ( ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) @ one_one_nat )
       => ( ( fcard_4742106318756258927_a_b_b @ Xs2 )
          = zero_zero_nat ) ) ) ).

% fcard0_if_le1_not_singleton
thf(fact_833_le__zero__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_834_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_real @ zero_zero_real @ one_one_real ).

% zero_less_one_class.zero_le_one
thf(fact_835_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one_class.zero_le_one
thf(fact_836_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_real @ zero_zero_real @ one_one_real ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_837_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_838_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_839_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_840_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_841_ranked__dtree_Omerge__singleton__sucs,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,T12: dtree_list_a_b,Rank: list_a > real] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( list_list_dtree_a_b @ ( node_list_a_b @ ( root_list_a_b @ T12 ) @ ( sucs_list_a_b @ T12 ) ) )
       => ( ( ( sucs_list_a_b @ T12 )
           != bot_bo2248824169281960260_a_b_b )
         => ? [T2: dtree_list_a_b,E2: b] :
              ( ( ranked_merge_a_b @ Rank @ Cmp @ T12 )
              = ( node_list_a_b @ ( root_list_a_b @ T12 ) @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ) ).

% ranked_dtree.merge_singleton_sucs
thf(fact_842_ranked__dtree_Omerge1__subtree__if__fcard__gt1,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,Rank: list_a > real,T12: dtree_list_a_b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ ( ranked_merge1_a_b @ Rank @ Cmp @ T12 ) )
       => ( ( ord_less_nat @ one_one_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) )
         => ? [Ys2: fset_P2153231429829016240_a_b_b] :
              ( ( ( ranked_merge1_a_b @ Rank @ Cmp @ ( node_list_a_b @ R2 @ Ys2 ) )
                = ( node_list_a_b @ R2 @ Xs2 ) )
              & ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Ys2 ) @ T12 )
              & ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) @ ( fcard_4742106318756258927_a_b_b @ Ys2 ) ) ) ) ) ) ).

% ranked_dtree.merge1_subtree_if_fcard_gt1
thf(fact_843_ranked__dtree_Omerge1__elem__in__img__if__fcard__gt1,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Rank: list_a > real,R2: list_a,Ys: fset_P2153231429829016240_a_b_b,Xs2: fset_P2153231429829016240_a_b_b,T22: dtree_list_a_b,E22: b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( ( ranked_merge1_a_b @ Rank @ Cmp @ ( node_list_a_b @ R2 @ Ys ) )
          = ( node_list_a_b @ R2 @ Xs2 ) )
       => ( ( ord_less_nat @ one_one_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) )
         => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
           => ? [T1: dtree_list_a_b] :
                ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Ys ) )
                & ( ( ranked_merge1_a_b @ Rank @ Cmp @ T1 )
                  = T22 ) ) ) ) ) ) ).

% ranked_dtree.merge1_elem_in_img_if_fcard_gt1
thf(fact_844_list__dtree__subset,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,Ys: fset_P2153231429829016240_a_b_b,R2: list_a] :
      ( ( ord_le8870638447146015504_a_b_b @ Xs2 @ Ys )
     => ( ( list_list_dtree_a_b @ ( node_list_a_b @ R2 @ Ys ) )
       => ( list_list_dtree_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ).

% list_dtree_subset
thf(fact_845_ranked__dtree_Omerge1__dverts__sub,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Rank: list_a > real,T12: dtree_list_a_b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ord_le8861187494160871172list_a @ ( dverts_list_a_b @ ( ranked_merge1_a_b @ Rank @ Cmp @ T12 ) ) @ ( dverts_list_a_b @ T12 ) ) ) ).

% ranked_dtree.merge1_dverts_sub
thf(fact_846_ranked__dtree_Omerge1__dverts__eq,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Rank: list_a > real] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( dverts_list_a_b @ ( ranked_merge1_a_b @ Rank @ Cmp @ T ) )
        = ( dverts_list_a_b @ T ) ) ) ).

% ranked_dtree.merge1_dverts_eq
thf(fact_847_list__dtree_Olist__dtree__rec,axiom,
    ! [T: dtree_list_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E: b] :
      ( ( list_list_dtree_a_b @ T )
     => ( ( ( node_list_a_b @ R2 @ Xs2 )
          = T )
       => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
         => ( list_list_dtree_a_b @ X ) ) ) ) ).

% list_dtree.list_dtree_rec
thf(fact_848_list__dtree_Olist__dtree__rec__suc,axiom,
    ! [T: dtree_list_a_b,X: dtree_list_a_b,E: b] :
      ( ( list_list_dtree_a_b @ T )
     => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) )
       => ( list_list_dtree_a_b @ X ) ) ) ).

% list_dtree.list_dtree_rec_suc
thf(fact_849_ranked__dtree_Osubtree__merge1__orig,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,Rank: list_a > real,T12: dtree_list_a_b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ ( ranked_merge1_a_b @ Rank @ Cmp @ T12 ) )
       => ? [Ys2: fset_P2153231429829016240_a_b_b] : ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Ys2 ) @ T12 ) ) ) ).

% ranked_dtree.subtree_merge1_orig
thf(fact_850_ranked__dtree_Omerge__empty__if__nwf,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,Rank: list_a > real] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ~ ( list_list_dtree_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
       => ( ( ranked_merge_a_b @ Rank @ Cmp @ ( node_list_a_b @ R2 @ Xs2 ) )
          = ( node_list_a_b @ R2 @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).

% ranked_dtree.merge_empty_if_nwf
thf(fact_851_ranked__dtree_Omerge__nempty__sucs,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,T12: dtree_list_a_b,Rank: list_a > real] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( list_list_dtree_a_b @ T12 )
       => ( ( ( sucs_list_a_b @ T12 )
           != bot_bo2248824169281960260_a_b_b )
         => ( ( sucs_list_a_b @ ( ranked_merge_a_b @ Rank @ Cmp @ T12 ) )
           != bot_bo2248824169281960260_a_b_b ) ) ) ) ).

% ranked_dtree.merge_nempty_sucs
thf(fact_852_ranked__dtree_Odverts__if__subtree__merge1,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,Rank: list_a > real,T12: dtree_list_a_b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ ( ranked_merge1_a_b @ Rank @ Cmp @ T12 ) )
       => ( member_list_a @ R2 @ ( dverts_list_a_b @ T12 ) ) ) ) ).

% ranked_dtree.dverts_if_subtree_merge1
thf(fact_853_ranked__dtree_Omerge__empty__if__nwf__sucs,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,T12: dtree_list_a_b,Rank: list_a > real] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ~ ( list_list_dtree_a_b @ T12 )
       => ( ( ranked_merge_a_b @ Rank @ Cmp @ T12 )
          = ( node_list_a_b @ ( root_list_a_b @ T12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).

% ranked_dtree.merge_empty_if_nwf_sucs
thf(fact_854_ranked__dtree_Omerge__nempty,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,Rank: list_a > real] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( list_list_dtree_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
       => ( ( Xs2 != bot_bo2248824169281960260_a_b_b )
         => ( ( sucs_list_a_b @ ( ranked_merge_a_b @ Rank @ Cmp @ ( node_list_a_b @ R2 @ Xs2 ) ) )
           != bot_bo2248824169281960260_a_b_b ) ) ) ) ).

% ranked_dtree.merge_nempty
thf(fact_855_ranked__dtree_Omerge1__child__in__orig,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Rank: list_a > real,R2: list_a,Ys: fset_P2153231429829016240_a_b_b,Xs2: fset_P2153231429829016240_a_b_b,T12: dtree_list_a_b,E1: b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( ( ranked_merge1_a_b @ Rank @ Cmp @ ( node_list_a_b @ R2 @ Ys ) )
          = ( node_list_a_b @ R2 @ Xs2 ) )
       => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
         => ? [T23: dtree_list_a_b] :
              ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Ys ) )
              & ( ( root_list_a_b @ T23 )
                = ( root_list_a_b @ T12 ) ) ) ) ) ) ).

% ranked_dtree.merge1_child_in_orig
thf(fact_856_ranked__dtree_Omerge1__fcard__le,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Rank: list_a > real,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ ( sucs_list_a_b @ ( ranked_merge1_a_b @ Rank @ Cmp @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) ) ) ).

% ranked_dtree.merge1_fcard_le
thf(fact_857_ranked__dtree_Omerge__disjoint__if__child,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Rank: list_a > real,R2: list_a,T12: dtree_list_a_b,E1: b,T22: dtree_list_a_b,E22: b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( ( ranked_merge_a_b @ Rank @ Cmp @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) )
          = ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ bot_bo2248824169281960260_a_b_b ) ) )
       => ( list_list_dtree_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ).

% ranked_dtree.merge_disjoint_if_child
thf(fact_858_ranked__dtree_Omerge__singleton,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,Rank: list_a > real] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( list_list_dtree_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
       => ( ( Xs2 != bot_bo2248824169281960260_a_b_b )
         => ? [T2: dtree_list_a_b,E2: b] :
              ( ( ranked_merge_a_b @ Rank @ Cmp @ ( node_list_a_b @ R2 @ Xs2 ) )
              = ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ) ).

% ranked_dtree.merge_singleton
thf(fact_859_ranked__dtree_Omerge1__subtree__if__mdeg__gt1,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,Rank: list_a > real,T12: dtree_list_a_b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ ( ranked_merge1_a_b @ Rank @ Cmp @ T12 ) )
       => ( ( ord_less_nat @ one_one_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
         => ? [Ys2: fset_P2153231429829016240_a_b_b] :
              ( ( ( ranked_merge1_a_b @ Rank @ Cmp @ ( node_list_a_b @ R2 @ Ys2 ) )
                = ( node_list_a_b @ R2 @ Xs2 ) )
              & ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Ys2 ) @ T12 ) ) ) ) ) ).

% ranked_dtree.merge1_subtree_if_mdeg_gt1
thf(fact_860_ranked__dtree_Omerge1__not__merge__if__fcard__gt1,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Rank: list_a > real,R2: list_a,Ys: fset_P2153231429829016240_a_b_b,Xs2: fset_P2153231429829016240_a_b_b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( ( ranked_merge1_a_b @ Rank @ Cmp @ ( node_list_a_b @ R2 @ Ys ) )
          = ( node_list_a_b @ R2 @ Xs2 ) )
       => ( ( ord_less_nat @ one_one_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) )
         => ( ( ranked_merge_a_b @ Rank @ Cmp @ ( node_list_a_b @ R2 @ Ys ) )
           != ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ) ).

% ranked_dtree.merge1_not_merge_if_fcard_gt1
thf(fact_861_ranked__dtree_Onot__single__subtree__if__nwf__sucs,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,T22: dtree_list_a_b,R12: list_a,T12: dtree_list_a_b,E1: b,Rank: list_a > real] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ~ ( list_list_dtree_a_b @ T22 )
       => ~ ( is_subtree_list_a_b @ ( node_list_a_b @ R12 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked_merge_a_b @ Rank @ Cmp @ T22 ) ) ) ) ).

% ranked_dtree.not_single_subtree_if_nwf_sucs
thf(fact_862_ranked__dtree_Onot__single__subtree__if__nwf,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,R12: list_a,T12: dtree_list_a_b,E1: b,Rank: list_a > real] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ~ ( list_list_dtree_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
       => ~ ( is_subtree_list_a_b @ ( node_list_a_b @ R12 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked_merge_a_b @ Rank @ Cmp @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ) ).

% ranked_dtree.not_single_subtree_if_nwf
thf(fact_863_less__eq__real__def,axiom,
    ( ord_less_eq_real
    = ( ^ [X4: real,Y3: real] :
          ( ( ord_less_real @ X4 @ Y3 )
          | ( X4 = Y3 ) ) ) ) ).

% less_eq_real_def
thf(fact_864_ranked__dtree_Omerge__strict__subtree__nocontr__sucs,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,T0: dtree_list_a_b,Rank: list_a > real,R12: list_a,T12: dtree_list_a_b,E1: b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ! [T23: dtree_list_a_b,R13: list_a,T1: dtree_list_a_b,E12: b] :
            ( ( member551035911493665803st_a_b @ T23 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T0 ) ) ) )
           => ( ( is_subtree_list_a_b @ ( node_list_a_b @ R13 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T23 )
             => ( ord_less_eq_real @ ( Rank @ ( rev_a @ R13 ) ) @ ( Rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) ) ) )
       => ( ( strict8995144569104247066st_a_b @ ( node_list_a_b @ R12 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked_merge_a_b @ Rank @ Cmp @ T0 ) )
         => ( ord_less_eq_real @ ( Rank @ ( rev_a @ R12 ) ) @ ( Rank @ ( rev_a @ ( root_list_a_b @ T12 ) ) ) ) ) ) ) ).

% ranked_dtree.merge_strict_subtree_nocontr_sucs
thf(fact_865_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z4 )
     => ~ ( ord_less_eq_real @ T @ X5 ) ) ).

% minf(8)
thf(fact_866_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z4 )
     => ~ ( ord_less_eq_nat @ T @ X5 ) ) ).

% minf(8)
thf(fact_867_image__eqI,axiom,
    ! [B: real,F: real > real,X: real,A3: set_real] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_real @ X @ A3 )
       => ( member_real @ B @ ( image_real_real @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_868_image__eqI,axiom,
    ! [B: real,F: list_a > real,X: list_a,A3: set_list_a] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_list_a @ X @ A3 )
       => ( member_real @ B @ ( image_list_a_real @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_869_image__eqI,axiom,
    ! [B: list_a,F: real > list_a,X: real,A3: set_real] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_real @ X @ A3 )
       => ( member_list_a @ B @ ( image_real_list_a @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_870_image__eqI,axiom,
    ! [B: list_a,F: list_a > list_a,X: list_a,A3: set_list_a] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_list_a @ X @ A3 )
       => ( member_list_a @ B @ ( image_list_a_list_a @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_871_image__eqI,axiom,
    ! [B: real,F: dtree_list_a_b > real,X: dtree_list_a_b,A3: set_dtree_list_a_b] :
      ( ( B
        = ( F @ X ) )
     => ( ( member551035911493665803st_a_b @ X @ A3 )
       => ( member_real @ B @ ( image_3232105638531924007b_real @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_872_image__eqI,axiom,
    ! [B: dtree_list_a_b,F: real > dtree_list_a_b,X: real,A3: set_real] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_real @ X @ A3 )
       => ( member551035911493665803st_a_b @ B @ ( image_3293520908065730343st_a_b @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_873_image__eqI,axiom,
    ! [B: dtree_list_a_b,F: list_a > dtree_list_a_b,X: list_a,A3: set_list_a] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_list_a @ X @ A3 )
       => ( member551035911493665803st_a_b @ B @ ( image_1909091145250899635st_a_b @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_874_image__eqI,axiom,
    ! [B: list_a,F: dtree_list_a_b > list_a,X: dtree_list_a_b,A3: set_dtree_list_a_b] :
      ( ( B
        = ( F @ X ) )
     => ( ( member551035911493665803st_a_b @ X @ A3 )
       => ( member_list_a @ B @ ( image_5727204212171164361list_a @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_875_image__eqI,axiom,
    ! [B: real,F: produc6499617310964463488_a_b_b > real,X: produc6499617310964463488_a_b_b,A3: set_Pr3443975907877334966_a_b_b] :
      ( ( B
        = ( F @ X ) )
     => ( ( member4695696432722591383_a_b_b @ X @ A3 )
       => ( member_real @ B @ ( image_3497077275575164339b_real @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_876_image__eqI,axiom,
    ! [B: produc6499617310964463488_a_b_b,F: real > produc6499617310964463488_a_b_b,X: real,A3: set_real] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_real @ X @ A3 )
       => ( member4695696432722591383_a_b_b @ B @ ( image_3400966383075225011_a_b_b @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_877_image__is__empty,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( ( image_nat_nat @ F @ A3 )
        = bot_bot_set_nat )
      = ( A3 = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_878_empty__is__image,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( bot_bot_set_nat
        = ( image_nat_nat @ F @ A3 ) )
      = ( A3 = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_879_image__empty,axiom,
    ! [F: nat > nat] :
      ( ( image_nat_nat @ F @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% image_empty
thf(fact_880_image__subsetI,axiom,
    ! [A3: set_real,F: real > real,B5: set_real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A3 )
         => ( member_real @ ( F @ X3 ) @ B5 ) )
     => ( ord_less_eq_set_real @ ( image_real_real @ F @ A3 ) @ B5 ) ) ).

% image_subsetI
thf(fact_881_image__subsetI,axiom,
    ! [A3: set_list_a,F: list_a > real,B5: set_real] :
      ( ! [X3: list_a] :
          ( ( member_list_a @ X3 @ A3 )
         => ( member_real @ ( F @ X3 ) @ B5 ) )
     => ( ord_less_eq_set_real @ ( image_list_a_real @ F @ A3 ) @ B5 ) ) ).

% image_subsetI
thf(fact_882_image__subsetI,axiom,
    ! [A3: set_real,F: real > list_a,B5: set_list_a] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A3 )
         => ( member_list_a @ ( F @ X3 ) @ B5 ) )
     => ( ord_le8861187494160871172list_a @ ( image_real_list_a @ F @ A3 ) @ B5 ) ) ).

% image_subsetI
thf(fact_883_image__subsetI,axiom,
    ! [A3: set_list_a,F: list_a > list_a,B5: set_list_a] :
      ( ! [X3: list_a] :
          ( ( member_list_a @ X3 @ A3 )
         => ( member_list_a @ ( F @ X3 ) @ B5 ) )
     => ( ord_le8861187494160871172list_a @ ( image_list_a_list_a @ F @ A3 ) @ B5 ) ) ).

% image_subsetI
thf(fact_884_image__subsetI,axiom,
    ! [A3: set_dtree_list_a_b,F: dtree_list_a_b > real,B5: set_real] :
      ( ! [X3: dtree_list_a_b] :
          ( ( member551035911493665803st_a_b @ X3 @ A3 )
         => ( member_real @ ( F @ X3 ) @ B5 ) )
     => ( ord_less_eq_set_real @ ( image_3232105638531924007b_real @ F @ A3 ) @ B5 ) ) ).

% image_subsetI
thf(fact_885_image__subsetI,axiom,
    ! [A3: set_real,F: real > dtree_list_a_b,B5: set_dtree_list_a_b] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A3 )
         => ( member551035911493665803st_a_b @ ( F @ X3 ) @ B5 ) )
     => ( ord_le7599451563663638410st_a_b @ ( image_3293520908065730343st_a_b @ F @ A3 ) @ B5 ) ) ).

% image_subsetI
thf(fact_886_image__subsetI,axiom,
    ! [A3: set_list_a,F: list_a > dtree_list_a_b,B5: set_dtree_list_a_b] :
      ( ! [X3: list_a] :
          ( ( member_list_a @ X3 @ A3 )
         => ( member551035911493665803st_a_b @ ( F @ X3 ) @ B5 ) )
     => ( ord_le7599451563663638410st_a_b @ ( image_1909091145250899635st_a_b @ F @ A3 ) @ B5 ) ) ).

% image_subsetI
thf(fact_887_image__subsetI,axiom,
    ! [A3: set_dtree_list_a_b,F: dtree_list_a_b > list_a,B5: set_list_a] :
      ( ! [X3: dtree_list_a_b] :
          ( ( member551035911493665803st_a_b @ X3 @ A3 )
         => ( member_list_a @ ( F @ X3 ) @ B5 ) )
     => ( ord_le8861187494160871172list_a @ ( image_5727204212171164361list_a @ F @ A3 ) @ B5 ) ) ).

% image_subsetI
thf(fact_888_image__subsetI,axiom,
    ! [A3: set_Pr3443975907877334966_a_b_b,F: produc6499617310964463488_a_b_b > real,B5: set_real] :
      ( ! [X3: produc6499617310964463488_a_b_b] :
          ( ( member4695696432722591383_a_b_b @ X3 @ A3 )
         => ( member_real @ ( F @ X3 ) @ B5 ) )
     => ( ord_less_eq_set_real @ ( image_3497077275575164339b_real @ F @ A3 ) @ B5 ) ) ).

% image_subsetI
thf(fact_889_image__subsetI,axiom,
    ! [A3: set_real,F: real > produc6499617310964463488_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A3 )
         => ( member4695696432722591383_a_b_b @ ( F @ X3 ) @ B5 ) )
     => ( ord_le1619362961161175062_a_b_b @ ( image_3400966383075225011_a_b_b @ F @ A3 ) @ B5 ) ) ).

% image_subsetI
thf(fact_890_darcs__child__subseteq,axiom,
    ! [X: dtree_list_a_b,Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
      ( ( member551035911493665803st_a_b @ X @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
     => ( ord_less_eq_set_b @ ( darcs_list_a_b @ X ) @ ( darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ).

% darcs_child_subseteq
thf(fact_891_dverts__child__subseteq,axiom,
    ! [X: dtree_list_a_b,Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
      ( ( member551035911493665803st_a_b @ X @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
     => ( ord_le8861187494160871172list_a @ ( dverts_list_a_b @ X ) @ ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ).

% dverts_child_subseteq
thf(fact_892_wf__darcs__rec,axiom,
    ! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T12: dtree_list_a_b] :
      ( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
     => ( ( member551035911493665803st_a_b @ T12 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
       => ( wf_darcs_list_a_b @ T12 ) ) ) ).

% wf_darcs_rec
thf(fact_893_subtree__if__suc,axiom,
    ! [T12: dtree_list_a_b,T22: dtree_list_a_b] :
      ( ( member551035911493665803st_a_b @ T12 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T22 ) ) ) )
     => ( is_subtree_list_a_b @ T12 @ T22 ) ) ).

% subtree_if_suc
thf(fact_894_child__uneq,axiom,
    ! [T: dtree_list_a_b,Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
      ( ( member551035911493665803st_a_b @ T @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
     => ( ( node_list_a_b @ R2 @ Xs2 )
       != T ) ) ).

% child_uneq
thf(fact_895_imageI,axiom,
    ! [X: real,A3: set_real,F: real > real] :
      ( ( member_real @ X @ A3 )
     => ( member_real @ ( F @ X ) @ ( image_real_real @ F @ A3 ) ) ) ).

% imageI
thf(fact_896_imageI,axiom,
    ! [X: list_a,A3: set_list_a,F: list_a > real] :
      ( ( member_list_a @ X @ A3 )
     => ( member_real @ ( F @ X ) @ ( image_list_a_real @ F @ A3 ) ) ) ).

% imageI
thf(fact_897_imageI,axiom,
    ! [X: real,A3: set_real,F: real > list_a] :
      ( ( member_real @ X @ A3 )
     => ( member_list_a @ ( F @ X ) @ ( image_real_list_a @ F @ A3 ) ) ) ).

% imageI
thf(fact_898_imageI,axiom,
    ! [X: list_a,A3: set_list_a,F: list_a > list_a] :
      ( ( member_list_a @ X @ A3 )
     => ( member_list_a @ ( F @ X ) @ ( image_list_a_list_a @ F @ A3 ) ) ) ).

% imageI
thf(fact_899_imageI,axiom,
    ! [X: dtree_list_a_b,A3: set_dtree_list_a_b,F: dtree_list_a_b > real] :
      ( ( member551035911493665803st_a_b @ X @ A3 )
     => ( member_real @ ( F @ X ) @ ( image_3232105638531924007b_real @ F @ A3 ) ) ) ).

% imageI
thf(fact_900_imageI,axiom,
    ! [X: real,A3: set_real,F: real > dtree_list_a_b] :
      ( ( member_real @ X @ A3 )
     => ( member551035911493665803st_a_b @ ( F @ X ) @ ( image_3293520908065730343st_a_b @ F @ A3 ) ) ) ).

% imageI
thf(fact_901_imageI,axiom,
    ! [X: list_a,A3: set_list_a,F: list_a > dtree_list_a_b] :
      ( ( member_list_a @ X @ A3 )
     => ( member551035911493665803st_a_b @ ( F @ X ) @ ( image_1909091145250899635st_a_b @ F @ A3 ) ) ) ).

% imageI
thf(fact_902_imageI,axiom,
    ! [X: dtree_list_a_b,A3: set_dtree_list_a_b,F: dtree_list_a_b > list_a] :
      ( ( member551035911493665803st_a_b @ X @ A3 )
     => ( member_list_a @ ( F @ X ) @ ( image_5727204212171164361list_a @ F @ A3 ) ) ) ).

% imageI
thf(fact_903_imageI,axiom,
    ! [X: produc6499617310964463488_a_b_b,A3: set_Pr3443975907877334966_a_b_b,F: produc6499617310964463488_a_b_b > real] :
      ( ( member4695696432722591383_a_b_b @ X @ A3 )
     => ( member_real @ ( F @ X ) @ ( image_3497077275575164339b_real @ F @ A3 ) ) ) ).

% imageI
thf(fact_904_imageI,axiom,
    ! [X: real,A3: set_real,F: real > produc6499617310964463488_a_b_b] :
      ( ( member_real @ X @ A3 )
     => ( member4695696432722591383_a_b_b @ ( F @ X ) @ ( image_3400966383075225011_a_b_b @ F @ A3 ) ) ) ).

% imageI
thf(fact_905_rev__image__eqI,axiom,
    ! [X: real,A3: set_real,B: real,F: real > real] :
      ( ( member_real @ X @ A3 )
     => ( ( B
          = ( F @ X ) )
       => ( member_real @ B @ ( image_real_real @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_906_rev__image__eqI,axiom,
    ! [X: list_a,A3: set_list_a,B: real,F: list_a > real] :
      ( ( member_list_a @ X @ A3 )
     => ( ( B
          = ( F @ X ) )
       => ( member_real @ B @ ( image_list_a_real @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_907_rev__image__eqI,axiom,
    ! [X: real,A3: set_real,B: list_a,F: real > list_a] :
      ( ( member_real @ X @ A3 )
     => ( ( B
          = ( F @ X ) )
       => ( member_list_a @ B @ ( image_real_list_a @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_908_rev__image__eqI,axiom,
    ! [X: list_a,A3: set_list_a,B: list_a,F: list_a > list_a] :
      ( ( member_list_a @ X @ A3 )
     => ( ( B
          = ( F @ X ) )
       => ( member_list_a @ B @ ( image_list_a_list_a @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_909_rev__image__eqI,axiom,
    ! [X: dtree_list_a_b,A3: set_dtree_list_a_b,B: real,F: dtree_list_a_b > real] :
      ( ( member551035911493665803st_a_b @ X @ A3 )
     => ( ( B
          = ( F @ X ) )
       => ( member_real @ B @ ( image_3232105638531924007b_real @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_910_rev__image__eqI,axiom,
    ! [X: real,A3: set_real,B: dtree_list_a_b,F: real > dtree_list_a_b] :
      ( ( member_real @ X @ A3 )
     => ( ( B
          = ( F @ X ) )
       => ( member551035911493665803st_a_b @ B @ ( image_3293520908065730343st_a_b @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_911_rev__image__eqI,axiom,
    ! [X: list_a,A3: set_list_a,B: dtree_list_a_b,F: list_a > dtree_list_a_b] :
      ( ( member_list_a @ X @ A3 )
     => ( ( B
          = ( F @ X ) )
       => ( member551035911493665803st_a_b @ B @ ( image_1909091145250899635st_a_b @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_912_rev__image__eqI,axiom,
    ! [X: dtree_list_a_b,A3: set_dtree_list_a_b,B: list_a,F: dtree_list_a_b > list_a] :
      ( ( member551035911493665803st_a_b @ X @ A3 )
     => ( ( B
          = ( F @ X ) )
       => ( member_list_a @ B @ ( image_5727204212171164361list_a @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_913_rev__image__eqI,axiom,
    ! [X: produc6499617310964463488_a_b_b,A3: set_Pr3443975907877334966_a_b_b,B: real,F: produc6499617310964463488_a_b_b > real] :
      ( ( member4695696432722591383_a_b_b @ X @ A3 )
     => ( ( B
          = ( F @ X ) )
       => ( member_real @ B @ ( image_3497077275575164339b_real @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_914_rev__image__eqI,axiom,
    ! [X: real,A3: set_real,B: produc6499617310964463488_a_b_b,F: real > produc6499617310964463488_a_b_b] :
      ( ( member_real @ X @ A3 )
     => ( ( B
          = ( F @ X ) )
       => ( member4695696432722591383_a_b_b @ B @ ( image_3400966383075225011_a_b_b @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_915_Compr__image__eq,axiom,
    ! [F: real > real,A3: set_real,P3: real > $o] :
      ( ( collect_real
        @ ^ [X4: real] :
            ( ( member_real @ X4 @ ( image_real_real @ F @ A3 ) )
            & ( P3 @ X4 ) ) )
      = ( image_real_real @ F
        @ ( collect_real
          @ ^ [X4: real] :
              ( ( member_real @ X4 @ A3 )
              & ( P3 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_916_Compr__image__eq,axiom,
    ! [F: real > list_a,A3: set_real,P3: list_a > $o] :
      ( ( collect_list_a
        @ ^ [X4: list_a] :
            ( ( member_list_a @ X4 @ ( image_real_list_a @ F @ A3 ) )
            & ( P3 @ X4 ) ) )
      = ( image_real_list_a @ F
        @ ( collect_real
          @ ^ [X4: real] :
              ( ( member_real @ X4 @ A3 )
              & ( P3 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_917_Compr__image__eq,axiom,
    ! [F: list_a > real,A3: set_list_a,P3: real > $o] :
      ( ( collect_real
        @ ^ [X4: real] :
            ( ( member_real @ X4 @ ( image_list_a_real @ F @ A3 ) )
            & ( P3 @ X4 ) ) )
      = ( image_list_a_real @ F
        @ ( collect_list_a
          @ ^ [X4: list_a] :
              ( ( member_list_a @ X4 @ A3 )
              & ( P3 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_918_Compr__image__eq,axiom,
    ! [F: list_a > list_a,A3: set_list_a,P3: list_a > $o] :
      ( ( collect_list_a
        @ ^ [X4: list_a] :
            ( ( member_list_a @ X4 @ ( image_list_a_list_a @ F @ A3 ) )
            & ( P3 @ X4 ) ) )
      = ( image_list_a_list_a @ F
        @ ( collect_list_a
          @ ^ [X4: list_a] :
              ( ( member_list_a @ X4 @ A3 )
              & ( P3 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_919_Compr__image__eq,axiom,
    ! [F: real > dtree_list_a_b,A3: set_real,P3: dtree_list_a_b > $o] :
      ( ( collec2944820760411501129st_a_b
        @ ^ [X4: dtree_list_a_b] :
            ( ( member551035911493665803st_a_b @ X4 @ ( image_3293520908065730343st_a_b @ F @ A3 ) )
            & ( P3 @ X4 ) ) )
      = ( image_3293520908065730343st_a_b @ F
        @ ( collect_real
          @ ^ [X4: real] :
              ( ( member_real @ X4 @ A3 )
              & ( P3 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_920_Compr__image__eq,axiom,
    ! [F: dtree_list_a_b > real,A3: set_dtree_list_a_b,P3: real > $o] :
      ( ( collect_real
        @ ^ [X4: real] :
            ( ( member_real @ X4 @ ( image_3232105638531924007b_real @ F @ A3 ) )
            & ( P3 @ X4 ) ) )
      = ( image_3232105638531924007b_real @ F
        @ ( collec2944820760411501129st_a_b
          @ ^ [X4: dtree_list_a_b] :
              ( ( member551035911493665803st_a_b @ X4 @ A3 )
              & ( P3 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_921_Compr__image__eq,axiom,
    ! [F: dtree_list_a_b > list_a,A3: set_dtree_list_a_b,P3: list_a > $o] :
      ( ( collect_list_a
        @ ^ [X4: list_a] :
            ( ( member_list_a @ X4 @ ( image_5727204212171164361list_a @ F @ A3 ) )
            & ( P3 @ X4 ) ) )
      = ( image_5727204212171164361list_a @ F
        @ ( collec2944820760411501129st_a_b
          @ ^ [X4: dtree_list_a_b] :
              ( ( member551035911493665803st_a_b @ X4 @ A3 )
              & ( P3 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_922_Compr__image__eq,axiom,
    ! [F: list_a > dtree_list_a_b,A3: set_list_a,P3: dtree_list_a_b > $o] :
      ( ( collec2944820760411501129st_a_b
        @ ^ [X4: dtree_list_a_b] :
            ( ( member551035911493665803st_a_b @ X4 @ ( image_1909091145250899635st_a_b @ F @ A3 ) )
            & ( P3 @ X4 ) ) )
      = ( image_1909091145250899635st_a_b @ F
        @ ( collect_list_a
          @ ^ [X4: list_a] :
              ( ( member_list_a @ X4 @ A3 )
              & ( P3 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_923_Compr__image__eq,axiom,
    ! [F: real > produc6499617310964463488_a_b_b,A3: set_real,P3: produc6499617310964463488_a_b_b > $o] :
      ( ( collec3241703154345475797_a_b_b
        @ ^ [X4: produc6499617310964463488_a_b_b] :
            ( ( member4695696432722591383_a_b_b @ X4 @ ( image_3400966383075225011_a_b_b @ F @ A3 ) )
            & ( P3 @ X4 ) ) )
      = ( image_3400966383075225011_a_b_b @ F
        @ ( collect_real
          @ ^ [X4: real] :
              ( ( member_real @ X4 @ A3 )
              & ( P3 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_924_Compr__image__eq,axiom,
    ! [F: produc6499617310964463488_a_b_b > real,A3: set_Pr3443975907877334966_a_b_b,P3: real > $o] :
      ( ( collect_real
        @ ^ [X4: real] :
            ( ( member_real @ X4 @ ( image_3497077275575164339b_real @ F @ A3 ) )
            & ( P3 @ X4 ) ) )
      = ( image_3497077275575164339b_real @ F
        @ ( collec3241703154345475797_a_b_b
          @ ^ [X4: produc6499617310964463488_a_b_b] :
              ( ( member4695696432722591383_a_b_b @ X4 @ A3 )
              & ( P3 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_925_imageE,axiom,
    ! [B: real,F: real > real,A3: set_real] :
      ( ( member_real @ B @ ( image_real_real @ F @ A3 ) )
     => ~ ! [X3: real] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_real @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_926_imageE,axiom,
    ! [B: list_a,F: real > list_a,A3: set_real] :
      ( ( member_list_a @ B @ ( image_real_list_a @ F @ A3 ) )
     => ~ ! [X3: real] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_real @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_927_imageE,axiom,
    ! [B: real,F: list_a > real,A3: set_list_a] :
      ( ( member_real @ B @ ( image_list_a_real @ F @ A3 ) )
     => ~ ! [X3: list_a] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_list_a @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_928_imageE,axiom,
    ! [B: list_a,F: list_a > list_a,A3: set_list_a] :
      ( ( member_list_a @ B @ ( image_list_a_list_a @ F @ A3 ) )
     => ~ ! [X3: list_a] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_list_a @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_929_imageE,axiom,
    ! [B: dtree_list_a_b,F: real > dtree_list_a_b,A3: set_real] :
      ( ( member551035911493665803st_a_b @ B @ ( image_3293520908065730343st_a_b @ F @ A3 ) )
     => ~ ! [X3: real] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_real @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_930_imageE,axiom,
    ! [B: real,F: dtree_list_a_b > real,A3: set_dtree_list_a_b] :
      ( ( member_real @ B @ ( image_3232105638531924007b_real @ F @ A3 ) )
     => ~ ! [X3: dtree_list_a_b] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member551035911493665803st_a_b @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_931_imageE,axiom,
    ! [B: list_a,F: dtree_list_a_b > list_a,A3: set_dtree_list_a_b] :
      ( ( member_list_a @ B @ ( image_5727204212171164361list_a @ F @ A3 ) )
     => ~ ! [X3: dtree_list_a_b] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member551035911493665803st_a_b @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_932_imageE,axiom,
    ! [B: dtree_list_a_b,F: list_a > dtree_list_a_b,A3: set_list_a] :
      ( ( member551035911493665803st_a_b @ B @ ( image_1909091145250899635st_a_b @ F @ A3 ) )
     => ~ ! [X3: list_a] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_list_a @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_933_imageE,axiom,
    ! [B: produc6499617310964463488_a_b_b,F: real > produc6499617310964463488_a_b_b,A3: set_real] :
      ( ( member4695696432722591383_a_b_b @ B @ ( image_3400966383075225011_a_b_b @ F @ A3 ) )
     => ~ ! [X3: real] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_real @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_934_imageE,axiom,
    ! [B: real,F: produc6499617310964463488_a_b_b > real,A3: set_Pr3443975907877334966_a_b_b] :
      ( ( member_real @ B @ ( image_3497077275575164339b_real @ F @ A3 ) )
     => ~ ! [X3: produc6499617310964463488_a_b_b] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member4695696432722591383_a_b_b @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_935_fst__eqD,axiom,
    ! [X: list_a,Y: list_a,A: list_a] :
      ( ( ( produc3698117735987127555list_a @ ( produc6837034575241423639list_a @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_936_fst__eqD,axiom,
    ! [X: list_a,Y: dtree_list_a_b,A: list_a] :
      ( ( ( produc318716513907093941st_a_b @ ( produc148520996349637281st_a_b @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_937_fst__eqD,axiom,
    ! [X: dtree_list_a_b,Y: b,A: dtree_list_a_b] :
      ( ( ( produc5948858871325780166_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_938_fst__eqD,axiom,
    ! [X: list_a,Y: produc111314985273491367st_a_b,A: list_a] :
      ( ( ( produc5507115380825155048st_a_b @ ( produc673257793671328980st_a_b @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_939_fst__conv,axiom,
    ! [X1: list_a,X2: list_a] :
      ( ( produc3698117735987127555list_a @ ( produc6837034575241423639list_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_940_fst__conv,axiom,
    ! [X1: list_a,X2: dtree_list_a_b] :
      ( ( produc318716513907093941st_a_b @ ( produc148520996349637281st_a_b @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_941_fst__conv,axiom,
    ! [X1: dtree_list_a_b,X2: b] :
      ( ( produc5948858871325780166_a_b_b @ ( produc7704165765595008946_a_b_b @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_942_fst__conv,axiom,
    ! [X1: list_a,X2: produc111314985273491367st_a_b] :
      ( ( produc5507115380825155048st_a_b @ ( produc673257793671328980st_a_b @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_943_suc__uneq,axiom,
    ! [T12: dtree_list_a_b,T: dtree_list_a_b] :
      ( ( member551035911493665803st_a_b @ T12 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) ) )
     => ( T != T12 ) ) ).

% suc_uneq
thf(fact_944_suc__uneq_H,axiom,
    ! [T12: dtree_list_a_b,T: dtree_list_a_b,V: list_a] :
      ( ( member551035911493665803st_a_b @ T12 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) ) )
     => ( T
       != ( node_list_a_b @ V @ ( sucs_list_a_b @ T12 ) ) ) ) ).

% suc_uneq'
thf(fact_945_child__uneq_H,axiom,
    ! [T: dtree_list_a_b,Xs2: fset_P2153231429829016240_a_b_b,R2: list_a,V: list_a] :
      ( ( member551035911493665803st_a_b @ T @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
     => ( ( node_list_a_b @ R2 @ Xs2 )
       != ( node_list_a_b @ V @ ( sucs_list_a_b @ T ) ) ) ) ).

% child_uneq'
thf(fact_946_subtree__if__child,axiom,
    ! [X: dtree_list_a_b,Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
      ( ( member551035911493665803st_a_b @ X @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
     => ( is_subtree_list_a_b @ X @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ).

% subtree_if_child
thf(fact_947_dverts__child__if__not__root,axiom,
    ! [V: list_a,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
      ( ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
     => ( ( V != R2 )
       => ? [X3: dtree_list_a_b] :
            ( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
            & ( member_list_a @ V @ ( dverts_list_a_b @ X3 ) ) ) ) ) ).

% dverts_child_if_not_root
thf(fact_948_mdeg__ge__child_H,axiom,
    ! [T12: dtree_list_a_b,Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
      ( ( member551035911493665803st_a_b @ T12 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
     => ( ord_less_eq_nat @ ( max_deg_list_a_b @ T12 ) @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ).

% mdeg_ge_child'
thf(fact_949_dverts__suc__subseteq,axiom,
    ! [X: dtree_list_a_b,T: dtree_list_a_b] :
      ( ( member551035911493665803st_a_b @ X @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) ) )
     => ( ord_le8861187494160871172list_a @ ( dverts_list_a_b @ X ) @ ( dverts_list_a_b @ T ) ) ) ).

% dverts_suc_subseteq
thf(fact_950_subtree__child__if__dvert__notroot,axiom,
    ! [V: list_a,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
      ( ( V != R2 )
     => ( ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
       => ? [R6: list_a,Ys2: fset_P2153231429829016240_a_b_b,Zs: fset_P2153231429829016240_a_b_b] :
            ( ( is_subtree_list_a_b @ ( node_list_a_b @ R6 @ Ys2 ) @ ( node_list_a_b @ R2 @ Xs2 ) )
            & ( member551035911493665803st_a_b @ ( node_list_a_b @ V @ Zs ) @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Ys2 ) ) ) ) ) ) ).

% subtree_child_if_dvert_notroot
thf(fact_951_wf__dtree_Oroot__not__subtree,axiom,
    ! [T: dtree_list_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b] :
      ( ( wf_dtree_list_a_b @ T )
     => ( ( ( node_list_a_b @ R2 @ Xs2 )
          = T )
       => ( ( member551035911493665803st_a_b @ X @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
         => ~ ( member_list_a @ R2 @ ( dverts_list_a_b @ X ) ) ) ) ) ).

% wf_dtree.root_not_subtree
thf(fact_952_dverts__suc__if__not__root,axiom,
    ! [V: list_a,T: dtree_list_a_b] :
      ( ( member_list_a @ V @ ( dverts_list_a_b @ T ) )
     => ( ( V
         != ( root_list_a_b @ T ) )
       => ? [X3: dtree_list_a_b] :
            ( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) ) )
            & ( member_list_a @ V @ ( dverts_list_a_b @ X3 ) ) ) ) ) ).

% dverts_suc_if_not_root
thf(fact_953_dtree__size__decr__aux_H,axiom,
    ! [T12: dtree_list_a_b,Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
      ( ( member551035911493665803st_a_b @ T12 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
     => ( ord_less_nat @ ( size_s415192292648992904st_a_b @ T12 ) @ ( size_s415192292648992904st_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ).

% dtree_size_decr_aux'
thf(fact_954_child__sub__if__strict__subtree,axiom,
    ! [T12: dtree_list_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
      ( ( strict8995144569104247066st_a_b @ T12 @ ( node_list_a_b @ R2 @ Xs2 ) )
     => ? [X3: dtree_list_a_b] :
          ( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
          & ( is_subtree_list_a_b @ T12 @ X3 ) ) ) ).

% child_sub_if_strict_subtree
thf(fact_955_subtree__child__if__strict__subtree,axiom,
    ! [T12: dtree_list_a_b,T22: dtree_list_a_b] :
      ( ( strict8995144569104247066st_a_b @ T12 @ T22 )
     => ? [R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
          ( ( is_subtree_list_a_b @ ( node_list_a_b @ R @ Xs ) @ T22 )
          & ( member551035911493665803st_a_b @ T12 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs ) ) ) ) ) ).

% subtree_child_if_strict_subtree
thf(fact_956_suc__sub__if__strict__subtree,axiom,
    ! [T12: dtree_list_a_b,T22: dtree_list_a_b] :
      ( ( strict8995144569104247066st_a_b @ T12 @ T22 )
     => ? [X3: dtree_list_a_b] :
          ( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T22 ) ) ) )
          & ( is_subtree_list_a_b @ T12 @ X3 ) ) ) ).

% suc_sub_if_strict_subtree
thf(fact_957_subtree__child__if__dvert__notelem,axiom,
    ! [V: list_a,T: dtree_list_a_b] :
      ( ( V
       != ( root_list_a_b @ T ) )
     => ( ( member_list_a @ V @ ( dverts_list_a_b @ T ) )
       => ? [R6: list_a,Ys2: fset_P2153231429829016240_a_b_b,Zs: fset_P2153231429829016240_a_b_b] :
            ( ( is_subtree_list_a_b @ ( node_list_a_b @ R6 @ Ys2 ) @ T )
            & ( member551035911493665803st_a_b @ ( node_list_a_b @ V @ Zs ) @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Ys2 ) ) ) ) ) ) ).

% subtree_child_if_dvert_notelem
thf(fact_958_wf__dtree_Odarcs__child__subset,axiom,
    ! [T: dtree_list_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b] :
      ( ( wf_dtree_list_a_b @ T )
     => ( ( ( node_list_a_b @ R2 @ Xs2 )
          = T )
       => ( ( member551035911493665803st_a_b @ X @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
         => ( ord_less_set_b @ ( darcs_list_a_b @ X ) @ ( darcs_list_a_b @ T ) ) ) ) ) ).

% wf_dtree.darcs_child_subset
thf(fact_959_wf__dtree_Odverts__child__subset,axiom,
    ! [T: dtree_list_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b] :
      ( ( wf_dtree_list_a_b @ T )
     => ( ( ( node_list_a_b @ R2 @ Xs2 )
          = T )
       => ( ( member551035911493665803st_a_b @ X @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
         => ( ord_less_set_list_a @ ( dverts_list_a_b @ X ) @ ( dverts_list_a_b @ T ) ) ) ) ) ).

% wf_dtree.dverts_child_subset
thf(fact_960_mdeg__child__if__gt__fcard,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
      ( ( ord_less_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
     => ? [X3: dtree_list_a_b] :
          ( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
          & ( ( max_deg_list_a_b @ X3 )
            = ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ) ).

% mdeg_child_if_gt_fcard
thf(fact_961_mdeg__fcard__if__fcard__ge__child_H,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
      ( ! [X3: dtree_list_a_b] :
          ( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
         => ( ord_less_eq_nat @ ( max_deg_list_a_b @ X3 ) @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) ) )
     => ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
        = ( fcard_4742106318756258927_a_b_b @ Xs2 ) ) ) ).

% mdeg_fcard_if_fcard_ge_child'
thf(fact_962_ranked__dtree_Overts__child__if__merge__child,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,T12: dtree_list_a_b,Rank: list_a > real,T0: dtree_list_a_b,X: list_a] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( member551035911493665803st_a_b @ T12 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ ( ranked_merge_a_b @ Rank @ Cmp @ T0 ) ) ) ) )
       => ( ( member_list_a @ X @ ( dverts_list_a_b @ T12 ) )
         => ? [X3: dtree_list_a_b] :
              ( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T0 ) ) ) )
              & ( member_list_a @ X @ ( dverts_list_a_b @ X3 ) ) ) ) ) ) ).

% ranked_dtree.verts_child_if_merge_child
thf(fact_963_singleton__suc,axiom,
    ! [T: dtree_list_a_b,R2: list_a,E: b] : ( member551035911493665803st_a_b @ T @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ) ).

% singleton_suc
thf(fact_964_mdeg__child__if__wedge,axiom,
    ! [N2: nat,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
      ( ( ord_less_nat @ N2 @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
     => ( ( ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) @ N2 )
          | ~ ! [X5: dtree_list_a_b] :
                ( ( member551035911493665803st_a_b @ X5 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
               => ( ord_less_eq_nat @ ( max_deg_list_a_b @ X5 ) @ N2 ) ) )
       => ? [X3: dtree_list_a_b] :
            ( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
            & ( ord_less_nat @ N2 @ ( max_deg_list_a_b @ X3 ) ) ) ) ) ).

% mdeg_child_if_wedge
thf(fact_965_complete__real,axiom,
    ! [S: set_real] :
      ( ? [X5: real] : ( member_real @ X5 @ S )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( member_real @ X3 @ S )
           => ( ord_less_eq_real @ X3 @ Z5 ) )
       => ? [Y4: real] :
            ( ! [X5: real] :
                ( ( member_real @ X5 @ S )
               => ( ord_less_eq_real @ X5 @ Y4 ) )
            & ! [Z5: real] :
                ( ! [X3: real] :
                    ( ( member_real @ X3 @ S )
                   => ( ord_less_eq_real @ X3 @ Z5 ) )
               => ( ord_less_eq_real @ Y4 @ Z5 ) ) ) ) ) ).

% complete_real
thf(fact_966_fcard__gt1__if__mdeg__gt__child1,axiom,
    ! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T12: dtree_list_a_b] :
      ( ( ord_less_nat @ one_one_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
     => ( ( member551035911493665803st_a_b @ T12 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
       => ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T12 ) @ one_one_nat )
         => ( ord_less_nat @ one_one_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) ) ) ) ) ).

% fcard_gt1_if_mdeg_gt_child1
thf(fact_967_fcard__gt1__if__mdeg__gt__suc1,axiom,
    ! [T22: dtree_list_a_b,T12: dtree_list_a_b] :
      ( ( ord_less_nat @ one_one_nat @ ( max_deg_list_a_b @ T22 ) )
     => ( ( member551035911493665803st_a_b @ T12 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T22 ) ) ) )
       => ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T12 ) @ one_one_nat )
         => ( ord_less_nat @ one_one_nat @ ( fcard_4742106318756258927_a_b_b @ ( sucs_list_a_b @ T22 ) ) ) ) ) ) ).

% fcard_gt1_if_mdeg_gt_suc1
thf(fact_968_fcard__gt1__if__mdeg__gt__child,axiom,
    ! [N2: nat,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T12: dtree_list_a_b] :
      ( ( ord_less_nat @ N2 @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
     => ( ( member551035911493665803st_a_b @ T12 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
       => ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T12 ) @ N2 )
         => ( ( N2 != zero_zero_nat )
           => ( ord_less_nat @ one_one_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) ) ) ) ) ) ).

% fcard_gt1_if_mdeg_gt_child
thf(fact_969_fcard__gt1__if__mdeg__gt__suc,axiom,
    ! [N2: nat,T22: dtree_list_a_b,T12: dtree_list_a_b] :
      ( ( ord_less_nat @ N2 @ ( max_deg_list_a_b @ T22 ) )
     => ( ( member551035911493665803st_a_b @ T12 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T22 ) ) ) )
       => ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T12 ) @ N2 )
         => ( ( N2 != zero_zero_nat )
           => ( ord_less_nat @ one_one_nat @ ( fcard_4742106318756258927_a_b_b @ ( sucs_list_a_b @ T22 ) ) ) ) ) ) ) ).

% fcard_gt1_if_mdeg_gt_suc
thf(fact_970_ranked__dtree_Omerge__single__root1,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,T22: dtree_list_a_b,Rank: list_a > real,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( member551035911493665803st_a_b @ T22 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ ( ranked_merge_a_b @ Rank @ Cmp @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ) )
       => ? [E23: b] :
            ( ( ranked_merge_a_b @ Rank @ Cmp @ ( node_list_a_b @ R2 @ Xs2 ) )
            = ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ).

% ranked_dtree.merge_single_root1
thf(fact_971_ranked__dtree_Odarcs__merge__eq,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Rank: list_a > real] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ! [X3: dtree_list_a_b] :
            ( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) ) )
           => ( ord_less_eq_nat @ ( max_deg_list_a_b @ X3 ) @ one_one_nat ) )
       => ( ( darcs_list_a_b @ ( ranked_merge_a_b @ Rank @ Cmp @ T ) )
          = ( darcs_list_a_b @ T ) ) ) ) ).

% ranked_dtree.darcs_merge_eq
thf(fact_972_ranked__dtree_Odverts__merge__eq,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Rank: list_a > real] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ! [X3: dtree_list_a_b] :
            ( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) ) )
           => ( ord_less_eq_nat @ ( max_deg_list_a_b @ X3 ) @ one_one_nat ) )
       => ( ( dverts_list_a_b @ ( ranked_merge_a_b @ Rank @ Cmp @ T ) )
          = ( dverts_list_a_b @ T ) ) ) ) ).

% ranked_dtree.dverts_merge_eq
thf(fact_973_ranked__dtree_Omerge1__childdeg__gt1__if__fcard__gt1,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Rank: list_a > real,T12: dtree_list_a_b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( ord_less_nat @ one_one_nat @ ( fcard_4742106318756258927_a_b_b @ ( sucs_list_a_b @ ( ranked_merge1_a_b @ Rank @ Cmp @ T12 ) ) ) )
       => ? [X3: dtree_list_a_b] :
            ( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T12 ) ) ) )
            & ( ord_less_nat @ one_one_nat @ ( max_deg_list_a_b @ X3 ) ) ) ) ) ).

% ranked_dtree.merge1_childdeg_gt1_if_fcard_gt1
thf(fact_974_ranked__dtree_Omerge__single__root1__sucs,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,T22: dtree_list_a_b,Rank: list_a > real,T12: dtree_list_a_b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( member551035911493665803st_a_b @ T22 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ ( ranked_merge_a_b @ Rank @ Cmp @ T12 ) ) ) ) )
       => ? [E23: b] :
            ( ( ranked_merge_a_b @ Rank @ Cmp @ T12 )
            = ( node_list_a_b @ ( root_list_a_b @ T12 ) @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ).

% ranked_dtree.merge_single_root1_sucs
thf(fact_975_ranked__dtree_Omerge1__childdeg__gt1__if__fcard__gt1__sub,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,Rank: list_a > real,T12: dtree_list_a_b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ ( ranked_merge1_a_b @ Rank @ Cmp @ T12 ) )
       => ( ( ord_less_nat @ one_one_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) )
         => ? [Ys2: fset_P2153231429829016240_a_b_b] :
              ( ( ( ranked_merge1_a_b @ Rank @ Cmp @ ( node_list_a_b @ R2 @ Ys2 ) )
                = ( node_list_a_b @ R2 @ Xs2 ) )
              & ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Ys2 ) @ T12 )
              & ? [X3: dtree_list_a_b] :
                  ( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Ys2 ) ) )
                  & ( ord_less_nat @ one_one_nat @ ( max_deg_list_a_b @ X3 ) ) ) ) ) ) ) ).

% ranked_dtree.merge1_childdeg_gt1_if_fcard_gt1_sub
thf(fact_976_ranked__dtree_Odverts__merge__sub,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,T0: dtree_list_a_b,Rank: list_a > real] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ! [X3: dtree_list_a_b] :
            ( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T0 ) ) ) )
           => ( ord_less_eq_nat @ ( max_deg_list_a_b @ X3 ) @ one_one_nat ) )
       => ( ord_le8861187494160871172list_a @ ( dverts_list_a_b @ ( ranked_merge_a_b @ Rank @ Cmp @ T0 ) ) @ ( dverts_list_a_b @ T0 ) ) ) ) ).

% ranked_dtree.dverts_merge_sub
thf(fact_977_ranked__dtree_Omerge1__wedge__if__uneq,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Rank: list_a > real,T12: dtree_list_a_b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( ( ranked_merge1_a_b @ Rank @ Cmp @ T12 )
         != T12 )
       => ? [R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
            ( ( is_subtree_list_a_b @ ( node_list_a_b @ R @ Xs ) @ T12 )
            & ( ord_less_nat @ one_one_nat @ ( fcard_4742106318756258927_a_b_b @ Xs ) )
            & ! [X5: dtree_list_a_b] :
                ( ( member551035911493665803st_a_b @ X5 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs ) ) )
               => ( ord_less_eq_nat @ ( max_deg_list_a_b @ X5 ) @ one_one_nat ) ) ) ) ) ).

% ranked_dtree.merge1_wedge_if_uneq
thf(fact_978_ranked__dtree_Omerge__subtree__dverts__supset,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,Rank: list_a > real] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ! [X3: dtree_list_a_b] :
            ( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) ) )
           => ( ord_less_eq_nat @ ( max_deg_list_a_b @ X3 ) @ one_one_nat ) )
       => ( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ ( ranked_merge_a_b @ Rank @ Cmp @ T ) )
         => ? [Ys2: fset_P2153231429829016240_a_b_b] :
              ( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Ys2 ) @ T )
              & ( ord_le8861187494160871172list_a @ ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Ys2 ) ) @ ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ) ) ) ).

% ranked_dtree.merge_subtree_dverts_supset
thf(fact_979_ranked__dtree_Omerge__strict__subtree__nocontr,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Xs2: fset_P2153231429829016240_a_b_b,Rank: list_a > real,R12: list_a,T12: dtree_list_a_b,E1: b,R2: list_a] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ! [T23: dtree_list_a_b,R13: list_a,T1: dtree_list_a_b,E12: b] :
            ( ( member551035911493665803st_a_b @ T23 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
           => ( ( is_subtree_list_a_b @ ( node_list_a_b @ R13 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T23 )
             => ( ord_less_eq_real @ ( Rank @ ( rev_a @ R13 ) ) @ ( Rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) ) ) )
       => ( ( strict8995144569104247066st_a_b @ ( node_list_a_b @ R12 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked_merge_a_b @ Rank @ Cmp @ ( node_list_a_b @ R2 @ Xs2 ) ) )
         => ( ord_less_eq_real @ ( Rank @ ( rev_a @ R12 ) ) @ ( Rank @ ( rev_a @ ( root_list_a_b @ T12 ) ) ) ) ) ) ) ).

% ranked_dtree.merge_strict_subtree_nocontr
thf(fact_980_ranked__dtree_Omerge__strict__subtree__dverts__sup,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,Rank: list_a > real] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ! [X3: dtree_list_a_b] :
            ( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) ) )
           => ( ord_less_eq_nat @ ( max_deg_list_a_b @ X3 ) @ one_one_nat ) )
       => ( ( strict8995144569104247066st_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ ( ranked_merge_a_b @ Rank @ Cmp @ T ) )
         => ? [Ys2: fset_P2153231429829016240_a_b_b] :
              ( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Ys2 ) @ T )
              & ( ord_le8861187494160871172list_a @ ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Ys2 ) ) @ ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ) ) ) ).

% ranked_dtree.merge_strict_subtree_dverts_sup
thf(fact_981_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z4 @ X5 )
     => ~ ( ord_less_eq_real @ X5 @ T ) ) ).

% pinf(6)
thf(fact_982_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z4 @ X5 )
     => ~ ( ord_less_eq_nat @ X5 @ T ) ) ).

% pinf(6)
thf(fact_983_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z4 @ X5 )
     => ( ord_less_eq_real @ T @ X5 ) ) ).

% pinf(8)
thf(fact_984_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z4 @ X5 )
     => ( ord_less_eq_nat @ T @ X5 ) ) ).

% pinf(8)
thf(fact_985_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z4 )
     => ( ord_less_eq_real @ X5 @ T ) ) ).

% minf(6)
thf(fact_986_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z4 )
     => ( ord_less_eq_nat @ X5 @ T ) ) ).

% minf(6)
thf(fact_987_ranked__dtree_Omerge__subtree__dlverts__supset,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,Rank: list_a > real] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ! [X3: dtree_list_a_b] :
            ( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) ) )
           => ( ord_less_eq_nat @ ( max_deg_list_a_b @ X3 ) @ one_one_nat ) )
       => ( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ ( ranked_merge_a_b @ Rank @ Cmp @ T ) )
         => ? [Ys2: fset_P2153231429829016240_a_b_b] :
              ( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Ys2 ) @ T )
              & ( ord_less_eq_set_a @ ( list_dlverts_a_b @ ( node_list_a_b @ R2 @ Ys2 ) ) @ ( list_dlverts_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ) ) ) ).

% ranked_dtree.merge_subtree_dlverts_supset
thf(fact_988_ranked__dtree_Odlverts__merge__eq,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,Rank: list_a > real] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ! [X3: dtree_list_a_b] :
            ( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) ) )
           => ( ord_less_eq_nat @ ( max_deg_list_a_b @ X3 ) @ one_one_nat ) )
       => ( ( list_dlverts_a_b @ ( ranked_merge_a_b @ Rank @ Cmp @ T ) )
          = ( list_dlverts_a_b @ T ) ) ) ) ).

% ranked_dtree.dlverts_merge_eq
thf(fact_989_fsts_Ocases,axiom,
    ! [A: dtree_list_a_b,P: produc6499617310964463488_a_b_b] :
      ( ( member551035911493665803st_a_b @ A @ ( basic_5528648086025252546_a_b_b @ P ) )
     => ( A
        = ( produc5948858871325780166_a_b_b @ P ) ) ) ).

% fsts.cases
thf(fact_990_child__in__dlverts,axiom,
    ! [T12: dtree_list_a_b,E: b,Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
      ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
     => ( ord_less_eq_set_a @ ( list_dlverts_a_b @ T12 ) @ ( list_dlverts_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ).

% child_in_dlverts
thf(fact_991_suc__in__dlverts,axiom,
    ! [T12: dtree_list_a_b,E: b,T22: dtree_list_a_b] :
      ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E ) @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T22 ) ) )
     => ( ord_less_eq_set_a @ ( list_dlverts_a_b @ T12 ) @ ( list_dlverts_a_b @ T22 ) ) ) ).

% suc_in_dlverts
thf(fact_992_ranked__dtree_Omerge1__subtree__dlverts__supset,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,Rank: list_a > real] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ ( ranked_merge1_a_b @ Rank @ Cmp @ T ) )
       => ? [Ys2: fset_P2153231429829016240_a_b_b] :
            ( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Ys2 ) @ T )
            & ( ord_less_eq_set_a @ ( list_dlverts_a_b @ ( node_list_a_b @ R2 @ Ys2 ) ) @ ( list_dlverts_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ) ) ).

% ranked_dtree.merge1_subtree_dlverts_supset
thf(fact_993_mdeg__img__empty__iff,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b] :
      ( ( ( image_5414249006199368395_b_nat @ max_deg_list_a_b @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
        = bot_bot_set_nat )
      = ( Xs2 = bot_bo2248824169281960260_a_b_b ) ) ).

% mdeg_img_empty_iff
thf(fact_994_suc__in__dlverts_H,axiom,
    ! [T12: dtree_list_a_b,T22: dtree_list_a_b] :
      ( ( member551035911493665803st_a_b @ T12 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T22 ) ) ) )
     => ( ord_less_eq_set_a @ ( list_dlverts_a_b @ T12 ) @ ( list_dlverts_a_b @ T22 ) ) ) ).

% suc_in_dlverts'
thf(fact_995_mdeg__img__nempty,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
      ( ( ord_less_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
     => ( ( image_5414249006199368395_b_nat @ max_deg_list_a_b @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
       != bot_bot_set_nat ) ) ).

% mdeg_img_nempty
thf(fact_996_fsts_Ointros,axiom,
    ! [P: produc6499617310964463488_a_b_b] : ( member551035911493665803st_a_b @ ( produc5948858871325780166_a_b_b @ P ) @ ( basic_5528648086025252546_a_b_b @ P ) ) ).

% fsts.intros
thf(fact_997_fsts_Osimps,axiom,
    ! [A: dtree_list_a_b,P: produc6499617310964463488_a_b_b] :
      ( ( member551035911493665803st_a_b @ A @ ( basic_5528648086025252546_a_b_b @ P ) )
      = ( A
        = ( produc5948858871325780166_a_b_b @ P ) ) ) ).

% fsts.simps
thf(fact_998_disjoint__dlverts__if__wf_H__aux,axiom,
    ! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T12: dtree_list_a_b,E1: b,T22: dtree_list_a_b,E22: b] :
      ( ( list_wf_dlverts_a_b2 @ ( node_list_a_b @ R2 @ Xs2 ) )
     => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
       => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
         => ( ( ( produc7704165765595008946_a_b_b @ T12 @ E1 )
             != ( produc7704165765595008946_a_b_b @ T22 @ E22 ) )
           => ( ( inf_inf_set_a @ ( list_dlverts_a_b @ T12 ) @ ( list_dlverts_a_b @ T22 ) )
              = bot_bot_set_a ) ) ) ) ) ).

% disjoint_dlverts_if_wf'_aux
thf(fact_999_x__not__root__strict__subtree,axiom,
    ! [X: a,T: dtree_list_a_b] :
      ( ( member_a @ X @ ( list_dlverts_a_b @ T ) )
     => ( ~ ( member_a @ X @ ( set_a2 @ ( root_list_a_b @ T ) ) )
       => ? [R: list_a,Xs: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b] :
            ( ( is_subtree_list_a_b @ ( node_list_a_b @ R @ Xs ) @ T )
            & ( member551035911493665803st_a_b @ T1 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs ) ) )
            & ( member_a @ X @ ( set_a2 @ ( root_list_a_b @ T1 ) ) ) ) ) ) ).

% x_not_root_strict_subtree
thf(fact_1000_set__rev,axiom,
    ! [Xs2: list_a] :
      ( ( set_a2 @ ( rev_a @ Xs2 ) )
      = ( set_a2 @ Xs2 ) ) ).

% set_rev
thf(fact_1001_subset__code_I1_J,axiom,
    ! [Xs2: list_list_a,B5: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs2 ) @ B5 )
      = ( ! [X4: list_a] :
            ( ( member_list_a @ X4 @ ( set_list_a2 @ Xs2 ) )
           => ( member_list_a @ X4 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_1002_subset__code_I1_J,axiom,
    ! [Xs2: list_P7479555583871319568_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
      ( ( ord_le1619362961161175062_a_b_b @ ( set_Pr3241176995924754219_a_b_b @ Xs2 ) @ B5 )
      = ( ! [X4: produc6499617310964463488_a_b_b] :
            ( ( member4695696432722591383_a_b_b @ X4 @ ( set_Pr3241176995924754219_a_b_b @ Xs2 ) )
           => ( member4695696432722591383_a_b_b @ X4 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_1003_subset__code_I1_J,axiom,
    ! [Xs2: list_dtree_list_a_b,B5: set_dtree_list_a_b] :
      ( ( ord_le7599451563663638410st_a_b @ ( set_dtree_list_a_b2 @ Xs2 ) @ B5 )
      = ( ! [X4: dtree_list_a_b] :
            ( ( member551035911493665803st_a_b @ X4 @ ( set_dtree_list_a_b2 @ Xs2 ) )
           => ( member551035911493665803st_a_b @ X4 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_1004_subset__code_I1_J,axiom,
    ! [Xs2: list_real,B5: set_real] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ B5 )
      = ( ! [X4: real] :
            ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
           => ( member_real @ X4 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_1005_list__in__verts__iff__lverts,axiom,
    ! [X: a,T: dtree_list_a_b] :
      ( ( member_a @ X @ ( list_dlverts_a_b @ T ) )
      = ( ? [X4: list_a] :
            ( ( member_list_a @ X4 @ ( dverts_list_a_b @ T ) )
            & ( member_a @ X @ ( set_a2 @ X4 ) ) ) ) ) ).

% list_in_verts_iff_lverts
thf(fact_1006_list__in__verts__if__lverts,axiom,
    ! [X: a,T: dtree_list_a_b] :
      ( ( member_a @ X @ ( list_dlverts_a_b @ T ) )
     => ? [X3: list_a] :
          ( ( member_list_a @ X3 @ ( dverts_list_a_b @ T ) )
          & ( member_a @ X @ ( set_a2 @ X3 ) ) ) ) ).

% list_in_verts_if_lverts
thf(fact_1007_lverts__if__in__verts,axiom,
    ! [V: list_a,T: dtree_list_a_b,X: a] :
      ( ( member_list_a @ V @ ( dverts_list_a_b @ T ) )
     => ( ( member_a @ X @ ( set_a2 @ V ) )
       => ( member_a @ X @ ( list_dlverts_a_b @ T ) ) ) ) ).

% lverts_if_in_verts
thf(fact_1008_subtree__root__if__dlverts,axiom,
    ! [X: a,T: dtree_list_a_b] :
      ( ( member_a @ X @ ( list_dlverts_a_b @ T ) )
     => ? [R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
          ( ( is_subtree_list_a_b @ ( node_list_a_b @ R @ Xs ) @ T )
          & ( member_a @ X @ ( set_a2 @ R ) ) ) ) ).

% subtree_root_if_dlverts
thf(fact_1009_disjoint__root__if__wf_H__aux,axiom,
    ! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T12: dtree_list_a_b,E1: b] :
      ( ( list_wf_dlverts_a_b2 @ ( node_list_a_b @ R2 @ Xs2 ) )
     => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
       => ( ( inf_inf_set_a @ ( set_a2 @ R2 ) @ ( list_dlverts_a_b @ T12 ) )
          = bot_bot_set_a ) ) ) ).

% disjoint_root_if_wf'_aux
thf(fact_1010_wf__dlverts_H__rec,axiom,
    ! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T12: dtree_list_a_b] :
      ( ( list_wf_dlverts_a_b2 @ ( node_list_a_b @ R2 @ Xs2 ) )
     => ( ( member551035911493665803st_a_b @ T12 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
       => ( list_wf_dlverts_a_b2 @ T12 ) ) ) ).

% wf_dlverts'_rec
thf(fact_1011_wf__dlverts_H__suc,axiom,
    ! [T: dtree_list_a_b,T12: dtree_list_a_b] :
      ( ( list_wf_dlverts_a_b2 @ T )
     => ( ( member551035911493665803st_a_b @ T12 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) ) )
       => ( list_wf_dlverts_a_b2 @ T12 ) ) ) ).

% wf_dlverts'_suc
thf(fact_1012_dtree__to__list__eq__dverts__single,axiom,
    ! [T: dtree_list_a_b,T12: dtree_list_a_b,E1: b] :
      ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T ) @ one_one_nat )
     => ( ( ( sucs_list_a_b @ T )
          = ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) )
       => ( ( image_4065140400681997415list_a @ product_fst_list_a_b @ ( set_Pr6156699709417984823st_a_b @ ( dtree_5522862299006258336st_a_b @ T ) ) )
          = ( dverts_list_a_b @ T12 ) ) ) ) ).

% dtree_to_list_eq_dverts_single
thf(fact_1013_ranked__dtree_Osucs__darcs__eq__dtree__list,axiom,
    ! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,T12: dtree_list_a_b,E1: b] :
      ( ( ranked_dtree_a_b @ T @ Cmp )
     => ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) )
       => ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T12 ) @ one_one_nat )
         => ( ( darcs_list_a_b @ ( node_list_a_b @ ( root_list_a_b @ T ) @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) )
            = ( image_4255047928038901090_a_b_b @ product_snd_list_a_b @ ( set_Pr6156699709417984823st_a_b @ ( dtree_5522862299006258336st_a_b @ ( node_list_a_b @ ( root_list_a_b @ T ) @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ) ) ) ) ).

% ranked_dtree.sucs_darcs_eq_dtree_list
thf(fact_1014_dtree__to__list__eq__dverts__sucs,axiom,
    ! [T: dtree_list_a_b] :
      ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T ) @ one_one_nat )
     => ( ( image_4065140400681997415list_a @ product_fst_list_a_b @ ( set_Pr6156699709417984823st_a_b @ ( dtree_5522862299006258336st_a_b @ T ) ) )
        = ( comple6928918032620976721list_a
          @ ( image_1454333736486480925list_a
            @ ^ [X4: produc6499617310964463488_a_b_b] : ( dverts_list_a_b @ ( produc5948858871325780166_a_b_b @ X4 ) )
            @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) ) ) ) ) ).

% dtree_to_list_eq_dverts_sucs
thf(fact_1015_prod_Ocollapse,axiom,
    ! [Prod: produc9164743771328383783list_a] :
      ( ( produc6837034575241423639list_a @ ( produc3698117735987127555list_a @ Prod ) @ ( produc8617614985401127493list_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_1016_prod_Ocollapse,axiom,
    ! [Prod: produc111314985273491367st_a_b] :
      ( ( produc148520996349637281st_a_b @ ( produc318716513907093941st_a_b @ Prod ) @ ( produc1857499627308065267st_a_b @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_1017_prod_Ocollapse,axiom,
    ! [Prod: produc6499617310964463488_a_b_b] :
      ( ( produc7704165765595008946_a_b_b @ ( produc5948858871325780166_a_b_b @ Prod ) @ ( produc5719641485658034180_a_b_b @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_1018_prod_Ocollapse,axiom,
    ! [Prod: produc7147531718898801626st_a_b] :
      ( ( produc673257793671328980st_a_b @ ( produc5507115380825155048st_a_b @ Prod ) @ ( produc6241816367173513510st_a_b @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_1019_cSUP__const,axiom,
    ! [A3: set_nat,C: nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( complete_Sup_Sup_nat
          @ ( image_nat_nat
            @ ^ [X4: nat] : C
            @ A3 ) )
        = C ) ) ).

% cSUP_const
thf(fact_1020_prod_Oexhaust__sel,axiom,
    ! [Prod: produc9164743771328383783list_a] :
      ( Prod
      = ( produc6837034575241423639list_a @ ( produc3698117735987127555list_a @ Prod ) @ ( produc8617614985401127493list_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_1021_prod_Oexhaust__sel,axiom,
    ! [Prod: produc111314985273491367st_a_b] :
      ( Prod
      = ( produc148520996349637281st_a_b @ ( produc318716513907093941st_a_b @ Prod ) @ ( produc1857499627308065267st_a_b @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_1022_prod_Oexhaust__sel,axiom,
    ! [Prod: produc6499617310964463488_a_b_b] :
      ( Prod
      = ( produc7704165765595008946_a_b_b @ ( produc5948858871325780166_a_b_b @ Prod ) @ ( produc5719641485658034180_a_b_b @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_1023_prod_Oexhaust__sel,axiom,
    ! [Prod: produc7147531718898801626st_a_b] :
      ( Prod
      = ( produc673257793671328980st_a_b @ ( produc5507115380825155048st_a_b @ Prod ) @ ( produc6241816367173513510st_a_b @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_1024_surjective__pairing,axiom,
    ! [T: produc9164743771328383783list_a] :
      ( T
      = ( produc6837034575241423639list_a @ ( produc3698117735987127555list_a @ T ) @ ( produc8617614985401127493list_a @ T ) ) ) ).

% surjective_pairing
thf(fact_1025_surjective__pairing,axiom,
    ! [T: produc111314985273491367st_a_b] :
      ( T
      = ( produc148520996349637281st_a_b @ ( produc318716513907093941st_a_b @ T ) @ ( produc1857499627308065267st_a_b @ T ) ) ) ).

% surjective_pairing
thf(fact_1026_surjective__pairing,axiom,
    ! [T: produc6499617310964463488_a_b_b] :
      ( T
      = ( produc7704165765595008946_a_b_b @ ( produc5948858871325780166_a_b_b @ T ) @ ( produc5719641485658034180_a_b_b @ T ) ) ) ).

% surjective_pairing
thf(fact_1027_surjective__pairing,axiom,
    ! [T: produc7147531718898801626st_a_b] :
      ( T
      = ( produc673257793671328980st_a_b @ ( produc5507115380825155048st_a_b @ T ) @ ( produc6241816367173513510st_a_b @ T ) ) ) ).

% surjective_pairing
thf(fact_1028_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P3: list_a > list_a > $o,X: list_a,Y: list_a,A: produc9164743771328383783list_a] :
      ( ( P3 @ X @ Y )
     => ( ( A
          = ( produc6837034575241423639list_a @ X @ Y ) )
       => ( P3 @ ( produc3698117735987127555list_a @ A ) @ ( produc8617614985401127493list_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1029_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P3: list_a > dtree_list_a_b > $o,X: list_a,Y: dtree_list_a_b,A: produc111314985273491367st_a_b] :
      ( ( P3 @ X @ Y )
     => ( ( A
          = ( produc148520996349637281st_a_b @ X @ Y ) )
       => ( P3 @ ( produc318716513907093941st_a_b @ A ) @ ( produc1857499627308065267st_a_b @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1030_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P3: dtree_list_a_b > b > $o,X: dtree_list_a_b,Y: b,A: produc6499617310964463488_a_b_b] :
      ( ( P3 @ X @ Y )
     => ( ( A
          = ( produc7704165765595008946_a_b_b @ X @ Y ) )
       => ( P3 @ ( produc5948858871325780166_a_b_b @ A ) @ ( produc5719641485658034180_a_b_b @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1031_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P3: list_a > produc111314985273491367st_a_b > $o,X: list_a,Y: produc111314985273491367st_a_b,A: produc7147531718898801626st_a_b] :
      ( ( P3 @ X @ Y )
     => ( ( A
          = ( produc673257793671328980st_a_b @ X @ Y ) )
       => ( P3 @ ( produc5507115380825155048st_a_b @ A ) @ ( produc6241816367173513510st_a_b @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1032_cSup__eq,axiom,
    ! [X6: set_real,A: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ X6 )
         => ( ord_less_eq_real @ X3 @ A ) )
     => ( ! [Y4: real] :
            ( ! [X5: real] :
                ( ( member_real @ X5 @ X6 )
               => ( ord_less_eq_real @ X5 @ Y4 ) )
           => ( ord_less_eq_real @ A @ Y4 ) )
       => ( ( comple1385675409528146559p_real @ X6 )
          = A ) ) ) ).

% cSup_eq
thf(fact_1033_cSup__eq__maximum,axiom,
    ! [Z3: real,X6: set_real] :
      ( ( member_real @ Z3 @ X6 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ X6 )
           => ( ord_less_eq_real @ X3 @ Z3 ) )
       => ( ( comple1385675409528146559p_real @ X6 )
          = Z3 ) ) ) ).

% cSup_eq_maximum
thf(fact_1034_cSup__eq__maximum,axiom,
    ! [Z3: nat,X6: set_nat] :
      ( ( member_nat @ Z3 @ X6 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X6 )
           => ( ord_less_eq_nat @ X3 @ Z3 ) )
       => ( ( complete_Sup_Sup_nat @ X6 )
          = Z3 ) ) ) ).

% cSup_eq_maximum
thf(fact_1035_snd__eqD,axiom,
    ! [X: list_a,Y: list_a,A: list_a] :
      ( ( ( produc8617614985401127493list_a @ ( produc6837034575241423639list_a @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_1036_snd__eqD,axiom,
    ! [X: list_a,Y: dtree_list_a_b,A: dtree_list_a_b] :
      ( ( ( produc1857499627308065267st_a_b @ ( produc148520996349637281st_a_b @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_1037_snd__eqD,axiom,
    ! [X: dtree_list_a_b,Y: b,A: b] :
      ( ( ( produc5719641485658034180_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_1038_snd__eqD,axiom,
    ! [X: list_a,Y: produc111314985273491367st_a_b,A: produc111314985273491367st_a_b] :
      ( ( ( produc6241816367173513510st_a_b @ ( produc673257793671328980st_a_b @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_1039_snd__conv,axiom,
    ! [X1: list_a,X2: list_a] :
      ( ( produc8617614985401127493list_a @ ( produc6837034575241423639list_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_1040_snd__conv,axiom,
    ! [X1: list_a,X2: dtree_list_a_b] :
      ( ( produc1857499627308065267st_a_b @ ( produc148520996349637281st_a_b @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_1041_snd__conv,axiom,
    ! [X1: dtree_list_a_b,X2: b] :
      ( ( produc5719641485658034180_a_b_b @ ( produc7704165765595008946_a_b_b @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_1042_snd__conv,axiom,
    ! [X1: list_a,X2: produc111314985273491367st_a_b] :
      ( ( produc6241816367173513510st_a_b @ ( produc673257793671328980st_a_b @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_1043_less__cSupD,axiom,
    ! [X6: set_real,Z3: real] :
      ( ( X6 != bot_bot_set_real )
     => ( ( ord_less_real @ Z3 @ ( comple1385675409528146559p_real @ X6 ) )
       => ? [X3: real] :
            ( ( member_real @ X3 @ X6 )
            & ( ord_less_real @ Z3 @ X3 ) ) ) ) ).

% less_cSupD
thf(fact_1044_less__cSupD,axiom,
    ! [X6: set_nat,Z3: nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ( ord_less_nat @ Z3 @ ( complete_Sup_Sup_nat @ X6 ) )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ X6 )
            & ( ord_less_nat @ Z3 @ X3 ) ) ) ) ).

% less_cSupD
thf(fact_1045_less__cSupE,axiom,
    ! [Y: real,X6: set_real] :
      ( ( ord_less_real @ Y @ ( comple1385675409528146559p_real @ X6 ) )
     => ( ( X6 != bot_bot_set_real )
       => ~ ! [X3: real] :
              ( ( member_real @ X3 @ X6 )
             => ~ ( ord_less_real @ Y @ X3 ) ) ) ) ).

% less_cSupE
thf(fact_1046_less__cSupE,axiom,
    ! [Y: nat,X6: set_nat] :
      ( ( ord_less_nat @ Y @ ( complete_Sup_Sup_nat @ X6 ) )
     => ( ( X6 != bot_bot_set_nat )
       => ~ ! [X3: nat] :
              ( ( member_nat @ X3 @ X6 )
             => ~ ( ord_less_nat @ Y @ X3 ) ) ) ) ).

% less_cSupE
thf(fact_1047_cSup__eq__non__empty,axiom,
    ! [X6: set_real,A: real] :
      ( ( X6 != bot_bot_set_real )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ X6 )
           => ( ord_less_eq_real @ X3 @ A ) )
       => ( ! [Y4: real] :
              ( ! [X5: real] :
                  ( ( member_real @ X5 @ X6 )
                 => ( ord_less_eq_real @ X5 @ Y4 ) )
             => ( ord_less_eq_real @ A @ Y4 ) )
         => ( ( comple1385675409528146559p_real @ X6 )
            = A ) ) ) ) ).

% cSup_eq_non_empty
thf(fact_1048_cSup__eq__non__empty,axiom,
    ! [X6: set_nat,A: nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X6 )
           => ( ord_less_eq_nat @ X3 @ A ) )
       => ( ! [Y4: nat] :
              ( ! [X5: nat] :
                  ( ( member_nat @ X5 @ X6 )
                 => ( ord_less_eq_nat @ X5 @ Y4 ) )
             => ( ord_less_eq_nat @ A @ Y4 ) )
         => ( ( complete_Sup_Sup_nat @ X6 )
            = A ) ) ) ) ).

% cSup_eq_non_empty
thf(fact_1049_cSup__least,axiom,
    ! [X6: set_real,Z3: real] :
      ( ( X6 != bot_bot_set_real )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ X6 )
           => ( ord_less_eq_real @ X3 @ Z3 ) )
       => ( ord_less_eq_real @ ( comple1385675409528146559p_real @ X6 ) @ Z3 ) ) ) ).

% cSup_least
thf(fact_1050_cSup__least,axiom,
    ! [X6: set_nat,Z3: nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X6 )
           => ( ord_less_eq_nat @ X3 @ Z3 ) )
       => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X6 ) @ Z3 ) ) ) ).

% cSup_least
thf(fact_1051_cSUP__least,axiom,
    ! [A3: set_list_a,F: list_a > real,M3: real] :
      ( ( A3 != bot_bot_set_list_a )
     => ( ! [X3: list_a] :
            ( ( member_list_a @ X3 @ A3 )
           => ( ord_less_eq_real @ ( F @ X3 ) @ M3 ) )
       => ( ord_less_eq_real @ ( comple1385675409528146559p_real @ ( image_list_a_real @ F @ A3 ) ) @ M3 ) ) ) ).

% cSUP_least
thf(fact_1052_cSUP__least,axiom,
    ! [A3: set_Pr3443975907877334966_a_b_b,F: produc6499617310964463488_a_b_b > real,M3: real] :
      ( ( A3 != bot_bo4897374000430069834_a_b_b )
     => ( ! [X3: produc6499617310964463488_a_b_b] :
            ( ( member4695696432722591383_a_b_b @ X3 @ A3 )
           => ( ord_less_eq_real @ ( F @ X3 ) @ M3 ) )
       => ( ord_less_eq_real @ ( comple1385675409528146559p_real @ ( image_3497077275575164339b_real @ F @ A3 ) ) @ M3 ) ) ) ).

% cSUP_least
thf(fact_1053_cSUP__least,axiom,
    ! [A3: set_dtree_list_a_b,F: dtree_list_a_b > real,M3: real] :
      ( ( A3 != bot_bo798015271861357502st_a_b )
     => ( ! [X3: dtree_list_a_b] :
            ( ( member551035911493665803st_a_b @ X3 @ A3 )
           => ( ord_less_eq_real @ ( F @ X3 ) @ M3 ) )
       => ( ord_less_eq_real @ ( comple1385675409528146559p_real @ ( image_3232105638531924007b_real @ F @ A3 ) ) @ M3 ) ) ) ).

% cSUP_least
thf(fact_1054_cSUP__least,axiom,
    ! [A3: set_real,F: real > real,M3: real] :
      ( ( A3 != bot_bot_set_real )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A3 )
           => ( ord_less_eq_real @ ( F @ X3 ) @ M3 ) )
       => ( ord_less_eq_real @ ( comple1385675409528146559p_real @ ( image_real_real @ F @ A3 ) ) @ M3 ) ) ) ).

% cSUP_least
thf(fact_1055_cSUP__least,axiom,
    ! [A3: set_nat,F: nat > real,M3: real] :
      ( ( A3 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ( ord_less_eq_real @ ( F @ X3 ) @ M3 ) )
       => ( ord_less_eq_real @ ( comple1385675409528146559p_real @ ( image_nat_real @ F @ A3 ) ) @ M3 ) ) ) ).

% cSUP_least
thf(fact_1056_cSUP__least,axiom,
    ! [A3: set_list_a,F: list_a > nat,M3: nat] :
      ( ( A3 != bot_bot_set_list_a )
     => ( ! [X3: list_a] :
            ( ( member_list_a @ X3 @ A3 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ M3 ) )
       => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_list_a_nat @ F @ A3 ) ) @ M3 ) ) ) ).

% cSUP_least
thf(fact_1057_cSUP__least,axiom,
    ! [A3: set_Pr3443975907877334966_a_b_b,F: produc6499617310964463488_a_b_b > nat,M3: nat] :
      ( ( A3 != bot_bo4897374000430069834_a_b_b )
     => ( ! [X3: produc6499617310964463488_a_b_b] :
            ( ( member4695696432722591383_a_b_b @ X3 @ A3 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ M3 ) )
       => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_8239705153813057111_b_nat @ F @ A3 ) ) @ M3 ) ) ) ).

% cSUP_least
thf(fact_1058_cSUP__least,axiom,
    ! [A3: set_dtree_list_a_b,F: dtree_list_a_b > nat,M3: nat] :
      ( ( A3 != bot_bo798015271861357502st_a_b )
     => ( ! [X3: dtree_list_a_b] :
            ( ( member551035911493665803st_a_b @ X3 @ A3 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ M3 ) )
       => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_5414249006199368395_b_nat @ F @ A3 ) ) @ M3 ) ) ) ).

% cSUP_least
thf(fact_1059_cSUP__least,axiom,
    ! [A3: set_real,F: real > nat,M3: nat] :
      ( ( A3 != bot_bot_set_real )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A3 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ M3 ) )
       => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_real_nat @ F @ A3 ) ) @ M3 ) ) ) ).

% cSUP_least
thf(fact_1060_cSUP__least,axiom,
    ! [A3: set_nat,F: nat > nat,M3: nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ M3 ) )
       => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A3 ) ) @ M3 ) ) ) ).

% cSUP_least
thf(fact_1061_dlverts__eq__dverts__union,axiom,
    ( list_dlverts_a_b
    = ( ^ [T3: dtree_list_a_b] : ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ set_a2 @ ( dverts_list_a_b @ T3 ) ) ) ) ) ).

% dlverts_eq_dverts_union
thf(fact_1062_dlverts__eq__dverts__union_H,axiom,
    ( list_dlverts_a_b
    = ( ^ [T3: dtree_list_a_b] : ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ set_a2 @ ( dverts_list_a_b @ T3 ) ) ) ) ) ).

% dlverts_eq_dverts_union'
thf(fact_1063_ccSUP__empty,axiom,
    ! [F: nat > set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) )
      = bot_bot_set_nat ) ).

% ccSUP_empty
thf(fact_1064_UN__constant,axiom,
    ! [A3: set_nat,C: set_nat] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [Y3: nat] : C
              @ A3 ) )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [Y3: nat] : C
              @ A3 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_1065_Sup__bot__conv_I1_J,axiom,
    ! [A3: set_set_nat] :
      ( ( ( comple7399068483239264473et_nat @ A3 )
        = bot_bot_set_nat )
      = ( ! [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A3 )
           => ( X4 = bot_bot_set_nat ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_1066_Sup__bot__conv_I2_J,axiom,
    ! [A3: set_set_nat] :
      ( ( bot_bot_set_nat
        = ( comple7399068483239264473et_nat @ A3 ) )
      = ( ! [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A3 )
           => ( X4 = bot_bot_set_nat ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_1067_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty
thf(fact_1068_SUP__identity__eq,axiom,
    ! [A3: set_nat] :
      ( ( complete_Sup_Sup_nat
        @ ( image_nat_nat
          @ ^ [X4: nat] : X4
          @ A3 ) )
      = ( complete_Sup_Sup_nat @ A3 ) ) ).

% SUP_identity_eq
thf(fact_1069_UN__I,axiom,
    ! [A: real,A3: set_real,B: real,B5: real > set_real] :
      ( ( member_real @ A @ A3 )
     => ( ( member_real @ B @ ( B5 @ A ) )
       => ( member_real @ B @ ( comple3096694443085538997t_real @ ( image_real_set_real @ B5 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_1070_UN__I,axiom,
    ! [A: list_a,A3: set_list_a,B: real,B5: list_a > set_real] :
      ( ( member_list_a @ A @ A3 )
     => ( ( member_real @ B @ ( B5 @ A ) )
       => ( member_real @ B @ ( comple3096694443085538997t_real @ ( image_8269827006147867297t_real @ B5 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_1071_UN__I,axiom,
    ! [A: real,A3: set_real,B: list_a,B5: real > set_list_a] :
      ( ( member_real @ A @ A3 )
     => ( ( member_list_a @ B @ ( B5 @ A ) )
       => ( member_list_a @ B @ ( comple6928918032620976721list_a @ ( image_3983288985320528369list_a @ B5 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_1072_UN__I,axiom,
    ! [A: list_a,A3: set_list_a,B: list_a,B5: list_a > set_list_a] :
      ( ( member_list_a @ A @ A3 )
     => ( ( member_list_a @ B @ ( B5 @ A ) )
       => ( member_list_a @ B @ ( comple6928918032620976721list_a @ ( image_5464838071766335845list_a @ B5 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_1073_UN__I,axiom,
    ! [A: dtree_list_a_b,A3: set_dtree_list_a_b,B: real,B5: dtree_list_a_b > set_real] :
      ( ( member551035911493665803st_a_b @ A @ A3 )
     => ( ( member_real @ B @ ( B5 @ A ) )
       => ( member_real @ B @ ( comple3096694443085538997t_real @ ( image_3381912758259792989t_real @ B5 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_1074_UN__I,axiom,
    ! [A: real,A3: set_real,B: dtree_list_a_b,B5: real > set_dtree_list_a_b] :
      ( ( member_real @ A @ A3 )
     => ( ( member551035911493665803st_a_b @ B @ ( B5 @ A ) )
       => ( member551035911493665803st_a_b @ B @ ( comple3787021748553583613st_a_b @ ( image_5051693651339568221st_a_b @ B5 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_1075_UN__I,axiom,
    ! [A: list_a,A3: set_list_a,B: dtree_list_a_b,B5: list_a > set_dtree_list_a_b] :
      ( ( member_list_a @ A @ A3 )
     => ( ( member551035911493665803st_a_b @ B @ ( B5 @ A ) )
       => ( member551035911493665803st_a_b @ B @ ( comple3787021748553583613st_a_b @ ( image_5859329506579205097st_a_b @ B5 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_1076_UN__I,axiom,
    ! [A: dtree_list_a_b,A3: set_dtree_list_a_b,B: list_a,B5: dtree_list_a_b > set_list_a] :
      ( ( member551035911493665803st_a_b @ A @ A3 )
     => ( ( member_list_a @ B @ ( B5 @ A ) )
       => ( member_list_a @ B @ ( comple6928918032620976721list_a @ ( image_7081116771595656873list_a @ B5 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_1077_UN__I,axiom,
    ! [A: produc6499617310964463488_a_b_b,A3: set_Pr3443975907877334966_a_b_b,B: real,B5: produc6499617310964463488_a_b_b > set_real] :
      ( ( member4695696432722591383_a_b_b @ A @ A3 )
     => ( ( member_real @ B @ ( B5 @ A ) )
       => ( member_real @ B @ ( comple3096694443085538997t_real @ ( image_7208769787748919785t_real @ B5 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_1078_UN__I,axiom,
    ! [A: real,A3: set_real,B: produc6499617310964463488_a_b_b,B5: real > set_Pr3443975907877334966_a_b_b] :
      ( ( member_real @ A @ A3 )
     => ( ( member4695696432722591383_a_b_b @ B @ ( B5 @ A ) )
       => ( member4695696432722591383_a_b_b @ B @ ( comple5737441310262272393_a_b_b @ ( image_5847990271261284841_a_b_b @ B5 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_1079_ccSup__empty,axiom,
    ( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
    = bot_bot_set_nat ) ).

% ccSup_empty
thf(fact_1080_Sup__empty,axiom,
    ( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
    = bot_bot_set_nat ) ).

% Sup_empty
thf(fact_1081_SUP__Sup__eq2,axiom,
    ! [S: set_se4091964020838898919list_a] :
      ( ( comple3554826314717667675st_a_o
        @ ( image_8564175721147914668st_a_o
          @ ^ [I: set_Pr4048851178543822343list_a,X4: list_a,Y3: list_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Y3 ) @ I )
          @ S ) )
      = ( ^ [X4: list_a,Y3: list_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Y3 ) @ ( comple1497117618345130356list_a @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_1082_SUP__Sup__eq2,axiom,
    ! [S: set_se3485313402130513683st_a_b] :
      ( ( comple396325037688036751_a_b_o
        @ ( image_3304428138862881312_a_b_o
          @ ^ [I: set_Pr5727314569254441693st_a_b,X4: list_a,Y3: dtree_list_a_b] : ( member6763472504040263486st_a_b @ ( produc148520996349637281st_a_b @ X4 @ Y3 ) @ I )
          @ S ) )
      = ( ^ [X4: list_a,Y3: dtree_list_a_b] : ( member6763472504040263486st_a_b @ ( produc148520996349637281st_a_b @ X4 @ Y3 ) @ ( comple8655260255396921136st_a_b @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_1083_SUP__Sup__eq2,axiom,
    ! [S: set_se486103598301167852_a_b_b] :
      ( ( comple5639742380414582692_b_b_o
        @ ( image_7324209900139855886_b_b_o
          @ ^ [I: set_Pr3443975907877334966_a_b_b,X4: dtree_list_a_b,Y3: b] : ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X4 @ Y3 ) @ I )
          @ S ) )
      = ( ^ [X4: dtree_list_a_b,Y3: b] : ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X4 @ Y3 ) @ ( comple5737441310262272393_a_b_b @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_1084_SUP__Sup__eq2,axiom,
    ! [S: set_se3213122578561875782st_a_b] :
      ( ( comple8439212109263825820_a_b_o
        @ ( image_1933873314476848224_a_b_o
          @ ^ [I: set_Pr8451570673126195472st_a_b,X4: list_a,Y3: produc111314985273491367st_a_b] : ( member4895573663636194545st_a_b @ ( produc673257793671328980st_a_b @ X4 @ Y3 ) @ I )
          @ S ) )
      = ( ^ [X4: list_a,Y3: produc111314985273491367st_a_b] : ( member4895573663636194545st_a_b @ ( produc673257793671328980st_a_b @ X4 @ Y3 ) @ ( comple3151476520729156835st_a_b @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_1085_Sup__set__def,axiom,
    ( comple6928918032620976721list_a
    = ( ^ [A6: set_set_list_a] :
          ( collect_list_a
          @ ^ [X4: list_a] : ( complete_Sup_Sup_o @ ( image_set_list_a_o @ ( member_list_a @ X4 ) @ A6 ) ) ) ) ) ).

% Sup_set_def
thf(fact_1086_Sup__set__def,axiom,
    ( comple5737441310262272393_a_b_b
    = ( ^ [A6: set_se486103598301167852_a_b_b] :
          ( collec3241703154345475797_a_b_b
          @ ^ [X4: produc6499617310964463488_a_b_b] : ( complete_Sup_Sup_o @ ( image_4582484477583286171_b_b_o @ ( member4695696432722591383_a_b_b @ X4 ) @ A6 ) ) ) ) ) ).

% Sup_set_def
thf(fact_1087_Sup__set__def,axiom,
    ( comple3787021748553583613st_a_b
    = ( ^ [A6: set_se4773973851007363424st_a_b] :
          ( collec2944820760411501129st_a_b
          @ ^ [X4: dtree_list_a_b] : ( complete_Sup_Sup_o @ ( image_6096855620777653543_a_b_o @ ( member551035911493665803st_a_b @ X4 ) @ A6 ) ) ) ) ) ).

% Sup_set_def
thf(fact_1088_Sup__set__def,axiom,
    ( comple3096694443085538997t_real
    = ( ^ [A6: set_set_real] :
          ( collect_real
          @ ^ [X4: real] : ( complete_Sup_Sup_o @ ( image_set_real_o @ ( member_real @ X4 ) @ A6 ) ) ) ) ) ).

% Sup_set_def
thf(fact_1089_Sup__SUP__eq,axiom,
    ( comple7218726029605528012st_a_o
    = ( ^ [S6: set_list_a_o,X4: list_a] : ( member_list_a @ X4 @ ( comple6928918032620976721list_a @ ( image_5820879363088598756list_a @ collect_list_a @ S6 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_1090_Sup__SUP__eq,axiom,
    ( comple1499699326375664188_b_b_o
    = ( ^ [S6: set_Pr5782749397791665583_b_b_o,X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ ( comple5737441310262272393_a_b_b @ ( image_5873563582397423296_a_b_b @ collec3241703154345475797_a_b_b @ S6 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_1091_Sup__SUP__eq,axiom,
    ( comple8325939764076033224_a_b_o
    = ( ^ [S6: set_dtree_list_a_b_o,X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ ( comple3787021748553583613st_a_b @ ( image_182262869802767808st_a_b @ collec2944820760411501129st_a_b @ S6 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_1092_Sup__SUP__eq,axiom,
    ( comple3015195443809154064real_o
    = ( ^ [S6: set_real_o,X4: real] : ( member_real @ X4 @ ( comple3096694443085538997t_real @ ( image_2734271470692514752t_real @ collect_real @ S6 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_1093_SUP__Sup__eq,axiom,
    ! [S: set_set_list_a] :
      ( ( comple7218726029605528012st_a_o
        @ ( image_5692628901704076184st_a_o
          @ ^ [I: set_list_a,X4: list_a] : ( member_list_a @ X4 @ I )
          @ S ) )
      = ( ^ [X4: list_a] : ( member_list_a @ X4 @ ( comple6928918032620976721list_a @ S ) ) ) ) ).

% SUP_Sup_eq
thf(fact_1094_SUP__Sup__eq,axiom,
    ! [S: set_se486103598301167852_a_b_b] :
      ( ( comple1499699326375664188_b_b_o
        @ ( image_780772297388590674_b_b_o
          @ ^ [I: set_Pr3443975907877334966_a_b_b,X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ I )
          @ S ) )
      = ( ^ [X4: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X4 @ ( comple5737441310262272393_a_b_b @ S ) ) ) ) ).

% SUP_Sup_eq
thf(fact_1095_SUP__Sup__eq,axiom,
    ! [S: set_se4773973851007363424st_a_b] :
      ( ( comple8325939764076033224_a_b_o
        @ ( image_430045828685645930_a_b_o
          @ ^ [I: set_dtree_list_a_b,X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ I )
          @ S ) )
      = ( ^ [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ ( comple3787021748553583613st_a_b @ S ) ) ) ) ).

% SUP_Sup_eq
thf(fact_1096_SUP__Sup__eq,axiom,
    ! [S: set_set_real] :
      ( ( comple3015195443809154064real_o
        @ ( image_5650221686686655994real_o
          @ ^ [I: set_real,X4: real] : ( member_real @ X4 @ I )
          @ S ) )
      = ( ^ [X4: real] : ( member_real @ X4 @ ( comple3096694443085538997t_real @ S ) ) ) ) ).

% SUP_Sup_eq
thf(fact_1097_wf__dtree_Odtail__root__in__set,axiom,
    ! [T: dtree_list_a_b,E: b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,Dt: b > list_a] :
      ( ( wf_dtree_list_a_b @ T )
     => ( ( member_b @ E @ ( darcs_list_a_b @ T ) )
       => ( ( T
            = ( node_list_a_b @ R2 @ Xs2 ) )
         => ( ( ( dtail_list_a_b @ T @ Dt @ E )
              = R2 )
           => ( member_b @ E @ ( image_4684437738885282872_b_b_b @ produc5719641485658034180_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) ) ) ) ) ) ).

% wf_dtree.dtail_root_in_set
thf(fact_1098_Union__empty,axiom,
    ( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
    = bot_bot_set_nat ) ).

% Union_empty
thf(fact_1099_Union__empty__conv,axiom,
    ! [A3: set_set_nat] :
      ( ( ( comple7399068483239264473et_nat @ A3 )
        = bot_bot_set_nat )
      = ( ! [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A3 )
           => ( X4 = bot_bot_set_nat ) ) ) ) ).

% Union_empty_conv
thf(fact_1100_empty__Union__conv,axiom,
    ! [A3: set_set_nat] :
      ( ( bot_bot_set_nat
        = ( comple7399068483239264473et_nat @ A3 ) )
      = ( ! [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A3 )
           => ( X4 = bot_bot_set_nat ) ) ) ) ).

% empty_Union_conv
thf(fact_1101_wf__dtree_Oout__arcs__in__subarcs__aux,axiom,
    ! [T: dtree_list_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,Dt: b > list_a,E: b] :
      ( ( wf_dtree_list_a_b @ T )
     => ( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ T )
       => ( ( ( dtail_list_a_b @ T @ Dt @ E )
            = R2 )
         => ( ( member_b @ E @ ( darcs_list_a_b @ T ) )
           => ( member_b @ E @ ( image_4684437738885282872_b_b_b @ produc5719641485658034180_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) ) ) ) ) ) ).

% wf_dtree.out_arcs_in_subarcs_aux
thf(fact_1102_UN__E,axiom,
    ! [B: real,B5: real > set_real,A3: set_real] :
      ( ( member_real @ B @ ( comple3096694443085538997t_real @ ( image_real_set_real @ B5 @ A3 ) ) )
     => ~ ! [X3: real] :
            ( ( member_real @ X3 @ A3 )
           => ~ ( member_real @ B @ ( B5 @ X3 ) ) ) ) ).

% UN_E
thf(fact_1103_UN__E,axiom,
    ! [B: list_a,B5: real > set_list_a,A3: set_real] :
      ( ( member_list_a @ B @ ( comple6928918032620976721list_a @ ( image_3983288985320528369list_a @ B5 @ A3 ) ) )
     => ~ ! [X3: real] :
            ( ( member_real @ X3 @ A3 )
           => ~ ( member_list_a @ B @ ( B5 @ X3 ) ) ) ) ).

% UN_E
thf(fact_1104_UN__E,axiom,
    ! [B: real,B5: list_a > set_real,A3: set_list_a] :
      ( ( member_real @ B @ ( comple3096694443085538997t_real @ ( image_8269827006147867297t_real @ B5 @ A3 ) ) )
     => ~ ! [X3: list_a] :
            ( ( member_list_a @ X3 @ A3 )
           => ~ ( member_real @ B @ ( B5 @ X3 ) ) ) ) ).

% UN_E
thf(fact_1105_UN__E,axiom,
    ! [B: list_a,B5: list_a > set_list_a,A3: set_list_a] :
      ( ( member_list_a @ B @ ( comple6928918032620976721list_a @ ( image_5464838071766335845list_a @ B5 @ A3 ) ) )
     => ~ ! [X3: list_a] :
            ( ( member_list_a @ X3 @ A3 )
           => ~ ( member_list_a @ B @ ( B5 @ X3 ) ) ) ) ).

% UN_E
thf(fact_1106_UN__E,axiom,
    ! [B: dtree_list_a_b,B5: real > set_dtree_list_a_b,A3: set_real] :
      ( ( member551035911493665803st_a_b @ B @ ( comple3787021748553583613st_a_b @ ( image_5051693651339568221st_a_b @ B5 @ A3 ) ) )
     => ~ ! [X3: real] :
            ( ( member_real @ X3 @ A3 )
           => ~ ( member551035911493665803st_a_b @ B @ ( B5 @ X3 ) ) ) ) ).

% UN_E
thf(fact_1107_UN__E,axiom,
    ! [B: real,B5: dtree_list_a_b > set_real,A3: set_dtree_list_a_b] :
      ( ( member_real @ B @ ( comple3096694443085538997t_real @ ( image_3381912758259792989t_real @ B5 @ A3 ) ) )
     => ~ ! [X3: dtree_list_a_b] :
            ( ( member551035911493665803st_a_b @ X3 @ A3 )
           => ~ ( member_real @ B @ ( B5 @ X3 ) ) ) ) ).

% UN_E
thf(fact_1108_UN__E,axiom,
    ! [B: list_a,B5: dtree_list_a_b > set_list_a,A3: set_dtree_list_a_b] :
      ( ( member_list_a @ B @ ( comple6928918032620976721list_a @ ( image_7081116771595656873list_a @ B5 @ A3 ) ) )
     => ~ ! [X3: dtree_list_a_b] :
            ( ( member551035911493665803st_a_b @ X3 @ A3 )
           => ~ ( member_list_a @ B @ ( B5 @ X3 ) ) ) ) ).

% UN_E
thf(fact_1109_UN__E,axiom,
    ! [B: dtree_list_a_b,B5: list_a > set_dtree_list_a_b,A3: set_list_a] :
      ( ( member551035911493665803st_a_b @ B @ ( comple3787021748553583613st_a_b @ ( image_5859329506579205097st_a_b @ B5 @ A3 ) ) )
     => ~ ! [X3: list_a] :
            ( ( member_list_a @ X3 @ A3 )
           => ~ ( member551035911493665803st_a_b @ B @ ( B5 @ X3 ) ) ) ) ).

% UN_E
thf(fact_1110_UN__E,axiom,
    ! [B: produc6499617310964463488_a_b_b,B5: real > set_Pr3443975907877334966_a_b_b,A3: set_real] :
      ( ( member4695696432722591383_a_b_b @ B @ ( comple5737441310262272393_a_b_b @ ( image_5847990271261284841_a_b_b @ B5 @ A3 ) ) )
     => ~ ! [X3: real] :
            ( ( member_real @ X3 @ A3 )
           => ~ ( member4695696432722591383_a_b_b @ B @ ( B5 @ X3 ) ) ) ) ).

% UN_E
thf(fact_1111_UN__E,axiom,
    ! [B: real,B5: produc6499617310964463488_a_b_b > set_real,A3: set_Pr3443975907877334966_a_b_b] :
      ( ( member_real @ B @ ( comple3096694443085538997t_real @ ( image_7208769787748919785t_real @ B5 @ A3 ) ) )
     => ~ ! [X3: produc6499617310964463488_a_b_b] :
            ( ( member4695696432722591383_a_b_b @ X3 @ A3 )
           => ~ ( member_real @ B @ ( B5 @ X3 ) ) ) ) ).

% UN_E
thf(fact_1112_Union__disjoint,axiom,
    ! [C3: set_set_nat,A3: set_nat] :
      ( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ C3 ) @ A3 )
        = bot_bot_set_nat )
      = ( ! [X4: set_nat] :
            ( ( member_set_nat @ X4 @ C3 )
           => ( ( inf_inf_set_nat @ X4 @ A3 )
              = bot_bot_set_nat ) ) ) ) ).

% Union_disjoint
thf(fact_1113_UN__empty,axiom,
    ! [B5: nat > set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ bot_bot_set_nat ) )
      = bot_bot_set_nat ) ).

% UN_empty
thf(fact_1114_SUP__constant,axiom,
    ! [A3: set_nat,C: set_nat] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [Y3: nat] : C
              @ A3 ) )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [Y3: nat] : C
              @ A3 ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_1115_SUP__empty,axiom,
    ! [F: nat > set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) )
      = bot_bot_set_nat ) ).

% SUP_empty
thf(fact_1116_vert__disjoint__if__to__list,axiom,
    ! [R2: list_a,T12: dtree_list_a_b,E1: b,V: list_a] :
      ( ( list_wf_dlverts_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) )
     => ( ( member_list_a @ V @ ( image_4065140400681997415list_a @ product_fst_list_a_b @ ( set_Pr6156699709417984823st_a_b @ ( dtree_5522862299006258336st_a_b @ T12 ) ) ) )
       => ( ( inf_inf_set_a @ ( set_a2 @ ( root_list_a_b @ T12 ) ) @ ( set_a2 @ V ) )
          = bot_bot_set_a ) ) ) ).

% vert_disjoint_if_to_list
thf(fact_1117_wf__dlverts__sub,axiom,
    ! [Xs2: fset_P2153231429829016240_a_b_b,Ys: fset_P2153231429829016240_a_b_b,R2: list_a] :
      ( ( ord_le8870638447146015504_a_b_b @ Xs2 @ Ys )
     => ( ( list_wf_dlverts_a_b @ ( node_list_a_b @ R2 @ Ys ) )
       => ( list_wf_dlverts_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ).

% wf_dlverts_sub
thf(fact_1118_dverts__same__if__set__wf,axiom,
    ! [T: dtree_list_a_b,V1: list_a,V22: list_a,X: a] :
      ( ( list_wf_dlverts_a_b @ T )
     => ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T ) )
       => ( ( member_list_a @ V22 @ ( dverts_list_a_b @ T ) )
         => ( ( member_a @ X @ ( set_a2 @ V1 ) )
           => ( ( member_a @ X @ ( set_a2 @ V22 ) )
             => ( V1 = V22 ) ) ) ) ) ) ).

% dverts_same_if_set_wf
thf(fact_1119_root__if__same__lvert__wf,axiom,
    ! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: a,V: list_a] :
      ( ( list_wf_dlverts_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
     => ( ( member_a @ X @ ( set_a2 @ R2 ) )
       => ( ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
         => ( ( member_a @ X @ ( set_a2 @ V ) )
           => ( V = R2 ) ) ) ) ) ).

% root_if_same_lvert_wf
thf(fact_1120_dverts__disj__if__wf__dlverts,axiom,
    ! [T: dtree_list_a_b,V1: list_a,V22: list_a] :
      ( ( list_wf_dlverts_a_b @ T )
     => ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T ) )
       => ( ( member_list_a @ V22 @ ( dverts_list_a_b @ T ) )
         => ( ( V1 != V22 )
           => ( ( inf_inf_set_a @ ( set_a2 @ V1 ) @ ( set_a2 @ V22 ) )
              = bot_bot_set_a ) ) ) ) ) ).

% dverts_disj_if_wf_dlverts
thf(fact_1121_dverts__inter__empty__if__verts__inter,axiom,
    ! [X: dtree_list_a_b,Y: dtree_list_a_b] :
      ( ( ( inf_inf_set_a @ ( list_dlverts_a_b @ X ) @ ( list_dlverts_a_b @ Y ) )
        = bot_bot_set_a )
     => ( ( list_wf_dlverts_a_b @ X )
       => ( ( inf_inf_set_list_a @ ( dverts_list_a_b @ X ) @ ( dverts_list_a_b @ Y ) )
          = bot_bot_set_list_a ) ) ) ).

% dverts_inter_empty_if_verts_inter
thf(fact_1122_wf__dlverts__suc,axiom,
    ! [T: dtree_list_a_b,T12: dtree_list_a_b] :
      ( ( list_wf_dlverts_a_b @ T )
     => ( ( member551035911493665803st_a_b @ T12 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) ) )
       => ( list_wf_dlverts_a_b @ T12 ) ) ) ).

% wf_dlverts_suc
thf(fact_1123_wf__dlverts__sucs,axiom,
    ! [T: dtree_list_a_b,X: produc6499617310964463488_a_b_b] :
      ( ( list_wf_dlverts_a_b @ T )
     => ( ( member4695696432722591383_a_b_b @ X @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) )
       => ( list_wf_dlverts_a_b @ ( node_list_a_b @ ( root_list_a_b @ T ) @ ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ).

% wf_dlverts_sucs
thf(fact_1124_notin__dlverts__child__if__wf__in__root,axiom,
    ! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: a,T: dtree_list_a_b] :
      ( ( list_wf_dlverts_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
     => ( ( member_a @ X @ ( set_a2 @ R2 ) )
       => ( ( member551035911493665803st_a_b @ T @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
         => ~ ( member_a @ X @ ( list_dlverts_a_b @ T ) ) ) ) ) ).

% notin_dlverts_child_if_wf_in_root
thf(fact_1125_dlverts__notin__root__sucs,axiom,
    ! [T: dtree_list_a_b,T12: dtree_list_a_b,X: a] :
      ( ( list_wf_dlverts_a_b @ T )
     => ( ( member551035911493665803st_a_b @ T12 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) ) )
       => ( ( member_a @ X @ ( list_dlverts_a_b @ T12 ) )
         => ~ ( member_a @ X @ ( set_a2 @ ( root_list_a_b @ T ) ) ) ) ) ) ).

% dlverts_notin_root_sucs
thf(fact_1126_notin__dlverts__suc__if__wf__in__root,axiom,
    ! [T12: dtree_list_a_b,X: a,T22: dtree_list_a_b] :
      ( ( list_wf_dlverts_a_b @ T12 )
     => ( ( member_a @ X @ ( set_a2 @ ( root_list_a_b @ T12 ) ) )
       => ( ( member551035911493665803st_a_b @ T22 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T12 ) ) ) )
         => ~ ( member_a @ X @ ( list_dlverts_a_b @ T22 ) ) ) ) ) ).

% notin_dlverts_suc_if_wf_in_root
thf(fact_1127_dtree__to__list__subtree__dverts__eq__fsts_H,axiom,
    ! [T: dtree_list_a_list_a,R2: list_a,Xs2: fset_P6656299774166858224list_a] :
      ( ( ord_less_eq_nat @ ( max_de8369102501826120024list_a @ T ) @ one_one_nat )
     => ( ( strict9149861969251335199list_a @ ( node_list_a_list_a @ R2 @ Xs2 ) @ T )
       => ? [As2: list_P321204300973800749list_a,E2: list_a,Bs: list_P321204300973800749list_a] :
            ( ( ( dtree_6750851072906405029list_a @ T )
              = ( append622457529216362434list_a @ As2 @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ R2 @ E2 ) @ Bs ) ) )
            & ( ( image_5971222721807453474list_a @ produc3698117735987127555list_a @ ( set_Pr2906193453920572092list_a @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ R2 @ E2 ) @ Bs ) ) )
              = ( dverts_list_a_list_a @ ( node_list_a_list_a @ R2 @ Xs2 ) ) ) ) ) ) ).

% dtree_to_list_subtree_dverts_eq_fsts'
thf(fact_1128_dtree__to__list__subtree__dverts__eq__fsts_H,axiom,
    ! [T: dtree_4875458026974816597st_a_b,R2: list_a,Xs2: fset_P6340920821931359400st_a_b] :
      ( ( ord_less_eq_nat @ ( max_de3954538496131799840st_a_b @ T ) @ one_one_nat )
     => ( ( strict4128509144933472153st_a_b @ ( node_l8355664270593351482st_a_b @ R2 @ Xs2 ) @ T )
       => ? [As2: list_P7455220050002504759st_a_b,E2: dtree_list_a_b,Bs: list_P7455220050002504759st_a_b] :
            ( ( ( dtree_547020913002174355st_a_b @ T )
              = ( append2144729903467096396st_a_b @ As2 @ ( cons_P1842004463258618993st_a_b @ ( produc148520996349637281st_a_b @ R2 @ E2 ) @ Bs ) ) )
            & ( ( image_5758649495165665878list_a @ produc318716513907093941st_a_b @ ( set_Pr7992201861846349266st_a_b @ ( cons_P1842004463258618993st_a_b @ ( produc148520996349637281st_a_b @ R2 @ E2 ) @ Bs ) ) )
              = ( dverts3100150328211173758st_a_b @ ( node_l8355664270593351482st_a_b @ R2 @ Xs2 ) ) ) ) ) ) ).

% dtree_to_list_subtree_dverts_eq_fsts'
thf(fact_1129_dtree__to__list__subtree__dverts__eq__fsts_H,axiom,
    ! [T: dtree_2614965411716861998_a_b_b,R2: dtree_list_a_b,Xs2: fset_P8033862543129508702_b_b_b] :
      ( ( ord_less_eq_nat @ ( max_de926933835450762161_a_b_b @ T ) @ one_one_nat )
     => ( ( strict7398043843113750698_a_b_b @ ( node_d3522988236360801483_a_b_b @ R2 @ Xs2 ) @ T )
       => ? [As2: list_P7479555583871319568_a_b_b,E2: b,Bs: list_P7479555583871319568_a_b_b] :
            ( ( ( dtree_5700413169342715300_a_b_b @ T )
              = ( append1383687441759991717_a_b_b @ As2 @ ( cons_P433472372565078346_a_b_b @ ( produc7704165765595008946_a_b_b @ R2 @ E2 ) @ Bs ) ) )
            & ( ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( set_Pr3241176995924754219_a_b_b @ ( cons_P433472372565078346_a_b_b @ ( produc7704165765595008946_a_b_b @ R2 @ E2 ) @ Bs ) ) )
              = ( dverts3279282023300981007_a_b_b @ ( node_d3522988236360801483_a_b_b @ R2 @ Xs2 ) ) ) ) ) ) ).

% dtree_to_list_subtree_dverts_eq_fsts'
thf(fact_1130_dtree__to__list__subtree__dverts__eq__fsts_H,axiom,
    ! [T: dtree_5894418989983070344st_a_b,R2: list_a,Xs2: fset_P1196402277819410702st_a_b] :
      ( ( ord_less_eq_nat @ ( max_de6115843874381134547st_a_b @ T ) @ one_one_nat )
     => ( ( strict2554826646234793420st_a_b @ ( node_l6315829471324094445st_a_b @ R2 @ Xs2 ) @ T )
       => ? [As2: list_P2775315667170835306st_a_b,E2: produc111314985273491367st_a_b,Bs: list_P2775315667170835306st_a_b] :
            ( ( ( dtree_3265670640512857798st_a_b @ T )
              = ( append7889034872329257471st_a_b @ As2 @ ( cons_P9012681797030764964st_a_b @ ( produc673257793671328980st_a_b @ R2 @ E2 ) @ Bs ) ) )
            & ( ( image_2865457199250190179list_a @ produc5507115380825155048st_a_b @ ( set_Pr2292375947623472517st_a_b @ ( cons_P9012681797030764964st_a_b @ ( produc673257793671328980st_a_b @ R2 @ E2 ) @ Bs ) ) )
              = ( dverts6436764938546248753st_a_b @ ( node_l6315829471324094445st_a_b @ R2 @ Xs2 ) ) ) ) ) ) ).

% dtree_to_list_subtree_dverts_eq_fsts'
thf(fact_1131_dtree__to__list__subtree__dverts__eq__fsts_H,axiom,
    ! [T: dtree_list_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
      ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T ) @ one_one_nat )
     => ( ( strict8995144569104247066st_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ T )
       => ? [As2: list_P2281458131032066856st_a_b,E2: b,Bs: list_P2281458131032066856st_a_b] :
            ( ( ( dtree_5522862299006258336st_a_b @ T )
              = ( append8668862642614166333st_a_b @ As2 @ ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ R2 @ E2 ) @ Bs ) ) )
            & ( ( image_4065140400681997415list_a @ product_fst_list_a_b @ ( set_Pr6156699709417984823st_a_b @ ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ R2 @ E2 ) @ Bs ) ) )
              = ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ) ) ).

% dtree_to_list_subtree_dverts_eq_fsts'
thf(fact_1132_dtree__to__list__split__subtree__dverts__eq__fsts_H,axiom,
    ! [T: dtree_list_a_list_a,As3: list_P321204300973800749list_a,R2: list_a,E: list_a,Bs2: list_P321204300973800749list_a] :
      ( ( ord_less_eq_nat @ ( max_de8369102501826120024list_a @ T ) @ one_one_nat )
     => ( ( ( append622457529216362434list_a @ As3 @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ R2 @ E ) @ Bs2 ) )
          = ( dtree_6750851072906405029list_a @ T ) )
       => ? [Xs: fset_P6656299774166858224list_a] :
            ( ( strict9149861969251335199list_a @ ( node_list_a_list_a @ R2 @ Xs ) @ T )
            & ( ( dverts_list_a_list_a @ ( node_list_a_list_a @ R2 @ Xs ) )
              = ( image_5971222721807453474list_a @ produc3698117735987127555list_a @ ( set_Pr2906193453920572092list_a @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ R2 @ E ) @ Bs2 ) ) ) ) ) ) ) ).

% dtree_to_list_split_subtree_dverts_eq_fsts'
thf(fact_1133_dtree__to__list__split__subtree__dverts__eq__fsts_H,axiom,
    ! [T: dtree_4875458026974816597st_a_b,As3: list_P7455220050002504759st_a_b,R2: list_a,E: dtree_list_a_b,Bs2: list_P7455220050002504759st_a_b] :
      ( ( ord_less_eq_nat @ ( max_de3954538496131799840st_a_b @ T ) @ one_one_nat )
     => ( ( ( append2144729903467096396st_a_b @ As3 @ ( cons_P1842004463258618993st_a_b @ ( produc148520996349637281st_a_b @ R2 @ E ) @ Bs2 ) )
          = ( dtree_547020913002174355st_a_b @ T ) )
       => ? [Xs: fset_P6340920821931359400st_a_b] :
            ( ( strict4128509144933472153st_a_b @ ( node_l8355664270593351482st_a_b @ R2 @ Xs ) @ T )
            & ( ( dverts3100150328211173758st_a_b @ ( node_l8355664270593351482st_a_b @ R2 @ Xs ) )
              = ( image_5758649495165665878list_a @ produc318716513907093941st_a_b @ ( set_Pr7992201861846349266st_a_b @ ( cons_P1842004463258618993st_a_b @ ( produc148520996349637281st_a_b @ R2 @ E ) @ Bs2 ) ) ) ) ) ) ) ).

% dtree_to_list_split_subtree_dverts_eq_fsts'
thf(fact_1134_dtree__to__list__split__subtree__dverts__eq__fsts_H,axiom,
    ! [T: dtree_2614965411716861998_a_b_b,As3: list_P7479555583871319568_a_b_b,R2: dtree_list_a_b,E: b,Bs2: list_P7479555583871319568_a_b_b] :
      ( ( ord_less_eq_nat @ ( max_de926933835450762161_a_b_b @ T ) @ one_one_nat )
     => ( ( ( append1383687441759991717_a_b_b @ As3 @ ( cons_P433472372565078346_a_b_b @ ( produc7704165765595008946_a_b_b @ R2 @ E ) @ Bs2 ) )
          = ( dtree_5700413169342715300_a_b_b @ T ) )
       => ? [Xs: fset_P8033862543129508702_b_b_b] :
            ( ( strict7398043843113750698_a_b_b @ ( node_d3522988236360801483_a_b_b @ R2 @ Xs ) @ T )
            & ( ( dverts3279282023300981007_a_b_b @ ( node_d3522988236360801483_a_b_b @ R2 @ Xs ) )
              = ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( set_Pr3241176995924754219_a_b_b @ ( cons_P433472372565078346_a_b_b @ ( produc7704165765595008946_a_b_b @ R2 @ E ) @ Bs2 ) ) ) ) ) ) ) ).

% dtree_to_list_split_subtree_dverts_eq_fsts'
thf(fact_1135_dtree__to__list__split__subtree__dverts__eq__fsts_H,axiom,
    ! [T: dtree_5894418989983070344st_a_b,As3: list_P2775315667170835306st_a_b,R2: list_a,E: produc111314985273491367st_a_b,Bs2: list_P2775315667170835306st_a_b] :
      ( ( ord_less_eq_nat @ ( max_de6115843874381134547st_a_b @ T ) @ one_one_nat )
     => ( ( ( append7889034872329257471st_a_b @ As3 @ ( cons_P9012681797030764964st_a_b @ ( produc673257793671328980st_a_b @ R2 @ E ) @ Bs2 ) )
          = ( dtree_3265670640512857798st_a_b @ T ) )
       => ? [Xs: fset_P1196402277819410702st_a_b] :
            ( ( strict2554826646234793420st_a_b @ ( node_l6315829471324094445st_a_b @ R2 @ Xs ) @ T )
            & ( ( dverts6436764938546248753st_a_b @ ( node_l6315829471324094445st_a_b @ R2 @ Xs ) )
              = ( image_2865457199250190179list_a @ produc5507115380825155048st_a_b @ ( set_Pr2292375947623472517st_a_b @ ( cons_P9012681797030764964st_a_b @ ( produc673257793671328980st_a_b @ R2 @ E ) @ Bs2 ) ) ) ) ) ) ) ).

% dtree_to_list_split_subtree_dverts_eq_fsts'
thf(fact_1136_dtree__to__list__split__subtree__dverts__eq__fsts_H,axiom,
    ! [T: dtree_list_a_b,As3: list_P2281458131032066856st_a_b,R2: list_a,E: b,Bs2: list_P2281458131032066856st_a_b] :
      ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T ) @ one_one_nat )
     => ( ( ( append8668862642614166333st_a_b @ As3 @ ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ R2 @ E ) @ Bs2 ) )
          = ( dtree_5522862299006258336st_a_b @ T ) )
       => ? [Xs: fset_P2153231429829016240_a_b_b] :
            ( ( strict8995144569104247066st_a_b @ ( node_list_a_b @ R2 @ Xs ) @ T )
            & ( ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs ) )
              = ( image_4065140400681997415list_a @ product_fst_list_a_b @ ( set_Pr6156699709417984823st_a_b @ ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ R2 @ E ) @ Bs2 ) ) ) ) ) ) ) ).

% dtree_to_list_split_subtree_dverts_eq_fsts'
thf(fact_1137_set__ConsD,axiom,
    ! [Y: list_a,X: list_a,Xs2: list_list_a] :
      ( ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X @ Xs2 ) ) )
     => ( ( Y = X )
        | ( member_list_a @ Y @ ( set_list_a2 @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_1138_set__ConsD,axiom,
    ! [Y: produc6499617310964463488_a_b_b,X: produc6499617310964463488_a_b_b,Xs2: list_P7479555583871319568_a_b_b] :
      ( ( member4695696432722591383_a_b_b @ Y @ ( set_Pr3241176995924754219_a_b_b @ ( cons_P433472372565078346_a_b_b @ X @ Xs2 ) ) )
     => ( ( Y = X )
        | ( member4695696432722591383_a_b_b @ Y @ ( set_Pr3241176995924754219_a_b_b @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_1139_set__ConsD,axiom,
    ! [Y: dtree_list_a_b,X: dtree_list_a_b,Xs2: list_dtree_list_a_b] :
      ( ( member551035911493665803st_a_b @ Y @ ( set_dtree_list_a_b2 @ ( cons_dtree_list_a_b @ X @ Xs2 ) ) )
     => ( ( Y = X )
        | ( member551035911493665803st_a_b @ Y @ ( set_dtree_list_a_b2 @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_1140_set__ConsD,axiom,
    ! [Y: real,X: real,Xs2: list_real] :
      ( ( member_real @ Y @ ( set_real2 @ ( cons_real @ X @ Xs2 ) ) )
     => ( ( Y = X )
        | ( member_real @ Y @ ( set_real2 @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_1141_list_Oset__cases,axiom,
    ! [E: list_a,A: list_list_a] :
      ( ( member_list_a @ E @ ( set_list_a2 @ A ) )
     => ( ! [Z2: list_list_a] :
            ( A
           != ( cons_list_a @ E @ Z2 ) )
       => ~ ! [Z1: list_a,Z2: list_list_a] :
              ( ( A
                = ( cons_list_a @ Z1 @ Z2 ) )
             => ~ ( member_list_a @ E @ ( set_list_a2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_1142_list_Oset__cases,axiom,
    ! [E: produc6499617310964463488_a_b_b,A: list_P7479555583871319568_a_b_b] :
      ( ( member4695696432722591383_a_b_b @ E @ ( set_Pr3241176995924754219_a_b_b @ A ) )
     => ( ! [Z2: list_P7479555583871319568_a_b_b] :
            ( A
           != ( cons_P433472372565078346_a_b_b @ E @ Z2 ) )
       => ~ ! [Z1: produc6499617310964463488_a_b_b,Z2: list_P7479555583871319568_a_b_b] :
              ( ( A
                = ( cons_P433472372565078346_a_b_b @ Z1 @ Z2 ) )
             => ~ ( member4695696432722591383_a_b_b @ E @ ( set_Pr3241176995924754219_a_b_b @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_1143_list_Oset__cases,axiom,
    ! [E: dtree_list_a_b,A: list_dtree_list_a_b] :
      ( ( member551035911493665803st_a_b @ E @ ( set_dtree_list_a_b2 @ A ) )
     => ( ! [Z2: list_dtree_list_a_b] :
            ( A
           != ( cons_dtree_list_a_b @ E @ Z2 ) )
       => ~ ! [Z1: dtree_list_a_b,Z2: list_dtree_list_a_b] :
              ( ( A
                = ( cons_dtree_list_a_b @ Z1 @ Z2 ) )
             => ~ ( member551035911493665803st_a_b @ E @ ( set_dtree_list_a_b2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_1144_list_Oset__cases,axiom,
    ! [E: real,A: list_real] :
      ( ( member_real @ E @ ( set_real2 @ A ) )
     => ( ! [Z2: list_real] :
            ( A
           != ( cons_real @ E @ Z2 ) )
       => ~ ! [Z1: real,Z2: list_real] :
              ( ( A
                = ( cons_real @ Z1 @ Z2 ) )
             => ~ ( member_real @ E @ ( set_real2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_1145_list_Oset__intros_I1_J,axiom,
    ! [X21: list_a,X222: list_list_a] : ( member_list_a @ X21 @ ( set_list_a2 @ ( cons_list_a @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_1146_list_Oset__intros_I1_J,axiom,
    ! [X21: produc6499617310964463488_a_b_b,X222: list_P7479555583871319568_a_b_b] : ( member4695696432722591383_a_b_b @ X21 @ ( set_Pr3241176995924754219_a_b_b @ ( cons_P433472372565078346_a_b_b @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_1147_list_Oset__intros_I1_J,axiom,
    ! [X21: dtree_list_a_b,X222: list_dtree_list_a_b] : ( member551035911493665803st_a_b @ X21 @ ( set_dtree_list_a_b2 @ ( cons_dtree_list_a_b @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_1148_list_Oset__intros_I1_J,axiom,
    ! [X21: real,X222: list_real] : ( member_real @ X21 @ ( set_real2 @ ( cons_real @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_1149_list_Oset__intros_I2_J,axiom,
    ! [Y: list_a,X222: list_list_a,X21: list_a] :
      ( ( member_list_a @ Y @ ( set_list_a2 @ X222 ) )
     => ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_1150_list_Oset__intros_I2_J,axiom,
    ! [Y: produc6499617310964463488_a_b_b,X222: list_P7479555583871319568_a_b_b,X21: produc6499617310964463488_a_b_b] :
      ( ( member4695696432722591383_a_b_b @ Y @ ( set_Pr3241176995924754219_a_b_b @ X222 ) )
     => ( member4695696432722591383_a_b_b @ Y @ ( set_Pr3241176995924754219_a_b_b @ ( cons_P433472372565078346_a_b_b @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_1151_list_Oset__intros_I2_J,axiom,
    ! [Y: dtree_list_a_b,X222: list_dtree_list_a_b,X21: dtree_list_a_b] :
      ( ( member551035911493665803st_a_b @ Y @ ( set_dtree_list_a_b2 @ X222 ) )
     => ( member551035911493665803st_a_b @ Y @ ( set_dtree_list_a_b2 @ ( cons_dtree_list_a_b @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_1152_list_Oset__intros_I2_J,axiom,
    ! [Y: real,X222: list_real,X21: real] :
      ( ( member_real @ Y @ ( set_real2 @ X222 ) )
     => ( member_real @ Y @ ( set_real2 @ ( cons_real @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_1153_split__list,axiom,
    ! [X: list_a,Xs2: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
     => ? [Ys2: list_list_a,Zs: list_list_a] :
          ( Xs2
          = ( append_list_a @ Ys2 @ ( cons_list_a @ X @ Zs ) ) ) ) ).

% split_list
thf(fact_1154_split__list,axiom,
    ! [X: produc6499617310964463488_a_b_b,Xs2: list_P7479555583871319568_a_b_b] :
      ( ( member4695696432722591383_a_b_b @ X @ ( set_Pr3241176995924754219_a_b_b @ Xs2 ) )
     => ? [Ys2: list_P7479555583871319568_a_b_b,Zs: list_P7479555583871319568_a_b_b] :
          ( Xs2
          = ( append1383687441759991717_a_b_b @ Ys2 @ ( cons_P433472372565078346_a_b_b @ X @ Zs ) ) ) ) ).

% split_list
thf(fact_1155_split__list,axiom,
    ! [X: dtree_list_a_b,Xs2: list_dtree_list_a_b] :
      ( ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Xs2 ) )
     => ? [Ys2: list_dtree_list_a_b,Zs: list_dtree_list_a_b] :
          ( Xs2
          = ( append2129087805494049561st_a_b @ Ys2 @ ( cons_dtree_list_a_b @ X @ Zs ) ) ) ) ).

% split_list
thf(fact_1156_split__list,axiom,
    ! [X: real,Xs2: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
     => ? [Ys2: list_real,Zs: list_real] :
          ( Xs2
          = ( append_real @ Ys2 @ ( cons_real @ X @ Zs ) ) ) ) ).

% split_list
thf(fact_1157_split__list__last,axiom,
    ! [X: list_a,Xs2: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
     => ? [Ys2: list_list_a,Zs: list_list_a] :
          ( ( Xs2
            = ( append_list_a @ Ys2 @ ( cons_list_a @ X @ Zs ) ) )
          & ~ ( member_list_a @ X @ ( set_list_a2 @ Zs ) ) ) ) ).

% split_list_last
thf(fact_1158_split__list__last,axiom,
    ! [X: produc6499617310964463488_a_b_b,Xs2: list_P7479555583871319568_a_b_b] :
      ( ( member4695696432722591383_a_b_b @ X @ ( set_Pr3241176995924754219_a_b_b @ Xs2 ) )
     => ? [Ys2: list_P7479555583871319568_a_b_b,Zs: list_P7479555583871319568_a_b_b] :
          ( ( Xs2
            = ( append1383687441759991717_a_b_b @ Ys2 @ ( cons_P433472372565078346_a_b_b @ X @ Zs ) ) )
          & ~ ( member4695696432722591383_a_b_b @ X @ ( set_Pr3241176995924754219_a_b_b @ Zs ) ) ) ) ).

% split_list_last
thf(fact_1159_split__list__last,axiom,
    ! [X: dtree_list_a_b,Xs2: list_dtree_list_a_b] :
      ( ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Xs2 ) )
     => ? [Ys2: list_dtree_list_a_b,Zs: list_dtree_list_a_b] :
          ( ( Xs2
            = ( append2129087805494049561st_a_b @ Ys2 @ ( cons_dtree_list_a_b @ X @ Zs ) ) )
          & ~ ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Zs ) ) ) ) ).

% split_list_last
thf(fact_1160_split__list__last,axiom,
    ! [X: real,Xs2: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
     => ? [Ys2: list_real,Zs: list_real] :
          ( ( Xs2
            = ( append_real @ Ys2 @ ( cons_real @ X @ Zs ) ) )
          & ~ ( member_real @ X @ ( set_real2 @ Zs ) ) ) ) ).

% split_list_last
thf(fact_1161_split__list__first,axiom,
    ! [X: list_a,Xs2: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
     => ? [Ys2: list_list_a,Zs: list_list_a] :
          ( ( Xs2
            = ( append_list_a @ Ys2 @ ( cons_list_a @ X @ Zs ) ) )
          & ~ ( member_list_a @ X @ ( set_list_a2 @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_1162_split__list__first,axiom,
    ! [X: produc6499617310964463488_a_b_b,Xs2: list_P7479555583871319568_a_b_b] :
      ( ( member4695696432722591383_a_b_b @ X @ ( set_Pr3241176995924754219_a_b_b @ Xs2 ) )
     => ? [Ys2: list_P7479555583871319568_a_b_b,Zs: list_P7479555583871319568_a_b_b] :
          ( ( Xs2
            = ( append1383687441759991717_a_b_b @ Ys2 @ ( cons_P433472372565078346_a_b_b @ X @ Zs ) ) )
          & ~ ( member4695696432722591383_a_b_b @ X @ ( set_Pr3241176995924754219_a_b_b @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_1163_split__list__first,axiom,
    ! [X: dtree_list_a_b,Xs2: list_dtree_list_a_b] :
      ( ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Xs2 ) )
     => ? [Ys2: list_dtree_list_a_b,Zs: list_dtree_list_a_b] :
          ( ( Xs2
            = ( append2129087805494049561st_a_b @ Ys2 @ ( cons_dtree_list_a_b @ X @ Zs ) ) )
          & ~ ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_1164_split__list__first,axiom,
    ! [X: real,Xs2: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
     => ? [Ys2: list_real,Zs: list_real] :
          ( ( Xs2
            = ( append_real @ Ys2 @ ( cons_real @ X @ Zs ) ) )
          & ~ ( member_real @ X @ ( set_real2 @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_1165_append__Cons__eq__iff,axiom,
    ! [X: list_a,Xs2: list_list_a,Ys: list_list_a,Xs3: list_list_a,Ys3: list_list_a] :
      ( ~ ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
     => ( ~ ( member_list_a @ X @ ( set_list_a2 @ Ys ) )
       => ( ( ( append_list_a @ Xs2 @ ( cons_list_a @ X @ Ys ) )
            = ( append_list_a @ Xs3 @ ( cons_list_a @ X @ Ys3 ) ) )
          = ( ( Xs2 = Xs3 )
            & ( Ys = Ys3 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_1166_append__Cons__eq__iff,axiom,
    ! [X: produc6499617310964463488_a_b_b,Xs2: list_P7479555583871319568_a_b_b,Ys: list_P7479555583871319568_a_b_b,Xs3: list_P7479555583871319568_a_b_b,Ys3: list_P7479555583871319568_a_b_b] :
      ( ~ ( member4695696432722591383_a_b_b @ X @ ( set_Pr3241176995924754219_a_b_b @ Xs2 ) )
     => ( ~ ( member4695696432722591383_a_b_b @ X @ ( set_Pr3241176995924754219_a_b_b @ Ys ) )
       => ( ( ( append1383687441759991717_a_b_b @ Xs2 @ ( cons_P433472372565078346_a_b_b @ X @ Ys ) )
            = ( append1383687441759991717_a_b_b @ Xs3 @ ( cons_P433472372565078346_a_b_b @ X @ Ys3 ) ) )
          = ( ( Xs2 = Xs3 )
            & ( Ys = Ys3 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_1167_append__Cons__eq__iff,axiom,
    ! [X: dtree_list_a_b,Xs2: list_dtree_list_a_b,Ys: list_dtree_list_a_b,Xs3: list_dtree_list_a_b,Ys3: list_dtree_list_a_b] :
      ( ~ ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Xs2 ) )
     => ( ~ ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Ys ) )
       => ( ( ( append2129087805494049561st_a_b @ Xs2 @ ( cons_dtree_list_a_b @ X @ Ys ) )
            = ( append2129087805494049561st_a_b @ Xs3 @ ( cons_dtree_list_a_b @ X @ Ys3 ) ) )
          = ( ( Xs2 = Xs3 )
            & ( Ys = Ys3 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_1168_append__Cons__eq__iff,axiom,
    ! [X: real,Xs2: list_real,Ys: list_real,Xs3: list_real,Ys3: list_real] :
      ( ~ ( member_real @ X @ ( set_real2 @ Xs2 ) )
     => ( ~ ( member_real @ X @ ( set_real2 @ Ys ) )
       => ( ( ( append_real @ Xs2 @ ( cons_real @ X @ Ys ) )
            = ( append_real @ Xs3 @ ( cons_real @ X @ Ys3 ) ) )
          = ( ( Xs2 = Xs3 )
            & ( Ys = Ys3 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_1169_in__set__conv__decomp,axiom,
    ! [X: list_a,Xs2: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
      = ( ? [Ys4: list_list_a,Zs2: list_list_a] :
            ( Xs2
            = ( append_list_a @ Ys4 @ ( cons_list_a @ X @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_1170_in__set__conv__decomp,axiom,
    ! [X: produc6499617310964463488_a_b_b,Xs2: list_P7479555583871319568_a_b_b] :
      ( ( member4695696432722591383_a_b_b @ X @ ( set_Pr3241176995924754219_a_b_b @ Xs2 ) )
      = ( ? [Ys4: list_P7479555583871319568_a_b_b,Zs2: list_P7479555583871319568_a_b_b] :
            ( Xs2
            = ( append1383687441759991717_a_b_b @ Ys4 @ ( cons_P433472372565078346_a_b_b @ X @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_1171_in__set__conv__decomp,axiom,
    ! [X: dtree_list_a_b,Xs2: list_dtree_list_a_b] :
      ( ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Xs2 ) )
      = ( ? [Ys4: list_dtree_list_a_b,Zs2: list_dtree_list_a_b] :
            ( Xs2
            = ( append2129087805494049561st_a_b @ Ys4 @ ( cons_dtree_list_a_b @ X @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_1172_in__set__conv__decomp,axiom,
    ! [X: real,Xs2: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
      = ( ? [Ys4: list_real,Zs2: list_real] :
            ( Xs2
            = ( append_real @ Ys4 @ ( cons_real @ X @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_1173_in__set__conv__decomp__last,axiom,
    ! [X: list_a,Xs2: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
      = ( ? [Ys4: list_list_a,Zs2: list_list_a] :
            ( ( Xs2
              = ( append_list_a @ Ys4 @ ( cons_list_a @ X @ Zs2 ) ) )
            & ~ ( member_list_a @ X @ ( set_list_a2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_1174_in__set__conv__decomp__last,axiom,
    ! [X: produc6499617310964463488_a_b_b,Xs2: list_P7479555583871319568_a_b_b] :
      ( ( member4695696432722591383_a_b_b @ X @ ( set_Pr3241176995924754219_a_b_b @ Xs2 ) )
      = ( ? [Ys4: list_P7479555583871319568_a_b_b,Zs2: list_P7479555583871319568_a_b_b] :
            ( ( Xs2
              = ( append1383687441759991717_a_b_b @ Ys4 @ ( cons_P433472372565078346_a_b_b @ X @ Zs2 ) ) )
            & ~ ( member4695696432722591383_a_b_b @ X @ ( set_Pr3241176995924754219_a_b_b @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_1175_in__set__conv__decomp__last,axiom,
    ! [X: dtree_list_a_b,Xs2: list_dtree_list_a_b] :
      ( ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Xs2 ) )
      = ( ? [Ys4: list_dtree_list_a_b,Zs2: list_dtree_list_a_b] :
            ( ( Xs2
              = ( append2129087805494049561st_a_b @ Ys4 @ ( cons_dtree_list_a_b @ X @ Zs2 ) ) )
            & ~ ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_1176_in__set__conv__decomp__last,axiom,
    ! [X: real,Xs2: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
      = ( ? [Ys4: list_real,Zs2: list_real] :
            ( ( Xs2
              = ( append_real @ Ys4 @ ( cons_real @ X @ Zs2 ) ) )
            & ~ ( member_real @ X @ ( set_real2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_1177_in__set__conv__decomp__first,axiom,
    ! [X: list_a,Xs2: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
      = ( ? [Ys4: list_list_a,Zs2: list_list_a] :
            ( ( Xs2
              = ( append_list_a @ Ys4 @ ( cons_list_a @ X @ Zs2 ) ) )
            & ~ ( member_list_a @ X @ ( set_list_a2 @ Ys4 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_1178_in__set__conv__decomp__first,axiom,
    ! [X: produc6499617310964463488_a_b_b,Xs2: list_P7479555583871319568_a_b_b] :
      ( ( member4695696432722591383_a_b_b @ X @ ( set_Pr3241176995924754219_a_b_b @ Xs2 ) )
      = ( ? [Ys4: list_P7479555583871319568_a_b_b,Zs2: list_P7479555583871319568_a_b_b] :
            ( ( Xs2
              = ( append1383687441759991717_a_b_b @ Ys4 @ ( cons_P433472372565078346_a_b_b @ X @ Zs2 ) ) )
            & ~ ( member4695696432722591383_a_b_b @ X @ ( set_Pr3241176995924754219_a_b_b @ Ys4 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_1179_in__set__conv__decomp__first,axiom,
    ! [X: dtree_list_a_b,Xs2: list_dtree_list_a_b] :
      ( ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Xs2 ) )
      = ( ? [Ys4: list_dtree_list_a_b,Zs2: list_dtree_list_a_b] :
            ( ( Xs2
              = ( append2129087805494049561st_a_b @ Ys4 @ ( cons_dtree_list_a_b @ X @ Zs2 ) ) )
            & ~ ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Ys4 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_1180_in__set__conv__decomp__first,axiom,
    ! [X: real,Xs2: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
      = ( ? [Ys4: list_real,Zs2: list_real] :
            ( ( Xs2
              = ( append_real @ Ys4 @ ( cons_real @ X @ Zs2 ) ) )
            & ~ ( member_real @ X @ ( set_real2 @ Ys4 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_1181_dtree__to__list__split__subtree,axiom,
    ! [As3: list_P2281458131032066856st_a_b,R2: list_a,E: b,Bs2: list_P2281458131032066856st_a_b,T: dtree_list_a_b] :
      ( ( ( append8668862642614166333st_a_b @ As3 @ ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ R2 @ E ) @ Bs2 ) )
        = ( dtree_5522862299006258336st_a_b @ T ) )
     => ? [Xs: fset_P2153231429829016240_a_b_b] :
          ( ( strict8995144569104247066st_a_b @ ( node_list_a_b @ R2 @ Xs ) @ T )
          & ( ( dtree_5522862299006258336st_a_b @ ( node_list_a_b @ R2 @ Xs ) )
            = Bs2 ) ) ) ).

% dtree_to_list_split_subtree
thf(fact_1182_dtree__to__list__split__subtree,axiom,
    ! [As3: list_P321204300973800749list_a,R2: list_a,E: list_a,Bs2: list_P321204300973800749list_a,T: dtree_list_a_list_a] :
      ( ( ( append622457529216362434list_a @ As3 @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ R2 @ E ) @ Bs2 ) )
        = ( dtree_6750851072906405029list_a @ T ) )
     => ? [Xs: fset_P6656299774166858224list_a] :
          ( ( strict9149861969251335199list_a @ ( node_list_a_list_a @ R2 @ Xs ) @ T )
          & ( ( dtree_6750851072906405029list_a @ ( node_list_a_list_a @ R2 @ Xs ) )
            = Bs2 ) ) ) ).

% dtree_to_list_split_subtree
thf(fact_1183_dtree__to__list__split__subtree,axiom,
    ! [As3: list_P7455220050002504759st_a_b,R2: list_a,E: dtree_list_a_b,Bs2: list_P7455220050002504759st_a_b,T: dtree_4875458026974816597st_a_b] :
      ( ( ( append2144729903467096396st_a_b @ As3 @ ( cons_P1842004463258618993st_a_b @ ( produc148520996349637281st_a_b @ R2 @ E ) @ Bs2 ) )
        = ( dtree_547020913002174355st_a_b @ T ) )
     => ? [Xs: fset_P6340920821931359400st_a_b] :
          ( ( strict4128509144933472153st_a_b @ ( node_l8355664270593351482st_a_b @ R2 @ Xs ) @ T )
          & ( ( dtree_547020913002174355st_a_b @ ( node_l8355664270593351482st_a_b @ R2 @ Xs ) )
            = Bs2 ) ) ) ).

% dtree_to_list_split_subtree
thf(fact_1184_dtree__to__list__split__subtree,axiom,
    ! [As3: list_P7479555583871319568_a_b_b,R2: dtree_list_a_b,E: b,Bs2: list_P7479555583871319568_a_b_b,T: dtree_2614965411716861998_a_b_b] :
      ( ( ( append1383687441759991717_a_b_b @ As3 @ ( cons_P433472372565078346_a_b_b @ ( produc7704165765595008946_a_b_b @ R2 @ E ) @ Bs2 ) )
        = ( dtree_5700413169342715300_a_b_b @ T ) )
     => ? [Xs: fset_P8033862543129508702_b_b_b] :
          ( ( strict7398043843113750698_a_b_b @ ( node_d3522988236360801483_a_b_b @ R2 @ Xs ) @ T )
          & ( ( dtree_5700413169342715300_a_b_b @ ( node_d3522988236360801483_a_b_b @ R2 @ Xs ) )
            = Bs2 ) ) ) ).

% dtree_to_list_split_subtree
thf(fact_1185_dtree__to__list__split__subtree,axiom,
    ! [As3: list_P2775315667170835306st_a_b,R2: list_a,E: produc111314985273491367st_a_b,Bs2: list_P2775315667170835306st_a_b,T: dtree_5894418989983070344st_a_b] :
      ( ( ( append7889034872329257471st_a_b @ As3 @ ( cons_P9012681797030764964st_a_b @ ( produc673257793671328980st_a_b @ R2 @ E ) @ Bs2 ) )
        = ( dtree_3265670640512857798st_a_b @ T ) )
     => ? [Xs: fset_P1196402277819410702st_a_b] :
          ( ( strict2554826646234793420st_a_b @ ( node_l6315829471324094445st_a_b @ R2 @ Xs ) @ T )
          & ( ( dtree_3265670640512857798st_a_b @ ( node_l6315829471324094445st_a_b @ R2 @ Xs ) )
            = Bs2 ) ) ) ).

% dtree_to_list_split_subtree
thf(fact_1186_dtree__to__list_Osimps_I1_J,axiom,
    ! [R2: list_a,T: dtree_list_a_list_a,E: list_a] :
      ( ( dtree_6750851072906405029list_a @ ( node_list_a_list_a @ R2 @ ( finser5913853959220252280list_a @ ( produc6276725351364364146list_a @ T @ E ) @ bot_bo7860097226086663300list_a ) ) )
      = ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ ( root_list_a_list_a @ T ) @ E ) @ ( dtree_6750851072906405029list_a @ T ) ) ) ).

% dtree_to_list.simps(1)
thf(fact_1187_dtree__to__list_Osimps_I1_J,axiom,
    ! [R2: list_a,T: dtree_4875458026974816597st_a_b,E: dtree_list_a_b] :
      ( ( dtree_547020913002174355st_a_b @ ( node_l8355664270593351482st_a_b @ R2 @ ( finser438930343355665584st_a_b @ ( produc4942817220686395122st_a_b @ T @ E ) @ bot_bo7273111426904488252st_a_b ) ) )
      = ( cons_P1842004463258618993st_a_b @ ( produc148520996349637281st_a_b @ ( root_l3034943007575730586st_a_b @ T ) @ E ) @ ( dtree_547020913002174355st_a_b @ T ) ) ) ).

% dtree_to_list.simps(1)
thf(fact_1188_dtree__to__list_Osimps_I1_J,axiom,
    ! [R2: dtree_list_a_b,T: dtree_2614965411716861998_a_b_b,E: b] :
      ( ( dtree_5700413169342715300_a_b_b @ ( node_d3522988236360801483_a_b_b @ R2 @ ( finser8863392264280748224_b_b_b @ ( produc7725859837237317096_b_b_b @ T @ E ) @ bot_bo1106383210820772810_b_b_b ) ) )
      = ( cons_P433472372565078346_a_b_b @ ( produc7704165765595008946_a_b_b @ ( root_d7852638473636976427_a_b_b @ T ) @ E ) @ ( dtree_5700413169342715300_a_b_b @ T ) ) ) ).

% dtree_to_list.simps(1)
thf(fact_1189_dtree__to__list_Osimps_I1_J,axiom,
    ! [R2: list_a,T: dtree_5894418989983070344st_a_b,E: produc111314985273491367st_a_b] :
      ( ( dtree_3265670640512857798st_a_b @ ( node_l6315829471324094445st_a_b @ R2 @ ( finser4675733205762001686st_a_b @ ( produc5223326772038911448st_a_b @ T @ E ) @ bot_bo2513736113629759906st_a_b ) ) )
      = ( cons_P9012681797030764964st_a_b @ ( produc673257793671328980st_a_b @ ( root_l4650499699236452941st_a_b @ T ) @ E ) @ ( dtree_3265670640512857798st_a_b @ T ) ) ) ).

% dtree_to_list.simps(1)
thf(fact_1190_dtree__to__list_Osimps_I1_J,axiom,
    ! [R2: list_a,T: dtree_list_a_b,E: b] :
      ( ( dtree_5522862299006258336st_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
      = ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ ( root_list_a_b @ T ) @ E ) @ ( dtree_5522862299006258336st_a_b @ T ) ) ) ).

% dtree_to_list.simps(1)
thf(fact_1191_dtree__to__list__sequence__subtree_H,axiom,
    ! [T: dtree_list_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
      ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T ) @ one_one_nat )
     => ( ( strict8995144569104247066st_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ T )
       => ? [As2: list_P2281458131032066856st_a_b,E2: b,Bs: list_P2281458131032066856st_a_b] :
            ( ( ( dtree_5522862299006258336st_a_b @ T )
              = ( append8668862642614166333st_a_b @ As2 @ ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ R2 @ E2 ) @ Bs ) ) )
            & ( ( dtree_5522862299006258336st_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
              = Bs ) ) ) ) ).

% dtree_to_list_sequence_subtree'
thf(fact_1192_dtree__to__list__sequence__subtree_H,axiom,
    ! [T: dtree_list_a_list_a,R2: list_a,Xs2: fset_P6656299774166858224list_a] :
      ( ( ord_less_eq_nat @ ( max_de8369102501826120024list_a @ T ) @ one_one_nat )
     => ( ( strict9149861969251335199list_a @ ( node_list_a_list_a @ R2 @ Xs2 ) @ T )
       => ? [As2: list_P321204300973800749list_a,E2: list_a,Bs: list_P321204300973800749list_a] :
            ( ( ( dtree_6750851072906405029list_a @ T )
              = ( append622457529216362434list_a @ As2 @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ R2 @ E2 ) @ Bs ) ) )
            & ( ( dtree_6750851072906405029list_a @ ( node_list_a_list_a @ R2 @ Xs2 ) )
              = Bs ) ) ) ) ).

% dtree_to_list_sequence_subtree'
thf(fact_1193_dtree__to__list__sequence__subtree_H,axiom,
    ! [T: dtree_4875458026974816597st_a_b,R2: list_a,Xs2: fset_P6340920821931359400st_a_b] :
      ( ( ord_less_eq_nat @ ( max_de3954538496131799840st_a_b @ T ) @ one_one_nat )
     => ( ( strict4128509144933472153st_a_b @ ( node_l8355664270593351482st_a_b @ R2 @ Xs2 ) @ T )
       => ? [As2: list_P7455220050002504759st_a_b,E2: dtree_list_a_b,Bs: list_P7455220050002504759st_a_b] :
            ( ( ( dtree_547020913002174355st_a_b @ T )
              = ( append2144729903467096396st_a_b @ As2 @ ( cons_P1842004463258618993st_a_b @ ( produc148520996349637281st_a_b @ R2 @ E2 ) @ Bs ) ) )
            & ( ( dtree_547020913002174355st_a_b @ ( node_l8355664270593351482st_a_b @ R2 @ Xs2 ) )
              = Bs ) ) ) ) ).

% dtree_to_list_sequence_subtree'
thf(fact_1194_dtree__to__list__sequence__subtree_H,axiom,
    ! [T: dtree_2614965411716861998_a_b_b,R2: dtree_list_a_b,Xs2: fset_P8033862543129508702_b_b_b] :
      ( ( ord_less_eq_nat @ ( max_de926933835450762161_a_b_b @ T ) @ one_one_nat )
     => ( ( strict7398043843113750698_a_b_b @ ( node_d3522988236360801483_a_b_b @ R2 @ Xs2 ) @ T )
       => ? [As2: list_P7479555583871319568_a_b_b,E2: b,Bs: list_P7479555583871319568_a_b_b] :
            ( ( ( dtree_5700413169342715300_a_b_b @ T )
              = ( append1383687441759991717_a_b_b @ As2 @ ( cons_P433472372565078346_a_b_b @ ( produc7704165765595008946_a_b_b @ R2 @ E2 ) @ Bs ) ) )
            & ( ( dtree_5700413169342715300_a_b_b @ ( node_d3522988236360801483_a_b_b @ R2 @ Xs2 ) )
              = Bs ) ) ) ) ).

% dtree_to_list_sequence_subtree'
thf(fact_1195_dtree__to__list__sequence__subtree_H,axiom,
    ! [T: dtree_5894418989983070344st_a_b,R2: list_a,Xs2: fset_P1196402277819410702st_a_b] :
      ( ( ord_less_eq_nat @ ( max_de6115843874381134547st_a_b @ T ) @ one_one_nat )
     => ( ( strict2554826646234793420st_a_b @ ( node_l6315829471324094445st_a_b @ R2 @ Xs2 ) @ T )
       => ? [As2: list_P2775315667170835306st_a_b,E2: produc111314985273491367st_a_b,Bs: list_P2775315667170835306st_a_b] :
            ( ( ( dtree_3265670640512857798st_a_b @ T )
              = ( append7889034872329257471st_a_b @ As2 @ ( cons_P9012681797030764964st_a_b @ ( produc673257793671328980st_a_b @ R2 @ E2 ) @ Bs ) ) )
            & ( ( dtree_3265670640512857798st_a_b @ ( node_l6315829471324094445st_a_b @ R2 @ Xs2 ) )
              = Bs ) ) ) ) ).

% dtree_to_list_sequence_subtree'
thf(fact_1196_dtree__to__list_Opsimps_I1_J,axiom,
    ! [R2: list_a,T: dtree_list_a_list_a,E: list_a] :
      ( ( accp_d2674248120870298800list_a @ dtree_4834894955182143892list_a @ ( node_list_a_list_a @ R2 @ ( finser5913853959220252280list_a @ ( produc6276725351364364146list_a @ T @ E ) @ bot_bo7860097226086663300list_a ) ) )
     => ( ( dtree_6750851072906405029list_a @ ( node_list_a_list_a @ R2 @ ( finser5913853959220252280list_a @ ( produc6276725351364364146list_a @ T @ E ) @ bot_bo7860097226086663300list_a ) ) )
        = ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ ( root_list_a_list_a @ T ) @ E ) @ ( dtree_6750851072906405029list_a @ T ) ) ) ) ).

% dtree_to_list.psimps(1)
thf(fact_1197_dtree__to__list_Opsimps_I1_J,axiom,
    ! [R2: list_a,T: dtree_4875458026974816597st_a_b,E: dtree_list_a_b] :
      ( ( accp_d9081407519697605342st_a_b @ dtree_4519695044402773988st_a_b @ ( node_l8355664270593351482st_a_b @ R2 @ ( finser438930343355665584st_a_b @ ( produc4942817220686395122st_a_b @ T @ E ) @ bot_bo7273111426904488252st_a_b ) ) )
     => ( ( dtree_547020913002174355st_a_b @ ( node_l8355664270593351482st_a_b @ R2 @ ( finser438930343355665584st_a_b @ ( produc4942817220686395122st_a_b @ T @ E ) @ bot_bo7273111426904488252st_a_b ) ) )
        = ( cons_P1842004463258618993st_a_b @ ( produc148520996349637281st_a_b @ ( root_l3034943007575730586st_a_b @ T ) @ E ) @ ( dtree_547020913002174355st_a_b @ T ) ) ) ) ).

% dtree_to_list.psimps(1)
thf(fact_1198_dtree__to__list_Opsimps_I1_J,axiom,
    ! [R2: dtree_list_a_b,T: dtree_2614965411716861998_a_b_b,E: b] :
      ( ( accp_d827264278867218999_a_b_b @ dtree_7331847154152954485_a_b_b @ ( node_d3522988236360801483_a_b_b @ R2 @ ( finser8863392264280748224_b_b_b @ ( produc7725859837237317096_b_b_b @ T @ E ) @ bot_bo1106383210820772810_b_b_b ) ) )
     => ( ( dtree_5700413169342715300_a_b_b @ ( node_d3522988236360801483_a_b_b @ R2 @ ( finser8863392264280748224_b_b_b @ ( produc7725859837237317096_b_b_b @ T @ E ) @ bot_bo1106383210820772810_b_b_b ) ) )
        = ( cons_P433472372565078346_a_b_b @ ( produc7704165765595008946_a_b_b @ ( root_d7852638473636976427_a_b_b @ T ) @ E ) @ ( dtree_5700413169342715300_a_b_b @ T ) ) ) ) ).

% dtree_to_list.psimps(1)
thf(fact_1199_dtree__to__list_Opsimps_I1_J,axiom,
    ! [R2: list_a,T: dtree_5894418989983070344st_a_b,E: produc111314985273491367st_a_b] :
      ( ( accp_d766546008032286865st_a_b @ dtree_3953530003412175255st_a_b @ ( node_l6315829471324094445st_a_b @ R2 @ ( finser4675733205762001686st_a_b @ ( produc5223326772038911448st_a_b @ T @ E ) @ bot_bo2513736113629759906st_a_b ) ) )
     => ( ( dtree_3265670640512857798st_a_b @ ( node_l6315829471324094445st_a_b @ R2 @ ( finser4675733205762001686st_a_b @ ( produc5223326772038911448st_a_b @ T @ E ) @ bot_bo2513736113629759906st_a_b ) ) )
        = ( cons_P9012681797030764964st_a_b @ ( produc673257793671328980st_a_b @ ( root_l4650499699236452941st_a_b @ T ) @ E ) @ ( dtree_3265670640512857798st_a_b @ T ) ) ) ) ).

% dtree_to_list.psimps(1)
thf(fact_1200_dtree__to__list_Opsimps_I1_J,axiom,
    ! [R2: list_a,T: dtree_list_a_b,E: b] :
      ( ( accp_dtree_list_a_b @ dtree_452016980859282703st_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
     => ( ( dtree_5522862299006258336st_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
        = ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ ( root_list_a_b @ T ) @ E ) @ ( dtree_5522862299006258336st_a_b @ T ) ) ) ) ).

% dtree_to_list.psimps(1)
thf(fact_1201_dtree__to__list__split__subtree__dverts__eq__fsts,axiom,
    ! [T: dtree_list_a_list_a,As3: list_P321204300973800749list_a,R2: list_a,E: list_a,Bs2: list_P321204300973800749list_a] :
      ( ( ord_less_eq_nat @ ( max_de8369102501826120024list_a @ T ) @ one_one_nat )
     => ( ( ( append622457529216362434list_a @ As3 @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ R2 @ E ) @ Bs2 ) )
          = ( dtree_6750851072906405029list_a @ T ) )
       => ? [Xs: fset_P6656299774166858224list_a] :
            ( ( strict9149861969251335199list_a @ ( node_list_a_list_a @ R2 @ Xs ) @ T )
            & ( ( dverts_list_a_list_a @ ( node_list_a_list_a @ R2 @ Xs ) )
              = ( insert_list_a @ R2 @ ( image_5971222721807453474list_a @ produc3698117735987127555list_a @ ( set_Pr2906193453920572092list_a @ Bs2 ) ) ) ) ) ) ) ).

% dtree_to_list_split_subtree_dverts_eq_fsts
thf(fact_1202_dtree__to__list__split__subtree__dverts__eq__fsts,axiom,
    ! [T: dtree_4875458026974816597st_a_b,As3: list_P7455220050002504759st_a_b,R2: list_a,E: dtree_list_a_b,Bs2: list_P7455220050002504759st_a_b] :
      ( ( ord_less_eq_nat @ ( max_de3954538496131799840st_a_b @ T ) @ one_one_nat )
     => ( ( ( append2144729903467096396st_a_b @ As3 @ ( cons_P1842004463258618993st_a_b @ ( produc148520996349637281st_a_b @ R2 @ E ) @ Bs2 ) )
          = ( dtree_547020913002174355st_a_b @ T ) )
       => ? [Xs: fset_P6340920821931359400st_a_b] :
            ( ( strict4128509144933472153st_a_b @ ( node_l8355664270593351482st_a_b @ R2 @ Xs ) @ T )
            & ( ( dverts3100150328211173758st_a_b @ ( node_l8355664270593351482st_a_b @ R2 @ Xs ) )
              = ( insert_list_a @ R2 @ ( image_5758649495165665878list_a @ produc318716513907093941st_a_b @ ( set_Pr7992201861846349266st_a_b @ Bs2 ) ) ) ) ) ) ) ).

% dtree_to_list_split_subtree_dverts_eq_fsts
thf(fact_1203_dtree__to__list__split__subtree__dverts__eq__fsts,axiom,
    ! [T: dtree_2614965411716861998_a_b_b,As3: list_P7479555583871319568_a_b_b,R2: dtree_list_a_b,E: b,Bs2: list_P7479555583871319568_a_b_b] :
      ( ( ord_less_eq_nat @ ( max_de926933835450762161_a_b_b @ T ) @ one_one_nat )
     => ( ( ( append1383687441759991717_a_b_b @ As3 @ ( cons_P433472372565078346_a_b_b @ ( produc7704165765595008946_a_b_b @ R2 @ E ) @ Bs2 ) )
          = ( dtree_5700413169342715300_a_b_b @ T ) )
       => ? [Xs: fset_P8033862543129508702_b_b_b] :
            ( ( strict7398043843113750698_a_b_b @ ( node_d3522988236360801483_a_b_b @ R2 @ Xs ) @ T )
            & ( ( dverts3279282023300981007_a_b_b @ ( node_d3522988236360801483_a_b_b @ R2 @ Xs ) )
              = ( insert7065945238938772516st_a_b @ R2 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( set_Pr3241176995924754219_a_b_b @ Bs2 ) ) ) ) ) ) ) ).

% dtree_to_list_split_subtree_dverts_eq_fsts
thf(fact_1204_dtree__to__list__split__subtree__dverts__eq__fsts,axiom,
    ! [T: dtree_5894418989983070344st_a_b,As3: list_P2775315667170835306st_a_b,R2: list_a,E: produc111314985273491367st_a_b,Bs2: list_P2775315667170835306st_a_b] :
      ( ( ord_less_eq_nat @ ( max_de6115843874381134547st_a_b @ T ) @ one_one_nat )
     => ( ( ( append7889034872329257471st_a_b @ As3 @ ( cons_P9012681797030764964st_a_b @ ( produc673257793671328980st_a_b @ R2 @ E ) @ Bs2 ) )
          = ( dtree_3265670640512857798st_a_b @ T ) )
       => ? [Xs: fset_P1196402277819410702st_a_b] :
            ( ( strict2554826646234793420st_a_b @ ( node_l6315829471324094445st_a_b @ R2 @ Xs ) @ T )
            & ( ( dverts6436764938546248753st_a_b @ ( node_l6315829471324094445st_a_b @ R2 @ Xs ) )
              = ( insert_list_a @ R2 @ ( image_2865457199250190179list_a @ produc5507115380825155048st_a_b @ ( set_Pr2292375947623472517st_a_b @ Bs2 ) ) ) ) ) ) ) ).

% dtree_to_list_split_subtree_dverts_eq_fsts
thf(fact_1205_dtree__to__list__split__subtree__dverts__eq__fsts,axiom,
    ! [T: dtree_list_a_b,As3: list_P2281458131032066856st_a_b,R2: list_a,E: b,Bs2: list_P2281458131032066856st_a_b] :
      ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T ) @ one_one_nat )
     => ( ( ( append8668862642614166333st_a_b @ As3 @ ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ R2 @ E ) @ Bs2 ) )
          = ( dtree_5522862299006258336st_a_b @ T ) )
       => ? [Xs: fset_P2153231429829016240_a_b_b] :
            ( ( strict8995144569104247066st_a_b @ ( node_list_a_b @ R2 @ Xs ) @ T )
            & ( ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs ) )
              = ( insert_list_a @ R2 @ ( image_4065140400681997415list_a @ product_fst_list_a_b @ ( set_Pr6156699709417984823st_a_b @ Bs2 ) ) ) ) ) ) ) ).

% dtree_to_list_split_subtree_dverts_eq_fsts
thf(fact_1206_dtree__to__list__subtree__dverts__eq__fsts,axiom,
    ! [T: dtree_list_a_list_a,R2: list_a,Xs2: fset_P6656299774166858224list_a] :
      ( ( ord_less_eq_nat @ ( max_de8369102501826120024list_a @ T ) @ one_one_nat )
     => ( ( strict9149861969251335199list_a @ ( node_list_a_list_a @ R2 @ Xs2 ) @ T )
       => ? [As2: list_P321204300973800749list_a,E2: list_a,Bs: list_P321204300973800749list_a] :
            ( ( ( dtree_6750851072906405029list_a @ T )
              = ( append622457529216362434list_a @ As2 @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ R2 @ E2 ) @ Bs ) ) )
            & ( ( insert_list_a @ R2 @ ( image_5971222721807453474list_a @ produc3698117735987127555list_a @ ( set_Pr2906193453920572092list_a @ Bs ) ) )
              = ( dverts_list_a_list_a @ ( node_list_a_list_a @ R2 @ Xs2 ) ) ) ) ) ) ).

% dtree_to_list_subtree_dverts_eq_fsts
thf(fact_1207_dtree__to__list__subtree__dverts__eq__fsts,axiom,
    ! [T: dtree_4875458026974816597st_a_b,R2: list_a,Xs2: fset_P6340920821931359400st_a_b] :
      ( ( ord_less_eq_nat @ ( max_de3954538496131799840st_a_b @ T ) @ one_one_nat )
     => ( ( strict4128509144933472153st_a_b @ ( node_l8355664270593351482st_a_b @ R2 @ Xs2 ) @ T )
       => ? [As2: list_P7455220050002504759st_a_b,E2: dtree_list_a_b,Bs: list_P7455220050002504759st_a_b] :
            ( ( ( dtree_547020913002174355st_a_b @ T )
              = ( append2144729903467096396st_a_b @ As2 @ ( cons_P1842004463258618993st_a_b @ ( produc148520996349637281st_a_b @ R2 @ E2 ) @ Bs ) ) )
            & ( ( insert_list_a @ R2 @ ( image_5758649495165665878list_a @ produc318716513907093941st_a_b @ ( set_Pr7992201861846349266st_a_b @ Bs ) ) )
              = ( dverts3100150328211173758st_a_b @ ( node_l8355664270593351482st_a_b @ R2 @ Xs2 ) ) ) ) ) ) ).

% dtree_to_list_subtree_dverts_eq_fsts
thf(fact_1208_dtree__to__list__subtree__dverts__eq__fsts,axiom,
    ! [T: dtree_2614965411716861998_a_b_b,R2: dtree_list_a_b,Xs2: fset_P8033862543129508702_b_b_b] :
      ( ( ord_less_eq_nat @ ( max_de926933835450762161_a_b_b @ T ) @ one_one_nat )
     => ( ( strict7398043843113750698_a_b_b @ ( node_d3522988236360801483_a_b_b @ R2 @ Xs2 ) @ T )
       => ? [As2: list_P7479555583871319568_a_b_b,E2: b,Bs: list_P7479555583871319568_a_b_b] :
            ( ( ( dtree_5700413169342715300_a_b_b @ T )
              = ( append1383687441759991717_a_b_b @ As2 @ ( cons_P433472372565078346_a_b_b @ ( produc7704165765595008946_a_b_b @ R2 @ E2 ) @ Bs ) ) )
            & ( ( insert7065945238938772516st_a_b @ R2 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( set_Pr3241176995924754219_a_b_b @ Bs ) ) )
              = ( dverts3279282023300981007_a_b_b @ ( node_d3522988236360801483_a_b_b @ R2 @ Xs2 ) ) ) ) ) ) ).

% dtree_to_list_subtree_dverts_eq_fsts
thf(fact_1209_dtree__to__list__subtree__dverts__eq__fsts,axiom,
    ! [T: dtree_5894418989983070344st_a_b,R2: list_a,Xs2: fset_P1196402277819410702st_a_b] :
      ( ( ord_less_eq_nat @ ( max_de6115843874381134547st_a_b @ T ) @ one_one_nat )
     => ( ( strict2554826646234793420st_a_b @ ( node_l6315829471324094445st_a_b @ R2 @ Xs2 ) @ T )
       => ? [As2: list_P2775315667170835306st_a_b,E2: produc111314985273491367st_a_b,Bs: list_P2775315667170835306st_a_b] :
            ( ( ( dtree_3265670640512857798st_a_b @ T )
              = ( append7889034872329257471st_a_b @ As2 @ ( cons_P9012681797030764964st_a_b @ ( produc673257793671328980st_a_b @ R2 @ E2 ) @ Bs ) ) )
            & ( ( insert_list_a @ R2 @ ( image_2865457199250190179list_a @ produc5507115380825155048st_a_b @ ( set_Pr2292375947623472517st_a_b @ Bs ) ) )
              = ( dverts6436764938546248753st_a_b @ ( node_l6315829471324094445st_a_b @ R2 @ Xs2 ) ) ) ) ) ) ).

% dtree_to_list_subtree_dverts_eq_fsts
thf(fact_1210_dtree__to__list__subtree__dverts__eq__fsts,axiom,
    ! [T: dtree_list_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
      ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T ) @ one_one_nat )
     => ( ( strict8995144569104247066st_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ T )
       => ? [As2: list_P2281458131032066856st_a_b,E2: b,Bs: list_P2281458131032066856st_a_b] :
            ( ( ( dtree_5522862299006258336st_a_b @ T )
              = ( append8668862642614166333st_a_b @ As2 @ ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ R2 @ E2 ) @ Bs ) ) )
            & ( ( insert_list_a @ R2 @ ( image_4065140400681997415list_a @ product_fst_list_a_b @ ( set_Pr6156699709417984823st_a_b @ Bs ) ) )
              = ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ) ) ).

% dtree_to_list_subtree_dverts_eq_fsts
thf(fact_1211_insert__iff,axiom,
    ! [A: list_a,B: list_a,A3: set_list_a] :
      ( ( member_list_a @ A @ ( insert_list_a @ B @ A3 ) )
      = ( ( A = B )
        | ( member_list_a @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_1212_insert__iff,axiom,
    ! [A: produc6499617310964463488_a_b_b,B: produc6499617310964463488_a_b_b,A3: set_Pr3443975907877334966_a_b_b] :
      ( ( member4695696432722591383_a_b_b @ A @ ( insert8720088738084019632_a_b_b @ B @ A3 ) )
      = ( ( A = B )
        | ( member4695696432722591383_a_b_b @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_1213_insert__iff,axiom,
    ! [A: dtree_list_a_b,B: dtree_list_a_b,A3: set_dtree_list_a_b] :
      ( ( member551035911493665803st_a_b @ A @ ( insert7065945238938772516st_a_b @ B @ A3 ) )
      = ( ( A = B )
        | ( member551035911493665803st_a_b @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_1214_insert__iff,axiom,
    ! [A: real,B: real,A3: set_real] :
      ( ( member_real @ A @ ( insert_real @ B @ A3 ) )
      = ( ( A = B )
        | ( member_real @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_1215_insertCI,axiom,
    ! [A: list_a,B5: set_list_a,B: list_a] :
      ( ( ~ ( member_list_a @ A @ B5 )
       => ( A = B ) )
     => ( member_list_a @ A @ ( insert_list_a @ B @ B5 ) ) ) ).

% insertCI
thf(fact_1216_insertCI,axiom,
    ! [A: produc6499617310964463488_a_b_b,B5: set_Pr3443975907877334966_a_b_b,B: produc6499617310964463488_a_b_b] :
      ( ( ~ ( member4695696432722591383_a_b_b @ A @ B5 )
       => ( A = B ) )
     => ( member4695696432722591383_a_b_b @ A @ ( insert8720088738084019632_a_b_b @ B @ B5 ) ) ) ).

% insertCI
thf(fact_1217_insertCI,axiom,
    ! [A: dtree_list_a_b,B5: set_dtree_list_a_b,B: dtree_list_a_b] :
      ( ( ~ ( member551035911493665803st_a_b @ A @ B5 )
       => ( A = B ) )
     => ( member551035911493665803st_a_b @ A @ ( insert7065945238938772516st_a_b @ B @ B5 ) ) ) ).

% insertCI
thf(fact_1218_insertCI,axiom,
    ! [A: real,B5: set_real,B: real] :
      ( ( ~ ( member_real @ A @ B5 )
       => ( A = B ) )
     => ( member_real @ A @ ( insert_real @ B @ B5 ) ) ) ).

% insertCI
thf(fact_1219_singletonI,axiom,
    ! [A: list_a] : ( member_list_a @ A @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).

% singletonI
thf(fact_1220_singletonI,axiom,
    ! [A: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ A @ ( insert8720088738084019632_a_b_b @ A @ bot_bo4897374000430069834_a_b_b ) ) ).

% singletonI
thf(fact_1221_singletonI,axiom,
    ! [A: dtree_list_a_b] : ( member551035911493665803st_a_b @ A @ ( insert7065945238938772516st_a_b @ A @ bot_bo798015271861357502st_a_b ) ) ).

% singletonI
thf(fact_1222_singletonI,axiom,
    ! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).

% singletonI
thf(fact_1223_singletonI,axiom,
    ! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_1224_insert__subset,axiom,
    ! [X: list_a,A3: set_list_a,B5: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( insert_list_a @ X @ A3 ) @ B5 )
      = ( ( member_list_a @ X @ B5 )
        & ( ord_le8861187494160871172list_a @ A3 @ B5 ) ) ) ).

% insert_subset
thf(fact_1225_insert__subset,axiom,
    ! [X: produc6499617310964463488_a_b_b,A3: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
      ( ( ord_le1619362961161175062_a_b_b @ ( insert8720088738084019632_a_b_b @ X @ A3 ) @ B5 )
      = ( ( member4695696432722591383_a_b_b @ X @ B5 )
        & ( ord_le1619362961161175062_a_b_b @ A3 @ B5 ) ) ) ).

% insert_subset
thf(fact_1226_insert__subset,axiom,
    ! [X: dtree_list_a_b,A3: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
      ( ( ord_le7599451563663638410st_a_b @ ( insert7065945238938772516st_a_b @ X @ A3 ) @ B5 )
      = ( ( member551035911493665803st_a_b @ X @ B5 )
        & ( ord_le7599451563663638410st_a_b @ A3 @ B5 ) ) ) ).

% insert_subset
thf(fact_1227_insert__subset,axiom,
    ! [X: real,A3: set_real,B5: set_real] :
      ( ( ord_less_eq_set_real @ ( insert_real @ X @ A3 ) @ B5 )
      = ( ( member_real @ X @ B5 )
        & ( ord_less_eq_set_real @ A3 @ B5 ) ) ) ).

% insert_subset
thf(fact_1228_Int__insert__left__if0,axiom,
    ! [A: list_a,C3: set_list_a,B5: set_list_a] :
      ( ~ ( member_list_a @ A @ C3 )
     => ( ( inf_inf_set_list_a @ ( insert_list_a @ A @ B5 ) @ C3 )
        = ( inf_inf_set_list_a @ B5 @ C3 ) ) ) ).

% Int_insert_left_if0
thf(fact_1229_Int__insert__left__if0,axiom,
    ! [A: produc6499617310964463488_a_b_b,C3: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
      ( ~ ( member4695696432722591383_a_b_b @ A @ C3 )
     => ( ( inf_in7962574851115124040_a_b_b @ ( insert8720088738084019632_a_b_b @ A @ B5 ) @ C3 )
        = ( inf_in7962574851115124040_a_b_b @ B5 @ C3 ) ) ) ).

% Int_insert_left_if0
thf(fact_1230_Int__insert__left__if0,axiom,
    ! [A: dtree_list_a_b,C3: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
      ( ~ ( member551035911493665803st_a_b @ A @ C3 )
     => ( ( inf_in3355993651213403836st_a_b @ ( insert7065945238938772516st_a_b @ A @ B5 ) @ C3 )
        = ( inf_in3355993651213403836st_a_b @ B5 @ C3 ) ) ) ).

% Int_insert_left_if0
thf(fact_1231_Int__insert__left__if0,axiom,
    ! [A: real,C3: set_real,B5: set_real] :
      ( ~ ( member_real @ A @ C3 )
     => ( ( inf_inf_set_real @ ( insert_real @ A @ B5 ) @ C3 )
        = ( inf_inf_set_real @ B5 @ C3 ) ) ) ).

% Int_insert_left_if0
thf(fact_1232_Int__insert__left__if1,axiom,
    ! [A: list_a,C3: set_list_a,B5: set_list_a] :
      ( ( member_list_a @ A @ C3 )
     => ( ( inf_inf_set_list_a @ ( insert_list_a @ A @ B5 ) @ C3 )
        = ( insert_list_a @ A @ ( inf_inf_set_list_a @ B5 @ C3 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1233_Int__insert__left__if1,axiom,
    ! [A: produc6499617310964463488_a_b_b,C3: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
      ( ( member4695696432722591383_a_b_b @ A @ C3 )
     => ( ( inf_in7962574851115124040_a_b_b @ ( insert8720088738084019632_a_b_b @ A @ B5 ) @ C3 )
        = ( insert8720088738084019632_a_b_b @ A @ ( inf_in7962574851115124040_a_b_b @ B5 @ C3 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1234_Int__insert__left__if1,axiom,
    ! [A: dtree_list_a_b,C3: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
      ( ( member551035911493665803st_a_b @ A @ C3 )
     => ( ( inf_in3355993651213403836st_a_b @ ( insert7065945238938772516st_a_b @ A @ B5 ) @ C3 )
        = ( insert7065945238938772516st_a_b @ A @ ( inf_in3355993651213403836st_a_b @ B5 @ C3 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1235_Int__insert__left__if1,axiom,
    ! [A: real,C3: set_real,B5: set_real] :
      ( ( member_real @ A @ C3 )
     => ( ( inf_inf_set_real @ ( insert_real @ A @ B5 ) @ C3 )
        = ( insert_real @ A @ ( inf_inf_set_real @ B5 @ C3 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1236_Int__insert__right__if0,axiom,
    ! [A: list_a,A3: set_list_a,B5: set_list_a] :
      ( ~ ( member_list_a @ A @ A3 )
     => ( ( inf_inf_set_list_a @ A3 @ ( insert_list_a @ A @ B5 ) )
        = ( inf_inf_set_list_a @ A3 @ B5 ) ) ) ).

% Int_insert_right_if0
thf(fact_1237_Int__insert__right__if0,axiom,
    ! [A: produc6499617310964463488_a_b_b,A3: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
      ( ~ ( member4695696432722591383_a_b_b @ A @ A3 )
     => ( ( inf_in7962574851115124040_a_b_b @ A3 @ ( insert8720088738084019632_a_b_b @ A @ B5 ) )
        = ( inf_in7962574851115124040_a_b_b @ A3 @ B5 ) ) ) ).

% Int_insert_right_if0
thf(fact_1238_Int__insert__right__if0,axiom,
    ! [A: dtree_list_a_b,A3: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
      ( ~ ( member551035911493665803st_a_b @ A @ A3 )
     => ( ( inf_in3355993651213403836st_a_b @ A3 @ ( insert7065945238938772516st_a_b @ A @ B5 ) )
        = ( inf_in3355993651213403836st_a_b @ A3 @ B5 ) ) ) ).

% Int_insert_right_if0
thf(fact_1239_Int__insert__right__if0,axiom,
    ! [A: real,A3: set_real,B5: set_real] :
      ( ~ ( member_real @ A @ A3 )
     => ( ( inf_inf_set_real @ A3 @ ( insert_real @ A @ B5 ) )
        = ( inf_inf_set_real @ A3 @ B5 ) ) ) ).

% Int_insert_right_if0
thf(fact_1240_Int__insert__right__if1,axiom,
    ! [A: list_a,A3: set_list_a,B5: set_list_a] :
      ( ( member_list_a @ A @ A3 )
     => ( ( inf_inf_set_list_a @ A3 @ ( insert_list_a @ A @ B5 ) )
        = ( insert_list_a @ A @ ( inf_inf_set_list_a @ A3 @ B5 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1241_Int__insert__right__if1,axiom,
    ! [A: produc6499617310964463488_a_b_b,A3: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
      ( ( member4695696432722591383_a_b_b @ A @ A3 )
     => ( ( inf_in7962574851115124040_a_b_b @ A3 @ ( insert8720088738084019632_a_b_b @ A @ B5 ) )
        = ( insert8720088738084019632_a_b_b @ A @ ( inf_in7962574851115124040_a_b_b @ A3 @ B5 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1242_Int__insert__right__if1,axiom,
    ! [A: dtree_list_a_b,A3: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
      ( ( member551035911493665803st_a_b @ A @ A3 )
     => ( ( inf_in3355993651213403836st_a_b @ A3 @ ( insert7065945238938772516st_a_b @ A @ B5 ) )
        = ( insert7065945238938772516st_a_b @ A @ ( inf_in3355993651213403836st_a_b @ A3 @ B5 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1243_Int__insert__right__if1,axiom,
    ! [A: real,A3: set_real,B5: set_real] :
      ( ( member_real @ A @ A3 )
     => ( ( inf_inf_set_real @ A3 @ ( insert_real @ A @ B5 ) )
        = ( insert_real @ A @ ( inf_inf_set_real @ A3 @ B5 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1244_singleton__conv,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : ( X4 = A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_1245_singleton__conv2,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ( ^ [Y5: nat,Z: nat] : ( Y5 = Z )
          @ A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_1246_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A3: set_nat,B: nat] :
      ( ( ( insert_nat @ A @ A3 )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1247_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A3: set_nat] :
      ( ( ( insert_nat @ B @ bot_bot_set_nat )
        = ( insert_nat @ A @ A3 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1248_cSup__singleton,axiom,
    ! [X: nat] :
      ( ( complete_Sup_Sup_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
      = X ) ).

% cSup_singleton
thf(fact_1249_insert__disjoint_I1_J,axiom,
    ! [A: list_a,A3: set_list_a,B5: set_list_a] :
      ( ( ( inf_inf_set_list_a @ ( insert_list_a @ A @ A3 ) @ B5 )
        = bot_bot_set_list_a )
      = ( ~ ( member_list_a @ A @ B5 )
        & ( ( inf_inf_set_list_a @ A3 @ B5 )
          = bot_bot_set_list_a ) ) ) ).

% insert_disjoint(1)
thf(fact_1250_insert__disjoint_I1_J,axiom,
    ! [A: produc6499617310964463488_a_b_b,A3: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
      ( ( ( inf_in7962574851115124040_a_b_b @ ( insert8720088738084019632_a_b_b @ A @ A3 ) @ B5 )
        = bot_bo4897374000430069834_a_b_b )
      = ( ~ ( member4695696432722591383_a_b_b @ A @ B5 )
        & ( ( inf_in7962574851115124040_a_b_b @ A3 @ B5 )
          = bot_bo4897374000430069834_a_b_b ) ) ) ).

% insert_disjoint(1)
thf(fact_1251_insert__disjoint_I1_J,axiom,
    ! [A: dtree_list_a_b,A3: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
      ( ( ( inf_in3355993651213403836st_a_b @ ( insert7065945238938772516st_a_b @ A @ A3 ) @ B5 )
        = bot_bo798015271861357502st_a_b )
      = ( ~ ( member551035911493665803st_a_b @ A @ B5 )
        & ( ( inf_in3355993651213403836st_a_b @ A3 @ B5 )
          = bot_bo798015271861357502st_a_b ) ) ) ).

% insert_disjoint(1)
thf(fact_1252_insert__disjoint_I1_J,axiom,
    ! [A: real,A3: set_real,B5: set_real] :
      ( ( ( inf_inf_set_real @ ( insert_real @ A @ A3 ) @ B5 )
        = bot_bot_set_real )
      = ( ~ ( member_real @ A @ B5 )
        & ( ( inf_inf_set_real @ A3 @ B5 )
          = bot_bot_set_real ) ) ) ).

% insert_disjoint(1)
thf(fact_1253_insert__disjoint_I1_J,axiom,
    ! [A: nat,A3: set_nat,B5: set_nat] :
      ( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A3 ) @ B5 )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A @ B5 )
        & ( ( inf_inf_set_nat @ A3 @ B5 )
          = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_1254_insert__disjoint_I2_J,axiom,
    ! [A: list_a,A3: set_list_a,B5: set_list_a] :
      ( ( bot_bot_set_list_a
        = ( inf_inf_set_list_a @ ( insert_list_a @ A @ A3 ) @ B5 ) )
      = ( ~ ( member_list_a @ A @ B5 )
        & ( bot_bot_set_list_a
          = ( inf_inf_set_list_a @ A3 @ B5 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1255_insert__disjoint_I2_J,axiom,
    ! [A: produc6499617310964463488_a_b_b,A3: set_Pr3443975907877334966_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
      ( ( bot_bo4897374000430069834_a_b_b
        = ( inf_in7962574851115124040_a_b_b @ ( insert8720088738084019632_a_b_b @ A @ A3 ) @ B5 ) )
      = ( ~ ( member4695696432722591383_a_b_b @ A @ B5 )
        & ( bot_bo4897374000430069834_a_b_b
          = ( inf_in7962574851115124040_a_b_b @ A3 @ B5 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1256_insert__disjoint_I2_J,axiom,
    ! [A: dtree_list_a_b,A3: set_dtree_list_a_b,B5: set_dtree_list_a_b] :
      ( ( bot_bo798015271861357502st_a_b
        = ( inf_in3355993651213403836st_a_b @ ( insert7065945238938772516st_a_b @ A @ A3 ) @ B5 ) )
      = ( ~ ( member551035911493665803st_a_b @ A @ B5 )
        & ( bot_bo798015271861357502st_a_b
          = ( inf_in3355993651213403836st_a_b @ A3 @ B5 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1257_insert__disjoint_I2_J,axiom,
    ! [A: real,A3: set_real,B5: set_real] :
      ( ( bot_bot_set_real
        = ( inf_inf_set_real @ ( insert_real @ A @ A3 ) @ B5 ) )
      = ( ~ ( member_real @ A @ B5 )
        & ( bot_bot_set_real
          = ( inf_inf_set_real @ A3 @ B5 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1258_insert__disjoint_I2_J,axiom,
    ! [A: nat,A3: set_nat,B5: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ ( insert_nat @ A @ A3 ) @ B5 ) )
      = ( ~ ( member_nat @ A @ B5 )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A3 @ B5 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1259_disjoint__insert_I1_J,axiom,
    ! [B5: set_list_a,A: list_a,A3: set_list_a] :
      ( ( ( inf_inf_set_list_a @ B5 @ ( insert_list_a @ A @ A3 ) )
        = bot_bot_set_list_a )
      = ( ~ ( member_list_a @ A @ B5 )
        & ( ( inf_inf_set_list_a @ B5 @ A3 )
          = bot_bot_set_list_a ) ) ) ).

% disjoint_insert(1)
thf(fact_1260_disjoint__insert_I1_J,axiom,
    ! [B5: set_Pr3443975907877334966_a_b_b,A: produc6499617310964463488_a_b_b,A3: set_Pr3443975907877334966_a_b_b] :
      ( ( ( inf_in7962574851115124040_a_b_b @ B5 @ ( insert8720088738084019632_a_b_b @ A @ A3 ) )
        = bot_bo4897374000430069834_a_b_b )
      = ( ~ ( member4695696432722591383_a_b_b @ A @ B5 )
        & ( ( inf_in7962574851115124040_a_b_b @ B5 @ A3 )
          = bot_bo4897374000430069834_a_b_b ) ) ) ).

% disjoint_insert(1)
thf(fact_1261_disjoint__insert_I1_J,axiom,
    ! [B5: set_dtree_list_a_b,A: dtree_list_a_b,A3: set_dtree_list_a_b] :
      ( ( ( inf_in3355993651213403836st_a_b @ B5 @ ( insert7065945238938772516st_a_b @ A @ A3 ) )
        = bot_bo798015271861357502st_a_b )
      = ( ~ ( member551035911493665803st_a_b @ A @ B5 )
        & ( ( inf_in3355993651213403836st_a_b @ B5 @ A3 )
          = bot_bo798015271861357502st_a_b ) ) ) ).

% disjoint_insert(1)
thf(fact_1262_disjoint__insert_I1_J,axiom,
    ! [B5: set_real,A: real,A3: set_real] :
      ( ( ( inf_inf_set_real @ B5 @ ( insert_real @ A @ A3 ) )
        = bot_bot_set_real )
      = ( ~ ( member_real @ A @ B5 )
        & ( ( inf_inf_set_real @ B5 @ A3 )
          = bot_bot_set_real ) ) ) ).

% disjoint_insert(1)
thf(fact_1263_disjoint__insert_I1_J,axiom,
    ! [B5: set_nat,A: nat,A3: set_nat] :
      ( ( ( inf_inf_set_nat @ B5 @ ( insert_nat @ A @ A3 ) )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A @ B5 )
        & ( ( inf_inf_set_nat @ B5 @ A3 )
          = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_1264_disjoint__insert_I2_J,axiom,
    ! [A3: set_list_a,B: list_a,B5: set_list_a] :
      ( ( bot_bot_set_list_a
        = ( inf_inf_set_list_a @ A3 @ ( insert_list_a @ B @ B5 ) ) )
      = ( ~ ( member_list_a @ B @ A3 )
        & ( bot_bot_set_list_a
          = ( inf_inf_set_list_a @ A3 @ B5 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1265_disjoint__insert_I2_J,axiom,
    ! [A3: set_Pr3443975907877334966_a_b_b,B: produc6499617310964463488_a_b_b,B5: set_Pr3443975907877334966_a_b_b] :
      ( ( bot_bo4897374000430069834_a_b_b
        = ( inf_in7962574851115124040_a_b_b @ A3 @ ( insert8720088738084019632_a_b_b @ B @ B5 ) ) )
      = ( ~ ( member4695696432722591383_a_b_b @ B @ A3 )
        & ( bot_bo4897374000430069834_a_b_b
          = ( inf_in7962574851115124040_a_b_b @ A3 @ B5 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1266_disjoint__insert_I2_J,axiom,
    ! [A3: set_dtree_list_a_b,B: dtree_list_a_b,B5: set_dtree_list_a_b] :
      ( ( bot_bo798015271861357502st_a_b
        = ( inf_in3355993651213403836st_a_b @ A3 @ ( insert7065945238938772516st_a_b @ B @ B5 ) ) )
      = ( ~ ( member551035911493665803st_a_b @ B @ A3 )
        & ( bot_bo798015271861357502st_a_b
          = ( inf_in3355993651213403836st_a_b @ A3 @ B5 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1267_disjoint__insert_I2_J,axiom,
    ! [A3: set_real,B: real,B5: set_real] :
      ( ( bot_bot_set_real
        = ( inf_inf_set_real @ A3 @ ( insert_real @ B @ B5 ) ) )
      = ( ~ ( member_real @ B @ A3 )
        & ( bot_bot_set_real
          = ( inf_inf_set_real @ A3 @ B5 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1268_disjoint__insert_I2_J,axiom,
    ! [A3: set_nat,B: nat,B5: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ A3 @ ( insert_nat @ B @ B5 ) ) )
      = ( ~ ( member_nat @ B @ A3 )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A3 @ B5 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1269_finsert_Orep__eq,axiom,
    ! [X: produc6499617310964463488_a_b_b,Xa2: fset_P2153231429829016240_a_b_b] :
      ( ( fset_P9138963618725001425_a_b_b @ ( finser2303212525150181944_a_b_b @ X @ Xa2 ) )
      = ( insert8720088738084019632_a_b_b @ X @ ( fset_P9138963618725001425_a_b_b @ Xa2 ) ) ) ).

% finsert.rep_eq

% Conjectures (1)
thf(conj_0,conjecture,
    member_list_a @ y @ ( dverts_list_a_b @ ( node_list_a_b @ r @ xs ) ) ).

%------------------------------------------------------------------------------