TPTP Problem File: SLH0014^1.p

View Solutions - Solve Problem 
%------------------------------------------------------------------------------
% 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    : FOL_Seq_Calc3/0007_Fair_Stream/prob_00053_001720__11911040_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1637 ( 605 unt; 344 typ;   0 def)
%            Number of atoms       : 3658 (1370 equ;   0 cnn)
%            Maximal formula atoms :   13 (   2 avg)
%            Number of connectives : 10799 ( 390   ~;  51   |; 273   &;8482   @)
%                                         (   0 <=>;1603  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   20 (   6 avg)
%            Number of types       :   28 (  27 usr)
%            Number of type conns  : 1290 (1290   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  320 ( 317 usr;  18 con; 0-4 aty)
%            Number of variables   : 3508 ( 322   ^;3085   !; 101   ?;3508   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 15:27:23.777
%------------------------------------------------------------------------------
% Could-be-implicit typings (27)
thf(ty_n_t__Stream__Ostream_It__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_J,type,
    stream8854267710728967580list_a: $tType ).

thf(ty_n_t__Stream__Ostream_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
    stream2255243159586646806list_a: $tType ).

thf(ty_n_t__Stream__Ostream_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
    stream8994515351414039536list_a: $tType ).

thf(ty_n_t__Stream__Ostream_It__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
    stream1977264251772372042list_a: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
    list_l7815035709764258326list_a: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
    set_list_list_list_a: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
    set_set_set_list_a: $tType ).

thf(ty_n_t__Stream__Ostream_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    stream_list_list_a: $tType ).

thf(ty_n_t__Stream__Ostream_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    stream_set_list_a: $tType ).

thf(ty_n_t__Stream__Ostream_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
    stream_list_set_a: $tType ).

thf(ty_n_t__Stream__Ostream_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
    stream_set_set_a: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    list_list_list_a: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    set_list_list_a: $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_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
    set_set_set_a: $tType ).

thf(ty_n_t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
    stream_list_a: $tType ).

thf(ty_n_t__Stream__Ostream_It__Set__Oset_Itf__a_J_J,type,
    stream_set_a: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    list_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__List__Olist_It__Set__Oset_Itf__a_J_J,type,
    list_set_a: $tType ).

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

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

thf(ty_n_t__Stream__Ostream_Itf__a_J,type,
    stream_a: $tType ).

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

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

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

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

% Explicit typings (317)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__List__Olist_Itf__a_J,type,
    bNF_Gr7042794125918077091list_a: set_list_list_a > list_a > set_list_list_a ).

thf(sy_c_BNF__Greatest__Fixpoint_OShift_001tf__a,type,
    bNF_Greatest_Shift_a: set_list_a > a > set_list_a ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__List__Olist_Itf__a_J,type,
    bNF_Gr4634511371912843295list_a: set_list_list_a > list_list_a > set_list_a ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001tf__a,type,
    bNF_Greatest_Succ_a: set_list_a > list_a > set_a ).

thf(sy_c_Finite__Set_OFpow_001t__List__Olist_Itf__a_J,type,
    finite_Fpow_list_a: set_list_a > set_set_list_a ).

thf(sy_c_Finite__Set_OFpow_001tf__a,type,
    finite_Fpow_a: set_a > set_set_a ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_Itf__a_J,type,
    finite_finite_list_a: set_list_a > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    finite5282473924520328461list_a: set_set_list_a > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
    finite_finite_set_a: set_set_a > $o ).

thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
    finite_finite_a: set_a > $o ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    inj_on_list_a_list_a: ( list_a > list_a ) > set_list_a > $o ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001tf__a,type,
    inj_on_list_a_a: ( list_a > a ) > set_list_a > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__List__Olist_Itf__a_J,type,
    inj_on_nat_list_a: ( nat > list_a ) > set_nat > $o ).

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

thf(sy_c_Fun_Oinj__on_001tf__a_001t__List__Olist_Itf__a_J,type,
    inj_on_a_list_a: ( a > list_a ) > set_a > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
    inj_on_a_a: ( a > a ) > set_a > $o ).

thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    monoto5743347545245725159list_a: set_set_list_a > ( set_list_a > set_list_a > $o ) > ( set_list_a > set_list_a > $o ) > ( set_list_a > set_list_a ) > $o ).

thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Set__Oset_Itf__a_J,type,
    monoto5794692711665054817_set_a: set_set_list_a > ( set_list_a > set_list_a > $o ) > ( set_a > set_a > $o ) > ( set_list_a > set_a ) > $o ).

thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    monoto7974761327354622061list_a: set_set_a > ( set_a > set_a > $o ) > ( set_list_a > set_list_a > $o ) > ( set_a > set_list_a ) > $o ).

thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
    monoto7172710143293369831_set_a: set_set_a > ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > ( set_a > set_a ) > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    minus_646659088055828811list_a: set_list_a > set_list_a > set_list_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
    minus_minus_set_a: set_a > set_a > set_a ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    uminus7925729386456332763list_a: set_list_a > set_list_a ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
    uminus_uminus_set_a: set_a > set_a ).

thf(sy_c_If_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    if_list_list_a: $o > list_list_a > list_list_a > list_list_a ).

thf(sy_c_If_001t__List__Olist_Itf__a_J,type,
    if_list_a: $o > list_a > list_a > list_a ).

thf(sy_c_If_001t__Stream__Ostream_It__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_J,type,
    if_str5554437364767477090list_a: $o > stream8854267710728967580list_a > stream8854267710728967580list_a > stream8854267710728967580list_a ).

thf(sy_c_If_001t__Stream__Ostream_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
    if_str8217234800680828380list_a: $o > stream2255243159586646806list_a > stream2255243159586646806list_a > stream2255243159586646806list_a ).

thf(sy_c_If_001t__Stream__Ostream_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    if_str7505741754068378070list_a: $o > stream_list_list_a > stream_list_list_a > stream_list_list_a ).

thf(sy_c_If_001t__Stream__Ostream_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
    if_stream_list_set_a: $o > stream_list_set_a > stream_list_set_a > stream_list_set_a ).

thf(sy_c_If_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
    if_stream_list_a: $o > stream_list_a > stream_list_a > stream_list_a ).

thf(sy_c_If_001t__Stream__Ostream_Itf__a_J,type,
    if_stream_a: $o > stream_a > stream_a > stream_a ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    inf_inf_set_list_a: set_list_a > set_list_a > set_list_a ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
    inf_inf_set_a: set_a > set_a > set_a ).

thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    semila4954572621509150738list_a: ( set_list_a > set_list_a > set_list_a ) > set_list_a > ( set_list_a > set_list_a > $o ) > ( set_list_a > set_list_a > $o ) > $o ).

thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Set__Oset_Itf__a_J,type,
    semila2496817875450240012_set_a: ( set_a > set_a > set_a ) > set_a > ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    sup_sup_set_list_a: set_list_a > set_list_a > set_list_a ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    sup_su4537662296134749976list_a: set_set_list_a > set_set_list_a > set_set_list_a ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    sup_sup_set_set_a: set_set_a > set_set_a > set_set_a ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
    sup_sup_set_a: set_a > set_a > set_a ).

thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    lattic2653631110915024129list_a: set_set_list_a > set_list_a ).

thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_Itf__a_J,type,
    lattic8209813465164889211_set_a: set_set_a > set_a ).

thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__List__Olist_Itf__a_J,type,
    lattic4064278158876714078list_a: ( list_a > list_a > list_a ) > ( list_a > list_a > $o ) > ( list_a > list_a > $o ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    lattic1671282250946225214list_a: ( set_list_a > set_list_a > set_list_a ) > ( set_list_a > set_list_a > $o ) > ( set_list_a > set_list_a > $o ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Set__Oset_Itf__a_J,type,
    lattic8986249270076014136_set_a: ( set_a > set_a > set_a ) > ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__order__set_001tf__a,type,
    lattic5078705180708912344_set_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > $o ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__set_001t__List__Olist_Itf__a_J,type,
    lattic3454955330975583506list_a: ( list_a > list_a > list_a ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__set_001tf__a,type,
    lattic5961991414251573132_set_a: ( a > a > a ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__set_OF_001t__List__Olist_Itf__a_J,type,
    lattic491866123428426078list_a: ( list_a > list_a > list_a ) > set_list_a > list_a ).

thf(sy_c_Lattices__Big_Osemilattice__set_OF_001tf__a,type,
    lattic5116578512385870296ce_F_a: ( a > a > a ) > set_a > a ).

thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    lattic2137287156484284315list_a: set_set_list_a > set_list_a ).

thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_Itf__a_J,type,
    lattic2918178356826803221_set_a: set_set_a > set_a ).

thf(sy_c_List_Oappend_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    append_list_list_a: list_list_list_a > list_list_list_a > list_list_list_a ).

thf(sy_c_List_Oappend_001t__List__Olist_Itf__a_J,type,
    append_list_a: list_list_a > list_list_a > list_list_a ).

thf(sy_c_List_Oappend_001tf__a,type,
    append_a: list_a > list_a > list_a ).

thf(sy_c_List_Obind_001t__List__Olist_It__List__Olist_Itf__a_J_J_001tf__a,type,
    bind_list_list_a_a: list_list_list_a > ( list_list_a > list_a ) > list_a ).

thf(sy_c_List_Obind_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    bind_list_a_list_a: list_list_a > ( list_a > list_list_a ) > list_list_a ).

thf(sy_c_List_Obind_001t__List__Olist_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
    bind_list_a_set_a: list_list_a > ( list_a > list_set_a ) > list_set_a ).

thf(sy_c_List_Obind_001t__List__Olist_Itf__a_J_001tf__a,type,
    bind_list_a_a: list_list_a > ( list_a > list_a ) > list_a ).

thf(sy_c_List_Obind_001t__Set__Oset_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    bind_set_a_list_a: list_set_a > ( set_a > list_list_a ) > list_list_a ).

thf(sy_c_List_Obind_001t__Set__Oset_Itf__a_J_001tf__a,type,
    bind_set_a_a: list_set_a > ( set_a > list_a ) > list_a ).

thf(sy_c_List_Obind_001tf__a_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    bind_a_list_list_a: list_a > ( a > list_list_list_a ) > list_list_list_a ).

thf(sy_c_List_Obind_001tf__a_001t__List__Olist_Itf__a_J,type,
    bind_a_list_a: list_a > ( a > list_list_a ) > list_list_a ).

thf(sy_c_List_Obind_001tf__a_001t__Set__Oset_Itf__a_J,type,
    bind_a_set_a: list_a > ( a > list_set_a ) > list_set_a ).

thf(sy_c_List_Obind_001tf__a_001tf__a,type,
    bind_a_a: list_a > ( a > list_a ) > list_a ).

thf(sy_c_List_Obutlast_001t__List__Olist_Itf__a_J,type,
    butlast_list_a: list_list_a > list_list_a ).

thf(sy_c_List_Obutlast_001tf__a,type,
    butlast_a: list_a > list_a ).

thf(sy_c_List_Ocan__select_001t__List__Olist_Itf__a_J,type,
    can_select_list_a: ( list_a > $o ) > set_list_a > $o ).

thf(sy_c_List_Ocan__select_001tf__a,type,
    can_select_a: ( a > $o ) > set_a > $o ).

thf(sy_c_List_Oconcat_001tf__a,type,
    concat_a: list_list_a > list_a ).

thf(sy_c_List_Ocoset_001t__List__Olist_Itf__a_J,type,
    coset_list_a: list_list_a > set_list_a ).

thf(sy_c_List_Ocoset_001tf__a,type,
    coset_a: list_a > set_a ).

thf(sy_c_List_Odistinct_001t__List__Olist_Itf__a_J,type,
    distinct_list_a: list_list_a > $o ).

thf(sy_c_List_Odistinct_001tf__a,type,
    distinct_a: list_a > $o ).

thf(sy_c_List_Odistinct__adj_001tf__a,type,
    distinct_adj_a: list_a > $o ).

thf(sy_c_List_Ofold_001tf__a_001t__List__Olist_Itf__a_J,type,
    fold_a_list_a: ( a > list_a > list_a ) > list_a > list_a > list_a ).

thf(sy_c_List_Ofoldr_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    foldr_list_a_list_a: ( list_a > list_a > list_a ) > list_list_a > list_a > list_a ).

thf(sy_c_List_Ogen__length_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    gen_le5572345604234726045list_a: nat > list_list_list_a > nat ).

thf(sy_c_List_Ogen__length_001t__List__Olist_Itf__a_J,type,
    gen_length_list_a: nat > list_list_a > nat ).

thf(sy_c_List_Ogen__length_001t__Set__Oset_Itf__a_J,type,
    gen_length_set_a: nat > list_set_a > nat ).

thf(sy_c_List_Ogen__length_001tf__a,type,
    gen_length_a: nat > list_a > nat ).

thf(sy_c_List_Oinsert_001t__List__Olist_Itf__a_J,type,
    insert_list_a: list_a > list_list_a > list_list_a ).

thf(sy_c_List_Oinsert_001tf__a,type,
    insert_a: a > list_a > list_a ).

thf(sy_c_List_Olast_001t__List__Olist_Itf__a_J,type,
    last_list_a: list_list_a > list_a ).

thf(sy_c_List_Olast_001tf__a,type,
    last_a: list_a > a ).

thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    cons_l8577867181958743494list_a: list_list_list_a > list_l7815035709764258326list_a > list_l7815035709764258326list_a ).

thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    cons_list_list_a: list_list_a > list_list_list_a > list_list_list_a ).

thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
    cons_list_a: list_a > list_list_a > list_list_a ).

thf(sy_c_List_Olist_OCons_001t__Set__Oset_Itf__a_J,type,
    cons_set_a: set_a > list_set_a > list_set_a ).

thf(sy_c_List_Olist_OCons_001tf__a,type,
    cons_a: a > list_a > list_a ).

thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    nil_list_list_list_a: list_l7815035709764258326list_a ).

thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    nil_list_list_a: list_list_list_a ).

thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
    nil_list_a: list_list_a ).

thf(sy_c_List_Olist_ONil_001t__Set__Oset_Itf__a_J,type,
    nil_set_a: list_set_a ).

thf(sy_c_List_Olist_ONil_001tf__a,type,
    nil_a: list_a ).

thf(sy_c_List_Olist_Ohd_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    hd_list_list_list_a: list_l7815035709764258326list_a > list_list_list_a ).

thf(sy_c_List_Olist_Ohd_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    hd_list_list_a: list_list_list_a > list_list_a ).

thf(sy_c_List_Olist_Ohd_001t__List__Olist_Itf__a_J,type,
    hd_list_a: list_list_a > list_a ).

thf(sy_c_List_Olist_Ohd_001t__Set__Oset_Itf__a_J,type,
    hd_set_a: list_set_a > set_a ).

thf(sy_c_List_Olist_Ohd_001tf__a,type,
    hd_a: list_a > a ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    set_list_list_a2: list_list_list_a > set_list_list_a ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__a_J,type,
    set_list_a2: list_list_a > set_list_a ).

thf(sy_c_List_Olist_Oset_001tf__a,type,
    set_a2: list_a > set_a ).

thf(sy_c_List_Olist_Otl_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    tl_list_list_list_a: list_l7815035709764258326list_a > list_l7815035709764258326list_a ).

thf(sy_c_List_Olist_Otl_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    tl_list_list_a: list_list_list_a > list_list_list_a ).

thf(sy_c_List_Olist_Otl_001t__List__Olist_Itf__a_J,type,
    tl_list_a: list_list_a > list_list_a ).

thf(sy_c_List_Olist_Otl_001t__Set__Oset_Itf__a_J,type,
    tl_set_a: list_set_a > list_set_a ).

thf(sy_c_List_Olist_Otl_001tf__a,type,
    tl_a: list_a > list_a ).

thf(sy_c_List_Olist__ex1_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    list_ex1_list_list_a: ( list_list_a > $o ) > list_list_list_a > $o ).

thf(sy_c_List_Olist__ex1_001t__List__Olist_Itf__a_J,type,
    list_ex1_list_a: ( list_a > $o ) > list_list_a > $o ).

thf(sy_c_List_Olist__ex1_001t__Set__Oset_Itf__a_J,type,
    list_ex1_set_a: ( set_a > $o ) > list_set_a > $o ).

thf(sy_c_List_Olist__ex1_001tf__a,type,
    list_ex1_a: ( a > $o ) > list_a > $o ).

thf(sy_c_List_Olistrelp_001tf__a_001tf__a,type,
    listrelp_a_a: ( a > a > $o ) > list_a > list_a > $o ).

thf(sy_c_List_Olists_001t__List__Olist_Itf__a_J,type,
    lists_list_a: set_list_a > set_list_list_a ).

thf(sy_c_List_Olists_001tf__a,type,
    lists_a: set_a > set_list_a ).

thf(sy_c_List_Olistset_001tf__a,type,
    listset_a: list_set_a > set_list_a ).

thf(sy_c_List_Omaps_001t__List__Olist_It__List__Olist_Itf__a_J_J_001tf__a,type,
    maps_list_list_a_a: ( list_list_a > list_a ) > list_list_list_a > list_a ).

thf(sy_c_List_Omaps_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    maps_list_a_list_a: ( list_a > list_list_a ) > list_list_a > list_list_a ).

thf(sy_c_List_Omaps_001t__List__Olist_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
    maps_list_a_set_a: ( list_a > list_set_a ) > list_list_a > list_set_a ).

thf(sy_c_List_Omaps_001t__List__Olist_Itf__a_J_001tf__a,type,
    maps_list_a_a: ( list_a > list_a ) > list_list_a > list_a ).

thf(sy_c_List_Omaps_001t__Set__Oset_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    maps_set_a_list_a: ( set_a > list_list_a ) > list_set_a > list_list_a ).

thf(sy_c_List_Omaps_001t__Set__Oset_Itf__a_J_001tf__a,type,
    maps_set_a_a: ( set_a > list_a ) > list_set_a > list_a ).

thf(sy_c_List_Omaps_001tf__a_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    maps_a_list_list_a: ( a > list_list_list_a ) > list_a > list_list_list_a ).

thf(sy_c_List_Omaps_001tf__a_001t__List__Olist_Itf__a_J,type,
    maps_a_list_a: ( a > list_list_a ) > list_a > list_list_a ).

thf(sy_c_List_Omaps_001tf__a_001t__Set__Oset_Itf__a_J,type,
    maps_a_set_a: ( a > list_set_a ) > list_a > list_set_a ).

thf(sy_c_List_Omaps_001tf__a_001tf__a,type,
    maps_a_a: ( a > list_a ) > list_a > list_a ).

thf(sy_c_List_Omember_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    member_list_list_a: list_list_list_a > list_list_a > $o ).

thf(sy_c_List_Omember_001t__List__Olist_Itf__a_J,type,
    member_list_a: list_list_a > list_a > $o ).

thf(sy_c_List_Omember_001t__Set__Oset_Itf__a_J,type,
    member_set_a: list_set_a > set_a > $o ).

thf(sy_c_List_Omember_001tf__a,type,
    member_a: list_a > a > $o ).

thf(sy_c_List_Onull_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    null_list_list_a: list_list_list_a > $o ).

thf(sy_c_List_Onull_001t__List__Olist_Itf__a_J,type,
    null_list_a: list_list_a > $o ).

thf(sy_c_List_Onull_001t__Set__Oset_Itf__a_J,type,
    null_set_a: list_set_a > $o ).

thf(sy_c_List_Onull_001tf__a,type,
    null_a: list_a > $o ).

thf(sy_c_List_Oord_Olexordp__eq_001tf__a,type,
    lexordp_eq_a: ( a > a > $o ) > list_a > list_a > $o ).

thf(sy_c_List_Oproduct__lists_001tf__a,type,
    product_lists_a: list_list_a > list_list_a ).

thf(sy_c_List_Oremdups__adj_001t__List__Olist_Itf__a_J,type,
    remdups_adj_list_a: list_list_a > list_list_a ).

thf(sy_c_List_Oremdups__adj_001tf__a,type,
    remdups_adj_a: list_a > list_a ).

thf(sy_c_List_Oremove1_001t__List__Olist_Itf__a_J,type,
    remove1_list_a: list_a > list_list_a > list_list_a ).

thf(sy_c_List_Oremove1_001tf__a,type,
    remove1_a: a > list_a > list_a ).

thf(sy_c_List_OremoveAll_001t__List__Olist_Itf__a_J,type,
    removeAll_list_a: list_a > list_list_a > list_list_a ).

thf(sy_c_List_OremoveAll_001tf__a,type,
    removeAll_a: a > list_a > list_a ).

thf(sy_c_List_Orev_001tf__a,type,
    rev_a: list_a > list_a ).

thf(sy_c_List_Orotate1_001tf__a,type,
    rotate1_a: list_a > list_a ).

thf(sy_c_List_Oshuffles_001tf__a,type,
    shuffles_a: list_a > list_a > set_list_a ).

thf(sy_c_List_Osubseqs_001t__List__Olist_Itf__a_J,type,
    subseqs_list_a: list_list_a > list_list_list_a ).

thf(sy_c_List_Osubseqs_001tf__a,type,
    subseqs_a: list_a > list_list_a ).

thf(sy_c_List_Osuccessively_001t__List__Olist_Itf__a_J,type,
    successively_list_a: ( list_a > list_a > $o ) > list_list_a > $o ).

thf(sy_c_List_Osuccessively_001tf__a,type,
    successively_a: ( a > a > $o ) > list_a > $o ).

thf(sy_c_List_Otranspose_001tf__a,type,
    transpose_a: list_list_a > list_list_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    bot_bot_set_list_a: set_list_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    bot_bo3186585308812441520list_a: set_set_list_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    bot_bot_set_set_a: set_set_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
    bot_bot_set_a: set_a ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    ord_less_set_list_a: set_list_a > set_list_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
    ord_less_set_a: set_a > set_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
    ord_le416739357993157210list_a: ( $o > $o > set_list_a ) > ( $o > $o > set_list_a ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mt__Set__Oset_Itf__a_J_J_J,type,
    ord_le5604070792465694036_set_a: ( $o > $o > set_a ) > ( $o > $o > set_a ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
    ord_le2535169540198774551list_a: ( $o > set_list_list_a ) > ( $o > set_list_list_a ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    ord_le354538097136858257list_a: ( $o > set_list_a ) > ( $o > set_list_a ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
    ord_le452127650934333553list_a: ( $o > set_set_list_a ) > ( $o > set_set_list_a ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
    ord_le2411852534652195563_set_a: ( $o > set_set_a ) > ( $o > set_set_a ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_Itf__a_J_J,type,
    ord_less_eq_o_set_a: ( $o > set_a ) > ( $o > set_a ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    ord_le8488217952732425610list_a: set_list_list_a > set_list_list_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    ord_le8861187494160871172list_a: set_list_a > set_list_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    ord_le8877086941679407844list_a: set_set_list_a > set_set_list_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
    ord_le7622928425522016964list_a: set_set_set_list_a > set_set_set_list_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
    ord_le5722252365846178494_set_a: set_set_set_a > set_set_set_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
    ord_less_eq_set_a: set_a > set_a > $o ).

thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_I_Eo_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    order_6341408326612538392list_a: ( ( $o > set_list_a ) > $o ) > $o > set_list_a ).

thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_I_Eo_Mt__Set__Oset_Itf__a_J_J,type,
    order_6114237596796908690_set_a: ( ( $o > set_a ) > $o ) > $o > set_a ).

thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    order_4834126633758709443list_a: ( set_list_list_a > $o ) > set_list_list_a ).

thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    order_733672244956367037list_a: ( set_list_a > $o ) > set_list_a ).

thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    order_8947689642956441757list_a: ( set_set_list_a > $o ) > set_set_list_a ).

thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    order_3565860530148683671_set_a: ( set_set_a > $o ) > set_set_a ).

thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_Itf__a_J,type,
    order_Greatest_set_a: ( set_a > $o ) > set_a ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    top_top_set_list_a: set_list_a ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    top_to7106483174946246804list_a: set_set_list_a ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    top_top_set_set_a: set_set_a ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
    top_top_set_a: set_a ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    collect_list_list_a: ( list_list_a > $o ) > set_list_list_a ).

thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
    collect_list_a: ( list_a > $o ) > set_list_a ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    collect_set_list_a: ( set_list_a > $o ) > set_set_list_a ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    collec3809296942973202735list_a: ( set_set_list_a > $o ) > set_set_set_list_a ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    collect_set_set_a: ( set_set_a > $o ) > set_set_set_a ).

thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
    collect_set_a: ( set_a > $o ) > set_set_a ).

thf(sy_c_Set_OCollect_001tf__a,type,
    collect_a: ( a > $o ) > set_a ).

thf(sy_c_Set_OPow_001t__List__Olist_Itf__a_J,type,
    pow_list_a: set_list_a > set_set_list_a ).

thf(sy_c_Set_OPow_001tf__a,type,
    pow_a: set_a > set_set_a ).

thf(sy_c_Set_Oimage_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    image_432481560377026271list_a: ( list_list_a > set_list_a ) > set_list_list_a > set_set_list_a ).

thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    image_list_a_list_a: ( list_a > list_a ) > set_list_a > set_list_a ).

thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
    image_list_a_set_a: ( list_a > set_a ) > set_list_a > set_set_a ).

thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001tf__a,type,
    image_list_a_a: ( list_a > a ) > set_list_a > set_a ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_Itf__a_J,type,
    image_nat_list_a: ( nat > list_a ) > set_nat > set_list_a ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
    image_nat_a: ( nat > a ) > set_nat > set_a ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    image_5749939591322298757list_a: ( set_list_a > set_list_a ) > set_set_list_a > set_set_list_a ).

thf(sy_c_Set_Oimage_001tf__a_001t__List__Olist_Itf__a_J,type,
    image_a_list_a: ( a > list_a ) > set_a > set_list_a ).

thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
    image_a_a: ( a > a ) > set_a > set_a ).

thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
    insert_list_a2: list_a > set_list_a > set_list_a ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    insert_set_list_a: set_list_a > set_set_list_a > set_set_list_a ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
    insert_set_a: set_a > set_set_a > set_set_a ).

thf(sy_c_Set_Oinsert_001tf__a,type,
    insert_a2: a > set_a > set_a ).

thf(sy_c_Set_Ois__singleton_001t__List__Olist_Itf__a_J,type,
    is_singleton_list_a: set_list_a > $o ).

thf(sy_c_Set_Opairwise_001t__List__Olist_Itf__a_J,type,
    pairwise_list_a: ( list_a > list_a > $o ) > set_list_a > $o ).

thf(sy_c_Set_Opairwise_001tf__a,type,
    pairwise_a: ( a > a > $o ) > set_a > $o ).

thf(sy_c_Set_Oremove_001t__List__Olist_Itf__a_J,type,
    remove_list_a: list_a > set_list_a > set_list_a ).

thf(sy_c_Set_Othe__elem_001tf__a,type,
    the_elem_a: set_a > a ).

thf(sy_c_Stream_Ocycle_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    cycle_list_list_a: list_list_list_a > stream_list_list_a ).

thf(sy_c_Stream_Ocycle_001t__List__Olist_Itf__a_J,type,
    cycle_list_a: list_list_a > stream_list_a ).

thf(sy_c_Stream_Ocycle_001tf__a,type,
    cycle_a: list_a > stream_a ).

thf(sy_c_Stream_Oflat_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    flat_l8549998914816956042list_a: stream8854267710728967580list_a > stream2255243159586646806list_a ).

thf(sy_c_Stream_Oflat_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    flat_list_list_a: stream2255243159586646806list_a > stream_list_list_a ).

thf(sy_c_Stream_Oflat_001t__List__Olist_Itf__a_J,type,
    flat_list_a: stream_list_list_a > stream_list_a ).

thf(sy_c_Stream_Oflat_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    flat_set_list_a: stream8994515351414039536list_a > stream_set_list_a ).

thf(sy_c_Stream_Oflat_001t__Set__Oset_Itf__a_J,type,
    flat_set_a: stream_list_set_a > stream_set_a ).

thf(sy_c_Stream_Oflat_001tf__a,type,
    flat_a: stream_list_a > stream_a ).

thf(sy_c_Stream_Osdrop__while_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    sdrop_8575598270869141876list_a: ( list_list_list_a > $o ) > stream2255243159586646806list_a > stream2255243159586646806list_a ).

thf(sy_c_Stream_Osdrop__while_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    sdrop_1072776591231973358list_a: ( list_list_a > $o ) > stream_list_list_a > stream_list_list_a ).

thf(sy_c_Stream_Osdrop__while_001t__List__Olist_Itf__a_J,type,
    sdrop_while_list_a: ( list_a > $o ) > stream_list_a > stream_list_a ).

thf(sy_c_Stream_Osdrop__while_001tf__a,type,
    sdrop_while_a: ( a > $o ) > stream_a > stream_a ).

thf(sy_c_Stream_Oshift_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    shift_2124064713455379641list_a: list_l7815035709764258326list_a > stream2255243159586646806list_a > stream2255243159586646806list_a ).

thf(sy_c_Stream_Oshift_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    shift_list_list_a: list_list_list_a > stream_list_list_a > stream_list_list_a ).

thf(sy_c_Stream_Oshift_001t__List__Olist_Itf__a_J,type,
    shift_list_a: list_list_a > stream_list_a > stream_list_a ).

thf(sy_c_Stream_Oshift_001t__Set__Oset_Itf__a_J,type,
    shift_set_a: list_set_a > stream_set_a > stream_set_a ).

thf(sy_c_Stream_Oshift_001tf__a,type,
    shift_a: list_a > stream_a > stream_a ).

thf(sy_c_Stream_Osinterleave_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    sinter1285254775629623405list_a: stream2255243159586646806list_a > stream2255243159586646806list_a > stream2255243159586646806list_a ).

thf(sy_c_Stream_Osinterleave_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    sinter664328039231274855list_a: stream_list_list_a > stream_list_list_a > stream_list_list_a ).

thf(sy_c_Stream_Osinterleave_001t__List__Olist_Itf__a_J,type,
    sinterleave_list_a: stream_list_a > stream_list_a > stream_list_a ).

thf(sy_c_Stream_Osinterleave_001tf__a,type,
    sinterleave_a: stream_a > stream_a > stream_a ).

thf(sy_c_Stream_Osmap2_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    smap2_6789832397309367756list_a: ( list_list_a > list_list_a > list_list_a ) > stream_list_list_a > stream_list_list_a > stream_list_list_a ).

thf(sy_c_Stream_Osmap2_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_Itf__a_J,type,
    smap2_7557085947764330182list_a: ( list_list_a > list_list_a > list_a ) > stream_list_list_a > stream_list_list_a > stream_list_a ).

thf(sy_c_Stream_Osmap2_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    smap2_2416595860148935890list_a: ( list_list_a > list_a > list_list_a ) > stream_list_list_a > stream_list_a > stream_list_list_a ).

thf(sy_c_Stream_Osmap2_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    smap2_9173189155220197580list_a: ( list_list_a > list_a > list_a ) > stream_list_list_a > stream_list_a > stream_list_a ).

thf(sy_c_Stream_Osmap2_001t__List__Olist_It__List__Olist_Itf__a_J_J_001tf__a_001tf__a,type,
    smap2_7313055693640098252_a_a_a: ( list_list_a > a > a ) > stream_list_list_a > stream_a > stream_a ).

thf(sy_c_Stream_Osmap2_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    smap2_1314303423829136838list_a: ( list_a > list_list_a > list_list_a ) > stream_list_a > stream_list_list_a > stream_list_list_a ).

thf(sy_c_Stream_Osmap2_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_Itf__a_J,type,
    smap2_9108955389640348608list_a: ( list_a > list_list_a > list_a ) > stream_list_a > stream_list_list_a > stream_list_a ).

thf(sy_c_Stream_Osmap2_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    smap2_3968465302024954316list_a: ( list_a > list_a > list_list_a ) > stream_list_a > stream_list_a > stream_list_list_a ).

thf(sy_c_Stream_Osmap2_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    smap2_6771825536533895878list_a: ( list_a > list_a > list_a ) > stream_list_a > stream_list_a > stream_list_a ).

thf(sy_c_Stream_Osmap2_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J_001tf__a,type,
    smap2_3658108627635373888st_a_a: ( list_a > list_a > a ) > stream_list_a > stream_list_a > stream_a ).

thf(sy_c_Stream_Osmap2_001t__List__Olist_Itf__a_J_001tf__a_001t__List__Olist_Itf__a_J,type,
    smap2_5547344383464640076list_a: ( list_a > a > list_a ) > stream_list_a > stream_a > stream_list_a ).

thf(sy_c_Stream_Osmap2_001t__List__Olist_Itf__a_J_001tf__a_001tf__a,type,
    smap2_list_a_a_a: ( list_a > a > a ) > stream_list_a > stream_a > stream_a ).

thf(sy_c_Stream_Osmap2_001tf__a_001t__List__Olist_It__List__Olist_Itf__a_J_J_001tf__a,type,
    smap2_4184781901917651636st_a_a: ( a > list_list_a > a ) > stream_a > stream_list_list_a > stream_a ).

thf(sy_c_Stream_Osmap2_001tf__a_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    smap2_5647228111304775744list_a: ( a > list_a > list_a ) > stream_a > stream_list_a > stream_list_a ).

thf(sy_c_Stream_Osmap2_001tf__a_001t__List__Olist_Itf__a_J_001tf__a,type,
    smap2_a_list_a_a: ( a > list_a > a ) > stream_a > stream_list_a > stream_a ).

thf(sy_c_Stream_Osmap2_001tf__a_001tf__a_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    smap2_3581788270250579916list_a: ( a > a > list_list_a ) > stream_a > stream_a > stream_list_list_a ).

thf(sy_c_Stream_Osmap2_001tf__a_001tf__a_001t__List__Olist_Itf__a_J,type,
    smap2_a_a_list_a: ( a > a > list_a ) > stream_a > stream_a > stream_list_a ).

thf(sy_c_Stream_Osmap2_001tf__a_001tf__a_001tf__a,type,
    smap2_a_a_a: ( a > a > a ) > stream_a > stream_a > stream_a ).

thf(sy_c_Stream_Osmember_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    smembe6049431907005952846list_a: list_list_list_a > stream2255243159586646806list_a > $o ).

thf(sy_c_Stream_Osmember_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    smember_list_list_a: list_list_a > stream_list_list_a > $o ).

thf(sy_c_Stream_Osmember_001t__List__Olist_Itf__a_J,type,
    smember_list_a: list_a > stream_list_a > $o ).

thf(sy_c_Stream_Osmember_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    smember_set_list_a: set_list_a > stream_set_list_a > $o ).

thf(sy_c_Stream_Osmember_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    smembe8028980443989764482list_a: set_set_list_a > stream1977264251772372042list_a > $o ).

thf(sy_c_Stream_Osmember_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    smember_set_set_a: set_set_a > stream_set_set_a > $o ).

thf(sy_c_Stream_Osmember_001t__Set__Oset_Itf__a_J,type,
    smember_set_a: set_a > stream_set_a > $o ).

thf(sy_c_Stream_Osmember_001tf__a,type,
    smember_a: a > stream_a > $o ).

thf(sy_c_Stream_Ostream_OSCons_001t__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
    sCons_3158304508880615213list_a: list_l7815035709764258326list_a > stream8854267710728967580list_a > stream8854267710728967580list_a ).

thf(sy_c_Stream_Ostream_OSCons_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    sCons_8165023923567507367list_a: list_list_list_a > stream2255243159586646806list_a > stream2255243159586646806list_a ).

thf(sy_c_Stream_Ostream_OSCons_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    sCons_list_list_a: list_list_a > stream_list_list_a > stream_list_list_a ).

thf(sy_c_Stream_Ostream_OSCons_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
    sCons_list_set_a: list_set_a > stream_list_set_a > stream_list_set_a ).

thf(sy_c_Stream_Ostream_OSCons_001t__List__Olist_Itf__a_J,type,
    sCons_list_a: list_a > stream_list_a > stream_list_a ).

thf(sy_c_Stream_Ostream_OSCons_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    sCons_set_list_a: set_list_a > stream_set_list_a > stream_set_list_a ).

thf(sy_c_Stream_Ostream_OSCons_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    sCons_set_set_list_a: set_set_list_a > stream1977264251772372042list_a > stream1977264251772372042list_a ).

thf(sy_c_Stream_Ostream_OSCons_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    sCons_set_set_a: set_set_a > stream_set_set_a > stream_set_set_a ).

thf(sy_c_Stream_Ostream_OSCons_001t__Set__Oset_Itf__a_J,type,
    sCons_set_a: set_a > stream_set_a > stream_set_a ).

thf(sy_c_Stream_Ostream_OSCons_001tf__a,type,
    sCons_a: a > stream_a > stream_a ).

thf(sy_c_Stream_Ostream_Oshd_001t__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
    shd_li1819241521678429890list_a: stream8854267710728967580list_a > list_l7815035709764258326list_a ).

thf(sy_c_Stream_Ostream_Oshd_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    shd_list_list_list_a: stream2255243159586646806list_a > list_list_list_a ).

thf(sy_c_Stream_Ostream_Oshd_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    shd_list_list_a: stream_list_list_a > list_list_a ).

thf(sy_c_Stream_Ostream_Oshd_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
    shd_list_set_a: stream_list_set_a > list_set_a ).

thf(sy_c_Stream_Ostream_Oshd_001t__List__Olist_Itf__a_J,type,
    shd_list_a: stream_list_a > list_a ).

thf(sy_c_Stream_Ostream_Oshd_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    shd_set_list_a: stream_set_list_a > set_list_a ).

thf(sy_c_Stream_Ostream_Oshd_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    shd_set_set_list_a: stream1977264251772372042list_a > set_set_list_a ).

thf(sy_c_Stream_Ostream_Oshd_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    shd_set_set_a: stream_set_set_a > set_set_a ).

thf(sy_c_Stream_Ostream_Oshd_001t__Set__Oset_Itf__a_J,type,
    shd_set_a: stream_set_a > set_a ).

thf(sy_c_Stream_Ostream_Oshd_001tf__a,type,
    shd_a: stream_a > a ).

thf(sy_c_Stream_Ostream_Osmap_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    smap_l7216702241007268101list_a: ( list_list_list_a > list_list_list_a ) > stream2255243159586646806list_a > stream2255243159586646806list_a ).

thf(sy_c_Stream_Ostream_Osmap_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    smap_l3391565068965508869list_a: ( list_list_a > list_list_a ) > stream_list_list_a > stream_list_list_a ).

thf(sy_c_Stream_Ostream_Osmap_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_Itf__a_J,type,
    smap_l1139969084374327423list_a: ( list_list_a > list_a ) > stream_list_list_a > stream_list_a ).

thf(sy_c_Stream_Ostream_Osmap_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    smap_l5222851033613708939list_a: ( list_a > list_list_a ) > stream_list_a > stream_list_list_a ).

thf(sy_c_Stream_Ostream_Osmap_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    smap_list_a_list_a: ( list_a > list_a ) > stream_list_a > stream_list_a ).

thf(sy_c_Stream_Ostream_Osmap_001t__List__Olist_Itf__a_J_001tf__a,type,
    smap_list_a_a: ( list_a > a ) > stream_list_a > stream_a ).

thf(sy_c_Stream_Ostream_Osmap_001tf__a_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    smap_a4630580661157078679list_a: ( a > list_list_list_a ) > stream_a > stream2255243159586646806list_a ).

thf(sy_c_Stream_Ostream_Osmap_001tf__a_001t__List__Olist_Itf__a_J,type,
    smap_a_list_a: ( a > list_a ) > stream_a > stream_list_a ).

thf(sy_c_Stream_Ostream_Osmap_001tf__a_001tf__a,type,
    smap_a_a: ( a > a ) > stream_a > stream_a ).

thf(sy_c_Stream_Ostream_Osset_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    sset_l5650713775164989220list_a: stream2255243159586646806list_a > set_list_list_list_a ).

thf(sy_c_Stream_Ostream_Osset_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    sset_list_list_a: stream_list_list_a > set_list_list_a ).

thf(sy_c_Stream_Ostream_Osset_001t__List__Olist_Itf__a_J,type,
    sset_list_a: stream_list_a > set_list_a ).

thf(sy_c_Stream_Ostream_Osset_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    sset_set_list_a: stream_set_list_a > set_set_list_a ).

thf(sy_c_Stream_Ostream_Osset_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    sset_set_set_list_a: stream1977264251772372042list_a > set_set_set_list_a ).

thf(sy_c_Stream_Ostream_Osset_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    sset_set_set_a: stream_set_set_a > set_set_set_a ).

thf(sy_c_Stream_Ostream_Osset_001t__Set__Oset_Itf__a_J,type,
    sset_set_a: stream_set_a > set_set_a ).

thf(sy_c_Stream_Ostream_Osset_001tf__a,type,
    sset_a: stream_a > set_a ).

thf(sy_c_Stream_Ostream_Ostl_001t__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
    stl_li4757347539825400518list_a: stream8854267710728967580list_a > stream8854267710728967580list_a ).

thf(sy_c_Stream_Ostream_Ostl_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    stl_list_list_list_a: stream2255243159586646806list_a > stream2255243159586646806list_a ).

thf(sy_c_Stream_Ostream_Ostl_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    stl_list_list_a: stream_list_list_a > stream_list_list_a ).

thf(sy_c_Stream_Ostream_Ostl_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    stl_list_set_list_a: stream8994515351414039536list_a > stream8994515351414039536list_a ).

thf(sy_c_Stream_Ostream_Ostl_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
    stl_list_set_a: stream_list_set_a > stream_list_set_a ).

thf(sy_c_Stream_Ostream_Ostl_001t__List__Olist_Itf__a_J,type,
    stl_list_a: stream_list_a > stream_list_a ).

thf(sy_c_Stream_Ostream_Ostl_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    stl_set_list_a: stream_set_list_a > stream_set_list_a ).

thf(sy_c_Stream_Ostream_Ostl_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    stl_set_set_list_a: stream1977264251772372042list_a > stream1977264251772372042list_a ).

thf(sy_c_Stream_Ostream_Ostl_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    stl_set_set_a: stream_set_set_a > stream_set_set_a ).

thf(sy_c_Stream_Ostream_Ostl_001t__Set__Oset_Itf__a_J,type,
    stl_set_a: stream_set_a > stream_set_a ).

thf(sy_c_Stream_Ostream_Ostl_001tf__a,type,
    stl_a: stream_a > stream_a ).

thf(sy_c_Zorn_Ochains_001t__List__Olist_Itf__a_J,type,
    chains_list_a: set_set_list_a > set_set_set_list_a ).

thf(sy_c_Zorn_Ochains_001tf__a,type,
    chains_a: set_set_a > set_set_set_a ).

thf(sy_c_member_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    member5342144027231129785list_a: list_list_list_a > set_list_list_list_a > $o ).

thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    member_list_list_a2: list_list_a > set_list_list_a > $o ).

thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
    member_list_a2: list_a > set_list_a > $o ).

thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    member_set_list_a: set_list_a > set_set_list_a > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    member8857465052274545133list_a: set_set_list_a > set_set_set_list_a > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    member_set_set_a: set_set_a > set_set_set_a > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
    member_set_a2: set_a > set_set_a > $o ).

thf(sy_c_member_001tf__a,type,
    member_a2: a > set_a > $o ).

thf(sy_v_s,type,
    s: stream_list_a ).

thf(sy_v_sa____,type,
    sa: stream_list_a ).

% Relevant facts (1275)
thf(fact_0_Suc_Oprems_I3_J,axiom,
    ! [X: list_a] :
      ( ( member_list_a2 @ X @ ( sset_list_a @ sa ) )
     => ( X != nil_a ) ) ).

% Suc.prems(3)
thf(fact_1_assms_I3_J,axiom,
    ! [X: list_a] :
      ( ( member_list_a2 @ X @ ( sset_list_a @ s ) )
     => ( X != nil_a ) ) ).

% assms(3)
thf(fact_2_stl__sset,axiom,
    ! [X2: set_list_a,A: stream_set_list_a] :
      ( ( member_set_list_a @ X2 @ ( sset_set_list_a @ ( stl_set_list_a @ A ) ) )
     => ( member_set_list_a @ X2 @ ( sset_set_list_a @ A ) ) ) ).

% stl_sset
thf(fact_3_stl__sset,axiom,
    ! [X2: set_set_list_a,A: stream1977264251772372042list_a] :
      ( ( member8857465052274545133list_a @ X2 @ ( sset_set_set_list_a @ ( stl_set_set_list_a @ A ) ) )
     => ( member8857465052274545133list_a @ X2 @ ( sset_set_set_list_a @ A ) ) ) ).

% stl_sset
thf(fact_4_stl__sset,axiom,
    ! [X2: set_set_a,A: stream_set_set_a] :
      ( ( member_set_set_a @ X2 @ ( sset_set_set_a @ ( stl_set_set_a @ A ) ) )
     => ( member_set_set_a @ X2 @ ( sset_set_set_a @ A ) ) ) ).

% stl_sset
thf(fact_5_stl__sset,axiom,
    ! [X2: set_a,A: stream_set_a] :
      ( ( member_set_a2 @ X2 @ ( sset_set_a @ ( stl_set_a @ A ) ) )
     => ( member_set_a2 @ X2 @ ( sset_set_a @ A ) ) ) ).

% stl_sset
thf(fact_6_stl__sset,axiom,
    ! [X2: list_list_list_a,A: stream2255243159586646806list_a] :
      ( ( member5342144027231129785list_a @ X2 @ ( sset_l5650713775164989220list_a @ ( stl_list_list_list_a @ A ) ) )
     => ( member5342144027231129785list_a @ X2 @ ( sset_l5650713775164989220list_a @ A ) ) ) ).

% stl_sset
thf(fact_7_stl__sset,axiom,
    ! [X2: list_list_a,A: stream_list_list_a] :
      ( ( member_list_list_a2 @ X2 @ ( sset_list_list_a @ ( stl_list_list_a @ A ) ) )
     => ( member_list_list_a2 @ X2 @ ( sset_list_list_a @ A ) ) ) ).

% stl_sset
thf(fact_8_stl__sset,axiom,
    ! [X2: a,A: stream_a] :
      ( ( member_a2 @ X2 @ ( sset_a @ ( stl_a @ A ) ) )
     => ( member_a2 @ X2 @ ( sset_a @ A ) ) ) ).

% stl_sset
thf(fact_9_stl__sset,axiom,
    ! [X2: list_a,A: stream_list_a] :
      ( ( member_list_a2 @ X2 @ ( sset_list_a @ ( stl_list_a @ A ) ) )
     => ( member_list_a2 @ X2 @ ( sset_list_a @ A ) ) ) ).

% stl_sset
thf(fact_10_Stream_Osmember__def,axiom,
    ( smember_set_list_a
    = ( ^ [X3: set_list_a,S: stream_set_list_a] : ( member_set_list_a @ X3 @ ( sset_set_list_a @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_11_Stream_Osmember__def,axiom,
    ( smembe8028980443989764482list_a
    = ( ^ [X3: set_set_list_a,S: stream1977264251772372042list_a] : ( member8857465052274545133list_a @ X3 @ ( sset_set_set_list_a @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_12_Stream_Osmember__def,axiom,
    ( smember_set_set_a
    = ( ^ [X3: set_set_a,S: stream_set_set_a] : ( member_set_set_a @ X3 @ ( sset_set_set_a @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_13_Stream_Osmember__def,axiom,
    ( smember_set_a
    = ( ^ [X3: set_a,S: stream_set_a] : ( member_set_a2 @ X3 @ ( sset_set_a @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_14_Stream_Osmember__def,axiom,
    ( smember_list_list_a
    = ( ^ [X3: list_list_a,S: stream_list_list_a] : ( member_list_list_a2 @ X3 @ ( sset_list_list_a @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_15_Stream_Osmember__def,axiom,
    ( smember_a
    = ( ^ [X3: a,S: stream_a] : ( member_a2 @ X3 @ ( sset_a @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_16_Stream_Osmember__def,axiom,
    ( smember_list_a
    = ( ^ [X3: list_a,S: stream_list_a] : ( member_list_a2 @ X3 @ ( sset_list_a @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_17_sset__induct,axiom,
    ! [Y: set_list_a,S2: stream_set_list_a,P: set_list_a > stream_set_list_a > $o] :
      ( ( member_set_list_a @ Y @ ( sset_set_list_a @ S2 ) )
     => ( ! [S3: stream_set_list_a] : ( P @ ( shd_set_list_a @ S3 ) @ S3 )
       => ( ! [S3: stream_set_list_a,Y2: set_list_a] :
              ( ( member_set_list_a @ Y2 @ ( sset_set_list_a @ ( stl_set_list_a @ S3 ) ) )
             => ( ( P @ Y2 @ ( stl_set_list_a @ S3 ) )
               => ( P @ Y2 @ S3 ) ) )
         => ( P @ Y @ S2 ) ) ) ) ).

% sset_induct
thf(fact_18_sset__induct,axiom,
    ! [Y: set_set_list_a,S2: stream1977264251772372042list_a,P: set_set_list_a > stream1977264251772372042list_a > $o] :
      ( ( member8857465052274545133list_a @ Y @ ( sset_set_set_list_a @ S2 ) )
     => ( ! [S3: stream1977264251772372042list_a] : ( P @ ( shd_set_set_list_a @ S3 ) @ S3 )
       => ( ! [S3: stream1977264251772372042list_a,Y2: set_set_list_a] :
              ( ( member8857465052274545133list_a @ Y2 @ ( sset_set_set_list_a @ ( stl_set_set_list_a @ S3 ) ) )
             => ( ( P @ Y2 @ ( stl_set_set_list_a @ S3 ) )
               => ( P @ Y2 @ S3 ) ) )
         => ( P @ Y @ S2 ) ) ) ) ).

% sset_induct
thf(fact_19_sset__induct,axiom,
    ! [Y: set_set_a,S2: stream_set_set_a,P: set_set_a > stream_set_set_a > $o] :
      ( ( member_set_set_a @ Y @ ( sset_set_set_a @ S2 ) )
     => ( ! [S3: stream_set_set_a] : ( P @ ( shd_set_set_a @ S3 ) @ S3 )
       => ( ! [S3: stream_set_set_a,Y2: set_set_a] :
              ( ( member_set_set_a @ Y2 @ ( sset_set_set_a @ ( stl_set_set_a @ S3 ) ) )
             => ( ( P @ Y2 @ ( stl_set_set_a @ S3 ) )
               => ( P @ Y2 @ S3 ) ) )
         => ( P @ Y @ S2 ) ) ) ) ).

% sset_induct
thf(fact_20_sset__induct,axiom,
    ! [Y: set_a,S2: stream_set_a,P: set_a > stream_set_a > $o] :
      ( ( member_set_a2 @ Y @ ( sset_set_a @ S2 ) )
     => ( ! [S3: stream_set_a] : ( P @ ( shd_set_a @ S3 ) @ S3 )
       => ( ! [S3: stream_set_a,Y2: set_a] :
              ( ( member_set_a2 @ Y2 @ ( sset_set_a @ ( stl_set_a @ S3 ) ) )
             => ( ( P @ Y2 @ ( stl_set_a @ S3 ) )
               => ( P @ Y2 @ S3 ) ) )
         => ( P @ Y @ S2 ) ) ) ) ).

% sset_induct
thf(fact_21_sset__induct,axiom,
    ! [Y: list_list_list_a,S2: stream2255243159586646806list_a,P: list_list_list_a > stream2255243159586646806list_a > $o] :
      ( ( member5342144027231129785list_a @ Y @ ( sset_l5650713775164989220list_a @ S2 ) )
     => ( ! [S3: stream2255243159586646806list_a] : ( P @ ( shd_list_list_list_a @ S3 ) @ S3 )
       => ( ! [S3: stream2255243159586646806list_a,Y2: list_list_list_a] :
              ( ( member5342144027231129785list_a @ Y2 @ ( sset_l5650713775164989220list_a @ ( stl_list_list_list_a @ S3 ) ) )
             => ( ( P @ Y2 @ ( stl_list_list_list_a @ S3 ) )
               => ( P @ Y2 @ S3 ) ) )
         => ( P @ Y @ S2 ) ) ) ) ).

% sset_induct
thf(fact_22_sset__induct,axiom,
    ! [Y: list_list_a,S2: stream_list_list_a,P: list_list_a > stream_list_list_a > $o] :
      ( ( member_list_list_a2 @ Y @ ( sset_list_list_a @ S2 ) )
     => ( ! [S3: stream_list_list_a] : ( P @ ( shd_list_list_a @ S3 ) @ S3 )
       => ( ! [S3: stream_list_list_a,Y2: list_list_a] :
              ( ( member_list_list_a2 @ Y2 @ ( sset_list_list_a @ ( stl_list_list_a @ S3 ) ) )
             => ( ( P @ Y2 @ ( stl_list_list_a @ S3 ) )
               => ( P @ Y2 @ S3 ) ) )
         => ( P @ Y @ S2 ) ) ) ) ).

% sset_induct
thf(fact_23_sset__induct,axiom,
    ! [Y: a,S2: stream_a,P: a > stream_a > $o] :
      ( ( member_a2 @ Y @ ( sset_a @ S2 ) )
     => ( ! [S3: stream_a] : ( P @ ( shd_a @ S3 ) @ S3 )
       => ( ! [S3: stream_a,Y2: a] :
              ( ( member_a2 @ Y2 @ ( sset_a @ ( stl_a @ S3 ) ) )
             => ( ( P @ Y2 @ ( stl_a @ S3 ) )
               => ( P @ Y2 @ S3 ) ) )
         => ( P @ Y @ S2 ) ) ) ) ).

% sset_induct
thf(fact_24_sset__induct,axiom,
    ! [Y: list_a,S2: stream_list_a,P: list_a > stream_list_a > $o] :
      ( ( member_list_a2 @ Y @ ( sset_list_a @ S2 ) )
     => ( ! [S3: stream_list_a] : ( P @ ( shd_list_a @ S3 ) @ S3 )
       => ( ! [S3: stream_list_a,Y2: list_a] :
              ( ( member_list_a2 @ Y2 @ ( sset_list_a @ ( stl_list_a @ S3 ) ) )
             => ( ( P @ Y2 @ ( stl_list_a @ S3 ) )
               => ( P @ Y2 @ S3 ) ) )
         => ( P @ Y @ S2 ) ) ) ) ).

% sset_induct
thf(fact_25_list__ex1__simps_I1_J,axiom,
    ! [P: list_list_a > $o] :
      ~ ( list_ex1_list_list_a @ P @ nil_list_list_a ) ).

% list_ex1_simps(1)
thf(fact_26_list__ex1__simps_I1_J,axiom,
    ! [P: set_a > $o] :
      ~ ( list_ex1_set_a @ P @ nil_set_a ) ).

% list_ex1_simps(1)
thf(fact_27_list__ex1__simps_I1_J,axiom,
    ! [P: list_a > $o] :
      ~ ( list_ex1_list_a @ P @ nil_list_a ) ).

% list_ex1_simps(1)
thf(fact_28_list__ex1__simps_I1_J,axiom,
    ! [P: a > $o] :
      ~ ( list_ex1_a @ P @ nil_a ) ).

% list_ex1_simps(1)
thf(fact_29_sset__flat__stl,axiom,
    ! [S2: stream_list_set_a] : ( ord_le3724670747650509150_set_a @ ( sset_set_a @ ( flat_set_a @ ( stl_list_set_a @ S2 ) ) ) @ ( sset_set_a @ ( flat_set_a @ S2 ) ) ) ).

% sset_flat_stl
thf(fact_30_sset__flat__stl,axiom,
    ! [S2: stream8994515351414039536list_a] : ( ord_le8877086941679407844list_a @ ( sset_set_list_a @ ( flat_set_list_a @ ( stl_list_set_list_a @ S2 ) ) ) @ ( sset_set_list_a @ ( flat_set_list_a @ S2 ) ) ) ).

% sset_flat_stl
thf(fact_31_sset__flat__stl,axiom,
    ! [S2: stream2255243159586646806list_a] : ( ord_le8488217952732425610list_a @ ( sset_list_list_a @ ( flat_list_list_a @ ( stl_list_list_list_a @ S2 ) ) ) @ ( sset_list_list_a @ ( flat_list_list_a @ S2 ) ) ) ).

% sset_flat_stl
thf(fact_32_sset__flat__stl,axiom,
    ! [S2: stream_list_a] : ( ord_less_eq_set_a @ ( sset_a @ ( flat_a @ ( stl_list_a @ S2 ) ) ) @ ( sset_a @ ( flat_a @ S2 ) ) ) ).

% sset_flat_stl
thf(fact_33_sset__flat__stl,axiom,
    ! [S2: stream_list_list_a] : ( ord_le8861187494160871172list_a @ ( sset_list_a @ ( flat_list_a @ ( stl_list_list_a @ S2 ) ) ) @ ( sset_list_a @ ( flat_list_a @ S2 ) ) ) ).

% sset_flat_stl
thf(fact_34_bind__simps_I1_J,axiom,
    ! [F: a > list_a] :
      ( ( bind_a_a @ nil_a @ F )
      = nil_a ) ).

% bind_simps(1)
thf(fact_35_bind__simps_I1_J,axiom,
    ! [F: a > list_list_a] :
      ( ( bind_a_list_a @ nil_a @ F )
      = nil_list_a ) ).

% bind_simps(1)
thf(fact_36_bind__simps_I1_J,axiom,
    ! [F: a > list_set_a] :
      ( ( bind_a_set_a @ nil_a @ F )
      = nil_set_a ) ).

% bind_simps(1)
thf(fact_37_bind__simps_I1_J,axiom,
    ! [F: list_a > list_a] :
      ( ( bind_list_a_a @ nil_list_a @ F )
      = nil_a ) ).

% bind_simps(1)
thf(fact_38_bind__simps_I1_J,axiom,
    ! [F: set_a > list_a] :
      ( ( bind_set_a_a @ nil_set_a @ F )
      = nil_a ) ).

% bind_simps(1)
thf(fact_39_bind__simps_I1_J,axiom,
    ! [F: a > list_list_list_a] :
      ( ( bind_a_list_list_a @ nil_a @ F )
      = nil_list_list_a ) ).

% bind_simps(1)
thf(fact_40_bind__simps_I1_J,axiom,
    ! [F: list_a > list_list_a] :
      ( ( bind_list_a_list_a @ nil_list_a @ F )
      = nil_list_a ) ).

% bind_simps(1)
thf(fact_41_bind__simps_I1_J,axiom,
    ! [F: list_a > list_set_a] :
      ( ( bind_list_a_set_a @ nil_list_a @ F )
      = nil_set_a ) ).

% bind_simps(1)
thf(fact_42_bind__simps_I1_J,axiom,
    ! [F: list_list_a > list_a] :
      ( ( bind_list_list_a_a @ nil_list_list_a @ F )
      = nil_a ) ).

% bind_simps(1)
thf(fact_43_bind__simps_I1_J,axiom,
    ! [F: set_a > list_list_a] :
      ( ( bind_set_a_list_a @ nil_set_a @ F )
      = nil_list_a ) ).

% bind_simps(1)
thf(fact_44_member__rec_I2_J,axiom,
    ! [Y: list_a] :
      ~ ( member_list_a @ nil_list_a @ Y ) ).

% member_rec(2)
thf(fact_45_member__rec_I2_J,axiom,
    ! [Y: list_list_a] :
      ~ ( member_list_list_a @ nil_list_list_a @ Y ) ).

% member_rec(2)
thf(fact_46_member__rec_I2_J,axiom,
    ! [Y: set_a] :
      ~ ( member_set_a @ nil_set_a @ Y ) ).

% member_rec(2)
thf(fact_47_member__rec_I2_J,axiom,
    ! [Y: a] :
      ~ ( member_a @ nil_a @ Y ) ).

% member_rec(2)
thf(fact_48_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_length_list_a @ N @ nil_list_a )
      = N ) ).

% gen_length_code(1)
thf(fact_49_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_le5572345604234726045list_a @ N @ nil_list_list_a )
      = N ) ).

% gen_length_code(1)
thf(fact_50_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_length_set_a @ N @ nil_set_a )
      = N ) ).

% gen_length_code(1)
thf(fact_51_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_length_a @ N @ nil_a )
      = N ) ).

% gen_length_code(1)
thf(fact_52_maps__simps_I2_J,axiom,
    ! [F: a > list_a] :
      ( ( maps_a_a @ F @ nil_a )
      = nil_a ) ).

% maps_simps(2)
thf(fact_53_maps__simps_I2_J,axiom,
    ! [F: a > list_list_a] :
      ( ( maps_a_list_a @ F @ nil_a )
      = nil_list_a ) ).

% maps_simps(2)
thf(fact_54_maps__simps_I2_J,axiom,
    ! [F: a > list_set_a] :
      ( ( maps_a_set_a @ F @ nil_a )
      = nil_set_a ) ).

% maps_simps(2)
thf(fact_55_maps__simps_I2_J,axiom,
    ! [F: list_a > list_a] :
      ( ( maps_list_a_a @ F @ nil_list_a )
      = nil_a ) ).

% maps_simps(2)
thf(fact_56_maps__simps_I2_J,axiom,
    ! [F: set_a > list_a] :
      ( ( maps_set_a_a @ F @ nil_set_a )
      = nil_a ) ).

% maps_simps(2)
thf(fact_57_maps__simps_I2_J,axiom,
    ! [F: a > list_list_list_a] :
      ( ( maps_a_list_list_a @ F @ nil_a )
      = nil_list_list_a ) ).

% maps_simps(2)
thf(fact_58_maps__simps_I2_J,axiom,
    ! [F: list_a > list_list_a] :
      ( ( maps_list_a_list_a @ F @ nil_list_a )
      = nil_list_a ) ).

% maps_simps(2)
thf(fact_59_maps__simps_I2_J,axiom,
    ! [F: list_a > list_set_a] :
      ( ( maps_list_a_set_a @ F @ nil_list_a )
      = nil_set_a ) ).

% maps_simps(2)
thf(fact_60_maps__simps_I2_J,axiom,
    ! [F: list_list_a > list_a] :
      ( ( maps_list_list_a_a @ F @ nil_list_list_a )
      = nil_a ) ).

% maps_simps(2)
thf(fact_61_maps__simps_I2_J,axiom,
    ! [F: set_a > list_list_a] :
      ( ( maps_set_a_list_a @ F @ nil_set_a )
      = nil_list_a ) ).

% maps_simps(2)
thf(fact_62_null__rec_I2_J,axiom,
    null_list_a @ nil_list_a ).

% null_rec(2)
thf(fact_63_null__rec_I2_J,axiom,
    null_list_list_a @ nil_list_list_a ).

% null_rec(2)
thf(fact_64_null__rec_I2_J,axiom,
    null_set_a @ nil_set_a ).

% null_rec(2)
thf(fact_65_null__rec_I2_J,axiom,
    null_a @ nil_a ).

% null_rec(2)
thf(fact_66_eq__Nil__null,axiom,
    ! [Xs: list_list_a] :
      ( ( Xs = nil_list_a )
      = ( null_list_a @ Xs ) ) ).

% eq_Nil_null
thf(fact_67_eq__Nil__null,axiom,
    ! [Xs: list_list_list_a] :
      ( ( Xs = nil_list_list_a )
      = ( null_list_list_a @ Xs ) ) ).

% eq_Nil_null
thf(fact_68_eq__Nil__null,axiom,
    ! [Xs: list_set_a] :
      ( ( Xs = nil_set_a )
      = ( null_set_a @ Xs ) ) ).

% eq_Nil_null
thf(fact_69_eq__Nil__null,axiom,
    ! [Xs: list_a] :
      ( ( Xs = nil_a )
      = ( null_a @ Xs ) ) ).

% eq_Nil_null
thf(fact_70_stream_Ocoinduct__strong,axiom,
    ! [R: stream_a > stream_a > $o,Stream: stream_a,Stream2: stream_a] :
      ( ( R @ Stream @ Stream2 )
     => ( ! [Stream3: stream_a,Stream4: stream_a] :
            ( ( R @ Stream3 @ Stream4 )
           => ( ( ( shd_a @ Stream3 )
                = ( shd_a @ Stream4 ) )
              & ( ( R @ ( stl_a @ Stream3 ) @ ( stl_a @ Stream4 ) )
                | ( ( stl_a @ Stream3 )
                  = ( stl_a @ Stream4 ) ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct_strong
thf(fact_71_stream_Ocoinduct__strong,axiom,
    ! [R: stream2255243159586646806list_a > stream2255243159586646806list_a > $o,Stream: stream2255243159586646806list_a,Stream2: stream2255243159586646806list_a] :
      ( ( R @ Stream @ Stream2 )
     => ( ! [Stream3: stream2255243159586646806list_a,Stream4: stream2255243159586646806list_a] :
            ( ( R @ Stream3 @ Stream4 )
           => ( ( ( shd_list_list_list_a @ Stream3 )
                = ( shd_list_list_list_a @ Stream4 ) )
              & ( ( R @ ( stl_list_list_list_a @ Stream3 ) @ ( stl_list_list_list_a @ Stream4 ) )
                | ( ( stl_list_list_list_a @ Stream3 )
                  = ( stl_list_list_list_a @ Stream4 ) ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct_strong
thf(fact_72_stream_Ocoinduct__strong,axiom,
    ! [R: stream_list_a > stream_list_a > $o,Stream: stream_list_a,Stream2: stream_list_a] :
      ( ( R @ Stream @ Stream2 )
     => ( ! [Stream3: stream_list_a,Stream4: stream_list_a] :
            ( ( R @ Stream3 @ Stream4 )
           => ( ( ( shd_list_a @ Stream3 )
                = ( shd_list_a @ Stream4 ) )
              & ( ( R @ ( stl_list_a @ Stream3 ) @ ( stl_list_a @ Stream4 ) )
                | ( ( stl_list_a @ Stream3 )
                  = ( stl_list_a @ Stream4 ) ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct_strong
thf(fact_73_stream_Ocoinduct__strong,axiom,
    ! [R: stream_list_list_a > stream_list_list_a > $o,Stream: stream_list_list_a,Stream2: stream_list_list_a] :
      ( ( R @ Stream @ Stream2 )
     => ( ! [Stream3: stream_list_list_a,Stream4: stream_list_list_a] :
            ( ( R @ Stream3 @ Stream4 )
           => ( ( ( shd_list_list_a @ Stream3 )
                = ( shd_list_list_a @ Stream4 ) )
              & ( ( R @ ( stl_list_list_a @ Stream3 ) @ ( stl_list_list_a @ Stream4 ) )
                | ( ( stl_list_list_a @ Stream3 )
                  = ( stl_list_list_a @ Stream4 ) ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct_strong
thf(fact_74_stream_Ocoinduct,axiom,
    ! [R: stream_a > stream_a > $o,Stream: stream_a,Stream2: stream_a] :
      ( ( R @ Stream @ Stream2 )
     => ( ! [Stream3: stream_a,Stream4: stream_a] :
            ( ( R @ Stream3 @ Stream4 )
           => ( ( ( shd_a @ Stream3 )
                = ( shd_a @ Stream4 ) )
              & ( R @ ( stl_a @ Stream3 ) @ ( stl_a @ Stream4 ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct
thf(fact_75_stream_Ocoinduct,axiom,
    ! [R: stream2255243159586646806list_a > stream2255243159586646806list_a > $o,Stream: stream2255243159586646806list_a,Stream2: stream2255243159586646806list_a] :
      ( ( R @ Stream @ Stream2 )
     => ( ! [Stream3: stream2255243159586646806list_a,Stream4: stream2255243159586646806list_a] :
            ( ( R @ Stream3 @ Stream4 )
           => ( ( ( shd_list_list_list_a @ Stream3 )
                = ( shd_list_list_list_a @ Stream4 ) )
              & ( R @ ( stl_list_list_list_a @ Stream3 ) @ ( stl_list_list_list_a @ Stream4 ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct
thf(fact_76_stream_Ocoinduct,axiom,
    ! [R: stream_list_a > stream_list_a > $o,Stream: stream_list_a,Stream2: stream_list_a] :
      ( ( R @ Stream @ Stream2 )
     => ( ! [Stream3: stream_list_a,Stream4: stream_list_a] :
            ( ( R @ Stream3 @ Stream4 )
           => ( ( ( shd_list_a @ Stream3 )
                = ( shd_list_a @ Stream4 ) )
              & ( R @ ( stl_list_a @ Stream3 ) @ ( stl_list_a @ Stream4 ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct
thf(fact_77_stream_Ocoinduct,axiom,
    ! [R: stream_list_list_a > stream_list_list_a > $o,Stream: stream_list_list_a,Stream2: stream_list_list_a] :
      ( ( R @ Stream @ Stream2 )
     => ( ! [Stream3: stream_list_list_a,Stream4: stream_list_list_a] :
            ( ( R @ Stream3 @ Stream4 )
           => ( ( ( shd_list_list_a @ Stream3 )
                = ( shd_list_list_a @ Stream4 ) )
              & ( R @ ( stl_list_list_a @ Stream3 ) @ ( stl_list_list_a @ Stream4 ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct
thf(fact_78_stream_Oexpand,axiom,
    ! [Stream: stream_a,Stream2: stream_a] :
      ( ( ( ( shd_a @ Stream )
          = ( shd_a @ Stream2 ) )
        & ( ( stl_a @ Stream )
          = ( stl_a @ Stream2 ) ) )
     => ( Stream = Stream2 ) ) ).

% stream.expand
thf(fact_79_stream_Oexpand,axiom,
    ! [Stream: stream2255243159586646806list_a,Stream2: stream2255243159586646806list_a] :
      ( ( ( ( shd_list_list_list_a @ Stream )
          = ( shd_list_list_list_a @ Stream2 ) )
        & ( ( stl_list_list_list_a @ Stream )
          = ( stl_list_list_list_a @ Stream2 ) ) )
     => ( Stream = Stream2 ) ) ).

% stream.expand
thf(fact_80_stream_Oexpand,axiom,
    ! [Stream: stream_list_a,Stream2: stream_list_a] :
      ( ( ( ( shd_list_a @ Stream )
          = ( shd_list_a @ Stream2 ) )
        & ( ( stl_list_a @ Stream )
          = ( stl_list_a @ Stream2 ) ) )
     => ( Stream = Stream2 ) ) ).

% stream.expand
thf(fact_81_stream_Oexpand,axiom,
    ! [Stream: stream_list_list_a,Stream2: stream_list_list_a] :
      ( ( ( ( shd_list_list_a @ Stream )
          = ( shd_list_list_a @ Stream2 ) )
        & ( ( stl_list_list_a @ Stream )
          = ( stl_list_list_a @ Stream2 ) ) )
     => ( Stream = Stream2 ) ) ).

% stream.expand
thf(fact_82_shd__sset,axiom,
    ! [A: stream_set_list_a] : ( member_set_list_a @ ( shd_set_list_a @ A ) @ ( sset_set_list_a @ A ) ) ).

% shd_sset
thf(fact_83_shd__sset,axiom,
    ! [A: stream1977264251772372042list_a] : ( member8857465052274545133list_a @ ( shd_set_set_list_a @ A ) @ ( sset_set_set_list_a @ A ) ) ).

% shd_sset
thf(fact_84_shd__sset,axiom,
    ! [A: stream_set_set_a] : ( member_set_set_a @ ( shd_set_set_a @ A ) @ ( sset_set_set_a @ A ) ) ).

% shd_sset
thf(fact_85_shd__sset,axiom,
    ! [A: stream_set_a] : ( member_set_a2 @ ( shd_set_a @ A ) @ ( sset_set_a @ A ) ) ).

% shd_sset
thf(fact_86_shd__sset,axiom,
    ! [A: stream2255243159586646806list_a] : ( member5342144027231129785list_a @ ( shd_list_list_list_a @ A ) @ ( sset_l5650713775164989220list_a @ A ) ) ).

% shd_sset
thf(fact_87_shd__sset,axiom,
    ! [A: stream_list_list_a] : ( member_list_list_a2 @ ( shd_list_list_a @ A ) @ ( sset_list_list_a @ A ) ) ).

% shd_sset
thf(fact_88_shd__sset,axiom,
    ! [A: stream_list_a] : ( member_list_a2 @ ( shd_list_a @ A ) @ ( sset_list_a @ A ) ) ).

% shd_sset
thf(fact_89_shd__sset,axiom,
    ! [A: stream_a] : ( member_a2 @ ( shd_a @ A ) @ ( sset_a @ A ) ) ).

% shd_sset
thf(fact_90_subsetI,axiom,
    ! [A2: set_set_set_list_a,B: set_set_set_list_a] :
      ( ! [X4: set_set_list_a] :
          ( ( member8857465052274545133list_a @ X4 @ A2 )
         => ( member8857465052274545133list_a @ X4 @ B ) )
     => ( ord_le7622928425522016964list_a @ A2 @ B ) ) ).

% subsetI
thf(fact_91_subsetI,axiom,
    ! [A2: set_set_set_a,B: set_set_set_a] :
      ( ! [X4: set_set_a] :
          ( ( member_set_set_a @ X4 @ A2 )
         => ( member_set_set_a @ X4 @ B ) )
     => ( ord_le5722252365846178494_set_a @ A2 @ B ) ) ).

% subsetI
thf(fact_92_subsetI,axiom,
    ! [A2: set_set_a,B: set_set_a] :
      ( ! [X4: set_a] :
          ( ( member_set_a2 @ X4 @ A2 )
         => ( member_set_a2 @ X4 @ B ) )
     => ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).

% subsetI
thf(fact_93_subsetI,axiom,
    ! [A2: set_set_list_a,B: set_set_list_a] :
      ( ! [X4: set_list_a] :
          ( ( member_set_list_a @ X4 @ A2 )
         => ( member_set_list_a @ X4 @ B ) )
     => ( ord_le8877086941679407844list_a @ A2 @ B ) ) ).

% subsetI
thf(fact_94_subsetI,axiom,
    ! [A2: set_list_list_a,B: set_list_list_a] :
      ( ! [X4: list_list_a] :
          ( ( member_list_list_a2 @ X4 @ A2 )
         => ( member_list_list_a2 @ X4 @ B ) )
     => ( ord_le8488217952732425610list_a @ A2 @ B ) ) ).

% subsetI
thf(fact_95_subsetI,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ! [X4: list_a] :
          ( ( member_list_a2 @ X4 @ A2 )
         => ( member_list_a2 @ X4 @ B ) )
     => ( ord_le8861187494160871172list_a @ A2 @ B ) ) ).

% subsetI
thf(fact_96_subsetI,axiom,
    ! [A2: set_a,B: set_a] :
      ( ! [X4: a] :
          ( ( member_a2 @ X4 @ A2 )
         => ( member_a2 @ X4 @ B ) )
     => ( ord_less_eq_set_a @ A2 @ B ) ) ).

% subsetI
thf(fact_97_subset__antisym,axiom,
    ! [A2: set_set_a,B: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ B )
     => ( ( ord_le3724670747650509150_set_a @ B @ A2 )
       => ( A2 = B ) ) ) ).

% subset_antisym
thf(fact_98_subset__antisym,axiom,
    ! [A2: set_set_list_a,B: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ A2 @ B )
     => ( ( ord_le8877086941679407844list_a @ B @ A2 )
       => ( A2 = B ) ) ) ).

% subset_antisym
thf(fact_99_subset__antisym,axiom,
    ! [A2: set_list_list_a,B: set_list_list_a] :
      ( ( ord_le8488217952732425610list_a @ A2 @ B )
     => ( ( ord_le8488217952732425610list_a @ B @ A2 )
       => ( A2 = B ) ) ) ).

% subset_antisym
thf(fact_100_subset__antisym,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B )
     => ( ( ord_le8861187494160871172list_a @ B @ A2 )
       => ( A2 = B ) ) ) ).

% subset_antisym
thf(fact_101_subset__antisym,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( ord_less_eq_set_a @ B @ A2 )
       => ( A2 = B ) ) ) ).

% subset_antisym
thf(fact_102_order__refl,axiom,
    ! [X2: $o > set_a] : ( ord_less_eq_o_set_a @ X2 @ X2 ) ).

% order_refl
thf(fact_103_order__refl,axiom,
    ! [X2: $o > set_list_a] : ( ord_le354538097136858257list_a @ X2 @ X2 ) ).

% order_refl
thf(fact_104_order__refl,axiom,
    ! [X2: set_set_a] : ( ord_le3724670747650509150_set_a @ X2 @ X2 ) ).

% order_refl
thf(fact_105_order__refl,axiom,
    ! [X2: set_set_list_a] : ( ord_le8877086941679407844list_a @ X2 @ X2 ) ).

% order_refl
thf(fact_106_order__refl,axiom,
    ! [X2: set_list_list_a] : ( ord_le8488217952732425610list_a @ X2 @ X2 ) ).

% order_refl
thf(fact_107_order__refl,axiom,
    ! [X2: set_list_a] : ( ord_le8861187494160871172list_a @ X2 @ X2 ) ).

% order_refl
thf(fact_108_order__refl,axiom,
    ! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).

% order_refl
thf(fact_109_dual__order_Orefl,axiom,
    ! [A: $o > set_a] : ( ord_less_eq_o_set_a @ A @ A ) ).

% dual_order.refl
thf(fact_110_dual__order_Orefl,axiom,
    ! [A: $o > set_list_a] : ( ord_le354538097136858257list_a @ A @ A ) ).

% dual_order.refl
thf(fact_111_dual__order_Orefl,axiom,
    ! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).

% dual_order.refl
thf(fact_112_dual__order_Orefl,axiom,
    ! [A: set_set_list_a] : ( ord_le8877086941679407844list_a @ A @ A ) ).

% dual_order.refl
thf(fact_113_dual__order_Orefl,axiom,
    ! [A: set_list_list_a] : ( ord_le8488217952732425610list_a @ A @ A ) ).

% dual_order.refl
thf(fact_114_dual__order_Orefl,axiom,
    ! [A: set_list_a] : ( ord_le8861187494160871172list_a @ A @ A ) ).

% dual_order.refl
thf(fact_115_dual__order_Orefl,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).

% dual_order.refl
thf(fact_116_flat__unfold,axiom,
    ! [Ws: stream_list_set_a] :
      ( ( ( shd_list_set_a @ Ws )
       != nil_set_a )
     => ( ( flat_set_a @ Ws )
        = ( shift_set_a @ ( shd_list_set_a @ Ws ) @ ( flat_set_a @ ( stl_list_set_a @ Ws ) ) ) ) ) ).

% flat_unfold
thf(fact_117_flat__unfold,axiom,
    ! [Ws: stream2255243159586646806list_a] :
      ( ( ( shd_list_list_list_a @ Ws )
       != nil_list_list_a )
     => ( ( flat_list_list_a @ Ws )
        = ( shift_list_list_a @ ( shd_list_list_list_a @ Ws ) @ ( flat_list_list_a @ ( stl_list_list_list_a @ Ws ) ) ) ) ) ).

% flat_unfold
thf(fact_118_flat__unfold,axiom,
    ! [Ws: stream_list_a] :
      ( ( ( shd_list_a @ Ws )
       != nil_a )
     => ( ( flat_a @ Ws )
        = ( shift_a @ ( shd_list_a @ Ws ) @ ( flat_a @ ( stl_list_a @ Ws ) ) ) ) ) ).

% flat_unfold
thf(fact_119_flat__unfold,axiom,
    ! [Ws: stream_list_list_a] :
      ( ( ( shd_list_list_a @ Ws )
       != nil_list_a )
     => ( ( flat_list_a @ Ws )
        = ( shift_list_a @ ( shd_list_list_a @ Ws ) @ ( flat_list_a @ ( stl_list_list_a @ Ws ) ) ) ) ) ).

% flat_unfold
thf(fact_120_sdrop__while_Osimps,axiom,
    ( sdrop_while_a
    = ( ^ [P2: a > $o,S: stream_a] : ( if_stream_a @ ( P2 @ ( shd_a @ S ) ) @ ( sdrop_while_a @ P2 @ ( stl_a @ S ) ) @ S ) ) ) ).

% sdrop_while.simps
thf(fact_121_sdrop__while_Osimps,axiom,
    ( sdrop_8575598270869141876list_a
    = ( ^ [P2: list_list_list_a > $o,S: stream2255243159586646806list_a] : ( if_str8217234800680828380list_a @ ( P2 @ ( shd_list_list_list_a @ S ) ) @ ( sdrop_8575598270869141876list_a @ P2 @ ( stl_list_list_list_a @ S ) ) @ S ) ) ) ).

% sdrop_while.simps
thf(fact_122_sdrop__while_Osimps,axiom,
    ( sdrop_while_list_a
    = ( ^ [P2: list_a > $o,S: stream_list_a] : ( if_stream_list_a @ ( P2 @ ( shd_list_a @ S ) ) @ ( sdrop_while_list_a @ P2 @ ( stl_list_a @ S ) ) @ S ) ) ) ).

% sdrop_while.simps
thf(fact_123_sdrop__while_Osimps,axiom,
    ( sdrop_1072776591231973358list_a
    = ( ^ [P2: list_list_a > $o,S: stream_list_list_a] : ( if_str7505741754068378070list_a @ ( P2 @ ( shd_list_list_a @ S ) ) @ ( sdrop_1072776591231973358list_a @ P2 @ ( stl_list_list_a @ S ) ) @ S ) ) ) ).

% sdrop_while.simps
thf(fact_124_in__mono,axiom,
    ! [A2: set_set_set_list_a,B: set_set_set_list_a,X2: set_set_list_a] :
      ( ( ord_le7622928425522016964list_a @ A2 @ B )
     => ( ( member8857465052274545133list_a @ X2 @ A2 )
       => ( member8857465052274545133list_a @ X2 @ B ) ) ) ).

% in_mono
thf(fact_125_in__mono,axiom,
    ! [A2: set_set_set_a,B: set_set_set_a,X2: set_set_a] :
      ( ( ord_le5722252365846178494_set_a @ A2 @ B )
     => ( ( member_set_set_a @ X2 @ A2 )
       => ( member_set_set_a @ X2 @ B ) ) ) ).

% in_mono
thf(fact_126_in__mono,axiom,
    ! [A2: set_set_a,B: set_set_a,X2: set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ B )
     => ( ( member_set_a2 @ X2 @ A2 )
       => ( member_set_a2 @ X2 @ B ) ) ) ).

% in_mono
thf(fact_127_in__mono,axiom,
    ! [A2: set_set_list_a,B: set_set_list_a,X2: set_list_a] :
      ( ( ord_le8877086941679407844list_a @ A2 @ B )
     => ( ( member_set_list_a @ X2 @ A2 )
       => ( member_set_list_a @ X2 @ B ) ) ) ).

% in_mono
thf(fact_128_in__mono,axiom,
    ! [A2: set_list_list_a,B: set_list_list_a,X2: list_list_a] :
      ( ( ord_le8488217952732425610list_a @ A2 @ B )
     => ( ( member_list_list_a2 @ X2 @ A2 )
       => ( member_list_list_a2 @ X2 @ B ) ) ) ).

% in_mono
thf(fact_129_in__mono,axiom,
    ! [A2: set_list_a,B: set_list_a,X2: list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B )
     => ( ( member_list_a2 @ X2 @ A2 )
       => ( member_list_a2 @ X2 @ B ) ) ) ).

% in_mono
thf(fact_130_in__mono,axiom,
    ! [A2: set_a,B: set_a,X2: a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( member_a2 @ X2 @ A2 )
       => ( member_a2 @ X2 @ B ) ) ) ).

% in_mono
thf(fact_131_subsetD,axiom,
    ! [A2: set_set_set_list_a,B: set_set_set_list_a,C: set_set_list_a] :
      ( ( ord_le7622928425522016964list_a @ A2 @ B )
     => ( ( member8857465052274545133list_a @ C @ A2 )
       => ( member8857465052274545133list_a @ C @ B ) ) ) ).

% subsetD
thf(fact_132_subsetD,axiom,
    ! [A2: set_set_set_a,B: set_set_set_a,C: set_set_a] :
      ( ( ord_le5722252365846178494_set_a @ A2 @ B )
     => ( ( member_set_set_a @ C @ A2 )
       => ( member_set_set_a @ C @ B ) ) ) ).

% subsetD
thf(fact_133_subsetD,axiom,
    ! [A2: set_set_a,B: set_set_a,C: set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ B )
     => ( ( member_set_a2 @ C @ A2 )
       => ( member_set_a2 @ C @ B ) ) ) ).

% subsetD
thf(fact_134_subsetD,axiom,
    ! [A2: set_set_list_a,B: set_set_list_a,C: set_list_a] :
      ( ( ord_le8877086941679407844list_a @ A2 @ B )
     => ( ( member_set_list_a @ C @ A2 )
       => ( member_set_list_a @ C @ B ) ) ) ).

% subsetD
thf(fact_135_subsetD,axiom,
    ! [A2: set_list_list_a,B: set_list_list_a,C: list_list_a] :
      ( ( ord_le8488217952732425610list_a @ A2 @ B )
     => ( ( member_list_list_a2 @ C @ A2 )
       => ( member_list_list_a2 @ C @ B ) ) ) ).

% subsetD
thf(fact_136_subsetD,axiom,
    ! [A2: set_list_a,B: set_list_a,C: list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B )
     => ( ( member_list_a2 @ C @ A2 )
       => ( member_list_a2 @ C @ B ) ) ) ).

% subsetD
thf(fact_137_subsetD,axiom,
    ! [A2: set_a,B: set_a,C: a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( member_a2 @ C @ A2 )
       => ( member_a2 @ C @ B ) ) ) ).

% subsetD
thf(fact_138_equalityE,axiom,
    ! [A2: set_set_a,B: set_set_a] :
      ( ( A2 = B )
     => ~ ( ( ord_le3724670747650509150_set_a @ A2 @ B )
         => ~ ( ord_le3724670747650509150_set_a @ B @ A2 ) ) ) ).

% equalityE
thf(fact_139_equalityE,axiom,
    ! [A2: set_set_list_a,B: set_set_list_a] :
      ( ( A2 = B )
     => ~ ( ( ord_le8877086941679407844list_a @ A2 @ B )
         => ~ ( ord_le8877086941679407844list_a @ B @ A2 ) ) ) ).

% equalityE
thf(fact_140_equalityE,axiom,
    ! [A2: set_list_list_a,B: set_list_list_a] :
      ( ( A2 = B )
     => ~ ( ( ord_le8488217952732425610list_a @ A2 @ B )
         => ~ ( ord_le8488217952732425610list_a @ B @ A2 ) ) ) ).

% equalityE
thf(fact_141_equalityE,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( A2 = B )
     => ~ ( ( ord_le8861187494160871172list_a @ A2 @ B )
         => ~ ( ord_le8861187494160871172list_a @ B @ A2 ) ) ) ).

% equalityE
thf(fact_142_equalityE,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( A2 = B )
     => ~ ( ( ord_less_eq_set_a @ A2 @ B )
         => ~ ( ord_less_eq_set_a @ B @ A2 ) ) ) ).

% equalityE
thf(fact_143_subset__eq,axiom,
    ( ord_le7622928425522016964list_a
    = ( ^ [A3: set_set_set_list_a,B2: set_set_set_list_a] :
        ! [X3: set_set_list_a] :
          ( ( member8857465052274545133list_a @ X3 @ A3 )
         => ( member8857465052274545133list_a @ X3 @ B2 ) ) ) ) ).

% subset_eq
thf(fact_144_subset__eq,axiom,
    ( ord_le5722252365846178494_set_a
    = ( ^ [A3: set_set_set_a,B2: set_set_set_a] :
        ! [X3: set_set_a] :
          ( ( member_set_set_a @ X3 @ A3 )
         => ( member_set_set_a @ X3 @ B2 ) ) ) ) ).

% subset_eq
thf(fact_145_subset__eq,axiom,
    ( ord_le3724670747650509150_set_a
    = ( ^ [A3: set_set_a,B2: set_set_a] :
        ! [X3: set_a] :
          ( ( member_set_a2 @ X3 @ A3 )
         => ( member_set_a2 @ X3 @ B2 ) ) ) ) ).

% subset_eq
thf(fact_146_subset__eq,axiom,
    ( ord_le8877086941679407844list_a
    = ( ^ [A3: set_set_list_a,B2: set_set_list_a] :
        ! [X3: set_list_a] :
          ( ( member_set_list_a @ X3 @ A3 )
         => ( member_set_list_a @ X3 @ B2 ) ) ) ) ).

% subset_eq
thf(fact_147_subset__eq,axiom,
    ( ord_le8488217952732425610list_a
    = ( ^ [A3: set_list_list_a,B2: set_list_list_a] :
        ! [X3: list_list_a] :
          ( ( member_list_list_a2 @ X3 @ A3 )
         => ( member_list_list_a2 @ X3 @ B2 ) ) ) ) ).

% subset_eq
thf(fact_148_subset__eq,axiom,
    ( ord_le8861187494160871172list_a
    = ( ^ [A3: set_list_a,B2: set_list_a] :
        ! [X3: list_a] :
          ( ( member_list_a2 @ X3 @ A3 )
         => ( member_list_a2 @ X3 @ B2 ) ) ) ) ).

% subset_eq
thf(fact_149_subset__eq,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A3: set_a,B2: set_a] :
        ! [X3: a] :
          ( ( member_a2 @ X3 @ A3 )
         => ( member_a2 @ X3 @ B2 ) ) ) ) ).

% subset_eq
thf(fact_150_equalityD1,axiom,
    ! [A2: set_set_a,B: set_set_a] :
      ( ( A2 = B )
     => ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).

% equalityD1
thf(fact_151_equalityD1,axiom,
    ! [A2: set_set_list_a,B: set_set_list_a] :
      ( ( A2 = B )
     => ( ord_le8877086941679407844list_a @ A2 @ B ) ) ).

% equalityD1
thf(fact_152_equalityD1,axiom,
    ! [A2: set_list_list_a,B: set_list_list_a] :
      ( ( A2 = B )
     => ( ord_le8488217952732425610list_a @ A2 @ B ) ) ).

% equalityD1
thf(fact_153_equalityD1,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( A2 = B )
     => ( ord_le8861187494160871172list_a @ A2 @ B ) ) ).

% equalityD1
thf(fact_154_equalityD1,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( A2 = B )
     => ( ord_less_eq_set_a @ A2 @ B ) ) ).

% equalityD1
thf(fact_155_shift__left__inj,axiom,
    ! [Xs: list_list_a,S1: stream_list_a,S22: stream_list_a] :
      ( ( ( shift_list_a @ Xs @ S1 )
        = ( shift_list_a @ Xs @ S22 ) )
      = ( S1 = S22 ) ) ).

% shift_left_inj
thf(fact_156_shift__left__inj,axiom,
    ! [Xs: list_a,S1: stream_a,S22: stream_a] :
      ( ( ( shift_a @ Xs @ S1 )
        = ( shift_a @ Xs @ S22 ) )
      = ( S1 = S22 ) ) ).

% shift_left_inj
thf(fact_157_shift__left__inj,axiom,
    ! [Xs: list_list_list_a,S1: stream_list_list_a,S22: stream_list_list_a] :
      ( ( ( shift_list_list_a @ Xs @ S1 )
        = ( shift_list_list_a @ Xs @ S22 ) )
      = ( S1 = S22 ) ) ).

% shift_left_inj
thf(fact_158_shift_Osimps_I1_J,axiom,
    ! [S2: stream_list_a] :
      ( ( shift_list_a @ nil_list_a @ S2 )
      = S2 ) ).

% shift.simps(1)
thf(fact_159_shift_Osimps_I1_J,axiom,
    ! [S2: stream_list_list_a] :
      ( ( shift_list_list_a @ nil_list_list_a @ S2 )
      = S2 ) ).

% shift.simps(1)
thf(fact_160_shift_Osimps_I1_J,axiom,
    ! [S2: stream_set_a] :
      ( ( shift_set_a @ nil_set_a @ S2 )
      = S2 ) ).

% shift.simps(1)
thf(fact_161_shift_Osimps_I1_J,axiom,
    ! [S2: stream_a] :
      ( ( shift_a @ nil_a @ S2 )
      = S2 ) ).

% shift.simps(1)
thf(fact_162_order__antisym__conv,axiom,
    ! [Y: $o > set_a,X2: $o > set_a] :
      ( ( ord_less_eq_o_set_a @ Y @ X2 )
     => ( ( ord_less_eq_o_set_a @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_163_order__antisym__conv,axiom,
    ! [Y: $o > set_list_a,X2: $o > set_list_a] :
      ( ( ord_le354538097136858257list_a @ Y @ X2 )
     => ( ( ord_le354538097136858257list_a @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_164_order__antisym__conv,axiom,
    ! [Y: set_set_a,X2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ Y @ X2 )
     => ( ( ord_le3724670747650509150_set_a @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_165_order__antisym__conv,axiom,
    ! [Y: set_set_list_a,X2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ Y @ X2 )
     => ( ( ord_le8877086941679407844list_a @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_166_order__antisym__conv,axiom,
    ! [Y: set_list_list_a,X2: set_list_list_a] :
      ( ( ord_le8488217952732425610list_a @ Y @ X2 )
     => ( ( ord_le8488217952732425610list_a @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_167_order__antisym__conv,axiom,
    ! [Y: set_list_a,X2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ Y @ X2 )
     => ( ( ord_le8861187494160871172list_a @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_168_order__antisym__conv,axiom,
    ! [Y: set_a,X2: set_a] :
      ( ( ord_less_eq_set_a @ Y @ X2 )
     => ( ( ord_less_eq_set_a @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_169_ord__le__eq__subst,axiom,
    ! [A: set_a,B3: set_a,F: set_a > set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_170_ord__le__eq__subst,axiom,
    ! [A: set_list_a,B3: set_list_a,F: set_list_a > set_a,C: set_a] :
      ( ( ord_le8861187494160871172list_a @ A @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X4: set_list_a,Y2: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_171_ord__le__eq__subst,axiom,
    ! [A: set_a,B3: set_a,F: set_a > set_list_a,C: set_list_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le8861187494160871172list_a @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_172_ord__le__eq__subst,axiom,
    ! [A: set_list_a,B3: set_list_a,F: set_list_a > set_list_a,C: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X4: set_list_a,Y2: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X4 @ Y2 )
             => ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le8861187494160871172list_a @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_173_ord__le__eq__subst,axiom,
    ! [A: set_a,B3: set_a,F: set_a > set_set_a,C: set_set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_174_ord__le__eq__subst,axiom,
    ! [A: set_set_a,B3: set_set_a,F: set_set_a > set_a,C: set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X4: set_set_a,Y2: set_set_a] :
              ( ( ord_le3724670747650509150_set_a @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_175_ord__le__eq__subst,axiom,
    ! [A: set_list_a,B3: set_list_a,F: set_list_a > set_set_a,C: set_set_a] :
      ( ( ord_le8861187494160871172list_a @ A @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X4: set_list_a,Y2: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X4 @ Y2 )
             => ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_176_ord__le__eq__subst,axiom,
    ! [A: set_a,B3: set_a,F: set_a > $o > set_a,C: $o > set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_less_eq_o_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_o_set_a @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_177_ord__le__eq__subst,axiom,
    ! [A: set_a,B3: set_a,F: set_a > set_set_list_a,C: set_set_list_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_le8877086941679407844list_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le8877086941679407844list_a @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_178_ord__le__eq__subst,axiom,
    ! [A: set_a,B3: set_a,F: set_a > set_list_list_a,C: set_list_list_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_le8488217952732425610list_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le8488217952732425610list_a @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_179_ord__eq__le__subst,axiom,
    ! [A: set_a,F: set_a > set_a,B3: set_a,C: set_a] :
      ( ( A
        = ( F @ B3 ) )
     => ( ( ord_less_eq_set_a @ B3 @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_180_ord__eq__le__subst,axiom,
    ! [A: set_a,F: set_list_a > set_a,B3: set_list_a,C: set_list_a] :
      ( ( A
        = ( F @ B3 ) )
     => ( ( ord_le8861187494160871172list_a @ B3 @ C )
       => ( ! [X4: set_list_a,Y2: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_181_ord__eq__le__subst,axiom,
    ! [A: set_list_a,F: set_a > set_list_a,B3: set_a,C: set_a] :
      ( ( A
        = ( F @ B3 ) )
     => ( ( ord_less_eq_set_a @ B3 @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le8861187494160871172list_a @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_182_ord__eq__le__subst,axiom,
    ! [A: set_list_a,F: set_list_a > set_list_a,B3: set_list_a,C: set_list_a] :
      ( ( A
        = ( F @ B3 ) )
     => ( ( ord_le8861187494160871172list_a @ B3 @ C )
       => ( ! [X4: set_list_a,Y2: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X4 @ Y2 )
             => ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le8861187494160871172list_a @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_183_ord__eq__le__subst,axiom,
    ! [A: set_set_a,F: set_a > set_set_a,B3: set_a,C: set_a] :
      ( ( A
        = ( F @ B3 ) )
     => ( ( ord_less_eq_set_a @ B3 @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_184_ord__eq__le__subst,axiom,
    ! [A: set_a,F: set_set_a > set_a,B3: set_set_a,C: set_set_a] :
      ( ( A
        = ( F @ B3 ) )
     => ( ( ord_le3724670747650509150_set_a @ B3 @ C )
       => ( ! [X4: set_set_a,Y2: set_set_a] :
              ( ( ord_le3724670747650509150_set_a @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_185_ord__eq__le__subst,axiom,
    ! [A: set_set_a,F: set_list_a > set_set_a,B3: set_list_a,C: set_list_a] :
      ( ( A
        = ( F @ B3 ) )
     => ( ( ord_le8861187494160871172list_a @ B3 @ C )
       => ( ! [X4: set_list_a,Y2: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X4 @ Y2 )
             => ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_186_ord__eq__le__subst,axiom,
    ! [A: $o > set_a,F: set_a > $o > set_a,B3: set_a,C: set_a] :
      ( ( A
        = ( F @ B3 ) )
     => ( ( ord_less_eq_set_a @ B3 @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_less_eq_o_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_o_set_a @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_187_ord__eq__le__subst,axiom,
    ! [A: set_set_list_a,F: set_a > set_set_list_a,B3: set_a,C: set_a] :
      ( ( A
        = ( F @ B3 ) )
     => ( ( ord_less_eq_set_a @ B3 @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_le8877086941679407844list_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le8877086941679407844list_a @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_188_ord__eq__le__subst,axiom,
    ! [A: set_list_list_a,F: set_a > set_list_list_a,B3: set_a,C: set_a] :
      ( ( A
        = ( F @ B3 ) )
     => ( ( ord_less_eq_set_a @ B3 @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_le8488217952732425610list_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le8488217952732425610list_a @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_189_order__eq__refl,axiom,
    ! [X2: $o > set_a,Y: $o > set_a] :
      ( ( X2 = Y )
     => ( ord_less_eq_o_set_a @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_190_order__eq__refl,axiom,
    ! [X2: $o > set_list_a,Y: $o > set_list_a] :
      ( ( X2 = Y )
     => ( ord_le354538097136858257list_a @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_191_order__eq__refl,axiom,
    ! [X2: set_set_a,Y: set_set_a] :
      ( ( X2 = Y )
     => ( ord_le3724670747650509150_set_a @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_192_order__eq__refl,axiom,
    ! [X2: set_set_list_a,Y: set_set_list_a] :
      ( ( X2 = Y )
     => ( ord_le8877086941679407844list_a @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_193_order__eq__refl,axiom,
    ! [X2: set_list_list_a,Y: set_list_list_a] :
      ( ( X2 = Y )
     => ( ord_le8488217952732425610list_a @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_194_order__eq__refl,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( X2 = Y )
     => ( ord_le8861187494160871172list_a @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_195_order__eq__refl,axiom,
    ! [X2: set_a,Y: set_a] :
      ( ( X2 = Y )
     => ( ord_less_eq_set_a @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_196_order__subst2,axiom,
    ! [A: set_a,B3: set_a,F: set_a > set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_197_order__subst2,axiom,
    ! [A: set_list_a,B3: set_list_a,F: set_list_a > set_a,C: set_a] :
      ( ( ord_le8861187494160871172list_a @ A @ B3 )
     => ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C )
       => ( ! [X4: set_list_a,Y2: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_198_order__subst2,axiom,
    ! [A: set_a,B3: set_a,F: set_a > set_list_a,C: set_list_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( ord_le8861187494160871172list_a @ ( F @ B3 ) @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le8861187494160871172list_a @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_199_order__subst2,axiom,
    ! [A: set_list_a,B3: set_list_a,F: set_list_a > set_list_a,C: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ B3 )
     => ( ( ord_le8861187494160871172list_a @ ( F @ B3 ) @ C )
       => ( ! [X4: set_list_a,Y2: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X4 @ Y2 )
             => ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le8861187494160871172list_a @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_200_order__subst2,axiom,
    ! [A: set_a,B3: set_a,F: set_a > set_set_a,C: set_set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( ord_le3724670747650509150_set_a @ ( F @ B3 ) @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_201_order__subst2,axiom,
    ! [A: set_set_a,B3: set_set_a,F: set_set_a > set_a,C: set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B3 )
     => ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C )
       => ( ! [X4: set_set_a,Y2: set_set_a] :
              ( ( ord_le3724670747650509150_set_a @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_202_order__subst2,axiom,
    ! [A: set_list_a,B3: set_list_a,F: set_list_a > set_set_a,C: set_set_a] :
      ( ( ord_le8861187494160871172list_a @ A @ B3 )
     => ( ( ord_le3724670747650509150_set_a @ ( F @ B3 ) @ C )
       => ( ! [X4: set_list_a,Y2: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X4 @ Y2 )
             => ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_203_order__subst2,axiom,
    ! [A: set_a,B3: set_a,F: set_a > $o > set_a,C: $o > set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( ord_less_eq_o_set_a @ ( F @ B3 ) @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_less_eq_o_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_o_set_a @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_204_order__subst2,axiom,
    ! [A: set_a,B3: set_a,F: set_a > set_set_list_a,C: set_set_list_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( ord_le8877086941679407844list_a @ ( F @ B3 ) @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_le8877086941679407844list_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le8877086941679407844list_a @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_205_order__subst2,axiom,
    ! [A: set_a,B3: set_a,F: set_a > set_list_list_a,C: set_list_list_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( ord_le8488217952732425610list_a @ ( F @ B3 ) @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_le8488217952732425610list_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le8488217952732425610list_a @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_206_order__subst1,axiom,
    ! [A: set_a,F: set_a > set_a,B3: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A @ ( F @ B3 ) )
     => ( ( ord_less_eq_set_a @ B3 @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_207_order__subst1,axiom,
    ! [A: set_list_a,F: set_a > set_list_a,B3: set_a,C: set_a] :
      ( ( ord_le8861187494160871172list_a @ A @ ( F @ B3 ) )
     => ( ( ord_less_eq_set_a @ B3 @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le8861187494160871172list_a @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_208_order__subst1,axiom,
    ! [A: set_a,F: set_list_a > set_a,B3: set_list_a,C: set_list_a] :
      ( ( ord_less_eq_set_a @ A @ ( F @ B3 ) )
     => ( ( ord_le8861187494160871172list_a @ B3 @ C )
       => ( ! [X4: set_list_a,Y2: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_209_order__subst1,axiom,
    ! [A: set_list_a,F: set_list_a > set_list_a,B3: set_list_a,C: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ ( F @ B3 ) )
     => ( ( ord_le8861187494160871172list_a @ B3 @ C )
       => ( ! [X4: set_list_a,Y2: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X4 @ Y2 )
             => ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le8861187494160871172list_a @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_210_order__subst1,axiom,
    ! [A: set_a,F: set_set_a > set_a,B3: set_set_a,C: set_set_a] :
      ( ( ord_less_eq_set_a @ A @ ( F @ B3 ) )
     => ( ( ord_le3724670747650509150_set_a @ B3 @ C )
       => ( ! [X4: set_set_a,Y2: set_set_a] :
              ( ( ord_le3724670747650509150_set_a @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_211_order__subst1,axiom,
    ! [A: set_set_a,F: set_a > set_set_a,B3: set_a,C: set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ ( F @ B3 ) )
     => ( ( ord_less_eq_set_a @ B3 @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_212_order__subst1,axiom,
    ! [A: set_list_a,F: set_set_a > set_list_a,B3: set_set_a,C: set_set_a] :
      ( ( ord_le8861187494160871172list_a @ A @ ( F @ B3 ) )
     => ( ( ord_le3724670747650509150_set_a @ B3 @ C )
       => ( ! [X4: set_set_a,Y2: set_set_a] :
              ( ( ord_le3724670747650509150_set_a @ X4 @ Y2 )
             => ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_le8861187494160871172list_a @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_213_order__subst1,axiom,
    ! [A: set_a,F: ( $o > set_a ) > set_a,B3: $o > set_a,C: $o > set_a] :
      ( ( ord_less_eq_set_a @ A @ ( F @ B3 ) )
     => ( ( ord_less_eq_o_set_a @ B3 @ C )
       => ( ! [X4: $o > set_a,Y2: $o > set_a] :
              ( ( ord_less_eq_o_set_a @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_214_order__subst1,axiom,
    ! [A: set_a,F: set_set_list_a > set_a,B3: set_set_list_a,C: set_set_list_a] :
      ( ( ord_less_eq_set_a @ A @ ( F @ B3 ) )
     => ( ( ord_le8877086941679407844list_a @ B3 @ C )
       => ( ! [X4: set_set_list_a,Y2: set_set_list_a] :
              ( ( ord_le8877086941679407844list_a @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_215_order__subst1,axiom,
    ! [A: set_a,F: set_list_list_a > set_a,B3: set_list_list_a,C: set_list_list_a] :
      ( ( ord_less_eq_set_a @ A @ ( F @ B3 ) )
     => ( ( ord_le8488217952732425610list_a @ B3 @ C )
       => ( ! [X4: set_list_list_a,Y2: set_list_list_a] :
              ( ( ord_le8488217952732425610list_a @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_216_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: $o > set_a,Z: $o > set_a] : ( Y3 = Z ) )
    = ( ^ [A4: $o > set_a,B4: $o > set_a] :
          ( ( ord_less_eq_o_set_a @ A4 @ B4 )
          & ( ord_less_eq_o_set_a @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_217_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: $o > set_list_a,Z: $o > set_list_a] : ( Y3 = Z ) )
    = ( ^ [A4: $o > set_list_a,B4: $o > set_list_a] :
          ( ( ord_le354538097136858257list_a @ A4 @ B4 )
          & ( ord_le354538097136858257list_a @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_218_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_set_a,Z: set_set_a] : ( Y3 = Z ) )
    = ( ^ [A4: set_set_a,B4: set_set_a] :
          ( ( ord_le3724670747650509150_set_a @ A4 @ B4 )
          & ( ord_le3724670747650509150_set_a @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_219_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_set_list_a,Z: set_set_list_a] : ( Y3 = Z ) )
    = ( ^ [A4: set_set_list_a,B4: set_set_list_a] :
          ( ( ord_le8877086941679407844list_a @ A4 @ B4 )
          & ( ord_le8877086941679407844list_a @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_220_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_list_list_a,Z: set_list_list_a] : ( Y3 = Z ) )
    = ( ^ [A4: set_list_list_a,B4: set_list_list_a] :
          ( ( ord_le8488217952732425610list_a @ A4 @ B4 )
          & ( ord_le8488217952732425610list_a @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_221_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_list_a,Z: set_list_a] : ( Y3 = Z ) )
    = ( ^ [A4: set_list_a,B4: set_list_a] :
          ( ( ord_le8861187494160871172list_a @ A4 @ B4 )
          & ( ord_le8861187494160871172list_a @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_222_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
    = ( ^ [A4: set_a,B4: set_a] :
          ( ( ord_less_eq_set_a @ A4 @ B4 )
          & ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_223_le__fun__def,axiom,
    ( ord_less_eq_o_set_a
    = ( ^ [F2: $o > set_a,G: $o > set_a] :
        ! [X3: $o] : ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).

% le_fun_def
thf(fact_224_le__fun__def,axiom,
    ( ord_le354538097136858257list_a
    = ( ^ [F2: $o > set_list_a,G: $o > set_list_a] :
        ! [X3: $o] : ( ord_le8861187494160871172list_a @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).

% le_fun_def
thf(fact_225_le__funI,axiom,
    ! [F: $o > set_list_a,G2: $o > set_list_a] :
      ( ! [X4: $o] : ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( G2 @ X4 ) )
     => ( ord_le354538097136858257list_a @ F @ G2 ) ) ).

% le_funI
thf(fact_226_le__funI,axiom,
    ! [F: $o > set_a,G2: $o > set_a] :
      ( ! [X4: $o] : ( ord_less_eq_set_a @ ( F @ X4 ) @ ( G2 @ X4 ) )
     => ( ord_less_eq_o_set_a @ F @ G2 ) ) ).

% le_funI
thf(fact_227_le__funE,axiom,
    ! [F: $o > set_a,G2: $o > set_a,X2: $o] :
      ( ( ord_less_eq_o_set_a @ F @ G2 )
     => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).

% le_funE
thf(fact_228_le__funE,axiom,
    ! [F: $o > set_list_a,G2: $o > set_list_a,X2: $o] :
      ( ( ord_le354538097136858257list_a @ F @ G2 )
     => ( ord_le8861187494160871172list_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).

% le_funE
thf(fact_229_le__funD,axiom,
    ! [F: $o > set_a,G2: $o > set_a,X2: $o] :
      ( ( ord_less_eq_o_set_a @ F @ G2 )
     => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).

% le_funD
thf(fact_230_le__funD,axiom,
    ! [F: $o > set_list_a,G2: $o > set_list_a,X2: $o] :
      ( ( ord_le354538097136858257list_a @ F @ G2 )
     => ( ord_le8861187494160871172list_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).

% le_funD
thf(fact_231_mem__Collect__eq,axiom,
    ! [A: set_list_a,P: set_list_a > $o] :
      ( ( member_set_list_a @ A @ ( collect_set_list_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_232_mem__Collect__eq,axiom,
    ! [A: set_set_list_a,P: set_set_list_a > $o] :
      ( ( member8857465052274545133list_a @ A @ ( collec3809296942973202735list_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_233_mem__Collect__eq,axiom,
    ! [A: set_set_a,P: set_set_a > $o] :
      ( ( member_set_set_a @ A @ ( collect_set_set_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_234_mem__Collect__eq,axiom,
    ! [A: set_a,P: set_a > $o] :
      ( ( member_set_a2 @ A @ ( collect_set_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_235_mem__Collect__eq,axiom,
    ! [A: a,P: a > $o] :
      ( ( member_a2 @ A @ ( collect_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_236_mem__Collect__eq,axiom,
    ! [A: list_a,P: list_a > $o] :
      ( ( member_list_a2 @ A @ ( collect_list_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_237_Collect__mem__eq,axiom,
    ! [A2: set_set_list_a] :
      ( ( collect_set_list_a
        @ ^ [X3: set_list_a] : ( member_set_list_a @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_238_Collect__mem__eq,axiom,
    ! [A2: set_set_set_list_a] :
      ( ( collec3809296942973202735list_a
        @ ^ [X3: set_set_list_a] : ( member8857465052274545133list_a @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_239_Collect__mem__eq,axiom,
    ! [A2: set_set_set_a] :
      ( ( collect_set_set_a
        @ ^ [X3: set_set_a] : ( member_set_set_a @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_240_Collect__mem__eq,axiom,
    ! [A2: set_set_a] :
      ( ( collect_set_a
        @ ^ [X3: set_a] : ( member_set_a2 @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_241_Collect__mem__eq,axiom,
    ! [A2: set_a] :
      ( ( collect_a
        @ ^ [X3: a] : ( member_a2 @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_242_Collect__mem__eq,axiom,
    ! [A2: set_list_a] :
      ( ( collect_list_a
        @ ^ [X3: list_a] : ( member_list_a2 @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_243_Collect__cong,axiom,
    ! [P: list_a > $o,Q: list_a > $o] :
      ( ! [X4: list_a] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_list_a @ P )
        = ( collect_list_a @ Q ) ) ) ).

% Collect_cong
thf(fact_244_Collect__cong,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ! [X4: a] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_a @ P )
        = ( collect_a @ Q ) ) ) ).

% Collect_cong
thf(fact_245_antisym,axiom,
    ! [A: $o > set_a,B3: $o > set_a] :
      ( ( ord_less_eq_o_set_a @ A @ B3 )
     => ( ( ord_less_eq_o_set_a @ B3 @ A )
       => ( A = B3 ) ) ) ).

% antisym
thf(fact_246_antisym,axiom,
    ! [A: $o > set_list_a,B3: $o > set_list_a] :
      ( ( ord_le354538097136858257list_a @ A @ B3 )
     => ( ( ord_le354538097136858257list_a @ B3 @ A )
       => ( A = B3 ) ) ) ).

% antisym
thf(fact_247_antisym,axiom,
    ! [A: set_set_a,B3: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B3 )
     => ( ( ord_le3724670747650509150_set_a @ B3 @ A )
       => ( A = B3 ) ) ) ).

% antisym
thf(fact_248_antisym,axiom,
    ! [A: set_set_list_a,B3: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ A @ B3 )
     => ( ( ord_le8877086941679407844list_a @ B3 @ A )
       => ( A = B3 ) ) ) ).

% antisym
thf(fact_249_antisym,axiom,
    ! [A: set_list_list_a,B3: set_list_list_a] :
      ( ( ord_le8488217952732425610list_a @ A @ B3 )
     => ( ( ord_le8488217952732425610list_a @ B3 @ A )
       => ( A = B3 ) ) ) ).

% antisym
thf(fact_250_antisym,axiom,
    ! [A: set_list_a,B3: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ B3 )
     => ( ( ord_le8861187494160871172list_a @ B3 @ A )
       => ( A = B3 ) ) ) ).

% antisym
thf(fact_251_antisym,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( ord_less_eq_set_a @ B3 @ A )
       => ( A = B3 ) ) ) ).

% antisym
thf(fact_252_dual__order_Otrans,axiom,
    ! [B3: $o > set_a,A: $o > set_a,C: $o > set_a] :
      ( ( ord_less_eq_o_set_a @ B3 @ A )
     => ( ( ord_less_eq_o_set_a @ C @ B3 )
       => ( ord_less_eq_o_set_a @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_253_dual__order_Otrans,axiom,
    ! [B3: $o > set_list_a,A: $o > set_list_a,C: $o > set_list_a] :
      ( ( ord_le354538097136858257list_a @ B3 @ A )
     => ( ( ord_le354538097136858257list_a @ C @ B3 )
       => ( ord_le354538097136858257list_a @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_254_dual__order_Otrans,axiom,
    ! [B3: set_set_a,A: set_set_a,C: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ B3 @ A )
     => ( ( ord_le3724670747650509150_set_a @ C @ B3 )
       => ( ord_le3724670747650509150_set_a @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_255_dual__order_Otrans,axiom,
    ! [B3: set_set_list_a,A: set_set_list_a,C: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ B3 @ A )
     => ( ( ord_le8877086941679407844list_a @ C @ B3 )
       => ( ord_le8877086941679407844list_a @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_256_dual__order_Otrans,axiom,
    ! [B3: set_list_list_a,A: set_list_list_a,C: set_list_list_a] :
      ( ( ord_le8488217952732425610list_a @ B3 @ A )
     => ( ( ord_le8488217952732425610list_a @ C @ B3 )
       => ( ord_le8488217952732425610list_a @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_257_dual__order_Otrans,axiom,
    ! [B3: set_list_a,A: set_list_a,C: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ B3 @ A )
     => ( ( ord_le8861187494160871172list_a @ C @ B3 )
       => ( ord_le8861187494160871172list_a @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_258_dual__order_Otrans,axiom,
    ! [B3: set_a,A: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ B3 @ A )
     => ( ( ord_less_eq_set_a @ C @ B3 )
       => ( ord_less_eq_set_a @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_259_dual__order_Oantisym,axiom,
    ! [B3: $o > set_a,A: $o > set_a] :
      ( ( ord_less_eq_o_set_a @ B3 @ A )
     => ( ( ord_less_eq_o_set_a @ A @ B3 )
       => ( A = B3 ) ) ) ).

% dual_order.antisym
thf(fact_260_dual__order_Oantisym,axiom,
    ! [B3: $o > set_list_a,A: $o > set_list_a] :
      ( ( ord_le354538097136858257list_a @ B3 @ A )
     => ( ( ord_le354538097136858257list_a @ A @ B3 )
       => ( A = B3 ) ) ) ).

% dual_order.antisym
thf(fact_261_dual__order_Oantisym,axiom,
    ! [B3: set_set_a,A: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ B3 @ A )
     => ( ( ord_le3724670747650509150_set_a @ A @ B3 )
       => ( A = B3 ) ) ) ).

% dual_order.antisym
thf(fact_262_dual__order_Oantisym,axiom,
    ! [B3: set_set_list_a,A: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ B3 @ A )
     => ( ( ord_le8877086941679407844list_a @ A @ B3 )
       => ( A = B3 ) ) ) ).

% dual_order.antisym
thf(fact_263_dual__order_Oantisym,axiom,
    ! [B3: set_list_list_a,A: set_list_list_a] :
      ( ( ord_le8488217952732425610list_a @ B3 @ A )
     => ( ( ord_le8488217952732425610list_a @ A @ B3 )
       => ( A = B3 ) ) ) ).

% dual_order.antisym
thf(fact_264_dual__order_Oantisym,axiom,
    ! [B3: set_list_a,A: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ B3 @ A )
     => ( ( ord_le8861187494160871172list_a @ A @ B3 )
       => ( A = B3 ) ) ) ).

% dual_order.antisym
thf(fact_265_dual__order_Oantisym,axiom,
    ! [B3: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ B3 @ A )
     => ( ( ord_less_eq_set_a @ A @ B3 )
       => ( A = B3 ) ) ) ).

% dual_order.antisym
thf(fact_266_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: $o > set_a,Z: $o > set_a] : ( Y3 = Z ) )
    = ( ^ [A4: $o > set_a,B4: $o > set_a] :
          ( ( ord_less_eq_o_set_a @ B4 @ A4 )
          & ( ord_less_eq_o_set_a @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_267_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: $o > set_list_a,Z: $o > set_list_a] : ( Y3 = Z ) )
    = ( ^ [A4: $o > set_list_a,B4: $o > set_list_a] :
          ( ( ord_le354538097136858257list_a @ B4 @ A4 )
          & ( ord_le354538097136858257list_a @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_268_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: set_set_a,Z: set_set_a] : ( Y3 = Z ) )
    = ( ^ [A4: set_set_a,B4: set_set_a] :
          ( ( ord_le3724670747650509150_set_a @ B4 @ A4 )
          & ( ord_le3724670747650509150_set_a @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_269_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: set_set_list_a,Z: set_set_list_a] : ( Y3 = Z ) )
    = ( ^ [A4: set_set_list_a,B4: set_set_list_a] :
          ( ( ord_le8877086941679407844list_a @ B4 @ A4 )
          & ( ord_le8877086941679407844list_a @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_270_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: set_list_list_a,Z: set_list_list_a] : ( Y3 = Z ) )
    = ( ^ [A4: set_list_list_a,B4: set_list_list_a] :
          ( ( ord_le8488217952732425610list_a @ B4 @ A4 )
          & ( ord_le8488217952732425610list_a @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_271_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: set_list_a,Z: set_list_a] : ( Y3 = Z ) )
    = ( ^ [A4: set_list_a,B4: set_list_a] :
          ( ( ord_le8861187494160871172list_a @ B4 @ A4 )
          & ( ord_le8861187494160871172list_a @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_272_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
    = ( ^ [A4: set_a,B4: set_a] :
          ( ( ord_less_eq_set_a @ B4 @ A4 )
          & ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_273_order__trans,axiom,
    ! [X2: $o > set_a,Y: $o > set_a,Z2: $o > set_a] :
      ( ( ord_less_eq_o_set_a @ X2 @ Y )
     => ( ( ord_less_eq_o_set_a @ Y @ Z2 )
       => ( ord_less_eq_o_set_a @ X2 @ Z2 ) ) ) ).

% order_trans
thf(fact_274_order__trans,axiom,
    ! [X2: $o > set_list_a,Y: $o > set_list_a,Z2: $o > set_list_a] :
      ( ( ord_le354538097136858257list_a @ X2 @ Y )
     => ( ( ord_le354538097136858257list_a @ Y @ Z2 )
       => ( ord_le354538097136858257list_a @ X2 @ Z2 ) ) ) ).

% order_trans
thf(fact_275_order__trans,axiom,
    ! [X2: set_set_a,Y: set_set_a,Z2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ X2 @ Y )
     => ( ( ord_le3724670747650509150_set_a @ Y @ Z2 )
       => ( ord_le3724670747650509150_set_a @ X2 @ Z2 ) ) ) ).

% order_trans
thf(fact_276_order__trans,axiom,
    ! [X2: set_set_list_a,Y: set_set_list_a,Z2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ X2 @ Y )
     => ( ( ord_le8877086941679407844list_a @ Y @ Z2 )
       => ( ord_le8877086941679407844list_a @ X2 @ Z2 ) ) ) ).

% order_trans
thf(fact_277_order__trans,axiom,
    ! [X2: set_list_list_a,Y: set_list_list_a,Z2: set_list_list_a] :
      ( ( ord_le8488217952732425610list_a @ X2 @ Y )
     => ( ( ord_le8488217952732425610list_a @ Y @ Z2 )
       => ( ord_le8488217952732425610list_a @ X2 @ Z2 ) ) ) ).

% order_trans
thf(fact_278_order__trans,axiom,
    ! [X2: set_list_a,Y: set_list_a,Z2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ X2 @ Y )
     => ( ( ord_le8861187494160871172list_a @ Y @ Z2 )
       => ( ord_le8861187494160871172list_a @ X2 @ Z2 ) ) ) ).

% order_trans
thf(fact_279_order__trans,axiom,
    ! [X2: set_a,Y: set_a,Z2: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ Y )
     => ( ( ord_less_eq_set_a @ Y @ Z2 )
       => ( ord_less_eq_set_a @ X2 @ Z2 ) ) ) ).

% order_trans
thf(fact_280_order_Otrans,axiom,
    ! [A: $o > set_a,B3: $o > set_a,C: $o > set_a] :
      ( ( ord_less_eq_o_set_a @ A @ B3 )
     => ( ( ord_less_eq_o_set_a @ B3 @ C )
       => ( ord_less_eq_o_set_a @ A @ C ) ) ) ).

% order.trans
thf(fact_281_order_Otrans,axiom,
    ! [A: $o > set_list_a,B3: $o > set_list_a,C: $o > set_list_a] :
      ( ( ord_le354538097136858257list_a @ A @ B3 )
     => ( ( ord_le354538097136858257list_a @ B3 @ C )
       => ( ord_le354538097136858257list_a @ A @ C ) ) ) ).

% order.trans
thf(fact_282_order_Otrans,axiom,
    ! [A: set_set_a,B3: set_set_a,C: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B3 )
     => ( ( ord_le3724670747650509150_set_a @ B3 @ C )
       => ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).

% order.trans
thf(fact_283_order_Otrans,axiom,
    ! [A: set_set_list_a,B3: set_set_list_a,C: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ A @ B3 )
     => ( ( ord_le8877086941679407844list_a @ B3 @ C )
       => ( ord_le8877086941679407844list_a @ A @ C ) ) ) ).

% order.trans
thf(fact_284_order_Otrans,axiom,
    ! [A: set_list_list_a,B3: set_list_list_a,C: set_list_list_a] :
      ( ( ord_le8488217952732425610list_a @ A @ B3 )
     => ( ( ord_le8488217952732425610list_a @ B3 @ C )
       => ( ord_le8488217952732425610list_a @ A @ C ) ) ) ).

% order.trans
thf(fact_285_order_Otrans,axiom,
    ! [A: set_list_a,B3: set_list_a,C: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ B3 )
     => ( ( ord_le8861187494160871172list_a @ B3 @ C )
       => ( ord_le8861187494160871172list_a @ A @ C ) ) ) ).

% order.trans
thf(fact_286_order_Otrans,axiom,
    ! [A: set_a,B3: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( ord_less_eq_set_a @ B3 @ C )
       => ( ord_less_eq_set_a @ A @ C ) ) ) ).

% order.trans
thf(fact_287_order__antisym,axiom,
    ! [X2: $o > set_a,Y: $o > set_a] :
      ( ( ord_less_eq_o_set_a @ X2 @ Y )
     => ( ( ord_less_eq_o_set_a @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_288_order__antisym,axiom,
    ! [X2: $o > set_list_a,Y: $o > set_list_a] :
      ( ( ord_le354538097136858257list_a @ X2 @ Y )
     => ( ( ord_le354538097136858257list_a @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_289_order__antisym,axiom,
    ! [X2: set_set_a,Y: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ X2 @ Y )
     => ( ( ord_le3724670747650509150_set_a @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_290_order__antisym,axiom,
    ! [X2: set_set_list_a,Y: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ X2 @ Y )
     => ( ( ord_le8877086941679407844list_a @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_291_order__antisym,axiom,
    ! [X2: set_list_list_a,Y: set_list_list_a] :
      ( ( ord_le8488217952732425610list_a @ X2 @ Y )
     => ( ( ord_le8488217952732425610list_a @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_292_order__antisym,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ X2 @ Y )
     => ( ( ord_le8861187494160871172list_a @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_293_order__antisym,axiom,
    ! [X2: set_a,Y: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ Y )
     => ( ( ord_less_eq_set_a @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_294_ord__le__eq__trans,axiom,
    ! [A: $o > set_a,B3: $o > set_a,C: $o > set_a] :
      ( ( ord_less_eq_o_set_a @ A @ B3 )
     => ( ( B3 = C )
       => ( ord_less_eq_o_set_a @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_295_ord__le__eq__trans,axiom,
    ! [A: $o > set_list_a,B3: $o > set_list_a,C: $o > set_list_a] :
      ( ( ord_le354538097136858257list_a @ A @ B3 )
     => ( ( B3 = C )
       => ( ord_le354538097136858257list_a @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_296_ord__le__eq__trans,axiom,
    ! [A: set_set_a,B3: set_set_a,C: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B3 )
     => ( ( B3 = C )
       => ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_297_ord__le__eq__trans,axiom,
    ! [A: set_set_list_a,B3: set_set_list_a,C: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ A @ B3 )
     => ( ( B3 = C )
       => ( ord_le8877086941679407844list_a @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_298_ord__le__eq__trans,axiom,
    ! [A: set_list_list_a,B3: set_list_list_a,C: set_list_list_a] :
      ( ( ord_le8488217952732425610list_a @ A @ B3 )
     => ( ( B3 = C )
       => ( ord_le8488217952732425610list_a @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_299_ord__le__eq__trans,axiom,
    ! [A: set_list_a,B3: set_list_a,C: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ B3 )
     => ( ( B3 = C )
       => ( ord_le8861187494160871172list_a @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_300_ord__le__eq__trans,axiom,
    ! [A: set_a,B3: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( B3 = C )
       => ( ord_less_eq_set_a @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_301_ord__eq__le__trans,axiom,
    ! [A: $o > set_a,B3: $o > set_a,C: $o > set_a] :
      ( ( A = B3 )
     => ( ( ord_less_eq_o_set_a @ B3 @ C )
       => ( ord_less_eq_o_set_a @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_302_ord__eq__le__trans,axiom,
    ! [A: $o > set_list_a,B3: $o > set_list_a,C: $o > set_list_a] :
      ( ( A = B3 )
     => ( ( ord_le354538097136858257list_a @ B3 @ C )
       => ( ord_le354538097136858257list_a @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_303_ord__eq__le__trans,axiom,
    ! [A: set_set_a,B3: set_set_a,C: set_set_a] :
      ( ( A = B3 )
     => ( ( ord_le3724670747650509150_set_a @ B3 @ C )
       => ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_304_ord__eq__le__trans,axiom,
    ! [A: set_set_list_a,B3: set_set_list_a,C: set_set_list_a] :
      ( ( A = B3 )
     => ( ( ord_le8877086941679407844list_a @ B3 @ C )
       => ( ord_le8877086941679407844list_a @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_305_ord__eq__le__trans,axiom,
    ! [A: set_list_list_a,B3: set_list_list_a,C: set_list_list_a] :
      ( ( A = B3 )
     => ( ( ord_le8488217952732425610list_a @ B3 @ C )
       => ( ord_le8488217952732425610list_a @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_306_ord__eq__le__trans,axiom,
    ! [A: set_list_a,B3: set_list_a,C: set_list_a] :
      ( ( A = B3 )
     => ( ( ord_le8861187494160871172list_a @ B3 @ C )
       => ( ord_le8861187494160871172list_a @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_307_ord__eq__le__trans,axiom,
    ! [A: set_a,B3: set_a,C: set_a] :
      ( ( A = B3 )
     => ( ( ord_less_eq_set_a @ B3 @ C )
       => ( ord_less_eq_set_a @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_308_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: $o > set_a,Z: $o > set_a] : ( Y3 = Z ) )
    = ( ^ [X3: $o > set_a,Y4: $o > set_a] :
          ( ( ord_less_eq_o_set_a @ X3 @ Y4 )
          & ( ord_less_eq_o_set_a @ Y4 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_309_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: $o > set_list_a,Z: $o > set_list_a] : ( Y3 = Z ) )
    = ( ^ [X3: $o > set_list_a,Y4: $o > set_list_a] :
          ( ( ord_le354538097136858257list_a @ X3 @ Y4 )
          & ( ord_le354538097136858257list_a @ Y4 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_310_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_set_a,Z: set_set_a] : ( Y3 = Z ) )
    = ( ^ [X3: set_set_a,Y4: set_set_a] :
          ( ( ord_le3724670747650509150_set_a @ X3 @ Y4 )
          & ( ord_le3724670747650509150_set_a @ Y4 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_311_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_set_list_a,Z: set_set_list_a] : ( Y3 = Z ) )
    = ( ^ [X3: set_set_list_a,Y4: set_set_list_a] :
          ( ( ord_le8877086941679407844list_a @ X3 @ Y4 )
          & ( ord_le8877086941679407844list_a @ Y4 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_312_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_list_list_a,Z: set_list_list_a] : ( Y3 = Z ) )
    = ( ^ [X3: set_list_list_a,Y4: set_list_list_a] :
          ( ( ord_le8488217952732425610list_a @ X3 @ Y4 )
          & ( ord_le8488217952732425610list_a @ Y4 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_313_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_list_a,Z: set_list_a] : ( Y3 = Z ) )
    = ( ^ [X3: set_list_a,Y4: set_list_a] :
          ( ( ord_le8861187494160871172list_a @ X3 @ Y4 )
          & ( ord_le8861187494160871172list_a @ Y4 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_314_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
    = ( ^ [X3: set_a,Y4: set_a] :
          ( ( ord_less_eq_set_a @ X3 @ Y4 )
          & ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_315_Collect__mono__iff,axiom,
    ! [P: set_a > $o,Q: set_a > $o] :
      ( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
      = ( ! [X3: set_a] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_316_Collect__mono__iff,axiom,
    ! [P: set_list_a > $o,Q: set_list_a > $o] :
      ( ( ord_le8877086941679407844list_a @ ( collect_set_list_a @ P ) @ ( collect_set_list_a @ Q ) )
      = ( ! [X3: set_list_a] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_317_Collect__mono__iff,axiom,
    ! [P: list_list_a > $o,Q: list_list_a > $o] :
      ( ( ord_le8488217952732425610list_a @ ( collect_list_list_a @ P ) @ ( collect_list_list_a @ Q ) )
      = ( ! [X3: list_list_a] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_318_Collect__mono__iff,axiom,
    ! [P: list_a > $o,Q: list_a > $o] :
      ( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) )
      = ( ! [X3: list_a] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_319_Collect__mono__iff,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
      = ( ! [X3: a] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_320_set__eq__subset,axiom,
    ( ( ^ [Y3: set_set_a,Z: set_set_a] : ( Y3 = Z ) )
    = ( ^ [A3: set_set_a,B2: set_set_a] :
          ( ( ord_le3724670747650509150_set_a @ A3 @ B2 )
          & ( ord_le3724670747650509150_set_a @ B2 @ A3 ) ) ) ) ).

% set_eq_subset
thf(fact_321_set__eq__subset,axiom,
    ( ( ^ [Y3: set_set_list_a,Z: set_set_list_a] : ( Y3 = Z ) )
    = ( ^ [A3: set_set_list_a,B2: set_set_list_a] :
          ( ( ord_le8877086941679407844list_a @ A3 @ B2 )
          & ( ord_le8877086941679407844list_a @ B2 @ A3 ) ) ) ) ).

% set_eq_subset
thf(fact_322_set__eq__subset,axiom,
    ( ( ^ [Y3: set_list_list_a,Z: set_list_list_a] : ( Y3 = Z ) )
    = ( ^ [A3: set_list_list_a,B2: set_list_list_a] :
          ( ( ord_le8488217952732425610list_a @ A3 @ B2 )
          & ( ord_le8488217952732425610list_a @ B2 @ A3 ) ) ) ) ).

% set_eq_subset
thf(fact_323_set__eq__subset,axiom,
    ( ( ^ [Y3: set_list_a,Z: set_list_a] : ( Y3 = Z ) )
    = ( ^ [A3: set_list_a,B2: set_list_a] :
          ( ( ord_le8861187494160871172list_a @ A3 @ B2 )
          & ( ord_le8861187494160871172list_a @ B2 @ A3 ) ) ) ) ).

% set_eq_subset
thf(fact_324_set__eq__subset,axiom,
    ( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
    = ( ^ [A3: set_a,B2: set_a] :
          ( ( ord_less_eq_set_a @ A3 @ B2 )
          & ( ord_less_eq_set_a @ B2 @ A3 ) ) ) ) ).

% set_eq_subset
thf(fact_325_subset__trans,axiom,
    ! [A2: set_set_a,B: set_set_a,C2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ B )
     => ( ( ord_le3724670747650509150_set_a @ B @ C2 )
       => ( ord_le3724670747650509150_set_a @ A2 @ C2 ) ) ) ).

% subset_trans
thf(fact_326_subset__trans,axiom,
    ! [A2: set_set_list_a,B: set_set_list_a,C2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ A2 @ B )
     => ( ( ord_le8877086941679407844list_a @ B @ C2 )
       => ( ord_le8877086941679407844list_a @ A2 @ C2 ) ) ) ).

% subset_trans
thf(fact_327_subset__trans,axiom,
    ! [A2: set_list_list_a,B: set_list_list_a,C2: set_list_list_a] :
      ( ( ord_le8488217952732425610list_a @ A2 @ B )
     => ( ( ord_le8488217952732425610list_a @ B @ C2 )
       => ( ord_le8488217952732425610list_a @ A2 @ C2 ) ) ) ).

% subset_trans
thf(fact_328_subset__trans,axiom,
    ! [A2: set_list_a,B: set_list_a,C2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B )
     => ( ( ord_le8861187494160871172list_a @ B @ C2 )
       => ( ord_le8861187494160871172list_a @ A2 @ C2 ) ) ) ).

% subset_trans
thf(fact_329_subset__trans,axiom,
    ! [A2: set_a,B: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( ord_less_eq_set_a @ B @ C2 )
       => ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).

% subset_trans
thf(fact_330_Collect__mono,axiom,
    ! [P: set_a > $o,Q: set_a > $o] :
      ( ! [X4: set_a] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).

% Collect_mono
thf(fact_331_Collect__mono,axiom,
    ! [P: set_list_a > $o,Q: set_list_a > $o] :
      ( ! [X4: set_list_a] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_le8877086941679407844list_a @ ( collect_set_list_a @ P ) @ ( collect_set_list_a @ Q ) ) ) ).

% Collect_mono
thf(fact_332_Collect__mono,axiom,
    ! [P: list_list_a > $o,Q: list_list_a > $o] :
      ( ! [X4: list_list_a] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_le8488217952732425610list_a @ ( collect_list_list_a @ P ) @ ( collect_list_list_a @ Q ) ) ) ).

% Collect_mono
thf(fact_333_Collect__mono,axiom,
    ! [P: list_a > $o,Q: list_a > $o] :
      ( ! [X4: list_a] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) ) ) ).

% Collect_mono
thf(fact_334_Collect__mono,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ! [X4: a] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).

% Collect_mono
thf(fact_335_subset__refl,axiom,
    ! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).

% subset_refl
thf(fact_336_subset__refl,axiom,
    ! [A2: set_set_list_a] : ( ord_le8877086941679407844list_a @ A2 @ A2 ) ).

% subset_refl
thf(fact_337_subset__refl,axiom,
    ! [A2: set_list_list_a] : ( ord_le8488217952732425610list_a @ A2 @ A2 ) ).

% subset_refl
thf(fact_338_subset__refl,axiom,
    ! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ A2 @ A2 ) ).

% subset_refl
thf(fact_339_subset__refl,axiom,
    ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).

% subset_refl
thf(fact_340_subset__iff,axiom,
    ( ord_le7622928425522016964list_a
    = ( ^ [A3: set_set_set_list_a,B2: set_set_set_list_a] :
        ! [T: set_set_list_a] :
          ( ( member8857465052274545133list_a @ T @ A3 )
         => ( member8857465052274545133list_a @ T @ B2 ) ) ) ) ).

% subset_iff
thf(fact_341_subset__iff,axiom,
    ( ord_le5722252365846178494_set_a
    = ( ^ [A3: set_set_set_a,B2: set_set_set_a] :
        ! [T: set_set_a] :
          ( ( member_set_set_a @ T @ A3 )
         => ( member_set_set_a @ T @ B2 ) ) ) ) ).

% subset_iff
thf(fact_342_subset__iff,axiom,
    ( ord_le3724670747650509150_set_a
    = ( ^ [A3: set_set_a,B2: set_set_a] :
        ! [T: set_a] :
          ( ( member_set_a2 @ T @ A3 )
         => ( member_set_a2 @ T @ B2 ) ) ) ) ).

% subset_iff
thf(fact_343_subset__iff,axiom,
    ( ord_le8877086941679407844list_a
    = ( ^ [A3: set_set_list_a,B2: set_set_list_a] :
        ! [T: set_list_a] :
          ( ( member_set_list_a @ T @ A3 )
         => ( member_set_list_a @ T @ B2 ) ) ) ) ).

% subset_iff
thf(fact_344_subset__iff,axiom,
    ( ord_le8488217952732425610list_a
    = ( ^ [A3: set_list_list_a,B2: set_list_list_a] :
        ! [T: list_list_a] :
          ( ( member_list_list_a2 @ T @ A3 )
         => ( member_list_list_a2 @ T @ B2 ) ) ) ) ).

% subset_iff
thf(fact_345_subset__iff,axiom,
    ( ord_le8861187494160871172list_a
    = ( ^ [A3: set_list_a,B2: set_list_a] :
        ! [T: list_a] :
          ( ( member_list_a2 @ T @ A3 )
         => ( member_list_a2 @ T @ B2 ) ) ) ) ).

% subset_iff
thf(fact_346_subset__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A3: set_a,B2: set_a] :
        ! [T: a] :
          ( ( member_a2 @ T @ A3 )
         => ( member_a2 @ T @ B2 ) ) ) ) ).

% subset_iff
thf(fact_347_equalityD2,axiom,
    ! [A2: set_set_a,B: set_set_a] :
      ( ( A2 = B )
     => ( ord_le3724670747650509150_set_a @ B @ A2 ) ) ).

% equalityD2
thf(fact_348_equalityD2,axiom,
    ! [A2: set_set_list_a,B: set_set_list_a] :
      ( ( A2 = B )
     => ( ord_le8877086941679407844list_a @ B @ A2 ) ) ).

% equalityD2
thf(fact_349_equalityD2,axiom,
    ! [A2: set_list_list_a,B: set_list_list_a] :
      ( ( A2 = B )
     => ( ord_le8488217952732425610list_a @ B @ A2 ) ) ).

% equalityD2
thf(fact_350_equalityD2,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( A2 = B )
     => ( ord_le8861187494160871172list_a @ B @ A2 ) ) ).

% equalityD2
thf(fact_351_equalityD2,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( A2 = B )
     => ( ord_less_eq_set_a @ B @ A2 ) ) ).

% equalityD2
thf(fact_352_flat__Stream,axiom,
    ! [Xs: list_set_a,Ws: stream_list_set_a] :
      ( ( Xs != nil_set_a )
     => ( ( flat_set_a @ ( sCons_list_set_a @ Xs @ Ws ) )
        = ( shift_set_a @ Xs @ ( flat_set_a @ Ws ) ) ) ) ).

% flat_Stream
thf(fact_353_flat__Stream,axiom,
    ! [Xs: list_list_list_a,Ws: stream2255243159586646806list_a] :
      ( ( Xs != nil_list_list_a )
     => ( ( flat_list_list_a @ ( sCons_8165023923567507367list_a @ Xs @ Ws ) )
        = ( shift_list_list_a @ Xs @ ( flat_list_list_a @ Ws ) ) ) ) ).

% flat_Stream
thf(fact_354_flat__Stream,axiom,
    ! [Xs: list_list_a,Ws: stream_list_list_a] :
      ( ( Xs != nil_list_a )
     => ( ( flat_list_a @ ( sCons_list_list_a @ Xs @ Ws ) )
        = ( shift_list_a @ Xs @ ( flat_list_a @ Ws ) ) ) ) ).

% flat_Stream
thf(fact_355_flat__Stream,axiom,
    ! [Xs: list_a,Ws: stream_list_a] :
      ( ( Xs != nil_a )
     => ( ( flat_a @ ( sCons_list_a @ Xs @ Ws ) )
        = ( shift_a @ Xs @ ( flat_a @ Ws ) ) ) ) ).

% flat_Stream
thf(fact_356_shift__simps_I1_J,axiom,
    ! [Xs: list_set_a,S2: stream_set_a] :
      ( ( ( Xs = nil_set_a )
       => ( ( shd_set_a @ ( shift_set_a @ Xs @ S2 ) )
          = ( shd_set_a @ S2 ) ) )
      & ( ( Xs != nil_set_a )
       => ( ( shd_set_a @ ( shift_set_a @ Xs @ S2 ) )
          = ( hd_set_a @ Xs ) ) ) ) ).

% shift_simps(1)
thf(fact_357_shift__simps_I1_J,axiom,
    ! [Xs: list_list_list_a,S2: stream_list_list_a] :
      ( ( ( Xs = nil_list_list_a )
       => ( ( shd_list_list_a @ ( shift_list_list_a @ Xs @ S2 ) )
          = ( shd_list_list_a @ S2 ) ) )
      & ( ( Xs != nil_list_list_a )
       => ( ( shd_list_list_a @ ( shift_list_list_a @ Xs @ S2 ) )
          = ( hd_list_list_a @ Xs ) ) ) ) ).

% shift_simps(1)
thf(fact_358_shift__simps_I1_J,axiom,
    ! [Xs: list_l7815035709764258326list_a,S2: stream2255243159586646806list_a] :
      ( ( ( Xs = nil_list_list_list_a )
       => ( ( shd_list_list_list_a @ ( shift_2124064713455379641list_a @ Xs @ S2 ) )
          = ( shd_list_list_list_a @ S2 ) ) )
      & ( ( Xs != nil_list_list_list_a )
       => ( ( shd_list_list_list_a @ ( shift_2124064713455379641list_a @ Xs @ S2 ) )
          = ( hd_list_list_list_a @ Xs ) ) ) ) ).

% shift_simps(1)
thf(fact_359_shift__simps_I1_J,axiom,
    ! [Xs: list_a,S2: stream_a] :
      ( ( ( Xs = nil_a )
       => ( ( shd_a @ ( shift_a @ Xs @ S2 ) )
          = ( shd_a @ S2 ) ) )
      & ( ( Xs != nil_a )
       => ( ( shd_a @ ( shift_a @ Xs @ S2 ) )
          = ( hd_a @ Xs ) ) ) ) ).

% shift_simps(1)
thf(fact_360_shift__simps_I1_J,axiom,
    ! [Xs: list_list_a,S2: stream_list_a] :
      ( ( ( Xs = nil_list_a )
       => ( ( shd_list_a @ ( shift_list_a @ Xs @ S2 ) )
          = ( shd_list_a @ S2 ) ) )
      & ( ( Xs != nil_list_a )
       => ( ( shd_list_a @ ( shift_list_a @ Xs @ S2 ) )
          = ( hd_list_a @ Xs ) ) ) ) ).

% shift_simps(1)
thf(fact_361_shift__simps_I2_J,axiom,
    ! [Xs: list_set_a,S2: stream_set_a] :
      ( ( ( Xs = nil_set_a )
       => ( ( stl_set_a @ ( shift_set_a @ Xs @ S2 ) )
          = ( stl_set_a @ S2 ) ) )
      & ( ( Xs != nil_set_a )
       => ( ( stl_set_a @ ( shift_set_a @ Xs @ S2 ) )
          = ( shift_set_a @ ( tl_set_a @ Xs ) @ S2 ) ) ) ) ).

% shift_simps(2)
thf(fact_362_shift__simps_I2_J,axiom,
    ! [Xs: list_l7815035709764258326list_a,S2: stream2255243159586646806list_a] :
      ( ( ( Xs = nil_list_list_list_a )
       => ( ( stl_list_list_list_a @ ( shift_2124064713455379641list_a @ Xs @ S2 ) )
          = ( stl_list_list_list_a @ S2 ) ) )
      & ( ( Xs != nil_list_list_list_a )
       => ( ( stl_list_list_list_a @ ( shift_2124064713455379641list_a @ Xs @ S2 ) )
          = ( shift_2124064713455379641list_a @ ( tl_list_list_list_a @ Xs ) @ S2 ) ) ) ) ).

% shift_simps(2)
thf(fact_363_shift__simps_I2_J,axiom,
    ! [Xs: list_a,S2: stream_a] :
      ( ( ( Xs = nil_a )
       => ( ( stl_a @ ( shift_a @ Xs @ S2 ) )
          = ( stl_a @ S2 ) ) )
      & ( ( Xs != nil_a )
       => ( ( stl_a @ ( shift_a @ Xs @ S2 ) )
          = ( shift_a @ ( tl_a @ Xs ) @ S2 ) ) ) ) ).

% shift_simps(2)
thf(fact_364_shift__simps_I2_J,axiom,
    ! [Xs: list_list_a,S2: stream_list_a] :
      ( ( ( Xs = nil_list_a )
       => ( ( stl_list_a @ ( shift_list_a @ Xs @ S2 ) )
          = ( stl_list_a @ S2 ) ) )
      & ( ( Xs != nil_list_a )
       => ( ( stl_list_a @ ( shift_list_a @ Xs @ S2 ) )
          = ( shift_list_a @ ( tl_list_a @ Xs ) @ S2 ) ) ) ) ).

% shift_simps(2)
thf(fact_365_shift__simps_I2_J,axiom,
    ! [Xs: list_list_list_a,S2: stream_list_list_a] :
      ( ( ( Xs = nil_list_list_a )
       => ( ( stl_list_list_a @ ( shift_list_list_a @ Xs @ S2 ) )
          = ( stl_list_list_a @ S2 ) ) )
      & ( ( Xs != nil_list_list_a )
       => ( ( stl_list_list_a @ ( shift_list_list_a @ Xs @ S2 ) )
          = ( shift_list_list_a @ ( tl_list_list_a @ Xs ) @ S2 ) ) ) ) ).

% shift_simps(2)
thf(fact_366_flat_Osimps_I1_J,axiom,
    ! [Ws: stream2255243159586646806list_a] :
      ( ( shd_list_list_a @ ( flat_list_list_a @ Ws ) )
      = ( hd_list_list_a @ ( shd_list_list_list_a @ Ws ) ) ) ).

% flat.simps(1)
thf(fact_367_flat_Osimps_I1_J,axiom,
    ! [Ws: stream8854267710728967580list_a] :
      ( ( shd_list_list_list_a @ ( flat_l8549998914816956042list_a @ Ws ) )
      = ( hd_list_list_list_a @ ( shd_li1819241521678429890list_a @ Ws ) ) ) ).

% flat.simps(1)
thf(fact_368_flat_Osimps_I1_J,axiom,
    ! [Ws: stream_list_a] :
      ( ( shd_a @ ( flat_a @ Ws ) )
      = ( hd_a @ ( shd_list_a @ Ws ) ) ) ).

% flat.simps(1)
thf(fact_369_flat_Osimps_I1_J,axiom,
    ! [Ws: stream_list_list_a] :
      ( ( shd_list_a @ ( flat_list_a @ Ws ) )
      = ( hd_list_a @ ( shd_list_list_a @ Ws ) ) ) ).

% flat.simps(1)
thf(fact_370_Greatest__equality,axiom,
    ! [P: ( $o > set_a ) > $o,X2: $o > set_a] :
      ( ( P @ X2 )
     => ( ! [Y2: $o > set_a] :
            ( ( P @ Y2 )
           => ( ord_less_eq_o_set_a @ Y2 @ X2 ) )
       => ( ( order_6114237596796908690_set_a @ P )
          = X2 ) ) ) ).

% Greatest_equality
thf(fact_371_Greatest__equality,axiom,
    ! [P: ( $o > set_list_a ) > $o,X2: $o > set_list_a] :
      ( ( P @ X2 )
     => ( ! [Y2: $o > set_list_a] :
            ( ( P @ Y2 )
           => ( ord_le354538097136858257list_a @ Y2 @ X2 ) )
       => ( ( order_6341408326612538392list_a @ P )
          = X2 ) ) ) ).

% Greatest_equality
thf(fact_372_Greatest__equality,axiom,
    ! [P: set_set_a > $o,X2: set_set_a] :
      ( ( P @ X2 )
     => ( ! [Y2: set_set_a] :
            ( ( P @ Y2 )
           => ( ord_le3724670747650509150_set_a @ Y2 @ X2 ) )
       => ( ( order_3565860530148683671_set_a @ P )
          = X2 ) ) ) ).

% Greatest_equality
thf(fact_373_Greatest__equality,axiom,
    ! [P: set_set_list_a > $o,X2: set_set_list_a] :
      ( ( P @ X2 )
     => ( ! [Y2: set_set_list_a] :
            ( ( P @ Y2 )
           => ( ord_le8877086941679407844list_a @ Y2 @ X2 ) )
       => ( ( order_8947689642956441757list_a @ P )
          = X2 ) ) ) ).

% Greatest_equality
thf(fact_374_Greatest__equality,axiom,
    ! [P: set_list_list_a > $o,X2: set_list_list_a] :
      ( ( P @ X2 )
     => ( ! [Y2: set_list_list_a] :
            ( ( P @ Y2 )
           => ( ord_le8488217952732425610list_a @ Y2 @ X2 ) )
       => ( ( order_4834126633758709443list_a @ P )
          = X2 ) ) ) ).

% Greatest_equality
thf(fact_375_Greatest__equality,axiom,
    ! [P: set_list_a > $o,X2: set_list_a] :
      ( ( P @ X2 )
     => ( ! [Y2: set_list_a] :
            ( ( P @ Y2 )
           => ( ord_le8861187494160871172list_a @ Y2 @ X2 ) )
       => ( ( order_733672244956367037list_a @ P )
          = X2 ) ) ) ).

% Greatest_equality
thf(fact_376_Greatest__equality,axiom,
    ! [P: set_a > $o,X2: set_a] :
      ( ( P @ X2 )
     => ( ! [Y2: set_a] :
            ( ( P @ Y2 )
           => ( ord_less_eq_set_a @ Y2 @ X2 ) )
       => ( ( order_Greatest_set_a @ P )
          = X2 ) ) ) ).

% Greatest_equality
thf(fact_377_GreatestI2__order,axiom,
    ! [P: ( $o > set_a ) > $o,X2: $o > set_a,Q: ( $o > set_a ) > $o] :
      ( ( P @ X2 )
     => ( ! [Y2: $o > set_a] :
            ( ( P @ Y2 )
           => ( ord_less_eq_o_set_a @ Y2 @ X2 ) )
       => ( ! [X4: $o > set_a] :
              ( ( P @ X4 )
             => ( ! [Y5: $o > set_a] :
                    ( ( P @ Y5 )
                   => ( ord_less_eq_o_set_a @ Y5 @ X4 ) )
               => ( Q @ X4 ) ) )
         => ( Q @ ( order_6114237596796908690_set_a @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_378_GreatestI2__order,axiom,
    ! [P: ( $o > set_list_a ) > $o,X2: $o > set_list_a,Q: ( $o > set_list_a ) > $o] :
      ( ( P @ X2 )
     => ( ! [Y2: $o > set_list_a] :
            ( ( P @ Y2 )
           => ( ord_le354538097136858257list_a @ Y2 @ X2 ) )
       => ( ! [X4: $o > set_list_a] :
              ( ( P @ X4 )
             => ( ! [Y5: $o > set_list_a] :
                    ( ( P @ Y5 )
                   => ( ord_le354538097136858257list_a @ Y5 @ X4 ) )
               => ( Q @ X4 ) ) )
         => ( Q @ ( order_6341408326612538392list_a @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_379_GreatestI2__order,axiom,
    ! [P: set_set_a > $o,X2: set_set_a,Q: set_set_a > $o] :
      ( ( P @ X2 )
     => ( ! [Y2: set_set_a] :
            ( ( P @ Y2 )
           => ( ord_le3724670747650509150_set_a @ Y2 @ X2 ) )
       => ( ! [X4: set_set_a] :
              ( ( P @ X4 )
             => ( ! [Y5: set_set_a] :
                    ( ( P @ Y5 )
                   => ( ord_le3724670747650509150_set_a @ Y5 @ X4 ) )
               => ( Q @ X4 ) ) )
         => ( Q @ ( order_3565860530148683671_set_a @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_380_GreatestI2__order,axiom,
    ! [P: set_set_list_a > $o,X2: set_set_list_a,Q: set_set_list_a > $o] :
      ( ( P @ X2 )
     => ( ! [Y2: set_set_list_a] :
            ( ( P @ Y2 )
           => ( ord_le8877086941679407844list_a @ Y2 @ X2 ) )
       => ( ! [X4: set_set_list_a] :
              ( ( P @ X4 )
             => ( ! [Y5: set_set_list_a] :
                    ( ( P @ Y5 )
                   => ( ord_le8877086941679407844list_a @ Y5 @ X4 ) )
               => ( Q @ X4 ) ) )
         => ( Q @ ( order_8947689642956441757list_a @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_381_GreatestI2__order,axiom,
    ! [P: set_list_list_a > $o,X2: set_list_list_a,Q: set_list_list_a > $o] :
      ( ( P @ X2 )
     => ( ! [Y2: set_list_list_a] :
            ( ( P @ Y2 )
           => ( ord_le8488217952732425610list_a @ Y2 @ X2 ) )
       => ( ! [X4: set_list_list_a] :
              ( ( P @ X4 )
             => ( ! [Y5: set_list_list_a] :
                    ( ( P @ Y5 )
                   => ( ord_le8488217952732425610list_a @ Y5 @ X4 ) )
               => ( Q @ X4 ) ) )
         => ( Q @ ( order_4834126633758709443list_a @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_382_GreatestI2__order,axiom,
    ! [P: set_list_a > $o,X2: set_list_a,Q: set_list_a > $o] :
      ( ( P @ X2 )
     => ( ! [Y2: set_list_a] :
            ( ( P @ Y2 )
           => ( ord_le8861187494160871172list_a @ Y2 @ X2 ) )
       => ( ! [X4: set_list_a] :
              ( ( P @ X4 )
             => ( ! [Y5: set_list_a] :
                    ( ( P @ Y5 )
                   => ( ord_le8861187494160871172list_a @ Y5 @ X4 ) )
               => ( Q @ X4 ) ) )
         => ( Q @ ( order_733672244956367037list_a @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_383_GreatestI2__order,axiom,
    ! [P: set_a > $o,X2: set_a,Q: set_a > $o] :
      ( ( P @ X2 )
     => ( ! [Y2: set_a] :
            ( ( P @ Y2 )
           => ( ord_less_eq_set_a @ Y2 @ X2 ) )
       => ( ! [X4: set_a] :
              ( ( P @ X4 )
             => ( ! [Y5: set_a] :
                    ( ( P @ Y5 )
                   => ( ord_less_eq_set_a @ Y5 @ X4 ) )
               => ( Q @ X4 ) ) )
         => ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_384_le__rel__bool__arg__iff,axiom,
    ( ord_le5604070792465694036_set_a
    = ( ^ [X5: $o > $o > set_a,Y6: $o > $o > set_a] :
          ( ( ord_less_eq_o_set_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
          & ( ord_less_eq_o_set_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).

% le_rel_bool_arg_iff
thf(fact_385_le__rel__bool__arg__iff,axiom,
    ( ord_le416739357993157210list_a
    = ( ^ [X5: $o > $o > set_list_a,Y6: $o > $o > set_list_a] :
          ( ( ord_le354538097136858257list_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
          & ( ord_le354538097136858257list_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).

% le_rel_bool_arg_iff
thf(fact_386_le__rel__bool__arg__iff,axiom,
    ( ord_le2411852534652195563_set_a
    = ( ^ [X5: $o > set_set_a,Y6: $o > set_set_a] :
          ( ( ord_le3724670747650509150_set_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
          & ( ord_le3724670747650509150_set_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).

% le_rel_bool_arg_iff
thf(fact_387_le__rel__bool__arg__iff,axiom,
    ( ord_le452127650934333553list_a
    = ( ^ [X5: $o > set_set_list_a,Y6: $o > set_set_list_a] :
          ( ( ord_le8877086941679407844list_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
          & ( ord_le8877086941679407844list_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).

% le_rel_bool_arg_iff
thf(fact_388_le__rel__bool__arg__iff,axiom,
    ( ord_le2535169540198774551list_a
    = ( ^ [X5: $o > set_list_list_a,Y6: $o > set_list_list_a] :
          ( ( ord_le8488217952732425610list_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
          & ( ord_le8488217952732425610list_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).

% le_rel_bool_arg_iff
thf(fact_389_le__rel__bool__arg__iff,axiom,
    ( ord_le354538097136858257list_a
    = ( ^ [X5: $o > set_list_a,Y6: $o > set_list_a] :
          ( ( ord_le8861187494160871172list_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
          & ( ord_le8861187494160871172list_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).

% le_rel_bool_arg_iff
thf(fact_390_le__rel__bool__arg__iff,axiom,
    ( ord_less_eq_o_set_a
    = ( ^ [X5: $o > set_a,Y6: $o > set_a] :
          ( ( ord_less_eq_set_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
          & ( ord_less_eq_set_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).

% le_rel_bool_arg_iff
thf(fact_391_verit__comp__simplify1_I2_J,axiom,
    ! [A: $o > set_a] : ( ord_less_eq_o_set_a @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_392_verit__comp__simplify1_I2_J,axiom,
    ! [A: $o > set_list_a] : ( ord_le354538097136858257list_a @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_393_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_394_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_set_list_a] : ( ord_le8877086941679407844list_a @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_395_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_list_list_a] : ( ord_le8488217952732425610list_a @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_396_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_list_a] : ( ord_le8861187494160871172list_a @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_397_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_398_stream_Oinject,axiom,
    ! [X1: list_a,X22: stream_list_a,Y1: list_a,Y22: stream_list_a] :
      ( ( ( sCons_list_a @ X1 @ X22 )
        = ( sCons_list_a @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% stream.inject
thf(fact_399_stream_Oinject,axiom,
    ! [X1: list_list_a,X22: stream_list_list_a,Y1: list_list_a,Y22: stream_list_list_a] :
      ( ( ( sCons_list_list_a @ X1 @ X22 )
        = ( sCons_list_list_a @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% stream.inject
thf(fact_400_stream_Oinject,axiom,
    ! [X1: a,X22: stream_a,Y1: a,Y22: stream_a] :
      ( ( ( sCons_a @ X1 @ X22 )
        = ( sCons_a @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% stream.inject
thf(fact_401_stream_Oinject,axiom,
    ! [X1: list_list_list_a,X22: stream2255243159586646806list_a,Y1: list_list_list_a,Y22: stream2255243159586646806list_a] :
      ( ( ( sCons_8165023923567507367list_a @ X1 @ X22 )
        = ( sCons_8165023923567507367list_a @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% stream.inject
thf(fact_402_smember__code,axiom,
    ! [X2: list_list_a,Y: list_list_a,S2: stream_list_list_a] :
      ( ( smember_list_list_a @ X2 @ ( sCons_list_list_a @ Y @ S2 ) )
      = ( ( X2 != Y )
       => ( smember_list_list_a @ X2 @ S2 ) ) ) ).

% smember_code
thf(fact_403_smember__code,axiom,
    ! [X2: a,Y: a,S2: stream_a] :
      ( ( smember_a @ X2 @ ( sCons_a @ Y @ S2 ) )
      = ( ( X2 != Y )
       => ( smember_a @ X2 @ S2 ) ) ) ).

% smember_code
thf(fact_404_smember__code,axiom,
    ! [X2: list_list_list_a,Y: list_list_list_a,S2: stream2255243159586646806list_a] :
      ( ( smembe6049431907005952846list_a @ X2 @ ( sCons_8165023923567507367list_a @ Y @ S2 ) )
      = ( ( X2 != Y )
       => ( smembe6049431907005952846list_a @ X2 @ S2 ) ) ) ).

% smember_code
thf(fact_405_smember__code,axiom,
    ! [X2: list_a,Y: list_a,S2: stream_list_a] :
      ( ( smember_list_a @ X2 @ ( sCons_list_a @ Y @ S2 ) )
      = ( ( X2 != Y )
       => ( smember_list_a @ X2 @ S2 ) ) ) ).

% smember_code
thf(fact_406_stream_Ocollapse,axiom,
    ! [Stream: stream_a] :
      ( ( sCons_a @ ( shd_a @ Stream ) @ ( stl_a @ Stream ) )
      = Stream ) ).

% stream.collapse
thf(fact_407_stream_Ocollapse,axiom,
    ! [Stream: stream2255243159586646806list_a] :
      ( ( sCons_8165023923567507367list_a @ ( shd_list_list_list_a @ Stream ) @ ( stl_list_list_list_a @ Stream ) )
      = Stream ) ).

% stream.collapse
thf(fact_408_stream_Ocollapse,axiom,
    ! [Stream: stream_list_a] :
      ( ( sCons_list_a @ ( shd_list_a @ Stream ) @ ( stl_list_a @ Stream ) )
      = Stream ) ).

% stream.collapse
thf(fact_409_stream_Ocollapse,axiom,
    ! [Stream: stream_list_list_a] :
      ( ( sCons_list_list_a @ ( shd_list_list_a @ Stream ) @ ( stl_list_list_a @ Stream ) )
      = Stream ) ).

% stream.collapse
thf(fact_410_stream_Oexhaust,axiom,
    ! [Y: stream_list_a] :
      ~ ! [X12: list_a,X23: stream_list_a] :
          ( Y
         != ( sCons_list_a @ X12 @ X23 ) ) ).

% stream.exhaust
thf(fact_411_stream_Oexhaust,axiom,
    ! [Y: stream_list_list_a] :
      ~ ! [X12: list_list_a,X23: stream_list_list_a] :
          ( Y
         != ( sCons_list_list_a @ X12 @ X23 ) ) ).

% stream.exhaust
thf(fact_412_stream_Oexhaust,axiom,
    ! [Y: stream_a] :
      ~ ! [X12: a,X23: stream_a] :
          ( Y
         != ( sCons_a @ X12 @ X23 ) ) ).

% stream.exhaust
thf(fact_413_stream_Oexhaust,axiom,
    ! [Y: stream2255243159586646806list_a] :
      ~ ! [X12: list_list_list_a,X23: stream2255243159586646806list_a] :
          ( Y
         != ( sCons_8165023923567507367list_a @ X12 @ X23 ) ) ).

% stream.exhaust
thf(fact_414_list_Oexpand,axiom,
    ! [List: list_list_a,List2: list_list_a] :
      ( ( ( List = nil_list_a )
        = ( List2 = nil_list_a ) )
     => ( ( ( List != nil_list_a )
         => ( ( List2 != nil_list_a )
           => ( ( ( hd_list_a @ List )
                = ( hd_list_a @ List2 ) )
              & ( ( tl_list_a @ List )
                = ( tl_list_a @ List2 ) ) ) ) )
       => ( List = List2 ) ) ) ).

% list.expand
thf(fact_415_list_Oexpand,axiom,
    ! [List: list_list_list_a,List2: list_list_list_a] :
      ( ( ( List = nil_list_list_a )
        = ( List2 = nil_list_list_a ) )
     => ( ( ( List != nil_list_list_a )
         => ( ( List2 != nil_list_list_a )
           => ( ( ( hd_list_list_a @ List )
                = ( hd_list_list_a @ List2 ) )
              & ( ( tl_list_list_a @ List )
                = ( tl_list_list_a @ List2 ) ) ) ) )
       => ( List = List2 ) ) ) ).

% list.expand
thf(fact_416_list_Oexpand,axiom,
    ! [List: list_set_a,List2: list_set_a] :
      ( ( ( List = nil_set_a )
        = ( List2 = nil_set_a ) )
     => ( ( ( List != nil_set_a )
         => ( ( List2 != nil_set_a )
           => ( ( ( hd_set_a @ List )
                = ( hd_set_a @ List2 ) )
              & ( ( tl_set_a @ List )
                = ( tl_set_a @ List2 ) ) ) ) )
       => ( List = List2 ) ) ) ).

% list.expand
thf(fact_417_list_Oexpand,axiom,
    ! [List: list_a,List2: list_a] :
      ( ( ( List = nil_a )
        = ( List2 = nil_a ) )
     => ( ( ( List != nil_a )
         => ( ( List2 != nil_a )
           => ( ( ( hd_a @ List )
                = ( hd_a @ List2 ) )
              & ( ( tl_a @ List )
                = ( tl_a @ List2 ) ) ) ) )
       => ( List = List2 ) ) ) ).

% list.expand
thf(fact_418_flat_Ocode,axiom,
    ( flat_set_a
    = ( ^ [Ws2: stream_list_set_a] :
          ( sCons_set_a @ ( hd_set_a @ ( shd_list_set_a @ Ws2 ) )
          @ ( flat_set_a
            @ ( if_stream_list_set_a
              @ ( ( tl_set_a @ ( shd_list_set_a @ Ws2 ) )
                = nil_set_a )
              @ ( stl_list_set_a @ Ws2 )
              @ ( sCons_list_set_a @ ( tl_set_a @ ( shd_list_set_a @ Ws2 ) ) @ ( stl_list_set_a @ Ws2 ) ) ) ) ) ) ) ).

% flat.code
thf(fact_419_flat_Ocode,axiom,
    ( flat_list_list_a
    = ( ^ [Ws2: stream2255243159586646806list_a] :
          ( sCons_list_list_a @ ( hd_list_list_a @ ( shd_list_list_list_a @ Ws2 ) )
          @ ( flat_list_list_a
            @ ( if_str8217234800680828380list_a
              @ ( ( tl_list_list_a @ ( shd_list_list_list_a @ Ws2 ) )
                = nil_list_list_a )
              @ ( stl_list_list_list_a @ Ws2 )
              @ ( sCons_8165023923567507367list_a @ ( tl_list_list_a @ ( shd_list_list_list_a @ Ws2 ) ) @ ( stl_list_list_list_a @ Ws2 ) ) ) ) ) ) ) ).

% flat.code
thf(fact_420_flat_Ocode,axiom,
    ( flat_l8549998914816956042list_a
    = ( ^ [Ws2: stream8854267710728967580list_a] :
          ( sCons_8165023923567507367list_a @ ( hd_list_list_list_a @ ( shd_li1819241521678429890list_a @ Ws2 ) )
          @ ( flat_l8549998914816956042list_a
            @ ( if_str5554437364767477090list_a
              @ ( ( tl_list_list_list_a @ ( shd_li1819241521678429890list_a @ Ws2 ) )
                = nil_list_list_list_a )
              @ ( stl_li4757347539825400518list_a @ Ws2 )
              @ ( sCons_3158304508880615213list_a @ ( tl_list_list_list_a @ ( shd_li1819241521678429890list_a @ Ws2 ) ) @ ( stl_li4757347539825400518list_a @ Ws2 ) ) ) ) ) ) ) ).

% flat.code
thf(fact_421_flat_Ocode,axiom,
    ( flat_a
    = ( ^ [Ws2: stream_list_a] :
          ( sCons_a @ ( hd_a @ ( shd_list_a @ Ws2 ) )
          @ ( flat_a
            @ ( if_stream_list_a
              @ ( ( tl_a @ ( shd_list_a @ Ws2 ) )
                = nil_a )
              @ ( stl_list_a @ Ws2 )
              @ ( sCons_list_a @ ( tl_a @ ( shd_list_a @ Ws2 ) ) @ ( stl_list_a @ Ws2 ) ) ) ) ) ) ) ).

% flat.code
thf(fact_422_flat_Ocode,axiom,
    ( flat_list_a
    = ( ^ [Ws2: stream_list_list_a] :
          ( sCons_list_a @ ( hd_list_a @ ( shd_list_list_a @ Ws2 ) )
          @ ( flat_list_a
            @ ( if_str7505741754068378070list_a
              @ ( ( tl_list_a @ ( shd_list_list_a @ Ws2 ) )
                = nil_list_a )
              @ ( stl_list_list_a @ Ws2 )
              @ ( sCons_list_list_a @ ( tl_list_a @ ( shd_list_list_a @ Ws2 ) ) @ ( stl_list_list_a @ Ws2 ) ) ) ) ) ) ) ).

% flat.code
thf(fact_423_list_Osel_I2_J,axiom,
    ( ( tl_list_a @ nil_list_a )
    = nil_list_a ) ).

% list.sel(2)
thf(fact_424_list_Osel_I2_J,axiom,
    ( ( tl_list_list_a @ nil_list_list_a )
    = nil_list_list_a ) ).

% list.sel(2)
thf(fact_425_list_Osel_I2_J,axiom,
    ( ( tl_set_a @ nil_set_a )
    = nil_set_a ) ).

% list.sel(2)
thf(fact_426_list_Osel_I2_J,axiom,
    ( ( tl_a @ nil_a )
    = nil_a ) ).

% list.sel(2)
thf(fact_427_stream_Osel_I2_J,axiom,
    ! [X1: a,X22: stream_a] :
      ( ( stl_a @ ( sCons_a @ X1 @ X22 ) )
      = X22 ) ).

% stream.sel(2)
thf(fact_428_stream_Osel_I2_J,axiom,
    ! [X1: list_list_list_a,X22: stream2255243159586646806list_a] :
      ( ( stl_list_list_list_a @ ( sCons_8165023923567507367list_a @ X1 @ X22 ) )
      = X22 ) ).

% stream.sel(2)
thf(fact_429_stream_Osel_I2_J,axiom,
    ! [X1: list_a,X22: stream_list_a] :
      ( ( stl_list_a @ ( sCons_list_a @ X1 @ X22 ) )
      = X22 ) ).

% stream.sel(2)
thf(fact_430_stream_Osel_I2_J,axiom,
    ! [X1: list_list_a,X22: stream_list_list_a] :
      ( ( stl_list_list_a @ ( sCons_list_list_a @ X1 @ X22 ) )
      = X22 ) ).

% stream.sel(2)
thf(fact_431_stream_Osel_I1_J,axiom,
    ! [X1: list_list_a,X22: stream_list_list_a] :
      ( ( shd_list_list_a @ ( sCons_list_list_a @ X1 @ X22 ) )
      = X1 ) ).

% stream.sel(1)
thf(fact_432_stream_Osel_I1_J,axiom,
    ! [X1: a,X22: stream_a] :
      ( ( shd_a @ ( sCons_a @ X1 @ X22 ) )
      = X1 ) ).

% stream.sel(1)
thf(fact_433_stream_Osel_I1_J,axiom,
    ! [X1: list_list_list_a,X22: stream2255243159586646806list_a] :
      ( ( shd_list_list_list_a @ ( sCons_8165023923567507367list_a @ X1 @ X22 ) )
      = X1 ) ).

% stream.sel(1)
thf(fact_434_stream_Osel_I1_J,axiom,
    ! [X1: list_a,X22: stream_list_a] :
      ( ( shd_list_a @ ( sCons_list_a @ X1 @ X22 ) )
      = X1 ) ).

% stream.sel(1)
thf(fact_435_stream_Oset__induct,axiom,
    ! [X2: set_list_a,A: stream_set_list_a,P: set_list_a > stream_set_list_a > $o] :
      ( ( member_set_list_a @ X2 @ ( sset_set_list_a @ A ) )
     => ( ! [Z1: set_list_a,Z22: stream_set_list_a] : ( P @ Z1 @ ( sCons_set_list_a @ Z1 @ Z22 ) )
       => ( ! [Z1: set_list_a,Z22: stream_set_list_a,Xa: set_list_a] :
              ( ( member_set_list_a @ Xa @ ( sset_set_list_a @ Z22 ) )
             => ( ( P @ Xa @ Z22 )
               => ( P @ Xa @ ( sCons_set_list_a @ Z1 @ Z22 ) ) ) )
         => ( P @ X2 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_436_stream_Oset__induct,axiom,
    ! [X2: set_set_list_a,A: stream1977264251772372042list_a,P: set_set_list_a > stream1977264251772372042list_a > $o] :
      ( ( member8857465052274545133list_a @ X2 @ ( sset_set_set_list_a @ A ) )
     => ( ! [Z1: set_set_list_a,Z22: stream1977264251772372042list_a] : ( P @ Z1 @ ( sCons_set_set_list_a @ Z1 @ Z22 ) )
       => ( ! [Z1: set_set_list_a,Z22: stream1977264251772372042list_a,Xa: set_set_list_a] :
              ( ( member8857465052274545133list_a @ Xa @ ( sset_set_set_list_a @ Z22 ) )
             => ( ( P @ Xa @ Z22 )
               => ( P @ Xa @ ( sCons_set_set_list_a @ Z1 @ Z22 ) ) ) )
         => ( P @ X2 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_437_stream_Oset__induct,axiom,
    ! [X2: set_set_a,A: stream_set_set_a,P: set_set_a > stream_set_set_a > $o] :
      ( ( member_set_set_a @ X2 @ ( sset_set_set_a @ A ) )
     => ( ! [Z1: set_set_a,Z22: stream_set_set_a] : ( P @ Z1 @ ( sCons_set_set_a @ Z1 @ Z22 ) )
       => ( ! [Z1: set_set_a,Z22: stream_set_set_a,Xa: set_set_a] :
              ( ( member_set_set_a @ Xa @ ( sset_set_set_a @ Z22 ) )
             => ( ( P @ Xa @ Z22 )
               => ( P @ Xa @ ( sCons_set_set_a @ Z1 @ Z22 ) ) ) )
         => ( P @ X2 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_438_stream_Oset__induct,axiom,
    ! [X2: set_a,A: stream_set_a,P: set_a > stream_set_a > $o] :
      ( ( member_set_a2 @ X2 @ ( sset_set_a @ A ) )
     => ( ! [Z1: set_a,Z22: stream_set_a] : ( P @ Z1 @ ( sCons_set_a @ Z1 @ Z22 ) )
       => ( ! [Z1: set_a,Z22: stream_set_a,Xa: set_a] :
              ( ( member_set_a2 @ Xa @ ( sset_set_a @ Z22 ) )
             => ( ( P @ Xa @ Z22 )
               => ( P @ Xa @ ( sCons_set_a @ Z1 @ Z22 ) ) ) )
         => ( P @ X2 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_439_stream_Oset__induct,axiom,
    ! [X2: list_list_a,A: stream_list_list_a,P: list_list_a > stream_list_list_a > $o] :
      ( ( member_list_list_a2 @ X2 @ ( sset_list_list_a @ A ) )
     => ( ! [Z1: list_list_a,Z22: stream_list_list_a] : ( P @ Z1 @ ( sCons_list_list_a @ Z1 @ Z22 ) )
       => ( ! [Z1: list_list_a,Z22: stream_list_list_a,Xa: list_list_a] :
              ( ( member_list_list_a2 @ Xa @ ( sset_list_list_a @ Z22 ) )
             => ( ( P @ Xa @ Z22 )
               => ( P @ Xa @ ( sCons_list_list_a @ Z1 @ Z22 ) ) ) )
         => ( P @ X2 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_440_stream_Oset__induct,axiom,
    ! [X2: list_list_list_a,A: stream2255243159586646806list_a,P: list_list_list_a > stream2255243159586646806list_a > $o] :
      ( ( member5342144027231129785list_a @ X2 @ ( sset_l5650713775164989220list_a @ A ) )
     => ( ! [Z1: list_list_list_a,Z22: stream2255243159586646806list_a] : ( P @ Z1 @ ( sCons_8165023923567507367list_a @ Z1 @ Z22 ) )
       => ( ! [Z1: list_list_list_a,Z22: stream2255243159586646806list_a,Xa: list_list_list_a] :
              ( ( member5342144027231129785list_a @ Xa @ ( sset_l5650713775164989220list_a @ Z22 ) )
             => ( ( P @ Xa @ Z22 )
               => ( P @ Xa @ ( sCons_8165023923567507367list_a @ Z1 @ Z22 ) ) ) )
         => ( P @ X2 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_441_stream_Oset__induct,axiom,
    ! [X2: list_a,A: stream_list_a,P: list_a > stream_list_a > $o] :
      ( ( member_list_a2 @ X2 @ ( sset_list_a @ A ) )
     => ( ! [Z1: list_a,Z22: stream_list_a] : ( P @ Z1 @ ( sCons_list_a @ Z1 @ Z22 ) )
       => ( ! [Z1: list_a,Z22: stream_list_a,Xa: list_a] :
              ( ( member_list_a2 @ Xa @ ( sset_list_a @ Z22 ) )
             => ( ( P @ Xa @ Z22 )
               => ( P @ Xa @ ( sCons_list_a @ Z1 @ Z22 ) ) ) )
         => ( P @ X2 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_442_stream_Oset__induct,axiom,
    ! [X2: a,A: stream_a,P: a > stream_a > $o] :
      ( ( member_a2 @ X2 @ ( sset_a @ A ) )
     => ( ! [Z1: a,Z22: stream_a] : ( P @ Z1 @ ( sCons_a @ Z1 @ Z22 ) )
       => ( ! [Z1: a,Z22: stream_a,Xa: a] :
              ( ( member_a2 @ Xa @ ( sset_a @ Z22 ) )
             => ( ( P @ Xa @ Z22 )
               => ( P @ Xa @ ( sCons_a @ Z1 @ Z22 ) ) ) )
         => ( P @ X2 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_443_stream_Oset__cases,axiom,
    ! [E: set_list_a,A: stream_set_list_a] :
      ( ( member_set_list_a @ E @ ( sset_set_list_a @ A ) )
     => ( ! [Z22: stream_set_list_a] :
            ( A
           != ( sCons_set_list_a @ E @ Z22 ) )
       => ~ ! [Z1: set_list_a,Z22: stream_set_list_a] :
              ( ( A
                = ( sCons_set_list_a @ Z1 @ Z22 ) )
             => ~ ( member_set_list_a @ E @ ( sset_set_list_a @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_444_stream_Oset__cases,axiom,
    ! [E: set_set_list_a,A: stream1977264251772372042list_a] :
      ( ( member8857465052274545133list_a @ E @ ( sset_set_set_list_a @ A ) )
     => ( ! [Z22: stream1977264251772372042list_a] :
            ( A
           != ( sCons_set_set_list_a @ E @ Z22 ) )
       => ~ ! [Z1: set_set_list_a,Z22: stream1977264251772372042list_a] :
              ( ( A
                = ( sCons_set_set_list_a @ Z1 @ Z22 ) )
             => ~ ( member8857465052274545133list_a @ E @ ( sset_set_set_list_a @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_445_stream_Oset__cases,axiom,
    ! [E: set_set_a,A: stream_set_set_a] :
      ( ( member_set_set_a @ E @ ( sset_set_set_a @ A ) )
     => ( ! [Z22: stream_set_set_a] :
            ( A
           != ( sCons_set_set_a @ E @ Z22 ) )
       => ~ ! [Z1: set_set_a,Z22: stream_set_set_a] :
              ( ( A
                = ( sCons_set_set_a @ Z1 @ Z22 ) )
             => ~ ( member_set_set_a @ E @ ( sset_set_set_a @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_446_stream_Oset__cases,axiom,
    ! [E: set_a,A: stream_set_a] :
      ( ( member_set_a2 @ E @ ( sset_set_a @ A ) )
     => ( ! [Z22: stream_set_a] :
            ( A
           != ( sCons_set_a @ E @ Z22 ) )
       => ~ ! [Z1: set_a,Z22: stream_set_a] :
              ( ( A
                = ( sCons_set_a @ Z1 @ Z22 ) )
             => ~ ( member_set_a2 @ E @ ( sset_set_a @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_447_stream_Oset__cases,axiom,
    ! [E: list_list_a,A: stream_list_list_a] :
      ( ( member_list_list_a2 @ E @ ( sset_list_list_a @ A ) )
     => ( ! [Z22: stream_list_list_a] :
            ( A
           != ( sCons_list_list_a @ E @ Z22 ) )
       => ~ ! [Z1: list_list_a,Z22: stream_list_list_a] :
              ( ( A
                = ( sCons_list_list_a @ Z1 @ Z22 ) )
             => ~ ( member_list_list_a2 @ E @ ( sset_list_list_a @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_448_stream_Oset__cases,axiom,
    ! [E: list_list_list_a,A: stream2255243159586646806list_a] :
      ( ( member5342144027231129785list_a @ E @ ( sset_l5650713775164989220list_a @ A ) )
     => ( ! [Z22: stream2255243159586646806list_a] :
            ( A
           != ( sCons_8165023923567507367list_a @ E @ Z22 ) )
       => ~ ! [Z1: list_list_list_a,Z22: stream2255243159586646806list_a] :
              ( ( A
                = ( sCons_8165023923567507367list_a @ Z1 @ Z22 ) )
             => ~ ( member5342144027231129785list_a @ E @ ( sset_l5650713775164989220list_a @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_449_stream_Oset__cases,axiom,
    ! [E: list_a,A: stream_list_a] :
      ( ( member_list_a2 @ E @ ( sset_list_a @ A ) )
     => ( ! [Z22: stream_list_a] :
            ( A
           != ( sCons_list_a @ E @ Z22 ) )
       => ~ ! [Z1: list_a,Z22: stream_list_a] :
              ( ( A
                = ( sCons_list_a @ Z1 @ Z22 ) )
             => ~ ( member_list_a2 @ E @ ( sset_list_a @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_450_stream_Oset__cases,axiom,
    ! [E: a,A: stream_a] :
      ( ( member_a2 @ E @ ( sset_a @ A ) )
     => ( ! [Z22: stream_a] :
            ( A
           != ( sCons_a @ E @ Z22 ) )
       => ~ ! [Z1: a,Z22: stream_a] :
              ( ( A
                = ( sCons_a @ Z1 @ Z22 ) )
             => ~ ( member_a2 @ E @ ( sset_a @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_451_stream_Oset__intros_I1_J,axiom,
    ! [X1: set_list_a,X22: stream_set_list_a] : ( member_set_list_a @ X1 @ ( sset_set_list_a @ ( sCons_set_list_a @ X1 @ X22 ) ) ) ).

% stream.set_intros(1)
thf(fact_452_stream_Oset__intros_I1_J,axiom,
    ! [X1: set_set_list_a,X22: stream1977264251772372042list_a] : ( member8857465052274545133list_a @ X1 @ ( sset_set_set_list_a @ ( sCons_set_set_list_a @ X1 @ X22 ) ) ) ).

% stream.set_intros(1)
thf(fact_453_stream_Oset__intros_I1_J,axiom,
    ! [X1: set_set_a,X22: stream_set_set_a] : ( member_set_set_a @ X1 @ ( sset_set_set_a @ ( sCons_set_set_a @ X1 @ X22 ) ) ) ).

% stream.set_intros(1)
thf(fact_454_stream_Oset__intros_I1_J,axiom,
    ! [X1: set_a,X22: stream_set_a] : ( member_set_a2 @ X1 @ ( sset_set_a @ ( sCons_set_a @ X1 @ X22 ) ) ) ).

% stream.set_intros(1)
thf(fact_455_stream_Oset__intros_I1_J,axiom,
    ! [X1: list_list_a,X22: stream_list_list_a] : ( member_list_list_a2 @ X1 @ ( sset_list_list_a @ ( sCons_list_list_a @ X1 @ X22 ) ) ) ).

% stream.set_intros(1)
thf(fact_456_stream_Oset__intros_I1_J,axiom,
    ! [X1: list_list_list_a,X22: stream2255243159586646806list_a] : ( member5342144027231129785list_a @ X1 @ ( sset_l5650713775164989220list_a @ ( sCons_8165023923567507367list_a @ X1 @ X22 ) ) ) ).

% stream.set_intros(1)
thf(fact_457_stream_Oset__intros_I1_J,axiom,
    ! [X1: list_a,X22: stream_list_a] : ( member_list_a2 @ X1 @ ( sset_list_a @ ( sCons_list_a @ X1 @ X22 ) ) ) ).

% stream.set_intros(1)
thf(fact_458_stream_Oset__intros_I1_J,axiom,
    ! [X1: a,X22: stream_a] : ( member_a2 @ X1 @ ( sset_a @ ( sCons_a @ X1 @ X22 ) ) ) ).

% stream.set_intros(1)
thf(fact_459_stream_Oset__intros_I2_J,axiom,
    ! [Y: set_list_a,X22: stream_set_list_a,X1: set_list_a] :
      ( ( member_set_list_a @ Y @ ( sset_set_list_a @ X22 ) )
     => ( member_set_list_a @ Y @ ( sset_set_list_a @ ( sCons_set_list_a @ X1 @ X22 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_460_stream_Oset__intros_I2_J,axiom,
    ! [Y: set_set_list_a,X22: stream1977264251772372042list_a,X1: set_set_list_a] :
      ( ( member8857465052274545133list_a @ Y @ ( sset_set_set_list_a @ X22 ) )
     => ( member8857465052274545133list_a @ Y @ ( sset_set_set_list_a @ ( sCons_set_set_list_a @ X1 @ X22 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_461_stream_Oset__intros_I2_J,axiom,
    ! [Y: set_set_a,X22: stream_set_set_a,X1: set_set_a] :
      ( ( member_set_set_a @ Y @ ( sset_set_set_a @ X22 ) )
     => ( member_set_set_a @ Y @ ( sset_set_set_a @ ( sCons_set_set_a @ X1 @ X22 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_462_stream_Oset__intros_I2_J,axiom,
    ! [Y: set_a,X22: stream_set_a,X1: set_a] :
      ( ( member_set_a2 @ Y @ ( sset_set_a @ X22 ) )
     => ( member_set_a2 @ Y @ ( sset_set_a @ ( sCons_set_a @ X1 @ X22 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_463_stream_Oset__intros_I2_J,axiom,
    ! [Y: list_list_a,X22: stream_list_list_a,X1: list_list_a] :
      ( ( member_list_list_a2 @ Y @ ( sset_list_list_a @ X22 ) )
     => ( member_list_list_a2 @ Y @ ( sset_list_list_a @ ( sCons_list_list_a @ X1 @ X22 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_464_stream_Oset__intros_I2_J,axiom,
    ! [Y: list_list_list_a,X22: stream2255243159586646806list_a,X1: list_list_list_a] :
      ( ( member5342144027231129785list_a @ Y @ ( sset_l5650713775164989220list_a @ X22 ) )
     => ( member5342144027231129785list_a @ Y @ ( sset_l5650713775164989220list_a @ ( sCons_8165023923567507367list_a @ X1 @ X22 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_465_stream_Oset__intros_I2_J,axiom,
    ! [Y: list_a,X22: stream_list_a,X1: list_a] :
      ( ( member_list_a2 @ Y @ ( sset_list_a @ X22 ) )
     => ( member_list_a2 @ Y @ ( sset_list_a @ ( sCons_list_a @ X1 @ X22 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_466_stream_Oset__intros_I2_J,axiom,
    ! [Y: a,X22: stream_a,X1: a] :
      ( ( member_a2 @ Y @ ( sset_a @ X22 ) )
     => ( member_a2 @ Y @ ( sset_a @ ( sCons_a @ X1 @ X22 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_467_sdrop__while__SCons,axiom,
    ! [P: list_a > $o,A: list_a,S2: stream_list_a] :
      ( ( ( P @ A )
       => ( ( sdrop_while_list_a @ P @ ( sCons_list_a @ A @ S2 ) )
          = ( sdrop_while_list_a @ P @ S2 ) ) )
      & ( ~ ( P @ A )
       => ( ( sdrop_while_list_a @ P @ ( sCons_list_a @ A @ S2 ) )
          = ( sCons_list_a @ A @ S2 ) ) ) ) ).

% sdrop_while_SCons
thf(fact_468_sdrop__while__SCons,axiom,
    ! [P: list_list_a > $o,A: list_list_a,S2: stream_list_list_a] :
      ( ( ( P @ A )
       => ( ( sdrop_1072776591231973358list_a @ P @ ( sCons_list_list_a @ A @ S2 ) )
          = ( sdrop_1072776591231973358list_a @ P @ S2 ) ) )
      & ( ~ ( P @ A )
       => ( ( sdrop_1072776591231973358list_a @ P @ ( sCons_list_list_a @ A @ S2 ) )
          = ( sCons_list_list_a @ A @ S2 ) ) ) ) ).

% sdrop_while_SCons
thf(fact_469_sdrop__while__SCons,axiom,
    ! [P: a > $o,A: a,S2: stream_a] :
      ( ( ( P @ A )
       => ( ( sdrop_while_a @ P @ ( sCons_a @ A @ S2 ) )
          = ( sdrop_while_a @ P @ S2 ) ) )
      & ( ~ ( P @ A )
       => ( ( sdrop_while_a @ P @ ( sCons_a @ A @ S2 ) )
          = ( sCons_a @ A @ S2 ) ) ) ) ).

% sdrop_while_SCons
thf(fact_470_sdrop__while__SCons,axiom,
    ! [P: list_list_list_a > $o,A: list_list_list_a,S2: stream2255243159586646806list_a] :
      ( ( ( P @ A )
       => ( ( sdrop_8575598270869141876list_a @ P @ ( sCons_8165023923567507367list_a @ A @ S2 ) )
          = ( sdrop_8575598270869141876list_a @ P @ S2 ) ) )
      & ( ~ ( P @ A )
       => ( ( sdrop_8575598270869141876list_a @ P @ ( sCons_8165023923567507367list_a @ A @ S2 ) )
          = ( sCons_8165023923567507367list_a @ A @ S2 ) ) ) ) ).

% sdrop_while_SCons
thf(fact_471_stream_Oexhaust__sel,axiom,
    ! [Stream: stream_a] :
      ( Stream
      = ( sCons_a @ ( shd_a @ Stream ) @ ( stl_a @ Stream ) ) ) ).

% stream.exhaust_sel
thf(fact_472_stream_Oexhaust__sel,axiom,
    ! [Stream: stream2255243159586646806list_a] :
      ( Stream
      = ( sCons_8165023923567507367list_a @ ( shd_list_list_list_a @ Stream ) @ ( stl_list_list_list_a @ Stream ) ) ) ).

% stream.exhaust_sel
thf(fact_473_stream_Oexhaust__sel,axiom,
    ! [Stream: stream_list_a] :
      ( Stream
      = ( sCons_list_a @ ( shd_list_a @ Stream ) @ ( stl_list_a @ Stream ) ) ) ).

% stream.exhaust_sel
thf(fact_474_stream_Oexhaust__sel,axiom,
    ! [Stream: stream_list_list_a] :
      ( Stream
      = ( sCons_list_list_a @ ( shd_list_list_a @ Stream ) @ ( stl_list_list_a @ Stream ) ) ) ).

% stream.exhaust_sel
thf(fact_475_flat_Osimps_I2_J,axiom,
    ! [Ws: stream_list_set_a] :
      ( ( stl_set_a @ ( flat_set_a @ Ws ) )
      = ( flat_set_a
        @ ( if_stream_list_set_a
          @ ( ( tl_set_a @ ( shd_list_set_a @ Ws ) )
            = nil_set_a )
          @ ( stl_list_set_a @ Ws )
          @ ( sCons_list_set_a @ ( tl_set_a @ ( shd_list_set_a @ Ws ) ) @ ( stl_list_set_a @ Ws ) ) ) ) ) ).

% flat.simps(2)
thf(fact_476_flat_Osimps_I2_J,axiom,
    ! [Ws: stream8854267710728967580list_a] :
      ( ( stl_list_list_list_a @ ( flat_l8549998914816956042list_a @ Ws ) )
      = ( flat_l8549998914816956042list_a
        @ ( if_str5554437364767477090list_a
          @ ( ( tl_list_list_list_a @ ( shd_li1819241521678429890list_a @ Ws ) )
            = nil_list_list_list_a )
          @ ( stl_li4757347539825400518list_a @ Ws )
          @ ( sCons_3158304508880615213list_a @ ( tl_list_list_list_a @ ( shd_li1819241521678429890list_a @ Ws ) ) @ ( stl_li4757347539825400518list_a @ Ws ) ) ) ) ) ).

% flat.simps(2)
thf(fact_477_flat_Osimps_I2_J,axiom,
    ! [Ws: stream_list_a] :
      ( ( stl_a @ ( flat_a @ Ws ) )
      = ( flat_a
        @ ( if_stream_list_a
          @ ( ( tl_a @ ( shd_list_a @ Ws ) )
            = nil_a )
          @ ( stl_list_a @ Ws )
          @ ( sCons_list_a @ ( tl_a @ ( shd_list_a @ Ws ) ) @ ( stl_list_a @ Ws ) ) ) ) ) ).

% flat.simps(2)
thf(fact_478_flat_Osimps_I2_J,axiom,
    ! [Ws: stream_list_list_a] :
      ( ( stl_list_a @ ( flat_list_a @ Ws ) )
      = ( flat_list_a
        @ ( if_str7505741754068378070list_a
          @ ( ( tl_list_a @ ( shd_list_list_a @ Ws ) )
            = nil_list_a )
          @ ( stl_list_list_a @ Ws )
          @ ( sCons_list_list_a @ ( tl_list_a @ ( shd_list_list_a @ Ws ) ) @ ( stl_list_list_a @ Ws ) ) ) ) ) ).

% flat.simps(2)
thf(fact_479_flat_Osimps_I2_J,axiom,
    ! [Ws: stream2255243159586646806list_a] :
      ( ( stl_list_list_a @ ( flat_list_list_a @ Ws ) )
      = ( flat_list_list_a
        @ ( if_str8217234800680828380list_a
          @ ( ( tl_list_list_a @ ( shd_list_list_list_a @ Ws ) )
            = nil_list_list_a )
          @ ( stl_list_list_list_a @ Ws )
          @ ( sCons_8165023923567507367list_a @ ( tl_list_list_a @ ( shd_list_list_list_a @ Ws ) ) @ ( stl_list_list_list_a @ Ws ) ) ) ) ) ).

% flat.simps(2)
thf(fact_480_sinterleave_Ocode,axiom,
    ( sinterleave_a
    = ( ^ [S12: stream_a,S23: stream_a] : ( sCons_a @ ( shd_a @ S12 ) @ ( sinterleave_a @ S23 @ ( stl_a @ S12 ) ) ) ) ) ).

% sinterleave.code
thf(fact_481_sinterleave_Ocode,axiom,
    ( sinter1285254775629623405list_a
    = ( ^ [S12: stream2255243159586646806list_a,S23: stream2255243159586646806list_a] : ( sCons_8165023923567507367list_a @ ( shd_list_list_list_a @ S12 ) @ ( sinter1285254775629623405list_a @ S23 @ ( stl_list_list_list_a @ S12 ) ) ) ) ) ).

% sinterleave.code
thf(fact_482_sinterleave_Ocode,axiom,
    ( sinterleave_list_a
    = ( ^ [S12: stream_list_a,S23: stream_list_a] : ( sCons_list_a @ ( shd_list_a @ S12 ) @ ( sinterleave_list_a @ S23 @ ( stl_list_a @ S12 ) ) ) ) ) ).

% sinterleave.code
thf(fact_483_sinterleave_Ocode,axiom,
    ( sinter664328039231274855list_a
    = ( ^ [S12: stream_list_list_a,S23: stream_list_list_a] : ( sCons_list_list_a @ ( shd_list_list_a @ S12 ) @ ( sinter664328039231274855list_a @ S23 @ ( stl_list_list_a @ S12 ) ) ) ) ) ).

% sinterleave.code
thf(fact_484_flat__Cons,axiom,
    ! [X2: set_a,Xs: list_set_a,Ws: stream_list_set_a] :
      ( ( flat_set_a @ ( sCons_list_set_a @ ( cons_set_a @ X2 @ Xs ) @ Ws ) )
      = ( sCons_set_a @ X2 @ ( flat_set_a @ ( if_stream_list_set_a @ ( Xs = nil_set_a ) @ Ws @ ( sCons_list_set_a @ Xs @ Ws ) ) ) ) ) ).

% flat_Cons
thf(fact_485_flat__Cons,axiom,
    ! [X2: list_list_list_a,Xs: list_l7815035709764258326list_a,Ws: stream8854267710728967580list_a] :
      ( ( flat_l8549998914816956042list_a @ ( sCons_3158304508880615213list_a @ ( cons_l8577867181958743494list_a @ X2 @ Xs ) @ Ws ) )
      = ( sCons_8165023923567507367list_a @ X2 @ ( flat_l8549998914816956042list_a @ ( if_str5554437364767477090list_a @ ( Xs = nil_list_list_list_a ) @ Ws @ ( sCons_3158304508880615213list_a @ Xs @ Ws ) ) ) ) ) ).

% flat_Cons
thf(fact_486_flat__Cons,axiom,
    ! [X2: list_list_a,Xs: list_list_list_a,Ws: stream2255243159586646806list_a] :
      ( ( flat_list_list_a @ ( sCons_8165023923567507367list_a @ ( cons_list_list_a @ X2 @ Xs ) @ Ws ) )
      = ( sCons_list_list_a @ X2 @ ( flat_list_list_a @ ( if_str8217234800680828380list_a @ ( Xs = nil_list_list_a ) @ Ws @ ( sCons_8165023923567507367list_a @ Xs @ Ws ) ) ) ) ) ).

% flat_Cons
thf(fact_487_flat__Cons,axiom,
    ! [X2: list_a,Xs: list_list_a,Ws: stream_list_list_a] :
      ( ( flat_list_a @ ( sCons_list_list_a @ ( cons_list_a @ X2 @ Xs ) @ Ws ) )
      = ( sCons_list_a @ X2 @ ( flat_list_a @ ( if_str7505741754068378070list_a @ ( Xs = nil_list_a ) @ Ws @ ( sCons_list_list_a @ Xs @ Ws ) ) ) ) ) ).

% flat_Cons
thf(fact_488_flat__Cons,axiom,
    ! [X2: a,Xs: list_a,Ws: stream_list_a] :
      ( ( flat_a @ ( sCons_list_a @ ( cons_a @ X2 @ Xs ) @ Ws ) )
      = ( sCons_a @ X2 @ ( flat_a @ ( if_stream_list_a @ ( Xs = nil_a ) @ Ws @ ( sCons_list_a @ Xs @ Ws ) ) ) ) ) ).

% flat_Cons
thf(fact_489_smap2_Ocode,axiom,
    ( smap2_a_a_a
    = ( ^ [F2: a > a > a,S12: stream_a,S23: stream_a] : ( sCons_a @ ( F2 @ ( shd_a @ S12 ) @ ( shd_a @ S23 ) ) @ ( smap2_a_a_a @ F2 @ ( stl_a @ S12 ) @ ( stl_a @ S23 ) ) ) ) ) ).

% smap2.code
thf(fact_490_smap2_Ocode,axiom,
    ( smap2_a_a_list_a
    = ( ^ [F2: a > a > list_a,S12: stream_a,S23: stream_a] : ( sCons_list_a @ ( F2 @ ( shd_a @ S12 ) @ ( shd_a @ S23 ) ) @ ( smap2_a_a_list_a @ F2 @ ( stl_a @ S12 ) @ ( stl_a @ S23 ) ) ) ) ) ).

% smap2.code
thf(fact_491_smap2_Ocode,axiom,
    ( smap2_list_a_a_a
    = ( ^ [F2: list_a > a > a,S12: stream_list_a,S23: stream_a] : ( sCons_a @ ( F2 @ ( shd_list_a @ S12 ) @ ( shd_a @ S23 ) ) @ ( smap2_list_a_a_a @ F2 @ ( stl_list_a @ S12 ) @ ( stl_a @ S23 ) ) ) ) ) ).

% smap2.code
thf(fact_492_smap2_Ocode,axiom,
    ( smap2_a_list_a_a
    = ( ^ [F2: a > list_a > a,S12: stream_a,S23: stream_list_a] : ( sCons_a @ ( F2 @ ( shd_a @ S12 ) @ ( shd_list_a @ S23 ) ) @ ( smap2_a_list_a_a @ F2 @ ( stl_a @ S12 ) @ ( stl_list_a @ S23 ) ) ) ) ) ).

% smap2.code
thf(fact_493_smap2_Ocode,axiom,
    ( smap2_5547344383464640076list_a
    = ( ^ [F2: list_a > a > list_a,S12: stream_list_a,S23: stream_a] : ( sCons_list_a @ ( F2 @ ( shd_list_a @ S12 ) @ ( shd_a @ S23 ) ) @ ( smap2_5547344383464640076list_a @ F2 @ ( stl_list_a @ S12 ) @ ( stl_a @ S23 ) ) ) ) ) ).

% smap2.code
thf(fact_494_smap2_Ocode,axiom,
    ( smap2_5647228111304775744list_a
    = ( ^ [F2: a > list_a > list_a,S12: stream_a,S23: stream_list_a] : ( sCons_list_a @ ( F2 @ ( shd_a @ S12 ) @ ( shd_list_a @ S23 ) ) @ ( smap2_5647228111304775744list_a @ F2 @ ( stl_a @ S12 ) @ ( stl_list_a @ S23 ) ) ) ) ) ).

% smap2.code
thf(fact_495_smap2_Ocode,axiom,
    ( smap2_3581788270250579916list_a
    = ( ^ [F2: a > a > list_list_a,S12: stream_a,S23: stream_a] : ( sCons_list_list_a @ ( F2 @ ( shd_a @ S12 ) @ ( shd_a @ S23 ) ) @ ( smap2_3581788270250579916list_a @ F2 @ ( stl_a @ S12 ) @ ( stl_a @ S23 ) ) ) ) ) ).

% smap2.code
thf(fact_496_smap2_Ocode,axiom,
    ( smap2_3658108627635373888st_a_a
    = ( ^ [F2: list_a > list_a > a,S12: stream_list_a,S23: stream_list_a] : ( sCons_a @ ( F2 @ ( shd_list_a @ S12 ) @ ( shd_list_a @ S23 ) ) @ ( smap2_3658108627635373888st_a_a @ F2 @ ( stl_list_a @ S12 ) @ ( stl_list_a @ S23 ) ) ) ) ) ).

% smap2.code
thf(fact_497_smap2_Ocode,axiom,
    ( smap2_7313055693640098252_a_a_a
    = ( ^ [F2: list_list_a > a > a,S12: stream_list_list_a,S23: stream_a] : ( sCons_a @ ( F2 @ ( shd_list_list_a @ S12 ) @ ( shd_a @ S23 ) ) @ ( smap2_7313055693640098252_a_a_a @ F2 @ ( stl_list_list_a @ S12 ) @ ( stl_a @ S23 ) ) ) ) ) ).

% smap2.code
thf(fact_498_smap2_Ocode,axiom,
    ( smap2_4184781901917651636st_a_a
    = ( ^ [F2: a > list_list_a > a,S12: stream_a,S23: stream_list_list_a] : ( sCons_a @ ( F2 @ ( shd_a @ S12 ) @ ( shd_list_list_a @ S23 ) ) @ ( smap2_4184781901917651636st_a_a @ F2 @ ( stl_a @ S12 ) @ ( stl_list_list_a @ S23 ) ) ) ) ) ).

% smap2.code
thf(fact_499_hd__Cons__tl,axiom,
    ! [Xs: list_set_a] :
      ( ( Xs != nil_set_a )
     => ( ( cons_set_a @ ( hd_set_a @ Xs ) @ ( tl_set_a @ Xs ) )
        = Xs ) ) ).

% hd_Cons_tl
thf(fact_500_hd__Cons__tl,axiom,
    ! [Xs: list_list_a] :
      ( ( Xs != nil_list_a )
     => ( ( cons_list_a @ ( hd_list_a @ Xs ) @ ( tl_list_a @ Xs ) )
        = Xs ) ) ).

% hd_Cons_tl
thf(fact_501_hd__Cons__tl,axiom,
    ! [Xs: list_list_list_a] :
      ( ( Xs != nil_list_list_a )
     => ( ( cons_list_list_a @ ( hd_list_list_a @ Xs ) @ ( tl_list_list_a @ Xs ) )
        = Xs ) ) ).

% hd_Cons_tl
thf(fact_502_hd__Cons__tl,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
     => ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
        = Xs ) ) ).

% hd_Cons_tl
thf(fact_503_list_Ocollapse,axiom,
    ! [List: list_set_a] :
      ( ( List != nil_set_a )
     => ( ( cons_set_a @ ( hd_set_a @ List ) @ ( tl_set_a @ List ) )
        = List ) ) ).

% list.collapse
thf(fact_504_list_Ocollapse,axiom,
    ! [List: list_list_a] :
      ( ( List != nil_list_a )
     => ( ( cons_list_a @ ( hd_list_a @ List ) @ ( tl_list_a @ List ) )
        = List ) ) ).

% list.collapse
thf(fact_505_list_Ocollapse,axiom,
    ! [List: list_list_list_a] :
      ( ( List != nil_list_list_a )
     => ( ( cons_list_list_a @ ( hd_list_list_a @ List ) @ ( tl_list_list_a @ List ) )
        = List ) ) ).

% list.collapse
thf(fact_506_list_Ocollapse,axiom,
    ! [List: list_a] :
      ( ( List != nil_a )
     => ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
        = List ) ) ).

% list.collapse
thf(fact_507_smap__ctr,axiom,
    ! [F: a > list_list_list_a,S2: stream_a,X2: list_list_list_a,S4: stream2255243159586646806list_a] :
      ( ( ( smap_a4630580661157078679list_a @ F @ S2 )
        = ( sCons_8165023923567507367list_a @ X2 @ S4 ) )
      = ( ( ( F @ ( shd_a @ S2 ) )
          = X2 )
        & ( ( smap_a4630580661157078679list_a @ F @ ( stl_a @ S2 ) )
          = S4 ) ) ) ).

% smap_ctr
thf(fact_508_smap__ctr,axiom,
    ! [F: list_list_list_a > list_list_list_a,S2: stream2255243159586646806list_a,X2: list_list_list_a,S4: stream2255243159586646806list_a] :
      ( ( ( smap_l7216702241007268101list_a @ F @ S2 )
        = ( sCons_8165023923567507367list_a @ X2 @ S4 ) )
      = ( ( ( F @ ( shd_list_list_list_a @ S2 ) )
          = X2 )
        & ( ( smap_l7216702241007268101list_a @ F @ ( stl_list_list_list_a @ S2 ) )
          = S4 ) ) ) ).

% smap_ctr
thf(fact_509_list_Oexhaust__sel,axiom,
    ! [List: list_a] :
      ( ( List != nil_a )
     => ( List
        = ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).

% list.exhaust_sel
thf(fact_510_cycle_Osimps_I1_J,axiom,
    ! [Xs: list_list_a] :
      ( ( shd_list_a @ ( cycle_list_a @ Xs ) )
      = ( hd_list_a @ Xs ) ) ).

% cycle.simps(1)
thf(fact_511_cycle__decomp,axiom,
    ! [U: list_a] :
      ( ( U != nil_a )
     => ( ( cycle_a @ U )
        = ( shift_a @ U @ ( cycle_a @ U ) ) ) ) ).

% cycle_decomp
thf(fact_512_stream_Omap__sel_I2_J,axiom,
    ! [F: list_a > list_a,A: stream_list_a] :
      ( ( stl_list_a @ ( smap_list_a_list_a @ F @ A ) )
      = ( smap_list_a_list_a @ F @ ( stl_list_a @ A ) ) ) ).

% stream.map_sel(2)
thf(fact_513_stream_Omap__sel_I2_J,axiom,
    ! [F: list_list_a > list_a,A: stream_list_list_a] :
      ( ( stl_list_a @ ( smap_l1139969084374327423list_a @ F @ A ) )
      = ( smap_l1139969084374327423list_a @ F @ ( stl_list_list_a @ A ) ) ) ).

% stream.map_sel(2)
thf(fact_514_stream_Omap__sel_I2_J,axiom,
    ! [F: list_a > list_list_a,A: stream_list_a] :
      ( ( stl_list_list_a @ ( smap_l5222851033613708939list_a @ F @ A ) )
      = ( smap_l5222851033613708939list_a @ F @ ( stl_list_a @ A ) ) ) ).

% stream.map_sel(2)
thf(fact_515_stream_Omap__sel_I2_J,axiom,
    ! [F: list_list_a > list_list_a,A: stream_list_list_a] :
      ( ( stl_list_list_a @ ( smap_l3391565068965508869list_a @ F @ A ) )
      = ( smap_l3391565068965508869list_a @ F @ ( stl_list_list_a @ A ) ) ) ).

% stream.map_sel(2)
thf(fact_516_stream_Omap__sel_I1_J,axiom,
    ! [F: list_a > list_a,A: stream_list_a] :
      ( ( shd_list_a @ ( smap_list_a_list_a @ F @ A ) )
      = ( F @ ( shd_list_a @ A ) ) ) ).

% stream.map_sel(1)
thf(fact_517_transpose_Ocases,axiom,
    ! [X2: list_list_a] :
      ( ( X2 != nil_list_a )
     => ( ! [Xss: list_list_a] :
            ( X2
           != ( cons_list_a @ nil_a @ Xss ) )
       => ~ ! [X4: a,Xs2: list_a,Xss: list_list_a] :
              ( X2
             != ( cons_list_a @ ( cons_a @ X4 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_518_list_Odistinct_I1_J,axiom,
    ! [X21: a,X222: list_a] :
      ( nil_a
     != ( cons_a @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_519_list_OdiscI,axiom,
    ! [List: list_a,X21: a,X222: list_a] :
      ( ( List
        = ( cons_a @ X21 @ X222 ) )
     => ( List != nil_a ) ) ).

% list.discI
thf(fact_520_list_Oexhaust,axiom,
    ! [Y: list_a] :
      ( ( Y != nil_a )
     => ~ ! [X212: a,X223: list_a] :
            ( Y
           != ( cons_a @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_521_remdups__adj_Ocases,axiom,
    ! [X2: list_a] :
      ( ( X2 != nil_a )
     => ( ! [X4: a] :
            ( X2
           != ( cons_a @ X4 @ nil_a ) )
       => ~ ! [X4: a,Y2: a,Xs2: list_a] :
              ( X2
             != ( cons_a @ X4 @ ( cons_a @ Y2 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_522_neq__Nil__conv,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
      = ( ? [Y4: a,Ys: list_a] :
            ( Xs
            = ( cons_a @ Y4 @ Ys ) ) ) ) ).

% neq_Nil_conv
thf(fact_523_list__induct2_H,axiom,
    ! [P: list_a > list_a > $o,Xs: list_a,Ys2: list_a] :
      ( ( P @ nil_a @ nil_a )
     => ( ! [X4: a,Xs2: list_a] : ( P @ ( cons_a @ X4 @ Xs2 ) @ nil_a )
       => ( ! [Y2: a,Ys3: list_a] : ( P @ nil_a @ ( cons_a @ Y2 @ Ys3 ) )
         => ( ! [X4: a,Xs2: list_a,Y2: a,Ys3: list_a] :
                ( ( P @ Xs2 @ Ys3 )
               => ( P @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y2 @ Ys3 ) ) )
           => ( P @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_524_list__nonempty__induct,axiom,
    ! [Xs: list_a,P: list_a > $o] :
      ( ( Xs != nil_a )
     => ( ! [X4: a] : ( P @ ( cons_a @ X4 @ nil_a ) )
       => ( ! [X4: a,Xs2: list_a] :
              ( ( Xs2 != nil_a )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_a @ X4 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_525_stream_Omap__ident__strong,axiom,
    ! [T2: stream_list_a,F: list_a > list_a] :
      ( ! [Z3: list_a] :
          ( ( member_list_a2 @ Z3 @ ( sset_list_a @ T2 ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( smap_list_a_list_a @ F @ T2 )
        = T2 ) ) ).

% stream.map_ident_strong
thf(fact_526_stream_Omap__ident__strong,axiom,
    ! [T2: stream_a,F: a > a] :
      ( ! [Z3: a] :
          ( ( member_a2 @ Z3 @ ( sset_a @ T2 ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( smap_a_a @ F @ T2 )
        = T2 ) ) ).

% stream.map_ident_strong
thf(fact_527_member__rec_I1_J,axiom,
    ! [X2: a,Xs: list_a,Y: a] :
      ( ( member_a @ ( cons_a @ X2 @ Xs ) @ Y )
      = ( ( X2 = Y )
        | ( member_a @ Xs @ Y ) ) ) ).

% member_rec(1)
thf(fact_528_smap2_Osimps_I2_J,axiom,
    ! [F: list_a > list_a > list_a,S1: stream_list_a,S22: stream_list_a] :
      ( ( stl_list_a @ ( smap2_6771825536533895878list_a @ F @ S1 @ S22 ) )
      = ( smap2_6771825536533895878list_a @ F @ ( stl_list_a @ S1 ) @ ( stl_list_a @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_529_smap2_Osimps_I2_J,axiom,
    ! [F: list_a > list_list_a > list_a,S1: stream_list_a,S22: stream_list_list_a] :
      ( ( stl_list_a @ ( smap2_9108955389640348608list_a @ F @ S1 @ S22 ) )
      = ( smap2_9108955389640348608list_a @ F @ ( stl_list_a @ S1 ) @ ( stl_list_list_a @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_530_smap2_Osimps_I2_J,axiom,
    ! [F: list_list_a > list_a > list_a,S1: stream_list_list_a,S22: stream_list_a] :
      ( ( stl_list_a @ ( smap2_9173189155220197580list_a @ F @ S1 @ S22 ) )
      = ( smap2_9173189155220197580list_a @ F @ ( stl_list_list_a @ S1 ) @ ( stl_list_a @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_531_smap2_Osimps_I2_J,axiom,
    ! [F: list_list_a > list_list_a > list_a,S1: stream_list_list_a,S22: stream_list_list_a] :
      ( ( stl_list_a @ ( smap2_7557085947764330182list_a @ F @ S1 @ S22 ) )
      = ( smap2_7557085947764330182list_a @ F @ ( stl_list_list_a @ S1 ) @ ( stl_list_list_a @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_532_smap2_Osimps_I2_J,axiom,
    ! [F: list_a > list_a > list_list_a,S1: stream_list_a,S22: stream_list_a] :
      ( ( stl_list_list_a @ ( smap2_3968465302024954316list_a @ F @ S1 @ S22 ) )
      = ( smap2_3968465302024954316list_a @ F @ ( stl_list_a @ S1 ) @ ( stl_list_a @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_533_smap2_Osimps_I2_J,axiom,
    ! [F: list_a > list_list_a > list_list_a,S1: stream_list_a,S22: stream_list_list_a] :
      ( ( stl_list_list_a @ ( smap2_1314303423829136838list_a @ F @ S1 @ S22 ) )
      = ( smap2_1314303423829136838list_a @ F @ ( stl_list_a @ S1 ) @ ( stl_list_list_a @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_534_smap2_Osimps_I2_J,axiom,
    ! [F: list_list_a > list_a > list_list_a,S1: stream_list_list_a,S22: stream_list_a] :
      ( ( stl_list_list_a @ ( smap2_2416595860148935890list_a @ F @ S1 @ S22 ) )
      = ( smap2_2416595860148935890list_a @ F @ ( stl_list_list_a @ S1 ) @ ( stl_list_a @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_535_smap2_Osimps_I2_J,axiom,
    ! [F: list_list_a > list_list_a > list_list_a,S1: stream_list_list_a,S22: stream_list_list_a] :
      ( ( stl_list_list_a @ ( smap2_6789832397309367756list_a @ F @ S1 @ S22 ) )
      = ( smap2_6789832397309367756list_a @ F @ ( stl_list_list_a @ S1 ) @ ( stl_list_list_a @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_536_smap2_Osimps_I1_J,axiom,
    ! [F: list_a > list_a > list_a,S1: stream_list_a,S22: stream_list_a] :
      ( ( shd_list_a @ ( smap2_6771825536533895878list_a @ F @ S1 @ S22 ) )
      = ( F @ ( shd_list_a @ S1 ) @ ( shd_list_a @ S22 ) ) ) ).

% smap2.simps(1)
thf(fact_537_tl__Nil,axiom,
    ! [Xs: list_a] :
      ( ( ( tl_a @ Xs )
        = nil_a )
      = ( ( Xs = nil_a )
        | ? [X3: a] :
            ( Xs
            = ( cons_a @ X3 @ nil_a ) ) ) ) ).

% tl_Nil
thf(fact_538_Nil__tl,axiom,
    ! [Xs: list_a] :
      ( ( nil_a
        = ( tl_a @ Xs ) )
      = ( ( Xs = nil_a )
        | ? [X3: a] :
            ( Xs
            = ( cons_a @ X3 @ nil_a ) ) ) ) ).

% Nil_tl
thf(fact_539_sinterleave_Osimps_I2_J,axiom,
    ! [S1: stream_list_a,S22: stream_list_a] :
      ( ( stl_list_a @ ( sinterleave_list_a @ S1 @ S22 ) )
      = ( sinterleave_list_a @ S22 @ ( stl_list_a @ S1 ) ) ) ).

% sinterleave.simps(2)
thf(fact_540_sinterleave_Osimps_I2_J,axiom,
    ! [S1: stream_list_list_a,S22: stream_list_list_a] :
      ( ( stl_list_list_a @ ( sinter664328039231274855list_a @ S1 @ S22 ) )
      = ( sinter664328039231274855list_a @ S22 @ ( stl_list_list_a @ S1 ) ) ) ).

% sinterleave.simps(2)
thf(fact_541_sinterleave_Osimps_I1_J,axiom,
    ! [S1: stream_list_a,S22: stream_list_a] :
      ( ( shd_list_a @ ( sinterleave_list_a @ S1 @ S22 ) )
      = ( shd_list_a @ S1 ) ) ).

% sinterleave.simps(1)
thf(fact_542_cycle__rotated,axiom,
    ! [V: list_a,U: list_a,S2: stream_a] :
      ( ( V != nil_a )
     => ( ( ( cycle_a @ U )
          = ( shift_a @ V @ S2 ) )
       => ( ( cycle_a @ ( append_a @ ( tl_a @ U ) @ ( cons_a @ ( hd_a @ U ) @ nil_a ) ) )
          = ( shift_a @ ( tl_a @ V ) @ S2 ) ) ) ) ).

% cycle_rotated
thf(fact_543_cycle_Ocode,axiom,
    ( cycle_a
    = ( ^ [Xs3: list_a] : ( sCons_a @ ( hd_a @ Xs3 ) @ ( cycle_a @ ( append_a @ ( tl_a @ Xs3 ) @ ( cons_a @ ( hd_a @ Xs3 ) @ nil_a ) ) ) ) ) ) ).

% cycle.code
thf(fact_544_cycle_Osimps_I2_J,axiom,
    ! [Xs: list_a] :
      ( ( stl_a @ ( cycle_a @ Xs ) )
      = ( cycle_a @ ( append_a @ ( tl_a @ Xs ) @ ( cons_a @ ( hd_a @ Xs ) @ nil_a ) ) ) ) ).

% cycle.simps(2)
thf(fact_545_cycle_Osimps_I2_J,axiom,
    ! [Xs: list_list_a] :
      ( ( stl_list_a @ ( cycle_list_a @ Xs ) )
      = ( cycle_list_a @ ( append_list_a @ ( tl_list_a @ Xs ) @ ( cons_list_a @ ( hd_list_a @ Xs ) @ nil_list_a ) ) ) ) ).

% cycle.simps(2)
thf(fact_546_cycle_Osimps_I2_J,axiom,
    ! [Xs: list_list_list_a] :
      ( ( stl_list_list_a @ ( cycle_list_list_a @ Xs ) )
      = ( cycle_list_list_a @ ( append_list_list_a @ ( tl_list_list_a @ Xs ) @ ( cons_list_list_a @ ( hd_list_list_a @ Xs ) @ nil_list_list_a ) ) ) ) ).

% cycle.simps(2)
thf(fact_547_insert__Nil,axiom,
    ! [X2: a] :
      ( ( insert_a @ X2 @ nil_a )
      = ( cons_a @ X2 @ nil_a ) ) ).

% insert_Nil
thf(fact_548_Cons__in__shuffles__iff,axiom,
    ! [Z2: a,Zs: list_a,Xs: list_a,Ys2: list_a] :
      ( ( member_list_a2 @ ( cons_a @ Z2 @ Zs ) @ ( shuffles_a @ Xs @ Ys2 ) )
      = ( ( ( Xs != nil_a )
          & ( ( hd_a @ Xs )
            = Z2 )
          & ( member_list_a2 @ Zs @ ( shuffles_a @ ( tl_a @ Xs ) @ Ys2 ) ) )
        | ( ( Ys2 != nil_a )
          & ( ( hd_a @ Ys2 )
            = Z2 )
          & ( member_list_a2 @ Zs @ ( shuffles_a @ Xs @ ( tl_a @ Ys2 ) ) ) ) ) ) ).

% Cons_in_shuffles_iff
thf(fact_549_cycle__Cons,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( cycle_a @ ( cons_a @ X2 @ Xs ) )
      = ( sCons_a @ X2 @ ( cycle_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) ) ) ) ).

% cycle_Cons
thf(fact_550_sset__cycle,axiom,
    ! [Xs: list_list_a] :
      ( ( Xs != nil_list_a )
     => ( ( sset_list_a @ ( cycle_list_a @ Xs ) )
        = ( set_list_a2 @ Xs ) ) ) ).

% sset_cycle
thf(fact_551_sset__cycle,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
     => ( ( sset_a @ ( cycle_a @ Xs ) )
        = ( set_a2 @ Xs ) ) ) ).

% sset_cycle
thf(fact_552_distinct__adj__Cons,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( distinct_adj_a @ ( cons_a @ X2 @ Xs ) )
      = ( ( Xs = nil_a )
        | ( ( X2
           != ( hd_a @ Xs ) )
          & ( distinct_adj_a @ Xs ) ) ) ) ).

% distinct_adj_Cons
thf(fact_553_append__is__Nil__conv,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( ( append_a @ Xs @ Ys2 )
        = nil_a )
      = ( ( Xs = nil_a )
        & ( Ys2 = nil_a ) ) ) ).

% append_is_Nil_conv
thf(fact_554_Nil__is__append__conv,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( nil_a
        = ( append_a @ Xs @ Ys2 ) )
      = ( ( Xs = nil_a )
        & ( Ys2 = nil_a ) ) ) ).

% Nil_is_append_conv
thf(fact_555_self__append__conv2,axiom,
    ! [Y: list_a,Xs: list_a] :
      ( ( Y
        = ( append_a @ Xs @ Y ) )
      = ( Xs = nil_a ) ) ).

% self_append_conv2
thf(fact_556_append__self__conv2,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( ( append_a @ Xs @ Ys2 )
        = Ys2 )
      = ( Xs = nil_a ) ) ).

% append_self_conv2
thf(fact_557_self__append__conv,axiom,
    ! [Y: list_a,Ys2: list_a] :
      ( ( Y
        = ( append_a @ Y @ Ys2 ) )
      = ( Ys2 = nil_a ) ) ).

% self_append_conv
thf(fact_558_append__self__conv,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( ( append_a @ Xs @ Ys2 )
        = Xs )
      = ( Ys2 = nil_a ) ) ).

% append_self_conv
thf(fact_559_append__Nil2,axiom,
    ! [Xs: list_a] :
      ( ( append_a @ Xs @ nil_a )
      = Xs ) ).

% append_Nil2
thf(fact_560_append_Oright__neutral,axiom,
    ! [A: list_a] :
      ( ( append_a @ A @ nil_a )
      = A ) ).

% append.right_neutral
thf(fact_561_Nil__in__shuffles,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( member_list_a2 @ nil_a @ ( shuffles_a @ Xs @ Ys2 ) )
      = ( ( Xs = nil_a )
        & ( Ys2 = nil_a ) ) ) ).

% Nil_in_shuffles
thf(fact_562_in__set__insert,axiom,
    ! [X2: list_a,Xs: list_list_a] :
      ( ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) )
     => ( ( insert_list_a @ X2 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_563_append1__eq__conv,axiom,
    ! [Xs: list_a,X2: a,Ys2: list_a,Y: a] :
      ( ( ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) )
        = ( append_a @ Ys2 @ ( cons_a @ Y @ nil_a ) ) )
      = ( ( Xs = Ys2 )
        & ( X2 = Y ) ) ) ).

% append1_eq_conv
thf(fact_564_tl__append2,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( Xs != nil_a )
     => ( ( tl_a @ ( append_a @ Xs @ Ys2 ) )
        = ( append_a @ ( tl_a @ Xs ) @ Ys2 ) ) ) ).

% tl_append2
thf(fact_565_hd__append2,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( Xs != nil_a )
     => ( ( hd_a @ ( append_a @ Xs @ Ys2 ) )
        = ( hd_a @ Xs ) ) ) ).

% hd_append2
thf(fact_566_not__in__set__insert,axiom,
    ! [X2: list_a,Xs: list_list_a] :
      ( ~ ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) )
     => ( ( insert_list_a @ X2 @ Xs )
        = ( cons_list_a @ X2 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_567_bind__simps_I2_J,axiom,
    ! [X2: a,Xs: list_a,F: a > list_a] :
      ( ( bind_a_a @ ( cons_a @ X2 @ Xs ) @ F )
      = ( append_a @ ( F @ X2 ) @ ( bind_a_a @ Xs @ F ) ) ) ).

% bind_simps(2)
thf(fact_568_split__list,axiom,
    ! [X2: list_a,Xs: list_list_a] :
      ( ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) )
     => ? [Ys3: list_list_a,Zs2: list_list_a] :
          ( Xs
          = ( append_list_a @ Ys3 @ ( cons_list_a @ X2 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_569_split__list__last,axiom,
    ! [X2: list_a,Xs: list_list_a] :
      ( ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) )
     => ? [Ys3: list_list_a,Zs2: list_list_a] :
          ( ( Xs
            = ( append_list_a @ Ys3 @ ( cons_list_a @ X2 @ Zs2 ) ) )
          & ~ ( member_list_a2 @ X2 @ ( set_list_a2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_570_split__list__first,axiom,
    ! [X2: list_a,Xs: list_list_a] :
      ( ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) )
     => ? [Ys3: list_list_a,Zs2: list_list_a] :
          ( ( Xs
            = ( append_list_a @ Ys3 @ ( cons_list_a @ X2 @ Zs2 ) ) )
          & ~ ( member_list_a2 @ X2 @ ( set_list_a2 @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_571_append__Cons__eq__iff,axiom,
    ! [X2: list_a,Xs: list_list_a,Ys2: list_list_a,Xs4: list_list_a,Ys4: list_list_a] :
      ( ~ ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) )
     => ( ~ ( member_list_a2 @ X2 @ ( set_list_a2 @ Ys2 ) )
       => ( ( ( append_list_a @ Xs @ ( cons_list_a @ X2 @ Ys2 ) )
            = ( append_list_a @ Xs4 @ ( cons_list_a @ X2 @ Ys4 ) ) )
          = ( ( Xs = Xs4 )
            & ( Ys2 = Ys4 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_572_in__set__conv__decomp,axiom,
    ! [X2: list_a,Xs: list_list_a] :
      ( ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) )
      = ( ? [Ys: list_list_a,Zs3: list_list_a] :
            ( Xs
            = ( append_list_a @ Ys @ ( cons_list_a @ X2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_573_in__set__conv__decomp__last,axiom,
    ! [X2: list_a,Xs: list_list_a] :
      ( ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) )
      = ( ? [Ys: list_list_a,Zs3: list_list_a] :
            ( ( Xs
              = ( append_list_a @ Ys @ ( cons_list_a @ X2 @ Zs3 ) ) )
            & ~ ( member_list_a2 @ X2 @ ( set_list_a2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_574_in__set__conv__decomp__first,axiom,
    ! [X2: list_a,Xs: list_list_a] :
      ( ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) )
      = ( ? [Ys: list_list_a,Zs3: list_list_a] :
            ( ( Xs
              = ( append_list_a @ Ys @ ( cons_list_a @ X2 @ Zs3 ) ) )
            & ~ ( member_list_a2 @ X2 @ ( set_list_a2 @ Ys ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_575_List_Oinsert__def,axiom,
    ( insert_list_a
    = ( ^ [X3: list_a,Xs3: list_list_a] : ( if_list_list_a @ ( member_list_a2 @ X3 @ ( set_list_a2 @ Xs3 ) ) @ Xs3 @ ( cons_list_a @ X3 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_576_eq__Nil__appendI,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( Xs = Ys2 )
     => ( Xs
        = ( append_a @ nil_a @ Ys2 ) ) ) ).

% eq_Nil_appendI
thf(fact_577_append_Oleft__neutral,axiom,
    ! [A: list_a] :
      ( ( append_a @ nil_a @ A )
      = A ) ).

% append.left_neutral
thf(fact_578_append__Nil,axiom,
    ! [Ys2: list_a] :
      ( ( append_a @ nil_a @ Ys2 )
      = Ys2 ) ).

% append_Nil
thf(fact_579_set__ConsD,axiom,
    ! [Y: list_a,X2: list_a,Xs: list_list_a] :
      ( ( member_list_a2 @ Y @ ( set_list_a2 @ ( cons_list_a @ X2 @ Xs ) ) )
     => ( ( Y = X2 )
        | ( member_list_a2 @ Y @ ( set_list_a2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_580_list_Oset__cases,axiom,
    ! [E: list_a,A: list_list_a] :
      ( ( member_list_a2 @ E @ ( set_list_a2 @ A ) )
     => ( ! [Z22: list_list_a] :
            ( A
           != ( cons_list_a @ E @ Z22 ) )
       => ~ ! [Z1: list_a,Z22: list_list_a] :
              ( ( A
                = ( cons_list_a @ Z1 @ Z22 ) )
             => ~ ( member_list_a2 @ E @ ( set_list_a2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_581_list_Oset__intros_I1_J,axiom,
    ! [X21: list_a,X222: list_list_a] : ( member_list_a2 @ X21 @ ( set_list_a2 @ ( cons_list_a @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_582_list_Oset__intros_I2_J,axiom,
    ! [Y: list_a,X222: list_list_a,X21: list_a] :
      ( ( member_list_a2 @ Y @ ( set_list_a2 @ X222 ) )
     => ( member_list_a2 @ Y @ ( set_list_a2 @ ( cons_list_a @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_583_subset__code_I1_J,axiom,
    ! [Xs: list_list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B )
      = ( ! [X3: list_a] :
            ( ( member_list_a2 @ X3 @ ( set_list_a2 @ Xs ) )
           => ( member_list_a2 @ X3 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_584_subset__code_I1_J,axiom,
    ! [Xs: list_a,B: set_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B )
      = ( ! [X3: a] :
            ( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
           => ( member_a2 @ X3 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_585_Cons__in__shuffles__rightI,axiom,
    ! [Zs: list_a,Xs: list_a,Ys2: list_a,Z2: a] :
      ( ( member_list_a2 @ Zs @ ( shuffles_a @ Xs @ Ys2 ) )
     => ( member_list_a2 @ ( cons_a @ Z2 @ Zs ) @ ( shuffles_a @ Xs @ ( cons_a @ Z2 @ Ys2 ) ) ) ) ).

% Cons_in_shuffles_rightI
thf(fact_586_Cons__in__shuffles__leftI,axiom,
    ! [Zs: list_a,Xs: list_a,Ys2: list_a,Z2: a] :
      ( ( member_list_a2 @ Zs @ ( shuffles_a @ Xs @ Ys2 ) )
     => ( member_list_a2 @ ( cons_a @ Z2 @ Zs ) @ ( shuffles_a @ ( cons_a @ Z2 @ Xs ) @ Ys2 ) ) ) ).

% Cons_in_shuffles_leftI
thf(fact_587_Nil__in__shufflesI,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( Xs = nil_a )
     => ( ( Ys2 = nil_a )
       => ( member_list_a2 @ nil_a @ ( shuffles_a @ Xs @ Ys2 ) ) ) ) ).

% Nil_in_shufflesI
thf(fact_588_distinct__adj__Nil,axiom,
    distinct_adj_a @ nil_a ).

% distinct_adj_Nil
thf(fact_589_list__bind__cong,axiom,
    ! [Xs: list_a,Ys2: list_a,F: a > list_a,G2: a > list_a] :
      ( ( Xs = Ys2 )
     => ( ! [X4: a] :
            ( ( member_a2 @ X4 @ ( set_a2 @ Xs ) )
           => ( ( F @ X4 )
              = ( G2 @ X4 ) ) )
       => ( ( bind_a_a @ Xs @ F )
          = ( bind_a_a @ Ys2 @ G2 ) ) ) ) ).

% list_bind_cong
thf(fact_590_list__ex1__iff,axiom,
    ( list_ex1_list_a
    = ( ^ [P2: list_a > $o,Xs3: list_list_a] :
        ? [X3: list_a] :
          ( ( member_list_a2 @ X3 @ ( set_list_a2 @ Xs3 ) )
          & ( P2 @ X3 )
          & ! [Y4: list_a] :
              ( ( ( member_list_a2 @ Y4 @ ( set_list_a2 @ Xs3 ) )
                & ( P2 @ Y4 ) )
             => ( Y4 = X3 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_591_list__ex1__iff,axiom,
    ( list_ex1_a
    = ( ^ [P2: a > $o,Xs3: list_a] :
        ? [X3: a] :
          ( ( member_a2 @ X3 @ ( set_a2 @ Xs3 ) )
          & ( P2 @ X3 )
          & ! [Y4: a] :
              ( ( ( member_a2 @ Y4 @ ( set_a2 @ Xs3 ) )
                & ( P2 @ Y4 ) )
             => ( Y4 = X3 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_592_in__set__member,axiom,
    ! [X2: list_a,Xs: list_list_a] :
      ( ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) )
      = ( member_list_a @ Xs @ X2 ) ) ).

% in_set_member
thf(fact_593_in__set__member,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
      = ( member_a @ Xs @ X2 ) ) ).

% in_set_member
thf(fact_594_rev__nonempty__induct,axiom,
    ! [Xs: list_a,P: list_a > $o] :
      ( ( Xs != nil_a )
     => ( ! [X4: a] : ( P @ ( cons_a @ X4 @ nil_a ) )
       => ( ! [X4: a,Xs2: list_a] :
              ( ( Xs2 != nil_a )
             => ( ( P @ Xs2 )
               => ( P @ ( append_a @ Xs2 @ ( cons_a @ X4 @ nil_a ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_595_append__eq__Cons__conv,axiom,
    ! [Ys2: list_a,Zs: list_a,X2: a,Xs: list_a] :
      ( ( ( append_a @ Ys2 @ Zs )
        = ( cons_a @ X2 @ Xs ) )
      = ( ( ( Ys2 = nil_a )
          & ( Zs
            = ( cons_a @ X2 @ Xs ) ) )
        | ? [Ys5: list_a] :
            ( ( Ys2
              = ( cons_a @ X2 @ Ys5 ) )
            & ( ( append_a @ Ys5 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_596_Cons__eq__append__conv,axiom,
    ! [X2: a,Xs: list_a,Ys2: list_a,Zs: list_a] :
      ( ( ( cons_a @ X2 @ Xs )
        = ( append_a @ Ys2 @ Zs ) )
      = ( ( ( Ys2 = nil_a )
          & ( ( cons_a @ X2 @ Xs )
            = Zs ) )
        | ? [Ys5: list_a] :
            ( ( ( cons_a @ X2 @ Ys5 )
              = Ys2 )
            & ( Xs
              = ( append_a @ Ys5 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_597_rev__exhaust,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
     => ~ ! [Ys3: list_a,Y2: a] :
            ( Xs
           != ( append_a @ Ys3 @ ( cons_a @ Y2 @ nil_a ) ) ) ) ).

% rev_exhaust
thf(fact_598_rev__induct,axiom,
    ! [P: list_a > $o,Xs: list_a] :
      ( ( P @ nil_a )
     => ( ! [X4: a,Xs2: list_a] :
            ( ( P @ Xs2 )
           => ( P @ ( append_a @ Xs2 @ ( cons_a @ X4 @ nil_a ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_599_set__subset__Cons,axiom,
    ! [Xs: list_list_a,X2: list_a] : ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ ( cons_list_a @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_600_set__subset__Cons,axiom,
    ! [Xs: list_a,X2: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_601_tl__append__if,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( ( Xs = nil_a )
       => ( ( tl_a @ ( append_a @ Xs @ Ys2 ) )
          = ( tl_a @ Ys2 ) ) )
      & ( ( Xs != nil_a )
       => ( ( tl_a @ ( append_a @ Xs @ Ys2 ) )
          = ( append_a @ ( tl_a @ Xs ) @ Ys2 ) ) ) ) ).

% tl_append_if
thf(fact_602_hd__append,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( ( Xs = nil_a )
       => ( ( hd_a @ ( append_a @ Xs @ Ys2 ) )
          = ( hd_a @ Ys2 ) ) )
      & ( ( Xs != nil_a )
       => ( ( hd_a @ ( append_a @ Xs @ Ys2 ) )
          = ( hd_a @ Xs ) ) ) ) ).

% hd_append
thf(fact_603_longest__common__prefix,axiom,
    ! [Xs: list_a,Ys2: list_a] :
    ? [Ps: list_a,Xs5: list_a,Ys6: list_a] :
      ( ( Xs
        = ( append_a @ Ps @ Xs5 ) )
      & ( Ys2
        = ( append_a @ Ps @ Ys6 ) )
      & ( ( Xs5 = nil_a )
        | ( Ys6 = nil_a )
        | ( ( hd_a @ Xs5 )
         != ( hd_a @ Ys6 ) ) ) ) ).

% longest_common_prefix
thf(fact_604_list_Oset__sel_I2_J,axiom,
    ! [A: list_list_a,X2: list_a] :
      ( ( A != nil_list_a )
     => ( ( member_list_a2 @ X2 @ ( set_list_a2 @ ( tl_list_a @ A ) ) )
       => ( member_list_a2 @ X2 @ ( set_list_a2 @ A ) ) ) ) ).

% list.set_sel(2)
thf(fact_605_list_Oset__sel_I2_J,axiom,
    ! [A: list_a,X2: a] :
      ( ( A != nil_a )
     => ( ( member_a2 @ X2 @ ( set_a2 @ ( tl_a @ A ) ) )
       => ( member_a2 @ X2 @ ( set_a2 @ A ) ) ) ) ).

% list.set_sel(2)
thf(fact_606_list_Oset__sel_I1_J,axiom,
    ! [A: list_list_a] :
      ( ( A != nil_list_a )
     => ( member_list_a2 @ ( hd_list_a @ A ) @ ( set_list_a2 @ A ) ) ) ).

% list.set_sel(1)
thf(fact_607_list_Oset__sel_I1_J,axiom,
    ! [A: list_a] :
      ( ( A != nil_a )
     => ( member_a2 @ ( hd_a @ A ) @ ( set_a2 @ A ) ) ) ).

% list.set_sel(1)
thf(fact_608_hd__in__set,axiom,
    ! [Xs: list_list_a] :
      ( ( Xs != nil_list_a )
     => ( member_list_a2 @ ( hd_list_a @ Xs ) @ ( set_list_a2 @ Xs ) ) ) ).

% hd_in_set
thf(fact_609_hd__in__set,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
     => ( member_a2 @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).

% hd_in_set
thf(fact_610_shufflesE,axiom,
    ! [Zs: list_a,Xs: list_a,Ys2: list_a] :
      ( ( member_list_a2 @ Zs @ ( shuffles_a @ Xs @ Ys2 ) )
     => ( ( ( Zs = Xs )
         => ( Ys2 != nil_a ) )
       => ( ( ( Zs = Ys2 )
           => ( Xs != nil_a ) )
         => ( ! [X4: a,Xs5: list_a] :
                ( ( Xs
                  = ( cons_a @ X4 @ Xs5 ) )
               => ! [Z3: a,Zs4: list_a] :
                    ( ( Zs
                      = ( cons_a @ Z3 @ Zs4 ) )
                   => ( ( X4 = Z3 )
                     => ~ ( member_list_a2 @ Zs4 @ ( shuffles_a @ Xs5 @ Ys2 ) ) ) ) )
           => ~ ! [Y2: a,Ys6: list_a] :
                  ( ( Ys2
                    = ( cons_a @ Y2 @ Ys6 ) )
                 => ! [Z3: a,Zs4: list_a] :
                      ( ( Zs
                        = ( cons_a @ Z3 @ Zs4 ) )
                     => ( ( Y2 = Z3 )
                       => ~ ( member_list_a2 @ Zs4 @ ( shuffles_a @ Xs @ Ys6 ) ) ) ) ) ) ) ) ) ).

% shufflesE
thf(fact_611_distinct__adj__singleton,axiom,
    ! [X2: a] : ( distinct_adj_a @ ( cons_a @ X2 @ nil_a ) ) ).

% distinct_adj_singleton
thf(fact_612_can__select__set__list__ex1,axiom,
    ! [P: a > $o,A2: list_a] :
      ( ( can_select_a @ P @ ( set_a2 @ A2 ) )
      = ( list_ex1_a @ P @ A2 ) ) ).

% can_select_set_list_ex1
thf(fact_613_the__elem__set,axiom,
    ! [X2: a] :
      ( ( the_elem_a @ ( set_a2 @ ( cons_a @ X2 @ nil_a ) ) )
      = X2 ) ).

% the_elem_set
thf(fact_614_rotate1__hd__tl,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
     => ( ( rotate1_a @ Xs )
        = ( append_a @ ( tl_a @ Xs ) @ ( cons_a @ ( hd_a @ Xs ) @ nil_a ) ) ) ) ).

% rotate1_hd_tl
thf(fact_615_product__lists_Osimps_I1_J,axiom,
    ( ( product_lists_a @ nil_list_a )
    = ( cons_list_a @ nil_a @ nil_list_a ) ) ).

% product_lists.simps(1)
thf(fact_616_subseqs_Osimps_I1_J,axiom,
    ( ( subseqs_a @ nil_a )
    = ( cons_list_a @ nil_a @ nil_list_a ) ) ).

% subseqs.simps(1)
thf(fact_617_subset__code_I3_J,axiom,
    ~ ( ord_le8861187494160871172list_a @ ( coset_list_a @ nil_list_a ) @ ( set_list_a2 @ nil_list_a ) ) ).

% subset_code(3)
thf(fact_618_subset__code_I3_J,axiom,
    ~ ( ord_less_eq_set_a @ ( coset_a @ nil_a ) @ ( set_a2 @ nil_a ) ) ).

% subset_code(3)
thf(fact_619_distinct__adj__append__iff,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( distinct_adj_a @ ( append_a @ Xs @ Ys2 ) )
      = ( ( distinct_adj_a @ Xs )
        & ( distinct_adj_a @ Ys2 )
        & ( ( Xs = nil_a )
          | ( Ys2 = nil_a )
          | ( ( last_a @ Xs )
           != ( hd_a @ Ys2 ) ) ) ) ) ).

% distinct_adj_append_iff
thf(fact_620_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_a] :
      ( ( ( rotate1_a @ Xs )
        = nil_a )
      = ( Xs = nil_a ) ) ).

% rotate1_is_Nil_conv
thf(fact_621_last__appendR,axiom,
    ! [Ys2: list_a,Xs: list_a] :
      ( ( Ys2 != nil_a )
     => ( ( last_a @ ( append_a @ Xs @ Ys2 ) )
        = ( last_a @ Ys2 ) ) ) ).

% last_appendR
thf(fact_622_last__appendL,axiom,
    ! [Ys2: list_a,Xs: list_a] :
      ( ( Ys2 = nil_a )
     => ( ( last_a @ ( append_a @ Xs @ Ys2 ) )
        = ( last_a @ Xs ) ) ) ).

% last_appendL
thf(fact_623_last__snoc,axiom,
    ! [Xs: list_a,X2: a] :
      ( ( last_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) )
      = X2 ) ).

% last_snoc
thf(fact_624_subseqs__refl,axiom,
    ! [Xs: list_a] : ( member_list_a2 @ Xs @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) ) ).

% subseqs_refl
thf(fact_625_can__select__def,axiom,
    ( can_select_list_a
    = ( ^ [P2: list_a > $o,A3: set_list_a] :
        ? [X3: list_a] :
          ( ( member_list_a2 @ X3 @ A3 )
          & ( P2 @ X3 )
          & ! [Y4: list_a] :
              ( ( ( member_list_a2 @ Y4 @ A3 )
                & ( P2 @ Y4 ) )
             => ( Y4 = X3 ) ) ) ) ) ).

% can_select_def
thf(fact_626_rotate1_Osimps_I1_J,axiom,
    ( ( rotate1_a @ nil_a )
    = nil_a ) ).

% rotate1.simps(1)
thf(fact_627_Cons__in__subseqsD,axiom,
    ! [Y: a,Ys2: list_a,Xs: list_a] :
      ( ( member_list_a2 @ ( cons_a @ Y @ Ys2 ) @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) )
     => ( member_list_a2 @ Ys2 @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_628_last__ConsR,axiom,
    ! [Xs: list_a,X2: a] :
      ( ( Xs != nil_a )
     => ( ( last_a @ ( cons_a @ X2 @ Xs ) )
        = ( last_a @ Xs ) ) ) ).

% last_ConsR
thf(fact_629_last__ConsL,axiom,
    ! [Xs: list_a,X2: a] :
      ( ( Xs = nil_a )
     => ( ( last_a @ ( cons_a @ X2 @ Xs ) )
        = X2 ) ) ).

% last_ConsL
thf(fact_630_last_Osimps,axiom,
    ! [Xs: list_a,X2: a] :
      ( ( ( Xs = nil_a )
       => ( ( last_a @ ( cons_a @ X2 @ Xs ) )
          = X2 ) )
      & ( ( Xs != nil_a )
       => ( ( last_a @ ( cons_a @ X2 @ Xs ) )
          = ( last_a @ Xs ) ) ) ) ).

% last.simps
thf(fact_631_last__in__set,axiom,
    ! [As: list_list_a] :
      ( ( As != nil_list_a )
     => ( member_list_a2 @ ( last_list_a @ As ) @ ( set_list_a2 @ As ) ) ) ).

% last_in_set
thf(fact_632_last__in__set,axiom,
    ! [As: list_a] :
      ( ( As != nil_a )
     => ( member_a2 @ ( last_a @ As ) @ ( set_a2 @ As ) ) ) ).

% last_in_set
thf(fact_633_last__append,axiom,
    ! [Ys2: list_a,Xs: list_a] :
      ( ( ( Ys2 = nil_a )
       => ( ( last_a @ ( append_a @ Xs @ Ys2 ) )
          = ( last_a @ Xs ) ) )
      & ( ( Ys2 != nil_a )
       => ( ( last_a @ ( append_a @ Xs @ Ys2 ) )
          = ( last_a @ Ys2 ) ) ) ) ).

% last_append
thf(fact_634_longest__common__suffix,axiom,
    ! [Xs: list_a,Ys2: list_a] :
    ? [Ss: list_a,Xs5: list_a,Ys6: list_a] :
      ( ( Xs
        = ( append_a @ Xs5 @ Ss ) )
      & ( Ys2
        = ( append_a @ Ys6 @ Ss ) )
      & ( ( Xs5 = nil_a )
        | ( Ys6 = nil_a )
        | ( ( last_a @ Xs5 )
         != ( last_a @ Ys6 ) ) ) ) ).

% longest_common_suffix
thf(fact_635_hd__Nil__eq__last,axiom,
    ( ( hd_a @ nil_a )
    = ( last_a @ nil_a ) ) ).

% hd_Nil_eq_last
thf(fact_636_last__tl,axiom,
    ! [Xs: list_a] :
      ( ( ( Xs = nil_a )
        | ( ( tl_a @ Xs )
         != nil_a ) )
     => ( ( last_a @ ( tl_a @ Xs ) )
        = ( last_a @ Xs ) ) ) ).

% last_tl
thf(fact_637_subset__code_I2_J,axiom,
    ! [A2: set_list_a,Ys2: list_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ ( coset_list_a @ Ys2 ) )
      = ( ! [X3: list_a] :
            ( ( member_list_a2 @ X3 @ ( set_list_a2 @ Ys2 ) )
           => ~ ( member_list_a2 @ X3 @ A2 ) ) ) ) ).

% subset_code(2)
thf(fact_638_subset__code_I2_J,axiom,
    ! [A2: set_a,Ys2: list_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( coset_a @ Ys2 ) )
      = ( ! [X3: a] :
            ( ( member_a2 @ X3 @ ( set_a2 @ Ys2 ) )
           => ~ ( member_a2 @ X3 @ A2 ) ) ) ) ).

% subset_code(2)
thf(fact_639_rotate1_Osimps_I2_J,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( rotate1_a @ ( cons_a @ X2 @ Xs ) )
      = ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) ) ).

% rotate1.simps(2)
thf(fact_640_append__butlast__last__id,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
     => ( ( append_a @ ( butlast_a @ Xs ) @ ( cons_a @ ( last_a @ Xs ) @ nil_a ) )
        = Xs ) ) ).

% append_butlast_last_id
thf(fact_641_snoc__eq__iff__butlast,axiom,
    ! [Xs: list_a,X2: a,Ys2: list_a] :
      ( ( ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) )
        = Ys2 )
      = ( ( Ys2 != nil_a )
        & ( ( butlast_a @ Ys2 )
          = Xs )
        & ( ( last_a @ Ys2 )
          = X2 ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_642_concat__eq__append__conv,axiom,
    ! [Xss2: list_list_a,Ys2: list_a,Zs: list_a] :
      ( ( ( concat_a @ Xss2 )
        = ( append_a @ Ys2 @ Zs ) )
      = ( ( ( Xss2 = nil_list_a )
         => ( ( Ys2 = nil_a )
            & ( Zs = nil_a ) ) )
        & ( ( Xss2 != nil_list_a )
         => ? [Xss1: list_list_a,Xs3: list_a,Xs6: list_a,Xss22: list_list_a] :
              ( ( Xss2
                = ( append_list_a @ Xss1 @ ( cons_list_a @ ( append_a @ Xs3 @ Xs6 ) @ Xss22 ) ) )
              & ( Ys2
                = ( append_a @ ( concat_a @ Xss1 ) @ Xs3 ) )
              & ( Zs
                = ( append_a @ Xs6 @ ( concat_a @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_643_successively__append__iff,axiom,
    ! [P: a > a > $o,Xs: list_a,Ys2: list_a] :
      ( ( successively_a @ P @ ( append_a @ Xs @ Ys2 ) )
      = ( ( successively_a @ P @ Xs )
        & ( successively_a @ P @ Ys2 )
        & ( ( Xs = nil_a )
          | ( Ys2 = nil_a )
          | ( P @ ( last_a @ Xs ) @ ( hd_a @ Ys2 ) ) ) ) ) ).

% successively_append_iff
thf(fact_644_butlast__snoc,axiom,
    ! [Xs: list_a,X2: a] :
      ( ( butlast_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) )
      = Xs ) ).

% butlast_snoc
thf(fact_645_remdups__adj__append_H,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( ( Xs = nil_a )
        | ( Ys2 = nil_a )
        | ( ( last_a @ Xs )
         != ( hd_a @ Ys2 ) ) )
     => ( ( remdups_adj_a @ ( append_a @ Xs @ Ys2 ) )
        = ( append_a @ ( remdups_adj_a @ Xs ) @ ( remdups_adj_a @ Ys2 ) ) ) ) ).

% remdups_adj_append'
thf(fact_646_member__remove,axiom,
    ! [X2: list_a,Y: list_a,A2: set_list_a] :
      ( ( member_list_a2 @ X2 @ ( remove_list_a @ Y @ A2 ) )
      = ( ( member_list_a2 @ X2 @ A2 )
        & ( X2 != Y ) ) ) ).

% member_remove
thf(fact_647_remdups__adj__Nil__iff,axiom,
    ! [Xs: list_a] :
      ( ( ( remdups_adj_a @ Xs )
        = nil_a )
      = ( Xs = nil_a ) ) ).

% remdups_adj_Nil_iff
thf(fact_648_Nil__eq__concat__conv,axiom,
    ! [Xss2: list_list_a] :
      ( ( nil_a
        = ( concat_a @ Xss2 ) )
      = ( ! [X3: list_a] :
            ( ( member_list_a2 @ X3 @ ( set_list_a2 @ Xss2 ) )
           => ( X3 = nil_a ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_649_concat__eq__Nil__conv,axiom,
    ! [Xss2: list_list_a] :
      ( ( ( concat_a @ Xss2 )
        = nil_a )
      = ( ! [X3: list_a] :
            ( ( member_list_a2 @ X3 @ ( set_list_a2 @ Xss2 ) )
           => ( X3 = nil_a ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_650_successively__remdups__adj__iff,axiom,
    ! [Xs: list_list_a,P: list_a > list_a > $o] :
      ( ! [X4: list_a] :
          ( ( member_list_a2 @ X4 @ ( set_list_a2 @ Xs ) )
         => ( P @ X4 @ X4 ) )
     => ( ( successively_list_a @ P @ ( remdups_adj_list_a @ Xs ) )
        = ( successively_list_a @ P @ Xs ) ) ) ).

% successively_remdups_adj_iff
thf(fact_651_remdups__adj_Osimps_I1_J,axiom,
    ( ( remdups_adj_a @ nil_a )
    = nil_a ) ).

% remdups_adj.simps(1)
thf(fact_652_successively_Osimps_I1_J,axiom,
    ! [P: a > a > $o] : ( successively_a @ P @ nil_a ) ).

% successively.simps(1)
thf(fact_653_successively__mono,axiom,
    ! [P: list_a > list_a > $o,Xs: list_list_a,Q: list_a > list_a > $o] :
      ( ( successively_list_a @ P @ Xs )
     => ( ! [X4: list_a,Y2: list_a] :
            ( ( member_list_a2 @ X4 @ ( set_list_a2 @ Xs ) )
           => ( ( member_list_a2 @ Y2 @ ( set_list_a2 @ Xs ) )
             => ( ( P @ X4 @ Y2 )
               => ( Q @ X4 @ Y2 ) ) ) )
       => ( successively_list_a @ Q @ Xs ) ) ) ).

% successively_mono
thf(fact_654_successively__cong,axiom,
    ! [Xs: list_list_a,P: list_a > list_a > $o,Q: list_a > list_a > $o,Ys2: list_list_a] :
      ( ! [X4: list_a,Y2: list_a] :
          ( ( member_list_a2 @ X4 @ ( set_list_a2 @ Xs ) )
         => ( ( member_list_a2 @ Y2 @ ( set_list_a2 @ Xs ) )
           => ( ( P @ X4 @ Y2 )
              = ( Q @ X4 @ Y2 ) ) ) )
     => ( ( Xs = Ys2 )
       => ( ( successively_list_a @ P @ Xs )
          = ( successively_list_a @ Q @ Ys2 ) ) ) ) ).

% successively_cong
thf(fact_655_butlast_Osimps_I1_J,axiom,
    ( ( butlast_a @ nil_a )
    = nil_a ) ).

% butlast.simps(1)
thf(fact_656_in__set__butlastD,axiom,
    ! [X2: list_a,Xs: list_list_a] :
      ( ( member_list_a2 @ X2 @ ( set_list_a2 @ ( butlast_list_a @ Xs ) ) )
     => ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) ) ) ).

% in_set_butlastD
thf(fact_657_remdups__adj_Osimps_I2_J,axiom,
    ! [X2: a] :
      ( ( remdups_adj_a @ ( cons_a @ X2 @ nil_a ) )
      = ( cons_a @ X2 @ nil_a ) ) ).

% remdups_adj.simps(2)
thf(fact_658_remdups__adj_Oelims,axiom,
    ! [X2: list_a,Y: list_a] :
      ( ( ( remdups_adj_a @ X2 )
        = Y )
     => ( ( ( X2 = nil_a )
         => ( Y != nil_a ) )
       => ( ! [X4: a] :
              ( ( X2
                = ( cons_a @ X4 @ nil_a ) )
             => ( Y
               != ( cons_a @ X4 @ nil_a ) ) )
         => ~ ! [X4: a,Y2: a,Xs2: list_a] :
                ( ( X2
                  = ( cons_a @ X4 @ ( cons_a @ Y2 @ Xs2 ) ) )
               => ~ ( ( ( X4 = Y2 )
                     => ( Y
                        = ( remdups_adj_a @ ( cons_a @ X4 @ Xs2 ) ) ) )
                    & ( ( X4 != Y2 )
                     => ( Y
                        = ( cons_a @ X4 @ ( remdups_adj_a @ ( cons_a @ Y2 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).

% remdups_adj.elims
thf(fact_659_successively_Oelims_I2_J,axiom,
    ! [X2: a > a > $o,Xa2: list_a] :
      ( ( successively_a @ X2 @ Xa2 )
     => ( ( Xa2 != nil_a )
       => ( ! [X4: a] :
              ( Xa2
             != ( cons_a @ X4 @ nil_a ) )
         => ~ ! [X4: a,Y2: a,Xs2: list_a] :
                ( ( Xa2
                  = ( cons_a @ X4 @ ( cons_a @ Y2 @ Xs2 ) ) )
               => ~ ( ( X2 @ X4 @ Y2 )
                    & ( successively_a @ X2 @ ( cons_a @ Y2 @ Xs2 ) ) ) ) ) ) ) ).

% successively.elims(2)
thf(fact_660_successively_Oelims_I1_J,axiom,
    ! [X2: a > a > $o,Xa2: list_a,Y: $o] :
      ( ( ( successively_a @ X2 @ Xa2 )
        = Y )
     => ( ( ( Xa2 = nil_a )
         => ~ Y )
       => ( ( ? [X4: a] :
                ( Xa2
                = ( cons_a @ X4 @ nil_a ) )
           => ~ Y )
         => ~ ! [X4: a,Y2: a,Xs2: list_a] :
                ( ( Xa2
                  = ( cons_a @ X4 @ ( cons_a @ Y2 @ Xs2 ) ) )
               => ( Y
                  = ( ~ ( ( X2 @ X4 @ Y2 )
                        & ( successively_a @ X2 @ ( cons_a @ Y2 @ Xs2 ) ) ) ) ) ) ) ) ) ).

% successively.elims(1)
thf(fact_661_successively_Osimps_I2_J,axiom,
    ! [P: a > a > $o,X2: a] : ( successively_a @ P @ ( cons_a @ X2 @ nil_a ) ) ).

% successively.simps(2)
thf(fact_662_concat_Osimps_I1_J,axiom,
    ( ( concat_a @ nil_list_a )
    = nil_a ) ).

% concat.simps(1)
thf(fact_663_butlast_Osimps_I2_J,axiom,
    ! [Xs: list_a,X2: a] :
      ( ( ( Xs = nil_a )
       => ( ( butlast_a @ ( cons_a @ X2 @ Xs ) )
          = nil_a ) )
      & ( ( Xs != nil_a )
       => ( ( butlast_a @ ( cons_a @ X2 @ Xs ) )
          = ( cons_a @ X2 @ ( butlast_a @ Xs ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_664_butlast__append,axiom,
    ! [Ys2: list_a,Xs: list_a] :
      ( ( ( Ys2 = nil_a )
       => ( ( butlast_a @ ( append_a @ Xs @ Ys2 ) )
          = ( butlast_a @ Xs ) ) )
      & ( ( Ys2 != nil_a )
       => ( ( butlast_a @ ( append_a @ Xs @ Ys2 ) )
          = ( append_a @ Xs @ ( butlast_a @ Ys2 ) ) ) ) ) ).

% butlast_append
thf(fact_665_in__set__butlast__appendI,axiom,
    ! [X2: list_a,Xs: list_list_a,Ys2: list_list_a] :
      ( ( ( member_list_a2 @ X2 @ ( set_list_a2 @ ( butlast_list_a @ Xs ) ) )
        | ( member_list_a2 @ X2 @ ( set_list_a2 @ ( butlast_list_a @ Ys2 ) ) ) )
     => ( member_list_a2 @ X2 @ ( set_list_a2 @ ( butlast_list_a @ ( append_list_a @ Xs @ Ys2 ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_666_remdups__adj__append__two,axiom,
    ! [Xs: list_a,X2: a,Y: a] :
      ( ( remdups_adj_a @ ( append_a @ Xs @ ( cons_a @ X2 @ ( cons_a @ Y @ nil_a ) ) ) )
      = ( append_a @ ( remdups_adj_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) ) @ ( if_list_a @ ( X2 = Y ) @ nil_a @ ( cons_a @ Y @ nil_a ) ) ) ) ).

% remdups_adj_append_two
thf(fact_667_successively__Cons,axiom,
    ! [P: a > a > $o,X2: a,Xs: list_a] :
      ( ( successively_a @ P @ ( cons_a @ X2 @ Xs ) )
      = ( ( Xs = nil_a )
        | ( ( P @ X2 @ ( hd_a @ Xs ) )
          & ( successively_a @ P @ Xs ) ) ) ) ).

% successively_Cons
thf(fact_668_hd__concat,axiom,
    ! [Xs: list_list_a] :
      ( ( Xs != nil_list_a )
     => ( ( ( hd_list_a @ Xs )
         != nil_a )
       => ( ( hd_a @ ( concat_a @ Xs ) )
          = ( hd_a @ ( hd_list_a @ Xs ) ) ) ) ) ).

% hd_concat
thf(fact_669_remdups__adj__append,axiom,
    ! [Xs_1: list_a,X2: a,Xs_2: list_a] :
      ( ( remdups_adj_a @ ( append_a @ Xs_1 @ ( cons_a @ X2 @ Xs_2 ) ) )
      = ( append_a @ ( remdups_adj_a @ ( append_a @ Xs_1 @ ( cons_a @ X2 @ nil_a ) ) ) @ ( tl_a @ ( remdups_adj_a @ ( cons_a @ X2 @ Xs_2 ) ) ) ) ) ).

% remdups_adj_append
thf(fact_670_subset__subseqs,axiom,
    ! [X6: set_list_a,Xs: list_list_a] :
      ( ( ord_le8861187494160871172list_a @ X6 @ ( set_list_a2 @ Xs ) )
     => ( member_set_list_a @ X6 @ ( image_432481560377026271list_a @ set_list_a2 @ ( set_list_list_a2 @ ( subseqs_list_a @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_671_subset__subseqs,axiom,
    ! [X6: set_a,Xs: list_a] :
      ( ( ord_less_eq_set_a @ X6 @ ( set_a2 @ Xs ) )
     => ( member_set_a2 @ X6 @ ( image_list_a_set_a @ set_a2 @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_672_transpose__empty,axiom,
    ! [Xs: list_list_a] :
      ( ( ( transpose_a @ Xs )
        = nil_list_a )
      = ( ! [X3: list_a] :
            ( ( member_list_a2 @ X3 @ ( set_list_a2 @ Xs ) )
           => ( X3 = nil_a ) ) ) ) ).

% transpose_empty
thf(fact_673_sset__shift,axiom,
    ! [Xs: list_list_a,S2: stream_list_a] :
      ( ( sset_list_a @ ( shift_list_a @ Xs @ S2 ) )
      = ( sup_sup_set_list_a @ ( set_list_a2 @ Xs ) @ ( sset_list_a @ S2 ) ) ) ).

% sset_shift
thf(fact_674_sset__shift,axiom,
    ! [Xs: list_a,S2: stream_a] :
      ( ( sset_a @ ( shift_a @ Xs @ S2 ) )
      = ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( sset_a @ S2 ) ) ) ).

% sset_shift
thf(fact_675_concat__conv__foldr,axiom,
    ( concat_a
    = ( ^ [Xss3: list_list_a] : ( foldr_list_a_list_a @ append_a @ Xss3 @ nil_a ) ) ) ).

% concat_conv_foldr
thf(fact_676_SuccI,axiom,
    ! [Kl: list_list_a,K: list_a,Kl2: set_list_list_a] :
      ( ( member_list_list_a2 @ ( append_list_a @ Kl @ ( cons_list_a @ K @ nil_list_a ) ) @ Kl2 )
     => ( member_list_a2 @ K @ ( bNF_Gr4634511371912843295list_a @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_677_SuccI,axiom,
    ! [Kl: list_a,K: a,Kl2: set_list_a] :
      ( ( member_list_a2 @ ( append_a @ Kl @ ( cons_a @ K @ nil_a ) ) @ Kl2 )
     => ( member_a2 @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_678_SuccD,axiom,
    ! [K: list_a,Kl2: set_list_list_a,Kl: list_list_a] :
      ( ( member_list_a2 @ K @ ( bNF_Gr4634511371912843295list_a @ Kl2 @ Kl ) )
     => ( member_list_list_a2 @ ( append_list_a @ Kl @ ( cons_list_a @ K @ nil_list_a ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_679_SuccD,axiom,
    ! [K: a,Kl2: set_list_a,Kl: list_a] :
      ( ( member_a2 @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) )
     => ( member_list_a2 @ ( append_a @ Kl @ ( cons_a @ K @ nil_a ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_680_remove1__split,axiom,
    ! [A: list_a,Xs: list_list_a,Ys2: list_list_a] :
      ( ( member_list_a2 @ A @ ( set_list_a2 @ Xs ) )
     => ( ( ( remove1_list_a @ A @ Xs )
          = Ys2 )
        = ( ? [Ls: list_list_a,Rs: list_list_a] :
              ( ( Xs
                = ( append_list_a @ Ls @ ( cons_list_a @ A @ Rs ) ) )
              & ~ ( member_list_a2 @ A @ ( set_list_a2 @ Ls ) )
              & ( Ys2
                = ( append_list_a @ Ls @ Rs ) ) ) ) ) ) ).

% remove1_split
thf(fact_681_image__eqI,axiom,
    ! [B3: list_a,F: list_a > list_a,X2: list_a,A2: set_list_a] :
      ( ( B3
        = ( F @ X2 ) )
     => ( ( member_list_a2 @ X2 @ A2 )
       => ( member_list_a2 @ B3 @ ( image_list_a_list_a @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_682_UnCI,axiom,
    ! [C: list_a,B: set_list_a,A2: set_list_a] :
      ( ( ~ ( member_list_a2 @ C @ B )
       => ( member_list_a2 @ C @ A2 ) )
     => ( member_list_a2 @ C @ ( sup_sup_set_list_a @ A2 @ B ) ) ) ).

% UnCI
thf(fact_683_Un__iff,axiom,
    ! [C: list_a,A2: set_list_a,B: set_list_a] :
      ( ( member_list_a2 @ C @ ( sup_sup_set_list_a @ A2 @ B ) )
      = ( ( member_list_a2 @ C @ A2 )
        | ( member_list_a2 @ C @ B ) ) ) ).

% Un_iff
thf(fact_684_Un__subset__iff,axiom,
    ! [A2: set_list_a,B: set_list_a,C2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A2 @ B ) @ C2 )
      = ( ( ord_le8861187494160871172list_a @ A2 @ C2 )
        & ( ord_le8861187494160871172list_a @ B @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_685_Un__subset__iff,axiom,
    ! [A2: set_a,B: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C2 )
      = ( ( ord_less_eq_set_a @ A2 @ C2 )
        & ( ord_less_eq_set_a @ B @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_686_in__set__remove1,axiom,
    ! [A: list_a,B3: list_a,Xs: list_list_a] :
      ( ( A != B3 )
     => ( ( member_list_a2 @ A @ ( set_list_a2 @ ( remove1_list_a @ B3 @ Xs ) ) )
        = ( member_list_a2 @ A @ ( set_list_a2 @ Xs ) ) ) ) ).

% in_set_remove1
thf(fact_687_UnE,axiom,
    ! [C: list_a,A2: set_list_a,B: set_list_a] :
      ( ( member_list_a2 @ C @ ( sup_sup_set_list_a @ A2 @ B ) )
     => ( ~ ( member_list_a2 @ C @ A2 )
       => ( member_list_a2 @ C @ B ) ) ) ).

% UnE
thf(fact_688_UnI1,axiom,
    ! [C: list_a,A2: set_list_a,B: set_list_a] :
      ( ( member_list_a2 @ C @ A2 )
     => ( member_list_a2 @ C @ ( sup_sup_set_list_a @ A2 @ B ) ) ) ).

% UnI1
thf(fact_689_UnI2,axiom,
    ! [C: list_a,B: set_list_a,A2: set_list_a] :
      ( ( member_list_a2 @ C @ B )
     => ( member_list_a2 @ C @ ( sup_sup_set_list_a @ A2 @ B ) ) ) ).

% UnI2
thf(fact_690_imageI,axiom,
    ! [X2: list_a,A2: set_list_a,F: list_a > list_a] :
      ( ( member_list_a2 @ X2 @ A2 )
     => ( member_list_a2 @ ( F @ X2 ) @ ( image_list_a_list_a @ F @ A2 ) ) ) ).

% imageI
thf(fact_691_rev__image__eqI,axiom,
    ! [X2: list_a,A2: set_list_a,B3: list_a,F: list_a > list_a] :
      ( ( member_list_a2 @ X2 @ A2 )
     => ( ( B3
          = ( F @ X2 ) )
       => ( member_list_a2 @ B3 @ ( image_list_a_list_a @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_692_subset__image__iff,axiom,
    ! [B: set_list_a,F: list_a > list_a,A2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ B @ ( image_list_a_list_a @ F @ A2 ) )
      = ( ? [AA: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ AA @ A2 )
            & ( B
              = ( image_list_a_list_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_693_subset__image__iff,axiom,
    ! [B: set_list_a,F: a > list_a,A2: set_a] :
      ( ( ord_le8861187494160871172list_a @ B @ ( image_a_list_a @ F @ A2 ) )
      = ( ? [AA: set_a] :
            ( ( ord_less_eq_set_a @ AA @ A2 )
            & ( B
              = ( image_a_list_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_694_subset__image__iff,axiom,
    ! [B: set_a,F: list_a > a,A2: set_list_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_list_a_a @ F @ A2 ) )
      = ( ? [AA: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ AA @ A2 )
            & ( B
              = ( image_list_a_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_695_subset__image__iff,axiom,
    ! [B: set_a,F: a > a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A2 ) )
      = ( ? [AA: set_a] :
            ( ( ord_less_eq_set_a @ AA @ A2 )
            & ( B
              = ( image_a_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_696_subset__imageE,axiom,
    ! [B: set_list_a,F: list_a > list_a,A2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ B @ ( image_list_a_list_a @ F @ A2 ) )
     => ~ ! [C3: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ C3 @ A2 )
           => ( B
             != ( image_list_a_list_a @ F @ C3 ) ) ) ) ).

% subset_imageE
thf(fact_697_subset__imageE,axiom,
    ! [B: set_list_a,F: a > list_a,A2: set_a] :
      ( ( ord_le8861187494160871172list_a @ B @ ( image_a_list_a @ F @ A2 ) )
     => ~ ! [C3: set_a] :
            ( ( ord_less_eq_set_a @ C3 @ A2 )
           => ( B
             != ( image_a_list_a @ F @ C3 ) ) ) ) ).

% subset_imageE
thf(fact_698_subset__imageE,axiom,
    ! [B: set_a,F: list_a > a,A2: set_list_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_list_a_a @ F @ A2 ) )
     => ~ ! [C3: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ C3 @ A2 )
           => ( B
             != ( image_list_a_a @ F @ C3 ) ) ) ) ).

% subset_imageE
thf(fact_699_subset__imageE,axiom,
    ! [B: set_a,F: a > a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A2 ) )
     => ~ ! [C3: set_a] :
            ( ( ord_less_eq_set_a @ C3 @ A2 )
           => ( B
             != ( image_a_a @ F @ C3 ) ) ) ) ).

% subset_imageE
thf(fact_700_image__subsetI,axiom,
    ! [A2: set_list_a,F: list_a > list_a,B: set_list_a] :
      ( ! [X4: list_a] :
          ( ( member_list_a2 @ X4 @ A2 )
         => ( member_list_a2 @ ( F @ X4 ) @ B ) )
     => ( ord_le8861187494160871172list_a @ ( image_list_a_list_a @ F @ A2 ) @ B ) ) ).

% image_subsetI
thf(fact_701_image__subsetI,axiom,
    ! [A2: set_list_a,F: list_a > a,B: set_a] :
      ( ! [X4: list_a] :
          ( ( member_list_a2 @ X4 @ A2 )
         => ( member_a2 @ ( F @ X4 ) @ B ) )
     => ( ord_less_eq_set_a @ ( image_list_a_a @ F @ A2 ) @ B ) ) ).

% image_subsetI
thf(fact_702_image__mono,axiom,
    ! [A2: set_list_a,B: set_list_a,F: list_a > list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B )
     => ( ord_le8861187494160871172list_a @ ( image_list_a_list_a @ F @ A2 ) @ ( image_list_a_list_a @ F @ B ) ) ) ).

% image_mono
thf(fact_703_image__mono,axiom,
    ! [A2: set_list_a,B: set_list_a,F: list_a > a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B )
     => ( ord_less_eq_set_a @ ( image_list_a_a @ F @ A2 ) @ ( image_list_a_a @ F @ B ) ) ) ).

% image_mono
thf(fact_704_image__mono,axiom,
    ! [A2: set_a,B: set_a,F: a > list_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ord_le8861187494160871172list_a @ ( image_a_list_a @ F @ A2 ) @ ( image_a_list_a @ F @ B ) ) ) ).

% image_mono
thf(fact_705_image__mono,axiom,
    ! [A2: set_a,B: set_a,F: a > a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B ) ) ) ).

% image_mono
thf(fact_706_subset__Un__eq,axiom,
    ( ord_le8861187494160871172list_a
    = ( ^ [A3: set_list_a,B2: set_list_a] :
          ( ( sup_sup_set_list_a @ A3 @ B2 )
          = B2 ) ) ) ).

% subset_Un_eq
thf(fact_707_subset__Un__eq,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A3: set_a,B2: set_a] :
          ( ( sup_sup_set_a @ A3 @ B2 )
          = B2 ) ) ) ).

% subset_Un_eq
thf(fact_708_subset__UnE,axiom,
    ! [C2: set_list_a,A2: set_list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ C2 @ ( sup_sup_set_list_a @ A2 @ B ) )
     => ~ ! [A5: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ A5 @ A2 )
           => ! [B5: set_list_a] :
                ( ( ord_le8861187494160871172list_a @ B5 @ B )
               => ( C2
                 != ( sup_sup_set_list_a @ A5 @ B5 ) ) ) ) ) ).

% subset_UnE
thf(fact_709_subset__UnE,axiom,
    ! [C2: set_a,A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) )
     => ~ ! [A5: set_a] :
            ( ( ord_less_eq_set_a @ A5 @ A2 )
           => ! [B5: set_a] :
                ( ( ord_less_eq_set_a @ B5 @ B )
               => ( C2
                 != ( sup_sup_set_a @ A5 @ B5 ) ) ) ) ) ).

% subset_UnE
thf(fact_710_Un__absorb2,axiom,
    ! [B: set_list_a,A2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ B @ A2 )
     => ( ( sup_sup_set_list_a @ A2 @ B )
        = A2 ) ) ).

% Un_absorb2
thf(fact_711_Un__absorb2,axiom,
    ! [B: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B @ A2 )
     => ( ( sup_sup_set_a @ A2 @ B )
        = A2 ) ) ).

% Un_absorb2
thf(fact_712_Un__absorb1,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B )
     => ( ( sup_sup_set_list_a @ A2 @ B )
        = B ) ) ).

% Un_absorb1
thf(fact_713_Un__absorb1,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( sup_sup_set_a @ A2 @ B )
        = B ) ) ).

% Un_absorb1
thf(fact_714_Un__upper2,axiom,
    ! [B: set_list_a,A2: set_list_a] : ( ord_le8861187494160871172list_a @ B @ ( sup_sup_set_list_a @ A2 @ B ) ) ).

% Un_upper2
thf(fact_715_Un__upper2,axiom,
    ! [B: set_a,A2: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A2 @ B ) ) ).

% Un_upper2
thf(fact_716_Un__upper1,axiom,
    ! [A2: set_list_a,B: set_list_a] : ( ord_le8861187494160871172list_a @ A2 @ ( sup_sup_set_list_a @ A2 @ B ) ) ).

% Un_upper1
thf(fact_717_Un__upper1,axiom,
    ! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) ) ).

% Un_upper1
thf(fact_718_Un__least,axiom,
    ! [A2: set_list_a,C2: set_list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ C2 )
     => ( ( ord_le8861187494160871172list_a @ B @ C2 )
       => ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A2 @ B ) @ C2 ) ) ) ).

% Un_least
thf(fact_719_Un__least,axiom,
    ! [A2: set_a,C2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ C2 )
     => ( ( ord_less_eq_set_a @ B @ C2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C2 ) ) ) ).

% Un_least
thf(fact_720_Un__mono,axiom,
    ! [A2: set_list_a,C2: set_list_a,B: set_list_a,D: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ C2 )
     => ( ( ord_le8861187494160871172list_a @ B @ D )
       => ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A2 @ B ) @ ( sup_sup_set_list_a @ C2 @ D ) ) ) ) ).

% Un_mono
thf(fact_721_Un__mono,axiom,
    ! [A2: set_a,C2: set_a,B: set_a,D: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ C2 )
     => ( ( ord_less_eq_set_a @ B @ D )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).

% Un_mono
thf(fact_722_remove1_Osimps_I1_J,axiom,
    ! [X2: a] :
      ( ( remove1_a @ X2 @ nil_a )
      = nil_a ) ).

% remove1.simps(1)
thf(fact_723_remove1__idem,axiom,
    ! [X2: list_a,Xs: list_list_a] :
      ( ~ ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) )
     => ( ( remove1_list_a @ X2 @ Xs )
        = Xs ) ) ).

% remove1_idem
thf(fact_724_notin__set__remove1,axiom,
    ! [X2: list_a,Xs: list_list_a,Y: list_a] :
      ( ~ ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) )
     => ~ ( member_list_a2 @ X2 @ ( set_list_a2 @ ( remove1_list_a @ Y @ Xs ) ) ) ) ).

% notin_set_remove1
thf(fact_725_set__shuffles,axiom,
    ! [Zs: list_a,Xs: list_a,Ys2: list_a] :
      ( ( member_list_a2 @ Zs @ ( shuffles_a @ Xs @ Ys2 ) )
     => ( ( set_a2 @ Zs )
        = ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys2 ) ) ) ) ).

% set_shuffles
thf(fact_726_stream_Oset__map,axiom,
    ! [F: list_a > list_a,V: stream_list_a] :
      ( ( sset_list_a @ ( smap_list_a_list_a @ F @ V ) )
      = ( image_list_a_list_a @ F @ ( sset_list_a @ V ) ) ) ).

% stream.set_map
thf(fact_727_stream_Oset__map,axiom,
    ! [F: a > list_a,V: stream_a] :
      ( ( sset_list_a @ ( smap_a_list_a @ F @ V ) )
      = ( image_a_list_a @ F @ ( sset_a @ V ) ) ) ).

% stream.set_map
thf(fact_728_stream_Oset__map,axiom,
    ! [F: list_a > a,V: stream_list_a] :
      ( ( sset_a @ ( smap_list_a_a @ F @ V ) )
      = ( image_list_a_a @ F @ ( sset_list_a @ V ) ) ) ).

% stream.set_map
thf(fact_729_stream_Oset__map,axiom,
    ! [F: a > a,V: stream_a] :
      ( ( sset_a @ ( smap_a_a @ F @ V ) )
      = ( image_a_a @ F @ ( sset_a @ V ) ) ) ).

% stream.set_map
thf(fact_730_set__remove1__subset,axiom,
    ! [X2: list_a,Xs: list_list_a] : ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( remove1_list_a @ X2 @ Xs ) ) @ ( set_list_a2 @ Xs ) ) ).

% set_remove1_subset
thf(fact_731_set__remove1__subset,axiom,
    ! [X2: a,Xs: list_a] : ( ord_less_eq_set_a @ ( set_a2 @ ( remove1_a @ X2 @ Xs ) ) @ ( set_a2 @ Xs ) ) ).

% set_remove1_subset
thf(fact_732_remove1__append,axiom,
    ! [X2: list_a,Xs: list_list_a,Ys2: list_list_a] :
      ( ( ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) )
       => ( ( remove1_list_a @ X2 @ ( append_list_a @ Xs @ Ys2 ) )
          = ( append_list_a @ ( remove1_list_a @ X2 @ Xs ) @ Ys2 ) ) )
      & ( ~ ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) )
       => ( ( remove1_list_a @ X2 @ ( append_list_a @ Xs @ Ys2 ) )
          = ( append_list_a @ Xs @ ( remove1_list_a @ X2 @ Ys2 ) ) ) ) ) ).

% remove1_append
thf(fact_733_sset__sinterleave,axiom,
    ! [S1: stream_list_a,S22: stream_list_a] :
      ( ( sset_list_a @ ( sinterleave_list_a @ S1 @ S22 ) )
      = ( sup_sup_set_list_a @ ( sset_list_a @ S1 ) @ ( sset_list_a @ S22 ) ) ) ).

% sset_sinterleave
thf(fact_734_sset__sinterleave,axiom,
    ! [S1: stream_a,S22: stream_a] :
      ( ( sset_a @ ( sinterleave_a @ S1 @ S22 ) )
      = ( sup_sup_set_a @ ( sset_a @ S1 ) @ ( sset_a @ S22 ) ) ) ).

% sset_sinterleave
thf(fact_735_transpose_Osimps_I2_J,axiom,
    ! [Xss2: list_list_a] :
      ( ( transpose_a @ ( cons_list_a @ nil_a @ Xss2 ) )
      = ( transpose_a @ Xss2 ) ) ).

% transpose.simps(2)
thf(fact_736_le__sup__iff,axiom,
    ! [X2: set_list_a,Y: set_list_a,Z2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ X2 @ Y ) @ Z2 )
      = ( ( ord_le8861187494160871172list_a @ X2 @ Z2 )
        & ( ord_le8861187494160871172list_a @ Y @ Z2 ) ) ) ).

% le_sup_iff
thf(fact_737_le__sup__iff,axiom,
    ! [X2: set_a,Y: set_a,Z2: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X2 @ Y ) @ Z2 )
      = ( ( ord_less_eq_set_a @ X2 @ Z2 )
        & ( ord_less_eq_set_a @ Y @ Z2 ) ) ) ).

% le_sup_iff
thf(fact_738_sup_Obounded__iff,axiom,
    ! [B3: set_list_a,C: set_list_a,A: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ B3 @ C ) @ A )
      = ( ( ord_le8861187494160871172list_a @ B3 @ A )
        & ( ord_le8861187494160871172list_a @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_739_sup_Obounded__iff,axiom,
    ! [B3: set_a,C: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B3 @ C ) @ A )
      = ( ( ord_less_eq_set_a @ B3 @ A )
        & ( ord_less_eq_set_a @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_740_empty__Shift,axiom,
    ! [Kl2: set_list_list_a,K: list_a] :
      ( ( member_list_list_a2 @ nil_list_a @ Kl2 )
     => ( ( member_list_a2 @ K @ ( bNF_Gr4634511371912843295list_a @ Kl2 @ nil_list_a ) )
       => ( member_list_list_a2 @ nil_list_a @ ( bNF_Gr7042794125918077091list_a @ Kl2 @ K ) ) ) ) ).

% empty_Shift
thf(fact_741_empty__Shift,axiom,
    ! [Kl2: set_list_a,K: a] :
      ( ( member_list_a2 @ nil_a @ Kl2 )
     => ( ( member_a2 @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ nil_a ) )
       => ( member_list_a2 @ nil_a @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) ) ) ) ).

% empty_Shift
thf(fact_742_Cons__shuffles__subset1,axiom,
    ! [X2: a,Xs: list_a,Ys2: list_a] : ( ord_le8861187494160871172list_a @ ( image_list_a_list_a @ ( cons_a @ X2 ) @ ( shuffles_a @ Xs @ Ys2 ) ) @ ( shuffles_a @ ( cons_a @ X2 @ Xs ) @ Ys2 ) ) ).

% Cons_shuffles_subset1
thf(fact_743_Cons__shuffles__subset2,axiom,
    ! [Y: a,Xs: list_a,Ys2: list_a] : ( ord_le8861187494160871172list_a @ ( image_list_a_list_a @ ( cons_a @ Y ) @ ( shuffles_a @ Xs @ Ys2 ) ) @ ( shuffles_a @ Xs @ ( cons_a @ Y @ Ys2 ) ) ) ).

% Cons_shuffles_subset2
thf(fact_744_ShiftD,axiom,
    ! [Kl: list_a,Kl2: set_list_a,K: a] :
      ( ( member_list_a2 @ Kl @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) )
     => ( member_list_a2 @ ( cons_a @ K @ Kl ) @ Kl2 ) ) ).

% ShiftD
thf(fact_745_sup_OcoboundedI2,axiom,
    ! [C: set_list_a,B3: set_list_a,A: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ C @ B3 )
     => ( ord_le8861187494160871172list_a @ C @ ( sup_sup_set_list_a @ A @ B3 ) ) ) ).

% sup.coboundedI2
thf(fact_746_sup_OcoboundedI2,axiom,
    ! [C: set_a,B3: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ C @ B3 )
     => ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B3 ) ) ) ).

% sup.coboundedI2
thf(fact_747_sup_OcoboundedI1,axiom,
    ! [C: set_list_a,A: set_list_a,B3: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ C @ A )
     => ( ord_le8861187494160871172list_a @ C @ ( sup_sup_set_list_a @ A @ B3 ) ) ) ).

% sup.coboundedI1
thf(fact_748_sup_OcoboundedI1,axiom,
    ! [C: set_a,A: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ C @ A )
     => ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B3 ) ) ) ).

% sup.coboundedI1
thf(fact_749_sup_Oabsorb__iff2,axiom,
    ( ord_le8861187494160871172list_a
    = ( ^ [A4: set_list_a,B4: set_list_a] :
          ( ( sup_sup_set_list_a @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_750_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A4: set_a,B4: set_a] :
          ( ( sup_sup_set_a @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_751_sup_Oabsorb__iff1,axiom,
    ( ord_le8861187494160871172list_a
    = ( ^ [B4: set_list_a,A4: set_list_a] :
          ( ( sup_sup_set_list_a @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_752_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_a
    = ( ^ [B4: set_a,A4: set_a] :
          ( ( sup_sup_set_a @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_753_sup_Ocobounded2,axiom,
    ! [B3: set_list_a,A: set_list_a] : ( ord_le8861187494160871172list_a @ B3 @ ( sup_sup_set_list_a @ A @ B3 ) ) ).

% sup.cobounded2
thf(fact_754_sup_Ocobounded2,axiom,
    ! [B3: set_a,A: set_a] : ( ord_less_eq_set_a @ B3 @ ( sup_sup_set_a @ A @ B3 ) ) ).

% sup.cobounded2
thf(fact_755_sup_Ocobounded1,axiom,
    ! [A: set_list_a,B3: set_list_a] : ( ord_le8861187494160871172list_a @ A @ ( sup_sup_set_list_a @ A @ B3 ) ) ).

% sup.cobounded1
thf(fact_756_sup_Ocobounded1,axiom,
    ! [A: set_a,B3: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B3 ) ) ).

% sup.cobounded1
thf(fact_757_sup_Oorder__iff,axiom,
    ( ord_le8861187494160871172list_a
    = ( ^ [B4: set_list_a,A4: set_list_a] :
          ( A4
          = ( sup_sup_set_list_a @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_758_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [B4: set_a,A4: set_a] :
          ( A4
          = ( sup_sup_set_a @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_759_sup_OboundedI,axiom,
    ! [B3: set_list_a,A: set_list_a,C: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ B3 @ A )
     => ( ( ord_le8861187494160871172list_a @ C @ A )
       => ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ B3 @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_760_sup_OboundedI,axiom,
    ! [B3: set_a,A: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ B3 @ A )
     => ( ( ord_less_eq_set_a @ C @ A )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ B3 @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_761_sup_OboundedE,axiom,
    ! [B3: set_list_a,C: set_list_a,A: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ B3 @ C ) @ A )
     => ~ ( ( ord_le8861187494160871172list_a @ B3 @ A )
         => ~ ( ord_le8861187494160871172list_a @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_762_sup_OboundedE,axiom,
    ! [B3: set_a,C: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B3 @ C ) @ A )
     => ~ ( ( ord_less_eq_set_a @ B3 @ A )
         => ~ ( ord_less_eq_set_a @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_763_sup__absorb2,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ X2 @ Y )
     => ( ( sup_sup_set_list_a @ X2 @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_764_sup__absorb2,axiom,
    ! [X2: set_a,Y: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ Y )
     => ( ( sup_sup_set_a @ X2 @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_765_sup__absorb1,axiom,
    ! [Y: set_list_a,X2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ Y @ X2 )
     => ( ( sup_sup_set_list_a @ X2 @ Y )
        = X2 ) ) ).

% sup_absorb1
thf(fact_766_sup__absorb1,axiom,
    ! [Y: set_a,X2: set_a] :
      ( ( ord_less_eq_set_a @ Y @ X2 )
     => ( ( sup_sup_set_a @ X2 @ Y )
        = X2 ) ) ).

% sup_absorb1
thf(fact_767_sup_Oabsorb2,axiom,
    ! [A: set_list_a,B3: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ B3 )
     => ( ( sup_sup_set_list_a @ A @ B3 )
        = B3 ) ) ).

% sup.absorb2
thf(fact_768_sup_Oabsorb2,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( sup_sup_set_a @ A @ B3 )
        = B3 ) ) ).

% sup.absorb2
thf(fact_769_sup_Oabsorb1,axiom,
    ! [B3: set_list_a,A: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ B3 @ A )
     => ( ( sup_sup_set_list_a @ A @ B3 )
        = A ) ) ).

% sup.absorb1
thf(fact_770_sup_Oabsorb1,axiom,
    ! [B3: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ B3 @ A )
     => ( ( sup_sup_set_a @ A @ B3 )
        = A ) ) ).

% sup.absorb1
thf(fact_771_sup__unique,axiom,
    ! [F: set_list_a > set_list_a > set_list_a,X2: set_list_a,Y: set_list_a] :
      ( ! [X4: set_list_a,Y2: set_list_a] : ( ord_le8861187494160871172list_a @ X4 @ ( F @ X4 @ Y2 ) )
     => ( ! [X4: set_list_a,Y2: set_list_a] : ( ord_le8861187494160871172list_a @ Y2 @ ( F @ X4 @ Y2 ) )
       => ( ! [X4: set_list_a,Y2: set_list_a,Z3: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ Y2 @ X4 )
             => ( ( ord_le8861187494160871172list_a @ Z3 @ X4 )
               => ( ord_le8861187494160871172list_a @ ( F @ Y2 @ Z3 ) @ X4 ) ) )
         => ( ( sup_sup_set_list_a @ X2 @ Y )
            = ( F @ X2 @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_772_sup__unique,axiom,
    ! [F: set_a > set_a > set_a,X2: set_a,Y: set_a] :
      ( ! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X4 @ ( F @ X4 @ Y2 ) )
     => ( ! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ Y2 @ ( F @ X4 @ Y2 ) )
       => ( ! [X4: set_a,Y2: set_a,Z3: set_a] :
              ( ( ord_less_eq_set_a @ Y2 @ X4 )
             => ( ( ord_less_eq_set_a @ Z3 @ X4 )
               => ( ord_less_eq_set_a @ ( F @ Y2 @ Z3 ) @ X4 ) ) )
         => ( ( sup_sup_set_a @ X2 @ Y )
            = ( F @ X2 @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_773_sup_OorderI,axiom,
    ! [A: set_list_a,B3: set_list_a] :
      ( ( A
        = ( sup_sup_set_list_a @ A @ B3 ) )
     => ( ord_le8861187494160871172list_a @ B3 @ A ) ) ).

% sup.orderI
thf(fact_774_sup_OorderI,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( A
        = ( sup_sup_set_a @ A @ B3 ) )
     => ( ord_less_eq_set_a @ B3 @ A ) ) ).

% sup.orderI
thf(fact_775_sup_OorderE,axiom,
    ! [B3: set_list_a,A: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ B3 @ A )
     => ( A
        = ( sup_sup_set_list_a @ A @ B3 ) ) ) ).

% sup.orderE
thf(fact_776_sup_OorderE,axiom,
    ! [B3: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ B3 @ A )
     => ( A
        = ( sup_sup_set_a @ A @ B3 ) ) ) ).

% sup.orderE
thf(fact_777_le__iff__sup,axiom,
    ( ord_le8861187494160871172list_a
    = ( ^ [X3: set_list_a,Y4: set_list_a] :
          ( ( sup_sup_set_list_a @ X3 @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_778_le__iff__sup,axiom,
    ( ord_less_eq_set_a
    = ( ^ [X3: set_a,Y4: set_a] :
          ( ( sup_sup_set_a @ X3 @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_779_sup__least,axiom,
    ! [Y: set_list_a,X2: set_list_a,Z2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ Y @ X2 )
     => ( ( ord_le8861187494160871172list_a @ Z2 @ X2 )
       => ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ Y @ Z2 ) @ X2 ) ) ) ).

% sup_least
thf(fact_780_sup__least,axiom,
    ! [Y: set_a,X2: set_a,Z2: set_a] :
      ( ( ord_less_eq_set_a @ Y @ X2 )
     => ( ( ord_less_eq_set_a @ Z2 @ X2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X2 ) ) ) ).

% sup_least
thf(fact_781_sup__mono,axiom,
    ! [A: set_list_a,C: set_list_a,B3: set_list_a,D2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ C )
     => ( ( ord_le8861187494160871172list_a @ B3 @ D2 )
       => ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A @ B3 ) @ ( sup_sup_set_list_a @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_782_sup__mono,axiom,
    ! [A: set_a,C: set_a,B3: set_a,D2: set_a] :
      ( ( ord_less_eq_set_a @ A @ C )
     => ( ( ord_less_eq_set_a @ B3 @ D2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B3 ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_783_sup_Omono,axiom,
    ! [C: set_list_a,A: set_list_a,D2: set_list_a,B3: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ C @ A )
     => ( ( ord_le8861187494160871172list_a @ D2 @ B3 )
       => ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ C @ D2 ) @ ( sup_sup_set_list_a @ A @ B3 ) ) ) ) ).

% sup.mono
thf(fact_784_sup_Omono,axiom,
    ! [C: set_a,A: set_a,D2: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ C @ A )
     => ( ( ord_less_eq_set_a @ D2 @ B3 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D2 ) @ ( sup_sup_set_a @ A @ B3 ) ) ) ) ).

% sup.mono
thf(fact_785_le__supI2,axiom,
    ! [X2: set_list_a,B3: set_list_a,A: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ X2 @ B3 )
     => ( ord_le8861187494160871172list_a @ X2 @ ( sup_sup_set_list_a @ A @ B3 ) ) ) ).

% le_supI2
thf(fact_786_le__supI2,axiom,
    ! [X2: set_a,B3: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ B3 )
     => ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A @ B3 ) ) ) ).

% le_supI2
thf(fact_787_le__supI1,axiom,
    ! [X2: set_list_a,A: set_list_a,B3: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ X2 @ A )
     => ( ord_le8861187494160871172list_a @ X2 @ ( sup_sup_set_list_a @ A @ B3 ) ) ) ).

% le_supI1
thf(fact_788_le__supI1,axiom,
    ! [X2: set_a,A: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ A )
     => ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A @ B3 ) ) ) ).

% le_supI1
thf(fact_789_sup__ge2,axiom,
    ! [Y: set_list_a,X2: set_list_a] : ( ord_le8861187494160871172list_a @ Y @ ( sup_sup_set_list_a @ X2 @ Y ) ) ).

% sup_ge2
thf(fact_790_sup__ge2,axiom,
    ! [Y: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X2 @ Y ) ) ).

% sup_ge2
thf(fact_791_sup__ge1,axiom,
    ! [X2: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ X2 @ ( sup_sup_set_list_a @ X2 @ Y ) ) ).

% sup_ge1
thf(fact_792_sup__ge1,axiom,
    ! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y ) ) ).

% sup_ge1
thf(fact_793_le__supI,axiom,
    ! [A: set_list_a,X2: set_list_a,B3: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ X2 )
     => ( ( ord_le8861187494160871172list_a @ B3 @ X2 )
       => ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A @ B3 ) @ X2 ) ) ) ).

% le_supI
thf(fact_794_le__supI,axiom,
    ! [A: set_a,X2: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ A @ X2 )
     => ( ( ord_less_eq_set_a @ B3 @ X2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B3 ) @ X2 ) ) ) ).

% le_supI
thf(fact_795_le__supE,axiom,
    ! [A: set_list_a,B3: set_list_a,X2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A @ B3 ) @ X2 )
     => ~ ( ( ord_le8861187494160871172list_a @ A @ X2 )
         => ~ ( ord_le8861187494160871172list_a @ B3 @ X2 ) ) ) ).

% le_supE
thf(fact_796_le__supE,axiom,
    ! [A: set_a,B3: set_a,X2: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B3 ) @ X2 )
     => ~ ( ( ord_less_eq_set_a @ A @ X2 )
         => ~ ( ord_less_eq_set_a @ B3 @ X2 ) ) ) ).

% le_supE
thf(fact_797_inf__sup__ord_I3_J,axiom,
    ! [X2: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ X2 @ ( sup_sup_set_list_a @ X2 @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_798_inf__sup__ord_I3_J,axiom,
    ! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_799_inf__sup__ord_I4_J,axiom,
    ! [Y: set_list_a,X2: set_list_a] : ( ord_le8861187494160871172list_a @ Y @ ( sup_sup_set_list_a @ X2 @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_800_inf__sup__ord_I4_J,axiom,
    ! [Y: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X2 @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_801_all__subset__image,axiom,
    ! [F: list_a > list_a,A2: set_list_a,P: set_list_a > $o] :
      ( ( ! [B2: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ B2 @ ( image_list_a_list_a @ F @ A2 ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ B2 @ A2 )
           => ( P @ ( image_list_a_list_a @ F @ B2 ) ) ) ) ) ).

% all_subset_image
thf(fact_802_all__subset__image,axiom,
    ! [F: a > list_a,A2: set_a,P: set_list_a > $o] :
      ( ( ! [B2: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ B2 @ ( image_a_list_a @ F @ A2 ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_a] :
            ( ( ord_less_eq_set_a @ B2 @ A2 )
           => ( P @ ( image_a_list_a @ F @ B2 ) ) ) ) ) ).

% all_subset_image
thf(fact_803_all__subset__image,axiom,
    ! [F: list_a > a,A2: set_list_a,P: set_a > $o] :
      ( ( ! [B2: set_a] :
            ( ( ord_less_eq_set_a @ B2 @ ( image_list_a_a @ F @ A2 ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ B2 @ A2 )
           => ( P @ ( image_list_a_a @ F @ B2 ) ) ) ) ) ).

% all_subset_image
thf(fact_804_all__subset__image,axiom,
    ! [F: a > a,A2: set_a,P: set_a > $o] :
      ( ( ! [B2: set_a] :
            ( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_a] :
            ( ( ord_less_eq_set_a @ B2 @ A2 )
           => ( P @ ( image_a_a @ F @ B2 ) ) ) ) ) ).

% all_subset_image
thf(fact_805_ord_Olexordp__eq__simps_I3_J,axiom,
    ! [Less: a > a > $o,X2: a,Xs: list_a] :
      ~ ( lexordp_eq_a @ Less @ ( cons_a @ X2 @ Xs ) @ nil_a ) ).

% ord.lexordp_eq_simps(3)
thf(fact_806_Pow__iff,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( member_set_list_a @ A2 @ ( pow_list_a @ B ) )
      = ( ord_le8861187494160871172list_a @ A2 @ B ) ) ).

% Pow_iff
thf(fact_807_Pow__iff,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( member_set_a2 @ A2 @ ( pow_a @ B ) )
      = ( ord_less_eq_set_a @ A2 @ B ) ) ).

% Pow_iff
thf(fact_808_PowI,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B )
     => ( member_set_list_a @ A2 @ ( pow_list_a @ B ) ) ) ).

% PowI
thf(fact_809_PowI,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( member_set_a2 @ A2 @ ( pow_a @ B ) ) ) ).

% PowI
thf(fact_810_ord_Olexordp__eq__simps_I2_J,axiom,
    ! [Less: a > a > $o,Xs: list_a] :
      ( ( lexordp_eq_a @ Less @ Xs @ nil_a )
      = ( Xs = nil_a ) ) ).

% ord.lexordp_eq_simps(2)
thf(fact_811_ord_Olexordp__eq__simps_I1_J,axiom,
    ! [Less: a > a > $o,Ys2: list_a] : ( lexordp_eq_a @ Less @ nil_a @ Ys2 ) ).

% ord.lexordp_eq_simps(1)
thf(fact_812_Pow__mono,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B )
     => ( ord_le8877086941679407844list_a @ ( pow_list_a @ A2 ) @ ( pow_list_a @ B ) ) ) ).

% Pow_mono
thf(fact_813_Pow__mono,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ord_le3724670747650509150_set_a @ ( pow_a @ A2 ) @ ( pow_a @ B ) ) ) ).

% Pow_mono
thf(fact_814_PowD,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( member_set_list_a @ A2 @ ( pow_list_a @ B ) )
     => ( ord_le8861187494160871172list_a @ A2 @ B ) ) ).

% PowD
thf(fact_815_PowD,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( member_set_a2 @ A2 @ ( pow_a @ B ) )
     => ( ord_less_eq_set_a @ A2 @ B ) ) ).

% PowD
thf(fact_816_ord_Olexordp__eq_ONil,axiom,
    ! [Less: a > a > $o,Ys2: list_a] : ( lexordp_eq_a @ Less @ nil_a @ Ys2 ) ).

% ord.lexordp_eq.Nil
thf(fact_817_ord_Olexordp__eq_Osimps,axiom,
    ( lexordp_eq_a
    = ( ^ [Less2: a > a > $o,A1: list_a,A22: list_a] :
          ( ? [Ys: list_a] :
              ( ( A1 = nil_a )
              & ( A22 = Ys ) )
          | ? [X3: a,Y4: a,Xs3: list_a,Ys: list_a] :
              ( ( A1
                = ( cons_a @ X3 @ Xs3 ) )
              & ( A22
                = ( cons_a @ Y4 @ Ys ) )
              & ( Less2 @ X3 @ Y4 ) )
          | ? [X3: a,Y4: a,Xs3: list_a,Ys: list_a] :
              ( ( A1
                = ( cons_a @ X3 @ Xs3 ) )
              & ( A22
                = ( cons_a @ Y4 @ Ys ) )
              & ~ ( Less2 @ X3 @ Y4 )
              & ~ ( Less2 @ Y4 @ X3 )
              & ( lexordp_eq_a @ Less2 @ Xs3 @ Ys ) ) ) ) ) ).

% ord.lexordp_eq.simps
thf(fact_818_ord_Olexordp__eq_Ocases,axiom,
    ! [Less: a > a > $o,A12: list_a,A23: list_a] :
      ( ( lexordp_eq_a @ Less @ A12 @ A23 )
     => ( ( A12 != nil_a )
       => ( ! [X4: a] :
              ( ? [Xs2: list_a] :
                  ( A12
                  = ( cons_a @ X4 @ Xs2 ) )
             => ! [Y2: a] :
                  ( ? [Ys3: list_a] :
                      ( A23
                      = ( cons_a @ Y2 @ Ys3 ) )
                 => ~ ( Less @ X4 @ Y2 ) ) )
         => ~ ! [X4: a,Y2: a,Xs2: list_a] :
                ( ( A12
                  = ( cons_a @ X4 @ Xs2 ) )
               => ! [Ys3: list_a] :
                    ( ( A23
                      = ( cons_a @ Y2 @ Ys3 ) )
                   => ( ~ ( Less @ X4 @ Y2 )
                     => ( ~ ( Less @ Y2 @ X4 )
                       => ~ ( lexordp_eq_a @ Less @ Xs2 @ Ys3 ) ) ) ) ) ) ) ) ).

% ord.lexordp_eq.cases
thf(fact_819_listrelp_ONil,axiom,
    ! [R2: a > a > $o] : ( listrelp_a_a @ R2 @ nil_a @ nil_a ) ).

% listrelp.Nil
thf(fact_820_Fpow__mono,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B )
     => ( ord_le8877086941679407844list_a @ ( finite_Fpow_list_a @ A2 ) @ ( finite_Fpow_list_a @ B ) ) ) ).

% Fpow_mono
thf(fact_821_Fpow__mono,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ord_le3724670747650509150_set_a @ ( finite_Fpow_a @ A2 ) @ ( finite_Fpow_a @ B ) ) ) ).

% Fpow_mono
thf(fact_822_listrelp_Ocases,axiom,
    ! [R2: a > a > $o,A12: list_a,A23: list_a] :
      ( ( listrelp_a_a @ R2 @ A12 @ A23 )
     => ( ( ( A12 = nil_a )
         => ( A23 != nil_a ) )
       => ~ ! [X4: a,Y2: a,Xs2: list_a] :
              ( ( A12
                = ( cons_a @ X4 @ Xs2 ) )
             => ! [Ys3: list_a] :
                  ( ( A23
                    = ( cons_a @ Y2 @ Ys3 ) )
                 => ( ( R2 @ X4 @ Y2 )
                   => ~ ( listrelp_a_a @ R2 @ Xs2 @ Ys3 ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_823_listrelp_Osimps,axiom,
    ( listrelp_a_a
    = ( ^ [R3: a > a > $o,A1: list_a,A22: list_a] :
          ( ( ( A1 = nil_a )
            & ( A22 = nil_a ) )
          | ? [X3: a,Y4: a,Xs3: list_a,Ys: list_a] :
              ( ( A1
                = ( cons_a @ X3 @ Xs3 ) )
              & ( A22
                = ( cons_a @ Y4 @ Ys ) )
              & ( R3 @ X3 @ Y4 )
              & ( listrelp_a_a @ R3 @ Xs3 @ Ys ) ) ) ) ) ).

% listrelp.simps
thf(fact_824_rev__eq__Cons__iff,axiom,
    ! [Xs: list_a,Y: a,Ys2: list_a] :
      ( ( ( rev_a @ Xs )
        = ( cons_a @ Y @ Ys2 ) )
      = ( Xs
        = ( append_a @ ( rev_a @ Ys2 ) @ ( cons_a @ Y @ nil_a ) ) ) ) ).

% rev_eq_Cons_iff
thf(fact_825_insertCI,axiom,
    ! [A: list_a,B: set_list_a,B3: list_a] :
      ( ( ~ ( member_list_a2 @ A @ B )
       => ( A = B3 ) )
     => ( member_list_a2 @ A @ ( insert_list_a2 @ B3 @ B ) ) ) ).

% insertCI
thf(fact_826_insert__iff,axiom,
    ! [A: list_a,B3: list_a,A2: set_list_a] :
      ( ( member_list_a2 @ A @ ( insert_list_a2 @ B3 @ A2 ) )
      = ( ( A = B3 )
        | ( member_list_a2 @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_827_insert__subset,axiom,
    ! [X2: list_a,A2: set_list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( insert_list_a2 @ X2 @ A2 ) @ B )
      = ( ( member_list_a2 @ X2 @ B )
        & ( ord_le8861187494160871172list_a @ A2 @ B ) ) ) ).

% insert_subset
thf(fact_828_insert__subset,axiom,
    ! [X2: a,A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ ( insert_a2 @ X2 @ A2 ) @ B )
      = ( ( member_a2 @ X2 @ B )
        & ( ord_less_eq_set_a @ A2 @ B ) ) ) ).

% insert_subset
thf(fact_829_Nil__is__rev__conv,axiom,
    ! [Xs: list_a] :
      ( ( nil_a
        = ( rev_a @ Xs ) )
      = ( Xs = nil_a ) ) ).

% Nil_is_rev_conv
thf(fact_830_rev__is__Nil__conv,axiom,
    ! [Xs: list_a] :
      ( ( ( rev_a @ Xs )
        = nil_a )
      = ( Xs = nil_a ) ) ).

% rev_is_Nil_conv
thf(fact_831_removeAll__id,axiom,
    ! [X2: list_a,Xs: list_list_a] :
      ( ~ ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) )
     => ( ( removeAll_list_a @ X2 @ Xs )
        = Xs ) ) ).

% removeAll_id
thf(fact_832_singleton__rev__conv,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( ( cons_a @ X2 @ nil_a )
        = ( rev_a @ Xs ) )
      = ( ( cons_a @ X2 @ nil_a )
        = Xs ) ) ).

% singleton_rev_conv
thf(fact_833_rev__singleton__conv,axiom,
    ! [Xs: list_a,X2: a] :
      ( ( ( rev_a @ Xs )
        = ( cons_a @ X2 @ nil_a ) )
      = ( Xs
        = ( cons_a @ X2 @ nil_a ) ) ) ).

% rev_singleton_conv
thf(fact_834_stream_Oset,axiom,
    ! [X1: list_a,X22: stream_list_a] :
      ( ( sset_list_a @ ( sCons_list_a @ X1 @ X22 ) )
      = ( insert_list_a2 @ X1 @ ( sset_list_a @ X22 ) ) ) ).

% stream.set
thf(fact_835_stream_Oset,axiom,
    ! [X1: a,X22: stream_a] :
      ( ( sset_a @ ( sCons_a @ X1 @ X22 ) )
      = ( insert_a2 @ X1 @ ( sset_a @ X22 ) ) ) ).

% stream.set
thf(fact_836_subset__insertI2,axiom,
    ! [A2: set_list_a,B: set_list_a,B3: list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B )
     => ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a2 @ B3 @ B ) ) ) ).

% subset_insertI2
thf(fact_837_subset__insertI2,axiom,
    ! [A2: set_a,B: set_a,B3: a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ord_less_eq_set_a @ A2 @ ( insert_a2 @ B3 @ B ) ) ) ).

% subset_insertI2
thf(fact_838_subset__insertI,axiom,
    ! [B: set_list_a,A: list_a] : ( ord_le8861187494160871172list_a @ B @ ( insert_list_a2 @ A @ B ) ) ).

% subset_insertI
thf(fact_839_subset__insertI,axiom,
    ! [B: set_a,A: a] : ( ord_less_eq_set_a @ B @ ( insert_a2 @ A @ B ) ) ).

% subset_insertI
thf(fact_840_subset__insert,axiom,
    ! [X2: list_a,A2: set_list_a,B: set_list_a] :
      ( ~ ( member_list_a2 @ X2 @ A2 )
     => ( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a2 @ X2 @ B ) )
        = ( ord_le8861187494160871172list_a @ A2 @ B ) ) ) ).

% subset_insert
thf(fact_841_subset__insert,axiom,
    ! [X2: a,A2: set_a,B: set_a] :
      ( ~ ( member_a2 @ X2 @ A2 )
     => ( ( ord_less_eq_set_a @ A2 @ ( insert_a2 @ X2 @ B ) )
        = ( ord_less_eq_set_a @ A2 @ B ) ) ) ).

% subset_insert
thf(fact_842_insert__mono,axiom,
    ! [C2: set_list_a,D: set_list_a,A: list_a] :
      ( ( ord_le8861187494160871172list_a @ C2 @ D )
     => ( ord_le8861187494160871172list_a @ ( insert_list_a2 @ A @ C2 ) @ ( insert_list_a2 @ A @ D ) ) ) ).

% insert_mono
thf(fact_843_insert__mono,axiom,
    ! [C2: set_a,D: set_a,A: a] :
      ( ( ord_less_eq_set_a @ C2 @ D )
     => ( ord_less_eq_set_a @ ( insert_a2 @ A @ C2 ) @ ( insert_a2 @ A @ D ) ) ) ).

% insert_mono
thf(fact_844_rev_Osimps_I1_J,axiom,
    ( ( rev_a @ nil_a )
    = nil_a ) ).

% rev.simps(1)
thf(fact_845_insertE,axiom,
    ! [A: list_a,B3: list_a,A2: set_list_a] :
      ( ( member_list_a2 @ A @ ( insert_list_a2 @ B3 @ A2 ) )
     => ( ( A != B3 )
       => ( member_list_a2 @ A @ A2 ) ) ) ).

% insertE
thf(fact_846_insertI1,axiom,
    ! [A: list_a,B: set_list_a] : ( member_list_a2 @ A @ ( insert_list_a2 @ A @ B ) ) ).

% insertI1
thf(fact_847_insertI2,axiom,
    ! [A: list_a,B: set_list_a,B3: list_a] :
      ( ( member_list_a2 @ A @ B )
     => ( member_list_a2 @ A @ ( insert_list_a2 @ B3 @ B ) ) ) ).

% insertI2
thf(fact_848_Set_Oset__insert,axiom,
    ! [X2: list_a,A2: set_list_a] :
      ( ( member_list_a2 @ X2 @ A2 )
     => ~ ! [B6: set_list_a] :
            ( ( A2
              = ( insert_list_a2 @ X2 @ B6 ) )
           => ( member_list_a2 @ X2 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_849_insert__ident,axiom,
    ! [X2: list_a,A2: set_list_a,B: set_list_a] :
      ( ~ ( member_list_a2 @ X2 @ A2 )
     => ( ~ ( member_list_a2 @ X2 @ B )
       => ( ( ( insert_list_a2 @ X2 @ A2 )
            = ( insert_list_a2 @ X2 @ B ) )
          = ( A2 = B ) ) ) ) ).

% insert_ident
thf(fact_850_insert__absorb,axiom,
    ! [A: list_a,A2: set_list_a] :
      ( ( member_list_a2 @ A @ A2 )
     => ( ( insert_list_a2 @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_851_insert__eq__iff,axiom,
    ! [A: list_a,A2: set_list_a,B3: list_a,B: set_list_a] :
      ( ~ ( member_list_a2 @ A @ A2 )
     => ( ~ ( member_list_a2 @ B3 @ B )
       => ( ( ( insert_list_a2 @ A @ A2 )
            = ( insert_list_a2 @ B3 @ B ) )
          = ( ( ( A = B3 )
             => ( A2 = B ) )
            & ( ( A != B3 )
             => ? [C4: set_list_a] :
                  ( ( A2
                    = ( insert_list_a2 @ B3 @ C4 ) )
                  & ~ ( member_list_a2 @ B3 @ C4 )
                  & ( B
                    = ( insert_list_a2 @ A @ C4 ) )
                  & ~ ( member_list_a2 @ A @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_852_mk__disjoint__insert,axiom,
    ! [A: list_a,A2: set_list_a] :
      ( ( member_list_a2 @ A @ A2 )
     => ? [B6: set_list_a] :
          ( ( A2
            = ( insert_list_a2 @ A @ B6 ) )
          & ~ ( member_list_a2 @ A @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_853_removeAll_Osimps_I1_J,axiom,
    ! [X2: a] :
      ( ( removeAll_a @ X2 @ nil_a )
      = nil_a ) ).

% removeAll.simps(1)
thf(fact_854_rev_Osimps_I2_J,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( rev_a @ ( cons_a @ X2 @ Xs ) )
      = ( append_a @ ( rev_a @ Xs ) @ ( cons_a @ X2 @ nil_a ) ) ) ).

% rev.simps(2)
thf(fact_855_insert__subsetI,axiom,
    ! [X2: list_a,A2: set_list_a,X6: set_list_a] :
      ( ( member_list_a2 @ X2 @ A2 )
     => ( ( ord_le8861187494160871172list_a @ X6 @ A2 )
       => ( ord_le8861187494160871172list_a @ ( insert_list_a2 @ X2 @ X6 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_856_insert__subsetI,axiom,
    ! [X2: a,A2: set_a,X6: set_a] :
      ( ( member_a2 @ X2 @ A2 )
     => ( ( ord_less_eq_set_a @ X6 @ A2 )
       => ( ord_less_eq_set_a @ ( insert_a2 @ X2 @ X6 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_857_shuffles_Oelims,axiom,
    ! [X2: list_a,Xa2: list_a,Y: set_list_a] :
      ( ( ( shuffles_a @ X2 @ Xa2 )
        = Y )
     => ( ( ( X2 = nil_a )
         => ( Y
           != ( insert_list_a2 @ Xa2 @ bot_bot_set_list_a ) ) )
       => ( ( ( Xa2 = nil_a )
           => ( Y
             != ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) ) )
         => ~ ! [X4: a,Xs2: list_a] :
                ( ( X2
                  = ( cons_a @ X4 @ Xs2 ) )
               => ! [Y2: a,Ys3: list_a] :
                    ( ( Xa2
                      = ( cons_a @ Y2 @ Ys3 ) )
                   => ( Y
                     != ( sup_sup_set_list_a @ ( image_list_a_list_a @ ( cons_a @ X4 ) @ ( shuffles_a @ Xs2 @ ( cons_a @ Y2 @ Ys3 ) ) ) @ ( image_list_a_list_a @ ( cons_a @ Y2 ) @ ( shuffles_a @ ( cons_a @ X4 @ Xs2 ) @ Ys3 ) ) ) ) ) ) ) ) ) ).

% shuffles.elims
thf(fact_858_all__not__in__conv,axiom,
    ! [A2: set_list_a] :
      ( ( ! [X3: list_a] :
            ~ ( member_list_a2 @ X3 @ A2 ) )
      = ( A2 = bot_bot_set_list_a ) ) ).

% all_not_in_conv
thf(fact_859_empty__iff,axiom,
    ! [C: list_a] :
      ~ ( member_list_a2 @ C @ bot_bot_set_list_a ) ).

% empty_iff
thf(fact_860_subset__empty,axiom,
    ! [A2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ bot_bot_set_list_a )
      = ( A2 = bot_bot_set_list_a ) ) ).

% subset_empty
thf(fact_861_subset__empty,axiom,
    ! [A2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
      = ( A2 = bot_bot_set_a ) ) ).

% subset_empty
thf(fact_862_empty__subsetI,axiom,
    ! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A2 ) ).

% empty_subsetI
thf(fact_863_empty__subsetI,axiom,
    ! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).

% empty_subsetI
thf(fact_864_singletonI,axiom,
    ! [A: list_a] : ( member_list_a2 @ A @ ( insert_list_a2 @ A @ bot_bot_set_list_a ) ) ).

% singletonI
thf(fact_865_singleton__insert__inj__eq,axiom,
    ! [B3: list_a,A: list_a,A2: set_list_a] :
      ( ( ( insert_list_a2 @ B3 @ bot_bot_set_list_a )
        = ( insert_list_a2 @ A @ A2 ) )
      = ( ( A = B3 )
        & ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a2 @ B3 @ bot_bot_set_list_a ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_866_singleton__insert__inj__eq,axiom,
    ! [B3: a,A: a,A2: set_a] :
      ( ( ( insert_a2 @ B3 @ bot_bot_set_a )
        = ( insert_a2 @ A @ A2 ) )
      = ( ( A = B3 )
        & ( ord_less_eq_set_a @ A2 @ ( insert_a2 @ B3 @ bot_bot_set_a ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_867_singleton__insert__inj__eq_H,axiom,
    ! [A: list_a,A2: set_list_a,B3: list_a] :
      ( ( ( insert_list_a2 @ A @ A2 )
        = ( insert_list_a2 @ B3 @ bot_bot_set_list_a ) )
      = ( ( A = B3 )
        & ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a2 @ B3 @ bot_bot_set_list_a ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_868_singleton__insert__inj__eq_H,axiom,
    ! [A: a,A2: set_a,B3: a] :
      ( ( ( insert_a2 @ A @ A2 )
        = ( insert_a2 @ B3 @ bot_bot_set_a ) )
      = ( ( A = B3 )
        & ( ord_less_eq_set_a @ A2 @ ( insert_a2 @ B3 @ bot_bot_set_a ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_869_set__empty,axiom,
    ! [Xs: list_a] :
      ( ( ( set_a2 @ Xs )
        = bot_bot_set_a )
      = ( Xs = nil_a ) ) ).

% set_empty
thf(fact_870_set__empty2,axiom,
    ! [Xs: list_a] :
      ( ( bot_bot_set_a
        = ( set_a2 @ Xs ) )
      = ( Xs = nil_a ) ) ).

% set_empty2
thf(fact_871_singletonD,axiom,
    ! [B3: list_a,A: list_a] :
      ( ( member_list_a2 @ B3 @ ( insert_list_a2 @ A @ bot_bot_set_list_a ) )
     => ( B3 = A ) ) ).

% singletonD
thf(fact_872_singleton__iff,axiom,
    ! [B3: list_a,A: list_a] :
      ( ( member_list_a2 @ B3 @ ( insert_list_a2 @ A @ bot_bot_set_list_a ) )
      = ( B3 = A ) ) ).

% singleton_iff
thf(fact_873_ex__in__conv,axiom,
    ! [A2: set_list_a] :
      ( ( ? [X3: list_a] : ( member_list_a2 @ X3 @ A2 ) )
      = ( A2 != bot_bot_set_list_a ) ) ).

% ex_in_conv
thf(fact_874_equals0I,axiom,
    ! [A2: set_list_a] :
      ( ! [Y2: list_a] :
          ~ ( member_list_a2 @ Y2 @ A2 )
     => ( A2 = bot_bot_set_list_a ) ) ).

% equals0I
thf(fact_875_equals0D,axiom,
    ! [A2: set_list_a,A: list_a] :
      ( ( A2 = bot_bot_set_list_a )
     => ~ ( member_list_a2 @ A @ A2 ) ) ).

% equals0D
thf(fact_876_emptyE,axiom,
    ! [A: list_a] :
      ~ ( member_list_a2 @ A @ bot_bot_set_list_a ) ).

% emptyE
thf(fact_877_subset__emptyI,axiom,
    ! [A2: set_list_a] :
      ( ! [X4: list_a] :
          ~ ( member_list_a2 @ X4 @ A2 )
     => ( ord_le8861187494160871172list_a @ A2 @ bot_bot_set_list_a ) ) ).

% subset_emptyI
thf(fact_878_subset__emptyI,axiom,
    ! [A2: set_a] :
      ( ! [X4: a] :
          ~ ( member_a2 @ X4 @ A2 )
     => ( ord_less_eq_set_a @ A2 @ bot_bot_set_a ) ) ).

% subset_emptyI
thf(fact_879_bot_Oextremum__uniqueI,axiom,
    ! [A: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ bot_bot_set_list_a )
     => ( A = bot_bot_set_list_a ) ) ).

% bot.extremum_uniqueI
thf(fact_880_bot_Oextremum__uniqueI,axiom,
    ! [A: set_a] :
      ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
     => ( A = bot_bot_set_a ) ) ).

% bot.extremum_uniqueI
thf(fact_881_bot_Oextremum__unique,axiom,
    ! [A: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ bot_bot_set_list_a )
      = ( A = bot_bot_set_list_a ) ) ).

% bot.extremum_unique
thf(fact_882_bot_Oextremum__unique,axiom,
    ! [A: set_a] :
      ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
      = ( A = bot_bot_set_a ) ) ).

% bot.extremum_unique
thf(fact_883_bot_Oextremum,axiom,
    ! [A: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A ) ).

% bot.extremum
thf(fact_884_bot_Oextremum,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).

% bot.extremum
thf(fact_885_Pow__set_I1_J,axiom,
    ( ( pow_a @ ( set_a2 @ nil_a ) )
    = ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) ).

% Pow_set(1)
thf(fact_886_subset__singletonD,axiom,
    ! [A2: set_list_a,X2: list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) )
     => ( ( A2 = bot_bot_set_list_a )
        | ( A2
          = ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) ) ) ) ).

% subset_singletonD
thf(fact_887_subset__singletonD,axiom,
    ! [A2: set_a,X2: a] :
      ( ( ord_less_eq_set_a @ A2 @ ( insert_a2 @ X2 @ bot_bot_set_a ) )
     => ( ( A2 = bot_bot_set_a )
        | ( A2
          = ( insert_a2 @ X2 @ bot_bot_set_a ) ) ) ) ).

% subset_singletonD
thf(fact_888_subset__singleton__iff,axiom,
    ! [X6: set_list_a,A: list_a] :
      ( ( ord_le8861187494160871172list_a @ X6 @ ( insert_list_a2 @ A @ bot_bot_set_list_a ) )
      = ( ( X6 = bot_bot_set_list_a )
        | ( X6
          = ( insert_list_a2 @ A @ bot_bot_set_list_a ) ) ) ) ).

% subset_singleton_iff
thf(fact_889_subset__singleton__iff,axiom,
    ! [X6: set_a,A: a] :
      ( ( ord_less_eq_set_a @ X6 @ ( insert_a2 @ A @ bot_bot_set_a ) )
      = ( ( X6 = bot_bot_set_a )
        | ( X6
          = ( insert_a2 @ A @ bot_bot_set_a ) ) ) ) ).

% subset_singleton_iff
thf(fact_890_empty__set,axiom,
    ( bot_bot_set_a
    = ( set_a2 @ nil_a ) ) ).

% empty_set
thf(fact_891_shuffles_Osimps_I2_J,axiom,
    ! [Xs: list_a] :
      ( ( shuffles_a @ Xs @ nil_a )
      = ( insert_list_a2 @ Xs @ bot_bot_set_list_a ) ) ).

% shuffles.simps(2)
thf(fact_892_shuffles_Osimps_I1_J,axiom,
    ! [Ys2: list_a] :
      ( ( shuffles_a @ nil_a @ Ys2 )
      = ( insert_list_a2 @ Ys2 @ bot_bot_set_list_a ) ) ).

% shuffles.simps(1)
thf(fact_893_listset_Osimps_I1_J,axiom,
    ( ( listset_a @ nil_set_a )
    = ( insert_list_a2 @ nil_a @ bot_bot_set_list_a ) ) ).

% listset.simps(1)
thf(fact_894_lists__empty,axiom,
    ( ( lists_a @ bot_bot_set_a )
    = ( insert_list_a2 @ nil_a @ bot_bot_set_list_a ) ) ).

% lists_empty
thf(fact_895_Cons__in__lists__iff,axiom,
    ! [X2: list_a,Xs: list_list_a,A2: set_list_a] :
      ( ( member_list_list_a2 @ ( cons_list_a @ X2 @ Xs ) @ ( lists_list_a @ A2 ) )
      = ( ( member_list_a2 @ X2 @ A2 )
        & ( member_list_list_a2 @ Xs @ ( lists_list_a @ A2 ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_896_Cons__in__lists__iff,axiom,
    ! [X2: a,Xs: list_a,A2: set_a] :
      ( ( member_list_a2 @ ( cons_a @ X2 @ Xs ) @ ( lists_a @ A2 ) )
      = ( ( member_a2 @ X2 @ A2 )
        & ( member_list_a2 @ Xs @ ( lists_a @ A2 ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_897_in__listsI,axiom,
    ! [Xs: list_a,A2: set_a] :
      ( ! [X4: a] :
          ( ( member_a2 @ X4 @ ( set_a2 @ Xs ) )
         => ( member_a2 @ X4 @ A2 ) )
     => ( member_list_a2 @ Xs @ ( lists_a @ A2 ) ) ) ).

% in_listsI
thf(fact_898_in__listsI,axiom,
    ! [Xs: list_list_a,A2: set_list_a] :
      ( ! [X4: list_a] :
          ( ( member_list_a2 @ X4 @ ( set_list_a2 @ Xs ) )
         => ( member_list_a2 @ X4 @ A2 ) )
     => ( member_list_list_a2 @ Xs @ ( lists_list_a @ A2 ) ) ) ).

% in_listsI
thf(fact_899_append__in__lists__conv,axiom,
    ! [Xs: list_a,Ys2: list_a,A2: set_a] :
      ( ( member_list_a2 @ ( append_a @ Xs @ Ys2 ) @ ( lists_a @ A2 ) )
      = ( ( member_list_a2 @ Xs @ ( lists_a @ A2 ) )
        & ( member_list_a2 @ Ys2 @ ( lists_a @ A2 ) ) ) ) ).

% append_in_lists_conv
thf(fact_900_lists_ONil,axiom,
    ! [A2: set_a] : ( member_list_a2 @ nil_a @ ( lists_a @ A2 ) ) ).

% lists.Nil
thf(fact_901_in__lists__conv__set,axiom,
    ! [Xs: list_list_a,A2: set_list_a] :
      ( ( member_list_list_a2 @ Xs @ ( lists_list_a @ A2 ) )
      = ( ! [X3: list_a] :
            ( ( member_list_a2 @ X3 @ ( set_list_a2 @ Xs ) )
           => ( member_list_a2 @ X3 @ A2 ) ) ) ) ).

% in_lists_conv_set
thf(fact_902_in__lists__conv__set,axiom,
    ! [Xs: list_a,A2: set_a] :
      ( ( member_list_a2 @ Xs @ ( lists_a @ A2 ) )
      = ( ! [X3: a] :
            ( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
           => ( member_a2 @ X3 @ A2 ) ) ) ) ).

% in_lists_conv_set
thf(fact_903_in__listsD,axiom,
    ! [Xs: list_list_a,A2: set_list_a] :
      ( ( member_list_list_a2 @ Xs @ ( lists_list_a @ A2 ) )
     => ! [X: list_a] :
          ( ( member_list_a2 @ X @ ( set_list_a2 @ Xs ) )
         => ( member_list_a2 @ X @ A2 ) ) ) ).

% in_listsD
thf(fact_904_in__listsD,axiom,
    ! [Xs: list_a,A2: set_a] :
      ( ( member_list_a2 @ Xs @ ( lists_a @ A2 ) )
     => ! [X: a] :
          ( ( member_a2 @ X @ ( set_a2 @ Xs ) )
         => ( member_a2 @ X @ A2 ) ) ) ).

% in_listsD
thf(fact_905_listsE,axiom,
    ! [X2: list_a,L: list_list_a,A2: set_list_a] :
      ( ( member_list_list_a2 @ ( cons_list_a @ X2 @ L ) @ ( lists_list_a @ A2 ) )
     => ~ ( ( member_list_a2 @ X2 @ A2 )
         => ~ ( member_list_list_a2 @ L @ ( lists_list_a @ A2 ) ) ) ) ).

% listsE
thf(fact_906_listsE,axiom,
    ! [X2: a,L: list_a,A2: set_a] :
      ( ( member_list_a2 @ ( cons_a @ X2 @ L ) @ ( lists_a @ A2 ) )
     => ~ ( ( member_a2 @ X2 @ A2 )
         => ~ ( member_list_a2 @ L @ ( lists_a @ A2 ) ) ) ) ).

% listsE
thf(fact_907_lists_OCons,axiom,
    ! [A: a,A2: set_a,L: list_a] :
      ( ( member_a2 @ A @ A2 )
     => ( ( member_list_a2 @ L @ ( lists_a @ A2 ) )
       => ( member_list_a2 @ ( cons_a @ A @ L ) @ ( lists_a @ A2 ) ) ) ) ).

% lists.Cons
thf(fact_908_lists_OCons,axiom,
    ! [A: list_a,A2: set_list_a,L: list_list_a] :
      ( ( member_list_a2 @ A @ A2 )
     => ( ( member_list_list_a2 @ L @ ( lists_list_a @ A2 ) )
       => ( member_list_list_a2 @ ( cons_list_a @ A @ L ) @ ( lists_list_a @ A2 ) ) ) ) ).

% lists.Cons
thf(fact_909_lists_Osimps,axiom,
    ! [A: list_list_a,A2: set_list_a] :
      ( ( member_list_list_a2 @ A @ ( lists_list_a @ A2 ) )
      = ( ( A = nil_list_a )
        | ? [A4: list_a,L2: list_list_a] :
            ( ( A
              = ( cons_list_a @ A4 @ L2 ) )
            & ( member_list_a2 @ A4 @ A2 )
            & ( member_list_list_a2 @ L2 @ ( lists_list_a @ A2 ) ) ) ) ) ).

% lists.simps
thf(fact_910_lists_Osimps,axiom,
    ! [A: list_a,A2: set_a] :
      ( ( member_list_a2 @ A @ ( lists_a @ A2 ) )
      = ( ( A = nil_a )
        | ? [A4: a,L2: list_a] :
            ( ( A
              = ( cons_a @ A4 @ L2 ) )
            & ( member_a2 @ A4 @ A2 )
            & ( member_list_a2 @ L2 @ ( lists_a @ A2 ) ) ) ) ) ).

% lists.simps
thf(fact_911_lists_Ocases,axiom,
    ! [A: list_list_a,A2: set_list_a] :
      ( ( member_list_list_a2 @ A @ ( lists_list_a @ A2 ) )
     => ( ( A != nil_list_a )
       => ~ ! [A6: list_a,L3: list_list_a] :
              ( ( A
                = ( cons_list_a @ A6 @ L3 ) )
             => ( ( member_list_a2 @ A6 @ A2 )
               => ~ ( member_list_list_a2 @ L3 @ ( lists_list_a @ A2 ) ) ) ) ) ) ).

% lists.cases
thf(fact_912_lists_Ocases,axiom,
    ! [A: list_a,A2: set_a] :
      ( ( member_list_a2 @ A @ ( lists_a @ A2 ) )
     => ( ( A != nil_a )
       => ~ ! [A6: a,L3: list_a] :
              ( ( A
                = ( cons_a @ A6 @ L3 ) )
             => ( ( member_a2 @ A6 @ A2 )
               => ~ ( member_list_a2 @ L3 @ ( lists_a @ A2 ) ) ) ) ) ) ).

% lists.cases
thf(fact_913_is__singletonI_H,axiom,
    ! [A2: set_list_a] :
      ( ( A2 != bot_bot_set_list_a )
     => ( ! [X4: list_a,Y2: list_a] :
            ( ( member_list_a2 @ X4 @ A2 )
           => ( ( member_list_a2 @ Y2 @ A2 )
             => ( X4 = Y2 ) ) )
       => ( is_singleton_list_a @ A2 ) ) ) ).

% is_singletonI'
thf(fact_914_lists__mono,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B )
     => ( ord_le8488217952732425610list_a @ ( lists_list_a @ A2 ) @ ( lists_list_a @ B ) ) ) ).

% lists_mono
thf(fact_915_lists__mono,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ord_le8861187494160871172list_a @ ( lists_a @ A2 ) @ ( lists_a @ B ) ) ) ).

% lists_mono
thf(fact_916_chains__extend,axiom,
    ! [C: set_set_list_a,S5: set_set_list_a,Z2: set_list_a] :
      ( ( member8857465052274545133list_a @ C @ ( chains_list_a @ S5 ) )
     => ( ( member_set_list_a @ Z2 @ S5 )
       => ( ! [X4: set_list_a] :
              ( ( member_set_list_a @ X4 @ C )
             => ( ord_le8861187494160871172list_a @ X4 @ Z2 ) )
         => ( member8857465052274545133list_a @ ( sup_su4537662296134749976list_a @ ( insert_set_list_a @ Z2 @ bot_bo3186585308812441520list_a ) @ C ) @ ( chains_list_a @ S5 ) ) ) ) ) ).

% chains_extend
thf(fact_917_chains__extend,axiom,
    ! [C: set_set_a,S5: set_set_a,Z2: set_a] :
      ( ( member_set_set_a @ C @ ( chains_a @ S5 ) )
     => ( ( member_set_a2 @ Z2 @ S5 )
       => ( ! [X4: set_a] :
              ( ( member_set_a2 @ X4 @ C )
             => ( ord_less_eq_set_a @ X4 @ Z2 ) )
         => ( member_set_set_a @ ( sup_sup_set_set_a @ ( insert_set_a @ Z2 @ bot_bot_set_set_a ) @ C ) @ ( chains_a @ S5 ) ) ) ) ) ).

% chains_extend
thf(fact_918_subset__Compl__singleton,axiom,
    ! [A2: set_list_a,B3: list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ ( uminus7925729386456332763list_a @ ( insert_list_a2 @ B3 @ bot_bot_set_list_a ) ) )
      = ( ~ ( member_list_a2 @ B3 @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_919_subset__Compl__singleton,axiom,
    ! [A2: set_a,B3: a] :
      ( ( ord_less_eq_set_a @ A2 @ ( uminus_uminus_set_a @ ( insert_a2 @ B3 @ bot_bot_set_a ) ) )
      = ( ~ ( member_a2 @ B3 @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_920_Diff__iff,axiom,
    ! [C: list_a,A2: set_list_a,B: set_list_a] :
      ( ( member_list_a2 @ C @ ( minus_646659088055828811list_a @ A2 @ B ) )
      = ( ( member_list_a2 @ C @ A2 )
        & ~ ( member_list_a2 @ C @ B ) ) ) ).

% Diff_iff
thf(fact_921_DiffI,axiom,
    ! [C: list_a,A2: set_list_a,B: set_list_a] :
      ( ( member_list_a2 @ C @ A2 )
     => ( ~ ( member_list_a2 @ C @ B )
       => ( member_list_a2 @ C @ ( minus_646659088055828811list_a @ A2 @ B ) ) ) ) ).

% DiffI
thf(fact_922_Compl__iff,axiom,
    ! [C: list_a,A2: set_list_a] :
      ( ( member_list_a2 @ C @ ( uminus7925729386456332763list_a @ A2 ) )
      = ( ~ ( member_list_a2 @ C @ A2 ) ) ) ).

% Compl_iff
thf(fact_923_ComplI,axiom,
    ! [C: list_a,A2: set_list_a] :
      ( ~ ( member_list_a2 @ C @ A2 )
     => ( member_list_a2 @ C @ ( uminus7925729386456332763list_a @ A2 ) ) ) ).

% ComplI
thf(fact_924_insert__Diff1,axiom,
    ! [X2: list_a,B: set_list_a,A2: set_list_a] :
      ( ( member_list_a2 @ X2 @ B )
     => ( ( minus_646659088055828811list_a @ ( insert_list_a2 @ X2 @ A2 ) @ B )
        = ( minus_646659088055828811list_a @ A2 @ B ) ) ) ).

% insert_Diff1
thf(fact_925_Diff__insert0,axiom,
    ! [X2: list_a,A2: set_list_a,B: set_list_a] :
      ( ~ ( member_list_a2 @ X2 @ A2 )
     => ( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a2 @ X2 @ B ) )
        = ( minus_646659088055828811list_a @ A2 @ B ) ) ) ).

% Diff_insert0
thf(fact_926_Compl__subset__Compl__iff,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( uminus7925729386456332763list_a @ A2 ) @ ( uminus7925729386456332763list_a @ B ) )
      = ( ord_le8861187494160871172list_a @ B @ A2 ) ) ).

% Compl_subset_Compl_iff
thf(fact_927_Compl__subset__Compl__iff,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A2 ) @ ( uminus_uminus_set_a @ B ) )
      = ( ord_less_eq_set_a @ B @ A2 ) ) ).

% Compl_subset_Compl_iff
thf(fact_928_Compl__anti__mono,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B )
     => ( ord_le8861187494160871172list_a @ ( uminus7925729386456332763list_a @ B ) @ ( uminus7925729386456332763list_a @ A2 ) ) ) ).

% Compl_anti_mono
thf(fact_929_Compl__anti__mono,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B ) @ ( uminus_uminus_set_a @ A2 ) ) ) ).

% Compl_anti_mono
thf(fact_930_Diff__eq__empty__iff,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( ( minus_646659088055828811list_a @ A2 @ B )
        = bot_bot_set_list_a )
      = ( ord_le8861187494160871172list_a @ A2 @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_931_Diff__eq__empty__iff,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ( minus_minus_set_a @ A2 @ B )
        = bot_bot_set_a )
      = ( ord_less_eq_set_a @ A2 @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_932_insert__Diff__if,axiom,
    ! [X2: list_a,B: set_list_a,A2: set_list_a] :
      ( ( ( member_list_a2 @ X2 @ B )
       => ( ( minus_646659088055828811list_a @ ( insert_list_a2 @ X2 @ A2 ) @ B )
          = ( minus_646659088055828811list_a @ A2 @ B ) ) )
      & ( ~ ( member_list_a2 @ X2 @ B )
       => ( ( minus_646659088055828811list_a @ ( insert_list_a2 @ X2 @ A2 ) @ B )
          = ( insert_list_a2 @ X2 @ ( minus_646659088055828811list_a @ A2 @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_933_DiffD2,axiom,
    ! [C: list_a,A2: set_list_a,B: set_list_a] :
      ( ( member_list_a2 @ C @ ( minus_646659088055828811list_a @ A2 @ B ) )
     => ~ ( member_list_a2 @ C @ B ) ) ).

% DiffD2
thf(fact_934_DiffD1,axiom,
    ! [C: list_a,A2: set_list_a,B: set_list_a] :
      ( ( member_list_a2 @ C @ ( minus_646659088055828811list_a @ A2 @ B ) )
     => ( member_list_a2 @ C @ A2 ) ) ).

% DiffD1
thf(fact_935_ComplD,axiom,
    ! [C: list_a,A2: set_list_a] :
      ( ( member_list_a2 @ C @ ( uminus7925729386456332763list_a @ A2 ) )
     => ~ ( member_list_a2 @ C @ A2 ) ) ).

% ComplD
thf(fact_936_DiffE,axiom,
    ! [C: list_a,A2: set_list_a,B: set_list_a] :
      ( ( member_list_a2 @ C @ ( minus_646659088055828811list_a @ A2 @ B ) )
     => ~ ( ( member_list_a2 @ C @ A2 )
         => ( member_list_a2 @ C @ B ) ) ) ).

% DiffE
thf(fact_937_Diff__mono,axiom,
    ! [A2: set_list_a,C2: set_list_a,D: set_list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ C2 )
     => ( ( ord_le8861187494160871172list_a @ D @ B )
       => ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ B ) @ ( minus_646659088055828811list_a @ C2 @ D ) ) ) ) ).

% Diff_mono
thf(fact_938_Diff__mono,axiom,
    ! [A2: set_a,C2: set_a,D: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ C2 )
     => ( ( ord_less_eq_set_a @ D @ B )
       => ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ C2 @ D ) ) ) ) ).

% Diff_mono
thf(fact_939_Diff__subset,axiom,
    ! [A2: set_list_a,B: set_list_a] : ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ B ) @ A2 ) ).

% Diff_subset
thf(fact_940_Diff__subset,axiom,
    ! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ A2 ) ).

% Diff_subset
thf(fact_941_double__diff,axiom,
    ! [A2: set_list_a,B: set_list_a,C2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B )
     => ( ( ord_le8861187494160871172list_a @ B @ C2 )
       => ( ( minus_646659088055828811list_a @ B @ ( minus_646659088055828811list_a @ C2 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_942_double__diff,axiom,
    ! [A2: set_a,B: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( ord_less_eq_set_a @ B @ C2 )
       => ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C2 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_943_Zorn__Lemma2,axiom,
    ! [A2: set_set_list_a] :
      ( ! [X4: set_set_list_a] :
          ( ( member8857465052274545133list_a @ X4 @ ( chains_list_a @ A2 ) )
         => ? [Xa3: set_list_a] :
              ( ( member_set_list_a @ Xa3 @ A2 )
              & ! [Xb: set_list_a] :
                  ( ( member_set_list_a @ Xb @ X4 )
                 => ( ord_le8861187494160871172list_a @ Xb @ Xa3 ) ) ) )
     => ? [X4: set_list_a] :
          ( ( member_set_list_a @ X4 @ A2 )
          & ! [Xa3: set_list_a] :
              ( ( member_set_list_a @ Xa3 @ A2 )
             => ( ( ord_le8861187494160871172list_a @ X4 @ Xa3 )
               => ( Xa3 = X4 ) ) ) ) ) ).

% Zorn_Lemma2
thf(fact_944_Zorn__Lemma2,axiom,
    ! [A2: set_set_a] :
      ( ! [X4: set_set_a] :
          ( ( member_set_set_a @ X4 @ ( chains_a @ A2 ) )
         => ? [Xa3: set_a] :
              ( ( member_set_a2 @ Xa3 @ A2 )
              & ! [Xb: set_a] :
                  ( ( member_set_a2 @ Xb @ X4 )
                 => ( ord_less_eq_set_a @ Xb @ Xa3 ) ) ) )
     => ? [X4: set_a] :
          ( ( member_set_a2 @ X4 @ A2 )
          & ! [Xa3: set_a] :
              ( ( member_set_a2 @ Xa3 @ A2 )
             => ( ( ord_less_eq_set_a @ X4 @ Xa3 )
               => ( Xa3 = X4 ) ) ) ) ) ).

% Zorn_Lemma2
thf(fact_945_chainsD,axiom,
    ! [C: set_set_list_a,S5: set_set_list_a,X2: set_list_a,Y: set_list_a] :
      ( ( member8857465052274545133list_a @ C @ ( chains_list_a @ S5 ) )
     => ( ( member_set_list_a @ X2 @ C )
       => ( ( member_set_list_a @ Y @ C )
         => ( ( ord_le8861187494160871172list_a @ X2 @ Y )
            | ( ord_le8861187494160871172list_a @ Y @ X2 ) ) ) ) ) ).

% chainsD
thf(fact_946_chainsD,axiom,
    ! [C: set_set_a,S5: set_set_a,X2: set_a,Y: set_a] :
      ( ( member_set_set_a @ C @ ( chains_a @ S5 ) )
     => ( ( member_set_a2 @ X2 @ C )
       => ( ( member_set_a2 @ Y @ C )
         => ( ( ord_less_eq_set_a @ X2 @ Y )
            | ( ord_less_eq_set_a @ Y @ X2 ) ) ) ) ) ).

% chainsD
thf(fact_947_insert__Diff,axiom,
    ! [A: list_a,A2: set_list_a] :
      ( ( member_list_a2 @ A @ A2 )
     => ( ( insert_list_a2 @ A @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a2 @ A @ bot_bot_set_list_a ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_948_Diff__insert__absorb,axiom,
    ! [X2: list_a,A2: set_list_a] :
      ( ~ ( member_list_a2 @ X2 @ A2 )
     => ( ( minus_646659088055828811list_a @ ( insert_list_a2 @ X2 @ A2 ) @ ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_949_subset__Compl__self__eq,axiom,
    ! [A2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ ( uminus7925729386456332763list_a @ A2 ) )
      = ( A2 = bot_bot_set_list_a ) ) ).

% subset_Compl_self_eq
thf(fact_950_subset__Compl__self__eq,axiom,
    ! [A2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( uminus_uminus_set_a @ A2 ) )
      = ( A2 = bot_bot_set_a ) ) ).

% subset_Compl_self_eq
thf(fact_951_subset__Diff__insert,axiom,
    ! [A2: set_list_a,B: set_list_a,X2: list_a,C2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ ( minus_646659088055828811list_a @ B @ ( insert_list_a2 @ X2 @ C2 ) ) )
      = ( ( ord_le8861187494160871172list_a @ A2 @ ( minus_646659088055828811list_a @ B @ C2 ) )
        & ~ ( member_list_a2 @ X2 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_952_subset__Diff__insert,axiom,
    ! [A2: set_a,B: set_a,X2: a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ ( insert_a2 @ X2 @ C2 ) ) )
      = ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ C2 ) )
        & ~ ( member_a2 @ X2 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_953_Diff__partition,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B )
     => ( ( sup_sup_set_list_a @ A2 @ ( minus_646659088055828811list_a @ B @ A2 ) )
        = B ) ) ).

% Diff_partition
thf(fact_954_Diff__partition,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
        = B ) ) ).

% Diff_partition
thf(fact_955_Diff__subset__conv,axiom,
    ! [A2: set_list_a,B: set_list_a,C2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ B ) @ C2 )
      = ( ord_le8861187494160871172list_a @ A2 @ ( sup_sup_set_list_a @ B @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_956_Diff__subset__conv,axiom,
    ! [A2: set_a,B: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ C2 )
      = ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ B @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_957_subset__insert__iff,axiom,
    ! [A2: set_list_a,X2: list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a2 @ X2 @ B ) )
      = ( ( ( member_list_a2 @ X2 @ A2 )
         => ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) ) @ B ) )
        & ( ~ ( member_list_a2 @ X2 @ A2 )
         => ( ord_le8861187494160871172list_a @ A2 @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_958_subset__insert__iff,axiom,
    ! [A2: set_a,X2: a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( insert_a2 @ X2 @ B ) )
      = ( ( ( member_a2 @ X2 @ A2 )
         => ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a2 @ X2 @ bot_bot_set_a ) ) @ B ) )
        & ( ~ ( member_a2 @ X2 @ A2 )
         => ( ord_less_eq_set_a @ A2 @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_959_Diff__single__insert,axiom,
    ! [A2: set_list_a,X2: list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) ) @ B )
     => ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a2 @ X2 @ B ) ) ) ).

% Diff_single_insert
thf(fact_960_Diff__single__insert,axiom,
    ! [A2: set_a,X2: a,B: set_a] :
      ( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a2 @ X2 @ bot_bot_set_a ) ) @ B )
     => ( ord_less_eq_set_a @ A2 @ ( insert_a2 @ X2 @ B ) ) ) ).

% Diff_single_insert
thf(fact_961_in__image__insert__iff,axiom,
    ! [B: set_set_list_a,X2: list_a,A2: set_list_a] :
      ( ! [C3: set_list_a] :
          ( ( member_set_list_a @ C3 @ B )
         => ~ ( member_list_a2 @ X2 @ C3 ) )
     => ( ( member_set_list_a @ A2 @ ( image_5749939591322298757list_a @ ( insert_list_a2 @ X2 ) @ B ) )
        = ( ( member_list_a2 @ X2 @ A2 )
          & ( member_set_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) ) @ B ) ) ) ) ).

% in_image_insert_iff
thf(fact_962_compl__le__compl__iff,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( uminus7925729386456332763list_a @ X2 ) @ ( uminus7925729386456332763list_a @ Y ) )
      = ( ord_le8861187494160871172list_a @ Y @ X2 ) ) ).

% compl_le_compl_iff
thf(fact_963_compl__le__compl__iff,axiom,
    ! [X2: set_a,Y: set_a] :
      ( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X2 ) @ ( uminus_uminus_set_a @ Y ) )
      = ( ord_less_eq_set_a @ Y @ X2 ) ) ).

% compl_le_compl_iff
thf(fact_964_diff__shunt__var,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( ( minus_646659088055828811list_a @ X2 @ Y )
        = bot_bot_set_list_a )
      = ( ord_le8861187494160871172list_a @ X2 @ Y ) ) ).

% diff_shunt_var
thf(fact_965_diff__shunt__var,axiom,
    ! [X2: set_a,Y: set_a] :
      ( ( ( minus_minus_set_a @ X2 @ Y )
        = bot_bot_set_a )
      = ( ord_less_eq_set_a @ X2 @ Y ) ) ).

% diff_shunt_var
thf(fact_966_compl__mono,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ X2 @ Y )
     => ( ord_le8861187494160871172list_a @ ( uminus7925729386456332763list_a @ Y ) @ ( uminus7925729386456332763list_a @ X2 ) ) ) ).

% compl_mono
thf(fact_967_compl__mono,axiom,
    ! [X2: set_a,Y: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ Y )
     => ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ ( uminus_uminus_set_a @ X2 ) ) ) ).

% compl_mono
thf(fact_968_compl__le__swap1,axiom,
    ! [Y: set_list_a,X2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ Y @ ( uminus7925729386456332763list_a @ X2 ) )
     => ( ord_le8861187494160871172list_a @ X2 @ ( uminus7925729386456332763list_a @ Y ) ) ) ).

% compl_le_swap1
thf(fact_969_compl__le__swap1,axiom,
    ! [Y: set_a,X2: set_a] :
      ( ( ord_less_eq_set_a @ Y @ ( uminus_uminus_set_a @ X2 ) )
     => ( ord_less_eq_set_a @ X2 @ ( uminus_uminus_set_a @ Y ) ) ) ).

% compl_le_swap1
thf(fact_970_compl__le__swap2,axiom,
    ! [Y: set_list_a,X2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( uminus7925729386456332763list_a @ Y ) @ X2 )
     => ( ord_le8861187494160871172list_a @ ( uminus7925729386456332763list_a @ X2 ) @ Y ) ) ).

% compl_le_swap2
thf(fact_971_compl__le__swap2,axiom,
    ! [Y: set_a,X2: set_a] :
      ( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ X2 )
     => ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X2 ) @ Y ) ) ).

% compl_le_swap2
thf(fact_972_psubset__insert__iff,axiom,
    ! [A2: set_list_a,X2: list_a,B: set_list_a] :
      ( ( ord_less_set_list_a @ A2 @ ( insert_list_a2 @ X2 @ B ) )
      = ( ( ( member_list_a2 @ X2 @ B )
         => ( ord_less_set_list_a @ A2 @ B ) )
        & ( ~ ( member_list_a2 @ X2 @ B )
         => ( ( ( member_list_a2 @ X2 @ A2 )
             => ( ord_less_set_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) ) @ B ) )
            & ( ~ ( member_list_a2 @ X2 @ A2 )
             => ( ord_le8861187494160871172list_a @ A2 @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_973_psubset__insert__iff,axiom,
    ! [A2: set_a,X2: a,B: set_a] :
      ( ( ord_less_set_a @ A2 @ ( insert_a2 @ X2 @ B ) )
      = ( ( ( member_a2 @ X2 @ B )
         => ( ord_less_set_a @ A2 @ B ) )
        & ( ~ ( member_a2 @ X2 @ B )
         => ( ( ( member_a2 @ X2 @ A2 )
             => ( ord_less_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a2 @ X2 @ bot_bot_set_a ) ) @ B ) )
            & ( ~ ( member_a2 @ X2 @ A2 )
             => ( ord_less_eq_set_a @ A2 @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_974_psubsetI,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B )
     => ( ( A2 != B )
       => ( ord_less_set_list_a @ A2 @ B ) ) ) ).

% psubsetI
thf(fact_975_psubsetI,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( A2 != B )
       => ( ord_less_set_a @ A2 @ B ) ) ) ).

% psubsetI
thf(fact_976_psubset__imp__ex__mem,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( ord_less_set_list_a @ A2 @ B )
     => ? [B7: list_a] : ( member_list_a2 @ B7 @ ( minus_646659088055828811list_a @ B @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_977_distinct_Osimps_I1_J,axiom,
    distinct_a @ nil_a ).

% distinct.simps(1)
thf(fact_978_pairwiseI,axiom,
    ! [S5: set_list_a,R: list_a > list_a > $o] :
      ( ! [X4: list_a,Y2: list_a] :
          ( ( member_list_a2 @ X4 @ S5 )
         => ( ( member_list_a2 @ Y2 @ S5 )
           => ( ( X4 != Y2 )
             => ( R @ X4 @ Y2 ) ) ) )
     => ( pairwise_list_a @ R @ S5 ) ) ).

% pairwiseI
thf(fact_979_pairwiseD,axiom,
    ! [R: list_a > list_a > $o,S5: set_list_a,X2: list_a,Y: list_a] :
      ( ( pairwise_list_a @ R @ S5 )
     => ( ( member_list_a2 @ X2 @ S5 )
       => ( ( member_list_a2 @ Y @ S5 )
         => ( ( X2 != Y )
           => ( R @ X2 @ Y ) ) ) ) ) ).

% pairwiseD
thf(fact_980_psubsetD,axiom,
    ! [A2: set_list_a,B: set_list_a,C: list_a] :
      ( ( ord_less_set_list_a @ A2 @ B )
     => ( ( member_list_a2 @ C @ A2 )
       => ( member_list_a2 @ C @ B ) ) ) ).

% psubsetD
thf(fact_981_psubsetE,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( ord_less_set_list_a @ A2 @ B )
     => ~ ( ( ord_le8861187494160871172list_a @ A2 @ B )
         => ( ord_le8861187494160871172list_a @ B @ A2 ) ) ) ).

% psubsetE
thf(fact_982_psubsetE,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ord_less_set_a @ A2 @ B )
     => ~ ( ( ord_less_eq_set_a @ A2 @ B )
         => ( ord_less_eq_set_a @ B @ A2 ) ) ) ).

% psubsetE
thf(fact_983_psubset__eq,axiom,
    ( ord_less_set_list_a
    = ( ^ [A3: set_list_a,B2: set_list_a] :
          ( ( ord_le8861187494160871172list_a @ A3 @ B2 )
          & ( A3 != B2 ) ) ) ) ).

% psubset_eq
thf(fact_984_psubset__eq,axiom,
    ( ord_less_set_a
    = ( ^ [A3: set_a,B2: set_a] :
          ( ( ord_less_eq_set_a @ A3 @ B2 )
          & ( A3 != B2 ) ) ) ) ).

% psubset_eq
thf(fact_985_psubset__imp__subset,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( ord_less_set_list_a @ A2 @ B )
     => ( ord_le8861187494160871172list_a @ A2 @ B ) ) ).

% psubset_imp_subset
thf(fact_986_psubset__imp__subset,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ord_less_set_a @ A2 @ B )
     => ( ord_less_eq_set_a @ A2 @ B ) ) ).

% psubset_imp_subset
thf(fact_987_psubset__subset__trans,axiom,
    ! [A2: set_list_a,B: set_list_a,C2: set_list_a] :
      ( ( ord_less_set_list_a @ A2 @ B )
     => ( ( ord_le8861187494160871172list_a @ B @ C2 )
       => ( ord_less_set_list_a @ A2 @ C2 ) ) ) ).

% psubset_subset_trans
thf(fact_988_psubset__subset__trans,axiom,
    ! [A2: set_a,B: set_a,C2: set_a] :
      ( ( ord_less_set_a @ A2 @ B )
     => ( ( ord_less_eq_set_a @ B @ C2 )
       => ( ord_less_set_a @ A2 @ C2 ) ) ) ).

% psubset_subset_trans
thf(fact_989_subset__not__subset__eq,axiom,
    ( ord_less_set_list_a
    = ( ^ [A3: set_list_a,B2: set_list_a] :
          ( ( ord_le8861187494160871172list_a @ A3 @ B2 )
          & ~ ( ord_le8861187494160871172list_a @ B2 @ A3 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_990_subset__not__subset__eq,axiom,
    ( ord_less_set_a
    = ( ^ [A3: set_a,B2: set_a] :
          ( ( ord_less_eq_set_a @ A3 @ B2 )
          & ~ ( ord_less_eq_set_a @ B2 @ A3 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_991_subset__psubset__trans,axiom,
    ! [A2: set_list_a,B: set_list_a,C2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B )
     => ( ( ord_less_set_list_a @ B @ C2 )
       => ( ord_less_set_list_a @ A2 @ C2 ) ) ) ).

% subset_psubset_trans
thf(fact_992_subset__psubset__trans,axiom,
    ! [A2: set_a,B: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( ord_less_set_a @ B @ C2 )
       => ( ord_less_set_a @ A2 @ C2 ) ) ) ).

% subset_psubset_trans
thf(fact_993_subset__iff__psubset__eq,axiom,
    ( ord_le8861187494160871172list_a
    = ( ^ [A3: set_list_a,B2: set_list_a] :
          ( ( ord_less_set_list_a @ A3 @ B2 )
          | ( A3 = B2 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_994_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A3: set_a,B2: set_a] :
          ( ( ord_less_set_a @ A3 @ B2 )
          | ( A3 = B2 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_995_leD,axiom,
    ! [Y: set_list_a,X2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ Y @ X2 )
     => ~ ( ord_less_set_list_a @ X2 @ Y ) ) ).

% leD
thf(fact_996_leD,axiom,
    ! [Y: set_a,X2: set_a] :
      ( ( ord_less_eq_set_a @ Y @ X2 )
     => ~ ( ord_less_set_a @ X2 @ Y ) ) ).

% leD
thf(fact_997_nless__le,axiom,
    ! [A: set_list_a,B3: set_list_a] :
      ( ( ~ ( ord_less_set_list_a @ A @ B3 ) )
      = ( ~ ( ord_le8861187494160871172list_a @ A @ B3 )
        | ( A = B3 ) ) ) ).

% nless_le
thf(fact_998_nless__le,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( ~ ( ord_less_set_a @ A @ B3 ) )
      = ( ~ ( ord_less_eq_set_a @ A @ B3 )
        | ( A = B3 ) ) ) ).

% nless_le
thf(fact_999_antisym__conv1,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ~ ( ord_less_set_list_a @ X2 @ Y )
     => ( ( ord_le8861187494160871172list_a @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% antisym_conv1
thf(fact_1000_antisym__conv1,axiom,
    ! [X2: set_a,Y: set_a] :
      ( ~ ( ord_less_set_a @ X2 @ Y )
     => ( ( ord_less_eq_set_a @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% antisym_conv1
thf(fact_1001_antisym__conv2,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ X2 @ Y )
     => ( ( ~ ( ord_less_set_list_a @ X2 @ Y ) )
        = ( X2 = Y ) ) ) ).

% antisym_conv2
thf(fact_1002_antisym__conv2,axiom,
    ! [X2: set_a,Y: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ Y )
     => ( ( ~ ( ord_less_set_a @ X2 @ Y ) )
        = ( X2 = Y ) ) ) ).

% antisym_conv2
thf(fact_1003_less__le__not__le,axiom,
    ( ord_less_set_list_a
    = ( ^ [X3: set_list_a,Y4: set_list_a] :
          ( ( ord_le8861187494160871172list_a @ X3 @ Y4 )
          & ~ ( ord_le8861187494160871172list_a @ Y4 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_1004_less__le__not__le,axiom,
    ( ord_less_set_a
    = ( ^ [X3: set_a,Y4: set_a] :
          ( ( ord_less_eq_set_a @ X3 @ Y4 )
          & ~ ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_1005_order_Oorder__iff__strict,axiom,
    ( ord_le8861187494160871172list_a
    = ( ^ [A4: set_list_a,B4: set_list_a] :
          ( ( ord_less_set_list_a @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1006_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A4: set_a,B4: set_a] :
          ( ( ord_less_set_a @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1007_order_Ostrict__iff__order,axiom,
    ( ord_less_set_list_a
    = ( ^ [A4: set_list_a,B4: set_list_a] :
          ( ( ord_le8861187494160871172list_a @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1008_order_Ostrict__iff__order,axiom,
    ( ord_less_set_a
    = ( ^ [A4: set_a,B4: set_a] :
          ( ( ord_less_eq_set_a @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1009_order_Ostrict__trans1,axiom,
    ! [A: set_list_a,B3: set_list_a,C: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ B3 )
     => ( ( ord_less_set_list_a @ B3 @ C )
       => ( ord_less_set_list_a @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1010_order_Ostrict__trans1,axiom,
    ! [A: set_a,B3: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( ord_less_set_a @ B3 @ C )
       => ( ord_less_set_a @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1011_order_Ostrict__trans2,axiom,
    ! [A: set_list_a,B3: set_list_a,C: set_list_a] :
      ( ( ord_less_set_list_a @ A @ B3 )
     => ( ( ord_le8861187494160871172list_a @ B3 @ C )
       => ( ord_less_set_list_a @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1012_order_Ostrict__trans2,axiom,
    ! [A: set_a,B3: set_a,C: set_a] :
      ( ( ord_less_set_a @ A @ B3 )
     => ( ( ord_less_eq_set_a @ B3 @ C )
       => ( ord_less_set_a @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1013_order_Ostrict__iff__not,axiom,
    ( ord_less_set_list_a
    = ( ^ [A4: set_list_a,B4: set_list_a] :
          ( ( ord_le8861187494160871172list_a @ A4 @ B4 )
          & ~ ( ord_le8861187494160871172list_a @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1014_order_Ostrict__iff__not,axiom,
    ( ord_less_set_a
    = ( ^ [A4: set_a,B4: set_a] :
          ( ( ord_less_eq_set_a @ A4 @ B4 )
          & ~ ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1015_dual__order_Oorder__iff__strict,axiom,
    ( ord_le8861187494160871172list_a
    = ( ^ [B4: set_list_a,A4: set_list_a] :
          ( ( ord_less_set_list_a @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1016_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_a
    = ( ^ [B4: set_a,A4: set_a] :
          ( ( ord_less_set_a @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1017_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_list_a
    = ( ^ [B4: set_list_a,A4: set_list_a] :
          ( ( ord_le8861187494160871172list_a @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1018_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_a
    = ( ^ [B4: set_a,A4: set_a] :
          ( ( ord_less_eq_set_a @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1019_dual__order_Ostrict__trans1,axiom,
    ! [B3: set_list_a,A: set_list_a,C: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ B3 @ A )
     => ( ( ord_less_set_list_a @ C @ B3 )
       => ( ord_less_set_list_a @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1020_dual__order_Ostrict__trans1,axiom,
    ! [B3: set_a,A: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ B3 @ A )
     => ( ( ord_less_set_a @ C @ B3 )
       => ( ord_less_set_a @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1021_dual__order_Ostrict__trans2,axiom,
    ! [B3: set_list_a,A: set_list_a,C: set_list_a] :
      ( ( ord_less_set_list_a @ B3 @ A )
     => ( ( ord_le8861187494160871172list_a @ C @ B3 )
       => ( ord_less_set_list_a @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1022_dual__order_Ostrict__trans2,axiom,
    ! [B3: set_a,A: set_a,C: set_a] :
      ( ( ord_less_set_a @ B3 @ A )
     => ( ( ord_less_eq_set_a @ C @ B3 )
       => ( ord_less_set_a @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1023_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_list_a
    = ( ^ [B4: set_list_a,A4: set_list_a] :
          ( ( ord_le8861187494160871172list_a @ B4 @ A4 )
          & ~ ( ord_le8861187494160871172list_a @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1024_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_a
    = ( ^ [B4: set_a,A4: set_a] :
          ( ( ord_less_eq_set_a @ B4 @ A4 )
          & ~ ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1025_order_Ostrict__implies__order,axiom,
    ! [A: set_list_a,B3: set_list_a] :
      ( ( ord_less_set_list_a @ A @ B3 )
     => ( ord_le8861187494160871172list_a @ A @ B3 ) ) ).

% order.strict_implies_order
thf(fact_1026_order_Ostrict__implies__order,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( ord_less_set_a @ A @ B3 )
     => ( ord_less_eq_set_a @ A @ B3 ) ) ).

% order.strict_implies_order
thf(fact_1027_dual__order_Ostrict__implies__order,axiom,
    ! [B3: set_list_a,A: set_list_a] :
      ( ( ord_less_set_list_a @ B3 @ A )
     => ( ord_le8861187494160871172list_a @ B3 @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1028_dual__order_Ostrict__implies__order,axiom,
    ! [B3: set_a,A: set_a] :
      ( ( ord_less_set_a @ B3 @ A )
     => ( ord_less_eq_set_a @ B3 @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1029_order__le__less,axiom,
    ( ord_le8861187494160871172list_a
    = ( ^ [X3: set_list_a,Y4: set_list_a] :
          ( ( ord_less_set_list_a @ X3 @ Y4 )
          | ( X3 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_1030_order__le__less,axiom,
    ( ord_less_eq_set_a
    = ( ^ [X3: set_a,Y4: set_a] :
          ( ( ord_less_set_a @ X3 @ Y4 )
          | ( X3 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_1031_order__less__le,axiom,
    ( ord_less_set_list_a
    = ( ^ [X3: set_list_a,Y4: set_list_a] :
          ( ( ord_le8861187494160871172list_a @ X3 @ Y4 )
          & ( X3 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_1032_order__less__le,axiom,
    ( ord_less_set_a
    = ( ^ [X3: set_a,Y4: set_a] :
          ( ( ord_less_eq_set_a @ X3 @ Y4 )
          & ( X3 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_1033_order__less__imp__le,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( ord_less_set_list_a @ X2 @ Y )
     => ( ord_le8861187494160871172list_a @ X2 @ Y ) ) ).

% order_less_imp_le
thf(fact_1034_order__less__imp__le,axiom,
    ! [X2: set_a,Y: set_a] :
      ( ( ord_less_set_a @ X2 @ Y )
     => ( ord_less_eq_set_a @ X2 @ Y ) ) ).

% order_less_imp_le
thf(fact_1035_order__le__neq__trans,axiom,
    ! [A: set_list_a,B3: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ B3 )
     => ( ( A != B3 )
       => ( ord_less_set_list_a @ A @ B3 ) ) ) ).

% order_le_neq_trans
thf(fact_1036_order__le__neq__trans,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( A != B3 )
       => ( ord_less_set_a @ A @ B3 ) ) ) ).

% order_le_neq_trans
thf(fact_1037_order__neq__le__trans,axiom,
    ! [A: set_list_a,B3: set_list_a] :
      ( ( A != B3 )
     => ( ( ord_le8861187494160871172list_a @ A @ B3 )
       => ( ord_less_set_list_a @ A @ B3 ) ) ) ).

% order_neq_le_trans
thf(fact_1038_order__neq__le__trans,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( A != B3 )
     => ( ( ord_less_eq_set_a @ A @ B3 )
       => ( ord_less_set_a @ A @ B3 ) ) ) ).

% order_neq_le_trans
thf(fact_1039_order__le__less__trans,axiom,
    ! [X2: set_list_a,Y: set_list_a,Z2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ X2 @ Y )
     => ( ( ord_less_set_list_a @ Y @ Z2 )
       => ( ord_less_set_list_a @ X2 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1040_order__le__less__trans,axiom,
    ! [X2: set_a,Y: set_a,Z2: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ Y )
     => ( ( ord_less_set_a @ Y @ Z2 )
       => ( ord_less_set_a @ X2 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1041_order__less__le__trans,axiom,
    ! [X2: set_list_a,Y: set_list_a,Z2: set_list_a] :
      ( ( ord_less_set_list_a @ X2 @ Y )
     => ( ( ord_le8861187494160871172list_a @ Y @ Z2 )
       => ( ord_less_set_list_a @ X2 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1042_order__less__le__trans,axiom,
    ! [X2: set_a,Y: set_a,Z2: set_a] :
      ( ( ord_less_set_a @ X2 @ Y )
     => ( ( ord_less_eq_set_a @ Y @ Z2 )
       => ( ord_less_set_a @ X2 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1043_order__le__less__subst2,axiom,
    ! [A: set_list_a,B3: set_list_a,F: set_list_a > set_list_a,C: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ B3 )
     => ( ( ord_less_set_list_a @ ( F @ B3 ) @ C )
       => ( ! [X4: set_list_a,Y2: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X4 @ Y2 )
             => ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_list_a @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1044_order__le__less__subst2,axiom,
    ! [A: set_list_a,B3: set_list_a,F: set_list_a > set_a,C: set_a] :
      ( ( ord_le8861187494160871172list_a @ A @ B3 )
     => ( ( ord_less_set_a @ ( F @ B3 ) @ C )
       => ( ! [X4: set_list_a,Y2: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1045_order__le__less__subst2,axiom,
    ! [A: set_a,B3: set_a,F: set_a > set_list_a,C: set_list_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( ord_less_set_list_a @ ( F @ B3 ) @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_list_a @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1046_order__le__less__subst2,axiom,
    ! [A: set_a,B3: set_a,F: set_a > set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( ord_less_set_a @ ( F @ B3 ) @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1047_order__less__le__subst1,axiom,
    ! [A: set_list_a,F: set_list_a > set_list_a,B3: set_list_a,C: set_list_a] :
      ( ( ord_less_set_list_a @ A @ ( F @ B3 ) )
     => ( ( ord_le8861187494160871172list_a @ B3 @ C )
       => ( ! [X4: set_list_a,Y2: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X4 @ Y2 )
             => ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_list_a @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1048_order__less__le__subst1,axiom,
    ! [A: set_a,F: set_list_a > set_a,B3: set_list_a,C: set_list_a] :
      ( ( ord_less_set_a @ A @ ( F @ B3 ) )
     => ( ( ord_le8861187494160871172list_a @ B3 @ C )
       => ( ! [X4: set_list_a,Y2: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1049_order__less__le__subst1,axiom,
    ! [A: set_list_a,F: set_a > set_list_a,B3: set_a,C: set_a] :
      ( ( ord_less_set_list_a @ A @ ( F @ B3 ) )
     => ( ( ord_less_eq_set_a @ B3 @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_le8861187494160871172list_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_list_a @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1050_order__less__le__subst1,axiom,
    ! [A: set_a,F: set_a > set_a,B3: set_a,C: set_a] :
      ( ( ord_less_set_a @ A @ ( F @ B3 ) )
     => ( ( ord_less_eq_set_a @ B3 @ C )
       => ( ! [X4: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1051_order__le__imp__less__or__eq,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ X2 @ Y )
     => ( ( ord_less_set_list_a @ X2 @ Y )
        | ( X2 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1052_order__le__imp__less__or__eq,axiom,
    ! [X2: set_a,Y: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ Y )
     => ( ( ord_less_set_a @ X2 @ Y )
        | ( X2 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1053_pairwise__insert,axiom,
    ! [R2: list_a > list_a > $o,X2: list_a,S2: set_list_a] :
      ( ( pairwise_list_a @ R2 @ ( insert_list_a2 @ X2 @ S2 ) )
      = ( ! [Y4: list_a] :
            ( ( ( member_list_a2 @ Y4 @ S2 )
              & ( Y4 != X2 ) )
           => ( ( R2 @ X2 @ Y4 )
              & ( R2 @ Y4 @ X2 ) ) )
        & ( pairwise_list_a @ R2 @ S2 ) ) ) ).

% pairwise_insert
thf(fact_1054_pairwise__subset,axiom,
    ! [P: list_a > list_a > $o,S5: set_list_a,T3: set_list_a] :
      ( ( pairwise_list_a @ P @ S5 )
     => ( ( ord_le8861187494160871172list_a @ T3 @ S5 )
       => ( pairwise_list_a @ P @ T3 ) ) ) ).

% pairwise_subset
thf(fact_1055_pairwise__subset,axiom,
    ! [P: a > a > $o,S5: set_a,T3: set_a] :
      ( ( pairwise_a @ P @ S5 )
     => ( ( ord_less_eq_set_a @ T3 @ S5 )
       => ( pairwise_a @ P @ T3 ) ) ) ).

% pairwise_subset
thf(fact_1056_pairwise__mono,axiom,
    ! [P: list_a > list_a > $o,A2: set_list_a,Q: list_a > list_a > $o,B: set_list_a] :
      ( ( pairwise_list_a @ P @ A2 )
     => ( ! [X4: list_a,Y2: list_a] :
            ( ( P @ X4 @ Y2 )
           => ( Q @ X4 @ Y2 ) )
       => ( ( ord_le8861187494160871172list_a @ B @ A2 )
         => ( pairwise_list_a @ Q @ B ) ) ) ) ).

% pairwise_mono
thf(fact_1057_pairwise__mono,axiom,
    ! [P: a > a > $o,A2: set_a,Q: a > a > $o,B: set_a] :
      ( ( pairwise_a @ P @ A2 )
     => ( ! [X4: a,Y2: a] :
            ( ( P @ X4 @ Y2 )
           => ( Q @ X4 @ Y2 ) )
       => ( ( ord_less_eq_set_a @ B @ A2 )
         => ( pairwise_a @ Q @ B ) ) ) ) ).

% pairwise_mono
thf(fact_1058_distinct__singleton,axiom,
    ! [X2: a] : ( distinct_a @ ( cons_a @ X2 @ nil_a ) ) ).

% distinct_singleton
thf(fact_1059_distinct_Osimps_I2_J,axiom,
    ! [X2: list_a,Xs: list_list_a] :
      ( ( distinct_list_a @ ( cons_list_a @ X2 @ Xs ) )
      = ( ~ ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) )
        & ( distinct_list_a @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_1060_subseqs__distinctD,axiom,
    ! [Ys2: list_a,Xs: list_a] :
      ( ( member_list_a2 @ Ys2 @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) )
     => ( ( distinct_a @ Xs )
       => ( distinct_a @ Ys2 ) ) ) ).

% subseqs_distinctD
thf(fact_1061_not__distinct__decomp,axiom,
    ! [Ws: list_a] :
      ( ~ ( distinct_a @ Ws )
     => ? [Xs2: list_a,Ys3: list_a,Zs2: list_a,Y2: a] :
          ( Ws
          = ( append_a @ Xs2 @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ ( append_a @ Ys3 @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Zs2 ) ) ) ) ) ) ).

% not_distinct_decomp
thf(fact_1062_not__distinct__conv__prefix,axiom,
    ! [As: list_list_a] :
      ( ( ~ ( distinct_list_a @ As ) )
      = ( ? [Xs3: list_list_a,Y4: list_a,Ys: list_list_a] :
            ( ( member_list_a2 @ Y4 @ ( set_list_a2 @ Xs3 ) )
            & ( distinct_list_a @ Xs3 )
            & ( As
              = ( append_list_a @ Xs3 @ ( cons_list_a @ Y4 @ Ys ) ) ) ) ) ) ).

% not_distinct_conv_prefix
thf(fact_1063_distinct__concat__iff,axiom,
    ! [Xs: list_list_a] :
      ( ( distinct_a @ ( concat_a @ Xs ) )
      = ( ( distinct_list_a @ ( removeAll_list_a @ nil_a @ Xs ) )
        & ! [Ys: list_a] :
            ( ( member_list_a2 @ Ys @ ( set_list_a2 @ Xs ) )
           => ( distinct_a @ Ys ) )
        & ! [Ys: list_a,Zs3: list_a] :
            ( ( ( member_list_a2 @ Ys @ ( set_list_a2 @ Xs ) )
              & ( member_list_a2 @ Zs3 @ ( set_list_a2 @ Xs ) )
              & ( Ys != Zs3 ) )
           => ( ( inf_inf_set_a @ ( set_a2 @ Ys ) @ ( set_a2 @ Zs3 ) )
              = bot_bot_set_a ) ) ) ) ).

% distinct_concat_iff
thf(fact_1064_distinct__concat,axiom,
    ! [Xs: list_list_a] :
      ( ( distinct_list_a @ Xs )
     => ( ! [Ys3: list_a] :
            ( ( member_list_a2 @ Ys3 @ ( set_list_a2 @ Xs ) )
           => ( distinct_a @ Ys3 ) )
       => ( ! [Ys3: list_a,Zs2: list_a] :
              ( ( member_list_a2 @ Ys3 @ ( set_list_a2 @ Xs ) )
             => ( ( member_list_a2 @ Zs2 @ ( set_list_a2 @ Xs ) )
               => ( ( Ys3 != Zs2 )
                 => ( ( inf_inf_set_a @ ( set_a2 @ Ys3 ) @ ( set_a2 @ Zs2 ) )
                    = bot_bot_set_a ) ) ) )
         => ( distinct_a @ ( concat_a @ Xs ) ) ) ) ) ).

% distinct_concat
thf(fact_1065_Int__iff,axiom,
    ! [C: list_a,A2: set_list_a,B: set_list_a] :
      ( ( member_list_a2 @ C @ ( inf_inf_set_list_a @ A2 @ B ) )
      = ( ( member_list_a2 @ C @ A2 )
        & ( member_list_a2 @ C @ B ) ) ) ).

% Int_iff
thf(fact_1066_IntI,axiom,
    ! [C: list_a,A2: set_list_a,B: set_list_a] :
      ( ( member_list_a2 @ C @ A2 )
     => ( ( member_list_a2 @ C @ B )
       => ( member_list_a2 @ C @ ( inf_inf_set_list_a @ A2 @ B ) ) ) ) ).

% IntI
thf(fact_1067_le__inf__iff,axiom,
    ! [X2: set_list_a,Y: set_list_a,Z2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ X2 @ ( inf_inf_set_list_a @ Y @ Z2 ) )
      = ( ( ord_le8861187494160871172list_a @ X2 @ Y )
        & ( ord_le8861187494160871172list_a @ X2 @ Z2 ) ) ) ).

% le_inf_iff
thf(fact_1068_le__inf__iff,axiom,
    ! [X2: set_a,Y: set_a,Z2: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z2 ) )
      = ( ( ord_less_eq_set_a @ X2 @ Y )
        & ( ord_less_eq_set_a @ X2 @ Z2 ) ) ) ).

% le_inf_iff
thf(fact_1069_inf_Obounded__iff,axiom,
    ! [A: set_list_a,B3: set_list_a,C: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ ( inf_inf_set_list_a @ B3 @ C ) )
      = ( ( ord_le8861187494160871172list_a @ A @ B3 )
        & ( ord_le8861187494160871172list_a @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_1070_inf_Obounded__iff,axiom,
    ! [A: set_a,B3: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B3 @ C ) )
      = ( ( ord_less_eq_set_a @ A @ B3 )
        & ( ord_less_eq_set_a @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_1071_Int__subset__iff,axiom,
    ! [C2: set_list_a,A2: set_list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ C2 @ ( inf_inf_set_list_a @ A2 @ B ) )
      = ( ( ord_le8861187494160871172list_a @ C2 @ A2 )
        & ( ord_le8861187494160871172list_a @ C2 @ B ) ) ) ).

% Int_subset_iff
thf(fact_1072_Int__subset__iff,axiom,
    ! [C2: set_a,A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
      = ( ( ord_less_eq_set_a @ C2 @ A2 )
        & ( ord_less_eq_set_a @ C2 @ B ) ) ) ).

% Int_subset_iff
thf(fact_1073_Int__insert__right__if1,axiom,
    ! [A: list_a,A2: set_list_a,B: set_list_a] :
      ( ( member_list_a2 @ A @ A2 )
     => ( ( inf_inf_set_list_a @ A2 @ ( insert_list_a2 @ A @ B ) )
        = ( insert_list_a2 @ A @ ( inf_inf_set_list_a @ A2 @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1074_Int__insert__right__if0,axiom,
    ! [A: list_a,A2: set_list_a,B: set_list_a] :
      ( ~ ( member_list_a2 @ A @ A2 )
     => ( ( inf_inf_set_list_a @ A2 @ ( insert_list_a2 @ A @ B ) )
        = ( inf_inf_set_list_a @ A2 @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_1075_Int__insert__left__if1,axiom,
    ! [A: list_a,C2: set_list_a,B: set_list_a] :
      ( ( member_list_a2 @ A @ C2 )
     => ( ( inf_inf_set_list_a @ ( insert_list_a2 @ A @ B ) @ C2 )
        = ( insert_list_a2 @ A @ ( inf_inf_set_list_a @ B @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1076_Int__insert__left__if0,axiom,
    ! [A: list_a,C2: set_list_a,B: set_list_a] :
      ( ~ ( member_list_a2 @ A @ C2 )
     => ( ( inf_inf_set_list_a @ ( insert_list_a2 @ A @ B ) @ C2 )
        = ( inf_inf_set_list_a @ B @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_1077_disjoint__insert_I2_J,axiom,
    ! [A2: set_list_a,B3: list_a,B: set_list_a] :
      ( ( bot_bot_set_list_a
        = ( inf_inf_set_list_a @ A2 @ ( insert_list_a2 @ B3 @ B ) ) )
      = ( ~ ( member_list_a2 @ B3 @ A2 )
        & ( bot_bot_set_list_a
          = ( inf_inf_set_list_a @ A2 @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1078_disjoint__insert_I1_J,axiom,
    ! [B: set_list_a,A: list_a,A2: set_list_a] :
      ( ( ( inf_inf_set_list_a @ B @ ( insert_list_a2 @ A @ A2 ) )
        = bot_bot_set_list_a )
      = ( ~ ( member_list_a2 @ A @ B )
        & ( ( inf_inf_set_list_a @ B @ A2 )
          = bot_bot_set_list_a ) ) ) ).

% disjoint_insert(1)
thf(fact_1079_insert__disjoint_I2_J,axiom,
    ! [A: list_a,A2: set_list_a,B: set_list_a] :
      ( ( bot_bot_set_list_a
        = ( inf_inf_set_list_a @ ( insert_list_a2 @ A @ A2 ) @ B ) )
      = ( ~ ( member_list_a2 @ A @ B )
        & ( bot_bot_set_list_a
          = ( inf_inf_set_list_a @ A2 @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1080_insert__disjoint_I1_J,axiom,
    ! [A: list_a,A2: set_list_a,B: set_list_a] :
      ( ( ( inf_inf_set_list_a @ ( insert_list_a2 @ A @ A2 ) @ B )
        = bot_bot_set_list_a )
      = ( ~ ( member_list_a2 @ A @ B )
        & ( ( inf_inf_set_list_a @ A2 @ B )
          = bot_bot_set_list_a ) ) ) ).

% insert_disjoint(1)
thf(fact_1081_distrib__inf__le,axiom,
    ! [X2: set_list_a,Y: set_list_a,Z2: set_list_a] : ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ X2 @ Y ) @ ( inf_inf_set_list_a @ X2 @ Z2 ) ) @ ( inf_inf_set_list_a @ X2 @ ( sup_sup_set_list_a @ Y @ Z2 ) ) ) ).

% distrib_inf_le
thf(fact_1082_distrib__inf__le,axiom,
    ! [X2: set_a,Y: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ ( inf_inf_set_a @ X2 @ Z2 ) ) @ ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).

% distrib_inf_le
thf(fact_1083_distrib__sup__le,axiom,
    ! [X2: set_list_a,Y: set_list_a,Z2: set_list_a] : ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ X2 @ ( inf_inf_set_list_a @ Y @ Z2 ) ) @ ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ X2 @ Y ) @ ( sup_sup_set_list_a @ X2 @ Z2 ) ) ) ).

% distrib_sup_le
thf(fact_1084_distrib__sup__le,axiom,
    ! [X2: set_a,Y: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z2 ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y ) @ ( sup_sup_set_a @ X2 @ Z2 ) ) ) ).

% distrib_sup_le
thf(fact_1085_Un__Int__assoc__eq,axiom,
    ! [A2: set_list_a,B: set_list_a,C2: set_list_a] :
      ( ( ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ A2 @ B ) @ C2 )
        = ( inf_inf_set_list_a @ A2 @ ( sup_sup_set_list_a @ B @ C2 ) ) )
      = ( ord_le8861187494160871172list_a @ C2 @ A2 ) ) ).

% Un_Int_assoc_eq
thf(fact_1086_Un__Int__assoc__eq,axiom,
    ! [A2: set_a,B: set_a,C2: set_a] :
      ( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C2 )
        = ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B @ C2 ) ) )
      = ( ord_less_eq_set_a @ C2 @ A2 ) ) ).

% Un_Int_assoc_eq
thf(fact_1087_Int__insert__left,axiom,
    ! [A: list_a,C2: set_list_a,B: set_list_a] :
      ( ( ( member_list_a2 @ A @ C2 )
       => ( ( inf_inf_set_list_a @ ( insert_list_a2 @ A @ B ) @ C2 )
          = ( insert_list_a2 @ A @ ( inf_inf_set_list_a @ B @ C2 ) ) ) )
      & ( ~ ( member_list_a2 @ A @ C2 )
       => ( ( inf_inf_set_list_a @ ( insert_list_a2 @ A @ B ) @ C2 )
          = ( inf_inf_set_list_a @ B @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_1088_Int__insert__right,axiom,
    ! [A: list_a,A2: set_list_a,B: set_list_a] :
      ( ( ( member_list_a2 @ A @ A2 )
       => ( ( inf_inf_set_list_a @ A2 @ ( insert_list_a2 @ A @ B ) )
          = ( insert_list_a2 @ A @ ( inf_inf_set_list_a @ A2 @ B ) ) ) )
      & ( ~ ( member_list_a2 @ A @ A2 )
       => ( ( inf_inf_set_list_a @ A2 @ ( insert_list_a2 @ A @ B ) )
          = ( inf_inf_set_list_a @ A2 @ B ) ) ) ) ).

% Int_insert_right
thf(fact_1089_lists__IntI,axiom,
    ! [L: list_a,A2: set_a,B: set_a] :
      ( ( member_list_a2 @ L @ ( lists_a @ A2 ) )
     => ( ( member_list_a2 @ L @ ( lists_a @ B ) )
       => ( member_list_a2 @ L @ ( lists_a @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ) ).

% lists_IntI
thf(fact_1090_IntD2,axiom,
    ! [C: list_a,A2: set_list_a,B: set_list_a] :
      ( ( member_list_a2 @ C @ ( inf_inf_set_list_a @ A2 @ B ) )
     => ( member_list_a2 @ C @ B ) ) ).

% IntD2
thf(fact_1091_IntD1,axiom,
    ! [C: list_a,A2: set_list_a,B: set_list_a] :
      ( ( member_list_a2 @ C @ ( inf_inf_set_list_a @ A2 @ B ) )
     => ( member_list_a2 @ C @ A2 ) ) ).

% IntD1
thf(fact_1092_IntE,axiom,
    ! [C: list_a,A2: set_list_a,B: set_list_a] :
      ( ( member_list_a2 @ C @ ( inf_inf_set_list_a @ A2 @ B ) )
     => ~ ( ( member_list_a2 @ C @ A2 )
         => ~ ( member_list_a2 @ C @ B ) ) ) ).

% IntE
thf(fact_1093_Int__mono,axiom,
    ! [A2: set_list_a,C2: set_list_a,B: set_list_a,D: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ C2 )
     => ( ( ord_le8861187494160871172list_a @ B @ D )
       => ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B ) @ ( inf_inf_set_list_a @ C2 @ D ) ) ) ) ).

% Int_mono
thf(fact_1094_Int__mono,axiom,
    ! [A2: set_a,C2: set_a,B: set_a,D: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ C2 )
     => ( ( ord_less_eq_set_a @ B @ D )
       => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).

% Int_mono
thf(fact_1095_Int__lower1,axiom,
    ! [A2: set_list_a,B: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B ) @ A2 ) ).

% Int_lower1
thf(fact_1096_Int__lower1,axiom,
    ! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ A2 ) ).

% Int_lower1
thf(fact_1097_Int__lower2,axiom,
    ! [A2: set_list_a,B: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B ) @ B ) ).

% Int_lower2
thf(fact_1098_Int__lower2,axiom,
    ! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ B ) ).

% Int_lower2
thf(fact_1099_Int__absorb1,axiom,
    ! [B: set_list_a,A2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ B @ A2 )
     => ( ( inf_inf_set_list_a @ A2 @ B )
        = B ) ) ).

% Int_absorb1
thf(fact_1100_Int__absorb1,axiom,
    ! [B: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B @ A2 )
     => ( ( inf_inf_set_a @ A2 @ B )
        = B ) ) ).

% Int_absorb1
thf(fact_1101_Int__absorb2,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B )
     => ( ( inf_inf_set_list_a @ A2 @ B )
        = A2 ) ) ).

% Int_absorb2
thf(fact_1102_Int__absorb2,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( inf_inf_set_a @ A2 @ B )
        = A2 ) ) ).

% Int_absorb2
thf(fact_1103_Int__greatest,axiom,
    ! [C2: set_list_a,A2: set_list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ C2 @ A2 )
     => ( ( ord_le8861187494160871172list_a @ C2 @ B )
       => ( ord_le8861187494160871172list_a @ C2 @ ( inf_inf_set_list_a @ A2 @ B ) ) ) ) ).

% Int_greatest
thf(fact_1104_Int__greatest,axiom,
    ! [C2: set_a,A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ C2 @ A2 )
     => ( ( ord_less_eq_set_a @ C2 @ B )
       => ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).

% Int_greatest
thf(fact_1105_Int__Collect__mono,axiom,
    ! [A2: set_list_a,B: set_list_a,P: list_a > $o,Q: list_a > $o] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B )
     => ( ! [X4: list_a] :
            ( ( member_list_a2 @ X4 @ A2 )
           => ( ( P @ X4 )
             => ( Q @ X4 ) ) )
       => ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ ( collect_list_a @ P ) ) @ ( inf_inf_set_list_a @ B @ ( collect_list_a @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1106_Int__Collect__mono,axiom,
    ! [A2: set_a,B: set_a,P: a > $o,Q: a > $o] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ! [X4: a] :
            ( ( member_a2 @ X4 @ A2 )
           => ( ( P @ X4 )
             => ( Q @ X4 ) ) )
       => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1107_inf__sup__ord_I2_J,axiom,
    ! [X2: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X2 @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_1108_inf__sup__ord_I2_J,axiom,
    ! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_1109_inf__sup__ord_I1_J,axiom,
    ! [X2: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X2 @ Y ) @ X2 ) ).

% inf_sup_ord(1)
thf(fact_1110_inf__sup__ord_I1_J,axiom,
    ! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ X2 ) ).

% inf_sup_ord(1)
thf(fact_1111_inf__le1,axiom,
    ! [X2: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X2 @ Y ) @ X2 ) ).

% inf_le1
thf(fact_1112_inf__le1,axiom,
    ! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ X2 ) ).

% inf_le1
thf(fact_1113_inf__le2,axiom,
    ! [X2: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X2 @ Y ) @ Y ) ).

% inf_le2
thf(fact_1114_inf__le2,axiom,
    ! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ Y ) ).

% inf_le2
thf(fact_1115_le__infE,axiom,
    ! [X2: set_list_a,A: set_list_a,B3: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ X2 @ ( inf_inf_set_list_a @ A @ B3 ) )
     => ~ ( ( ord_le8861187494160871172list_a @ X2 @ A )
         => ~ ( ord_le8861187494160871172list_a @ X2 @ B3 ) ) ) ).

% le_infE
thf(fact_1116_le__infE,axiom,
    ! [X2: set_a,A: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A @ B3 ) )
     => ~ ( ( ord_less_eq_set_a @ X2 @ A )
         => ~ ( ord_less_eq_set_a @ X2 @ B3 ) ) ) ).

% le_infE
thf(fact_1117_le__infI,axiom,
    ! [X2: set_list_a,A: set_list_a,B3: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ X2 @ A )
     => ( ( ord_le8861187494160871172list_a @ X2 @ B3 )
       => ( ord_le8861187494160871172list_a @ X2 @ ( inf_inf_set_list_a @ A @ B3 ) ) ) ) ).

% le_infI
thf(fact_1118_le__infI,axiom,
    ! [X2: set_a,A: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ A )
     => ( ( ord_less_eq_set_a @ X2 @ B3 )
       => ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A @ B3 ) ) ) ) ).

% le_infI
thf(fact_1119_inf__mono,axiom,
    ! [A: set_list_a,C: set_list_a,B3: set_list_a,D2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ C )
     => ( ( ord_le8861187494160871172list_a @ B3 @ D2 )
       => ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B3 ) @ ( inf_inf_set_list_a @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1120_inf__mono,axiom,
    ! [A: set_a,C: set_a,B3: set_a,D2: set_a] :
      ( ( ord_less_eq_set_a @ A @ C )
     => ( ( ord_less_eq_set_a @ B3 @ D2 )
       => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B3 ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1121_le__infI1,axiom,
    ! [A: set_list_a,X2: set_list_a,B3: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ X2 )
     => ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B3 ) @ X2 ) ) ).

% le_infI1
thf(fact_1122_le__infI1,axiom,
    ! [A: set_a,X2: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ A @ X2 )
     => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B3 ) @ X2 ) ) ).

% le_infI1
thf(fact_1123_le__infI2,axiom,
    ! [B3: set_list_a,X2: set_list_a,A: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ B3 @ X2 )
     => ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B3 ) @ X2 ) ) ).

% le_infI2
thf(fact_1124_le__infI2,axiom,
    ! [B3: set_a,X2: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ B3 @ X2 )
     => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B3 ) @ X2 ) ) ).

% le_infI2
thf(fact_1125_inf_OorderE,axiom,
    ! [A: set_list_a,B3: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ B3 )
     => ( A
        = ( inf_inf_set_list_a @ A @ B3 ) ) ) ).

% inf.orderE
thf(fact_1126_inf_OorderE,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( A
        = ( inf_inf_set_a @ A @ B3 ) ) ) ).

% inf.orderE
thf(fact_1127_inf_OorderI,axiom,
    ! [A: set_list_a,B3: set_list_a] :
      ( ( A
        = ( inf_inf_set_list_a @ A @ B3 ) )
     => ( ord_le8861187494160871172list_a @ A @ B3 ) ) ).

% inf.orderI
thf(fact_1128_inf_OorderI,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( A
        = ( inf_inf_set_a @ A @ B3 ) )
     => ( ord_less_eq_set_a @ A @ B3 ) ) ).

% inf.orderI
thf(fact_1129_inf__unique,axiom,
    ! [F: set_list_a > set_list_a > set_list_a,X2: set_list_a,Y: set_list_a] :
      ( ! [X4: set_list_a,Y2: set_list_a] : ( ord_le8861187494160871172list_a @ ( F @ X4 @ Y2 ) @ X4 )
     => ( ! [X4: set_list_a,Y2: set_list_a] : ( ord_le8861187494160871172list_a @ ( F @ X4 @ Y2 ) @ Y2 )
       => ( ! [X4: set_list_a,Y2: set_list_a,Z3: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X4 @ Y2 )
             => ( ( ord_le8861187494160871172list_a @ X4 @ Z3 )
               => ( ord_le8861187494160871172list_a @ X4 @ ( F @ Y2 @ Z3 ) ) ) )
         => ( ( inf_inf_set_list_a @ X2 @ Y )
            = ( F @ X2 @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_1130_inf__unique,axiom,
    ! [F: set_a > set_a > set_a,X2: set_a,Y: set_a] :
      ( ! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y2 ) @ X4 )
     => ( ! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y2 ) @ Y2 )
       => ( ! [X4: set_a,Y2: set_a,Z3: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y2 )
             => ( ( ord_less_eq_set_a @ X4 @ Z3 )
               => ( ord_less_eq_set_a @ X4 @ ( F @ Y2 @ Z3 ) ) ) )
         => ( ( inf_inf_set_a @ X2 @ Y )
            = ( F @ X2 @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_1131_le__iff__inf,axiom,
    ( ord_le8861187494160871172list_a
    = ( ^ [X3: set_list_a,Y4: set_list_a] :
          ( ( inf_inf_set_list_a @ X3 @ Y4 )
          = X3 ) ) ) ).

% le_iff_inf
thf(fact_1132_le__iff__inf,axiom,
    ( ord_less_eq_set_a
    = ( ^ [X3: set_a,Y4: set_a] :
          ( ( inf_inf_set_a @ X3 @ Y4 )
          = X3 ) ) ) ).

% le_iff_inf
thf(fact_1133_inf_Oabsorb1,axiom,
    ! [A: set_list_a,B3: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ B3 )
     => ( ( inf_inf_set_list_a @ A @ B3 )
        = A ) ) ).

% inf.absorb1
thf(fact_1134_inf_Oabsorb1,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( inf_inf_set_a @ A @ B3 )
        = A ) ) ).

% inf.absorb1
thf(fact_1135_inf_Oabsorb2,axiom,
    ! [B3: set_list_a,A: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ B3 @ A )
     => ( ( inf_inf_set_list_a @ A @ B3 )
        = B3 ) ) ).

% inf.absorb2
thf(fact_1136_inf_Oabsorb2,axiom,
    ! [B3: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ B3 @ A )
     => ( ( inf_inf_set_a @ A @ B3 )
        = B3 ) ) ).

% inf.absorb2
thf(fact_1137_inf__absorb1,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ X2 @ Y )
     => ( ( inf_inf_set_list_a @ X2 @ Y )
        = X2 ) ) ).

% inf_absorb1
thf(fact_1138_inf__absorb1,axiom,
    ! [X2: set_a,Y: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ Y )
     => ( ( inf_inf_set_a @ X2 @ Y )
        = X2 ) ) ).

% inf_absorb1
thf(fact_1139_inf__absorb2,axiom,
    ! [Y: set_list_a,X2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ Y @ X2 )
     => ( ( inf_inf_set_list_a @ X2 @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1140_inf__absorb2,axiom,
    ! [Y: set_a,X2: set_a] :
      ( ( ord_less_eq_set_a @ Y @ X2 )
     => ( ( inf_inf_set_a @ X2 @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1141_inf_OboundedE,axiom,
    ! [A: set_list_a,B3: set_list_a,C: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ ( inf_inf_set_list_a @ B3 @ C ) )
     => ~ ( ( ord_le8861187494160871172list_a @ A @ B3 )
         => ~ ( ord_le8861187494160871172list_a @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_1142_inf_OboundedE,axiom,
    ! [A: set_a,B3: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B3 @ C ) )
     => ~ ( ( ord_less_eq_set_a @ A @ B3 )
         => ~ ( ord_less_eq_set_a @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_1143_inf_OboundedI,axiom,
    ! [A: set_list_a,B3: set_list_a,C: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ B3 )
     => ( ( ord_le8861187494160871172list_a @ A @ C )
       => ( ord_le8861187494160871172list_a @ A @ ( inf_inf_set_list_a @ B3 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_1144_inf_OboundedI,axiom,
    ! [A: set_a,B3: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( ord_less_eq_set_a @ A @ C )
       => ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B3 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_1145_inf__greatest,axiom,
    ! [X2: set_list_a,Y: set_list_a,Z2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ X2 @ Y )
     => ( ( ord_le8861187494160871172list_a @ X2 @ Z2 )
       => ( ord_le8861187494160871172list_a @ X2 @ ( inf_inf_set_list_a @ Y @ Z2 ) ) ) ) ).

% inf_greatest
thf(fact_1146_inf__greatest,axiom,
    ! [X2: set_a,Y: set_a,Z2: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ Y )
     => ( ( ord_less_eq_set_a @ X2 @ Z2 )
       => ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ) ).

% inf_greatest
thf(fact_1147_inf_Oorder__iff,axiom,
    ( ord_le8861187494160871172list_a
    = ( ^ [A4: set_list_a,B4: set_list_a] :
          ( A4
          = ( inf_inf_set_list_a @ A4 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_1148_inf_Oorder__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A4: set_a,B4: set_a] :
          ( A4
          = ( inf_inf_set_a @ A4 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_1149_inf_Ocobounded1,axiom,
    ! [A: set_list_a,B3: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B3 ) @ A ) ).

% inf.cobounded1
thf(fact_1150_inf_Ocobounded1,axiom,
    ! [A: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B3 ) @ A ) ).

% inf.cobounded1
thf(fact_1151_inf_Ocobounded2,axiom,
    ! [A: set_list_a,B3: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B3 ) @ B3 ) ).

% inf.cobounded2
thf(fact_1152_inf_Ocobounded2,axiom,
    ! [A: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B3 ) @ B3 ) ).

% inf.cobounded2
thf(fact_1153_inf_Oabsorb__iff1,axiom,
    ( ord_le8861187494160871172list_a
    = ( ^ [A4: set_list_a,B4: set_list_a] :
          ( ( inf_inf_set_list_a @ A4 @ B4 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_1154_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A4: set_a,B4: set_a] :
          ( ( inf_inf_set_a @ A4 @ B4 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_1155_inf_Oabsorb__iff2,axiom,
    ( ord_le8861187494160871172list_a
    = ( ^ [B4: set_list_a,A4: set_list_a] :
          ( ( inf_inf_set_list_a @ A4 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_1156_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_a
    = ( ^ [B4: set_a,A4: set_a] :
          ( ( inf_inf_set_a @ A4 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_1157_inf_OcoboundedI1,axiom,
    ! [A: set_list_a,C: set_list_a,B3: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ C )
     => ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B3 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_1158_inf_OcoboundedI1,axiom,
    ! [A: set_a,C: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ A @ C )
     => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B3 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_1159_inf_OcoboundedI2,axiom,
    ! [B3: set_list_a,C: set_list_a,A: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ B3 @ C )
     => ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B3 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_1160_inf_OcoboundedI2,axiom,
    ! [B3: set_a,C: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ B3 @ C )
     => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B3 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_1161_Int__emptyI,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ! [X4: list_a] :
          ( ( member_list_a2 @ X4 @ A2 )
         => ~ ( member_list_a2 @ X4 @ B ) )
     => ( ( inf_inf_set_list_a @ A2 @ B )
        = bot_bot_set_list_a ) ) ).

% Int_emptyI
thf(fact_1162_disjoint__iff,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( ( inf_inf_set_list_a @ A2 @ B )
        = bot_bot_set_list_a )
      = ( ! [X3: list_a] :
            ( ( member_list_a2 @ X3 @ A2 )
           => ~ ( member_list_a2 @ X3 @ B ) ) ) ) ).

% disjoint_iff
thf(fact_1163_inf__shunt,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( ( inf_inf_set_list_a @ X2 @ Y )
        = bot_bot_set_list_a )
      = ( ord_le8861187494160871172list_a @ X2 @ ( uminus7925729386456332763list_a @ Y ) ) ) ).

% inf_shunt
thf(fact_1164_inf__shunt,axiom,
    ! [X2: set_a,Y: set_a] :
      ( ( ( inf_inf_set_a @ X2 @ Y )
        = bot_bot_set_a )
      = ( ord_less_eq_set_a @ X2 @ ( uminus_uminus_set_a @ Y ) ) ) ).

% inf_shunt
thf(fact_1165_sup__neg__inf,axiom,
    ! [P3: set_list_a,Q2: set_list_a,R2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ P3 @ ( sup_sup_set_list_a @ Q2 @ R2 ) )
      = ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ P3 @ ( uminus7925729386456332763list_a @ Q2 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_1166_sup__neg__inf,axiom,
    ! [P3: set_a,Q2: set_a,R2: set_a] :
      ( ( ord_less_eq_set_a @ P3 @ ( sup_sup_set_a @ Q2 @ R2 ) )
      = ( ord_less_eq_set_a @ ( inf_inf_set_a @ P3 @ ( uminus_uminus_set_a @ Q2 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_1167_shunt2,axiom,
    ! [X2: set_list_a,Y: set_list_a,Z2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X2 @ ( uminus7925729386456332763list_a @ Y ) ) @ Z2 )
      = ( ord_le8861187494160871172list_a @ X2 @ ( sup_sup_set_list_a @ Y @ Z2 ) ) ) ).

% shunt2
thf(fact_1168_shunt2,axiom,
    ! [X2: set_a,Y: set_a,Z2: set_a] :
      ( ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ ( uminus_uminus_set_a @ Y ) ) @ Z2 )
      = ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).

% shunt2
thf(fact_1169_shunt1,axiom,
    ! [X2: set_list_a,Y: set_list_a,Z2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X2 @ Y ) @ Z2 )
      = ( ord_le8861187494160871172list_a @ X2 @ ( sup_sup_set_list_a @ ( uminus7925729386456332763list_a @ Y ) @ Z2 ) ) ) ).

% shunt1
thf(fact_1170_shunt1,axiom,
    ! [X2: set_a,Y: set_a,Z2: set_a] :
      ( ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ Z2 )
      = ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ ( uminus_uminus_set_a @ Y ) @ Z2 ) ) ) ).

% shunt1
thf(fact_1171_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( ( inf_inf_set_list_a @ A2 @ B )
        = bot_bot_set_list_a )
      = ( ord_le8861187494160871172list_a @ A2 @ ( uminus7925729386456332763list_a @ B ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_1172_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ( inf_inf_set_a @ A2 @ B )
        = bot_bot_set_a )
      = ( ord_less_eq_set_a @ A2 @ ( uminus_uminus_set_a @ B ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_1173_distinct__disjoint__shuffles,axiom,
    ! [Xs: list_a,Ys2: list_a,Zs: list_a] :
      ( ( distinct_a @ Xs )
     => ( ( distinct_a @ Ys2 )
       => ( ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys2 ) )
            = bot_bot_set_a )
         => ( ( member_list_a2 @ Zs @ ( shuffles_a @ Xs @ Ys2 ) )
           => ( distinct_a @ Zs ) ) ) ) ) ).

% distinct_disjoint_shuffles
thf(fact_1174_finite__remove__induct,axiom,
    ! [B: set_list_a,P: set_list_a > $o] :
      ( ( finite_finite_list_a @ B )
     => ( ( P @ bot_bot_set_list_a )
       => ( ! [A7: set_list_a] :
              ( ( finite_finite_list_a @ A7 )
             => ( ( A7 != bot_bot_set_list_a )
               => ( ( ord_le8861187494160871172list_a @ A7 @ B )
                 => ( ! [X: list_a] :
                        ( ( member_list_a2 @ X @ A7 )
                       => ( P @ ( minus_646659088055828811list_a @ A7 @ ( insert_list_a2 @ X @ bot_bot_set_list_a ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_1175_finite__remove__induct,axiom,
    ! [B: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ B )
     => ( ( P @ bot_bot_set_a )
       => ( ! [A7: set_a] :
              ( ( finite_finite_a @ A7 )
             => ( ( A7 != bot_bot_set_a )
               => ( ( ord_less_eq_set_a @ A7 @ B )
                 => ( ! [X: a] :
                        ( ( member_a2 @ X @ A7 )
                       => ( P @ ( minus_minus_set_a @ A7 @ ( insert_a2 @ X @ bot_bot_set_a ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_1176_remove__induct,axiom,
    ! [P: set_list_a > $o,B: set_list_a] :
      ( ( P @ bot_bot_set_list_a )
     => ( ( ~ ( finite_finite_list_a @ B )
         => ( P @ B ) )
       => ( ! [A7: set_list_a] :
              ( ( finite_finite_list_a @ A7 )
             => ( ( A7 != bot_bot_set_list_a )
               => ( ( ord_le8861187494160871172list_a @ A7 @ B )
                 => ( ! [X: list_a] :
                        ( ( member_list_a2 @ X @ A7 )
                       => ( P @ ( minus_646659088055828811list_a @ A7 @ ( insert_list_a2 @ X @ bot_bot_set_list_a ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_1177_remove__induct,axiom,
    ! [P: set_a > $o,B: set_a] :
      ( ( P @ bot_bot_set_a )
     => ( ( ~ ( finite_finite_a @ B )
         => ( P @ B ) )
       => ( ! [A7: set_a] :
              ( ( finite_finite_a @ A7 )
             => ( ( A7 != bot_bot_set_a )
               => ( ( ord_less_eq_set_a @ A7 @ B )
                 => ( ! [X: a] :
                        ( ( member_a2 @ X @ A7 )
                       => ( P @ ( minus_minus_set_a @ A7 @ ( insert_a2 @ X @ bot_bot_set_a ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_1178_finite__has__maximal2,axiom,
    ! [A2: set_set_list_a,A: set_list_a] :
      ( ( finite5282473924520328461list_a @ A2 )
     => ( ( member_set_list_a @ A @ A2 )
       => ? [X4: set_list_a] :
            ( ( member_set_list_a @ X4 @ A2 )
            & ( ord_le8861187494160871172list_a @ A @ X4 )
            & ! [Xa3: set_list_a] :
                ( ( member_set_list_a @ Xa3 @ A2 )
               => ( ( ord_le8861187494160871172list_a @ X4 @ Xa3 )
                 => ( X4 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1179_finite__has__maximal2,axiom,
    ! [A2: set_set_a,A: set_a] :
      ( ( finite_finite_set_a @ A2 )
     => ( ( member_set_a2 @ A @ A2 )
       => ? [X4: set_a] :
            ( ( member_set_a2 @ X4 @ A2 )
            & ( ord_less_eq_set_a @ A @ X4 )
            & ! [Xa3: set_a] :
                ( ( member_set_a2 @ Xa3 @ A2 )
               => ( ( ord_less_eq_set_a @ X4 @ Xa3 )
                 => ( X4 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1180_finite__has__minimal2,axiom,
    ! [A2: set_set_list_a,A: set_list_a] :
      ( ( finite5282473924520328461list_a @ A2 )
     => ( ( member_set_list_a @ A @ A2 )
       => ? [X4: set_list_a] :
            ( ( member_set_list_a @ X4 @ A2 )
            & ( ord_le8861187494160871172list_a @ X4 @ A )
            & ! [Xa3: set_list_a] :
                ( ( member_set_list_a @ Xa3 @ A2 )
               => ( ( ord_le8861187494160871172list_a @ Xa3 @ X4 )
                 => ( X4 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1181_finite__has__minimal2,axiom,
    ! [A2: set_set_a,A: set_a] :
      ( ( finite_finite_set_a @ A2 )
     => ( ( member_set_a2 @ A @ A2 )
       => ? [X4: set_a] :
            ( ( member_set_a2 @ X4 @ A2 )
            & ( ord_less_eq_set_a @ X4 @ A )
            & ! [Xa3: set_a] :
                ( ( member_set_a2 @ Xa3 @ A2 )
               => ( ( ord_less_eq_set_a @ Xa3 @ X4 )
                 => ( X4 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1182_finite__subset,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B )
     => ( ( finite_finite_list_a @ B )
       => ( finite_finite_list_a @ A2 ) ) ) ).

% finite_subset
thf(fact_1183_finite__subset,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( finite_finite_a @ B )
       => ( finite_finite_a @ A2 ) ) ) ).

% finite_subset
thf(fact_1184_infinite__super,axiom,
    ! [S5: set_list_a,T3: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ S5 @ T3 )
     => ( ~ ( finite_finite_list_a @ S5 )
       => ~ ( finite_finite_list_a @ T3 ) ) ) ).

% infinite_super
thf(fact_1185_infinite__super,axiom,
    ! [S5: set_a,T3: set_a] :
      ( ( ord_less_eq_set_a @ S5 @ T3 )
     => ( ~ ( finite_finite_a @ S5 )
       => ~ ( finite_finite_a @ T3 ) ) ) ).

% infinite_super
thf(fact_1186_rev__finite__subset,axiom,
    ! [B: set_list_a,A2: set_list_a] :
      ( ( finite_finite_list_a @ B )
     => ( ( ord_le8861187494160871172list_a @ A2 @ B )
       => ( finite_finite_list_a @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_1187_rev__finite__subset,axiom,
    ! [B: set_a,A2: set_a] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_eq_set_a @ A2 @ B )
       => ( finite_finite_a @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_1188_all__finite__subset__image,axiom,
    ! [F: list_a > list_a,A2: set_list_a,P: set_list_a > $o] :
      ( ( ! [B2: set_list_a] :
            ( ( ( finite_finite_list_a @ B2 )
              & ( ord_le8861187494160871172list_a @ B2 @ ( image_list_a_list_a @ F @ A2 ) ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_list_a] :
            ( ( ( finite_finite_list_a @ B2 )
              & ( ord_le8861187494160871172list_a @ B2 @ A2 ) )
           => ( P @ ( image_list_a_list_a @ F @ B2 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1189_all__finite__subset__image,axiom,
    ! [F: a > list_a,A2: set_a,P: set_list_a > $o] :
      ( ( ! [B2: set_list_a] :
            ( ( ( finite_finite_list_a @ B2 )
              & ( ord_le8861187494160871172list_a @ B2 @ ( image_a_list_a @ F @ A2 ) ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_a] :
            ( ( ( finite_finite_a @ B2 )
              & ( ord_less_eq_set_a @ B2 @ A2 ) )
           => ( P @ ( image_a_list_a @ F @ B2 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1190_all__finite__subset__image,axiom,
    ! [F: list_a > a,A2: set_list_a,P: set_a > $o] :
      ( ( ! [B2: set_a] :
            ( ( ( finite_finite_a @ B2 )
              & ( ord_less_eq_set_a @ B2 @ ( image_list_a_a @ F @ A2 ) ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_list_a] :
            ( ( ( finite_finite_list_a @ B2 )
              & ( ord_le8861187494160871172list_a @ B2 @ A2 ) )
           => ( P @ ( image_list_a_a @ F @ B2 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1191_all__finite__subset__image,axiom,
    ! [F: a > a,A2: set_a,P: set_a > $o] :
      ( ( ! [B2: set_a] :
            ( ( ( finite_finite_a @ B2 )
              & ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_a] :
            ( ( ( finite_finite_a @ B2 )
              & ( ord_less_eq_set_a @ B2 @ A2 ) )
           => ( P @ ( image_a_a @ F @ B2 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1192_ex__finite__subset__image,axiom,
    ! [F: list_a > list_a,A2: set_list_a,P: set_list_a > $o] :
      ( ( ? [B2: set_list_a] :
            ( ( finite_finite_list_a @ B2 )
            & ( ord_le8861187494160871172list_a @ B2 @ ( image_list_a_list_a @ F @ A2 ) )
            & ( P @ B2 ) ) )
      = ( ? [B2: set_list_a] :
            ( ( finite_finite_list_a @ B2 )
            & ( ord_le8861187494160871172list_a @ B2 @ A2 )
            & ( P @ ( image_list_a_list_a @ F @ B2 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1193_ex__finite__subset__image,axiom,
    ! [F: a > list_a,A2: set_a,P: set_list_a > $o] :
      ( ( ? [B2: set_list_a] :
            ( ( finite_finite_list_a @ B2 )
            & ( ord_le8861187494160871172list_a @ B2 @ ( image_a_list_a @ F @ A2 ) )
            & ( P @ B2 ) ) )
      = ( ? [B2: set_a] :
            ( ( finite_finite_a @ B2 )
            & ( ord_less_eq_set_a @ B2 @ A2 )
            & ( P @ ( image_a_list_a @ F @ B2 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1194_ex__finite__subset__image,axiom,
    ! [F: list_a > a,A2: set_list_a,P: set_a > $o] :
      ( ( ? [B2: set_a] :
            ( ( finite_finite_a @ B2 )
            & ( ord_less_eq_set_a @ B2 @ ( image_list_a_a @ F @ A2 ) )
            & ( P @ B2 ) ) )
      = ( ? [B2: set_list_a] :
            ( ( finite_finite_list_a @ B2 )
            & ( ord_le8861187494160871172list_a @ B2 @ A2 )
            & ( P @ ( image_list_a_a @ F @ B2 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1195_ex__finite__subset__image,axiom,
    ! [F: a > a,A2: set_a,P: set_a > $o] :
      ( ( ? [B2: set_a] :
            ( ( finite_finite_a @ B2 )
            & ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
            & ( P @ B2 ) ) )
      = ( ? [B2: set_a] :
            ( ( finite_finite_a @ B2 )
            & ( ord_less_eq_set_a @ B2 @ A2 )
            & ( P @ ( image_a_a @ F @ B2 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1196_finite__subset__image,axiom,
    ! [B: set_list_a,F: list_a > list_a,A2: set_list_a] :
      ( ( finite_finite_list_a @ B )
     => ( ( ord_le8861187494160871172list_a @ B @ ( image_list_a_list_a @ F @ A2 ) )
       => ? [C3: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ C3 @ A2 )
            & ( finite_finite_list_a @ C3 )
            & ( B
              = ( image_list_a_list_a @ F @ C3 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1197_finite__subset__image,axiom,
    ! [B: set_list_a,F: a > list_a,A2: set_a] :
      ( ( finite_finite_list_a @ B )
     => ( ( ord_le8861187494160871172list_a @ B @ ( image_a_list_a @ F @ A2 ) )
       => ? [C3: set_a] :
            ( ( ord_less_eq_set_a @ C3 @ A2 )
            & ( finite_finite_a @ C3 )
            & ( B
              = ( image_a_list_a @ F @ C3 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1198_finite__subset__image,axiom,
    ! [B: set_a,F: list_a > a,A2: set_list_a] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_eq_set_a @ B @ ( image_list_a_a @ F @ A2 ) )
       => ? [C3: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ C3 @ A2 )
            & ( finite_finite_list_a @ C3 )
            & ( B
              = ( image_list_a_a @ F @ C3 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1199_finite__subset__image,axiom,
    ! [B: set_a,F: a > a,A2: set_a] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A2 ) )
       => ? [C3: set_a] :
            ( ( ord_less_eq_set_a @ C3 @ A2 )
            & ( finite_finite_a @ C3 )
            & ( B
              = ( image_a_a @ F @ C3 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1200_finite__has__minimal,axiom,
    ! [A2: set_set_list_a] :
      ( ( finite5282473924520328461list_a @ A2 )
     => ( ( A2 != bot_bo3186585308812441520list_a )
       => ? [X4: set_list_a] :
            ( ( member_set_list_a @ X4 @ A2 )
            & ! [Xa3: set_list_a] :
                ( ( member_set_list_a @ Xa3 @ A2 )
               => ( ( ord_le8861187494160871172list_a @ Xa3 @ X4 )
                 => ( X4 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1201_finite__has__minimal,axiom,
    ! [A2: set_set_a] :
      ( ( finite_finite_set_a @ A2 )
     => ( ( A2 != bot_bot_set_set_a )
       => ? [X4: set_a] :
            ( ( member_set_a2 @ X4 @ A2 )
            & ! [Xa3: set_a] :
                ( ( member_set_a2 @ Xa3 @ A2 )
               => ( ( ord_less_eq_set_a @ Xa3 @ X4 )
                 => ( X4 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1202_finite__has__maximal,axiom,
    ! [A2: set_set_list_a] :
      ( ( finite5282473924520328461list_a @ A2 )
     => ( ( A2 != bot_bo3186585308812441520list_a )
       => ? [X4: set_list_a] :
            ( ( member_set_list_a @ X4 @ A2 )
            & ! [Xa3: set_list_a] :
                ( ( member_set_list_a @ Xa3 @ A2 )
               => ( ( ord_le8861187494160871172list_a @ X4 @ Xa3 )
                 => ( X4 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1203_finite__has__maximal,axiom,
    ! [A2: set_set_a] :
      ( ( finite_finite_set_a @ A2 )
     => ( ( A2 != bot_bot_set_set_a )
       => ? [X4: set_a] :
            ( ( member_set_a2 @ X4 @ A2 )
            & ! [Xa3: set_a] :
                ( ( member_set_a2 @ Xa3 @ A2 )
               => ( ( ord_less_eq_set_a @ X4 @ Xa3 )
                 => ( X4 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1204_finite__subset__induct_H,axiom,
    ! [F3: set_list_a,A2: set_list_a,P: set_list_a > $o] :
      ( ( finite_finite_list_a @ F3 )
     => ( ( ord_le8861187494160871172list_a @ F3 @ A2 )
       => ( ( P @ bot_bot_set_list_a )
         => ( ! [A6: list_a,F4: set_list_a] :
                ( ( finite_finite_list_a @ F4 )
               => ( ( member_list_a2 @ A6 @ A2 )
                 => ( ( ord_le8861187494160871172list_a @ F4 @ A2 )
                   => ( ~ ( member_list_a2 @ A6 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_list_a2 @ A6 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1205_finite__subset__induct_H,axiom,
    ! [F3: set_a,A2: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ F3 )
     => ( ( ord_less_eq_set_a @ F3 @ A2 )
       => ( ( P @ bot_bot_set_a )
         => ( ! [A6: a,F4: set_a] :
                ( ( finite_finite_a @ F4 )
               => ( ( member_a2 @ A6 @ A2 )
                 => ( ( ord_less_eq_set_a @ F4 @ A2 )
                   => ( ~ ( member_a2 @ A6 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_a2 @ A6 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1206_finite__subset__induct,axiom,
    ! [F3: set_list_a,A2: set_list_a,P: set_list_a > $o] :
      ( ( finite_finite_list_a @ F3 )
     => ( ( ord_le8861187494160871172list_a @ F3 @ A2 )
       => ( ( P @ bot_bot_set_list_a )
         => ( ! [A6: list_a,F4: set_list_a] :
                ( ( finite_finite_list_a @ F4 )
               => ( ( member_list_a2 @ A6 @ A2 )
                 => ( ~ ( member_list_a2 @ A6 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_list_a2 @ A6 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1207_finite__subset__induct,axiom,
    ! [F3: set_a,A2: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ F3 )
     => ( ( ord_less_eq_set_a @ F3 @ A2 )
       => ( ( P @ bot_bot_set_a )
         => ( ! [A6: a,F4: set_a] :
                ( ( finite_finite_a @ F4 )
               => ( ( member_a2 @ A6 @ A2 )
                 => ( ~ ( member_a2 @ A6 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_a2 @ A6 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1208_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic1671282250946225214list_a @ inf_inf_set_list_a @ ord_le8861187494160871172list_a @ ord_less_set_list_a ).

% Inf_fin.semilattice_order_set_axioms
thf(fact_1209_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic8986249270076014136_set_a @ inf_inf_set_a @ ord_less_eq_set_a @ ord_less_set_a ).

% Inf_fin.semilattice_order_set_axioms
thf(fact_1210_Sup__fin_OcoboundedI,axiom,
    ! [A2: set_set_list_a,A: set_list_a] :
      ( ( finite5282473924520328461list_a @ A2 )
     => ( ( member_set_list_a @ A @ A2 )
       => ( ord_le8861187494160871172list_a @ A @ ( lattic2137287156484284315list_a @ A2 ) ) ) ) ).

% Sup_fin.coboundedI
thf(fact_1211_Sup__fin_OcoboundedI,axiom,
    ! [A2: set_set_a,A: set_a] :
      ( ( finite_finite_set_a @ A2 )
     => ( ( member_set_a2 @ A @ A2 )
       => ( ord_less_eq_set_a @ A @ ( lattic2918178356826803221_set_a @ A2 ) ) ) ) ).

% Sup_fin.coboundedI
thf(fact_1212_Sup__fin_Obounded__iff,axiom,
    ! [A2: set_set_list_a,X2: set_list_a] :
      ( ( finite5282473924520328461list_a @ A2 )
     => ( ( A2 != bot_bo3186585308812441520list_a )
       => ( ( ord_le8861187494160871172list_a @ ( lattic2137287156484284315list_a @ A2 ) @ X2 )
          = ( ! [X3: set_list_a] :
                ( ( member_set_list_a @ X3 @ A2 )
               => ( ord_le8861187494160871172list_a @ X3 @ X2 ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_1213_Sup__fin_Obounded__iff,axiom,
    ! [A2: set_set_a,X2: set_a] :
      ( ( finite_finite_set_a @ A2 )
     => ( ( A2 != bot_bot_set_set_a )
       => ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ X2 )
          = ( ! [X3: set_a] :
                ( ( member_set_a2 @ X3 @ A2 )
               => ( ord_less_eq_set_a @ X3 @ X2 ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_1214_Sup__fin_OboundedI,axiom,
    ! [A2: set_set_list_a,X2: set_list_a] :
      ( ( finite5282473924520328461list_a @ A2 )
     => ( ( A2 != bot_bo3186585308812441520list_a )
       => ( ! [A6: set_list_a] :
              ( ( member_set_list_a @ A6 @ A2 )
             => ( ord_le8861187494160871172list_a @ A6 @ X2 ) )
         => ( ord_le8861187494160871172list_a @ ( lattic2137287156484284315list_a @ A2 ) @ X2 ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_1215_Sup__fin_OboundedI,axiom,
    ! [A2: set_set_a,X2: set_a] :
      ( ( finite_finite_set_a @ A2 )
     => ( ( A2 != bot_bot_set_set_a )
       => ( ! [A6: set_a] :
              ( ( member_set_a2 @ A6 @ A2 )
             => ( ord_less_eq_set_a @ A6 @ X2 ) )
         => ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ X2 ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_1216_Sup__fin_OboundedE,axiom,
    ! [A2: set_set_list_a,X2: set_list_a] :
      ( ( finite5282473924520328461list_a @ A2 )
     => ( ( A2 != bot_bo3186585308812441520list_a )
       => ( ( ord_le8861187494160871172list_a @ ( lattic2137287156484284315list_a @ A2 ) @ X2 )
         => ! [A8: set_list_a] :
              ( ( member_set_list_a @ A8 @ A2 )
             => ( ord_le8861187494160871172list_a @ A8 @ X2 ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_1217_Sup__fin_OboundedE,axiom,
    ! [A2: set_set_a,X2: set_a] :
      ( ( finite_finite_set_a @ A2 )
     => ( ( A2 != bot_bot_set_set_a )
       => ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ X2 )
         => ! [A8: set_a] :
              ( ( member_set_a2 @ A8 @ A2 )
             => ( ord_less_eq_set_a @ A8 @ X2 ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_1218_Sup__fin_Osubset__imp,axiom,
    ! [A2: set_set_list_a,B: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ A2 @ B )
     => ( ( A2 != bot_bo3186585308812441520list_a )
       => ( ( finite5282473924520328461list_a @ B )
         => ( ord_le8861187494160871172list_a @ ( lattic2137287156484284315list_a @ A2 ) @ ( lattic2137287156484284315list_a @ B ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_1219_Sup__fin_Osubset__imp,axiom,
    ! [A2: set_set_a,B: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ B )
     => ( ( A2 != bot_bot_set_set_a )
       => ( ( finite_finite_set_a @ B )
         => ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ ( lattic2918178356826803221_set_a @ B ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_1220_semilattice__order__set_Osubset__imp,axiom,
    ! [F: list_a > list_a > list_a,Less_eq: list_a > list_a > $o,Less: list_a > list_a > $o,A2: set_list_a,B: set_list_a] :
      ( ( lattic4064278158876714078list_a @ F @ Less_eq @ Less )
     => ( ( ord_le8861187494160871172list_a @ A2 @ B )
       => ( ( A2 != bot_bot_set_list_a )
         => ( ( finite_finite_list_a @ B )
           => ( Less_eq @ ( lattic491866123428426078list_a @ F @ B ) @ ( lattic491866123428426078list_a @ F @ A2 ) ) ) ) ) ) ).

% semilattice_order_set.subset_imp
thf(fact_1221_semilattice__order__set_Osubset__imp,axiom,
    ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A2: set_a,B: set_a] :
      ( ( lattic5078705180708912344_set_a @ F @ Less_eq @ Less )
     => ( ( ord_less_eq_set_a @ A2 @ B )
       => ( ( A2 != bot_bot_set_a )
         => ( ( finite_finite_a @ B )
           => ( Less_eq @ ( lattic5116578512385870296ce_F_a @ F @ B ) @ ( lattic5116578512385870296ce_F_a @ F @ A2 ) ) ) ) ) ) ).

% semilattice_order_set.subset_imp
thf(fact_1222_Inf__fin__le__Sup__fin,axiom,
    ! [A2: set_set_list_a] :
      ( ( finite5282473924520328461list_a @ A2 )
     => ( ( A2 != bot_bo3186585308812441520list_a )
       => ( ord_le8861187494160871172list_a @ ( lattic2653631110915024129list_a @ A2 ) @ ( lattic2137287156484284315list_a @ A2 ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_1223_Inf__fin__le__Sup__fin,axiom,
    ! [A2: set_set_a] :
      ( ( finite_finite_set_a @ A2 )
     => ( ( A2 != bot_bot_set_set_a )
       => ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ A2 ) @ ( lattic2918178356826803221_set_a @ A2 ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_1224_Inf__fin_OcoboundedI,axiom,
    ! [A2: set_set_list_a,A: set_list_a] :
      ( ( finite5282473924520328461list_a @ A2 )
     => ( ( member_set_list_a @ A @ A2 )
       => ( ord_le8861187494160871172list_a @ ( lattic2653631110915024129list_a @ A2 ) @ A ) ) ) ).

% Inf_fin.coboundedI
thf(fact_1225_Inf__fin_OcoboundedI,axiom,
    ! [A2: set_set_a,A: set_a] :
      ( ( finite_finite_set_a @ A2 )
     => ( ( member_set_a2 @ A @ A2 )
       => ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ A2 ) @ A ) ) ) ).

% Inf_fin.coboundedI
thf(fact_1226_Inf__fin_OboundedE,axiom,
    ! [A2: set_set_list_a,X2: set_list_a] :
      ( ( finite5282473924520328461list_a @ A2 )
     => ( ( A2 != bot_bo3186585308812441520list_a )
       => ( ( ord_le8861187494160871172list_a @ X2 @ ( lattic2653631110915024129list_a @ A2 ) )
         => ! [A8: set_list_a] :
              ( ( member_set_list_a @ A8 @ A2 )
             => ( ord_le8861187494160871172list_a @ X2 @ A8 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_1227_Inf__fin_OboundedE,axiom,
    ! [A2: set_set_a,X2: set_a] :
      ( ( finite_finite_set_a @ A2 )
     => ( ( A2 != bot_bot_set_set_a )
       => ( ( ord_less_eq_set_a @ X2 @ ( lattic8209813465164889211_set_a @ A2 ) )
         => ! [A8: set_a] :
              ( ( member_set_a2 @ A8 @ A2 )
             => ( ord_less_eq_set_a @ X2 @ A8 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_1228_Inf__fin_OboundedI,axiom,
    ! [A2: set_set_list_a,X2: set_list_a] :
      ( ( finite5282473924520328461list_a @ A2 )
     => ( ( A2 != bot_bo3186585308812441520list_a )
       => ( ! [A6: set_list_a] :
              ( ( member_set_list_a @ A6 @ A2 )
             => ( ord_le8861187494160871172list_a @ X2 @ A6 ) )
         => ( ord_le8861187494160871172list_a @ X2 @ ( lattic2653631110915024129list_a @ A2 ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_1229_Inf__fin_OboundedI,axiom,
    ! [A2: set_set_a,X2: set_a] :
      ( ( finite_finite_set_a @ A2 )
     => ( ( A2 != bot_bot_set_set_a )
       => ( ! [A6: set_a] :
              ( ( member_set_a2 @ A6 @ A2 )
             => ( ord_less_eq_set_a @ X2 @ A6 ) )
         => ( ord_less_eq_set_a @ X2 @ ( lattic8209813465164889211_set_a @ A2 ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_1230_Inf__fin_Obounded__iff,axiom,
    ! [A2: set_set_list_a,X2: set_list_a] :
      ( ( finite5282473924520328461list_a @ A2 )
     => ( ( A2 != bot_bo3186585308812441520list_a )
       => ( ( ord_le8861187494160871172list_a @ X2 @ ( lattic2653631110915024129list_a @ A2 ) )
          = ( ! [X3: set_list_a] :
                ( ( member_set_list_a @ X3 @ A2 )
               => ( ord_le8861187494160871172list_a @ X2 @ X3 ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_1231_Inf__fin_Obounded__iff,axiom,
    ! [A2: set_set_a,X2: set_a] :
      ( ( finite_finite_set_a @ A2 )
     => ( ( A2 != bot_bot_set_set_a )
       => ( ( ord_less_eq_set_a @ X2 @ ( lattic8209813465164889211_set_a @ A2 ) )
          = ( ! [X3: set_a] :
                ( ( member_set_a2 @ X3 @ A2 )
               => ( ord_less_eq_set_a @ X2 @ X3 ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_1232_Inf__fin_Osubset__imp,axiom,
    ! [A2: set_set_list_a,B: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ A2 @ B )
     => ( ( A2 != bot_bo3186585308812441520list_a )
       => ( ( finite5282473924520328461list_a @ B )
         => ( ord_le8861187494160871172list_a @ ( lattic2653631110915024129list_a @ B ) @ ( lattic2653631110915024129list_a @ A2 ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_1233_Inf__fin_Osubset__imp,axiom,
    ! [A2: set_set_a,B: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ B )
     => ( ( A2 != bot_bot_set_set_a )
       => ( ( finite_finite_set_a @ B )
         => ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ B ) @ ( lattic8209813465164889211_set_a @ A2 ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_1234_semilattice__set_Osubset,axiom,
    ! [F: list_a > list_a > list_a,A2: set_list_a,B: set_list_a] :
      ( ( lattic3454955330975583506list_a @ F )
     => ( ( finite_finite_list_a @ A2 )
       => ( ( B != bot_bot_set_list_a )
         => ( ( ord_le8861187494160871172list_a @ B @ A2 )
           => ( ( F @ ( lattic491866123428426078list_a @ F @ B ) @ ( lattic491866123428426078list_a @ F @ A2 ) )
              = ( lattic491866123428426078list_a @ F @ A2 ) ) ) ) ) ) ).

% semilattice_set.subset
thf(fact_1235_semilattice__set_Osubset,axiom,
    ! [F: a > a > a,A2: set_a,B: set_a] :
      ( ( lattic5961991414251573132_set_a @ F )
     => ( ( finite_finite_a @ A2 )
       => ( ( B != bot_bot_set_a )
         => ( ( ord_less_eq_set_a @ B @ A2 )
           => ( ( F @ ( lattic5116578512385870296ce_F_a @ F @ B ) @ ( lattic5116578512385870296ce_F_a @ F @ A2 ) )
              = ( lattic5116578512385870296ce_F_a @ F @ A2 ) ) ) ) ) ) ).

% semilattice_set.subset
thf(fact_1236_rev__conv__fold,axiom,
    ( rev_a
    = ( ^ [Xs3: list_a] : ( fold_a_list_a @ cons_a @ Xs3 @ nil_a ) ) ) ).

% rev_conv_fold
thf(fact_1237_finite__surj__inj,axiom,
    ! [A2: set_list_a,F: list_a > list_a] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( ord_le8861187494160871172list_a @ A2 @ ( image_list_a_list_a @ F @ A2 ) )
       => ( inj_on_list_a_list_a @ F @ A2 ) ) ) ).

% finite_surj_inj
thf(fact_1238_finite__surj__inj,axiom,
    ! [A2: set_a,F: a > a] :
      ( ( finite_finite_a @ A2 )
     => ( ( ord_less_eq_set_a @ A2 @ ( image_a_a @ F @ A2 ) )
       => ( inj_on_a_a @ F @ A2 ) ) ) ).

% finite_surj_inj
thf(fact_1239_endo__inj__surj,axiom,
    ! [A2: set_list_a,F: list_a > list_a] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( ord_le8861187494160871172list_a @ ( image_list_a_list_a @ F @ A2 ) @ A2 )
       => ( ( inj_on_list_a_list_a @ F @ A2 )
         => ( ( image_list_a_list_a @ F @ A2 )
            = A2 ) ) ) ) ).

% endo_inj_surj
thf(fact_1240_endo__inj__surj,axiom,
    ! [A2: set_a,F: a > a] :
      ( ( finite_finite_a @ A2 )
     => ( ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ A2 )
       => ( ( inj_on_a_a @ F @ A2 )
         => ( ( image_a_a @ F @ A2 )
            = A2 ) ) ) ) ).

% endo_inj_surj
thf(fact_1241_inj__on__image__mem__iff,axiom,
    ! [F: list_a > list_a,B: set_list_a,A: list_a,A2: set_list_a] :
      ( ( inj_on_list_a_list_a @ F @ B )
     => ( ( member_list_a2 @ A @ B )
       => ( ( ord_le8861187494160871172list_a @ A2 @ B )
         => ( ( member_list_a2 @ ( F @ A ) @ ( image_list_a_list_a @ F @ A2 ) )
            = ( member_list_a2 @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1242_inj__on__image__mem__iff,axiom,
    ! [F: a > list_a,B: set_a,A: a,A2: set_a] :
      ( ( inj_on_a_list_a @ F @ B )
     => ( ( member_a2 @ A @ B )
       => ( ( ord_less_eq_set_a @ A2 @ B )
         => ( ( member_list_a2 @ ( F @ A ) @ ( image_a_list_a @ F @ A2 ) )
            = ( member_a2 @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1243_subset__image__inj,axiom,
    ! [S5: set_list_a,F: list_a > list_a,T3: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ S5 @ ( image_list_a_list_a @ F @ T3 ) )
      = ( ? [U2: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ U2 @ T3 )
            & ( inj_on_list_a_list_a @ F @ U2 )
            & ( S5
              = ( image_list_a_list_a @ F @ U2 ) ) ) ) ) ).

% subset_image_inj
thf(fact_1244_subset__image__inj,axiom,
    ! [S5: set_list_a,F: a > list_a,T3: set_a] :
      ( ( ord_le8861187494160871172list_a @ S5 @ ( image_a_list_a @ F @ T3 ) )
      = ( ? [U2: set_a] :
            ( ( ord_less_eq_set_a @ U2 @ T3 )
            & ( inj_on_a_list_a @ F @ U2 )
            & ( S5
              = ( image_a_list_a @ F @ U2 ) ) ) ) ) ).

% subset_image_inj
thf(fact_1245_subset__image__inj,axiom,
    ! [S5: set_a,F: list_a > a,T3: set_list_a] :
      ( ( ord_less_eq_set_a @ S5 @ ( image_list_a_a @ F @ T3 ) )
      = ( ? [U2: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ U2 @ T3 )
            & ( inj_on_list_a_a @ F @ U2 )
            & ( S5
              = ( image_list_a_a @ F @ U2 ) ) ) ) ) ).

% subset_image_inj
thf(fact_1246_subset__image__inj,axiom,
    ! [S5: set_a,F: a > a,T3: set_a] :
      ( ( ord_less_eq_set_a @ S5 @ ( image_a_a @ F @ T3 ) )
      = ( ? [U2: set_a] :
            ( ( ord_less_eq_set_a @ U2 @ T3 )
            & ( inj_on_a_a @ F @ U2 )
            & ( S5
              = ( image_a_a @ F @ U2 ) ) ) ) ) ).

% subset_image_inj
thf(fact_1247_UNIV__I,axiom,
    ! [X2: list_a] : ( member_list_a2 @ X2 @ top_top_set_list_a ) ).

% UNIV_I
thf(fact_1248_infinite__iff__countable__subset,axiom,
    ! [S5: set_list_a] :
      ( ( ~ ( finite_finite_list_a @ S5 ) )
      = ( ? [F2: nat > list_a] :
            ( ( inj_on_nat_list_a @ F2 @ top_top_set_nat )
            & ( ord_le8861187494160871172list_a @ ( image_nat_list_a @ F2 @ top_top_set_nat ) @ S5 ) ) ) ) ).

% infinite_iff_countable_subset
thf(fact_1249_infinite__iff__countable__subset,axiom,
    ! [S5: set_a] :
      ( ( ~ ( finite_finite_a @ S5 ) )
      = ( ? [F2: nat > a] :
            ( ( inj_on_nat_a @ F2 @ top_top_set_nat )
            & ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ top_top_set_nat ) @ S5 ) ) ) ) ).

% infinite_iff_countable_subset
thf(fact_1250_infinite__countable__subset,axiom,
    ! [S5: set_list_a] :
      ( ~ ( finite_finite_list_a @ S5 )
     => ? [F5: nat > list_a] :
          ( ( inj_on_nat_list_a @ F5 @ top_top_set_nat )
          & ( ord_le8861187494160871172list_a @ ( image_nat_list_a @ F5 @ top_top_set_nat ) @ S5 ) ) ) ).

% infinite_countable_subset
thf(fact_1251_infinite__countable__subset,axiom,
    ! [S5: set_a] :
      ( ~ ( finite_finite_a @ S5 )
     => ? [F5: nat > a] :
          ( ( inj_on_nat_a @ F5 @ top_top_set_nat )
          & ( ord_less_eq_set_a @ ( image_nat_a @ F5 @ top_top_set_nat ) @ S5 ) ) ) ).

% infinite_countable_subset
thf(fact_1252_inj__image__subset__iff,axiom,
    ! [F: list_a > list_a,A2: set_list_a,B: set_list_a] :
      ( ( inj_on_list_a_list_a @ F @ top_top_set_list_a )
     => ( ( ord_le8861187494160871172list_a @ ( image_list_a_list_a @ F @ A2 ) @ ( image_list_a_list_a @ F @ B ) )
        = ( ord_le8861187494160871172list_a @ A2 @ B ) ) ) ).

% inj_image_subset_iff
thf(fact_1253_inj__image__subset__iff,axiom,
    ! [F: a > list_a,A2: set_a,B: set_a] :
      ( ( inj_on_a_list_a @ F @ top_top_set_a )
     => ( ( ord_le8861187494160871172list_a @ ( image_a_list_a @ F @ A2 ) @ ( image_a_list_a @ F @ B ) )
        = ( ord_less_eq_set_a @ A2 @ B ) ) ) ).

% inj_image_subset_iff
thf(fact_1254_inj__image__subset__iff,axiom,
    ! [F: list_a > a,A2: set_list_a,B: set_list_a] :
      ( ( inj_on_list_a_a @ F @ top_top_set_list_a )
     => ( ( ord_less_eq_set_a @ ( image_list_a_a @ F @ A2 ) @ ( image_list_a_a @ F @ B ) )
        = ( ord_le8861187494160871172list_a @ A2 @ B ) ) ) ).

% inj_image_subset_iff
thf(fact_1255_inj__image__subset__iff,axiom,
    ! [F: a > a,A2: set_a,B: set_a] :
      ( ( inj_on_a_a @ F @ top_top_set_a )
     => ( ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B ) )
        = ( ord_less_eq_set_a @ A2 @ B ) ) ) ).

% inj_image_subset_iff
thf(fact_1256_top__greatest,axiom,
    ! [A: set_list_a] : ( ord_le8861187494160871172list_a @ A @ top_top_set_list_a ) ).

% top_greatest
thf(fact_1257_top__greatest,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).

% top_greatest
thf(fact_1258_top_Oextremum__unique,axiom,
    ! [A: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ top_top_set_list_a @ A )
      = ( A = top_top_set_list_a ) ) ).

% top.extremum_unique
thf(fact_1259_top_Oextremum__unique,axiom,
    ! [A: set_a] :
      ( ( ord_less_eq_set_a @ top_top_set_a @ A )
      = ( A = top_top_set_a ) ) ).

% top.extremum_unique
thf(fact_1260_top_Oextremum__uniqueI,axiom,
    ! [A: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ top_top_set_list_a @ A )
     => ( A = top_top_set_list_a ) ) ).

% top.extremum_uniqueI
thf(fact_1261_top_Oextremum__uniqueI,axiom,
    ! [A: set_a] :
      ( ( ord_less_eq_set_a @ top_top_set_a @ A )
     => ( A = top_top_set_a ) ) ).

% top.extremum_uniqueI
thf(fact_1262_subset__UNIV,axiom,
    ! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ A2 @ top_top_set_list_a ) ).

% subset_UNIV
thf(fact_1263_subset__UNIV,axiom,
    ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ top_top_set_a ) ).

% subset_UNIV
thf(fact_1264_UNIV__eq__I,axiom,
    ! [A2: set_list_a] :
      ( ! [X4: list_a] : ( member_list_a2 @ X4 @ A2 )
     => ( top_top_set_list_a = A2 ) ) ).

% UNIV_eq_I
thf(fact_1265_UNIV__witness,axiom,
    ? [X4: list_a] : ( member_list_a2 @ X4 @ top_top_set_list_a ) ).

% UNIV_witness
thf(fact_1266_UNIV__coset,axiom,
    ( top_top_set_a
    = ( coset_a @ nil_a ) ) ).

% UNIV_coset
thf(fact_1267_sup__shunt,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( ( sup_sup_set_list_a @ X2 @ Y )
        = top_top_set_list_a )
      = ( ord_le8861187494160871172list_a @ ( uminus7925729386456332763list_a @ X2 ) @ Y ) ) ).

% sup_shunt
thf(fact_1268_sup__shunt,axiom,
    ! [X2: set_a,Y: set_a] :
      ( ( ( sup_sup_set_a @ X2 @ Y )
        = top_top_set_a )
      = ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X2 ) @ Y ) ) ).

% sup_shunt
thf(fact_1269_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila4954572621509150738list_a @ inf_inf_set_list_a @ top_top_set_list_a @ ord_le8861187494160871172list_a @ ord_less_set_list_a ).

% inf_top.semilattice_neutr_order_axioms
thf(fact_1270_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila2496817875450240012_set_a @ inf_inf_set_a @ top_top_set_a @ ord_less_eq_set_a @ ord_less_set_a ).

% inf_top.semilattice_neutr_order_axioms
thf(fact_1271_mono__Int,axiom,
    ! [F: set_list_a > set_list_a,A2: set_list_a,B: set_list_a] :
      ( ( monoto5743347545245725159list_a @ top_to7106483174946246804list_a @ ord_le8861187494160871172list_a @ ord_le8861187494160871172list_a @ F )
     => ( ord_le8861187494160871172list_a @ ( F @ ( inf_inf_set_list_a @ A2 @ B ) ) @ ( inf_inf_set_list_a @ ( F @ A2 ) @ ( F @ B ) ) ) ) ).

% mono_Int
thf(fact_1272_mono__Int,axiom,
    ! [F: set_list_a > set_a,A2: set_list_a,B: set_list_a] :
      ( ( monoto5794692711665054817_set_a @ top_to7106483174946246804list_a @ ord_le8861187494160871172list_a @ ord_less_eq_set_a @ F )
     => ( ord_less_eq_set_a @ ( F @ ( inf_inf_set_list_a @ A2 @ B ) ) @ ( inf_inf_set_a @ ( F @ A2 ) @ ( F @ B ) ) ) ) ).

% mono_Int
thf(fact_1273_mono__Int,axiom,
    ! [F: set_a > set_list_a,A2: set_a,B: set_a] :
      ( ( monoto7974761327354622061list_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_le8861187494160871172list_a @ F )
     => ( ord_le8861187494160871172list_a @ ( F @ ( inf_inf_set_a @ A2 @ B ) ) @ ( inf_inf_set_list_a @ ( F @ A2 ) @ ( F @ B ) ) ) ) ).

% mono_Int
thf(fact_1274_mono__Int,axiom,
    ! [F: set_a > set_a,A2: set_a,B: set_a] :
      ( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
     => ( ord_less_eq_set_a @ ( F @ ( inf_inf_set_a @ A2 @ B ) ) @ ( inf_inf_set_a @ ( F @ A2 ) @ ( F @ B ) ) ) ) ).

% mono_Int

% Helper facts (17)
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
    ! [X2: list_a,Y: list_a] :
      ( ( if_list_a @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
    ! [X2: list_a,Y: list_a] :
      ( ( if_list_a @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_2_1_If_001t__Stream__Ostream_Itf__a_J_T,axiom,
    ! [X2: stream_a,Y: stream_a] :
      ( ( if_stream_a @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Stream__Ostream_Itf__a_J_T,axiom,
    ! [X2: stream_a,Y: stream_a] :
      ( ( if_stream_a @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_Itf__a_J_J_T,axiom,
    ! [X2: list_list_a,Y: list_list_a] :
      ( ( if_list_list_a @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_Itf__a_J_J_T,axiom,
    ! [X2: list_list_a,Y: list_list_a] :
      ( ( if_list_list_a @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_2_1_If_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J_T,axiom,
    ! [X2: stream_list_a,Y: stream_list_a] :
      ( ( if_stream_list_a @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J_T,axiom,
    ! [X2: stream_list_a,Y: stream_list_a] :
      ( ( if_stream_list_a @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_2_1_If_001t__Stream__Ostream_It__List__Olist_It__Set__Oset_Itf__a_J_J_J_T,axiom,
    ! [X2: stream_list_set_a,Y: stream_list_set_a] :
      ( ( if_stream_list_set_a @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Stream__Ostream_It__List__Olist_It__Set__Oset_Itf__a_J_J_J_T,axiom,
    ! [X2: stream_list_set_a,Y: stream_list_set_a] :
      ( ( if_stream_list_set_a @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_2_1_If_001t__Stream__Ostream_It__List__Olist_It__List__Olist_Itf__a_J_J_J_T,axiom,
    ! [X2: stream_list_list_a,Y: stream_list_list_a] :
      ( ( if_str7505741754068378070list_a @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Stream__Ostream_It__List__Olist_It__List__Olist_Itf__a_J_J_J_T,axiom,
    ! [X2: stream_list_list_a,Y: stream_list_list_a] :
      ( ( if_str7505741754068378070list_a @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_2_1_If_001t__Stream__Ostream_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_T,axiom,
    ! [X2: stream2255243159586646806list_a,Y: stream2255243159586646806list_a] :
      ( ( if_str8217234800680828380list_a @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Stream__Ostream_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_T,axiom,
    ! [X2: stream2255243159586646806list_a,Y: stream2255243159586646806list_a] :
      ( ( if_str8217234800680828380list_a @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_3_1_If_001t__Stream__Ostream_It__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Stream__Ostream_It__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_J_T,axiom,
    ! [X2: stream8854267710728967580list_a,Y: stream8854267710728967580list_a] :
      ( ( if_str5554437364767477090list_a @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Stream__Ostream_It__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_J_T,axiom,
    ! [X2: stream8854267710728967580list_a,Y: stream8854267710728967580list_a] :
      ( ( if_str5554437364767477090list_a @ $true @ X2 @ Y )
      = X2 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ! [X4: list_a] :
      ( ( member_list_a2 @ X4 @ ( sset_list_a @ ( stl_list_a @ sa ) ) )
     => ( X4 != nil_a ) ) ).

%------------------------------------------------------------------------------