%------------------------------------------------------------------------------
% 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    : Rewrite_Properties_Reduction/0017_Replace_Constant/prob_00202_008735__13907132_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1840 ( 564 unt; 587 typ;   0 def)
%            Number of atoms       : 3640 (1680 equ;   0 cnn)
%            Maximal formula atoms :   20 (   2 avg)
%            Number of connectives : 13715 ( 719   ~;  82   |; 277   &;10931   @)
%                                         (   0 <=>;1706  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   8 avg)
%            Number of types       :  106 ( 105 usr)
%            Number of type conns  : 1786 (1786   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  485 ( 482 usr;  28 con; 0-4 aty)
%            Number of variables   : 4332 ( 145   ^;3931   !; 256   ?;4332   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:18:37.915
%------------------------------------------------------------------------------
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    list_P7736225833432154391st_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    list_P5364314822750548887at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Nat__Onat_J_J,type,
    set_Pr6708357783813671845_b_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    set_Pr4549835640365387557rm_a_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    set_Pr698032948827214833st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    set_Pr7549493975000385137at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_Eo_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc4226810134323546766st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    set_Pr1357592988741419896rm_a_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mtf__b_J_J,type,
    set_Pr3369833000147915158_a_b_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
    produc2720409071189015237_b_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    produc1234881154892807749rm_a_b: $tType ).

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

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

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

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

thf(ty_n_t__List__Olist_I_062_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Nat__Onat_J_J,type,
    list_term_a_b_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    produc6551080308567045442rm_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mtf__b_J,type,
    produc6510227487322075744_a_b_b: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    list_list_term_a_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    set_Pr4934435412358123699_a_nat: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    set_list_term_a_b: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
    set_list_list_nat: $tType ).

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

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

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

thf(ty_n_t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    list_term_a_b: $tType ).

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

thf(ty_n_t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    set_term_a_b: $tType ).

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

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

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

thf(ty_n_t__Term__Oterm_Itf__a_Mtf__b_J,type,
    term_a_b: $tType ).

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

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

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

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

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

% Explicit typings (482)
thf(sy_c_BNF__Def_OGrp_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    bNF_Gr4926885435746096120rm_a_b: set_term_a_b > ( term_a_b > term_a_b ) > term_a_b > term_a_b > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
    bNF_re3851489860074285745at_nat: ( nat > nat > $o ) > ( list_nat > nat > $o ) > ( nat > list_nat ) > ( nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    bNF_re1612532812912000561st_nat: ( nat > nat > $o ) > ( nat > list_nat > $o ) > ( nat > nat ) > ( nat > list_nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    bNF_re2975845484123888120rm_a_b: ( nat > nat > $o ) > ( nat > produc357393685978478089rm_a_b > $o ) > ( nat > nat ) > ( nat > produc357393685978478089rm_a_b ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__Nat__Onat,type,
    bNF_re917291077122903834_b_nat: ( nat > nat > $o ) > ( produc357393685978478089rm_a_b > nat > $o ) > ( nat > produc357393685978478089rm_a_b ) > ( nat > nat ) > $o ).

thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__List__Olist_It__Nat__Onat_J,type,
    bNF_Gr9051742241863529473st_nat: set_list_list_nat > list_nat > set_list_list_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Nat__Onat,type,
    bNF_Gr1872714664788909425ft_nat: set_list_nat > nat > set_list_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    bNF_Gr369565347726608424rm_a_b: set_li7311725489111463791rm_a_b > produc357393685978478089rm_a_b > set_li7311725489111463791rm_a_b ).

thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    bNF_Gr2285504805956674357rm_a_b: set_list_term_a_b > term_a_b > set_list_term_a_b ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__List__Olist_It__Nat__Onat_J,type,
    bNF_Gr3053708287304744325st_nat: set_list_list_nat > list_list_nat > set_list_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Nat__Onat,type,
    bNF_Gr6352880689984616693cc_nat: set_list_nat > list_nat > set_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    bNF_Gr7233806482624125860rm_a_b: set_li7311725489111463791rm_a_b > list_P8875379029341186191rm_a_b > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    bNF_Gr5510842888252665017rm_a_b: set_list_term_a_b > list_term_a_b > set_term_a_b ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__List__Olist_It__Nat__Onat_J_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    bNF_Gr89567240463286756rm_a_b: set_list_nat > ( list_nat > term_a_b ) > ( list_nat > term_a_b ) > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Nat__Onat_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    bNF_Gr3221002532544744404rm_a_b: set_nat > ( nat > term_a_b ) > ( nat > term_a_b ) > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    bNF_Gr4099196499516361305rm_a_b: set_Pr4386577575007340137rm_a_b > ( produc357393685978478089rm_a_b > term_a_b ) > ( produc357393685978478089rm_a_b > term_a_b ) > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Basic__Utils_Oadd__elem__list__lists_001t__Nat__Onat,type,
    basic_4874698711677410535ts_nat: nat > list_nat > list_list_nat ).

thf(sy_c_Basic__Utils_Oadd__elem__list__lists_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    basic_1593220722155286443rm_a_b: term_a_b > list_term_a_b > list_list_term_a_b ).

thf(sy_c_Basic__Utils_Olist__of__permutation__element__n_001t__Nat__Onat,type,
    basic_7079635023375748421_n_nat: nat > nat > list_nat > list_list_nat ).

thf(sy_c_Basic__Utils_Olist__of__permutation__element__n_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    basic_7447718476515137801rm_a_b: term_a_b > nat > list_term_a_b > list_list_term_a_b ).

thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
    finite_Fpow_nat: set_nat > set_set_nat ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    comp_t8072462138577781308rm_a_b: ( term_a_b > term_a_b ) > ( produc357393685978478089rm_a_b > term_a_b ) > produc357393685978478089rm_a_b > term_a_b ).

thf(sy_c_Fun_Oid_001t__List__Olist_It__Nat__Onat_J,type,
    id_list_nat: list_nat > list_nat ).

thf(sy_c_Fun_Oid_001t__Nat__Onat,type,
    id_nat: nat > nat ).

thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    id_Pro3718135800935220218rm_a_b: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b ).

thf(sy_c_Fun_Oid_001t__Set__Oset_It__Nat__Onat_J,type,
    id_set_nat: set_nat > set_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
    minus_minus_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    minus_5192120951422937424rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
    plus_plus_nat: nat > nat > nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

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

thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
    lattic8265883725875713057ax_nat: set_nat > nat ).

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

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

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    append1152621114427393060rm_a_b: list_P8875379029341186191rm_a_b > list_P8875379029341186191rm_a_b > list_P8875379029341186191rm_a_b ).

thf(sy_c_List_Oappend_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    append_term_a_b: list_term_a_b > list_term_a_b > list_term_a_b ).

thf(sy_c_List_Obind_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    bind_l7796378977173581257st_nat: list_list_nat > ( list_nat > list_list_nat ) > list_list_nat ).

thf(sy_c_List_Obind_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
    bind_list_nat_nat: list_list_nat > ( list_nat > list_nat ) > list_nat ).

thf(sy_c_List_Obind_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    bind_nat_list_nat: list_nat > ( nat > list_list_nat ) > list_list_nat ).

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

thf(sy_c_List_Obind_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bind_n1878750130520726888at_nat: list_nat > ( nat > list_P6011104703257516679at_nat ) > list_P6011104703257516679at_nat ).

thf(sy_c_List_Obind_001t__Nat__Onat_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    bind_nat_term_a_b: list_nat > ( nat > list_term_a_b ) > list_term_a_b ).

thf(sy_c_List_Obind_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    bind_P7742074774332787594at_nat: list_P6011104703257516679at_nat > ( product_prod_nat_nat > list_nat ) > list_nat ).

thf(sy_c_List_Obind_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__List__Olist_It__Nat__Onat_J,type,
    bind_t442032754801615613st_nat: list_term_a_b > ( term_a_b > list_list_nat ) > list_list_nat ).

thf(sy_c_List_Obind_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
    bind_term_a_b_nat: list_term_a_b > ( term_a_b > list_nat ) > list_nat ).

thf(sy_c_List_Obind_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    bind_t2899167355749536305rm_a_b: list_term_a_b > ( term_a_b > list_term_a_b ) > list_term_a_b ).

thf(sy_c_List_Ocan__select_001t__List__Olist_It__Nat__Onat_J,type,
    can_select_list_nat: ( list_nat > $o ) > set_list_nat > $o ).

thf(sy_c_List_Ocan__select_001t__Nat__Onat,type,
    can_select_nat: ( nat > $o ) > set_nat > $o ).

thf(sy_c_List_Ocan__select_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    can_se4942192940725369094rm_a_b: ( produc357393685978478089rm_a_b > $o ) > set_Pr4386577575007340137rm_a_b > $o ).

thf(sy_c_List_Ocoset_001t__List__Olist_It__Nat__Onat_J,type,
    coset_list_nat: list_list_nat > set_list_nat ).

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

thf(sy_c_List_Ocoset_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    coset_9213161874213406204rm_a_b: list_P8875379029341186191rm_a_b > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_List_Ocoset_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    coset_term_a_b: list_term_a_b > set_term_a_b ).

thf(sy_c_List_Odrop_001t__List__Olist_It__Nat__Onat_J,type,
    drop_list_nat: nat > list_list_nat > list_list_nat ).

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

thf(sy_c_List_Odrop_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    drop_P2146130716461001295rm_a_b: nat > list_P8875379029341186191rm_a_b > list_P8875379029341186191rm_a_b ).

thf(sy_c_List_Odrop_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    drop_term_a_b: nat > list_term_a_b > list_term_a_b ).

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

thf(sy_c_List_Oenumerate_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    enumerate_term_a_b: nat > list_term_a_b > list_P2364656488115551307rm_a_b ).

thf(sy_c_List_Ogen__length_001_062_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
    gen_le5771528551073234224_b_nat: nat > list_term_a_b_nat > nat ).

thf(sy_c_List_Ogen__length_001t__List__Olist_It__Nat__Onat_J,type,
    gen_length_list_nat: nat > list_list_nat > nat ).

thf(sy_c_List_Ogen__length_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    gen_le4942244902447077073rm_a_b: nat > list_list_term_a_b > nat ).

thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
    gen_length_nat: nat > list_nat > nat ).

thf(sy_c_List_Ogen__length_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    gen_le2383899666085517716at_nat: nat > list_P6011104703257516679at_nat > nat ).

thf(sy_c_List_Ogen__length_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    gen_le1467257176787788120rm_a_b: nat > list_P2364656488115551307rm_a_b > nat ).

thf(sy_c_List_Ogen__length_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    gen_length_term_a_b: nat > list_term_a_b > nat ).

thf(sy_c_List_Olenlex_001t__Nat__Onat,type,
    lenlex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_List_Olenlex_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    lenlex_term_a_b: set_Pr4386577575007340137rm_a_b > set_Pr8564414093027780873rm_a_b ).

thf(sy_c_List_Olex_001t__Nat__Onat,type,
    lex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_List_Olex_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    lex_term_a_b: set_Pr4386577575007340137rm_a_b > set_Pr8564414093027780873rm_a_b ).

thf(sy_c_List_Olexord_001t__List__Olist_It__Nat__Onat_J,type,
    lexord_list_nat: set_Pr3451248702717554689st_nat > set_Pr1190453367779242145st_nat ).

thf(sy_c_List_Olexord_001t__Nat__Onat,type,
    lexord_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_List_Olexord_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    lexord1555492263668129660rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr3870824916597072583rm_a_b ).

thf(sy_c_List_Olexord_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    lexord_term_a_b: set_Pr4386577575007340137rm_a_b > set_Pr8564414093027780873rm_a_b ).

thf(sy_c_List_Olist_OCons_001_062_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
    cons_term_a_b_nat: ( term_a_b > nat ) > list_term_a_b_nat > list_term_a_b_nat ).

thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
    cons_list_nat: list_nat > list_list_nat > list_list_nat ).

thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    cons_list_term_a_b: list_term_a_b > list_list_term_a_b > list_list_term_a_b ).

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

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    cons_P6512896166579812791at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    cons_P3536305108106557631rm_a_b: produc357393685978478089rm_a_b > list_P8875379029341186191rm_a_b > list_P8875379029341186191rm_a_b ).

thf(sy_c_List_Olist_OCons_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    cons_term_a_b: term_a_b > list_term_a_b > list_term_a_b ).

thf(sy_c_List_Olist_ONil_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    nil_nat_nat: list_nat_nat ).

thf(sy_c_List_Olist_ONil_001_062_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
    nil_term_a_b_nat: list_term_a_b_nat ).

thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
    nil_list_nat: list_list_nat ).

thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    nil_list_term_a_b: list_list_term_a_b ).

thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    nil_Pr5478986624290739719at_nat: list_P6011104703257516679at_nat ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    nil_Pr4197086119080101963rm_a_b: list_P2364656488115551307rm_a_b ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
    nil_Pr5682614035376309451_b_nat: list_P4523178631563835595_b_nat ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    nil_Pr6942174756412032271rm_a_b: list_P8875379029341186191rm_a_b ).

thf(sy_c_List_Olist_ONil_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    nil_term_a_b: list_term_a_b ).

thf(sy_c_List_Olist_Ohd_001t__List__Olist_It__Nat__Onat_J,type,
    hd_list_nat: list_list_nat > list_nat ).

thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
    hd_nat: list_nat > nat ).

thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    hd_Pro3460610213475200108at_nat: list_P6011104703257516679at_nat > product_prod_nat_nat ).

thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    hd_Pro4077529047277824048rm_a_b: list_P2364656488115551307rm_a_b > produc1234881154892807749rm_a_b ).

thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
    hd_Pro5563056963574031536_b_nat: list_P4523178631563835595_b_nat > produc2720409071189015237_b_nat ).

thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    hd_Pro6001814652670619252rm_a_b: list_P8875379029341186191rm_a_b > produc357393685978478089rm_a_b ).

thf(sy_c_List_Olist_Ohd_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    hd_term_a_b: list_term_a_b > term_a_b ).

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

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    set_list_term_a_b2: list_list_term_a_b > set_list_term_a_b ).

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

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

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    set_Pr7715406932454250534at_nat: list_P5364314822750548887at_nat > set_Pr7549493975000385137at_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    set_Pr7273799732993181129rm_a_b: list_P3366363762698041518rm_a_b > set_Pr1829413238147913556rm_a_b ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J,type,
    set_Pr1526418413117253030st_nat: list_P7736225833432154391st_nat > set_Pr698032948827214833st_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    set_Pr4325273992971601369rm_a_b: list_P1092504657957737918rm_a_b > set_Pr1899728212265713508rm_a_b ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    set_Pr3323217615300463043st_nat: list_P4196053138373943208st_nat > set_Pr2659102613823815246st_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Nat__Onat_J,type,
    set_Pr225064623739089587_b_nat: list_P3607324908554941848_b_nat > set_Pr4414548462862917438_b_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    set_Pr1141850553043368956rm_a_b: list_P6146127666372278893rm_a_b > set_Pr2972776593051762503rm_a_b ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    set_Pr2370880052973118494rm_a_b: list_P8875379029341186191rm_a_b > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_List_Olist_Oset_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    set_term_a_b2: list_term_a_b > set_term_a_b ).

thf(sy_c_List_Olist_Osize__list_001t__List__Olist_It__Nat__Onat_J,type,
    size_list_list_nat: ( list_nat > nat ) > list_list_nat > nat ).

thf(sy_c_List_Olist_Osize__list_001t__Nat__Onat,type,
    size_list_nat: ( nat > nat ) > list_nat > nat ).

thf(sy_c_List_Olist_Osize__list_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    size_l3466988539524548625rm_a_b: ( produc357393685978478089rm_a_b > nat ) > list_P8875379029341186191rm_a_b > nat ).

thf(sy_c_List_Olist_Osize__list_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    size_list_term_a_b: ( term_a_b > nat ) > list_term_a_b > nat ).

thf(sy_c_List_Olist_Otl_001t__List__Olist_It__Nat__Onat_J,type,
    tl_list_nat: list_list_nat > list_list_nat ).

thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
    tl_nat: list_nat > list_nat ).

thf(sy_c_List_Olist_Otl_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    tl_Pro9011950709736066168rm_a_b: list_P8875379029341186191rm_a_b > list_P8875379029341186191rm_a_b ).

thf(sy_c_List_Olist_Otl_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    tl_term_a_b: list_term_a_b > list_term_a_b ).

thf(sy_c_List_Olist__ex1_001_062_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
    list_e8867884845635725434_b_nat: ( ( term_a_b > nat ) > $o ) > list_term_a_b_nat > $o ).

thf(sy_c_List_Olist__ex1_001t__List__Olist_It__Nat__Onat_J,type,
    list_ex1_list_nat: ( list_nat > $o ) > list_list_nat > $o ).

thf(sy_c_List_Olist__ex1_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    list_e1122455069120957467rm_a_b: ( list_term_a_b > $o ) > list_list_term_a_b > $o ).

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

thf(sy_c_List_Olist__ex1_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    list_e8644085759156585930at_nat: ( product_prod_nat_nat > $o ) > list_P6011104703257516679at_nat > $o ).

thf(sy_c_List_Olist__ex1_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    list_e8606123503008938638rm_a_b: ( produc1234881154892807749rm_a_b > $o ) > list_P2364656488115551307rm_a_b > $o ).

thf(sy_c_List_Olist__ex1_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    list_e2893281867963234770rm_a_b: ( produc357393685978478089rm_a_b > $o ) > list_P8875379029341186191rm_a_b > $o ).

thf(sy_c_List_Olist__ex1_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    list_ex1_term_a_b: ( term_a_b > $o ) > list_term_a_b > $o ).

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

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

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    list_u4211539095652206455rm_a_b: list_P8875379029341186191rm_a_b > nat > produc357393685978478089rm_a_b > list_P8875379029341186191rm_a_b ).

thf(sy_c_List_Olist__update_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    list_update_term_a_b: list_term_a_b > nat > term_a_b > list_term_a_b ).

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

thf(sy_c_List_Olistrel1_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    listrel1_term_a_b: set_Pr4386577575007340137rm_a_b > set_Pr8564414093027780873rm_a_b ).

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

thf(sy_c_List_Olistrel_001t__Nat__Onat_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    listrel_nat_term_a_b: set_Pr4549835640365387557rm_a_b > set_Pr2653742144321748037rm_a_b ).

thf(sy_c_List_Olistrel_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
    listrel_term_a_b_nat: set_Pr6708357783813671845_b_nat > set_Pr5972536988916469445st_nat ).

thf(sy_c_List_Olistrel_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    listre1194016999521814427rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr8564414093027780873rm_a_b ).

thf(sy_c_List_Olists_001t__List__Olist_It__Nat__Onat_J,type,
    lists_list_nat: set_list_nat > set_list_list_nat ).

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

thf(sy_c_List_Olists_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    lists_3936190290964281341rm_a_b: set_Pr4386577575007340137rm_a_b > set_li7311725489111463791rm_a_b ).

thf(sy_c_List_Olists_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    lists_term_a_b: set_term_a_b > set_list_term_a_b ).

thf(sy_c_List_Omap__tailrec__rev_001t__Nat__Onat_001t__Nat__Onat,type,
    map_ta7164188454487880599at_nat: ( nat > nat ) > list_nat > list_nat > list_nat ).

thf(sy_c_List_Omap__tailrec__rev_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
    map_ta4339121960748780251_b_nat: ( term_a_b > nat ) > list_term_a_b > list_nat > list_nat ).

thf(sy_c_List_Omap__tailrec__rev__rel_001t__Nat__Onat_001t__Nat__Onat,type,
    map_ta8615873517111064934at_nat: produc1616951275169580055st_nat > produc1616951275169580055st_nat > $o ).

thf(sy_c_List_Omap__tailrec__rev__rel_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
    map_ta8187459354307304746_b_nat: produc6275736337513660055st_nat > produc6275736337513660055st_nat > $o ).

thf(sy_c_List_Omaps_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    maps_l5785965478274863235st_nat: ( list_nat > list_list_nat ) > list_list_nat > list_list_nat ).

thf(sy_c_List_Omaps_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
    maps_list_nat_nat: ( list_nat > list_nat ) > list_list_nat > list_nat ).

thf(sy_c_List_Omaps_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    maps_nat_list_nat: ( nat > list_list_nat ) > list_nat > list_list_nat ).

thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Nat__Onat,type,
    maps_nat_nat: ( nat > list_nat ) > list_nat > list_nat ).

thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    maps_n1297257645151936558at_nat: ( nat > list_P6011104703257516679at_nat ) > list_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    maps_nat_term_a_b: ( nat > list_term_a_b ) > list_nat > list_term_a_b ).

thf(sy_c_List_Omaps_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    maps_P7160582288963997264at_nat: ( product_prod_nat_nat > list_nat ) > list_P6011104703257516679at_nat > list_nat ).

thf(sy_c_List_Omaps_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__List__Olist_It__Nat__Onat_J,type,
    maps_t7654991292757673399st_nat: ( term_a_b > list_list_nat ) > list_term_a_b > list_list_nat ).

thf(sy_c_List_Omaps_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
    maps_term_a_b_nat: ( term_a_b > list_nat ) > list_term_a_b > list_nat ).

thf(sy_c_List_Omaps_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    maps_t888753856850818283rm_a_b: ( term_a_b > list_term_a_b ) > list_term_a_b > list_term_a_b ).

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

thf(sy_c_List_Omeasures_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    measures_term_a_b: list_term_a_b_nat > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_List_Omember_001_062_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
    member_term_a_b_nat: list_term_a_b_nat > ( term_a_b > nat ) > $o ).

thf(sy_c_List_Omember_001t__List__Olist_It__Nat__Onat_J,type,
    member_list_nat: list_list_nat > list_nat > $o ).

thf(sy_c_List_Omember_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    member_list_term_a_b: list_list_term_a_b > list_term_a_b > $o ).

thf(sy_c_List_Omember_001t__Nat__Onat,type,
    member_nat: list_nat > nat > $o ).

thf(sy_c_List_Omember_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member6104210405413575452at_nat: list_P6011104703257516679at_nat > product_prod_nat_nat > $o ).

thf(sy_c_List_Omember_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    member2625873403759427296rm_a_b: list_P2364656488115551307rm_a_b > produc1234881154892807749rm_a_b > $o ).

thf(sy_c_List_Omember_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    member1202727356028726564rm_a_b: list_P8875379029341186191rm_a_b > produc357393685978478089rm_a_b > $o ).

thf(sy_c_List_Omember_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    member_term_a_b: list_term_a_b > term_a_b > $o ).

thf(sy_c_List_On__lists_001t__Nat__Onat,type,
    n_lists_nat: nat > list_nat > list_list_nat ).

thf(sy_c_List_On__lists_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    n_lists_term_a_b: nat > list_term_a_b > list_list_term_a_b ).

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

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    nth_Pr1098598711982373392rm_a_b: list_P8875379029341186191rm_a_b > nat > produc357393685978478089rm_a_b ).

thf(sy_c_List_Onth_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    nth_term_a_b: list_term_a_b > nat > term_a_b ).

thf(sy_c_List_Onull_001_062_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
    null_term_a_b_nat: list_term_a_b_nat > $o ).

thf(sy_c_List_Onull_001t__List__Olist_It__Nat__Onat_J,type,
    null_list_nat: list_list_nat > $o ).

thf(sy_c_List_Onull_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    null_list_term_a_b: list_list_term_a_b > $o ).

thf(sy_c_List_Onull_001t__Nat__Onat,type,
    null_nat: list_nat > $o ).

thf(sy_c_List_Onull_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    null_P7301981044126162959at_nat: list_P6011104703257516679at_nat > $o ).

thf(sy_c_List_Onull_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    null_P4232181139178827347rm_a_b: list_P2364656488115551307rm_a_b > $o ).

thf(sy_c_List_Onull_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    null_term_a_b: list_term_a_b > $o ).

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

thf(sy_c_List_Oproduct__lists_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    produc17669015410068453rm_a_b: list_list_term_a_b > list_list_term_a_b ).

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

thf(sy_c_List_Orotate1_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    rotate1_term_a_b: list_term_a_b > list_term_a_b ).

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

thf(sy_c_List_Osubseqs_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    subseqs_term_a_b: list_term_a_b > list_list_term_a_b ).

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

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

thf(sy_c_List_Otake_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    take_P7698341771522494711rm_a_b: nat > list_P8875379029341186191rm_a_b > list_P8875379029341186191rm_a_b ).

thf(sy_c_List_Otake_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    take_term_a_b: nat > list_term_a_b > list_term_a_b ).

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

thf(sy_c_List_Ozip_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
    zip_list_nat_nat: list_list_nat > list_nat > list_P5364314822750548887at_nat ).

thf(sy_c_List_Ozip_001t__List__Olist_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    zip_li1978623282951132976rm_a_b: list_list_nat > list_P8875379029341186191rm_a_b > list_P3366363762698041518rm_a_b ).

thf(sy_c_List_Ozip_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    zip_nat_list_nat: list_nat > list_list_nat > list_P7736225833432154391st_nat ).

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

thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    zip_na8725512192174737984rm_a_b: list_nat > list_P8875379029341186191rm_a_b > list_P1092504657957737918rm_a_b ).

thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    zip_nat_term_a_b: list_nat > list_term_a_b > list_P2364656488115551307rm_a_b ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__List__Olist_It__Nat__Onat_J,type,
    zip_Pr4019727599664654706st_nat: list_P8875379029341186191rm_a_b > list_list_nat > list_P4196053138373943208st_nat ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__Nat__Onat,type,
    zip_Pr6666957785173753698_b_nat: list_P8875379029341186191rm_a_b > list_nat > list_P3607324908554941848_b_nat ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    zip_Pr5674138466097917431rm_a_b: list_P8875379029341186191rm_a_b > list_P8875379029341186191rm_a_b > list_P6146127666372278893rm_a_b ).

thf(sy_c_List_Ozip_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
    zip_term_a_b_nat: list_term_a_b > list_nat > list_P4523178631563835595_b_nat ).

thf(sy_c_List_Ozip_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    zip_te2260252107881041761rm_a_b: list_term_a_b > list_term_a_b > list_P8875379029341186191rm_a_b ).

thf(sy_c_Missing__List_Oconcat__lists_001t__Nat__Onat,type,
    missin4567272213201432058ts_nat: list_list_nat > list_list_nat ).

thf(sy_c_Missing__List_Oconcat__lists_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    missin8060632096978918206rm_a_b: list_list_term_a_b > list_list_term_a_b ).

thf(sy_c_Missing__List_Ofilter2_001t__Nat__Onat_001t__Nat__Onat,type,
    missin7656342814955748836at_nat: ( nat > nat > $o ) > list_nat > list_nat > produc1828647624359046049st_nat ).

thf(sy_c_Missing__List_Ofilter2_001t__Nat__Onat_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    missin7648764293464682024rm_a_b: ( nat > term_a_b > $o ) > list_nat > list_term_a_b > produc9043357390500214885rm_a_b ).

thf(sy_c_Missing__List_Ofilter2_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
    missin8610218420629654184_b_nat: ( term_a_b > nat > $o ) > list_term_a_b > list_nat > produc895009549562661605st_nat ).

thf(sy_c_Missing__List_Ofilter2_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    missin5478484803207743724rm_a_b: ( term_a_b > term_a_b > $o ) > list_term_a_b > list_term_a_b > produc51424535725745577rm_a_b ).

thf(sy_c_Missing__List_Ofilter2__rel_001t__Nat__Onat_001t__Nat__Onat,type,
    missin8708480906560974681at_nat: produc4787317212837456354st_nat > produc4787317212837456354st_nat > $o ).

thf(sy_c_Missing__List_Ofilter2__rel_001t__Nat__Onat_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    missin2721041770646806813rm_a_b: produc8734217775477389290rm_a_b > produc8734217775477389290rm_a_b > $o ).

thf(sy_c_Missing__List_Ofilter2__rel_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
    missin3682495897811778973_b_nat: produc6817702157447538794st_nat > produc6817702157447538794st_nat > $o ).

thf(sy_c_Missing__List_Ofilter2__rel_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    missin5945364489896066401rm_a_b: produc8215350257458196082rm_a_b > produc8215350257458196082rm_a_b > $o ).

thf(sy_c_Missing__List_Olist__inter_001t__List__Olist_It__Nat__Onat_J,type,
    missin6532874241183986279st_nat: list_list_nat > list_list_nat > list_list_nat ).

thf(sy_c_Missing__List_Olist__inter_001t__Nat__Onat,type,
    missin6377695591745783511er_nat: list_nat > list_nat > list_nat ).

thf(sy_c_Missing__List_Olist__inter_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    missin3381965633867963714rm_a_b: list_P8875379029341186191rm_a_b > list_P8875379029341186191rm_a_b > list_P8875379029341186191rm_a_b ).

thf(sy_c_Missing__List_Olist__inter_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    missin8990008842131906971rm_a_b: list_term_a_b > list_term_a_b > list_term_a_b ).

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    size_s8600120110137556219rm_a_b: list_P8875379029341186191rm_a_b > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    size_s8906293707977694520rm_a_b: list_term_a_b > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    size_size_term_a_b: term_a_b > nat ).

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    ord_le1190675801316882794st_nat: set_list_nat > set_list_nat > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    ord_le4883060154188259093rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    ord_le118470702582115849rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    ord_le2705286416250468010rm_a_b: set_term_a_b > set_term_a_b > $o ).

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

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    top_top_list_nat_o: list_nat > $o ).

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

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_M_Eo_J,type,
    top_to1262317115727556708_a_b_o: produc357393685978478089rm_a_b > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    top_top_set_list_nat: set_list_nat ).

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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    top_to1314267278846557113rm_a_b: set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    top_top_set_term_a_b: set_term_a_b ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    produc3074213772457683607_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > produc5270577607013733799_nat_o ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc4727192421694094319st_nat: ( nat > nat > $o ) > list_nat > produc254973753779126261st_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc3127733452865184594st_nat: ( nat > nat > $o ) > produc1828647624359046049st_nat > produc4787317212837456354st_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Term__Oterm_Itf__a_Mtf__b_J_M_Eo_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    produc2750092164434624090rm_a_b: ( nat > term_a_b > $o ) > produc9043357390500214885rm_a_b > produc8734217775477389290rm_a_b ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_Eo_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc8587622027977423880st_nat: ( nat > $o ) > list_nat > produc4226810134323546766st_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc5770335208449155351at_nat: ( nat > nat ) > ( nat > nat ) > produc1932156733058919263at_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc4626581765195395529st_nat: ( nat > nat ) > produc1828647624359046049st_nat > produc1616951275169580055st_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Term__Oterm_Itf__a_Mtf__b_J_M_062_It__Nat__Onat_M_Eo_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc7813177934582050522st_nat: ( term_a_b > nat > $o ) > produc895009549562661605st_nat > produc6817702157447538794st_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Term__Oterm_Itf__a_Mtf__b_J_M_062_It__Term__Oterm_Itf__a_Mtf__b_J_M_Eo_J_J_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
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thf(sy_c_Product__Type_Oprod_Oswap_001t__Term__Oterm_Itf__a_Mtf__b_J_001tf__b,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    member3205530567375370445st_nat: produc5438398690206989622st_nat > set_Pr1385048790419436780st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    member4140859439665155893rm_a_b: produc6373727562496775070rm_a_b > set_Pr1829413238147913556rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
    member2631635837537579280rm_a_b: produc8018406874456678119rm_a_b > set_Pr3870824916597072583rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member8889939629616814094st_nat: produc895009549562661605st_nat > set_Pr5972536988916469445st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    member4405265420456397394rm_a_b: produc51424535725745577rm_a_b > set_Pr8564414093027780873rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member1631564025489475386st_nat: produc4575160907756185873st_nat > set_Pr698032948827214833st_nat > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    member755457025140340037rm_a_b: produc8160904860966076974rm_a_b > set_Pr1899728212265713508rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    member7514656309480873070rm_a_b: produc1234881154892807749rm_a_b > set_Pr4549835640365387557rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_J,type,
    member6918694168860644693_nat_o: produc4754350896526429996_nat_o > set_Pr7697833614269146252_nat_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member6277506213002785702at_nat: produc2297559132824252175at_nat > set_Pr5877781423743409989at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member6135396219256978631st_nat: produc8368264342088597808st_nat > set_Pr3833137770001865702st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member190277321972437807st_nat: produc2423145444804056984st_nat > set_Pr2659102613823815246st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Nat__Onat_J,type,
    member5878619692762604063_b_nat: produc4060695491733565192_b_nat > set_Pr4414548462862917438_b_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    member8417600551952982416rm_a_b: produc4523971788519308903rm_a_b > set_Pr2972776593051762503rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
    member9000184225777080558_b_nat: produc2720409071189015237_b_nat > set_Pr6708357783813671845_b_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    member5869715511025134514rm_a_b: produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mtf__b_J,type,
    member6959280461167840375_a_b_b: produc6510227487322075744_a_b_b > set_Pr3369833000147915158_a_b_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    member7000133282412810073rm_a_b: produc6551080308567045442rm_a_b > set_Pr1357592988741419896rm_a_b > $o ).

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

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

thf(sy_c_member_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    member_term_a_b2: term_a_b > set_term_a_b > $o ).

thf(sy_v_U,type,
    u: set_Pr4386577575007340137rm_a_b ).

thf(sy_v_V,type,
    v: set_Pr4386577575007340137rm_a_b ).

thf(sy_v_c,type,
    c: a ).

thf(sy_v_p____,type,
    p: list_nat ).

thf(sy_v_s____,type,
    s: term_a_b ).

thf(sy_v_t____,type,
    t: term_a_b ).

thf(sy_v_u____,type,
    u2: term_a_b ).

thf(sy_v_w____,type,
    w: term_a_b ).

% Relevant facts (1249)
thf(fact_0_w_I1_J,axiom,
    member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ s @ w ) @ u ).

% w(1)
thf(fact_1_w_I2_J,axiom,
    member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ w @ t ) @ v ).

% w(2)
thf(fact_2_local_Oconst_I1_J,axiom,
    replac8618238927709742794ed_a_b @ c @ u ).

% local.const(1)
thf(fact_3_local_Oconst_I2_J,axiom,
    replac8618238927709742794ed_a_b @ c @ v ).

% local.const(2)
thf(fact_4_wit,axiom,
    member_list_nat2 @ p @ ( terms_7168686267159881682rm_a_b @ ( fun_a_b @ c @ nil_term_a_b ) @ s ) ).

% wit
thf(fact_5_local_Ostep,axiom,
    member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ s @ t ) @ ( relcom370159955682700863rm_a_b @ u @ v ) ).

% local.step
thf(fact_6_b,axiom,
    ? [Q: list_nat] :
      ( ( member_list_nat2 @ Q @ ( terms_7168686267159881682rm_a_b @ ( fun_a_b @ c @ nil_term_a_b ) @ w ) )
      & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( term_r6860082780075436317at_a_b @ s @ p @ u2 ) @ ( term_r6860082780075436317at_a_b @ w @ Q @ u2 ) ) @ u ) ) ).

% b
thf(fact_7__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062w_O_A_092_060lbrakk_062_Is_M_Aw_J_A_092_060in_062_AU_059_A_Iw_M_At_J_A_092_060in_062_AV_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [W: term_a_b] :
        ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ s @ W ) @ u )
       => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ W @ t ) @ v ) ) ).

% \<open>\<And>thesis. (\<And>w. \<lbrakk>(s, w) \<in> U; (w, t) \<in> V\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_8__092_060open_062_092_060And_062u_O_A_Is_091p_A_092_060leftarrow_062_Au_093_M_Aw_J_A_092_060in_062_AU_A_092_060or_062_A_I_092_060exists_062q_O_Aq_A_092_060in_062_Aposs__of__term_A_IconstT_Ac_J_Aw_A_092_060and_062_A_Is_091p_A_092_060leftarrow_062_Au_093_M_Aw_091q_A_092_060leftarrow_062_Au_093_J_A_092_060in_062_AU_J_092_060close_062,axiom,
    ! [U: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( term_r6860082780075436317at_a_b @ s @ p @ U ) @ w ) @ u )
      | ? [Q: list_nat] :
          ( ( member_list_nat2 @ Q @ ( terms_7168686267159881682rm_a_b @ ( fun_a_b @ c @ nil_term_a_b ) @ w ) )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( term_r6860082780075436317at_a_b @ s @ p @ U ) @ ( term_r6860082780075436317at_a_b @ w @ Q @ U ) ) @ u ) ) ) ).

% \<open>\<And>u. (s[p \<leftarrow> u], w) \<in> U \<or> (\<exists>q. q \<in> poss_of_term (constT c) w \<and> (s[p \<leftarrow> u], w[q \<leftarrow> u]) \<in> U)\<close>
thf(fact_9__092_060open_062_092_060And_062thesis_O_A_092_060lbrakk_062_Is_091p_A_092_060leftarrow_062_Au_093_M_Aw_J_A_092_060in_062_AU_A_092_060Longrightarrow_062_Athesis_059_A_092_060exists_062q_O_Aq_A_092_060in_062_Aposs__of__term_A_IconstT_Ac_J_Aw_A_092_060and_062_A_Is_091p_A_092_060leftarrow_062_Au_093_M_Aw_091q_A_092_060leftarrow_062_Au_093_J_A_092_060in_062_AU_A_092_060Longrightarrow_062_Athesis_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( term_r6860082780075436317at_a_b @ s @ p @ u2 ) @ w ) @ u )
   => ? [Q: list_nat] :
        ( ( member_list_nat2 @ Q @ ( terms_7168686267159881682rm_a_b @ ( fun_a_b @ c @ nil_term_a_b ) @ w ) )
        & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( term_r6860082780075436317at_a_b @ s @ p @ u2 ) @ ( term_r6860082780075436317at_a_b @ w @ Q @ u2 ) ) @ u ) ) ) ).

% \<open>\<And>thesis. \<lbrakk>(s[p \<leftarrow> u], w) \<in> U \<Longrightarrow> thesis; \<exists>q. q \<in> poss_of_term (constT c) w \<and> (s[p \<leftarrow> u], w[q \<leftarrow> u]) \<in> U \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis\<close>
thf(fact_10_const__replace__closedD,axiom,
    ! [C: a,U2: set_Pr4386577575007340137rm_a_b,P: list_nat,S: term_a_b,T: term_a_b,U: term_a_b] :
      ( ( replac8618238927709742794ed_a_b @ C @ U2 )
     => ( ( member_list_nat2 @ P @ ( terms_7168686267159881682rm_a_b @ ( fun_a_b @ C @ nil_term_a_b ) @ S ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ U2 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( term_r6860082780075436317at_a_b @ S @ P @ U ) @ T ) @ U2 )
            | ? [Q: list_nat] :
                ( ( member_list_nat2 @ Q @ ( terms_7168686267159881682rm_a_b @ ( fun_a_b @ C @ nil_term_a_b ) @ T ) )
                & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( term_r6860082780075436317at_a_b @ S @ P @ U ) @ ( term_r6860082780075436317at_a_b @ T @ Q @ U ) ) @ U2 ) ) ) ) ) ) ).

% const_replace_closedD
thf(fact_11_const__replace__closedI,axiom,
    ! [C: a,U2: set_Pr4386577575007340137rm_a_b] :
      ( ! [S2: term_a_b,T2: term_a_b,U3: term_a_b,P2: list_nat] :
          ( ( member_list_nat2 @ P2 @ ( terms_7168686267159881682rm_a_b @ ( fun_a_b @ C @ nil_term_a_b ) @ S2 ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S2 @ T2 ) @ U2 )
           => ( ? [Q2: list_nat] :
                  ( ( member_list_nat2 @ Q2 @ ( terms_7168686267159881682rm_a_b @ ( fun_a_b @ C @ nil_term_a_b ) @ T2 ) )
                  & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( term_r6860082780075436317at_a_b @ S2 @ P2 @ U3 ) @ ( term_r6860082780075436317at_a_b @ T2 @ Q2 @ U3 ) ) @ U2 ) )
              | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( term_r6860082780075436317at_a_b @ S2 @ P2 @ U3 ) @ T2 ) @ U2 ) ) ) )
     => ( replac8618238927709742794ed_a_b @ C @ U2 ) ) ).

% const_replace_closedI
thf(fact_12_const__replace__closed__def,axiom,
    ( replac8618238927709742794ed_a_b
    = ( ^ [C2: a,U4: set_Pr4386577575007340137rm_a_b] :
        ! [S3: term_a_b,T3: term_a_b,U5: term_a_b,P3: list_nat] :
          ( ( member_list_nat2 @ P3 @ ( terms_7168686267159881682rm_a_b @ ( fun_a_b @ C2 @ nil_term_a_b ) @ S3 ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S3 @ T3 ) @ U4 )
           => ( ? [Q3: list_nat] :
                  ( ( member_list_nat2 @ Q3 @ ( terms_7168686267159881682rm_a_b @ ( fun_a_b @ C2 @ nil_term_a_b ) @ T3 ) )
                  & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( term_r6860082780075436317at_a_b @ S3 @ P3 @ U5 ) @ ( term_r6860082780075436317at_a_b @ T3 @ Q3 @ U5 ) ) @ U4 ) )
              | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( term_r6860082780075436317at_a_b @ S3 @ P3 @ U5 ) @ T3 ) @ U4 ) ) ) ) ) ) ).

% const_replace_closed_def
thf(fact_13_replace__term__at__same__pos,axiom,
    ! [S: term_a_b,P: list_nat,U: term_a_b,T: term_a_b] :
      ( ( term_r6860082780075436317at_a_b @ ( term_r6860082780075436317at_a_b @ S @ P @ U ) @ P @ T )
      = ( term_r6860082780075436317at_a_b @ S @ P @ T ) ) ).

% replace_term_at_same_pos
thf(fact_14_term_Oinject_I2_J,axiom,
    ! [X21: a,X22: list_term_a_b,Y21: a,Y22: list_term_a_b] :
      ( ( ( fun_a_b @ X21 @ X22 )
        = ( fun_a_b @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% term.inject(2)
thf(fact_15_prod_Oinject,axiom,
    ! [X1: nat > nat,X2: produc1828647624359046049st_nat,Y1: nat > nat,Y2: produc1828647624359046049st_nat] :
      ( ( ( produc4626581765195395529st_nat @ X1 @ X2 )
        = ( produc4626581765195395529st_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_16_prod_Oinject,axiom,
    ! [X1: nat > term_a_b > $o,X2: produc9043357390500214885rm_a_b,Y1: nat > term_a_b > $o,Y2: produc9043357390500214885rm_a_b] :
      ( ( ( produc2750092164434624090rm_a_b @ X1 @ X2 )
        = ( produc2750092164434624090rm_a_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_17_prod_Oinject,axiom,
    ! [X1: nat > nat > $o,X2: produc1828647624359046049st_nat,Y1: nat > nat > $o,Y2: produc1828647624359046049st_nat] :
      ( ( ( produc3127733452865184594st_nat @ X1 @ X2 )
        = ( produc3127733452865184594st_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_18_prod_Oinject,axiom,
    ! [X1: nat > nat > $o,X2: list_nat,Y1: nat > nat > $o,Y2: list_nat] :
      ( ( ( produc4727192421694094319st_nat @ X1 @ X2 )
        = ( produc4727192421694094319st_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_19_prod_Oinject,axiom,
    ! [X1: b,X2: term_a_b,Y1: b,Y2: term_a_b] :
      ( ( ( produc1437816968797971900rm_a_b @ X1 @ X2 )
        = ( produc1437816968797971900rm_a_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_20_prod_Oinject,axiom,
    ! [X1: term_a_b,X2: term_a_b,Y1: term_a_b,Y2: term_a_b] :
      ( ( ( produc7020197800436672577rm_a_b @ X1 @ X2 )
        = ( produc7020197800436672577rm_a_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_21_old_Oprod_Oinject,axiom,
    ! [A: nat > nat,B: produc1828647624359046049st_nat,A2: nat > nat,B2: produc1828647624359046049st_nat] :
      ( ( ( produc4626581765195395529st_nat @ A @ B )
        = ( produc4626581765195395529st_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_22_old_Oprod_Oinject,axiom,
    ! [A: nat > term_a_b > $o,B: produc9043357390500214885rm_a_b,A2: nat > term_a_b > $o,B2: produc9043357390500214885rm_a_b] :
      ( ( ( produc2750092164434624090rm_a_b @ A @ B )
        = ( produc2750092164434624090rm_a_b @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_23_old_Oprod_Oinject,axiom,
    ! [A: nat > nat > $o,B: produc1828647624359046049st_nat,A2: nat > nat > $o,B2: produc1828647624359046049st_nat] :
      ( ( ( produc3127733452865184594st_nat @ A @ B )
        = ( produc3127733452865184594st_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_24_old_Oprod_Oinject,axiom,
    ! [A: nat > nat > $o,B: list_nat,A2: nat > nat > $o,B2: list_nat] :
      ( ( ( produc4727192421694094319st_nat @ A @ B )
        = ( produc4727192421694094319st_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_25_old_Oprod_Oinject,axiom,
    ! [A: b,B: term_a_b,A2: b,B2: term_a_b] :
      ( ( ( produc1437816968797971900rm_a_b @ A @ B )
        = ( produc1437816968797971900rm_a_b @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_26_old_Oprod_Oinject,axiom,
    ! [A: term_a_b,B: term_a_b,A2: term_a_b,B2: term_a_b] :
      ( ( ( produc7020197800436672577rm_a_b @ A @ B )
        = ( produc7020197800436672577rm_a_b @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_27_relcomp_Ocases,axiom,
    ! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( relcom370159955682700863rm_a_b @ R @ S ) )
     => ~ ! [B3: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B3 ) @ R )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_28_relcomp_Ocases,axiom,
    ! [A1: nat,A22: nat,R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( relcomp_nat_nat_nat @ R @ S ) )
     => ~ ! [B3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B3 ) @ R )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_29_relcomp_Ocases,axiom,
    ! [A1: b,A22: term_a_b,R: set_Product_prod_b_b,S: set_Pr1357592988741419896rm_a_b] :
      ( ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ A1 @ A22 ) @ ( relcomp_b_b_term_a_b @ R @ S ) )
     => ~ ! [B3: b] :
            ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A1 @ B3 ) @ R )
           => ~ ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_30_relcomp_Ocases,axiom,
    ! [A1: term_a_b,A22: term_a_b,R: set_Pr3369833000147915158_a_b_b,S: set_Pr1357592988741419896rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( relcom5423879875627813950rm_a_b @ R @ S ) )
     => ~ ! [B3: b] :
            ( ( member6959280461167840375_a_b_b @ ( produc1666658391573147538_a_b_b @ A1 @ B3 ) @ R )
           => ~ ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_31_relcomp_Ocases,axiom,
    ! [A1: b,A22: term_a_b,R: set_Pr1357592988741419896rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ A1 @ A22 ) @ ( relcom6871247487278447976rm_a_b @ R @ S ) )
     => ~ ! [B3: term_a_b] :
            ( ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ A1 @ B3 ) @ R )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_32_relcomp_Ocases,axiom,
    ! [A1: nat > nat > $o,A22: list_nat,R: set_Pr7072801126362145067st_nat,S: set_Pr3451248702717554689st_nat] :
      ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ A1 @ A22 ) @ ( relcom6714777091229659495st_nat @ R @ S ) )
     => ~ ! [B3: list_nat] :
            ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ A1 @ B3 ) @ R )
           => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_33_relcomp_Ocases,axiom,
    ! [A1: nat > nat,A22: produc1828647624359046049st_nat,R: set_Pr7682762132356531903at_nat,S: set_Pr3595532568945684557st_nat] :
      ( ( member5596898355348113582st_nat @ ( produc4626581765195395529st_nat @ A1 @ A22 ) @ ( relcom2190433716068671200st_nat @ R @ S ) )
     => ~ ! [B3: nat > nat] :
            ( ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A1 @ B3 ) @ R )
           => ~ ( member5596898355348113582st_nat @ ( produc4626581765195395529st_nat @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_34_relcomp_Ocases,axiom,
    ! [A1: nat > nat > $o,A22: produc1828647624359046049st_nat,R: set_Pr7072801126362145067st_nat,S: set_Pr1385048790419436780st_nat] :
      ( ( member6951660485171671051st_nat @ ( produc3127733452865184594st_nat @ A1 @ A22 ) @ ( relcom2919347406790136794st_nat @ R @ S ) )
     => ~ ! [B3: list_nat] :
            ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ A1 @ B3 ) @ R )
           => ~ ( member3205530567375370445st_nat @ ( produc4487115339913071592st_nat @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_35_relcomp_Ocases,axiom,
    ! [A1: nat > nat > $o,A22: list_nat,R: set_Pr1489038555860665479_nat_o,S: set_Pr7072801126362145067st_nat] :
      ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ A1 @ A22 ) @ ( relcom9058974518648859105st_nat @ R @ S ) )
     => ~ ! [B3: nat > nat > $o] :
            ( ( member498802742439182032_nat_o @ ( produc3074213772457683607_nat_o @ A1 @ B3 ) @ R )
           => ~ ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_36_relcomp_Ocases,axiom,
    ! [A1: nat > nat > $o,A22: list_nat,R: set_Pr4817715314677154882st_nat,S: set_Pr3833137770001865702st_nat] :
      ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ A1 @ A22 ) @ ( relcom210944645552855580st_nat @ R @ S ) )
     => ~ ! [B3: produc1828647624359046049st_nat] :
            ( ( member6951660485171671051st_nat @ ( produc3127733452865184594st_nat @ A1 @ B3 ) @ R )
           => ~ ( member6135396219256978631st_nat @ ( produc1778712578675790378st_nat @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_37_relcomp_Osimps,axiom,
    ! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( relcom370159955682700863rm_a_b @ R @ S ) )
      = ( ? [A3: term_a_b,B4: term_a_b,C2: term_a_b] :
            ( ( A1 = A3 )
            & ( A22 = C2 )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B4 ) @ R )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C2 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_38_relcomp_Osimps,axiom,
    ! [A1: nat,A22: nat,R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( relcomp_nat_nat_nat @ R @ S ) )
      = ( ? [A3: nat,B4: nat,C2: nat] :
            ( ( A1 = A3 )
            & ( A22 = C2 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ R )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C2 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_39_relcomp_Osimps,axiom,
    ! [A1: b,A22: term_a_b,R: set_Product_prod_b_b,S: set_Pr1357592988741419896rm_a_b] :
      ( ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ A1 @ A22 ) @ ( relcomp_b_b_term_a_b @ R @ S ) )
      = ( ? [A3: b,B4: b,C2: term_a_b] :
            ( ( A1 = A3 )
            & ( A22 = C2 )
            & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A3 @ B4 ) @ R )
            & ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ B4 @ C2 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_40_relcomp_Osimps,axiom,
    ! [A1: term_a_b,A22: term_a_b,R: set_Pr3369833000147915158_a_b_b,S: set_Pr1357592988741419896rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( relcom5423879875627813950rm_a_b @ R @ S ) )
      = ( ? [A3: term_a_b,B4: b,C2: term_a_b] :
            ( ( A1 = A3 )
            & ( A22 = C2 )
            & ( member6959280461167840375_a_b_b @ ( produc1666658391573147538_a_b_b @ A3 @ B4 ) @ R )
            & ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ B4 @ C2 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_41_relcomp_Osimps,axiom,
    ! [A1: b,A22: term_a_b,R: set_Pr1357592988741419896rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ A1 @ A22 ) @ ( relcom6871247487278447976rm_a_b @ R @ S ) )
      = ( ? [A3: b,B4: term_a_b,C2: term_a_b] :
            ( ( A1 = A3 )
            & ( A22 = C2 )
            & ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ A3 @ B4 ) @ R )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C2 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_42_relcomp_Osimps,axiom,
    ! [A1: nat > nat > $o,A22: list_nat,R: set_Pr7072801126362145067st_nat,S: set_Pr3451248702717554689st_nat] :
      ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ A1 @ A22 ) @ ( relcom6714777091229659495st_nat @ R @ S ) )
      = ( ? [A3: nat > nat > $o,B4: list_nat,C2: list_nat] :
            ( ( A1 = A3 )
            & ( A22 = C2 )
            & ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ A3 @ B4 ) @ R )
            & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B4 @ C2 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_43_relcomp_Osimps,axiom,
    ! [A1: nat > nat,A22: produc1828647624359046049st_nat,R: set_Pr7682762132356531903at_nat,S: set_Pr3595532568945684557st_nat] :
      ( ( member5596898355348113582st_nat @ ( produc4626581765195395529st_nat @ A1 @ A22 ) @ ( relcom2190433716068671200st_nat @ R @ S ) )
      = ( ? [A3: nat > nat,B4: nat > nat,C2: produc1828647624359046049st_nat] :
            ( ( A1 = A3 )
            & ( A22 = C2 )
            & ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A3 @ B4 ) @ R )
            & ( member5596898355348113582st_nat @ ( produc4626581765195395529st_nat @ B4 @ C2 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_44_relcomp_Osimps,axiom,
    ! [A1: nat > nat > $o,A22: produc1828647624359046049st_nat,R: set_Pr7072801126362145067st_nat,S: set_Pr1385048790419436780st_nat] :
      ( ( member6951660485171671051st_nat @ ( produc3127733452865184594st_nat @ A1 @ A22 ) @ ( relcom2919347406790136794st_nat @ R @ S ) )
      = ( ? [A3: nat > nat > $o,B4: list_nat,C2: produc1828647624359046049st_nat] :
            ( ( A1 = A3 )
            & ( A22 = C2 )
            & ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ A3 @ B4 ) @ R )
            & ( member3205530567375370445st_nat @ ( produc4487115339913071592st_nat @ B4 @ C2 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_45_relcomp_Osimps,axiom,
    ! [A1: nat > nat > $o,A22: list_nat,R: set_Pr1489038555860665479_nat_o,S: set_Pr7072801126362145067st_nat] :
      ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ A1 @ A22 ) @ ( relcom9058974518648859105st_nat @ R @ S ) )
      = ( ? [A3: nat > nat > $o,B4: nat > nat > $o,C2: list_nat] :
            ( ( A1 = A3 )
            & ( A22 = C2 )
            & ( member498802742439182032_nat_o @ ( produc3074213772457683607_nat_o @ A3 @ B4 ) @ R )
            & ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ B4 @ C2 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_46_relcomp_Osimps,axiom,
    ! [A1: nat > nat > $o,A22: list_nat,R: set_Pr4817715314677154882st_nat,S: set_Pr3833137770001865702st_nat] :
      ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ A1 @ A22 ) @ ( relcom210944645552855580st_nat @ R @ S ) )
      = ( ? [A3: nat > nat > $o,B4: produc1828647624359046049st_nat,C2: list_nat] :
            ( ( A1 = A3 )
            & ( A22 = C2 )
            & ( member6951660485171671051st_nat @ ( produc3127733452865184594st_nat @ A3 @ B4 ) @ R )
            & ( member6135396219256978631st_nat @ ( produc1778712578675790378st_nat @ B4 @ C2 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_47_relcomp_OrelcompI,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ S )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( relcom370159955682700863rm_a_b @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_48_relcomp_OrelcompI,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C: nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ S )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C ) @ ( relcomp_nat_nat_nat @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_49_relcomp_OrelcompI,axiom,
    ! [A: b,B: b,R: set_Product_prod_b_b,C: term_a_b,S: set_Pr1357592988741419896rm_a_b] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ R )
     => ( ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ B @ C ) @ S )
       => ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ A @ C ) @ ( relcomp_b_b_term_a_b @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_50_relcomp_OrelcompI,axiom,
    ! [A: term_a_b,B: b,R: set_Pr3369833000147915158_a_b_b,C: term_a_b,S: set_Pr1357592988741419896rm_a_b] :
      ( ( member6959280461167840375_a_b_b @ ( produc1666658391573147538_a_b_b @ A @ B ) @ R )
     => ( ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ B @ C ) @ S )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( relcom5423879875627813950rm_a_b @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_51_relcomp_OrelcompI,axiom,
    ! [A: b,B: term_a_b,R: set_Pr1357592988741419896rm_a_b,C: term_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ A @ B ) @ R )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ S )
       => ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ A @ C ) @ ( relcom6871247487278447976rm_a_b @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_52_relcomp_OrelcompI,axiom,
    ! [A: nat > nat > $o,B: list_nat,R: set_Pr7072801126362145067st_nat,C: list_nat,S: set_Pr3451248702717554689st_nat] :
      ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ A @ B ) @ R )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ C ) @ S )
       => ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ A @ C ) @ ( relcom6714777091229659495st_nat @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_53_relcomp_OrelcompI,axiom,
    ! [A: nat > nat,B: nat > nat,R: set_Pr7682762132356531903at_nat,C: produc1828647624359046049st_nat,S: set_Pr3595532568945684557st_nat] :
      ( ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A @ B ) @ R )
     => ( ( member5596898355348113582st_nat @ ( produc4626581765195395529st_nat @ B @ C ) @ S )
       => ( member5596898355348113582st_nat @ ( produc4626581765195395529st_nat @ A @ C ) @ ( relcom2190433716068671200st_nat @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_54_relcomp_OrelcompI,axiom,
    ! [A: nat > nat > $o,B: nat > nat > $o,R: set_Pr1489038555860665479_nat_o,C: list_nat,S: set_Pr7072801126362145067st_nat] :
      ( ( member498802742439182032_nat_o @ ( produc3074213772457683607_nat_o @ A @ B ) @ R )
     => ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ B @ C ) @ S )
       => ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ A @ C ) @ ( relcom9058974518648859105st_nat @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_55_relcomp_OrelcompI,axiom,
    ! [A: nat > nat > $o,B: produc1828647624359046049st_nat,R: set_Pr4817715314677154882st_nat,C: list_nat,S: set_Pr3833137770001865702st_nat] :
      ( ( member6951660485171671051st_nat @ ( produc3127733452865184594st_nat @ A @ B ) @ R )
     => ( ( member6135396219256978631st_nat @ ( produc1778712578675790378st_nat @ B @ C ) @ S )
       => ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ A @ C ) @ ( relcom210944645552855580st_nat @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_56_relcomp_OrelcompI,axiom,
    ! [A: nat > nat > $o,B: list_nat,R: set_Pr7072801126362145067st_nat,C: produc1828647624359046049st_nat,S: set_Pr1385048790419436780st_nat] :
      ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ A @ B ) @ R )
     => ( ( member3205530567375370445st_nat @ ( produc4487115339913071592st_nat @ B @ C ) @ S )
       => ( member6951660485171671051st_nat @ ( produc3127733452865184594st_nat @ A @ C ) @ ( relcom2919347406790136794st_nat @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_57_relcompE,axiom,
    ! [Xz: produc357393685978478089rm_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ Xz @ ( relcom370159955682700863rm_a_b @ R @ S ) )
     => ~ ! [X: term_a_b,Y: term_a_b,Z: term_a_b] :
            ( ( Xz
              = ( produc7020197800436672577rm_a_b @ X @ Z ) )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_58_relcompE,axiom,
    ! [Xz: product_prod_nat_nat,R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Xz @ ( relcomp_nat_nat_nat @ R @ S ) )
     => ~ ! [X: nat,Y: nat,Z: nat] :
            ( ( Xz
              = ( product_Pair_nat_nat @ X @ Z ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_59_relcompE,axiom,
    ! [Xz: produc6551080308567045442rm_a_b,R: set_Product_prod_b_b,S: set_Pr1357592988741419896rm_a_b] :
      ( ( member7000133282412810073rm_a_b @ Xz @ ( relcomp_b_b_term_a_b @ R @ S ) )
     => ~ ! [X: b,Y: b,Z: term_a_b] :
            ( ( Xz
              = ( produc1437816968797971900rm_a_b @ X @ Z ) )
           => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y ) @ R )
             => ~ ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ Y @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_60_relcompE,axiom,
    ! [Xz: produc357393685978478089rm_a_b,R: set_Pr3369833000147915158_a_b_b,S: set_Pr1357592988741419896rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ Xz @ ( relcom5423879875627813950rm_a_b @ R @ S ) )
     => ~ ! [X: term_a_b,Y: b,Z: term_a_b] :
            ( ( Xz
              = ( produc7020197800436672577rm_a_b @ X @ Z ) )
           => ( ( member6959280461167840375_a_b_b @ ( produc1666658391573147538_a_b_b @ X @ Y ) @ R )
             => ~ ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ Y @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_61_relcompE,axiom,
    ! [Xz: produc6551080308567045442rm_a_b,R: set_Pr1357592988741419896rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member7000133282412810073rm_a_b @ Xz @ ( relcom6871247487278447976rm_a_b @ R @ S ) )
     => ~ ! [X: b,Y: term_a_b,Z: term_a_b] :
            ( ( Xz
              = ( produc1437816968797971900rm_a_b @ X @ Z ) )
           => ( ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ X @ Y ) @ R )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_62_relcompE,axiom,
    ! [Xz: produc254973753779126261st_nat,R: set_Pr7072801126362145067st_nat,S: set_Pr3451248702717554689st_nat] :
      ( ( member5350383351084882060st_nat @ Xz @ ( relcom6714777091229659495st_nat @ R @ S ) )
     => ~ ! [X: nat > nat > $o,Y: list_nat,Z: list_nat] :
            ( ( Xz
              = ( produc4727192421694094319st_nat @ X @ Z ) )
           => ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ X @ Y ) @ R )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_63_relcompE,axiom,
    ! [Xz: produc1616951275169580055st_nat,R: set_Pr7682762132356531903at_nat,S: set_Pr3595532568945684557st_nat] :
      ( ( member5596898355348113582st_nat @ Xz @ ( relcom2190433716068671200st_nat @ R @ S ) )
     => ~ ! [X: nat > nat,Y: nat > nat,Z: produc1828647624359046049st_nat] :
            ( ( Xz
              = ( produc4626581765195395529st_nat @ X @ Z ) )
           => ( ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ X @ Y ) @ R )
             => ~ ( member5596898355348113582st_nat @ ( produc4626581765195395529st_nat @ Y @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_64_relcompE,axiom,
    ! [Xz: produc4787317212837456354st_nat,R: set_Pr7072801126362145067st_nat,S: set_Pr1385048790419436780st_nat] :
      ( ( member6951660485171671051st_nat @ Xz @ ( relcom2919347406790136794st_nat @ R @ S ) )
     => ~ ! [X: nat > nat > $o,Y: list_nat,Z: produc1828647624359046049st_nat] :
            ( ( Xz
              = ( produc3127733452865184594st_nat @ X @ Z ) )
           => ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ X @ Y ) @ R )
             => ~ ( member3205530567375370445st_nat @ ( produc4487115339913071592st_nat @ Y @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_65_relcompE,axiom,
    ! [Xz: produc254973753779126261st_nat,R: set_Pr1489038555860665479_nat_o,S: set_Pr7072801126362145067st_nat] :
      ( ( member5350383351084882060st_nat @ Xz @ ( relcom9058974518648859105st_nat @ R @ S ) )
     => ~ ! [X: nat > nat > $o,Y: nat > nat > $o,Z: list_nat] :
            ( ( Xz
              = ( produc4727192421694094319st_nat @ X @ Z ) )
           => ( ( member498802742439182032_nat_o @ ( produc3074213772457683607_nat_o @ X @ Y ) @ R )
             => ~ ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ Y @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_66_relcompE,axiom,
    ! [Xz: produc254973753779126261st_nat,R: set_Pr4817715314677154882st_nat,S: set_Pr3833137770001865702st_nat] :
      ( ( member5350383351084882060st_nat @ Xz @ ( relcom210944645552855580st_nat @ R @ S ) )
     => ~ ! [X: nat > nat > $o,Y: produc1828647624359046049st_nat,Z: list_nat] :
            ( ( Xz
              = ( produc4727192421694094319st_nat @ X @ Z ) )
           => ( ( member6951660485171671051st_nat @ ( produc3127733452865184594st_nat @ X @ Y ) @ R )
             => ~ ( member6135396219256978631st_nat @ ( produc1778712578675790378st_nat @ Y @ Z ) @ S ) ) ) ) ).

% relcompE
thf(fact_67_prod__induct3,axiom,
    ! [P4: produc1616951275169580055st_nat > $o,X3: produc1616951275169580055st_nat] :
      ( ! [A4: nat > nat,B3: list_nat,C3: list_nat] : ( P4 @ ( produc4626581765195395529st_nat @ A4 @ ( produc2694037385005941721st_nat @ B3 @ C3 ) ) )
     => ( P4 @ X3 ) ) ).

% prod_induct3
thf(fact_68_prod__induct3,axiom,
    ! [P4: produc8734217775477389290rm_a_b > $o,X3: produc8734217775477389290rm_a_b] :
      ( ! [A4: nat > term_a_b > $o,B3: list_nat,C3: list_term_a_b] : ( P4 @ ( produc2750092164434624090rm_a_b @ A4 @ ( produc4336470666470712605rm_a_b @ B3 @ C3 ) ) )
     => ( P4 @ X3 ) ) ).

% prod_induct3
thf(fact_69_prod__induct3,axiom,
    ! [P4: produc4787317212837456354st_nat > $o,X3: produc4787317212837456354st_nat] :
      ( ! [A4: nat > nat > $o,B3: list_nat,C3: list_nat] : ( P4 @ ( produc3127733452865184594st_nat @ A4 @ ( produc2694037385005941721st_nat @ B3 @ C3 ) ) )
     => ( P4 @ X3 ) ) ).

% prod_induct3
thf(fact_70_prod__cases3,axiom,
    ! [Y3: produc1616951275169580055st_nat] :
      ~ ! [A4: nat > nat,B3: list_nat,C3: list_nat] :
          ( Y3
         != ( produc4626581765195395529st_nat @ A4 @ ( produc2694037385005941721st_nat @ B3 @ C3 ) ) ) ).

% prod_cases3
thf(fact_71_prod__cases3,axiom,
    ! [Y3: produc8734217775477389290rm_a_b] :
      ~ ! [A4: nat > term_a_b > $o,B3: list_nat,C3: list_term_a_b] :
          ( Y3
         != ( produc2750092164434624090rm_a_b @ A4 @ ( produc4336470666470712605rm_a_b @ B3 @ C3 ) ) ) ).

% prod_cases3
thf(fact_72_prod__cases3,axiom,
    ! [Y3: produc4787317212837456354st_nat] :
      ~ ! [A4: nat > nat > $o,B3: list_nat,C3: list_nat] :
          ( Y3
         != ( produc3127733452865184594st_nat @ A4 @ ( produc2694037385005941721st_nat @ B3 @ C3 ) ) ) ).

% prod_cases3
thf(fact_73_Pair__inject,axiom,
    ! [A: nat > nat,B: produc1828647624359046049st_nat,A2: nat > nat,B2: produc1828647624359046049st_nat] :
      ( ( ( produc4626581765195395529st_nat @ A @ B )
        = ( produc4626581765195395529st_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_74_Pair__inject,axiom,
    ! [A: nat > term_a_b > $o,B: produc9043357390500214885rm_a_b,A2: nat > term_a_b > $o,B2: produc9043357390500214885rm_a_b] :
      ( ( ( produc2750092164434624090rm_a_b @ A @ B )
        = ( produc2750092164434624090rm_a_b @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_75_Pair__inject,axiom,
    ! [A: nat > nat > $o,B: produc1828647624359046049st_nat,A2: nat > nat > $o,B2: produc1828647624359046049st_nat] :
      ( ( ( produc3127733452865184594st_nat @ A @ B )
        = ( produc3127733452865184594st_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_76_Pair__inject,axiom,
    ! [A: nat > nat > $o,B: list_nat,A2: nat > nat > $o,B2: list_nat] :
      ( ( ( produc4727192421694094319st_nat @ A @ B )
        = ( produc4727192421694094319st_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_77_Pair__inject,axiom,
    ! [A: b,B: term_a_b,A2: b,B2: term_a_b] :
      ( ( ( produc1437816968797971900rm_a_b @ A @ B )
        = ( produc1437816968797971900rm_a_b @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_78_Pair__inject,axiom,
    ! [A: term_a_b,B: term_a_b,A2: term_a_b,B2: term_a_b] :
      ( ( ( produc7020197800436672577rm_a_b @ A @ B )
        = ( produc7020197800436672577rm_a_b @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_79_prod__cases,axiom,
    ! [P4: produc1616951275169580055st_nat > $o,P: produc1616951275169580055st_nat] :
      ( ! [A4: nat > nat,B3: produc1828647624359046049st_nat] : ( P4 @ ( produc4626581765195395529st_nat @ A4 @ B3 ) )
     => ( P4 @ P ) ) ).

% prod_cases
thf(fact_80_prod__cases,axiom,
    ! [P4: produc8734217775477389290rm_a_b > $o,P: produc8734217775477389290rm_a_b] :
      ( ! [A4: nat > term_a_b > $o,B3: produc9043357390500214885rm_a_b] : ( P4 @ ( produc2750092164434624090rm_a_b @ A4 @ B3 ) )
     => ( P4 @ P ) ) ).

% prod_cases
thf(fact_81_prod__cases,axiom,
    ! [P4: produc4787317212837456354st_nat > $o,P: produc4787317212837456354st_nat] :
      ( ! [A4: nat > nat > $o,B3: produc1828647624359046049st_nat] : ( P4 @ ( produc3127733452865184594st_nat @ A4 @ B3 ) )
     => ( P4 @ P ) ) ).

% prod_cases
thf(fact_82_prod__cases,axiom,
    ! [P4: produc254973753779126261st_nat > $o,P: produc254973753779126261st_nat] :
      ( ! [A4: nat > nat > $o,B3: list_nat] : ( P4 @ ( produc4727192421694094319st_nat @ A4 @ B3 ) )
     => ( P4 @ P ) ) ).

% prod_cases
thf(fact_83_prod__cases,axiom,
    ! [P4: produc6551080308567045442rm_a_b > $o,P: produc6551080308567045442rm_a_b] :
      ( ! [A4: b,B3: term_a_b] : ( P4 @ ( produc1437816968797971900rm_a_b @ A4 @ B3 ) )
     => ( P4 @ P ) ) ).

% prod_cases
thf(fact_84_prod__cases,axiom,
    ! [P4: produc357393685978478089rm_a_b > $o,P: produc357393685978478089rm_a_b] :
      ( ! [A4: term_a_b,B3: term_a_b] : ( P4 @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) )
     => ( P4 @ P ) ) ).

% prod_cases
thf(fact_85_surj__pair,axiom,
    ! [P: produc1616951275169580055st_nat] :
    ? [X: nat > nat,Y: produc1828647624359046049st_nat] :
      ( P
      = ( produc4626581765195395529st_nat @ X @ Y ) ) ).

% surj_pair
thf(fact_86_surj__pair,axiom,
    ! [P: produc8734217775477389290rm_a_b] :
    ? [X: nat > term_a_b > $o,Y: produc9043357390500214885rm_a_b] :
      ( P
      = ( produc2750092164434624090rm_a_b @ X @ Y ) ) ).

% surj_pair
thf(fact_87_surj__pair,axiom,
    ! [P: produc4787317212837456354st_nat] :
    ? [X: nat > nat > $o,Y: produc1828647624359046049st_nat] :
      ( P
      = ( produc3127733452865184594st_nat @ X @ Y ) ) ).

% surj_pair
thf(fact_88_surj__pair,axiom,
    ! [P: produc254973753779126261st_nat] :
    ? [X: nat > nat > $o,Y: list_nat] :
      ( P
      = ( produc4727192421694094319st_nat @ X @ Y ) ) ).

% surj_pair
thf(fact_89_surj__pair,axiom,
    ! [P: produc6551080308567045442rm_a_b] :
    ? [X: b,Y: term_a_b] :
      ( P
      = ( produc1437816968797971900rm_a_b @ X @ Y ) ) ).

% surj_pair
thf(fact_90_surj__pair,axiom,
    ! [P: produc357393685978478089rm_a_b] :
    ? [X: term_a_b,Y: term_a_b] :
      ( P
      = ( produc7020197800436672577rm_a_b @ X @ Y ) ) ).

% surj_pair
thf(fact_91_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc1616951275169580055st_nat] :
      ~ ! [A4: nat > nat,B3: produc1828647624359046049st_nat] :
          ( Y3
         != ( produc4626581765195395529st_nat @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_92_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc8734217775477389290rm_a_b] :
      ~ ! [A4: nat > term_a_b > $o,B3: produc9043357390500214885rm_a_b] :
          ( Y3
         != ( produc2750092164434624090rm_a_b @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_93_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc4787317212837456354st_nat] :
      ~ ! [A4: nat > nat > $o,B3: produc1828647624359046049st_nat] :
          ( Y3
         != ( produc3127733452865184594st_nat @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_94_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc254973753779126261st_nat] :
      ~ ! [A4: nat > nat > $o,B3: list_nat] :
          ( Y3
         != ( produc4727192421694094319st_nat @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_95_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc6551080308567045442rm_a_b] :
      ~ ! [A4: b,B3: term_a_b] :
          ( Y3
         != ( produc1437816968797971900rm_a_b @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_96_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc357393685978478089rm_a_b] :
      ~ ! [A4: term_a_b,B3: term_a_b] :
          ( Y3
         != ( produc7020197800436672577rm_a_b @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_97_O__assoc,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,T4: set_Pr4386577575007340137rm_a_b] :
      ( ( relcom370159955682700863rm_a_b @ ( relcom370159955682700863rm_a_b @ R2 @ S4 ) @ T4 )
      = ( relcom370159955682700863rm_a_b @ R2 @ ( relcom370159955682700863rm_a_b @ S4 @ T4 ) ) ) ).

% O_assoc
thf(fact_98_relcompEpair,axiom,
    ! [A: term_a_b,C: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( relcom370159955682700863rm_a_b @ R @ S ) )
     => ~ ! [B3: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ R )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_99_relcompEpair,axiom,
    ! [A: nat,C: nat,R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C ) @ ( relcomp_nat_nat_nat @ R @ S ) )
     => ~ ! [B3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B3 ) @ R )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_100_relcompEpair,axiom,
    ! [A: b,C: term_a_b,R: set_Product_prod_b_b,S: set_Pr1357592988741419896rm_a_b] :
      ( ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ A @ C ) @ ( relcomp_b_b_term_a_b @ R @ S ) )
     => ~ ! [B3: b] :
            ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B3 ) @ R )
           => ~ ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_101_relcompEpair,axiom,
    ! [A: term_a_b,C: term_a_b,R: set_Pr3369833000147915158_a_b_b,S: set_Pr1357592988741419896rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( relcom5423879875627813950rm_a_b @ R @ S ) )
     => ~ ! [B3: b] :
            ( ( member6959280461167840375_a_b_b @ ( produc1666658391573147538_a_b_b @ A @ B3 ) @ R )
           => ~ ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_102_relcompEpair,axiom,
    ! [A: b,C: term_a_b,R: set_Pr1357592988741419896rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ A @ C ) @ ( relcom6871247487278447976rm_a_b @ R @ S ) )
     => ~ ! [B3: term_a_b] :
            ( ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ A @ B3 ) @ R )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_103_relcompEpair,axiom,
    ! [A: nat > nat > $o,C: list_nat,R: set_Pr7072801126362145067st_nat,S: set_Pr3451248702717554689st_nat] :
      ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ A @ C ) @ ( relcom6714777091229659495st_nat @ R @ S ) )
     => ~ ! [B3: list_nat] :
            ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ A @ B3 ) @ R )
           => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_104_relcompEpair,axiom,
    ! [A: nat > nat,C: produc1828647624359046049st_nat,R: set_Pr7682762132356531903at_nat,S: set_Pr3595532568945684557st_nat] :
      ( ( member5596898355348113582st_nat @ ( produc4626581765195395529st_nat @ A @ C ) @ ( relcom2190433716068671200st_nat @ R @ S ) )
     => ~ ! [B3: nat > nat] :
            ( ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A @ B3 ) @ R )
           => ~ ( member5596898355348113582st_nat @ ( produc4626581765195395529st_nat @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_105_relcompEpair,axiom,
    ! [A: nat > nat > $o,C: produc1828647624359046049st_nat,R: set_Pr7072801126362145067st_nat,S: set_Pr1385048790419436780st_nat] :
      ( ( member6951660485171671051st_nat @ ( produc3127733452865184594st_nat @ A @ C ) @ ( relcom2919347406790136794st_nat @ R @ S ) )
     => ~ ! [B3: list_nat] :
            ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ A @ B3 ) @ R )
           => ~ ( member3205530567375370445st_nat @ ( produc4487115339913071592st_nat @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_106_relcompEpair,axiom,
    ! [A: nat > nat > $o,C: list_nat,R: set_Pr1489038555860665479_nat_o,S: set_Pr7072801126362145067st_nat] :
      ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ A @ C ) @ ( relcom9058974518648859105st_nat @ R @ S ) )
     => ~ ! [B3: nat > nat > $o] :
            ( ( member498802742439182032_nat_o @ ( produc3074213772457683607_nat_o @ A @ B3 ) @ R )
           => ~ ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_107_relcompEpair,axiom,
    ! [A: nat > nat > $o,C: list_nat,R: set_Pr4817715314677154882st_nat,S: set_Pr3833137770001865702st_nat] :
      ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ A @ C ) @ ( relcom210944645552855580st_nat @ R @ S ) )
     => ~ ! [B3: produc1828647624359046049st_nat] :
            ( ( member6951660485171671051st_nat @ ( produc3127733452865184594st_nat @ A @ B3 ) @ R )
           => ~ ( member6135396219256978631st_nat @ ( produc1778712578675790378st_nat @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_108_relcomp3__I,axiom,
    ! [T: term_a_b,U: term_a_b,A5: set_Pr4386577575007340137rm_a_b,S: term_a_b,B5: set_Pr4386577575007340137rm_a_b,V: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ A5 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ B5 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ V ) @ B5 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ V ) @ ( relcom370159955682700863rm_a_b @ B5 @ ( relcom370159955682700863rm_a_b @ A5 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_109_relcomp3__I,axiom,
    ! [T: nat,U: nat,A5: set_Pr1261947904930325089at_nat,S: nat,B5: set_Pr1261947904930325089at_nat,V: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T @ U ) @ A5 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ S @ T ) @ B5 )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ B5 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ S @ V ) @ ( relcomp_nat_nat_nat @ B5 @ ( relcomp_nat_nat_nat @ A5 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_110_relcomp3__I,axiom,
    ! [T: term_a_b,U: b,A5: set_Pr3369833000147915158_a_b_b,S: b,B5: set_Pr1357592988741419896rm_a_b,V: term_a_b] :
      ( ( member6959280461167840375_a_b_b @ ( produc1666658391573147538_a_b_b @ T @ U ) @ A5 )
     => ( ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ S @ T ) @ B5 )
       => ( ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ U @ V ) @ B5 )
         => ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ S @ V ) @ ( relcom6871247487278447976rm_a_b @ B5 @ ( relcom5423879875627813950rm_a_b @ A5 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_111_relcomp3__I,axiom,
    ! [T: b,U: term_a_b,A5: set_Pr1357592988741419896rm_a_b,S: term_a_b,B5: set_Pr3369833000147915158_a_b_b,V: b] :
      ( ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ T @ U ) @ A5 )
     => ( ( member6959280461167840375_a_b_b @ ( produc1666658391573147538_a_b_b @ S @ T ) @ B5 )
       => ( ( member6959280461167840375_a_b_b @ ( produc1666658391573147538_a_b_b @ U @ V ) @ B5 )
         => ( member6959280461167840375_a_b_b @ ( produc1666658391573147538_a_b_b @ S @ V ) @ ( relcomp_term_a_b_b_b @ B5 @ ( relcomp_b_term_a_b_b @ A5 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_112_relcomp3__I,axiom,
    ! [T: list_nat,U: nat > nat > $o,A5: set_Pr4291780605709124401_nat_o,S: nat > nat > $o,B5: set_Pr7072801126362145067st_nat,V: list_nat] :
      ( ( member6626851962243112338_nat_o @ ( produc2887701473640236333_nat_o @ T @ U ) @ A5 )
     => ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ S @ T ) @ B5 )
       => ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ U @ V ) @ B5 )
         => ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ S @ V ) @ ( relcom6714777091229659495st_nat @ B5 @ ( relcom4328489779883936677st_nat @ A5 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_113_relcomp3__I,axiom,
    ! [T: nat > nat > $o,U: list_nat,A5: set_Pr7072801126362145067st_nat,S: list_nat,B5: set_Pr4291780605709124401_nat_o,V: nat > nat > $o] :
      ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ T @ U ) @ A5 )
     => ( ( member6626851962243112338_nat_o @ ( produc2887701473640236333_nat_o @ S @ T ) @ B5 )
       => ( ( member6626851962243112338_nat_o @ ( produc2887701473640236333_nat_o @ U @ V ) @ B5 )
         => ( member6626851962243112338_nat_o @ ( produc2887701473640236333_nat_o @ S @ V ) @ ( relcom4787998327250395233_nat_o @ B5 @ ( relcom7219483570595001119_nat_o @ A5 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_114_relcomp3__I,axiom,
    ! [T: produc1828647624359046049st_nat,U: nat > nat,A5: set_Pr5877781423743409989at_nat,S: nat > nat,B5: set_Pr3595532568945684557st_nat,V: produc1828647624359046049st_nat] :
      ( ( member6277506213002785702at_nat @ ( produc2376817894703970953at_nat @ T @ U ) @ A5 )
     => ( ( member5596898355348113582st_nat @ ( produc4626581765195395529st_nat @ S @ T ) @ B5 )
       => ( ( member5596898355348113582st_nat @ ( produc4626581765195395529st_nat @ U @ V ) @ B5 )
         => ( member5596898355348113582st_nat @ ( produc4626581765195395529st_nat @ S @ V ) @ ( relcom1616210961554918272st_nat @ B5 @ ( relcom7999779220937822912st_nat @ A5 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_115_relcomp3__I,axiom,
    ! [T: nat > nat,U: produc1828647624359046049st_nat,A5: set_Pr3595532568945684557st_nat,S: produc1828647624359046049st_nat,B5: set_Pr5877781423743409989at_nat,V: nat > nat] :
      ( ( member5596898355348113582st_nat @ ( produc4626581765195395529st_nat @ T @ U ) @ A5 )
     => ( ( member6277506213002785702at_nat @ ( produc2376817894703970953at_nat @ S @ T ) @ B5 )
       => ( ( member6277506213002785702at_nat @ ( produc2376817894703970953at_nat @ U @ V ) @ B5 )
         => ( member6277506213002785702at_nat @ ( produc2376817894703970953at_nat @ S @ V ) @ ( relcom3979775500800512096at_nat @ B5 @ ( relcom9164041882432022432at_nat @ A5 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_116_relcomp3__I,axiom,
    ! [T: produc1828647624359046049st_nat,U: nat > nat > $o,A5: set_Pr7697833614269146252_nat_o,S: nat > nat > $o,B5: set_Pr4817715314677154882st_nat,V: produc1828647624359046049st_nat] :
      ( ( member6918694168860644693_nat_o @ ( produc3563911201077956380_nat_o @ T @ U ) @ A5 )
     => ( ( member6951660485171671051st_nat @ ( produc3127733452865184594st_nat @ S @ T ) @ B5 )
       => ( ( member6951660485171671051st_nat @ ( produc3127733452865184594st_nat @ U @ V ) @ B5 )
         => ( member6951660485171671051st_nat @ ( produc3127733452865184594st_nat @ S @ V ) @ ( relcom2952420654723421029st_nat @ B5 @ ( relcom3383388244270204187st_nat @ A5 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_117_relcomp3__I,axiom,
    ! [T: nat > nat > $o,U: produc1828647624359046049st_nat,A5: set_Pr4817715314677154882st_nat,S: produc1828647624359046049st_nat,B5: set_Pr7697833614269146252_nat_o,V: nat > nat > $o] :
      ( ( member6951660485171671051st_nat @ ( produc3127733452865184594st_nat @ T @ U ) @ A5 )
     => ( ( member6918694168860644693_nat_o @ ( produc3563911201077956380_nat_o @ S @ T ) @ B5 )
       => ( ( member6918694168860644693_nat_o @ ( produc3563911201077956380_nat_o @ U @ V ) @ B5 )
         => ( member6918694168860644693_nat_o @ ( produc3563911201077956380_nat_o @ S @ V ) @ ( relcom5352553830001889120_nat_o @ B5 @ ( relcom5425406725469424554_nat_o @ A5 @ B5 ) ) ) ) ) ) ).

% relcomp3_I
thf(fact_118_pos__diff__itself,axiom,
    ! [P: list_P2364656488115551307rm_a_b] :
      ( ( term_p4159419441187170626rm_a_b @ P @ P )
      = nil_Pr4197086119080101963rm_a_b ) ).

% pos_diff_itself
thf(fact_119_pos__diff__itself,axiom,
    ! [P: list_P6011104703257516679at_nat] :
      ( ( term_p3376976900432600702at_nat @ P @ P )
      = nil_Pr5478986624290739719at_nat ) ).

% pos_diff_itself
thf(fact_120_pos__diff__itself,axiom,
    ! [P: list_list_term_a_b] :
      ( ( term_p6300577740959955047rm_a_b @ P @ P )
      = nil_list_term_a_b ) ).

% pos_diff_itself
thf(fact_121_pos__diff__itself,axiom,
    ! [P: list_list_nat] :
      ( ( term_p7564741194569991203st_nat @ P @ P )
      = nil_list_nat ) ).

% pos_diff_itself
thf(fact_122_pos__diff__itself,axiom,
    ! [P: list_term_a_b_nat] :
      ( ( term_p4502178508558846150_b_nat @ P @ P )
      = nil_term_a_b_nat ) ).

% pos_diff_itself
thf(fact_123_pos__diff__itself,axiom,
    ! [P: list_term_a_b] :
      ( ( term_p798503758663136087rm_a_b @ P @ P )
      = nil_term_a_b ) ).

% pos_diff_itself
thf(fact_124_pos__diff__itself,axiom,
    ! [P: list_nat] :
      ( ( term_pos_diff_nat @ P @ P )
      = nil_nat ) ).

% pos_diff_itself
thf(fact_125_ssubst__Pair__rhs,axiom,
    ! [R: nat,S: nat,R2: set_Pr1261947904930325089at_nat,S5: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S ) @ R2 )
     => ( ( S5 = S )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S5 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_126_ssubst__Pair__rhs,axiom,
    ! [R: nat > nat,S: produc1828647624359046049st_nat,R2: set_Pr3595532568945684557st_nat,S5: produc1828647624359046049st_nat] :
      ( ( member5596898355348113582st_nat @ ( produc4626581765195395529st_nat @ R @ S ) @ R2 )
     => ( ( S5 = S )
       => ( member5596898355348113582st_nat @ ( produc4626581765195395529st_nat @ R @ S5 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_127_ssubst__Pair__rhs,axiom,
    ! [R: nat > term_a_b > $o,S: produc9043357390500214885rm_a_b,R2: set_Pr1861077313142575434rm_a_b,S5: produc9043357390500214885rm_a_b] :
      ( ( member5734712557143010579rm_a_b @ ( produc2750092164434624090rm_a_b @ R @ S ) @ R2 )
     => ( ( S5 = S )
       => ( member5734712557143010579rm_a_b @ ( produc2750092164434624090rm_a_b @ R @ S5 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_128_ssubst__Pair__rhs,axiom,
    ! [R: nat > nat > $o,S: produc1828647624359046049st_nat,R2: set_Pr4817715314677154882st_nat,S5: produc1828647624359046049st_nat] :
      ( ( member6951660485171671051st_nat @ ( produc3127733452865184594st_nat @ R @ S ) @ R2 )
     => ( ( S5 = S )
       => ( member6951660485171671051st_nat @ ( produc3127733452865184594st_nat @ R @ S5 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_129_ssubst__Pair__rhs,axiom,
    ! [R: nat > nat > $o,S: list_nat,R2: set_Pr7072801126362145067st_nat,S5: list_nat] :
      ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ R @ S ) @ R2 )
     => ( ( S5 = S )
       => ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ R @ S5 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_130_ssubst__Pair__rhs,axiom,
    ! [R: b,S: term_a_b,R2: set_Pr1357592988741419896rm_a_b,S5: term_a_b] :
      ( ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ R @ S ) @ R2 )
     => ( ( S5 = S )
       => ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ R @ S5 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_131_ssubst__Pair__rhs,axiom,
    ! [R: term_a_b,S: term_a_b,R2: set_Pr4386577575007340137rm_a_b,S5: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R @ S ) @ R2 )
     => ( ( S5 = S )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R @ S5 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_132_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P4: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P4 ) )
      = ( P4 @ A ) ) ).

% mem_Collect_eq
thf(fact_133_mem__Collect__eq,axiom,
    ! [A: term_a_b,P4: term_a_b > $o] :
      ( ( member_term_a_b2 @ A @ ( collect_term_a_b @ P4 ) )
      = ( P4 @ A ) ) ).

% mem_Collect_eq
thf(fact_134_mem__Collect__eq,axiom,
    ! [A: list_P8875379029341186191rm_a_b,P4: list_P8875379029341186191rm_a_b > $o] :
      ( ( member7646937072540562872rm_a_b @ A @ ( collec4396703598834874874rm_a_b @ P4 ) )
      = ( P4 @ A ) ) ).

% mem_Collect_eq
thf(fact_135_mem__Collect__eq,axiom,
    ! [A: list_term_a_b,P4: list_term_a_b > $o] :
      ( ( member_list_term_a_b2 @ A @ ( collec2649484519605854073rm_a_b @ P4 ) )
      = ( P4 @ A ) ) ).

% mem_Collect_eq
thf(fact_136_mem__Collect__eq,axiom,
    ! [A: list_list_nat,P4: list_list_nat > $o] :
      ( ( member_list_list_nat @ A @ ( collec5989764272469232197st_nat @ P4 ) )
      = ( P4 @ A ) ) ).

% mem_Collect_eq
thf(fact_137_mem__Collect__eq,axiom,
    ! [A: list_nat,P4: list_nat > $o] :
      ( ( member_list_nat2 @ A @ ( collect_list_nat @ P4 ) )
      = ( P4 @ A ) ) ).

% mem_Collect_eq
thf(fact_138_mem__Collect__eq,axiom,
    ! [A: produc357393685978478089rm_a_b,P4: produc357393685978478089rm_a_b > $o] :
      ( ( member5869715511025134514rm_a_b @ A @ ( collec99177395925924084rm_a_b @ P4 ) )
      = ( P4 @ A ) ) ).

% mem_Collect_eq
thf(fact_139_mem__Collect__eq,axiom,
    ! [A: nat,P4: nat > $o] :
      ( ( member_nat2 @ A @ ( collect_nat @ P4 ) )
      = ( P4 @ A ) ) ).

% mem_Collect_eq
thf(fact_140_Collect__mem__eq,axiom,
    ! [A5: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_141_Collect__mem__eq,axiom,
    ! [A5: set_term_a_b] :
      ( ( collect_term_a_b
        @ ^ [X4: term_a_b] : ( member_term_a_b2 @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_142_Collect__mem__eq,axiom,
    ! [A5: set_li7311725489111463791rm_a_b] :
      ( ( collec4396703598834874874rm_a_b
        @ ^ [X4: list_P8875379029341186191rm_a_b] : ( member7646937072540562872rm_a_b @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_143_Collect__mem__eq,axiom,
    ! [A5: set_list_term_a_b] :
      ( ( collec2649484519605854073rm_a_b
        @ ^ [X4: list_term_a_b] : ( member_list_term_a_b2 @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_144_Collect__mem__eq,axiom,
    ! [A5: set_list_list_nat] :
      ( ( collec5989764272469232197st_nat
        @ ^ [X4: list_list_nat] : ( member_list_list_nat @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_145_Collect__mem__eq,axiom,
    ! [A5: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] : ( member_list_nat2 @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_146_Collect__mem__eq,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b] :
      ( ( collec99177395925924084rm_a_b
        @ ^ [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_147_Collect__mem__eq,axiom,
    ! [A5: set_nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : ( member_nat2 @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_148_Collect__cong,axiom,
    ! [P4: nat > $o,Q4: nat > $o] :
      ( ! [X: nat] :
          ( ( P4 @ X )
          = ( Q4 @ X ) )
     => ( ( collect_nat @ P4 )
        = ( collect_nat @ Q4 ) ) ) ).

% Collect_cong
thf(fact_149_Collect__cong,axiom,
    ! [P4: produc357393685978478089rm_a_b > $o,Q4: produc357393685978478089rm_a_b > $o] :
      ( ! [X: produc357393685978478089rm_a_b] :
          ( ( P4 @ X )
          = ( Q4 @ X ) )
     => ( ( collec99177395925924084rm_a_b @ P4 )
        = ( collec99177395925924084rm_a_b @ Q4 ) ) ) ).

% Collect_cong
thf(fact_150_Collect__cong,axiom,
    ! [P4: list_nat > $o,Q4: list_nat > $o] :
      ( ! [X: list_nat] :
          ( ( P4 @ X )
          = ( Q4 @ X ) )
     => ( ( collect_list_nat @ P4 )
        = ( collect_list_nat @ Q4 ) ) ) ).

% Collect_cong
thf(fact_151_curry__conv,axiom,
    ( produc6008550485034061667_a_b_o
    = ( ^ [F: produc357393685978478089rm_a_b > $o,A3: term_a_b,B4: term_a_b] : ( F @ ( produc7020197800436672577rm_a_b @ A3 @ B4 ) ) ) ) ).

% curry_conv
thf(fact_152_curryI,axiom,
    ! [F2: produc1616951275169580055st_nat > $o,A: nat > nat,B: produc1828647624359046049st_nat] :
      ( ( F2 @ ( produc4626581765195395529st_nat @ A @ B ) )
     => ( produc4347078155291544769_nat_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_153_curryI,axiom,
    ! [F2: produc8734217775477389290rm_a_b > $o,A: nat > term_a_b > $o,B: produc9043357390500214885rm_a_b] :
      ( ( F2 @ ( produc2750092164434624090rm_a_b @ A @ B ) )
     => ( produc6219099037331169930_a_b_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_154_curryI,axiom,
    ! [F2: produc4787317212837456354st_nat > $o,A: nat > nat > $o,B: produc1828647624359046049st_nat] :
      ( ( F2 @ ( produc3127733452865184594st_nat @ A @ B ) )
     => ( produc2023121225571165330_nat_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_155_curryI,axiom,
    ! [F2: produc254973753779126261st_nat > $o,A: nat > nat > $o,B: list_nat] :
      ( ( F2 @ ( produc4727192421694094319st_nat @ A @ B ) )
     => ( produc1492399064212916507_nat_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_156_curryI,axiom,
    ! [F2: produc6551080308567045442rm_a_b > $o,A: b,B: term_a_b] :
      ( ( F2 @ ( produc1437816968797971900rm_a_b @ A @ B ) )
     => ( produc863636840395689358_a_b_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_157_curryI,axiom,
    ! [F2: produc357393685978478089rm_a_b > $o,A: term_a_b,B: term_a_b] :
      ( ( F2 @ ( produc7020197800436672577rm_a_b @ A @ B ) )
     => ( produc6008550485034061667_a_b_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_158_in__inv__image,axiom,
    ! [X3: nat,Y3: nat,R: set_Pr1261947904930325089at_nat,F2: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( inv_image_nat_nat @ R @ F2 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ R ) ) ).

% in_inv_image
thf(fact_159_in__inv__image,axiom,
    ! [X3: nat,Y3: nat,R: set_Pr4386577575007340137rm_a_b,F2: nat > term_a_b] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( inv_im2609263929585225528_b_nat @ R @ F2 ) )
      = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ R ) ) ).

% in_inv_image
thf(fact_160_in__inv__image,axiom,
    ! [X3: term_a_b,Y3: term_a_b,R: set_Pr1261947904930325089at_nat,F2: term_a_b > nat] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( inv_im1647809802420253368rm_a_b @ R @ F2 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ R ) ) ).

% in_inv_image
thf(fact_161_in__inv__image,axiom,
    ! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( inv_im3004224898869642620rm_a_b @ R @ F2 ) )
      = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ R ) ) ).

% in_inv_image
thf(fact_162_swap__simp,axiom,
    ! [X3: term_a_b,Y3: term_a_b] :
      ( ( produc2541846788875228641rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) )
      = ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) ) ).

% swap_simp
thf(fact_163_swap__simp,axiom,
    ! [X3: term_a_b,Y3: b] :
      ( ( produc1428861184864852466_a_b_b @ ( produc1666658391573147538_a_b_b @ X3 @ Y3 ) )
      = ( produc1437816968797971900rm_a_b @ Y3 @ X3 ) ) ).

% swap_simp
thf(fact_164_swap__simp,axiom,
    ! [X3: b,Y3: term_a_b] :
      ( ( produc1200019762089676828rm_a_b @ ( produc1437816968797971900rm_a_b @ X3 @ Y3 ) )
      = ( produc1666658391573147538_a_b_b @ Y3 @ X3 ) ) ).

% swap_simp
thf(fact_165_swap__simp,axiom,
    ! [X3: list_nat,Y3: nat > nat > $o] :
      ( ( produc1534639687815544717_nat_o @ ( produc2887701473640236333_nat_o @ X3 @ Y3 ) )
      = ( produc4727192421694094319st_nat @ Y3 @ X3 ) ) ).

% swap_simp
thf(fact_166_swap__simp,axiom,
    ! [X3: nat > nat > $o,Y3: list_nat] :
      ( ( produc3374130635869402703st_nat @ ( produc4727192421694094319st_nat @ X3 @ Y3 ) )
      = ( produc2887701473640236333_nat_o @ Y3 @ X3 ) ) ).

% swap_simp
thf(fact_167_swap__simp,axiom,
    ! [X3: produc1828647624359046049st_nat,Y3: nat > nat] :
      ( ( produc1697598133222388969at_nat @ ( produc2376817894703970953at_nat @ X3 @ Y3 ) )
      = ( produc4626581765195395529st_nat @ Y3 @ X3 ) ) ).

% swap_simp
thf(fact_168_swap__simp,axiom,
    ! [X3: nat > nat,Y3: produc1828647624359046049st_nat] :
      ( ( produc3947362003713813545st_nat @ ( produc4626581765195395529st_nat @ X3 @ Y3 ) )
      = ( produc2376817894703970953at_nat @ Y3 @ X3 ) ) ).

% swap_simp
thf(fact_169_swap__simp,axiom,
    ! [X3: produc1828647624359046049st_nat,Y3: nat > nat > $o] :
      ( ( produc1311334868336790716_nat_o @ ( produc3563911201077956380_nat_o @ X3 @ Y3 ) )
      = ( produc3127733452865184594st_nat @ Y3 @ X3 ) ) ).

% swap_simp
thf(fact_170_swap__simp,axiom,
    ! [X3: nat > nat > $o,Y3: produc1828647624359046049st_nat] :
      ( ( produc875157120124018930st_nat @ ( produc3127733452865184594st_nat @ X3 @ Y3 ) )
      = ( produc3563911201077956380_nat_o @ Y3 @ X3 ) ) ).

% swap_simp
thf(fact_171_swap__simp,axiom,
    ! [X3: produc9043357390500214885rm_a_b,Y3: nat > term_a_b > $o] :
      ( ( produc2333045541586062900_a_b_o @ ( produc5450814272218574996_a_b_o @ X3 @ Y3 ) )
      = ( produc2750092164434624090rm_a_b @ Y3 @ X3 ) ) ).

% swap_simp
thf(fact_172_swap__swap,axiom,
    ! [P: produc357393685978478089rm_a_b] :
      ( ( produc2541846788875228641rm_a_b @ ( produc2541846788875228641rm_a_b @ P ) )
      = P ) ).

% swap_swap
thf(fact_173_curryD,axiom,
    ! [F2: produc1616951275169580055st_nat > $o,A: nat > nat,B: produc1828647624359046049st_nat] :
      ( ( produc4347078155291544769_nat_o @ F2 @ A @ B )
     => ( F2 @ ( produc4626581765195395529st_nat @ A @ B ) ) ) ).

% curryD
thf(fact_174_curryD,axiom,
    ! [F2: produc8734217775477389290rm_a_b > $o,A: nat > term_a_b > $o,B: produc9043357390500214885rm_a_b] :
      ( ( produc6219099037331169930_a_b_o @ F2 @ A @ B )
     => ( F2 @ ( produc2750092164434624090rm_a_b @ A @ B ) ) ) ).

% curryD
thf(fact_175_curryD,axiom,
    ! [F2: produc4787317212837456354st_nat > $o,A: nat > nat > $o,B: produc1828647624359046049st_nat] :
      ( ( produc2023121225571165330_nat_o @ F2 @ A @ B )
     => ( F2 @ ( produc3127733452865184594st_nat @ A @ B ) ) ) ).

% curryD
thf(fact_176_curryD,axiom,
    ! [F2: produc254973753779126261st_nat > $o,A: nat > nat > $o,B: list_nat] :
      ( ( produc1492399064212916507_nat_o @ F2 @ A @ B )
     => ( F2 @ ( produc4727192421694094319st_nat @ A @ B ) ) ) ).

% curryD
thf(fact_177_curryD,axiom,
    ! [F2: produc6551080308567045442rm_a_b > $o,A: b,B: term_a_b] :
      ( ( produc863636840395689358_a_b_o @ F2 @ A @ B )
     => ( F2 @ ( produc1437816968797971900rm_a_b @ A @ B ) ) ) ).

% curryD
thf(fact_178_curryD,axiom,
    ! [F2: produc357393685978478089rm_a_b > $o,A: term_a_b,B: term_a_b] :
      ( ( produc6008550485034061667_a_b_o @ F2 @ A @ B )
     => ( F2 @ ( produc7020197800436672577rm_a_b @ A @ B ) ) ) ).

% curryD
thf(fact_179_curryE,axiom,
    ! [F2: produc1616951275169580055st_nat > $o,A: nat > nat,B: produc1828647624359046049st_nat] :
      ( ( produc4347078155291544769_nat_o @ F2 @ A @ B )
     => ( F2 @ ( produc4626581765195395529st_nat @ A @ B ) ) ) ).

% curryE
thf(fact_180_curryE,axiom,
    ! [F2: produc8734217775477389290rm_a_b > $o,A: nat > term_a_b > $o,B: produc9043357390500214885rm_a_b] :
      ( ( produc6219099037331169930_a_b_o @ F2 @ A @ B )
     => ( F2 @ ( produc2750092164434624090rm_a_b @ A @ B ) ) ) ).

% curryE
thf(fact_181_curryE,axiom,
    ! [F2: produc4787317212837456354st_nat > $o,A: nat > nat > $o,B: produc1828647624359046049st_nat] :
      ( ( produc2023121225571165330_nat_o @ F2 @ A @ B )
     => ( F2 @ ( produc3127733452865184594st_nat @ A @ B ) ) ) ).

% curryE
thf(fact_182_curryE,axiom,
    ! [F2: produc254973753779126261st_nat > $o,A: nat > nat > $o,B: list_nat] :
      ( ( produc1492399064212916507_nat_o @ F2 @ A @ B )
     => ( F2 @ ( produc4727192421694094319st_nat @ A @ B ) ) ) ).

% curryE
thf(fact_183_curryE,axiom,
    ! [F2: produc6551080308567045442rm_a_b > $o,A: b,B: term_a_b] :
      ( ( produc863636840395689358_a_b_o @ F2 @ A @ B )
     => ( F2 @ ( produc1437816968797971900rm_a_b @ A @ B ) ) ) ).

% curryE
thf(fact_184_curryE,axiom,
    ! [F2: produc357393685978478089rm_a_b > $o,A: term_a_b,B: term_a_b] :
      ( ( produc6008550485034061667_a_b_o @ F2 @ A @ B )
     => ( F2 @ ( produc7020197800436672577rm_a_b @ A @ B ) ) ) ).

% curryE
thf(fact_185_position__diff__Nil,axiom,
    ! [Q5: list_P2364656488115551307rm_a_b] :
      ( ( term_p4159419441187170626rm_a_b @ Q5 @ nil_Pr4197086119080101963rm_a_b )
      = Q5 ) ).

% position_diff_Nil
thf(fact_186_position__diff__Nil,axiom,
    ! [Q5: list_P6011104703257516679at_nat] :
      ( ( term_p3376976900432600702at_nat @ Q5 @ nil_Pr5478986624290739719at_nat )
      = Q5 ) ).

% position_diff_Nil
thf(fact_187_position__diff__Nil,axiom,
    ! [Q5: list_list_term_a_b] :
      ( ( term_p6300577740959955047rm_a_b @ Q5 @ nil_list_term_a_b )
      = Q5 ) ).

% position_diff_Nil
thf(fact_188_position__diff__Nil,axiom,
    ! [Q5: list_list_nat] :
      ( ( term_p7564741194569991203st_nat @ Q5 @ nil_list_nat )
      = Q5 ) ).

% position_diff_Nil
thf(fact_189_position__diff__Nil,axiom,
    ! [Q5: list_term_a_b_nat] :
      ( ( term_p4502178508558846150_b_nat @ Q5 @ nil_term_a_b_nat )
      = Q5 ) ).

% position_diff_Nil
thf(fact_190_position__diff__Nil,axiom,
    ! [Q5: list_term_a_b] :
      ( ( term_p798503758663136087rm_a_b @ Q5 @ nil_term_a_b )
      = Q5 ) ).

% position_diff_Nil
thf(fact_191_position__diff__Nil,axiom,
    ! [Q5: list_nat] :
      ( ( term_pos_diff_nat @ Q5 @ nil_nat )
      = Q5 ) ).

% position_diff_Nil
thf(fact_192_list__ex1__simps_I1_J,axiom,
    ! [P4: produc1234881154892807749rm_a_b > $o] :
      ~ ( list_e8606123503008938638rm_a_b @ P4 @ nil_Pr4197086119080101963rm_a_b ) ).

% list_ex1_simps(1)
thf(fact_193_list__ex1__simps_I1_J,axiom,
    ! [P4: product_prod_nat_nat > $o] :
      ~ ( list_e8644085759156585930at_nat @ P4 @ nil_Pr5478986624290739719at_nat ) ).

% list_ex1_simps(1)
thf(fact_194_list__ex1__simps_I1_J,axiom,
    ! [P4: list_term_a_b > $o] :
      ~ ( list_e1122455069120957467rm_a_b @ P4 @ nil_list_term_a_b ) ).

% list_ex1_simps(1)
thf(fact_195_list__ex1__simps_I1_J,axiom,
    ! [P4: ( term_a_b > nat ) > $o] :
      ~ ( list_e8867884845635725434_b_nat @ P4 @ nil_term_a_b_nat ) ).

% list_ex1_simps(1)
thf(fact_196_list__ex1__simps_I1_J,axiom,
    ! [P4: produc357393685978478089rm_a_b > $o] :
      ~ ( list_e2893281867963234770rm_a_b @ P4 @ nil_Pr6942174756412032271rm_a_b ) ).

% list_ex1_simps(1)
thf(fact_197_list__ex1__simps_I1_J,axiom,
    ! [P4: list_nat > $o] :
      ~ ( list_ex1_list_nat @ P4 @ nil_list_nat ) ).

% list_ex1_simps(1)
thf(fact_198_list__ex1__simps_I1_J,axiom,
    ! [P4: term_a_b > $o] :
      ~ ( list_ex1_term_a_b @ P4 @ nil_term_a_b ) ).

% list_ex1_simps(1)
thf(fact_199_list__ex1__simps_I1_J,axiom,
    ! [P4: nat > $o] :
      ~ ( list_ex1_nat @ P4 @ nil_nat ) ).

% list_ex1_simps(1)
thf(fact_200_replace__term__at_Osimps_I1_J,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( term_r6860082780075436317at_a_b @ S @ nil_nat @ T )
      = T ) ).

% replace_term_at.simps(1)
thf(fact_201_bind__simps_I1_J,axiom,
    ! [F2: nat > list_nat] :
      ( ( bind_nat_nat @ nil_nat @ F2 )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_202_bind__simps_I1_J,axiom,
    ! [F2: term_a_b > list_nat] :
      ( ( bind_term_a_b_nat @ nil_term_a_b @ F2 )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_203_bind__simps_I1_J,axiom,
    ! [F2: nat > list_term_a_b] :
      ( ( bind_nat_term_a_b @ nil_nat @ F2 )
      = nil_term_a_b ) ).

% bind_simps(1)
thf(fact_204_bind__simps_I1_J,axiom,
    ! [F2: term_a_b > list_term_a_b] :
      ( ( bind_t2899167355749536305rm_a_b @ nil_term_a_b @ F2 )
      = nil_term_a_b ) ).

% bind_simps(1)
thf(fact_205_bind__simps_I1_J,axiom,
    ! [F2: nat > list_list_nat] :
      ( ( bind_nat_list_nat @ nil_nat @ F2 )
      = nil_list_nat ) ).

% bind_simps(1)
thf(fact_206_bind__simps_I1_J,axiom,
    ! [F2: list_nat > list_nat] :
      ( ( bind_list_nat_nat @ nil_list_nat @ F2 )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_207_bind__simps_I1_J,axiom,
    ! [F2: nat > list_P6011104703257516679at_nat] :
      ( ( bind_n1878750130520726888at_nat @ nil_nat @ F2 )
      = nil_Pr5478986624290739719at_nat ) ).

% bind_simps(1)
thf(fact_208_bind__simps_I1_J,axiom,
    ! [F2: product_prod_nat_nat > list_nat] :
      ( ( bind_P7742074774332787594at_nat @ nil_Pr5478986624290739719at_nat @ F2 )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_209_bind__simps_I1_J,axiom,
    ! [F2: list_nat > list_list_nat] :
      ( ( bind_l7796378977173581257st_nat @ nil_list_nat @ F2 )
      = nil_list_nat ) ).

% bind_simps(1)
thf(fact_210_bind__simps_I1_J,axiom,
    ! [F2: term_a_b > list_list_nat] :
      ( ( bind_t442032754801615613st_nat @ nil_term_a_b @ F2 )
      = nil_list_nat ) ).

% bind_simps(1)
thf(fact_211_member__rec_I2_J,axiom,
    ! [Y3: produc357393685978478089rm_a_b] :
      ~ ( member1202727356028726564rm_a_b @ nil_Pr6942174756412032271rm_a_b @ Y3 ) ).

% member_rec(2)
thf(fact_212_member__rec_I2_J,axiom,
    ! [Y3: produc1234881154892807749rm_a_b] :
      ~ ( member2625873403759427296rm_a_b @ nil_Pr4197086119080101963rm_a_b @ Y3 ) ).

% member_rec(2)
thf(fact_213_member__rec_I2_J,axiom,
    ! [Y3: product_prod_nat_nat] :
      ~ ( member6104210405413575452at_nat @ nil_Pr5478986624290739719at_nat @ Y3 ) ).

% member_rec(2)
thf(fact_214_member__rec_I2_J,axiom,
    ! [Y3: list_term_a_b] :
      ~ ( member_list_term_a_b @ nil_list_term_a_b @ Y3 ) ).

% member_rec(2)
thf(fact_215_member__rec_I2_J,axiom,
    ! [Y3: list_nat] :
      ~ ( member_list_nat @ nil_list_nat @ Y3 ) ).

% member_rec(2)
thf(fact_216_member__rec_I2_J,axiom,
    ! [Y3: term_a_b > nat] :
      ~ ( member_term_a_b_nat @ nil_term_a_b_nat @ Y3 ) ).

% member_rec(2)
thf(fact_217_member__rec_I2_J,axiom,
    ! [Y3: term_a_b] :
      ~ ( member_term_a_b @ nil_term_a_b @ Y3 ) ).

% member_rec(2)
thf(fact_218_member__rec_I2_J,axiom,
    ! [Y3: nat] :
      ~ ( member_nat @ nil_nat @ Y3 ) ).

% member_rec(2)
thf(fact_219_in__measures_I1_J,axiom,
    ! [X3: nat,Y3: nat] :
      ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( measures_nat @ nil_nat_nat ) ) ).

% in_measures(1)
thf(fact_220_in__measures_I1_J,axiom,
    ! [X3: term_a_b,Y3: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( measures_term_a_b @ nil_term_a_b_nat ) ) ).

% in_measures(1)
thf(fact_221_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_le1467257176787788120rm_a_b @ N @ nil_Pr4197086119080101963rm_a_b )
      = N ) ).

% gen_length_code(1)
thf(fact_222_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_le2383899666085517716at_nat @ N @ nil_Pr5478986624290739719at_nat )
      = N ) ).

% gen_length_code(1)
thf(fact_223_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_le4942244902447077073rm_a_b @ N @ nil_list_term_a_b )
      = N ) ).

% gen_length_code(1)
thf(fact_224_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_length_list_nat @ N @ nil_list_nat )
      = N ) ).

% gen_length_code(1)
thf(fact_225_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_le5771528551073234224_b_nat @ N @ nil_term_a_b_nat )
      = N ) ).

% gen_length_code(1)
thf(fact_226_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_length_term_a_b @ N @ nil_term_a_b )
      = N ) ).

% gen_length_code(1)
thf(fact_227_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_length_nat @ N @ nil_nat )
      = N ) ).

% gen_length_code(1)
thf(fact_228_maps__simps_I2_J,axiom,
    ! [F2: nat > list_nat] :
      ( ( maps_nat_nat @ F2 @ nil_nat )
      = nil_nat ) ).

% maps_simps(2)
thf(fact_229_maps__simps_I2_J,axiom,
    ! [F2: term_a_b > list_nat] :
      ( ( maps_term_a_b_nat @ F2 @ nil_term_a_b )
      = nil_nat ) ).

% maps_simps(2)
thf(fact_230_maps__simps_I2_J,axiom,
    ! [F2: nat > list_term_a_b] :
      ( ( maps_nat_term_a_b @ F2 @ nil_nat )
      = nil_term_a_b ) ).

% maps_simps(2)
thf(fact_231_maps__simps_I2_J,axiom,
    ! [F2: term_a_b > list_term_a_b] :
      ( ( maps_t888753856850818283rm_a_b @ F2 @ nil_term_a_b )
      = nil_term_a_b ) ).

% maps_simps(2)
thf(fact_232_maps__simps_I2_J,axiom,
    ! [F2: nat > list_list_nat] :
      ( ( maps_nat_list_nat @ F2 @ nil_nat )
      = nil_list_nat ) ).

% maps_simps(2)
thf(fact_233_maps__simps_I2_J,axiom,
    ! [F2: list_nat > list_nat] :
      ( ( maps_list_nat_nat @ F2 @ nil_list_nat )
      = nil_nat ) ).

% maps_simps(2)
thf(fact_234_maps__simps_I2_J,axiom,
    ! [F2: nat > list_P6011104703257516679at_nat] :
      ( ( maps_n1297257645151936558at_nat @ F2 @ nil_nat )
      = nil_Pr5478986624290739719at_nat ) ).

% maps_simps(2)
thf(fact_235_maps__simps_I2_J,axiom,
    ! [F2: product_prod_nat_nat > list_nat] :
      ( ( maps_P7160582288963997264at_nat @ F2 @ nil_Pr5478986624290739719at_nat )
      = nil_nat ) ).

% maps_simps(2)
thf(fact_236_maps__simps_I2_J,axiom,
    ! [F2: list_nat > list_list_nat] :
      ( ( maps_l5785965478274863235st_nat @ F2 @ nil_list_nat )
      = nil_list_nat ) ).

% maps_simps(2)
thf(fact_237_maps__simps_I2_J,axiom,
    ! [F2: term_a_b > list_list_nat] :
      ( ( maps_t7654991292757673399st_nat @ F2 @ nil_term_a_b )
      = nil_list_nat ) ).

% maps_simps(2)
thf(fact_238_pair__in__swap__image,axiom,
    ! [Y3: term_a_b,X3: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ ( image_3126328586747194693rm_a_b @ produc2541846788875228641rm_a_b @ A5 ) )
      = ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_239_pair__in__swap__image,axiom,
    ! [Y3: nat,X3: nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X3 ) @ ( image_5168914502847457605at_nat @ product_swap_nat_nat @ A5 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_240_pair__in__swap__image,axiom,
    ! [Y3: term_a_b,X3: b,A5: set_Pr1357592988741419896rm_a_b] :
      ( ( member6959280461167840375_a_b_b @ ( produc1666658391573147538_a_b_b @ Y3 @ X3 ) @ ( image_7745387974640170793_a_b_b @ produc1200019762089676828rm_a_b @ A5 ) )
      = ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ X3 @ Y3 ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_241_pair__in__swap__image,axiom,
    ! [Y3: b,X3: term_a_b,A5: set_Pr3369833000147915158_a_b_b] :
      ( ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ Y3 @ X3 ) @ ( image_6915049758239296041rm_a_b @ produc1428861184864852466_a_b_b @ A5 ) )
      = ( member6959280461167840375_a_b_b @ ( produc1666658391573147538_a_b_b @ X3 @ Y3 ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_242_pair__in__swap__image,axiom,
    ! [Y3: list_nat,X3: nat > nat > $o,A5: set_Pr7072801126362145067st_nat] :
      ( ( member6626851962243112338_nat_o @ ( produc2887701473640236333_nat_o @ Y3 @ X3 ) @ ( image_4755272365918978807_nat_o @ produc3374130635869402703st_nat @ A5 ) )
      = ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ X3 @ Y3 ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_243_pair__in__swap__image,axiom,
    ! [Y3: nat > nat > $o,X3: list_nat,A5: set_Pr4291780605709124401_nat_o] :
      ( ( member5350383351084882060st_nat @ ( produc4727192421694094319st_nat @ Y3 @ X3 ) @ ( image_7223566576988150519st_nat @ produc1534639687815544717_nat_o @ A5 ) )
      = ( member6626851962243112338_nat_o @ ( produc2887701473640236333_nat_o @ X3 @ Y3 ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_244_pair__in__swap__image,axiom,
    ! [Y3: produc1828647624359046049st_nat,X3: nat > nat,A5: set_Pr3595532568945684557st_nat] :
      ( ( member6277506213002785702at_nat @ ( produc2376817894703970953at_nat @ Y3 @ X3 ) @ ( image_3216512029154347309at_nat @ produc3947362003713813545st_nat @ A5 ) )
      = ( member5596898355348113582st_nat @ ( produc4626581765195395529st_nat @ X3 @ Y3 ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_245_pair__in__swap__image,axiom,
    ! [Y3: nat > nat,X3: produc1828647624359046049st_nat,A5: set_Pr5877781423743409989at_nat] :
      ( ( member5596898355348113582st_nat @ ( produc4626581765195395529st_nat @ Y3 @ X3 ) @ ( image_1413504709866899757st_nat @ produc1697598133222388969at_nat @ A5 ) )
      = ( member6277506213002785702at_nat @ ( produc2376817894703970953at_nat @ X3 @ Y3 ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_246_pair__in__swap__image,axiom,
    ! [Y3: produc1828647624359046049st_nat,X3: nat > nat > $o,A5: set_Pr4817715314677154882st_nat] :
      ( ( member6918694168860644693_nat_o @ ( produc3563911201077956380_nat_o @ Y3 @ X3 ) @ ( image_5834055821832758479_nat_o @ produc875157120124018930st_nat @ A5 ) )
      = ( member6951660485171671051st_nat @ ( produc3127733452865184594st_nat @ X3 @ Y3 ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_247_pair__in__swap__image,axiom,
    ! [Y3: nat > nat > $o,X3: produc1828647624359046049st_nat,A5: set_Pr7697833614269146252_nat_o] :
      ( ( member6951660485171671051st_nat @ ( produc3127733452865184594st_nat @ Y3 @ X3 ) @ ( image_7126609314878544571st_nat @ produc1311334868336790716_nat_o @ A5 ) )
      = ( member6918694168860644693_nat_o @ ( produc3563911201077956380_nat_o @ X3 @ Y3 ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_248_null__rec_I2_J,axiom,
    null_P4232181139178827347rm_a_b @ nil_Pr4197086119080101963rm_a_b ).

% null_rec(2)
thf(fact_249_null__rec_I2_J,axiom,
    null_P7301981044126162959at_nat @ nil_Pr5478986624290739719at_nat ).

% null_rec(2)
thf(fact_250_null__rec_I2_J,axiom,
    null_list_term_a_b @ nil_list_term_a_b ).

% null_rec(2)
thf(fact_251_null__rec_I2_J,axiom,
    null_list_nat @ nil_list_nat ).

% null_rec(2)
thf(fact_252_null__rec_I2_J,axiom,
    null_term_a_b_nat @ nil_term_a_b_nat ).

% null_rec(2)
thf(fact_253_null__rec_I2_J,axiom,
    null_term_a_b @ nil_term_a_b ).

% null_rec(2)
thf(fact_254_null__rec_I2_J,axiom,
    null_nat @ nil_nat ).

% null_rec(2)
thf(fact_255_eq__Nil__null,axiom,
    ! [Xs: list_P2364656488115551307rm_a_b] :
      ( ( Xs = nil_Pr4197086119080101963rm_a_b )
      = ( null_P4232181139178827347rm_a_b @ Xs ) ) ).

% eq_Nil_null
thf(fact_256_eq__Nil__null,axiom,
    ! [Xs: list_P6011104703257516679at_nat] :
      ( ( Xs = nil_Pr5478986624290739719at_nat )
      = ( null_P7301981044126162959at_nat @ Xs ) ) ).

% eq_Nil_null
thf(fact_257_eq__Nil__null,axiom,
    ! [Xs: list_list_term_a_b] :
      ( ( Xs = nil_list_term_a_b )
      = ( null_list_term_a_b @ Xs ) ) ).

% eq_Nil_null
thf(fact_258_eq__Nil__null,axiom,
    ! [Xs: list_list_nat] :
      ( ( Xs = nil_list_nat )
      = ( null_list_nat @ Xs ) ) ).

% eq_Nil_null
thf(fact_259_eq__Nil__null,axiom,
    ! [Xs: list_term_a_b_nat] :
      ( ( Xs = nil_term_a_b_nat )
      = ( null_term_a_b_nat @ Xs ) ) ).

% eq_Nil_null
thf(fact_260_eq__Nil__null,axiom,
    ! [Xs: list_term_a_b] :
      ( ( Xs = nil_term_a_b )
      = ( null_term_a_b @ Xs ) ) ).

% eq_Nil_null
thf(fact_261_eq__Nil__null,axiom,
    ! [Xs: list_nat] :
      ( ( Xs = nil_nat )
      = ( null_nat @ Xs ) ) ).

% eq_Nil_null
thf(fact_262_image__eqI,axiom,
    ! [B: nat,F2: nat > nat,X3: nat,A5: set_nat] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member_nat2 @ X3 @ A5 )
       => ( member_nat2 @ B @ ( image_nat_nat @ F2 @ A5 ) ) ) ) ).

% image_eqI
thf(fact_263_image__eqI,axiom,
    ! [B: nat,F2: list_nat > nat,X3: list_nat,A5: set_list_nat] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member_list_nat2 @ X3 @ A5 )
       => ( member_nat2 @ B @ ( image_list_nat_nat @ F2 @ A5 ) ) ) ) ).

% image_eqI
thf(fact_264_image__eqI,axiom,
    ! [B: list_nat,F2: nat > list_nat,X3: nat,A5: set_nat] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member_nat2 @ X3 @ A5 )
       => ( member_list_nat2 @ B @ ( image_nat_list_nat @ F2 @ A5 ) ) ) ) ).

% image_eqI
thf(fact_265_image__eqI,axiom,
    ! [B: list_nat,F2: list_nat > list_nat,X3: list_nat,A5: set_list_nat] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member_list_nat2 @ X3 @ A5 )
       => ( member_list_nat2 @ B @ ( image_7976474329151083847st_nat @ F2 @ A5 ) ) ) ) ).

% image_eqI
thf(fact_266_image__eqI,axiom,
    ! [B: nat,F2: produc357393685978478089rm_a_b > nat,X3: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member5869715511025134514rm_a_b @ X3 @ A5 )
       => ( member_nat2 @ B @ ( image_5783597275426814868_b_nat @ F2 @ A5 ) ) ) ) ).

% image_eqI
thf(fact_267_image__eqI,axiom,
    ! [B: produc357393685978478089rm_a_b,F2: nat > produc357393685978478089rm_a_b,X3: nat,A5: set_nat] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member_nat2 @ X3 @ A5 )
       => ( member5869715511025134514rm_a_b @ B @ ( image_7842151682427799154rm_a_b @ F2 @ A5 ) ) ) ) ).

% image_eqI
thf(fact_268_image__eqI,axiom,
    ! [B: produc357393685978478089rm_a_b,F2: list_nat > produc357393685978478089rm_a_b,X3: list_nat,A5: set_list_nat] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member_list_nat2 @ X3 @ A5 )
       => ( member5869715511025134514rm_a_b @ B @ ( image_6727437281251480162rm_a_b @ F2 @ A5 ) ) ) ) ).

% image_eqI
thf(fact_269_image__eqI,axiom,
    ! [B: list_nat,F2: produc357393685978478089rm_a_b > list_nat,X3: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member5869715511025134514rm_a_b @ X3 @ A5 )
       => ( member_list_nat2 @ B @ ( image_8768541597965001892st_nat @ F2 @ A5 ) ) ) ) ).

% image_eqI
thf(fact_270_image__eqI,axiom,
    ! [B: produc357393685978478089rm_a_b,F2: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b,X3: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member5869715511025134514rm_a_b @ X3 @ A5 )
       => ( member5869715511025134514rm_a_b @ B @ ( image_3126328586747194693rm_a_b @ F2 @ A5 ) ) ) ) ).

% image_eqI
thf(fact_271_image__eqI,axiom,
    ! [B: set_nat,F2: set_nat > set_nat,X3: set_nat,A5: set_set_nat] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member_set_nat @ X3 @ A5 )
       => ( member_set_nat @ B @ ( image_7916887816326733075et_nat @ F2 @ A5 ) ) ) ) ).

% image_eqI
thf(fact_272_surj__swap,axiom,
    ( ( image_3126328586747194693rm_a_b @ produc2541846788875228641rm_a_b @ top_to1314267278846557113rm_a_b )
    = top_to1314267278846557113rm_a_b ) ).

% surj_swap
thf(fact_273_Nil__not__par_I1_J,axiom,
    ! [P: list_P2364656488115551307rm_a_b] :
      ~ ( term_p6424153574643673891rm_a_b @ nil_Pr4197086119080101963rm_a_b @ P ) ).

% Nil_not_par(1)
thf(fact_274_Nil__not__par_I1_J,axiom,
    ! [P: list_P6011104703257516679at_nat] :
      ~ ( term_p8419326880412058847at_nat @ nil_Pr5478986624290739719at_nat @ P ) ).

% Nil_not_par(1)
thf(fact_275_Nil__not__par_I1_J,axiom,
    ! [P: list_list_term_a_b] :
      ~ ( term_p3834017231281530950rm_a_b @ nil_list_term_a_b @ P ) ).

% Nil_not_par(1)
thf(fact_276_Nil__not__par_I1_J,axiom,
    ! [P: list_list_nat] :
      ~ ( term_p4950861579910180738st_nat @ nil_list_nat @ P ) ).

% Nil_not_par(1)
thf(fact_277_Nil__not__par_I1_J,axiom,
    ! [P: list_term_a_b_nat] :
      ~ ( term_p700130527931300261_b_nat @ nil_term_a_b_nat @ P ) ).

% Nil_not_par(1)
thf(fact_278_Nil__not__par_I1_J,axiom,
    ! [P: list_term_a_b] :
      ~ ( term_p7407996180858101430rm_a_b @ nil_term_a_b @ P ) ).

% Nil_not_par(1)
thf(fact_279_Nil__not__par_I1_J,axiom,
    ! [P: list_nat] :
      ~ ( term_p5017330785391824242ar_nat @ nil_nat @ P ) ).

% Nil_not_par(1)
thf(fact_280_Nil__not__par_I2_J,axiom,
    ! [P: list_P2364656488115551307rm_a_b] :
      ~ ( term_p6424153574643673891rm_a_b @ P @ nil_Pr4197086119080101963rm_a_b ) ).

% Nil_not_par(2)
thf(fact_281_Nil__not__par_I2_J,axiom,
    ! [P: list_P6011104703257516679at_nat] :
      ~ ( term_p8419326880412058847at_nat @ P @ nil_Pr5478986624290739719at_nat ) ).

% Nil_not_par(2)
thf(fact_282_Nil__not__par_I2_J,axiom,
    ! [P: list_list_term_a_b] :
      ~ ( term_p3834017231281530950rm_a_b @ P @ nil_list_term_a_b ) ).

% Nil_not_par(2)
thf(fact_283_Nil__not__par_I2_J,axiom,
    ! [P: list_list_nat] :
      ~ ( term_p4950861579910180738st_nat @ P @ nil_list_nat ) ).

% Nil_not_par(2)
thf(fact_284_Nil__not__par_I2_J,axiom,
    ! [P: list_term_a_b_nat] :
      ~ ( term_p700130527931300261_b_nat @ P @ nil_term_a_b_nat ) ).

% Nil_not_par(2)
thf(fact_285_Nil__not__par_I2_J,axiom,
    ! [P: list_term_a_b] :
      ~ ( term_p7407996180858101430rm_a_b @ P @ nil_term_a_b ) ).

% Nil_not_par(2)
thf(fact_286_Nil__not__par_I2_J,axiom,
    ! [P: list_nat] :
      ~ ( term_p5017330785391824242ar_nat @ P @ nil_nat ) ).

% Nil_not_par(2)
thf(fact_287_Inf_OINF__cong,axiom,
    ! [A5: set_set_nat,B5: set_set_nat,C4: set_nat > set_nat,D: set_nat > set_nat,Inf: set_set_nat > set_nat] :
      ( ( A5 = B5 )
     => ( ! [X: set_nat] :
            ( ( member_set_nat @ X @ B5 )
           => ( ( C4 @ X )
              = ( D @ X ) ) )
       => ( ( Inf @ ( image_7916887816326733075et_nat @ C4 @ A5 ) )
          = ( Inf @ ( image_7916887816326733075et_nat @ D @ B5 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_288_Inf_OINF__cong,axiom,
    ! [A5: set_list_nat,B5: set_list_nat,C4: list_nat > list_nat,D: list_nat > list_nat,Inf: set_list_nat > list_nat] :
      ( ( A5 = B5 )
     => ( ! [X: list_nat] :
            ( ( member_list_nat2 @ X @ B5 )
           => ( ( C4 @ X )
              = ( D @ X ) ) )
       => ( ( Inf @ ( image_7976474329151083847st_nat @ C4 @ A5 ) )
          = ( Inf @ ( image_7976474329151083847st_nat @ D @ B5 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_289_Inf_OINF__cong,axiom,
    ! [A5: set_list_nat,B5: set_list_nat,C4: list_nat > nat,D: list_nat > nat,Inf: set_nat > nat] :
      ( ( A5 = B5 )
     => ( ! [X: list_nat] :
            ( ( member_list_nat2 @ X @ B5 )
           => ( ( C4 @ X )
              = ( D @ X ) ) )
       => ( ( Inf @ ( image_list_nat_nat @ C4 @ A5 ) )
          = ( Inf @ ( image_list_nat_nat @ D @ B5 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_290_Inf_OINF__cong,axiom,
    ! [A5: set_nat,B5: set_nat,C4: nat > produc357393685978478089rm_a_b,D: nat > produc357393685978478089rm_a_b,Inf: set_Pr4386577575007340137rm_a_b > produc357393685978478089rm_a_b] :
      ( ( A5 = B5 )
     => ( ! [X: nat] :
            ( ( member_nat2 @ X @ B5 )
           => ( ( C4 @ X )
              = ( D @ X ) ) )
       => ( ( Inf @ ( image_7842151682427799154rm_a_b @ C4 @ A5 ) )
          = ( Inf @ ( image_7842151682427799154rm_a_b @ D @ B5 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_291_Inf_OINF__cong,axiom,
    ! [A5: set_nat,B5: set_nat,C4: nat > list_nat,D: nat > list_nat,Inf: set_list_nat > list_nat] :
      ( ( A5 = B5 )
     => ( ! [X: nat] :
            ( ( member_nat2 @ X @ B5 )
           => ( ( C4 @ X )
              = ( D @ X ) ) )
       => ( ( Inf @ ( image_nat_list_nat @ C4 @ A5 ) )
          = ( Inf @ ( image_nat_list_nat @ D @ B5 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_292_Inf_OINF__cong,axiom,
    ! [A5: set_nat,B5: set_nat,C4: nat > nat,D: nat > nat,Inf: set_nat > nat] :
      ( ( A5 = B5 )
     => ( ! [X: nat] :
            ( ( member_nat2 @ X @ B5 )
           => ( ( C4 @ X )
              = ( D @ X ) ) )
       => ( ( Inf @ ( image_nat_nat @ C4 @ A5 ) )
          = ( Inf @ ( image_nat_nat @ D @ B5 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_293_Sup_OSUP__cong,axiom,
    ! [A5: set_set_nat,B5: set_set_nat,C4: set_nat > set_nat,D: set_nat > set_nat,Sup: set_set_nat > set_nat] :
      ( ( A5 = B5 )
     => ( ! [X: set_nat] :
            ( ( member_set_nat @ X @ B5 )
           => ( ( C4 @ X )
              = ( D @ X ) ) )
       => ( ( Sup @ ( image_7916887816326733075et_nat @ C4 @ A5 ) )
          = ( Sup @ ( image_7916887816326733075et_nat @ D @ B5 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_294_Sup_OSUP__cong,axiom,
    ! [A5: set_list_nat,B5: set_list_nat,C4: list_nat > list_nat,D: list_nat > list_nat,Sup: set_list_nat > list_nat] :
      ( ( A5 = B5 )
     => ( ! [X: list_nat] :
            ( ( member_list_nat2 @ X @ B5 )
           => ( ( C4 @ X )
              = ( D @ X ) ) )
       => ( ( Sup @ ( image_7976474329151083847st_nat @ C4 @ A5 ) )
          = ( Sup @ ( image_7976474329151083847st_nat @ D @ B5 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_295_Sup_OSUP__cong,axiom,
    ! [A5: set_list_nat,B5: set_list_nat,C4: list_nat > nat,D: list_nat > nat,Sup: set_nat > nat] :
      ( ( A5 = B5 )
     => ( ! [X: list_nat] :
            ( ( member_list_nat2 @ X @ B5 )
           => ( ( C4 @ X )
              = ( D @ X ) ) )
       => ( ( Sup @ ( image_list_nat_nat @ C4 @ A5 ) )
          = ( Sup @ ( image_list_nat_nat @ D @ B5 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_296_Sup_OSUP__cong,axiom,
    ! [A5: set_nat,B5: set_nat,C4: nat > produc357393685978478089rm_a_b,D: nat > produc357393685978478089rm_a_b,Sup: set_Pr4386577575007340137rm_a_b > produc357393685978478089rm_a_b] :
      ( ( A5 = B5 )
     => ( ! [X: nat] :
            ( ( member_nat2 @ X @ B5 )
           => ( ( C4 @ X )
              = ( D @ X ) ) )
       => ( ( Sup @ ( image_7842151682427799154rm_a_b @ C4 @ A5 ) )
          = ( Sup @ ( image_7842151682427799154rm_a_b @ D @ B5 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_297_Sup_OSUP__cong,axiom,
    ! [A5: set_nat,B5: set_nat,C4: nat > list_nat,D: nat > list_nat,Sup: set_list_nat > list_nat] :
      ( ( A5 = B5 )
     => ( ! [X: nat] :
            ( ( member_nat2 @ X @ B5 )
           => ( ( C4 @ X )
              = ( D @ X ) ) )
       => ( ( Sup @ ( image_nat_list_nat @ C4 @ A5 ) )
          = ( Sup @ ( image_nat_list_nat @ D @ B5 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_298_Sup_OSUP__cong,axiom,
    ! [A5: set_nat,B5: set_nat,C4: nat > nat,D: nat > nat,Sup: set_nat > nat] :
      ( ( A5 = B5 )
     => ( ! [X: nat] :
            ( ( member_nat2 @ X @ B5 )
           => ( ( C4 @ X )
              = ( D @ X ) ) )
       => ( ( Sup @ ( image_nat_nat @ C4 @ A5 ) )
          = ( Sup @ ( image_nat_nat @ D @ B5 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_299_imageI,axiom,
    ! [X3: nat,A5: set_nat,F2: nat > nat] :
      ( ( member_nat2 @ X3 @ A5 )
     => ( member_nat2 @ ( F2 @ X3 ) @ ( image_nat_nat @ F2 @ A5 ) ) ) ).

% imageI
thf(fact_300_imageI,axiom,
    ! [X3: list_nat,A5: set_list_nat,F2: list_nat > nat] :
      ( ( member_list_nat2 @ X3 @ A5 )
     => ( member_nat2 @ ( F2 @ X3 ) @ ( image_list_nat_nat @ F2 @ A5 ) ) ) ).

% imageI
thf(fact_301_imageI,axiom,
    ! [X3: nat,A5: set_nat,F2: nat > list_nat] :
      ( ( member_nat2 @ X3 @ A5 )
     => ( member_list_nat2 @ ( F2 @ X3 ) @ ( image_nat_list_nat @ F2 @ A5 ) ) ) ).

% imageI
thf(fact_302_imageI,axiom,
    ! [X3: list_nat,A5: set_list_nat,F2: list_nat > list_nat] :
      ( ( member_list_nat2 @ X3 @ A5 )
     => ( member_list_nat2 @ ( F2 @ X3 ) @ ( image_7976474329151083847st_nat @ F2 @ A5 ) ) ) ).

% imageI
thf(fact_303_imageI,axiom,
    ! [X3: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,F2: produc357393685978478089rm_a_b > nat] :
      ( ( member5869715511025134514rm_a_b @ X3 @ A5 )
     => ( member_nat2 @ ( F2 @ X3 ) @ ( image_5783597275426814868_b_nat @ F2 @ A5 ) ) ) ).

% imageI
thf(fact_304_imageI,axiom,
    ! [X3: nat,A5: set_nat,F2: nat > produc357393685978478089rm_a_b] :
      ( ( member_nat2 @ X3 @ A5 )
     => ( member5869715511025134514rm_a_b @ ( F2 @ X3 ) @ ( image_7842151682427799154rm_a_b @ F2 @ A5 ) ) ) ).

% imageI
thf(fact_305_imageI,axiom,
    ! [X3: list_nat,A5: set_list_nat,F2: list_nat > produc357393685978478089rm_a_b] :
      ( ( member_list_nat2 @ X3 @ A5 )
     => ( member5869715511025134514rm_a_b @ ( F2 @ X3 ) @ ( image_6727437281251480162rm_a_b @ F2 @ A5 ) ) ) ).

% imageI
thf(fact_306_imageI,axiom,
    ! [X3: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,F2: produc357393685978478089rm_a_b > list_nat] :
      ( ( member5869715511025134514rm_a_b @ X3 @ A5 )
     => ( member_list_nat2 @ ( F2 @ X3 ) @ ( image_8768541597965001892st_nat @ F2 @ A5 ) ) ) ).

% imageI
thf(fact_307_imageI,axiom,
    ! [X3: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,F2: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X3 @ A5 )
     => ( member5869715511025134514rm_a_b @ ( F2 @ X3 ) @ ( image_3126328586747194693rm_a_b @ F2 @ A5 ) ) ) ).

% imageI
thf(fact_308_imageI,axiom,
    ! [X3: set_nat,A5: set_set_nat,F2: set_nat > set_nat] :
      ( ( member_set_nat @ X3 @ A5 )
     => ( member_set_nat @ ( F2 @ X3 ) @ ( image_7916887816326733075et_nat @ F2 @ A5 ) ) ) ).

% imageI
thf(fact_309_image__iff,axiom,
    ! [Z2: set_nat,F2: set_nat > set_nat,A5: set_set_nat] :
      ( ( member_set_nat @ Z2 @ ( image_7916887816326733075et_nat @ F2 @ A5 ) )
      = ( ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A5 )
            & ( Z2
              = ( F2 @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_310_image__iff,axiom,
    ! [Z2: list_nat,F2: list_nat > list_nat,A5: set_list_nat] :
      ( ( member_list_nat2 @ Z2 @ ( image_7976474329151083847st_nat @ F2 @ A5 ) )
      = ( ? [X4: list_nat] :
            ( ( member_list_nat2 @ X4 @ A5 )
            & ( Z2
              = ( F2 @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_311_image__iff,axiom,
    ! [Z2: list_nat,F2: nat > list_nat,A5: set_nat] :
      ( ( member_list_nat2 @ Z2 @ ( image_nat_list_nat @ F2 @ A5 ) )
      = ( ? [X4: nat] :
            ( ( member_nat2 @ X4 @ A5 )
            & ( Z2
              = ( F2 @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_312_image__iff,axiom,
    ! [Z2: produc357393685978478089rm_a_b,F2: nat > produc357393685978478089rm_a_b,A5: set_nat] :
      ( ( member5869715511025134514rm_a_b @ Z2 @ ( image_7842151682427799154rm_a_b @ F2 @ A5 ) )
      = ( ? [X4: nat] :
            ( ( member_nat2 @ X4 @ A5 )
            & ( Z2
              = ( F2 @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_313_image__iff,axiom,
    ! [Z2: nat,F2: list_nat > nat,A5: set_list_nat] :
      ( ( member_nat2 @ Z2 @ ( image_list_nat_nat @ F2 @ A5 ) )
      = ( ? [X4: list_nat] :
            ( ( member_list_nat2 @ X4 @ A5 )
            & ( Z2
              = ( F2 @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_314_image__iff,axiom,
    ! [Z2: nat,F2: nat > nat,A5: set_nat] :
      ( ( member_nat2 @ Z2 @ ( image_nat_nat @ F2 @ A5 ) )
      = ( ? [X4: nat] :
            ( ( member_nat2 @ X4 @ A5 )
            & ( Z2
              = ( F2 @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_315_bex__imageD,axiom,
    ! [F2: nat > produc357393685978478089rm_a_b,A5: set_nat,P4: produc357393685978478089rm_a_b > $o] :
      ( ? [X5: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X5 @ ( image_7842151682427799154rm_a_b @ F2 @ A5 ) )
          & ( P4 @ X5 ) )
     => ? [X: nat] :
          ( ( member_nat2 @ X @ A5 )
          & ( P4 @ ( F2 @ X ) ) ) ) ).

% bex_imageD
thf(fact_316_bex__imageD,axiom,
    ! [F2: nat > list_nat,A5: set_nat,P4: list_nat > $o] :
      ( ? [X5: list_nat] :
          ( ( member_list_nat2 @ X5 @ ( image_nat_list_nat @ F2 @ A5 ) )
          & ( P4 @ X5 ) )
     => ? [X: nat] :
          ( ( member_nat2 @ X @ A5 )
          & ( P4 @ ( F2 @ X ) ) ) ) ).

% bex_imageD
thf(fact_317_bex__imageD,axiom,
    ! [F2: nat > nat,A5: set_nat,P4: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat2 @ X5 @ ( image_nat_nat @ F2 @ A5 ) )
          & ( P4 @ X5 ) )
     => ? [X: nat] :
          ( ( member_nat2 @ X @ A5 )
          & ( P4 @ ( F2 @ X ) ) ) ) ).

% bex_imageD
thf(fact_318_image__cong,axiom,
    ! [M: set_nat,N2: set_nat,F2: nat > nat,G: nat > nat] :
      ( ( M = N2 )
     => ( ! [X: nat] :
            ( ( member_nat2 @ X @ N2 )
           => ( ( F2 @ X )
              = ( G @ X ) ) )
       => ( ( image_nat_nat @ F2 @ M )
          = ( image_nat_nat @ G @ N2 ) ) ) ) ).

% image_cong
thf(fact_319_UNIV__I,axiom,
    ! [X3: list_nat] : ( member_list_nat2 @ X3 @ top_top_set_list_nat ) ).

% UNIV_I
thf(fact_320_UNIV__I,axiom,
    ! [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ top_to1314267278846557113rm_a_b ) ).

% UNIV_I
thf(fact_321_UNIV__I,axiom,
    ! [X3: nat] : ( member_nat2 @ X3 @ top_top_set_nat ) ).

% UNIV_I
thf(fact_322_UNIV__witness,axiom,
    ? [X: list_nat] : ( member_list_nat2 @ X @ top_top_set_list_nat ) ).

% UNIV_witness
thf(fact_323_UNIV__witness,axiom,
    ? [X: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X @ top_to1314267278846557113rm_a_b ) ).

% UNIV_witness
thf(fact_324_UNIV__witness,axiom,
    ? [X: nat] : ( member_nat2 @ X @ top_top_set_nat ) ).

% UNIV_witness
thf(fact_325_UNIV__eq__I,axiom,
    ! [A5: set_list_nat] :
      ( ! [X: list_nat] : ( member_list_nat2 @ X @ A5 )
     => ( top_top_set_list_nat = A5 ) ) ).

% UNIV_eq_I
thf(fact_326_UNIV__eq__I,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b] :
      ( ! [X: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X @ A5 )
     => ( top_to1314267278846557113rm_a_b = A5 ) ) ).

% UNIV_eq_I
thf(fact_327_UNIV__eq__I,axiom,
    ! [A5: set_nat] :
      ( ! [X: nat] : ( member_nat2 @ X @ A5 )
     => ( top_top_set_nat = A5 ) ) ).

% UNIV_eq_I
thf(fact_328_range__eqI,axiom,
    ! [B: nat,F2: nat > nat,X3: nat] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( member_nat2 @ B @ ( image_nat_nat @ F2 @ top_top_set_nat ) ) ) ).

% range_eqI
thf(fact_329_rangeI,axiom,
    ! [F2: nat > nat,X3: nat] : ( member_nat2 @ ( F2 @ X3 ) @ ( image_nat_nat @ F2 @ top_top_set_nat ) ) ).

% rangeI
thf(fact_330_parallel__replace__term__commute,axiom,
    ! [P: list_nat,Q5: list_nat,S: term_a_b,T: term_a_b,U: term_a_b] :
      ( ( term_p5017330785391824242ar_nat @ P @ Q5 )
     => ( ( term_r6860082780075436317at_a_b @ ( term_r6860082780075436317at_a_b @ S @ P @ T ) @ Q5 @ U )
        = ( term_r6860082780075436317at_a_b @ ( term_r6860082780075436317at_a_b @ S @ Q5 @ U ) @ P @ T ) ) ) ).

% parallel_replace_term_commute
thf(fact_331_rev__image__eqI,axiom,
    ! [X3: list_nat,A5: set_list_nat,B: list_nat,F2: list_nat > list_nat] :
      ( ( member_list_nat2 @ X3 @ A5 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member_list_nat2 @ B @ ( image_7976474329151083847st_nat @ F2 @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_332_rev__image__eqI,axiom,
    ! [X3: list_nat,A5: set_list_nat,B: produc357393685978478089rm_a_b,F2: list_nat > produc357393685978478089rm_a_b] :
      ( ( member_list_nat2 @ X3 @ A5 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member5869715511025134514rm_a_b @ B @ ( image_6727437281251480162rm_a_b @ F2 @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_333_rev__image__eqI,axiom,
    ! [X3: list_nat,A5: set_list_nat,B: nat,F2: list_nat > nat] :
      ( ( member_list_nat2 @ X3 @ A5 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member_nat2 @ B @ ( image_list_nat_nat @ F2 @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_334_rev__image__eqI,axiom,
    ! [X3: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B: list_nat,F2: produc357393685978478089rm_a_b > list_nat] :
      ( ( member5869715511025134514rm_a_b @ X3 @ A5 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member_list_nat2 @ B @ ( image_8768541597965001892st_nat @ F2 @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_335_rev__image__eqI,axiom,
    ! [X3: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b,F2: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X3 @ A5 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member5869715511025134514rm_a_b @ B @ ( image_3126328586747194693rm_a_b @ F2 @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_336_rev__image__eqI,axiom,
    ! [X3: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B: nat,F2: produc357393685978478089rm_a_b > nat] :
      ( ( member5869715511025134514rm_a_b @ X3 @ A5 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member_nat2 @ B @ ( image_5783597275426814868_b_nat @ F2 @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_337_rev__image__eqI,axiom,
    ! [X3: nat,A5: set_nat,B: list_nat,F2: nat > list_nat] :
      ( ( member_nat2 @ X3 @ A5 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member_list_nat2 @ B @ ( image_nat_list_nat @ F2 @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_338_rev__image__eqI,axiom,
    ! [X3: nat,A5: set_nat,B: produc357393685978478089rm_a_b,F2: nat > produc357393685978478089rm_a_b] :
      ( ( member_nat2 @ X3 @ A5 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member5869715511025134514rm_a_b @ B @ ( image_7842151682427799154rm_a_b @ F2 @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_339_rev__image__eqI,axiom,
    ! [X3: nat,A5: set_nat,B: nat,F2: nat > nat] :
      ( ( member_nat2 @ X3 @ A5 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member_nat2 @ B @ ( image_nat_nat @ F2 @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_340_ball__imageD,axiom,
    ! [F2: nat > nat,A5: set_nat,P4: nat > $o] :
      ( ! [X: nat] :
          ( ( member_nat2 @ X @ ( image_nat_nat @ F2 @ A5 ) )
         => ( P4 @ X ) )
     => ! [X5: nat] :
          ( ( member_nat2 @ X5 @ A5 )
         => ( P4 @ ( F2 @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_341_iso__tuple__UNIV__I,axiom,
    ! [X3: list_nat] : ( member_list_nat2 @ X3 @ top_top_set_list_nat ) ).

% iso_tuple_UNIV_I
thf(fact_342_iso__tuple__UNIV__I,axiom,
    ! [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ top_to1314267278846557113rm_a_b ) ).

% iso_tuple_UNIV_I
thf(fact_343_iso__tuple__UNIV__I,axiom,
    ! [X3: nat] : ( member_nat2 @ X3 @ top_top_set_nat ) ).

% iso_tuple_UNIV_I
thf(fact_344_surj__def,axiom,
    ! [F2: nat > nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
      = ( ! [Y4: nat] :
          ? [X4: nat] :
            ( Y4
            = ( F2 @ X4 ) ) ) ) ).

% surj_def
thf(fact_345_surjI,axiom,
    ! [G: nat > nat,F2: nat > nat] :
      ( ! [X: nat] :
          ( ( G @ ( F2 @ X ) )
          = X )
     => ( ( image_nat_nat @ G @ top_top_set_nat )
        = top_top_set_nat ) ) ).

% surjI
thf(fact_346_surjE,axiom,
    ! [F2: nat > nat,Y3: nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
     => ~ ! [X: nat] :
            ( Y3
           != ( F2 @ X ) ) ) ).

% surjE
thf(fact_347_surjD,axiom,
    ! [F2: nat > nat,Y3: nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
     => ? [X: nat] :
          ( Y3
          = ( F2 @ X ) ) ) ).

% surjD
thf(fact_348_map__prod__surj,axiom,
    ! [F2: nat > nat,G: nat > nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ( image_nat_nat @ G @ top_top_set_nat )
          = top_top_set_nat )
       => ( ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F2 @ G ) @ top_to4669805908274784177at_nat )
          = top_to4669805908274784177at_nat ) ) ) ).

% map_prod_surj
thf(fact_349_asymI,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ! [X: term_a_b,Y: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
         => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R ) )
     => ( asym_on_term_a_b @ top_top_set_term_a_b @ R ) ) ).

% asymI
thf(fact_350_map__prod__simp,axiom,
    ! [F2: term_a_b > term_a_b,G: term_a_b > term_a_b,A: term_a_b,B: term_a_b] :
      ( ( produc4972970123731894602rm_a_b @ F2 @ G @ ( produc7020197800436672577rm_a_b @ A @ B ) )
      = ( produc7020197800436672577rm_a_b @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_351_asym__onI,axiom,
    ! [A5: set_list_nat,R: set_Pr3451248702717554689st_nat] :
      ( ! [X: list_nat,Y: list_nat] :
          ( ( member_list_nat2 @ X @ A5 )
         => ( ( member_list_nat2 @ Y @ A5 )
           => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y @ X ) @ R ) ) ) )
     => ( asym_on_list_nat @ A5 @ R ) ) ).

% asym_onI
thf(fact_352_asym__onI,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X @ A5 )
         => ( ( member5869715511025134514rm_a_b @ Y @ A5 )
           => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R )
             => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ X ) @ R ) ) ) )
     => ( asym_o1443355727950249531rm_a_b @ A5 @ R ) ) ).

% asym_onI
thf(fact_353_asym__onI,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ! [X: nat,Y: nat] :
          ( ( member_nat2 @ X @ A5 )
         => ( ( member_nat2 @ Y @ A5 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R ) ) ) )
     => ( asym_on_nat @ A5 @ R ) ) ).

% asym_onI
thf(fact_354_asym__onI,axiom,
    ! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ! [X: term_a_b,Y: term_a_b] :
          ( ( member_term_a_b2 @ X @ A5 )
         => ( ( member_term_a_b2 @ Y @ A5 )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
             => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R ) ) ) )
     => ( asym_on_term_a_b @ A5 @ R ) ) ).

% asym_onI
thf(fact_355_prod_Ocollapse,axiom,
    ! [Prod: produc357393685978478089rm_a_b] :
      ( ( produc7020197800436672577rm_a_b @ ( produc5709096180074326061rm_a_b @ Prod ) @ ( produc968600491044949615rm_a_b @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_356_map__prod__imageI,axiom,
    ! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b,G: term_a_b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_3126328586747194693rm_a_b @ ( produc4972970123731894602rm_a_b @ F2 @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_357_surjective__pairing,axiom,
    ! [T: produc357393685978478089rm_a_b] :
      ( T
      = ( produc7020197800436672577rm_a_b @ ( produc5709096180074326061rm_a_b @ T ) @ ( produc968600491044949615rm_a_b @ T ) ) ) ).

% surjective_pairing
thf(fact_358_prod_Oexhaust__sel,axiom,
    ! [Prod: produc357393685978478089rm_a_b] :
      ( Prod
      = ( produc7020197800436672577rm_a_b @ ( produc5709096180074326061rm_a_b @ Prod ) @ ( produc968600491044949615rm_a_b @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_359_top__empty__eq,axiom,
    ( top_top_list_nat_o
    = ( ^ [X4: list_nat] : ( member_list_nat2 @ X4 @ top_top_set_list_nat ) ) ) ).

% top_empty_eq
thf(fact_360_top__empty__eq,axiom,
    ( top_to1262317115727556708_a_b_o
    = ( ^ [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ top_to1314267278846557113rm_a_b ) ) ) ).

% top_empty_eq
thf(fact_361_top__empty__eq,axiom,
    ( top_top_nat_o
    = ( ^ [X4: nat] : ( member_nat2 @ X4 @ top_top_set_nat ) ) ) ).

% top_empty_eq
thf(fact_362_prod_Oswap__def,axiom,
    ( produc2541846788875228641rm_a_b
    = ( ^ [P3: produc357393685978478089rm_a_b] : ( produc7020197800436672577rm_a_b @ ( produc968600491044949615rm_a_b @ P3 ) @ ( produc5709096180074326061rm_a_b @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_363_snd__conv,axiom,
    ! [X1: term_a_b,X2: term_a_b] :
      ( ( produc968600491044949615rm_a_b @ ( produc7020197800436672577rm_a_b @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_364_snd__eqD,axiom,
    ! [X3: term_a_b,Y3: term_a_b,A: term_a_b] :
      ( ( ( produc968600491044949615rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_365_fst__conv,axiom,
    ! [X1: term_a_b,X2: term_a_b] :
      ( ( produc5709096180074326061rm_a_b @ ( produc7020197800436672577rm_a_b @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_366_fst__eqD,axiom,
    ! [X3: term_a_b,Y3: term_a_b,A: term_a_b] :
      ( ( ( produc5709096180074326061rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_367_asym__on__def,axiom,
    ( asym_on_term_a_b
    = ( ^ [A6: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
        ! [X4: term_a_b] :
          ( ( member_term_a_b2 @ X4 @ A6 )
         => ! [Y4: term_a_b] :
              ( ( member_term_a_b2 @ Y4 @ A6 )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y4 ) @ R3 )
               => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ X4 ) @ R3 ) ) ) ) ) ) ).

% asym_on_def
thf(fact_368_asym__onD,axiom,
    ! [A5: set_list_nat,R: set_Pr3451248702717554689st_nat,X3: list_nat,Y3: list_nat] :
      ( ( asym_on_list_nat @ A5 @ R )
     => ( ( member_list_nat2 @ X3 @ A5 )
       => ( ( member_list_nat2 @ Y3 @ A5 )
         => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ R )
           => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y3 @ X3 ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_369_asym__onD,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
      ( ( asym_o1443355727950249531rm_a_b @ A5 @ R )
     => ( ( member5869715511025134514rm_a_b @ X3 @ A5 )
       => ( ( member5869715511025134514rm_a_b @ Y3 @ A5 )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ R )
           => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ X3 ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_370_asym__onD,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X3: nat,Y3: nat] :
      ( ( asym_on_nat @ A5 @ R )
     => ( ( member_nat2 @ X3 @ A5 )
       => ( ( member_nat2 @ Y3 @ A5 )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X3 ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_371_asym__onD,axiom,
    ! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b] :
      ( ( asym_on_term_a_b @ A5 @ R )
     => ( ( member_term_a_b2 @ X3 @ A5 )
       => ( ( member_term_a_b2 @ Y3 @ A5 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_372_prod__fun__imageE,axiom,
    ! [C: produc357393685978478089rm_a_b,F2: term_a_b > term_a_b,G: term_a_b > term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ C @ ( image_3126328586747194693rm_a_b @ ( produc4972970123731894602rm_a_b @ F2 @ G ) @ R2 ) )
     => ~ ! [X: term_a_b,Y: term_a_b] :
            ( ( C
              = ( produc7020197800436672577rm_a_b @ ( F2 @ X ) @ ( G @ Y ) ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_373_asymD,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b] :
      ( ( asym_on_term_a_b @ top_top_set_term_a_b @ R )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
       => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ R ) ) ) ).

% asymD
thf(fact_374_asym__iff,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( asym_on_term_a_b @ top_top_set_term_a_b @ R )
      = ( ! [X4: term_a_b,Y4: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y4 ) @ R )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ X4 ) @ R ) ) ) ) ).

% asym_iff
thf(fact_375_exI__realizer,axiom,
    ! [P4: term_a_b > term_a_b > $o,Y3: term_a_b,X3: term_a_b] :
      ( ( P4 @ Y3 @ X3 )
     => ( P4 @ ( produc968600491044949615rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) ) @ ( produc5709096180074326061rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_376_conjI__realizer,axiom,
    ! [P4: term_a_b > $o,P: term_a_b,Q4: term_a_b > $o,Q5: term_a_b] :
      ( ( P4 @ P )
     => ( ( Q4 @ Q5 )
       => ( ( P4 @ ( produc5709096180074326061rm_a_b @ ( produc7020197800436672577rm_a_b @ P @ Q5 ) ) )
          & ( Q4 @ ( produc968600491044949615rm_a_b @ ( produc7020197800436672577rm_a_b @ P @ Q5 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_377_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P4: term_a_b > term_a_b > $o,X3: term_a_b,Y3: term_a_b,A: produc357393685978478089rm_a_b] :
      ( ( P4 @ X3 @ Y3 )
     => ( ( A
          = ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) )
       => ( P4 @ ( produc5709096180074326061rm_a_b @ A ) @ ( produc968600491044949615rm_a_b @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_378_eq__snd__iff,axiom,
    ! [B: term_a_b,P: produc357393685978478089rm_a_b] :
      ( ( B
        = ( produc968600491044949615rm_a_b @ P ) )
      = ( ? [A3: term_a_b] :
            ( P
            = ( produc7020197800436672577rm_a_b @ A3 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_379_sndI,axiom,
    ! [X3: produc357393685978478089rm_a_b,Y3: term_a_b,Z2: term_a_b] :
      ( ( X3
        = ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) )
     => ( ( produc968600491044949615rm_a_b @ X3 )
        = Z2 ) ) ).

% sndI
thf(fact_380_eq__fst__iff,axiom,
    ! [A: term_a_b,P: produc357393685978478089rm_a_b] :
      ( ( A
        = ( produc5709096180074326061rm_a_b @ P ) )
      = ( ? [B4: term_a_b] :
            ( P
            = ( produc7020197800436672577rm_a_b @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_381_fstI,axiom,
    ! [X3: produc357393685978478089rm_a_b,Y3: term_a_b,Z2: term_a_b] :
      ( ( X3
        = ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) )
     => ( ( produc5709096180074326061rm_a_b @ X3 )
        = Y3 ) ) ).

% fstI
thf(fact_382_UNIV__coset,axiom,
    ( top_top_set_term_a_b
    = ( coset_term_a_b @ nil_term_a_b ) ) ).

% UNIV_coset
thf(fact_383_UNIV__coset,axiom,
    ( top_top_set_nat
    = ( coset_nat @ nil_nat ) ) ).

% UNIV_coset
thf(fact_384_image2__eqI,axiom,
    ! [B: term_a_b,F2: list_nat > term_a_b,X3: list_nat,C: term_a_b,G: list_nat > term_a_b,A5: set_list_nat] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member_list_nat2 @ X3 @ A5 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( bNF_Gr89567240463286756rm_a_b @ A5 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_385_image2__eqI,axiom,
    ! [B: term_a_b,F2: produc357393685978478089rm_a_b > term_a_b,X3: produc357393685978478089rm_a_b,C: term_a_b,G: produc357393685978478089rm_a_b > term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member5869715511025134514rm_a_b @ X3 @ A5 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( bNF_Gr4099196499516361305rm_a_b @ A5 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_386_image2__eqI,axiom,
    ! [B: term_a_b,F2: nat > term_a_b,X3: nat,C: term_a_b,G: nat > term_a_b,A5: set_nat] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member_nat2 @ X3 @ A5 )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( bNF_Gr3221002532544744404rm_a_b @ A5 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_387_in__lex__prod,axiom,
    ! [A: term_a_b,B: term_a_b,A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) ) @ ( lex_pr6163557265797435481rm_a_b @ R @ S ) )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_388_Nil__lenlex__iff1,axiom,
    ! [Ns: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ nil_term_a_b @ Ns ) @ ( lenlex_term_a_b @ R ) )
      = ( Ns != nil_term_a_b ) ) ).

% Nil_lenlex_iff1
thf(fact_389_Nil__lenlex__iff1,axiom,
    ! [Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ns ) @ ( lenlex_nat @ R ) )
      = ( Ns != nil_nat ) ) ).

% Nil_lenlex_iff1
thf(fact_390_mem__case__prodE,axiom,
    ! [Z2: list_nat,C: term_a_b > term_a_b > set_list_nat,P: produc357393685978478089rm_a_b] :
      ( ( member_list_nat2 @ Z2 @ ( produc7523824612178879596st_nat @ C @ P ) )
     => ~ ! [X: term_a_b,Y: term_a_b] :
            ( ( P
              = ( produc7020197800436672577rm_a_b @ X @ Y ) )
           => ~ ( member_list_nat2 @ Z2 @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_391_mem__case__prodE,axiom,
    ! [Z2: produc357393685978478089rm_a_b,C: term_a_b > term_a_b > set_Pr4386577575007340137rm_a_b,P: produc357393685978478089rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ Z2 @ ( produc782875466020721299rm_a_b @ C @ P ) )
     => ~ ! [X: term_a_b,Y: term_a_b] :
            ( ( P
              = ( produc7020197800436672577rm_a_b @ X @ Y ) )
           => ~ ( member5869715511025134514rm_a_b @ Z2 @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_392_mem__case__prodE,axiom,
    ! [Z2: nat,C: term_a_b > term_a_b > set_nat,P: produc357393685978478089rm_a_b] :
      ( ( member_nat2 @ Z2 @ ( produc6068407349128148444et_nat @ C @ P ) )
     => ~ ! [X: term_a_b,Y: term_a_b] :
            ( ( P
              = ( produc7020197800436672577rm_a_b @ X @ Y ) )
           => ~ ( member_nat2 @ Z2 @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_393_lenlex__irreflexive,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b] :
      ( ! [X: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R )
     => ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Xs ) @ ( lenlex_term_a_b @ R ) ) ) ).

% lenlex_irreflexive
thf(fact_394_Nil__lenlex__iff2,axiom,
    ! [Ns: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ns @ nil_term_a_b ) @ ( lenlex_term_a_b @ R ) ) ).

% Nil_lenlex_iff2
thf(fact_395_Nil__lenlex__iff2,axiom,
    ! [Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ns @ nil_nat ) @ ( lenlex_nat @ R ) ) ).

% Nil_lenlex_iff2
thf(fact_396_Product__Type_OCollect__case__prodD,axiom,
    ! [X3: produc357393685978478089rm_a_b,A5: term_a_b > term_a_b > $o] :
      ( ( member5869715511025134514rm_a_b @ X3 @ ( collec99177395925924084rm_a_b @ ( produc6251846659952913346_a_b_o @ A5 ) ) )
     => ( A5 @ ( produc5709096180074326061rm_a_b @ X3 ) @ ( produc968600491044949615rm_a_b @ X3 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_397_relcomp3__transI,axiom,
    ! [B5: set_Pr4386577575007340137rm_a_b,T: term_a_b,U: term_a_b,A5: set_Pr4386577575007340137rm_a_b,S: term_a_b,V: term_a_b] :
      ( ( trans_on_term_a_b @ top_top_set_term_a_b @ B5 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ B5 @ ( relcom370159955682700863rm_a_b @ A5 @ B5 ) ) )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ B5 )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ V ) @ B5 )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ V ) @ ( relcom370159955682700863rm_a_b @ B5 @ ( relcom370159955682700863rm_a_b @ A5 @ B5 ) ) ) ) ) ) ) ).

% relcomp3_transI
thf(fact_398_subsetI,axiom,
    ! [A5: set_list_nat,B5: set_list_nat] :
      ( ! [X: list_nat] :
          ( ( member_list_nat2 @ X @ A5 )
         => ( member_list_nat2 @ X @ B5 ) )
     => ( ord_le6045566169113846134st_nat @ A5 @ B5 ) ) ).

% subsetI
thf(fact_399_subsetI,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ! [X: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X @ A5 )
         => ( member5869715511025134514rm_a_b @ X @ B5 ) )
     => ( ord_le118470702582115849rm_a_b @ A5 @ B5 ) ) ).

% subsetI
thf(fact_400_subsetI,axiom,
    ! [A5: set_nat,B5: set_nat] :
      ( ! [X: nat] :
          ( ( member_nat2 @ X @ A5 )
         => ( member_nat2 @ X @ B5 ) )
     => ( ord_less_eq_set_nat @ A5 @ B5 ) ) ).

% subsetI
thf(fact_401_in__mono,axiom,
    ! [A5: set_list_nat,B5: set_list_nat,X3: list_nat] :
      ( ( ord_le6045566169113846134st_nat @ A5 @ B5 )
     => ( ( member_list_nat2 @ X3 @ A5 )
       => ( member_list_nat2 @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_402_in__mono,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,X3: produc357393685978478089rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ A5 @ B5 )
     => ( ( member5869715511025134514rm_a_b @ X3 @ A5 )
       => ( member5869715511025134514rm_a_b @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_403_in__mono,axiom,
    ! [A5: set_nat,B5: set_nat,X3: nat] :
      ( ( ord_less_eq_set_nat @ A5 @ B5 )
     => ( ( member_nat2 @ X3 @ A5 )
       => ( member_nat2 @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_404_subsetD,axiom,
    ! [A5: set_list_nat,B5: set_list_nat,C: list_nat] :
      ( ( ord_le6045566169113846134st_nat @ A5 @ B5 )
     => ( ( member_list_nat2 @ C @ A5 )
       => ( member_list_nat2 @ C @ B5 ) ) ) ).

% subsetD
thf(fact_405_subsetD,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ A5 @ B5 )
     => ( ( member5869715511025134514rm_a_b @ C @ A5 )
       => ( member5869715511025134514rm_a_b @ C @ B5 ) ) ) ).

% subsetD
thf(fact_406_subsetD,axiom,
    ! [A5: set_nat,B5: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A5 @ B5 )
     => ( ( member_nat2 @ C @ A5 )
       => ( member_nat2 @ C @ B5 ) ) ) ).

% subsetD
thf(fact_407_subset__eq,axiom,
    ( ord_le6045566169113846134st_nat
    = ( ^ [A6: set_list_nat,B6: set_list_nat] :
        ! [X4: list_nat] :
          ( ( member_list_nat2 @ X4 @ A6 )
         => ( member_list_nat2 @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_408_subset__eq,axiom,
    ( ord_le118470702582115849rm_a_b
    = ( ^ [A6: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
        ! [X4: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X4 @ A6 )
         => ( member5869715511025134514rm_a_b @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_409_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [X4: nat] :
          ( ( member_nat2 @ X4 @ A6 )
         => ( member_nat2 @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_410_subset__iff,axiom,
    ( ord_le6045566169113846134st_nat
    = ( ^ [A6: set_list_nat,B6: set_list_nat] :
        ! [T3: list_nat] :
          ( ( member_list_nat2 @ T3 @ A6 )
         => ( member_list_nat2 @ T3 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_411_subset__iff,axiom,
    ( ord_le118470702582115849rm_a_b
    = ( ^ [A6: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
        ! [T3: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ T3 @ A6 )
         => ( member5869715511025134514rm_a_b @ T3 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_412_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [T3: nat] :
          ( ( member_nat2 @ T3 @ A6 )
         => ( member_nat2 @ T3 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_413_trans__O__subset,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( trans_on_term_a_b @ top_top_set_term_a_b @ R )
     => ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R @ R ) @ R ) ) ).

% trans_O_subset
thf(fact_414_trans__O__iff,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b] :
      ( ( trans_on_term_a_b @ top_top_set_term_a_b @ A5 )
      = ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ A5 @ A5 ) @ A5 ) ) ).

% trans_O_iff
thf(fact_415_Sup_OSUP__image,axiom,
    ! [Sup: set_nat > nat,G: nat > nat,F2: nat > nat,A5: set_nat] :
      ( ( Sup @ ( image_nat_nat @ G @ ( image_nat_nat @ F2 @ A5 ) ) )
      = ( Sup @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F2 ) @ A5 ) ) ) ).

% Sup.SUP_image
thf(fact_416_Inf_OINF__image,axiom,
    ! [Inf: set_nat > nat,G: nat > nat,F2: nat > nat,A5: set_nat] :
      ( ( Inf @ ( image_nat_nat @ G @ ( image_nat_nat @ F2 @ A5 ) ) )
      = ( Inf @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F2 ) @ A5 ) ) ) ).

% Inf.INF_image
thf(fact_417_image__eq__imp__comp,axiom,
    ! [F2: nat > nat,A5: set_nat,G: nat > nat,B5: set_nat,H: nat > nat] :
      ( ( ( image_nat_nat @ F2 @ A5 )
        = ( image_nat_nat @ G @ B5 ) )
     => ( ( image_nat_nat @ ( comp_nat_nat_nat @ H @ F2 ) @ A5 )
        = ( image_nat_nat @ ( comp_nat_nat_nat @ H @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_418_image__comp,axiom,
    ! [F2: nat > nat,G: nat > nat,R: set_nat] :
      ( ( image_nat_nat @ F2 @ ( image_nat_nat @ G @ R ) )
      = ( image_nat_nat @ ( comp_nat_nat_nat @ F2 @ G ) @ R ) ) ).

% image_comp
thf(fact_419_trans__onD,axiom,
    ! [A5: set_list_nat,R: set_Pr3451248702717554689st_nat,X3: list_nat,Y3: list_nat,Z2: list_nat] :
      ( ( trans_on_list_nat @ A5 @ R )
     => ( ( member_list_nat2 @ X3 @ A5 )
       => ( ( member_list_nat2 @ Y3 @ A5 )
         => ( ( member_list_nat2 @ Z2 @ A5 )
           => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ R )
             => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y3 @ Z2 ) @ R )
               => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Z2 ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_420_trans__onD,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b] :
      ( ( trans_5404704112166290345rm_a_b @ A5 @ R )
     => ( ( member5869715511025134514rm_a_b @ X3 @ A5 )
       => ( ( member5869715511025134514rm_a_b @ Y3 @ A5 )
         => ( ( member5869715511025134514rm_a_b @ Z2 @ A5 )
           => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ R )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Z2 ) @ R )
               => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Z2 ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_421_trans__onD,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X3: nat,Y3: nat,Z2: nat] :
      ( ( trans_on_nat @ A5 @ R )
     => ( ( member_nat2 @ X3 @ A5 )
       => ( ( member_nat2 @ Y3 @ A5 )
         => ( ( member_nat2 @ Z2 @ A5 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ R )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z2 ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_422_trans__onD,axiom,
    ! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b,Z2: term_a_b] :
      ( ( trans_on_term_a_b @ A5 @ R )
     => ( ( member_term_a_b2 @ X3 @ A5 )
       => ( ( member_term_a_b2 @ Y3 @ A5 )
         => ( ( member_term_a_b2 @ Z2 @ A5 )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ R )
               => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_423_trans__onI,axiom,
    ! [A5: set_list_nat,R: set_Pr3451248702717554689st_nat] :
      ( ! [X: list_nat,Y: list_nat,Z: list_nat] :
          ( ( member_list_nat2 @ X @ A5 )
         => ( ( member_list_nat2 @ Y @ A5 )
           => ( ( member_list_nat2 @ Z @ A5 )
             => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R )
               => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y @ Z ) @ R )
                 => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Z ) @ R ) ) ) ) ) )
     => ( trans_on_list_nat @ A5 @ R ) ) ).

% trans_onI
thf(fact_424_trans__onI,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X @ A5 )
         => ( ( member5869715511025134514rm_a_b @ Y @ A5 )
           => ( ( member5869715511025134514rm_a_b @ Z @ A5 )
             => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R )
               => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z ) @ R )
                 => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z ) @ R ) ) ) ) ) )
     => ( trans_5404704112166290345rm_a_b @ A5 @ R ) ) ).

% trans_onI
thf(fact_425_trans__onI,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ! [X: nat,Y: nat,Z: nat] :
          ( ( member_nat2 @ X @ A5 )
         => ( ( member_nat2 @ Y @ A5 )
           => ( ( member_nat2 @ Z @ A5 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
               => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ R )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R ) ) ) ) ) )
     => ( trans_on_nat @ A5 @ R ) ) ).

% trans_onI
thf(fact_426_trans__onI,axiom,
    ! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ! [X: term_a_b,Y: term_a_b,Z: term_a_b] :
          ( ( member_term_a_b2 @ X @ A5 )
         => ( ( member_term_a_b2 @ Y @ A5 )
           => ( ( member_term_a_b2 @ Z @ A5 )
             => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
               => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ R )
                 => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ R ) ) ) ) ) )
     => ( trans_on_term_a_b @ A5 @ R ) ) ).

% trans_onI
thf(fact_427_trans__on__def,axiom,
    ( trans_on_term_a_b
    = ( ^ [A6: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
        ! [X4: term_a_b] :
          ( ( member_term_a_b2 @ X4 @ A6 )
         => ! [Y4: term_a_b] :
              ( ( member_term_a_b2 @ Y4 @ A6 )
             => ! [Z3: term_a_b] :
                  ( ( member_term_a_b2 @ Z3 @ A6 )
                 => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y4 ) @ R3 )
                   => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Z3 ) @ R3 )
                     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Z3 ) @ R3 ) ) ) ) ) ) ) ) ).

% trans_on_def
thf(fact_428_subrelI,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ! [X: term_a_b,Y: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ S ) )
     => ( ord_le118470702582115849rm_a_b @ R @ S ) ) ).

% subrelI
thf(fact_429_lists_ONil,axiom,
    ! [A5: set_term_a_b] : ( member_list_term_a_b2 @ nil_term_a_b @ ( lists_term_a_b @ A5 ) ) ).

% lists.Nil
thf(fact_430_lists_ONil,axiom,
    ! [A5: set_nat] : ( member_list_nat2 @ nil_nat @ ( lists_nat @ A5 ) ) ).

% lists.Nil
thf(fact_431_subset__image__iff,axiom,
    ! [B5: set_nat,F2: nat > nat,A5: set_nat] :
      ( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F2 @ A5 ) )
      = ( ? [AA: set_nat] :
            ( ( ord_less_eq_set_nat @ AA @ A5 )
            & ( B5
              = ( image_nat_nat @ F2 @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_432_image__subset__iff,axiom,
    ! [F2: nat > nat,A5: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A5 ) @ B5 )
      = ( ! [X4: nat] :
            ( ( member_nat2 @ X4 @ A5 )
           => ( member_nat2 @ ( F2 @ X4 ) @ B5 ) ) ) ) ).

% image_subset_iff
thf(fact_433_subset__imageE,axiom,
    ! [B5: set_nat,F2: nat > nat,A5: set_nat] :
      ( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F2 @ A5 ) )
     => ~ ! [C5: set_nat] :
            ( ( ord_less_eq_set_nat @ C5 @ A5 )
           => ( B5
             != ( image_nat_nat @ F2 @ C5 ) ) ) ) ).

% subset_imageE
thf(fact_434_image__subsetI,axiom,
    ! [A5: set_list_nat,F2: list_nat > list_nat,B5: set_list_nat] :
      ( ! [X: list_nat] :
          ( ( member_list_nat2 @ X @ A5 )
         => ( member_list_nat2 @ ( F2 @ X ) @ B5 ) )
     => ( ord_le6045566169113846134st_nat @ ( image_7976474329151083847st_nat @ F2 @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_435_image__subsetI,axiom,
    ! [A5: set_list_nat,F2: list_nat > produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ! [X: list_nat] :
          ( ( member_list_nat2 @ X @ A5 )
         => ( member5869715511025134514rm_a_b @ ( F2 @ X ) @ B5 ) )
     => ( ord_le118470702582115849rm_a_b @ ( image_6727437281251480162rm_a_b @ F2 @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_436_image__subsetI,axiom,
    ! [A5: set_list_nat,F2: list_nat > nat,B5: set_nat] :
      ( ! [X: list_nat] :
          ( ( member_list_nat2 @ X @ A5 )
         => ( member_nat2 @ ( F2 @ X ) @ B5 ) )
     => ( ord_less_eq_set_nat @ ( image_list_nat_nat @ F2 @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_437_image__subsetI,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,F2: produc357393685978478089rm_a_b > list_nat,B5: set_list_nat] :
      ( ! [X: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X @ A5 )
         => ( member_list_nat2 @ ( F2 @ X ) @ B5 ) )
     => ( ord_le6045566169113846134st_nat @ ( image_8768541597965001892st_nat @ F2 @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_438_image__subsetI,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,F2: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ! [X: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X @ A5 )
         => ( member5869715511025134514rm_a_b @ ( F2 @ X ) @ B5 ) )
     => ( ord_le118470702582115849rm_a_b @ ( image_3126328586747194693rm_a_b @ F2 @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_439_image__subsetI,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,F2: produc357393685978478089rm_a_b > nat,B5: set_nat] :
      ( ! [X: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X @ A5 )
         => ( member_nat2 @ ( F2 @ X ) @ B5 ) )
     => ( ord_less_eq_set_nat @ ( image_5783597275426814868_b_nat @ F2 @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_440_image__subsetI,axiom,
    ! [A5: set_nat,F2: nat > list_nat,B5: set_list_nat] :
      ( ! [X: nat] :
          ( ( member_nat2 @ X @ A5 )
         => ( member_list_nat2 @ ( F2 @ X ) @ B5 ) )
     => ( ord_le6045566169113846134st_nat @ ( image_nat_list_nat @ F2 @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_441_image__subsetI,axiom,
    ! [A5: set_nat,F2: nat > produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ! [X: nat] :
          ( ( member_nat2 @ X @ A5 )
         => ( member5869715511025134514rm_a_b @ ( F2 @ X ) @ B5 ) )
     => ( ord_le118470702582115849rm_a_b @ ( image_7842151682427799154rm_a_b @ F2 @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_442_image__subsetI,axiom,
    ! [A5: set_nat,F2: nat > nat,B5: set_nat] :
      ( ! [X: nat] :
          ( ( member_nat2 @ X @ A5 )
         => ( member_nat2 @ ( F2 @ X ) @ B5 ) )
     => ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_443_image__mono,axiom,
    ! [A5: set_nat,B5: set_nat,F2: nat > nat] :
      ( ( ord_less_eq_set_nat @ A5 @ B5 )
     => ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A5 ) @ ( image_nat_nat @ F2 @ B5 ) ) ) ).

% image_mono
thf(fact_444_O__mono1,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R2 @ R4 )
     => ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ R2 ) @ ( relcom370159955682700863rm_a_b @ S4 @ R4 ) ) ) ).

% O_mono1
thf(fact_445_O__mono2,axiom,
    ! [R2: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b,T4: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R2 @ R4 )
     => ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R2 @ T4 ) @ ( relcom370159955682700863rm_a_b @ R4 @ T4 ) ) ) ).

% O_mono2
thf(fact_446_relcomp__mono,axiom,
    ! [R5: set_Pr4386577575007340137rm_a_b,R: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R5 @ R )
     => ( ( ord_le118470702582115849rm_a_b @ S5 @ S )
       => ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R5 @ S5 ) @ ( relcom370159955682700863rm_a_b @ R @ S ) ) ) ) ).

% relcomp_mono
thf(fact_447_lexord__linear,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,X3: list_term_a_b,Y3: list_term_a_b] :
      ( ! [A4: term_a_b,B3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ R )
          | ( A4 = B3 )
          | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A4 ) @ R ) )
     => ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X3 @ Y3 ) @ ( lexord_term_a_b @ R ) )
        | ( X3 = Y3 )
        | ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Y3 @ X3 ) @ ( lexord_term_a_b @ R ) ) ) ) ).

% lexord_linear
thf(fact_448_lexord__irreflexive,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b] :
      ( ! [X: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R )
     => ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Xs ) @ ( lexord_term_a_b @ R ) ) ) ).

% lexord_irreflexive
thf(fact_449_lexord__Nil__right,axiom,
    ! [X3: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X3 @ nil_term_a_b ) @ ( lexord_term_a_b @ R ) ) ).

% lexord_Nil_right
thf(fact_450_lexord__Nil__right,axiom,
    ! [X3: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ nil_nat ) @ ( lexord_nat @ R ) ) ).

% lexord_Nil_right
thf(fact_451_Nil2__notin__lex,axiom,
    ! [Xs: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ nil_term_a_b ) @ ( lex_term_a_b @ R ) ) ).

% Nil2_notin_lex
thf(fact_452_Nil2__notin__lex,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( lex_nat @ R ) ) ).

% Nil2_notin_lex
thf(fact_453_Nil__notin__lex,axiom,
    ! [Ys: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ nil_term_a_b @ Ys ) @ ( lex_term_a_b @ R ) ) ).

% Nil_notin_lex
thf(fact_454_Nil__notin__lex,axiom,
    ! [Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys ) @ ( lex_nat @ R ) ) ).

% Nil_notin_lex
thf(fact_455_comp__surj,axiom,
    ! [F2: nat > nat,G: nat > nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ( image_nat_nat @ G @ top_top_set_nat )
          = top_top_set_nat )
       => ( ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F2 ) @ top_top_set_nat )
          = top_top_set_nat ) ) ) ).

% comp_surj
thf(fact_456_transI,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ! [X: term_a_b,Y: term_a_b,Z: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ R )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ R ) ) )
     => ( trans_on_term_a_b @ top_top_set_term_a_b @ R ) ) ).

% transI
thf(fact_457_transE,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b,Z2: term_a_b] :
      ( ( trans_on_term_a_b @ top_top_set_term_a_b @ R )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ R )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ R ) ) ) ) ).

% transE
thf(fact_458_transD,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b,Z2: term_a_b] :
      ( ( trans_on_term_a_b @ top_top_set_term_a_b @ R )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ R )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ R ) ) ) ) ).

% transD
thf(fact_459_range__subsetD,axiom,
    ! [F2: nat > nat,B5: set_nat,I: nat] :
      ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ top_top_set_nat ) @ B5 )
     => ( member_nat2 @ ( F2 @ I ) @ B5 ) ) ).

% range_subsetD
thf(fact_460_fun_Oset__map,axiom,
    ! [F2: nat > nat,V: nat > nat] :
      ( ( image_nat_nat @ ( comp_nat_nat_nat @ F2 @ V ) @ top_top_set_nat )
      = ( image_nat_nat @ F2 @ ( image_nat_nat @ V @ top_top_set_nat ) ) ) ).

% fun.set_map
thf(fact_461_fun_Omap__ident__strong,axiom,
    ! [T: nat > nat,F2: nat > nat] :
      ( ! [Z: nat] :
          ( ( member_nat2 @ Z @ ( image_nat_nat @ T @ top_top_set_nat ) )
         => ( ( F2 @ Z )
            = Z ) )
     => ( ( comp_nat_nat_nat @ F2 @ T )
        = T ) ) ).

% fun.map_ident_strong
thf(fact_462_all__subset__image,axiom,
    ! [F2: nat > nat,A5: set_nat,P4: set_nat > $o] :
      ( ( ! [B6: set_nat] :
            ( ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F2 @ A5 ) )
           => ( P4 @ B6 ) ) )
      = ( ! [B6: set_nat] :
            ( ( ord_less_eq_set_nat @ B6 @ A5 )
           => ( P4 @ ( image_nat_nat @ F2 @ B6 ) ) ) ) ) ).

% all_subset_image
thf(fact_463_apfst__conv,axiom,
    ! [F2: term_a_b > term_a_b,X3: term_a_b,Y3: term_a_b] :
      ( ( produc4217944709291808634rm_a_b @ F2 @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) )
      = ( produc7020197800436672577rm_a_b @ ( F2 @ X3 ) @ Y3 ) ) ).

% apfst_conv
thf(fact_464_apsnd__conv,axiom,
    ! [F2: term_a_b > term_a_b,X3: term_a_b,Y3: term_a_b] :
      ( ( produc6963505556121627836rm_a_b @ F2 @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) )
      = ( produc7020197800436672577rm_a_b @ X3 @ ( F2 @ Y3 ) ) ) ).

% apsnd_conv
thf(fact_465_listrel__Nil2,axiom,
    ! [Xs: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ nil_term_a_b ) @ ( listre1194016999521814427rm_a_b @ R ) )
     => ( Xs = nil_term_a_b ) ) ).

% listrel_Nil2
thf(fact_466_listrel__Nil2,axiom,
    ! [Xs: list_nat,R: set_Pr4549835640365387557rm_a_b] :
      ( ( member7814915433699591566rm_a_b @ ( produc4336470666470712605rm_a_b @ Xs @ nil_term_a_b ) @ ( listrel_nat_term_a_b @ R ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil2
thf(fact_467_listrel__Nil2,axiom,
    ! [Xs: list_term_a_b,R: set_Pr6708357783813671845_b_nat] :
      ( ( member8889939629616814094st_nat @ ( produc5386604813764302749st_nat @ Xs @ nil_nat ) @ ( listrel_term_a_b_nat @ R ) )
     => ( Xs = nil_term_a_b ) ) ).

% listrel_Nil2
thf(fact_468_listrel__Nil2,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel_nat_nat @ R ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil2
thf(fact_469_listrel__Nil1,axiom,
    ! [Xs: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ nil_term_a_b @ Xs ) @ ( listre1194016999521814427rm_a_b @ R ) )
     => ( Xs = nil_term_a_b ) ) ).

% listrel_Nil1
thf(fact_470_listrel__Nil1,axiom,
    ! [Xs: list_nat,R: set_Pr6708357783813671845_b_nat] :
      ( ( member8889939629616814094st_nat @ ( produc5386604813764302749st_nat @ nil_term_a_b @ Xs ) @ ( listrel_term_a_b_nat @ R ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil1
thf(fact_471_listrel__Nil1,axiom,
    ! [Xs: list_term_a_b,R: set_Pr4549835640365387557rm_a_b] :
      ( ( member7814915433699591566rm_a_b @ ( produc4336470666470712605rm_a_b @ nil_nat @ Xs ) @ ( listrel_nat_term_a_b @ R ) )
     => ( Xs = nil_term_a_b ) ) ).

% listrel_Nil1
thf(fact_472_listrel__Nil1,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel_nat_nat @ R ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil1
thf(fact_473_listrel_ONil,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] : ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ nil_term_a_b @ nil_term_a_b ) @ ( listre1194016999521814427rm_a_b @ R ) ) ).

% listrel.Nil
thf(fact_474_listrel_ONil,axiom,
    ! [R: set_Pr6708357783813671845_b_nat] : ( member8889939629616814094st_nat @ ( produc5386604813764302749st_nat @ nil_term_a_b @ nil_nat ) @ ( listrel_term_a_b_nat @ R ) ) ).

% listrel.Nil
thf(fact_475_listrel_ONil,axiom,
    ! [R: set_Pr4549835640365387557rm_a_b] : ( member7814915433699591566rm_a_b @ ( produc4336470666470712605rm_a_b @ nil_nat @ nil_term_a_b ) @ ( listrel_nat_term_a_b @ R ) ) ).

% listrel.Nil
thf(fact_476_listrel_ONil,axiom,
    ! [R: set_Pr1261947904930325089at_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) @ ( listrel_nat_nat @ R ) ) ).

% listrel.Nil
thf(fact_477_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: produc357393685978478089rm_a_b,A5: set_term_a_b,F2: term_a_b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ Z2 @ ( collec99177395925924084rm_a_b @ ( produc6251846659952913346_a_b_o @ ( bNF_Gr4926885435746096120rm_a_b @ A5 @ F2 ) ) ) )
     => ( ( comp_t8072462138577781308rm_a_b @ F2 @ produc5709096180074326061rm_a_b @ Z2 )
        = ( produc968600491044949615rm_a_b @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_478_same__fstI,axiom,
    ! [P4: term_a_b > $o,X3: term_a_b,Y5: term_a_b,Y3: term_a_b,R2: term_a_b > set_Pr4386577575007340137rm_a_b] :
      ( ( P4 @ X3 )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ Y3 ) @ ( R2 @ X3 ) )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y5 ) @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) ) @ ( same_f843731131676418198rm_a_b @ P4 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_479_case__prod__Pair,axiom,
    ( ( produc2089504961790700083rm_a_b @ produc7020197800436672577rm_a_b )
    = id_Pro3718135800935220218rm_a_b ) ).

% case_prod_Pair
thf(fact_480_image__id,axiom,
    ( ( image_nat_nat @ id_nat )
    = id_set_nat ) ).

% image_id
thf(fact_481_Inf_OINF__id__eq,axiom,
    ! [Inf: set_nat > nat,A5: set_nat] :
      ( ( Inf @ ( image_nat_nat @ id_nat @ A5 ) )
      = ( Inf @ A5 ) ) ).

% Inf.INF_id_eq
thf(fact_482_Sup_OSUP__id__eq,axiom,
    ! [Sup: set_nat > nat,A5: set_nat] :
      ( ( Sup @ ( image_nat_nat @ id_nat @ A5 ) )
      = ( Sup @ A5 ) ) ).

% Sup.SUP_id_eq
thf(fact_483_fun_Orel__refl__strong,axiom,
    ! [X3: nat > nat,Ra: nat > nat > $o] :
      ( ! [Z: nat] :
          ( ( member_nat2 @ Z @ ( image_nat_nat @ X3 @ top_top_set_nat ) )
         => ( Ra @ Z @ Z ) )
     => ( bNF_re5653821019739307937at_nat
        @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
        @ Ra
        @ X3
        @ X3 ) ) ).

% fun.rel_refl_strong
thf(fact_484_fun_Orel__mono__strong,axiom,
    ! [R2: list_nat > nat > $o,X3: nat > list_nat,Y3: nat > nat,Ra: list_nat > nat > $o] :
      ( ( bNF_re3851489860074285745at_nat
        @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
        @ R2
        @ X3
        @ Y3 )
     => ( ! [Z: list_nat,Yb: nat] :
            ( ( member_list_nat2 @ Z @ ( image_nat_list_nat @ X3 @ top_top_set_nat ) )
           => ( ( member_nat2 @ Yb @ ( image_nat_nat @ Y3 @ top_top_set_nat ) )
             => ( ( R2 @ Z @ Yb )
               => ( Ra @ Z @ Yb ) ) ) )
       => ( bNF_re3851489860074285745at_nat
          @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
          @ Ra
          @ X3
          @ Y3 ) ) ) ).

% fun.rel_mono_strong
thf(fact_485_fun_Orel__mono__strong,axiom,
    ! [R2: produc357393685978478089rm_a_b > nat > $o,X3: nat > produc357393685978478089rm_a_b,Y3: nat > nat,Ra: produc357393685978478089rm_a_b > nat > $o] :
      ( ( bNF_re917291077122903834_b_nat
        @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
        @ R2
        @ X3
        @ Y3 )
     => ( ! [Z: produc357393685978478089rm_a_b,Yb: nat] :
            ( ( member5869715511025134514rm_a_b @ Z @ ( image_7842151682427799154rm_a_b @ X3 @ top_top_set_nat ) )
           => ( ( member_nat2 @ Yb @ ( image_nat_nat @ Y3 @ top_top_set_nat ) )
             => ( ( R2 @ Z @ Yb )
               => ( Ra @ Z @ Yb ) ) ) )
       => ( bNF_re917291077122903834_b_nat
          @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
          @ Ra
          @ X3
          @ Y3 ) ) ) ).

% fun.rel_mono_strong
thf(fact_486_fun_Orel__mono__strong,axiom,
    ! [R2: nat > list_nat > $o,X3: nat > nat,Y3: nat > list_nat,Ra: nat > list_nat > $o] :
      ( ( bNF_re1612532812912000561st_nat
        @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
        @ R2
        @ X3
        @ Y3 )
     => ( ! [Z: nat,Yb: list_nat] :
            ( ( member_nat2 @ Z @ ( image_nat_nat @ X3 @ top_top_set_nat ) )
           => ( ( member_list_nat2 @ Yb @ ( image_nat_list_nat @ Y3 @ top_top_set_nat ) )
             => ( ( R2 @ Z @ Yb )
               => ( Ra @ Z @ Yb ) ) ) )
       => ( bNF_re1612532812912000561st_nat
          @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
          @ Ra
          @ X3
          @ Y3 ) ) ) ).

% fun.rel_mono_strong
thf(fact_487_fun_Orel__mono__strong,axiom,
    ! [R2: nat > produc357393685978478089rm_a_b > $o,X3: nat > nat,Y3: nat > produc357393685978478089rm_a_b,Ra: nat > produc357393685978478089rm_a_b > $o] :
      ( ( bNF_re2975845484123888120rm_a_b
        @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
        @ R2
        @ X3
        @ Y3 )
     => ( ! [Z: nat,Yb: produc357393685978478089rm_a_b] :
            ( ( member_nat2 @ Z @ ( image_nat_nat @ X3 @ top_top_set_nat ) )
           => ( ( member5869715511025134514rm_a_b @ Yb @ ( image_7842151682427799154rm_a_b @ Y3 @ top_top_set_nat ) )
             => ( ( R2 @ Z @ Yb )
               => ( Ra @ Z @ Yb ) ) ) )
       => ( bNF_re2975845484123888120rm_a_b
          @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
          @ Ra
          @ X3
          @ Y3 ) ) ) ).

% fun.rel_mono_strong
thf(fact_488_fun_Orel__mono__strong,axiom,
    ! [R2: nat > nat > $o,X3: nat > nat,Y3: nat > nat,Ra: nat > nat > $o] :
      ( ( bNF_re5653821019739307937at_nat
        @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
        @ R2
        @ X3
        @ Y3 )
     => ( ! [Z: nat,Yb: nat] :
            ( ( member_nat2 @ Z @ ( image_nat_nat @ X3 @ top_top_set_nat ) )
           => ( ( member_nat2 @ Yb @ ( image_nat_nat @ Y3 @ top_top_set_nat ) )
             => ( ( R2 @ Z @ Yb )
               => ( Ra @ Z @ Yb ) ) ) )
       => ( bNF_re5653821019739307937at_nat
          @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
          @ Ra
          @ X3
          @ Y3 ) ) ) ).

% fun.rel_mono_strong
thf(fact_489_fun_Orel__cong,axiom,
    ! [X3: nat > list_nat,Ya: nat > list_nat,Y3: nat > nat,Xa: nat > nat,R2: list_nat > nat > $o,Ra: list_nat > nat > $o] :
      ( ( X3 = Ya )
     => ( ( Y3 = Xa )
       => ( ! [Z: list_nat,Yb: nat] :
              ( ( member_list_nat2 @ Z @ ( image_nat_list_nat @ Ya @ top_top_set_nat ) )
             => ( ( member_nat2 @ Yb @ ( image_nat_nat @ Xa @ top_top_set_nat ) )
               => ( ( R2 @ Z @ Yb )
                  = ( Ra @ Z @ Yb ) ) ) )
         => ( ( bNF_re3851489860074285745at_nat
              @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
              @ R2
              @ X3
              @ Y3 )
            = ( bNF_re3851489860074285745at_nat
              @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
              @ Ra
              @ Ya
              @ Xa ) ) ) ) ) ).

% fun.rel_cong
thf(fact_490_fun_Orel__cong,axiom,
    ! [X3: nat > produc357393685978478089rm_a_b,Ya: nat > produc357393685978478089rm_a_b,Y3: nat > nat,Xa: nat > nat,R2: produc357393685978478089rm_a_b > nat > $o,Ra: produc357393685978478089rm_a_b > nat > $o] :
      ( ( X3 = Ya )
     => ( ( Y3 = Xa )
       => ( ! [Z: produc357393685978478089rm_a_b,Yb: nat] :
              ( ( member5869715511025134514rm_a_b @ Z @ ( image_7842151682427799154rm_a_b @ Ya @ top_top_set_nat ) )
             => ( ( member_nat2 @ Yb @ ( image_nat_nat @ Xa @ top_top_set_nat ) )
               => ( ( R2 @ Z @ Yb )
                  = ( Ra @ Z @ Yb ) ) ) )
         => ( ( bNF_re917291077122903834_b_nat
              @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
              @ R2
              @ X3
              @ Y3 )
            = ( bNF_re917291077122903834_b_nat
              @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
              @ Ra
              @ Ya
              @ Xa ) ) ) ) ) ).

% fun.rel_cong
thf(fact_491_fun_Orel__cong,axiom,
    ! [X3: nat > nat,Ya: nat > nat,Y3: nat > list_nat,Xa: nat > list_nat,R2: nat > list_nat > $o,Ra: nat > list_nat > $o] :
      ( ( X3 = Ya )
     => ( ( Y3 = Xa )
       => ( ! [Z: nat,Yb: list_nat] :
              ( ( member_nat2 @ Z @ ( image_nat_nat @ Ya @ top_top_set_nat ) )
             => ( ( member_list_nat2 @ Yb @ ( image_nat_list_nat @ Xa @ top_top_set_nat ) )
               => ( ( R2 @ Z @ Yb )
                  = ( Ra @ Z @ Yb ) ) ) )
         => ( ( bNF_re1612532812912000561st_nat
              @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
              @ R2
              @ X3
              @ Y3 )
            = ( bNF_re1612532812912000561st_nat
              @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
              @ Ra
              @ Ya
              @ Xa ) ) ) ) ) ).

% fun.rel_cong
thf(fact_492_fun_Orel__cong,axiom,
    ! [X3: nat > nat,Ya: nat > nat,Y3: nat > produc357393685978478089rm_a_b,Xa: nat > produc357393685978478089rm_a_b,R2: nat > produc357393685978478089rm_a_b > $o,Ra: nat > produc357393685978478089rm_a_b > $o] :
      ( ( X3 = Ya )
     => ( ( Y3 = Xa )
       => ( ! [Z: nat,Yb: produc357393685978478089rm_a_b] :
              ( ( member_nat2 @ Z @ ( image_nat_nat @ Ya @ top_top_set_nat ) )
             => ( ( member5869715511025134514rm_a_b @ Yb @ ( image_7842151682427799154rm_a_b @ Xa @ top_top_set_nat ) )
               => ( ( R2 @ Z @ Yb )
                  = ( Ra @ Z @ Yb ) ) ) )
         => ( ( bNF_re2975845484123888120rm_a_b
              @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
              @ R2
              @ X3
              @ Y3 )
            = ( bNF_re2975845484123888120rm_a_b
              @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
              @ Ra
              @ Ya
              @ Xa ) ) ) ) ) ).

% fun.rel_cong
thf(fact_493_fun_Orel__cong,axiom,
    ! [X3: nat > nat,Ya: nat > nat,Y3: nat > nat,Xa: nat > nat,R2: nat > nat > $o,Ra: nat > nat > $o] :
      ( ( X3 = Ya )
     => ( ( Y3 = Xa )
       => ( ! [Z: nat,Yb: nat] :
              ( ( member_nat2 @ Z @ ( image_nat_nat @ Ya @ top_top_set_nat ) )
             => ( ( member_nat2 @ Yb @ ( image_nat_nat @ Xa @ top_top_set_nat ) )
               => ( ( R2 @ Z @ Yb )
                  = ( Ra @ Z @ Yb ) ) ) )
         => ( ( bNF_re5653821019739307937at_nat
              @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
              @ R2
              @ X3
              @ Y3 )
            = ( bNF_re5653821019739307937at_nat
              @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
              @ Ra
              @ Ya
              @ Xa ) ) ) ) ) ).

% fun.rel_cong
thf(fact_494_surj__id,axiom,
    ( ( image_nat_nat @ id_nat @ top_top_set_nat )
    = top_top_set_nat ) ).

% surj_id
thf(fact_495_Collect__case__prod__Grp__in,axiom,
    ! [Z2: produc357393685978478089rm_a_b,A5: set_term_a_b,F2: term_a_b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ Z2 @ ( collec99177395925924084rm_a_b @ ( produc6251846659952913346_a_b_o @ ( bNF_Gr4926885435746096120rm_a_b @ A5 @ F2 ) ) ) )
     => ( member_term_a_b2 @ ( produc5709096180074326061rm_a_b @ Z2 ) @ A5 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_496_DEADID_Oin__rel,axiom,
    ( ( ^ [Y6: list_nat,Z4: list_nat] : ( Y6 = Z4 ) )
    = ( ^ [A3: list_nat,B4: list_nat] :
        ? [Z3: list_nat] :
          ( ( member_list_nat2 @ Z3 @ top_top_set_list_nat )
          & ( ( id_list_nat @ Z3 )
            = A3 )
          & ( ( id_list_nat @ Z3 )
            = B4 ) ) ) ) ).

% DEADID.in_rel
thf(fact_497_DEADID_Oin__rel,axiom,
    ( ( ^ [Y6: produc357393685978478089rm_a_b,Z4: produc357393685978478089rm_a_b] : ( Y6 = Z4 ) )
    = ( ^ [A3: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b] :
        ? [Z3: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ Z3 @ top_to1314267278846557113rm_a_b )
          & ( ( id_Pro3718135800935220218rm_a_b @ Z3 )
            = A3 )
          & ( ( id_Pro3718135800935220218rm_a_b @ Z3 )
            = B4 ) ) ) ) ).

% DEADID.in_rel
thf(fact_498_DEADID_Oin__rel,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    = ( ^ [A3: nat,B4: nat] :
        ? [Z3: nat] :
          ( ( member_nat2 @ Z3 @ top_top_set_nat )
          & ( ( id_nat @ Z3 )
            = A3 )
          & ( ( id_nat @ Z3 )
            = B4 ) ) ) ) ).

% DEADID.in_rel
thf(fact_499_listrel_Osimps,axiom,
    ! [A1: list_term_a_b,A22: list_nat,R: set_Pr6708357783813671845_b_nat] :
      ( ( member8889939629616814094st_nat @ ( produc5386604813764302749st_nat @ A1 @ A22 ) @ ( listrel_term_a_b_nat @ R ) )
      = ( ( ( A1 = nil_term_a_b )
          & ( A22 = nil_nat ) )
        | ? [X4: term_a_b,Y4: nat,Xs2: list_term_a_b,Ys2: list_nat] :
            ( ( A1
              = ( cons_term_a_b @ X4 @ Xs2 ) )
            & ( A22
              = ( cons_nat @ Y4 @ Ys2 ) )
            & ( member9000184225777080558_b_nat @ ( produc2478027105211390077_b_nat @ X4 @ Y4 ) @ R )
            & ( member8889939629616814094st_nat @ ( produc5386604813764302749st_nat @ Xs2 @ Ys2 ) @ ( listrel_term_a_b_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_500_listrel_Osimps,axiom,
    ! [A1: list_nat,A22: list_term_a_b,R: set_Pr4549835640365387557rm_a_b] :
      ( ( member7814915433699591566rm_a_b @ ( produc4336470666470712605rm_a_b @ A1 @ A22 ) @ ( listrel_nat_term_a_b @ R ) )
      = ( ( ( A1 = nil_nat )
          & ( A22 = nil_term_a_b ) )
        | ? [X4: nat,Y4: term_a_b,Xs2: list_nat,Ys2: list_term_a_b] :
            ( ( A1
              = ( cons_nat @ X4 @ Xs2 ) )
            & ( A22
              = ( cons_term_a_b @ Y4 @ Ys2 ) )
            & ( member7514656309480873070rm_a_b @ ( produc1516572978046417917rm_a_b @ X4 @ Y4 ) @ R )
            & ( member7814915433699591566rm_a_b @ ( produc4336470666470712605rm_a_b @ Xs2 @ Ys2 ) @ ( listrel_nat_term_a_b @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_501_listrel_Osimps,axiom,
    ! [A1: list_nat,A22: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R ) )
      = ( ( ( A1 = nil_nat )
          & ( A22 = nil_nat ) )
        | ? [X4: nat,Y4: nat,Xs2: list_nat,Ys2: list_nat] :
            ( ( A1
              = ( cons_nat @ X4 @ Xs2 ) )
            & ( A22
              = ( cons_nat @ Y4 @ Ys2 ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R )
            & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_502_listrel_Osimps,axiom,
    ! [A1: list_term_a_b,A22: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A1 @ A22 ) @ ( listre1194016999521814427rm_a_b @ R ) )
      = ( ( ( A1 = nil_term_a_b )
          & ( A22 = nil_term_a_b ) )
        | ? [X4: term_a_b,Y4: term_a_b,Xs2: list_term_a_b,Ys2: list_term_a_b] :
            ( ( A1
              = ( cons_term_a_b @ X4 @ Xs2 ) )
            & ( A22
              = ( cons_term_a_b @ Y4 @ Ys2 ) )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y4 ) @ R )
            & ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs2 @ Ys2 ) @ ( listre1194016999521814427rm_a_b @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_503_listrel_Ocases,axiom,
    ! [A1: list_term_a_b,A22: list_nat,R: set_Pr6708357783813671845_b_nat] :
      ( ( member8889939629616814094st_nat @ ( produc5386604813764302749st_nat @ A1 @ A22 ) @ ( listrel_term_a_b_nat @ R ) )
     => ( ( ( A1 = nil_term_a_b )
         => ( A22 != nil_nat ) )
       => ~ ! [X: term_a_b,Y: nat,Xs3: list_term_a_b] :
              ( ( A1
                = ( cons_term_a_b @ X @ Xs3 ) )
             => ! [Ys3: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y @ Ys3 ) )
                 => ( ( member9000184225777080558_b_nat @ ( produc2478027105211390077_b_nat @ X @ Y ) @ R )
                   => ~ ( member8889939629616814094st_nat @ ( produc5386604813764302749st_nat @ Xs3 @ Ys3 ) @ ( listrel_term_a_b_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_504_listrel_Ocases,axiom,
    ! [A1: list_nat,A22: list_term_a_b,R: set_Pr4549835640365387557rm_a_b] :
      ( ( member7814915433699591566rm_a_b @ ( produc4336470666470712605rm_a_b @ A1 @ A22 ) @ ( listrel_nat_term_a_b @ R ) )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_term_a_b ) )
       => ~ ! [X: nat,Y: term_a_b,Xs3: list_nat] :
              ( ( A1
                = ( cons_nat @ X @ Xs3 ) )
             => ! [Ys3: list_term_a_b] :
                  ( ( A22
                    = ( cons_term_a_b @ Y @ Ys3 ) )
                 => ( ( member7514656309480873070rm_a_b @ ( produc1516572978046417917rm_a_b @ X @ Y ) @ R )
                   => ~ ( member7814915433699591566rm_a_b @ ( produc4336470666470712605rm_a_b @ Xs3 @ Ys3 ) @ ( listrel_nat_term_a_b @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_505_listrel_Ocases,axiom,
    ! [A1: list_nat,A22: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R ) )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_nat ) )
       => ~ ! [X: nat,Y: nat,Xs3: list_nat] :
              ( ( A1
                = ( cons_nat @ X @ Xs3 ) )
             => ! [Ys3: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y @ Ys3 ) )
                 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
                   => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys3 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_506_listrel_Ocases,axiom,
    ! [A1: list_term_a_b,A22: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A1 @ A22 ) @ ( listre1194016999521814427rm_a_b @ R ) )
     => ( ( ( A1 = nil_term_a_b )
         => ( A22 != nil_term_a_b ) )
       => ~ ! [X: term_a_b,Y: term_a_b,Xs3: list_term_a_b] :
              ( ( A1
                = ( cons_term_a_b @ X @ Xs3 ) )
             => ! [Ys3: list_term_a_b] :
                  ( ( A22
                    = ( cons_term_a_b @ Y @ Ys3 ) )
                 => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
                   => ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs3 @ Ys3 ) @ ( listre1194016999521814427rm_a_b @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_507_lexord__Nil__left,axiom,
    ! [Y3: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ nil_term_a_b @ Y3 ) @ ( lexord_term_a_b @ R ) )
      = ( ? [A3: term_a_b,X4: list_term_a_b] :
            ( Y3
            = ( cons_term_a_b @ A3 @ X4 ) ) ) ) ).

% lexord_Nil_left
thf(fact_508_lexord__Nil__left,axiom,
    ! [Y3: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Y3 ) @ ( lexord_nat @ R ) )
      = ( ? [A3: nat,X4: list_nat] :
            ( Y3
            = ( cons_nat @ A3 @ X4 ) ) ) ) ).

% lexord_Nil_left
thf(fact_509_list_Oinject,axiom,
    ! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
      ( ( ( cons_nat @ X21 @ X22 )
        = ( cons_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_510_Cons__in__lists__iff,axiom,
    ! [X3: list_nat,Xs: list_list_nat,A5: set_list_nat] :
      ( ( member_list_list_nat @ ( cons_list_nat @ X3 @ Xs ) @ ( lists_list_nat @ A5 ) )
      = ( ( member_list_nat2 @ X3 @ A5 )
        & ( member_list_list_nat @ Xs @ ( lists_list_nat @ A5 ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_511_Cons__in__lists__iff,axiom,
    ! [X3: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member7646937072540562872rm_a_b @ ( cons_P3536305108106557631rm_a_b @ X3 @ Xs ) @ ( lists_3936190290964281341rm_a_b @ A5 ) )
      = ( ( member5869715511025134514rm_a_b @ X3 @ A5 )
        & ( member7646937072540562872rm_a_b @ Xs @ ( lists_3936190290964281341rm_a_b @ A5 ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_512_Cons__in__lists__iff,axiom,
    ! [X3: nat,Xs: list_nat,A5: set_nat] :
      ( ( member_list_nat2 @ ( cons_nat @ X3 @ Xs ) @ ( lists_nat @ A5 ) )
      = ( ( member_nat2 @ X3 @ A5 )
        & ( member_list_nat2 @ Xs @ ( lists_nat @ A5 ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_513_position__diff__Cons,axiom,
    ! [I: nat,Ps: list_nat,Qs: list_nat] :
      ( ( term_pos_diff_nat @ ( cons_nat @ I @ Ps ) @ ( cons_nat @ I @ Qs ) )
      = ( term_pos_diff_nat @ Ps @ Qs ) ) ).

% position_diff_Cons
thf(fact_514_lexord__cons__cons,axiom,
    ! [A: nat,X3: list_nat,B: nat,Y3: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ A @ X3 ) @ ( cons_nat @ B @ Y3 ) ) @ ( lexord_nat @ R ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
        | ( ( A = B )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ ( lexord_nat @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_515_lexord__cons__cons,axiom,
    ! [A: term_a_b,X3: list_term_a_b,B: term_a_b,Y3: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ A @ X3 ) @ ( cons_term_a_b @ B @ Y3 ) ) @ ( lexord_term_a_b @ R ) )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
        | ( ( A = B )
          & ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X3 @ Y3 ) @ ( lexord_term_a_b @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_516_not__Cons__self2,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( cons_nat @ X3 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_517_transpose_Ocases,axiom,
    ! [X3: list_list_term_a_b] :
      ( ( X3 != nil_list_term_a_b )
     => ( ! [Xss: list_list_term_a_b] :
            ( X3
           != ( cons_list_term_a_b @ nil_term_a_b @ Xss ) )
       => ~ ! [X: term_a_b,Xs3: list_term_a_b,Xss: list_list_term_a_b] :
              ( X3
             != ( cons_list_term_a_b @ ( cons_term_a_b @ X @ Xs3 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_518_transpose_Ocases,axiom,
    ! [X3: list_list_nat] :
      ( ( X3 != nil_list_nat )
     => ( ! [Xss: list_list_nat] :
            ( X3
           != ( cons_list_nat @ nil_nat @ Xss ) )
       => ~ ! [X: nat,Xs3: list_nat,Xss: list_list_nat] :
              ( X3
             != ( cons_list_nat @ ( cons_nat @ X @ Xs3 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_519_list_Odistinct_I1_J,axiom,
    ! [X21: term_a_b,X22: list_term_a_b] :
      ( nil_term_a_b
     != ( cons_term_a_b @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_520_list_Odistinct_I1_J,axiom,
    ! [X21: nat,X22: list_nat] :
      ( nil_nat
     != ( cons_nat @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_521_list_OdiscI,axiom,
    ! [List: list_term_a_b,X21: term_a_b,X22: list_term_a_b] :
      ( ( List
        = ( cons_term_a_b @ X21 @ X22 ) )
     => ( List != nil_term_a_b ) ) ).

% list.discI
thf(fact_522_list_OdiscI,axiom,
    ! [List: list_nat,X21: nat,X22: list_nat] :
      ( ( List
        = ( cons_nat @ X21 @ X22 ) )
     => ( List != nil_nat ) ) ).

% list.discI
thf(fact_523_list_Oexhaust,axiom,
    ! [Y3: list_term_a_b] :
      ( ( Y3 != nil_term_a_b )
     => ~ ! [X212: term_a_b,X222: list_term_a_b] :
            ( Y3
           != ( cons_term_a_b @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_524_list_Oexhaust,axiom,
    ! [Y3: list_nat] :
      ( ( Y3 != nil_nat )
     => ~ ! [X212: nat,X222: list_nat] :
            ( Y3
           != ( cons_nat @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_525_List_Omin__list_Ocases,axiom,
    ! [X3: list_nat] :
      ( ! [X: nat,Xs3: list_nat] :
          ( X3
         != ( cons_nat @ X @ Xs3 ) )
     => ( X3 = nil_nat ) ) ).

% List.min_list.cases
thf(fact_526_remdups__adj_Ocases,axiom,
    ! [X3: list_term_a_b] :
      ( ( X3 != nil_term_a_b )
     => ( ! [X: term_a_b] :
            ( X3
           != ( cons_term_a_b @ X @ nil_term_a_b ) )
       => ~ ! [X: term_a_b,Y: term_a_b,Xs3: list_term_a_b] :
              ( X3
             != ( cons_term_a_b @ X @ ( cons_term_a_b @ Y @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_527_remdups__adj_Ocases,axiom,
    ! [X3: list_nat] :
      ( ( X3 != nil_nat )
     => ( ! [X: nat] :
            ( X3
           != ( cons_nat @ X @ nil_nat ) )
       => ~ ! [X: nat,Y: nat,Xs3: list_nat] :
              ( X3
             != ( cons_nat @ X @ ( cons_nat @ Y @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_528_neq__Nil__conv,axiom,
    ! [Xs: list_term_a_b] :
      ( ( Xs != nil_term_a_b )
      = ( ? [Y4: term_a_b,Ys2: list_term_a_b] :
            ( Xs
            = ( cons_term_a_b @ Y4 @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_529_neq__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
      = ( ? [Y4: nat,Ys2: list_nat] :
            ( Xs
            = ( cons_nat @ Y4 @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_530_list__induct2_H,axiom,
    ! [P4: list_term_a_b > list_term_a_b > $o,Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( P4 @ nil_term_a_b @ nil_term_a_b )
     => ( ! [X: term_a_b,Xs3: list_term_a_b] : ( P4 @ ( cons_term_a_b @ X @ Xs3 ) @ nil_term_a_b )
       => ( ! [Y: term_a_b,Ys3: list_term_a_b] : ( P4 @ nil_term_a_b @ ( cons_term_a_b @ Y @ Ys3 ) )
         => ( ! [X: term_a_b,Xs3: list_term_a_b,Y: term_a_b,Ys3: list_term_a_b] :
                ( ( P4 @ Xs3 @ Ys3 )
               => ( P4 @ ( cons_term_a_b @ X @ Xs3 ) @ ( cons_term_a_b @ Y @ Ys3 ) ) )
           => ( P4 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_531_list__induct2_H,axiom,
    ! [P4: list_term_a_b > list_nat > $o,Xs: list_term_a_b,Ys: list_nat] :
      ( ( P4 @ nil_term_a_b @ nil_nat )
     => ( ! [X: term_a_b,Xs3: list_term_a_b] : ( P4 @ ( cons_term_a_b @ X @ Xs3 ) @ nil_nat )
       => ( ! [Y: nat,Ys3: list_nat] : ( P4 @ nil_term_a_b @ ( cons_nat @ Y @ Ys3 ) )
         => ( ! [X: term_a_b,Xs3: list_term_a_b,Y: nat,Ys3: list_nat] :
                ( ( P4 @ Xs3 @ Ys3 )
               => ( P4 @ ( cons_term_a_b @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys3 ) ) )
           => ( P4 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_532_list__induct2_H,axiom,
    ! [P4: list_nat > list_term_a_b > $o,Xs: list_nat,Ys: list_term_a_b] :
      ( ( P4 @ nil_nat @ nil_term_a_b )
     => ( ! [X: nat,Xs3: list_nat] : ( P4 @ ( cons_nat @ X @ Xs3 ) @ nil_term_a_b )
       => ( ! [Y: term_a_b,Ys3: list_term_a_b] : ( P4 @ nil_nat @ ( cons_term_a_b @ Y @ Ys3 ) )
         => ( ! [X: nat,Xs3: list_nat,Y: term_a_b,Ys3: list_term_a_b] :
                ( ( P4 @ Xs3 @ Ys3 )
               => ( P4 @ ( cons_nat @ X @ Xs3 ) @ ( cons_term_a_b @ Y @ Ys3 ) ) )
           => ( P4 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_533_list__induct2_H,axiom,
    ! [P4: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
      ( ( P4 @ nil_nat @ nil_nat )
     => ( ! [X: nat,Xs3: list_nat] : ( P4 @ ( cons_nat @ X @ Xs3 ) @ nil_nat )
       => ( ! [Y: nat,Ys3: list_nat] : ( P4 @ nil_nat @ ( cons_nat @ Y @ Ys3 ) )
         => ( ! [X: nat,Xs3: list_nat,Y: nat,Ys3: list_nat] :
                ( ( P4 @ Xs3 @ Ys3 )
               => ( P4 @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys3 ) ) )
           => ( P4 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_534_list__nonempty__induct,axiom,
    ! [Xs: list_term_a_b,P4: list_term_a_b > $o] :
      ( ( Xs != nil_term_a_b )
     => ( ! [X: term_a_b] : ( P4 @ ( cons_term_a_b @ X @ nil_term_a_b ) )
       => ( ! [X: term_a_b,Xs3: list_term_a_b] :
              ( ( Xs3 != nil_term_a_b )
             => ( ( P4 @ Xs3 )
               => ( P4 @ ( cons_term_a_b @ X @ Xs3 ) ) ) )
         => ( P4 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_535_list__nonempty__induct,axiom,
    ! [Xs: list_nat,P4: list_nat > $o] :
      ( ( Xs != nil_nat )
     => ( ! [X: nat] : ( P4 @ ( cons_nat @ X @ nil_nat ) )
       => ( ! [X: nat,Xs3: list_nat] :
              ( ( Xs3 != nil_nat )
             => ( ( P4 @ Xs3 )
               => ( P4 @ ( cons_nat @ X @ Xs3 ) ) ) )
         => ( P4 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_536_successively_Ocases,axiom,
    ! [X3: produc8847469151730319409rm_a_b] :
      ( ! [P5: term_a_b > term_a_b > $o] :
          ( X3
         != ( produc5931190720334929579rm_a_b @ P5 @ nil_term_a_b ) )
     => ( ! [P5: term_a_b > term_a_b > $o,X: term_a_b] :
            ( X3
           != ( produc5931190720334929579rm_a_b @ P5 @ ( cons_term_a_b @ X @ nil_term_a_b ) ) )
       => ~ ! [P5: term_a_b > term_a_b > $o,X: term_a_b,Y: term_a_b,Xs3: list_term_a_b] :
              ( X3
             != ( produc5931190720334929579rm_a_b @ P5 @ ( cons_term_a_b @ X @ ( cons_term_a_b @ Y @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_537_successively_Ocases,axiom,
    ! [X3: produc254973753779126261st_nat] :
      ( ! [P5: nat > nat > $o] :
          ( X3
         != ( produc4727192421694094319st_nat @ P5 @ nil_nat ) )
     => ( ! [P5: nat > nat > $o,X: nat] :
            ( X3
           != ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X @ nil_nat ) ) )
       => ~ ! [P5: nat > nat > $o,X: nat,Y: nat,Xs3: list_nat] :
              ( X3
             != ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X @ ( cons_nat @ Y @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_538_lists_OCons,axiom,
    ! [A: list_nat,A5: set_list_nat,L: list_list_nat] :
      ( ( member_list_nat2 @ A @ A5 )
     => ( ( member_list_list_nat @ L @ ( lists_list_nat @ A5 ) )
       => ( member_list_list_nat @ ( cons_list_nat @ A @ L ) @ ( lists_list_nat @ A5 ) ) ) ) ).

% lists.Cons
thf(fact_539_lists_OCons,axiom,
    ! [A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,L: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ A @ A5 )
     => ( ( member7646937072540562872rm_a_b @ L @ ( lists_3936190290964281341rm_a_b @ A5 ) )
       => ( member7646937072540562872rm_a_b @ ( cons_P3536305108106557631rm_a_b @ A @ L ) @ ( lists_3936190290964281341rm_a_b @ A5 ) ) ) ) ).

% lists.Cons
thf(fact_540_lists_OCons,axiom,
    ! [A: nat,A5: set_nat,L: list_nat] :
      ( ( member_nat2 @ A @ A5 )
     => ( ( member_list_nat2 @ L @ ( lists_nat @ A5 ) )
       => ( member_list_nat2 @ ( cons_nat @ A @ L ) @ ( lists_nat @ A5 ) ) ) ) ).

% lists.Cons
thf(fact_541_listsE,axiom,
    ! [X3: list_nat,L: list_list_nat,A5: set_list_nat] :
      ( ( member_list_list_nat @ ( cons_list_nat @ X3 @ L ) @ ( lists_list_nat @ A5 ) )
     => ~ ( ( member_list_nat2 @ X3 @ A5 )
         => ~ ( member_list_list_nat @ L @ ( lists_list_nat @ A5 ) ) ) ) ).

% listsE
thf(fact_542_listsE,axiom,
    ! [X3: produc357393685978478089rm_a_b,L: list_P8875379029341186191rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member7646937072540562872rm_a_b @ ( cons_P3536305108106557631rm_a_b @ X3 @ L ) @ ( lists_3936190290964281341rm_a_b @ A5 ) )
     => ~ ( ( member5869715511025134514rm_a_b @ X3 @ A5 )
         => ~ ( member7646937072540562872rm_a_b @ L @ ( lists_3936190290964281341rm_a_b @ A5 ) ) ) ) ).

% listsE
thf(fact_543_listsE,axiom,
    ! [X3: nat,L: list_nat,A5: set_nat] :
      ( ( member_list_nat2 @ ( cons_nat @ X3 @ L ) @ ( lists_nat @ A5 ) )
     => ~ ( ( member_nat2 @ X3 @ A5 )
         => ~ ( member_list_nat2 @ L @ ( lists_nat @ A5 ) ) ) ) ).

% listsE
thf(fact_544_irrefl__onD,axiom,
    ! [A5: set_list_nat,R: set_Pr3451248702717554689st_nat,A: list_nat] :
      ( ( irrefl_on_list_nat @ A5 @ R )
     => ( ( member_list_nat2 @ A @ A5 )
       => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ A ) @ R ) ) ) ).

% irrefl_onD
thf(fact_545_irrefl__onD,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b] :
      ( ( irrefl6620156600162922239rm_a_b @ A5 @ R )
     => ( ( member5869715511025134514rm_a_b @ A @ A5 )
       => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ A ) @ R ) ) ) ).

% irrefl_onD
thf(fact_546_irrefl__onD,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( irrefl_on_nat @ A5 @ R )
     => ( ( member_nat2 @ A @ A5 )
       => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R ) ) ) ).

% irrefl_onD
thf(fact_547_irrefl__onD,axiom,
    ! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
      ( ( irrefl_on_term_a_b @ A5 @ R )
     => ( ( member_term_a_b2 @ A @ A5 )
       => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R ) ) ) ).

% irrefl_onD
thf(fact_548_irrefl__onI,axiom,
    ! [A5: set_list_nat,R: set_Pr3451248702717554689st_nat] :
      ( ! [A4: list_nat] :
          ( ( member_list_nat2 @ A4 @ A5 )
         => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A4 @ A4 ) @ R ) )
     => ( irrefl_on_list_nat @ A5 @ R ) ) ).

% irrefl_onI
thf(fact_549_irrefl__onI,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
      ( ! [A4: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ A4 @ A5 )
         => ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ A4 ) @ R ) )
     => ( irrefl6620156600162922239rm_a_b @ A5 @ R ) ) ).

% irrefl_onI
thf(fact_550_irrefl__onI,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ! [A4: nat] :
          ( ( member_nat2 @ A4 @ A5 )
         => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A4 ) @ R ) )
     => ( irrefl_on_nat @ A5 @ R ) ) ).

% irrefl_onI
thf(fact_551_irrefl__onI,axiom,
    ! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ! [A4: term_a_b] :
          ( ( member_term_a_b2 @ A4 @ A5 )
         => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ A4 ) @ R ) )
     => ( irrefl_on_term_a_b @ A5 @ R ) ) ).

% irrefl_onI
thf(fact_552_irrefl__on__def,axiom,
    ( irrefl_on_term_a_b
    = ( ^ [A6: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
        ! [X4: term_a_b] :
          ( ( member_term_a_b2 @ X4 @ A6 )
         => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ X4 ) @ R3 ) ) ) ) ).

% irrefl_on_def
thf(fact_553_par__Cons__iff,axiom,
    ! [I: nat,Ps: list_nat,J: nat,Qs: list_nat] :
      ( ( term_p5017330785391824242ar_nat @ ( cons_nat @ I @ Ps ) @ ( cons_nat @ J @ Qs ) )
      = ( ( I != J )
        | ( term_p5017330785391824242ar_nat @ Ps @ Qs ) ) ) ).

% par_Cons_iff
thf(fact_554_par__pos__prefix,axiom,
    ! [I: nat,P: list_nat,Q5: list_nat] :
      ( ( term_p5017330785391824242ar_nat @ ( cons_nat @ I @ P ) @ ( cons_nat @ I @ Q5 ) )
     => ( term_p5017330785391824242ar_nat @ P @ Q5 ) ) ).

% par_pos_prefix
thf(fact_555_null__rec_I1_J,axiom,
    ! [X3: nat,Xs: list_nat] :
      ~ ( null_nat @ ( cons_nat @ X3 @ Xs ) ) ).

% null_rec(1)
thf(fact_556_member__rec_I1_J,axiom,
    ! [X3: nat,Xs: list_nat,Y3: nat] :
      ( ( member_nat @ ( cons_nat @ X3 @ Xs ) @ Y3 )
      = ( ( X3 = Y3 )
        | ( member_nat @ Xs @ Y3 ) ) ) ).

% member_rec(1)
thf(fact_557_shuffles_Ocases,axiom,
    ! [X3: produc51424535725745577rm_a_b] :
      ( ! [Ys3: list_term_a_b] :
          ( X3
         != ( produc4885699992713594593rm_a_b @ nil_term_a_b @ Ys3 ) )
     => ( ! [Xs3: list_term_a_b] :
            ( X3
           != ( produc4885699992713594593rm_a_b @ Xs3 @ nil_term_a_b ) )
       => ~ ! [X: term_a_b,Xs3: list_term_a_b,Y: term_a_b,Ys3: list_term_a_b] :
              ( X3
             != ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X @ Xs3 ) @ ( cons_term_a_b @ Y @ Ys3 ) ) ) ) ) ).

% shuffles.cases
thf(fact_558_shuffles_Ocases,axiom,
    ! [X3: produc1828647624359046049st_nat] :
      ( ! [Ys3: list_nat] :
          ( X3
         != ( produc2694037385005941721st_nat @ nil_nat @ Ys3 ) )
     => ( ! [Xs3: list_nat] :
            ( X3
           != ( produc2694037385005941721st_nat @ Xs3 @ nil_nat ) )
       => ~ ! [X: nat,Xs3: list_nat,Y: nat,Ys3: list_nat] :
              ( X3
             != ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys3 ) ) ) ) ) ).

% shuffles.cases
thf(fact_559_remove__prefix_Ocases,axiom,
    ! [X3: produc51424535725745577rm_a_b] :
      ( ! [X: term_a_b,Xs3: list_term_a_b,Y: term_a_b,Ys3: list_term_a_b] :
          ( X3
         != ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X @ Xs3 ) @ ( cons_term_a_b @ Y @ Ys3 ) ) )
     => ( ! [Ys3: list_term_a_b] :
            ( X3
           != ( produc4885699992713594593rm_a_b @ nil_term_a_b @ Ys3 ) )
       => ~ ! [V2: term_a_b,Va: list_term_a_b] :
              ( X3
             != ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ V2 @ Va ) @ nil_term_a_b ) ) ) ) ).

% remove_prefix.cases
thf(fact_560_remove__prefix_Ocases,axiom,
    ! [X3: produc1828647624359046049st_nat] :
      ( ! [X: nat,Xs3: list_nat,Y: nat,Ys3: list_nat] :
          ( X3
         != ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys3 ) ) )
     => ( ! [Ys3: list_nat] :
            ( X3
           != ( produc2694037385005941721st_nat @ nil_nat @ Ys3 ) )
       => ~ ! [V2: nat,Va: list_nat] :
              ( X3
             != ( produc2694037385005941721st_nat @ ( cons_nat @ V2 @ Va ) @ nil_nat ) ) ) ) ).

% remove_prefix.cases
thf(fact_561_lists_Osimps,axiom,
    ! [A: list_list_nat,A5: set_list_nat] :
      ( ( member_list_list_nat @ A @ ( lists_list_nat @ A5 ) )
      = ( ( A = nil_list_nat )
        | ? [A3: list_nat,L2: list_list_nat] :
            ( ( A
              = ( cons_list_nat @ A3 @ L2 ) )
            & ( member_list_nat2 @ A3 @ A5 )
            & ( member_list_list_nat @ L2 @ ( lists_list_nat @ A5 ) ) ) ) ) ).

% lists.simps
thf(fact_562_lists_Osimps,axiom,
    ! [A: list_P8875379029341186191rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member7646937072540562872rm_a_b @ A @ ( lists_3936190290964281341rm_a_b @ A5 ) )
      = ( ( A = nil_Pr6942174756412032271rm_a_b )
        | ? [A3: produc357393685978478089rm_a_b,L2: list_P8875379029341186191rm_a_b] :
            ( ( A
              = ( cons_P3536305108106557631rm_a_b @ A3 @ L2 ) )
            & ( member5869715511025134514rm_a_b @ A3 @ A5 )
            & ( member7646937072540562872rm_a_b @ L2 @ ( lists_3936190290964281341rm_a_b @ A5 ) ) ) ) ) ).

% lists.simps
thf(fact_563_lists_Osimps,axiom,
    ! [A: list_term_a_b,A5: set_term_a_b] :
      ( ( member_list_term_a_b2 @ A @ ( lists_term_a_b @ A5 ) )
      = ( ( A = nil_term_a_b )
        | ? [A3: term_a_b,L2: list_term_a_b] :
            ( ( A
              = ( cons_term_a_b @ A3 @ L2 ) )
            & ( member_term_a_b2 @ A3 @ A5 )
            & ( member_list_term_a_b2 @ L2 @ ( lists_term_a_b @ A5 ) ) ) ) ) ).

% lists.simps
thf(fact_564_lists_Osimps,axiom,
    ! [A: list_nat,A5: set_nat] :
      ( ( member_list_nat2 @ A @ ( lists_nat @ A5 ) )
      = ( ( A = nil_nat )
        | ? [A3: nat,L2: list_nat] :
            ( ( A
              = ( cons_nat @ A3 @ L2 ) )
            & ( member_nat2 @ A3 @ A5 )
            & ( member_list_nat2 @ L2 @ ( lists_nat @ A5 ) ) ) ) ) ).

% lists.simps
thf(fact_565_lists_Ocases,axiom,
    ! [A: list_list_nat,A5: set_list_nat] :
      ( ( member_list_list_nat @ A @ ( lists_list_nat @ A5 ) )
     => ( ( A != nil_list_nat )
       => ~ ! [A4: list_nat,L3: list_list_nat] :
              ( ( A
                = ( cons_list_nat @ A4 @ L3 ) )
             => ( ( member_list_nat2 @ A4 @ A5 )
               => ~ ( member_list_list_nat @ L3 @ ( lists_list_nat @ A5 ) ) ) ) ) ) ).

% lists.cases
thf(fact_566_lists_Ocases,axiom,
    ! [A: list_P8875379029341186191rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member7646937072540562872rm_a_b @ A @ ( lists_3936190290964281341rm_a_b @ A5 ) )
     => ( ( A != nil_Pr6942174756412032271rm_a_b )
       => ~ ! [A4: produc357393685978478089rm_a_b,L3: list_P8875379029341186191rm_a_b] :
              ( ( A
                = ( cons_P3536305108106557631rm_a_b @ A4 @ L3 ) )
             => ( ( member5869715511025134514rm_a_b @ A4 @ A5 )
               => ~ ( member7646937072540562872rm_a_b @ L3 @ ( lists_3936190290964281341rm_a_b @ A5 ) ) ) ) ) ) ).

% lists.cases
thf(fact_567_lists_Ocases,axiom,
    ! [A: list_term_a_b,A5: set_term_a_b] :
      ( ( member_list_term_a_b2 @ A @ ( lists_term_a_b @ A5 ) )
     => ( ( A != nil_term_a_b )
       => ~ ! [A4: term_a_b,L3: list_term_a_b] :
              ( ( A
                = ( cons_term_a_b @ A4 @ L3 ) )
             => ( ( member_term_a_b2 @ A4 @ A5 )
               => ~ ( member_list_term_a_b2 @ L3 @ ( lists_term_a_b @ A5 ) ) ) ) ) ) ).

% lists.cases
thf(fact_568_lists_Ocases,axiom,
    ! [A: list_nat,A5: set_nat] :
      ( ( member_list_nat2 @ A @ ( lists_nat @ A5 ) )
     => ( ( A != nil_nat )
       => ~ ! [A4: nat,L3: list_nat] :
              ( ( A
                = ( cons_nat @ A4 @ L3 ) )
             => ( ( member_nat2 @ A4 @ A5 )
               => ~ ( member_list_nat2 @ L3 @ ( lists_nat @ A5 ) ) ) ) ) ) ).

% lists.cases
thf(fact_569_measures__lesseq,axiom,
    ! [F2: term_a_b > nat,X3: term_a_b,Y3: term_a_b,Fs: list_term_a_b_nat] :
      ( ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( measures_term_a_b @ Fs ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( measures_term_a_b @ ( cons_term_a_b_nat @ F2 @ Fs ) ) ) ) ) ).

% measures_lesseq
thf(fact_570_irreflI,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ! [A4: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ A4 ) @ R )
     => ( irrefl_on_term_a_b @ top_top_set_term_a_b @ R ) ) ).

% irreflI
thf(fact_571_irreflD,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b] :
      ( ( irrefl_on_term_a_b @ top_top_set_term_a_b @ R )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ R ) ) ).

% irreflD
thf(fact_572_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y3: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel_nat_nat @ R ) )
     => ~ ! [X: nat,Xs3: list_nat] :
            ( ( Xs
              = ( cons_nat @ X @ Xs3 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_573_listrel__Cons2,axiom,
    ! [Xs: list_term_a_b,Y3: term_a_b,Ys: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ ( cons_term_a_b @ Y3 @ Ys ) ) @ ( listre1194016999521814427rm_a_b @ R ) )
     => ~ ! [X: term_a_b,Xs3: list_term_a_b] :
            ( ( Xs
              = ( cons_term_a_b @ X @ Xs3 ) )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R )
             => ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs3 @ Ys ) @ ( listre1194016999521814427rm_a_b @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_574_listrel__Cons1,axiom,
    ! [Y3: nat,Ys: list_nat,Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y3 @ Ys ) @ Xs ) @ ( listrel_nat_nat @ R ) )
     => ~ ! [Y: nat,Ys3: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y @ Ys3 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Y ) @ R )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Ys3 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_575_listrel__Cons1,axiom,
    ! [Y3: term_a_b,Ys: list_term_a_b,Xs: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ Y3 @ Ys ) @ Xs ) @ ( listre1194016999521814427rm_a_b @ R ) )
     => ~ ! [Y: term_a_b,Ys3: list_term_a_b] :
            ( ( Xs
              = ( cons_term_a_b @ Y @ Ys3 ) )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Y ) @ R )
             => ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ys @ Ys3 ) @ ( listre1194016999521814427rm_a_b @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_576_listrel_OCons,axiom,
    ! [X3: nat,Y3: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel_nat_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_577_listrel_OCons,axiom,
    ! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
     => ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listre1194016999521814427rm_a_b @ R ) )
       => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X3 @ Xs ) @ ( cons_term_a_b @ Y3 @ Ys ) ) @ ( listre1194016999521814427rm_a_b @ R ) ) ) ) ).

% listrel.Cons
thf(fact_578_filter2_Ocases,axiom,
    ! [X3: produc8215350257458196082rm_a_b] :
      ( ! [P5: term_a_b > term_a_b > $o,Uu: list_term_a_b] :
          ( X3
         != ( produc1278856658843369954rm_a_b @ P5 @ ( produc4885699992713594593rm_a_b @ nil_term_a_b @ Uu ) ) )
     => ( ! [P5: term_a_b > term_a_b > $o,V2: term_a_b,Va: list_term_a_b] :
            ( X3
           != ( produc1278856658843369954rm_a_b @ P5 @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ V2 @ Va ) @ nil_term_a_b ) ) )
       => ~ ! [P5: term_a_b > term_a_b > $o,A4: term_a_b,As: list_term_a_b,B3: term_a_b,Bs: list_term_a_b] :
              ( X3
             != ( produc1278856658843369954rm_a_b @ P5 @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ A4 @ As ) @ ( cons_term_a_b @ B3 @ Bs ) ) ) ) ) ) ).

% filter2.cases
thf(fact_579_filter2_Ocases,axiom,
    ! [X3: produc6817702157447538794st_nat] :
      ( ! [P5: term_a_b > nat > $o,Uu: list_nat] :
          ( X3
         != ( produc7813177934582050522st_nat @ P5 @ ( produc5386604813764302749st_nat @ nil_term_a_b @ Uu ) ) )
     => ( ! [P5: term_a_b > nat > $o,V2: term_a_b,Va: list_term_a_b] :
            ( X3
           != ( produc7813177934582050522st_nat @ P5 @ ( produc5386604813764302749st_nat @ ( cons_term_a_b @ V2 @ Va ) @ nil_nat ) ) )
       => ~ ! [P5: term_a_b > nat > $o,A4: term_a_b,As: list_term_a_b,B3: nat,Bs: list_nat] :
              ( X3
             != ( produc7813177934582050522st_nat @ P5 @ ( produc5386604813764302749st_nat @ ( cons_term_a_b @ A4 @ As ) @ ( cons_nat @ B3 @ Bs ) ) ) ) ) ) ).

% filter2.cases
thf(fact_580_filter2_Ocases,axiom,
    ! [X3: produc8734217775477389290rm_a_b] :
      ( ! [P5: nat > term_a_b > $o,Uu: list_term_a_b] :
          ( X3
         != ( produc2750092164434624090rm_a_b @ P5 @ ( produc4336470666470712605rm_a_b @ nil_nat @ Uu ) ) )
     => ( ! [P5: nat > term_a_b > $o,V2: nat,Va: list_nat] :
            ( X3
           != ( produc2750092164434624090rm_a_b @ P5 @ ( produc4336470666470712605rm_a_b @ ( cons_nat @ V2 @ Va ) @ nil_term_a_b ) ) )
       => ~ ! [P5: nat > term_a_b > $o,A4: nat,As: list_nat,B3: term_a_b,Bs: list_term_a_b] :
              ( X3
             != ( produc2750092164434624090rm_a_b @ P5 @ ( produc4336470666470712605rm_a_b @ ( cons_nat @ A4 @ As ) @ ( cons_term_a_b @ B3 @ Bs ) ) ) ) ) ) ).

% filter2.cases
thf(fact_581_filter2_Ocases,axiom,
    ! [X3: produc4787317212837456354st_nat] :
      ( ! [P5: nat > nat > $o,Uu: list_nat] :
          ( X3
         != ( produc3127733452865184594st_nat @ P5 @ ( produc2694037385005941721st_nat @ nil_nat @ Uu ) ) )
     => ( ! [P5: nat > nat > $o,V2: nat,Va: list_nat] :
            ( X3
           != ( produc3127733452865184594st_nat @ P5 @ ( produc2694037385005941721st_nat @ ( cons_nat @ V2 @ Va ) @ nil_nat ) ) )
       => ~ ! [P5: nat > nat > $o,A4: nat,As: list_nat,B3: nat,Bs: list_nat] :
              ( X3
             != ( produc3127733452865184594st_nat @ P5 @ ( produc2694037385005941721st_nat @ ( cons_nat @ A4 @ As ) @ ( cons_nat @ B3 @ Bs ) ) ) ) ) ) ).

% filter2.cases
thf(fact_582_list__inter_Ocases,axiom,
    ! [X3: produc51424535725745577rm_a_b] :
      ( ! [Bs: list_term_a_b] :
          ( X3
         != ( produc4885699992713594593rm_a_b @ nil_term_a_b @ Bs ) )
     => ~ ! [A4: term_a_b,As: list_term_a_b,Bs: list_term_a_b] :
            ( X3
           != ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ A4 @ As ) @ Bs ) ) ) ).

% list_inter.cases
thf(fact_583_list__inter_Ocases,axiom,
    ! [X3: produc1828647624359046049st_nat] :
      ( ! [Bs: list_nat] :
          ( X3
         != ( produc2694037385005941721st_nat @ nil_nat @ Bs ) )
     => ~ ! [A4: nat,As: list_nat,Bs: list_nat] :
            ( X3
           != ( produc2694037385005941721st_nat @ ( cons_nat @ A4 @ As ) @ Bs ) ) ) ).

% list_inter.cases
thf(fact_584_union__list__sorted_Ocases,axiom,
    ! [X3: produc1828647624359046049st_nat] :
      ( ! [X: nat,Xs3: list_nat,Y: nat,Ys3: list_nat] :
          ( X3
         != ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys3 ) ) )
     => ( ! [Ys3: list_nat] :
            ( X3
           != ( produc2694037385005941721st_nat @ nil_nat @ Ys3 ) )
       => ~ ! [V2: nat,Va: list_nat] :
              ( X3
             != ( produc2694037385005941721st_nat @ ( cons_nat @ V2 @ Va ) @ nil_nat ) ) ) ) ).

% union_list_sorted.cases
thf(fact_585_subtract__list__sorted_Ocases,axiom,
    ! [X3: produc1828647624359046049st_nat] :
      ( ! [X: nat,Xs3: list_nat,Y: nat,Ys3: list_nat] :
          ( X3
         != ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys3 ) ) )
     => ( ! [Ys3: list_nat] :
            ( X3
           != ( produc2694037385005941721st_nat @ nil_nat @ Ys3 ) )
       => ~ ! [V2: nat,Va: list_nat] :
              ( X3
             != ( produc2694037385005941721st_nat @ ( cons_nat @ V2 @ Va ) @ nil_nat ) ) ) ) ).

% subtract_list_sorted.cases
thf(fact_586_mem__idx_Ocases,axiom,
    ! [X3: produc5674375379913900953rm_a_b] :
      ( ! [Uu: term_a_b] :
          ( X3
         != ( produc967488655884398673rm_a_b @ Uu @ nil_term_a_b ) )
     => ~ ! [X: term_a_b,A4: term_a_b,As: list_term_a_b] :
            ( X3
           != ( produc967488655884398673rm_a_b @ X @ ( cons_term_a_b @ A4 @ As ) ) ) ) ).

% mem_idx.cases
thf(fact_587_mem__idx_Ocases,axiom,
    ! [X3: produc4575160907756185873st_nat] :
      ( ! [Uu: nat] :
          ( X3
         != ( produc8282810413953273033st_nat @ Uu @ nil_nat ) )
     => ~ ! [X: nat,A4: nat,As: list_nat] :
            ( X3
           != ( produc8282810413953273033st_nat @ X @ ( cons_nat @ A4 @ As ) ) ) ) ).

% mem_idx.cases
thf(fact_588_list__4__cases,axiom,
    ! [Xs: list_term_a_b] :
      ( ( Xs != nil_term_a_b )
     => ( ! [X: term_a_b] :
            ( Xs
           != ( cons_term_a_b @ X @ nil_term_a_b ) )
       => ( ! [X: term_a_b,Y: term_a_b] :
              ( Xs
             != ( cons_term_a_b @ X @ ( cons_term_a_b @ Y @ nil_term_a_b ) ) )
         => ~ ! [X: term_a_b,Y: term_a_b,Z: term_a_b,Zs: list_term_a_b] :
                ( Xs
               != ( cons_term_a_b @ X @ ( cons_term_a_b @ Y @ ( cons_term_a_b @ Z @ Zs ) ) ) ) ) ) ) ).

% list_4_cases
thf(fact_589_list__4__cases,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ( ! [X: nat] :
            ( Xs
           != ( cons_nat @ X @ nil_nat ) )
       => ( ! [X: nat,Y: nat] :
              ( Xs
             != ( cons_nat @ X @ ( cons_nat @ Y @ nil_nat ) ) )
         => ~ ! [X: nat,Y: nat,Z: nat,Zs: list_nat] :
                ( Xs
               != ( cons_nat @ X @ ( cons_nat @ Y @ ( cons_nat @ Z @ Zs ) ) ) ) ) ) ) ).

% list_4_cases
thf(fact_590_span_Ocases,axiom,
    ! [X3: produc5260046862718186894rm_a_b] :
      ( ! [P5: term_a_b > $o,X: term_a_b,Xs3: list_term_a_b] :
          ( X3
         != ( produc2136336272451886600rm_a_b @ P5 @ ( cons_term_a_b @ X @ Xs3 ) ) )
     => ~ ! [Uu: term_a_b > $o] :
            ( X3
           != ( produc2136336272451886600rm_a_b @ Uu @ nil_term_a_b ) ) ) ).

% span.cases
thf(fact_591_span_Ocases,axiom,
    ! [X3: produc4226810134323546766st_nat] :
      ( ! [P5: nat > $o,X: nat,Xs3: list_nat] :
          ( X3
         != ( produc8587622027977423880st_nat @ P5 @ ( cons_nat @ X @ Xs3 ) ) )
     => ~ ! [Uu: nat > $o] :
            ( X3
           != ( produc8587622027977423880st_nat @ Uu @ nil_nat ) ) ) ).

% span.cases
thf(fact_592_distinct__eq_Ocases,axiom,
    ! [X3: produc8847469151730319409rm_a_b] :
      ( ! [Uu: term_a_b > term_a_b > $o] :
          ( X3
         != ( produc5931190720334929579rm_a_b @ Uu @ nil_term_a_b ) )
     => ~ ! [Eq: term_a_b > term_a_b > $o,X: term_a_b,Xs3: list_term_a_b] :
            ( X3
           != ( produc5931190720334929579rm_a_b @ Eq @ ( cons_term_a_b @ X @ Xs3 ) ) ) ) ).

% distinct_eq.cases
thf(fact_593_distinct__eq_Ocases,axiom,
    ! [X3: produc254973753779126261st_nat] :
      ( ! [Uu: nat > nat > $o] :
          ( X3
         != ( produc4727192421694094319st_nat @ Uu @ nil_nat ) )
     => ~ ! [Eq: nat > nat > $o,X: nat,Xs3: list_nat] :
            ( X3
           != ( produc4727192421694094319st_nat @ Eq @ ( cons_nat @ X @ Xs3 ) ) ) ) ).

% distinct_eq.cases
thf(fact_594_Missing__List_Omin__list_Ocases,axiom,
    ! [X3: list_nat] :
      ( ! [X: nat] :
          ( X3
         != ( cons_nat @ X @ nil_nat ) )
     => ( ! [X: nat,V2: nat,Va: list_nat] :
            ( X3
           != ( cons_nat @ X @ ( cons_nat @ V2 @ Va ) ) )
       => ( X3 = nil_nat ) ) ) ).

% Missing_List.min_list.cases
thf(fact_595_list__3__cases,axiom,
    ! [Xs: list_term_a_b] :
      ( ( Xs != nil_term_a_b )
     => ( ! [X: term_a_b] :
            ( Xs
           != ( cons_term_a_b @ X @ nil_term_a_b ) )
       => ~ ! [X: term_a_b,Y: term_a_b,Ys3: list_term_a_b] :
              ( Xs
             != ( cons_term_a_b @ X @ ( cons_term_a_b @ Y @ Ys3 ) ) ) ) ) ).

% list_3_cases
thf(fact_596_list__3__cases,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ( ! [X: nat] :
            ( Xs
           != ( cons_nat @ X @ nil_nat ) )
       => ~ ! [X: nat,Y: nat,Ys3: list_nat] :
              ( Xs
             != ( cons_nat @ X @ ( cons_nat @ Y @ Ys3 ) ) ) ) ) ).

% list_3_cases
thf(fact_597_add__elem__list__lists_Osimps_I1_J,axiom,
    ! [X3: term_a_b] :
      ( ( basic_1593220722155286443rm_a_b @ X3 @ nil_term_a_b )
      = ( cons_list_term_a_b @ ( cons_term_a_b @ X3 @ nil_term_a_b ) @ nil_list_term_a_b ) ) ).

% add_elem_list_lists.simps(1)
thf(fact_598_add__elem__list__lists_Osimps_I1_J,axiom,
    ! [X3: nat] :
      ( ( basic_4874698711677410535ts_nat @ X3 @ nil_nat )
      = ( cons_list_nat @ ( cons_nat @ X3 @ nil_nat ) @ nil_list_nat ) ) ).

% add_elem_list_lists.simps(1)
thf(fact_599_product__lists_Osimps_I1_J,axiom,
    ( ( produc17669015410068453rm_a_b @ nil_list_term_a_b )
    = ( cons_list_term_a_b @ nil_term_a_b @ nil_list_term_a_b ) ) ).

% product_lists.simps(1)
thf(fact_600_product__lists_Osimps_I1_J,axiom,
    ( ( product_lists_nat @ nil_list_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% product_lists.simps(1)
thf(fact_601_concat__lists_Osimps_I1_J,axiom,
    ( ( missin8060632096978918206rm_a_b @ nil_list_term_a_b )
    = ( cons_list_term_a_b @ nil_term_a_b @ nil_list_term_a_b ) ) ).

% concat_lists.simps(1)
thf(fact_602_concat__lists_Osimps_I1_J,axiom,
    ( ( missin4567272213201432058ts_nat @ nil_list_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% concat_lists.simps(1)
thf(fact_603_in__measures_I2_J,axiom,
    ! [X3: term_a_b,Y3: term_a_b,F2: term_a_b > nat,Fs: list_term_a_b_nat] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( measures_term_a_b @ ( cons_term_a_b_nat @ F2 @ Fs ) ) )
      = ( ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
        | ( ( ( F2 @ X3 )
            = ( F2 @ Y3 ) )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( measures_term_a_b @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_604_append_Oright__neutral,axiom,
    ! [A: list_term_a_b] :
      ( ( append_term_a_b @ A @ nil_term_a_b )
      = A ) ).

% append.right_neutral
thf(fact_605_append_Oright__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ A @ nil_nat )
      = A ) ).

% append.right_neutral
thf(fact_606_append__Nil2,axiom,
    ! [Xs: list_term_a_b] :
      ( ( append_term_a_b @ Xs @ nil_term_a_b )
      = Xs ) ).

% append_Nil2
thf(fact_607_append__Nil2,axiom,
    ! [Xs: list_nat] :
      ( ( append_nat @ Xs @ nil_nat )
      = Xs ) ).

% append_Nil2
thf(fact_608_append__self__conv,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( ( append_term_a_b @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_term_a_b ) ) ).

% append_self_conv
thf(fact_609_append__self__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_nat ) ) ).

% append_self_conv
thf(fact_610_self__append__conv,axiom,
    ! [Y3: list_term_a_b,Ys: list_term_a_b] :
      ( ( Y3
        = ( append_term_a_b @ Y3 @ Ys ) )
      = ( Ys = nil_term_a_b ) ) ).

% self_append_conv
thf(fact_611_self__append__conv,axiom,
    ! [Y3: list_nat,Ys: list_nat] :
      ( ( Y3
        = ( append_nat @ Y3 @ Ys ) )
      = ( Ys = nil_nat ) ) ).

% self_append_conv
thf(fact_612_append__self__conv2,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( ( append_term_a_b @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_term_a_b ) ) ).

% append_self_conv2
thf(fact_613_append__self__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_nat ) ) ).

% append_self_conv2
thf(fact_614_self__append__conv2,axiom,
    ! [Y3: list_term_a_b,Xs: list_term_a_b] :
      ( ( Y3
        = ( append_term_a_b @ Xs @ Y3 ) )
      = ( Xs = nil_term_a_b ) ) ).

% self_append_conv2
thf(fact_615_self__append__conv2,axiom,
    ! [Y3: list_nat,Xs: list_nat] :
      ( ( Y3
        = ( append_nat @ Xs @ Y3 ) )
      = ( Xs = nil_nat ) ) ).

% self_append_conv2
thf(fact_616_Nil__is__append__conv,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( nil_term_a_b
        = ( append_term_a_b @ Xs @ Ys ) )
      = ( ( Xs = nil_term_a_b )
        & ( Ys = nil_term_a_b ) ) ) ).

% Nil_is_append_conv
thf(fact_617_Nil__is__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( nil_nat
        = ( append_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_618_append__is__Nil__conv,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( ( append_term_a_b @ Xs @ Ys )
        = nil_term_a_b )
      = ( ( Xs = nil_term_a_b )
        & ( Ys = nil_term_a_b ) ) ) ).

% append_is_Nil_conv
thf(fact_619_append__is__Nil__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = nil_nat )
      = ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_620_append__in__lists__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat,A5: set_nat] :
      ( ( member_list_nat2 @ ( append_nat @ Xs @ Ys ) @ ( lists_nat @ A5 ) )
      = ( ( member_list_nat2 @ Xs @ ( lists_nat @ A5 ) )
        & ( member_list_nat2 @ Ys @ ( lists_nat @ A5 ) ) ) ) ).

% append_in_lists_conv
thf(fact_621_append1__eq__conv,axiom,
    ! [Xs: list_term_a_b,X3: term_a_b,Ys: list_term_a_b,Y3: term_a_b] :
      ( ( ( append_term_a_b @ Xs @ ( cons_term_a_b @ X3 @ nil_term_a_b ) )
        = ( append_term_a_b @ Ys @ ( cons_term_a_b @ Y3 @ nil_term_a_b ) ) )
      = ( ( Xs = Ys )
        & ( X3 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_622_append1__eq__conv,axiom,
    ! [Xs: list_nat,X3: nat,Ys: list_nat,Y3: nat] :
      ( ( ( append_nat @ Xs @ ( cons_nat @ X3 @ nil_nat ) )
        = ( append_nat @ Ys @ ( cons_nat @ Y3 @ nil_nat ) ) )
      = ( ( Xs = Ys )
        & ( X3 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_623_Cons__eq__appendI,axiom,
    ! [X3: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs2: list_nat] :
      ( ( ( cons_nat @ X3 @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_nat @ Xs1 @ Zs2 ) )
       => ( ( cons_nat @ X3 @ Xs )
          = ( append_nat @ Ys @ Zs2 ) ) ) ) ).

% Cons_eq_appendI
thf(fact_624_append__Cons,axiom,
    ! [X3: nat,Xs: list_nat,Ys: list_nat] :
      ( ( append_nat @ ( cons_nat @ X3 @ Xs ) @ Ys )
      = ( cons_nat @ X3 @ ( append_nat @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_625_eq__Nil__appendI,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_term_a_b @ nil_term_a_b @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_626_eq__Nil__appendI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_nat @ nil_nat @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_627_append_Oleft__neutral,axiom,
    ! [A: list_term_a_b] :
      ( ( append_term_a_b @ nil_term_a_b @ A )
      = A ) ).

% append.left_neutral
thf(fact_628_append_Oleft__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ nil_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_629_append__Nil,axiom,
    ! [Ys: list_term_a_b] :
      ( ( append_term_a_b @ nil_term_a_b @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_630_append__Nil,axiom,
    ! [Ys: list_nat] :
      ( ( append_nat @ nil_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_631_order__le__imp__less__or__eq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ X3 @ Y3 )
        | ( X3 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_632_linorder__le__less__linear,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
      | ( ord_less_nat @ Y3 @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_633_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_634_order__le__less__subst1,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_635_order__less__le__trans,axiom,
    ! [X3: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z2 )
       => ( ord_less_nat @ X3 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_636_order__le__less__trans,axiom,
    ! [X3: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z2 )
       => ( ord_less_nat @ X3 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_637_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_638_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_639_order__less__imp__le,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ord_less_eq_nat @ X3 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_640_linorder__not__less,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
      = ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% linorder_not_less
thf(fact_641_linorder__not__le,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ~ ( ord_less_eq_nat @ X3 @ Y3 ) )
      = ( ord_less_nat @ Y3 @ X3 ) ) ).

% linorder_not_le
thf(fact_642_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_643_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y4: nat] :
          ( ( ord_less_nat @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_644_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_645_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_646_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B4 @ A3 )
          & ~ ( ord_less_eq_nat @ A3 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_647_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_648_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_649_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B4 @ A3 )
          & ( A3 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_650_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A3: nat] :
          ( ( ord_less_nat @ B4 @ A3 )
          | ( A3 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_651_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A3 @ B4 )
          & ~ ( ord_less_eq_nat @ B4 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_652_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_653_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_654_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A3 @ B4 )
          & ( A3 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_655_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B4: nat] :
          ( ( ord_less_nat @ A3 @ B4 )
          | ( A3 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_656_not__le__imp__less,axiom,
    ! [Y3: nat,X3: nat] :
      ( ~ ( ord_less_eq_nat @ Y3 @ X3 )
     => ( ord_less_nat @ X3 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_657_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y4 )
          & ~ ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_658_antisym__conv2,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_659_antisym__conv1,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_660_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_661_leI,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% leI
thf(fact_662_leD,axiom,
    ! [Y3: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X3 )
     => ~ ( ord_less_nat @ X3 @ Y3 ) ) ).

% leD
thf(fact_663_gt__ex,axiom,
    ! [X3: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).

% gt_ex
thf(fact_664_less__imp__neq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% less_imp_neq
thf(fact_665_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_666_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_667_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_668_less__induct,axiom,
    ! [P4: nat > $o,A: nat] :
      ( ! [X: nat] :
          ( ! [Y7: nat] :
              ( ( ord_less_nat @ Y7 @ X )
             => ( P4 @ Y7 ) )
         => ( P4 @ X ) )
     => ( P4 @ A ) ) ).

% less_induct
thf(fact_669_antisym__conv3,axiom,
    ! [Y3: nat,X3: nat] :
      ( ~ ( ord_less_nat @ Y3 @ X3 )
     => ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_670_linorder__cases,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y3 )
     => ( ( X3 != Y3 )
       => ( ord_less_nat @ Y3 @ X3 ) ) ) ).

% linorder_cases
thf(fact_671_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_672_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_673_exists__least__iff,axiom,
    ( ( ^ [P6: nat > $o] :
        ? [X6: nat] : ( P6 @ X6 ) )
    = ( ^ [P7: nat > $o] :
        ? [N3: nat] :
          ( ( P7 @ N3 )
          & ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N3 )
             => ~ ( P7 @ M2 ) ) ) ) ) ).

% exists_least_iff
thf(fact_674_linorder__less__wlog,axiom,
    ! [P4: nat > nat > $o,A: nat,B: nat] :
      ( ! [A4: nat,B3: nat] :
          ( ( ord_less_nat @ A4 @ B3 )
         => ( P4 @ A4 @ B3 ) )
     => ( ! [A4: nat] : ( P4 @ A4 @ A4 )
       => ( ! [A4: nat,B3: nat] :
              ( ( P4 @ B3 @ A4 )
             => ( P4 @ A4 @ B3 ) )
         => ( P4 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_675_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_676_not__less__iff__gr__or__eq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
      = ( ( ord_less_nat @ Y3 @ X3 )
        | ( X3 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_677_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_678_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_679_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_680_linorder__neqE,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_nat @ X3 @ Y3 )
       => ( ord_less_nat @ Y3 @ X3 ) ) ) ).

% linorder_neqE
thf(fact_681_order__less__asym,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).

% order_less_asym
thf(fact_682_linorder__neq__iff,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( X3 != Y3 )
      = ( ( ord_less_nat @ X3 @ Y3 )
        | ( ord_less_nat @ Y3 @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_683_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_684_order__less__trans,axiom,
    ! [X3: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z2 )
       => ( ord_less_nat @ X3 @ Z2 ) ) ) ).

% order_less_trans
thf(fact_685_ord__eq__less__subst,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_686_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F2 @ B )
          = C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_687_order__less__irrefl,axiom,
    ! [X3: nat] :
      ~ ( ord_less_nat @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_688_order__less__subst1,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_689_order__less__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ ( F2 @ B ) @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_690_order__less__not__sym,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).

% order_less_not_sym
thf(fact_691_order__less__imp__triv,axiom,
    ! [X3: nat,Y3: nat,P4: $o] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ X3 )
       => P4 ) ) ).

% order_less_imp_triv
thf(fact_692_linorder__less__linear,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
      | ( X3 = Y3 )
      | ( ord_less_nat @ Y3 @ X3 ) ) ).

% linorder_less_linear
thf(fact_693_order__less__imp__not__eq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_694_order__less__imp__not__eq2,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( Y3 != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_695_order__less__imp__not__less,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_696_rev__nonempty__induct,axiom,
    ! [Xs: list_term_a_b,P4: list_term_a_b > $o] :
      ( ( Xs != nil_term_a_b )
     => ( ! [X: term_a_b] : ( P4 @ ( cons_term_a_b @ X @ nil_term_a_b ) )
       => ( ! [X: term_a_b,Xs3: list_term_a_b] :
              ( ( Xs3 != nil_term_a_b )
             => ( ( P4 @ Xs3 )
               => ( P4 @ ( append_term_a_b @ Xs3 @ ( cons_term_a_b @ X @ nil_term_a_b ) ) ) ) )
         => ( P4 @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_697_rev__nonempty__induct,axiom,
    ! [Xs: list_nat,P4: list_nat > $o] :
      ( ( Xs != nil_nat )
     => ( ! [X: nat] : ( P4 @ ( cons_nat @ X @ nil_nat ) )
       => ( ! [X: nat,Xs3: list_nat] :
              ( ( Xs3 != nil_nat )
             => ( ( P4 @ Xs3 )
               => ( P4 @ ( append_nat @ Xs3 @ ( cons_nat @ X @ nil_nat ) ) ) ) )
         => ( P4 @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_698_append__eq__Cons__conv,axiom,
    ! [Ys: list_term_a_b,Zs2: list_term_a_b,X3: term_a_b,Xs: list_term_a_b] :
      ( ( ( append_term_a_b @ Ys @ Zs2 )
        = ( cons_term_a_b @ X3 @ Xs ) )
      = ( ( ( Ys = nil_term_a_b )
          & ( Zs2
            = ( cons_term_a_b @ X3 @ Xs ) ) )
        | ? [Ys4: list_term_a_b] :
            ( ( Ys
              = ( cons_term_a_b @ X3 @ Ys4 ) )
            & ( ( append_term_a_b @ Ys4 @ Zs2 )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_699_append__eq__Cons__conv,axiom,
    ! [Ys: list_nat,Zs2: list_nat,X3: nat,Xs: list_nat] :
      ( ( ( append_nat @ Ys @ Zs2 )
        = ( cons_nat @ X3 @ Xs ) )
      = ( ( ( Ys = nil_nat )
          & ( Zs2
            = ( cons_nat @ X3 @ Xs ) ) )
        | ? [Ys4: list_nat] :
            ( ( Ys
              = ( cons_nat @ X3 @ Ys4 ) )
            & ( ( append_nat @ Ys4 @ Zs2 )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_700_Cons__eq__append__conv,axiom,
    ! [X3: term_a_b,Xs: list_term_a_b,Ys: list_term_a_b,Zs2: list_term_a_b] :
      ( ( ( cons_term_a_b @ X3 @ Xs )
        = ( append_term_a_b @ Ys @ Zs2 ) )
      = ( ( ( Ys = nil_term_a_b )
          & ( ( cons_term_a_b @ X3 @ Xs )
            = Zs2 ) )
        | ? [Ys4: list_term_a_b] :
            ( ( ( cons_term_a_b @ X3 @ Ys4 )
              = Ys )
            & ( Xs
              = ( append_term_a_b @ Ys4 @ Zs2 ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_701_Cons__eq__append__conv,axiom,
    ! [X3: nat,Xs: list_nat,Ys: list_nat,Zs2: list_nat] :
      ( ( ( cons_nat @ X3 @ Xs )
        = ( append_nat @ Ys @ Zs2 ) )
      = ( ( ( Ys = nil_nat )
          & ( ( cons_nat @ X3 @ Xs )
            = Zs2 ) )
        | ? [Ys4: list_nat] :
            ( ( ( cons_nat @ X3 @ Ys4 )
              = Ys )
            & ( Xs
              = ( append_nat @ Ys4 @ Zs2 ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_702_rev__exhaust,axiom,
    ! [Xs: list_term_a_b] :
      ( ( Xs != nil_term_a_b )
     => ~ ! [Ys3: list_term_a_b,Y: term_a_b] :
            ( Xs
           != ( append_term_a_b @ Ys3 @ ( cons_term_a_b @ Y @ nil_term_a_b ) ) ) ) ).

% rev_exhaust
thf(fact_703_rev__exhaust,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ~ ! [Ys3: list_nat,Y: nat] :
            ( Xs
           != ( append_nat @ Ys3 @ ( cons_nat @ Y @ nil_nat ) ) ) ) ).

% rev_exhaust
thf(fact_704_rev__induct,axiom,
    ! [P4: list_term_a_b > $o,Xs: list_term_a_b] :
      ( ( P4 @ nil_term_a_b )
     => ( ! [X: term_a_b,Xs3: list_term_a_b] :
            ( ( P4 @ Xs3 )
           => ( P4 @ ( append_term_a_b @ Xs3 @ ( cons_term_a_b @ X @ nil_term_a_b ) ) ) )
       => ( P4 @ Xs ) ) ) ).

% rev_induct
thf(fact_705_rev__induct,axiom,
    ! [P4: list_nat > $o,Xs: list_nat] :
      ( ( P4 @ nil_nat )
     => ( ! [X: nat,Xs3: list_nat] :
            ( ( P4 @ Xs3 )
           => ( P4 @ ( append_nat @ Xs3 @ ( cons_nat @ X @ nil_nat ) ) ) )
       => ( P4 @ Xs ) ) ) ).

% rev_induct
thf(fact_706_lexord__append__leftD,axiom,
    ! [X3: list_term_a_b,U: list_term_a_b,V: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ X3 @ U ) @ ( append_term_a_b @ X3 @ V ) ) @ ( lexord_term_a_b @ R ) )
     => ( ! [A4: term_a_b] :
            ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ A4 ) @ R )
       => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ U @ V ) @ ( lexord_term_a_b @ R ) ) ) ) ).

% lexord_append_leftD
thf(fact_707_lexord__append__rightI,axiom,
    ! [Y3: list_nat,X3: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ? [B7: nat,Z5: list_nat] :
          ( Y3
          = ( cons_nat @ B7 @ Z5 ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ ( append_nat @ X3 @ Y3 ) ) @ ( lexord_nat @ R ) ) ) ).

% lexord_append_rightI
thf(fact_708_lex__append__left__iff,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b,Ys: list_term_a_b,Zs2: list_term_a_b] :
      ( ! [X: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R )
     => ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ Xs @ Ys ) @ ( append_term_a_b @ Xs @ Zs2 ) ) @ ( lex_term_a_b @ R ) )
        = ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ys @ Zs2 ) @ ( lex_term_a_b @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_709_lex__append__leftD,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b,Ys: list_term_a_b,Zs2: list_term_a_b] :
      ( ! [X: term_a_b] :
          ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R )
     => ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ Xs @ Ys ) @ ( append_term_a_b @ Xs @ Zs2 ) ) @ ( lex_term_a_b @ R ) )
       => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ys @ Zs2 ) @ ( lex_term_a_b @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_710_lexord__append__left__rightI,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,U: list_nat,X3: list_nat,Y3: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ U @ ( cons_nat @ A @ X3 ) ) @ ( append_nat @ U @ ( cons_nat @ B @ Y3 ) ) ) @ ( lexord_nat @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_711_lexord__append__left__rightI,axiom,
    ! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,U: list_term_a_b,X3: list_term_a_b,Y3: list_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
     => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ U @ ( cons_term_a_b @ A @ X3 ) ) @ ( append_term_a_b @ U @ ( cons_term_a_b @ B @ Y3 ) ) ) @ ( lexord_term_a_b @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_712_measures__less,axiom,
    ! [F2: term_a_b > nat,X3: term_a_b,Y3: term_a_b,Fs: list_term_a_b_nat] :
      ( ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( measures_term_a_b @ ( cons_term_a_b_nat @ F2 @ Fs ) ) ) ) ).

% measures_less
thf(fact_713_in__measure,axiom,
    ! [X3: term_a_b,Y3: term_a_b,F2: term_a_b > nat] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( measure_term_a_b @ F2 ) )
      = ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ).

% in_measure
thf(fact_714_SuccI,axiom,
    ! [Kl: list_list_nat,K: list_nat,Kl2: set_list_list_nat] :
      ( ( member_list_list_nat @ ( append_list_nat @ Kl @ ( cons_list_nat @ K @ nil_list_nat ) ) @ Kl2 )
     => ( member_list_nat2 @ K @ ( bNF_Gr3053708287304744325st_nat @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_715_SuccI,axiom,
    ! [Kl: list_P8875379029341186191rm_a_b,K: produc357393685978478089rm_a_b,Kl2: set_li7311725489111463791rm_a_b] :
      ( ( member7646937072540562872rm_a_b @ ( append1152621114427393060rm_a_b @ Kl @ ( cons_P3536305108106557631rm_a_b @ K @ nil_Pr6942174756412032271rm_a_b ) ) @ Kl2 )
     => ( member5869715511025134514rm_a_b @ K @ ( bNF_Gr7233806482624125860rm_a_b @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_716_SuccI,axiom,
    ! [Kl: list_term_a_b,K: term_a_b,Kl2: set_list_term_a_b] :
      ( ( member_list_term_a_b2 @ ( append_term_a_b @ Kl @ ( cons_term_a_b @ K @ nil_term_a_b ) ) @ Kl2 )
     => ( member_term_a_b2 @ K @ ( bNF_Gr5510842888252665017rm_a_b @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_717_SuccI,axiom,
    ! [Kl: list_nat,K: nat,Kl2: set_list_nat] :
      ( ( member_list_nat2 @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 )
     => ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_718_SuccD,axiom,
    ! [K: list_nat,Kl2: set_list_list_nat,Kl: list_list_nat] :
      ( ( member_list_nat2 @ K @ ( bNF_Gr3053708287304744325st_nat @ Kl2 @ Kl ) )
     => ( member_list_list_nat @ ( append_list_nat @ Kl @ ( cons_list_nat @ K @ nil_list_nat ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_719_SuccD,axiom,
    ! [K: produc357393685978478089rm_a_b,Kl2: set_li7311725489111463791rm_a_b,Kl: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ K @ ( bNF_Gr7233806482624125860rm_a_b @ Kl2 @ Kl ) )
     => ( member7646937072540562872rm_a_b @ ( append1152621114427393060rm_a_b @ Kl @ ( cons_P3536305108106557631rm_a_b @ K @ nil_Pr6942174756412032271rm_a_b ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_720_SuccD,axiom,
    ! [K: term_a_b,Kl2: set_list_term_a_b,Kl: list_term_a_b] :
      ( ( member_term_a_b2 @ K @ ( bNF_Gr5510842888252665017rm_a_b @ Kl2 @ Kl ) )
     => ( member_list_term_a_b2 @ ( append_term_a_b @ Kl @ ( cons_term_a_b @ K @ nil_term_a_b ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_721_SuccD,axiom,
    ! [K: nat,Kl2: set_list_nat,Kl: list_nat] :
      ( ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) )
     => ( member_list_nat2 @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_722_replace__term__at__above,axiom,
    ! [P: list_nat,Q5: list_nat,S: term_a_b,T: term_a_b,U: term_a_b] :
      ( ( term_p3503116865373065078eq_nat @ P @ Q5 )
     => ( ( term_r6860082780075436317at_a_b @ ( term_r6860082780075436317at_a_b @ S @ Q5 @ T ) @ P @ U )
        = ( term_r6860082780075436317at_a_b @ S @ P @ U ) ) ) ).

% replace_term_at_above
thf(fact_723_replace__term__at__below,axiom,
    ! [P: list_nat,Q5: list_nat,S: term_a_b,T: term_a_b,U: term_a_b] :
      ( ( ( P != Q5 )
        & ( term_p3503116865373065078eq_nat @ P @ Q5 ) )
     => ( ( term_r6860082780075436317at_a_b @ ( term_r6860082780075436317at_a_b @ S @ P @ T ) @ Q5 @ U )
        = ( term_r6860082780075436317at_a_b @ S @ P @ ( term_r6860082780075436317at_a_b @ T @ ( term_pos_diff_nat @ Q5 @ P ) @ U ) ) ) ) ).

% replace_term_at_below
thf(fact_724_position__less__eq__Cons,axiom,
    ! [I: nat,Ps: list_nat,J: nat,Qs: list_nat] :
      ( ( term_p3503116865373065078eq_nat @ ( cons_nat @ I @ Ps ) @ ( cons_nat @ J @ Qs ) )
      = ( ( I = J )
        & ( term_p3503116865373065078eq_nat @ Ps @ Qs ) ) ) ).

% position_less_eq_Cons
thf(fact_725_position__less__Nil__is__bot2,axiom,
    ! [P: list_term_a_b] :
      ( ( term_p8391561492822560442rm_a_b @ P @ nil_term_a_b )
      = ( P = nil_term_a_b ) ) ).

% position_less_Nil_is_bot2
thf(fact_726_position__less__Nil__is__bot2,axiom,
    ! [P: list_nat] :
      ( ( term_p3503116865373065078eq_nat @ P @ nil_nat )
      = ( P = nil_nat ) ) ).

% position_less_Nil_is_bot2
thf(fact_727_position__less__Nil__is__bot,axiom,
    ! [P: list_term_a_b] : ( term_p8391561492822560442rm_a_b @ nil_term_a_b @ P ) ).

% position_less_Nil_is_bot
thf(fact_728_position__less__Nil__is__bot,axiom,
    ! [P: list_nat] : ( term_p3503116865373065078eq_nat @ nil_nat @ P ) ).

% position_less_Nil_is_bot
thf(fact_729_max__list_Ocases,axiom,
    ! [X3: list_nat] :
      ( ( X3 != nil_nat )
     => ~ ! [X: nat,Xs3: list_nat] :
            ( X3
           != ( cons_nat @ X @ Xs3 ) ) ) ).

% max_list.cases
thf(fact_730_empty__Shift,axiom,
    ! [Kl2: set_list_list_nat,K: list_nat] :
      ( ( member_list_list_nat @ nil_list_nat @ Kl2 )
     => ( ( member_list_nat2 @ K @ ( bNF_Gr3053708287304744325st_nat @ Kl2 @ nil_list_nat ) )
       => ( member_list_list_nat @ nil_list_nat @ ( bNF_Gr9051742241863529473st_nat @ Kl2 @ K ) ) ) ) ).

% empty_Shift
thf(fact_731_empty__Shift,axiom,
    ! [Kl2: set_li7311725489111463791rm_a_b,K: produc357393685978478089rm_a_b] :
      ( ( member7646937072540562872rm_a_b @ nil_Pr6942174756412032271rm_a_b @ Kl2 )
     => ( ( member5869715511025134514rm_a_b @ K @ ( bNF_Gr7233806482624125860rm_a_b @ Kl2 @ nil_Pr6942174756412032271rm_a_b ) )
       => ( member7646937072540562872rm_a_b @ nil_Pr6942174756412032271rm_a_b @ ( bNF_Gr369565347726608424rm_a_b @ Kl2 @ K ) ) ) ) ).

% empty_Shift
thf(fact_732_empty__Shift,axiom,
    ! [Kl2: set_list_term_a_b,K: term_a_b] :
      ( ( member_list_term_a_b2 @ nil_term_a_b @ Kl2 )
     => ( ( member_term_a_b2 @ K @ ( bNF_Gr5510842888252665017rm_a_b @ Kl2 @ nil_term_a_b ) )
       => ( member_list_term_a_b2 @ nil_term_a_b @ ( bNF_Gr2285504805956674357rm_a_b @ Kl2 @ K ) ) ) ) ).

% empty_Shift
thf(fact_733_empty__Shift,axiom,
    ! [Kl2: set_list_nat,K: nat] :
      ( ( member_list_nat2 @ nil_nat @ Kl2 )
     => ( ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ nil_nat ) )
       => ( member_list_nat2 @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) ) ) ) ).

% empty_Shift
thf(fact_734_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_nat,X3: nat,Ys: list_nat,Y3: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ ( cons_nat @ X3 @ nil_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y3 @ nil_nat ) ) ) @ ( listrel1_nat @ R ) )
      = ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
          & ( X3 = Y3 ) )
        | ( ( Xs = Ys )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_735_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_term_a_b,X3: term_a_b,Ys: list_term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ X3 @ nil_term_a_b ) ) @ ( append_term_a_b @ Ys @ ( cons_term_a_b @ Y3 @ nil_term_a_b ) ) ) @ ( listrel1_term_a_b @ R ) )
      = ( ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listrel1_term_a_b @ R ) )
          & ( X3 = Y3 ) )
        | ( ( Xs = Ys )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_736_Cons__lenlex__iff,axiom,
    ! [M3: nat,Ms: list_nat,N: nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M3 @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R ) )
      = ( ( ord_less_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) )
        | ( ( ( size_size_list_nat @ Ms )
            = ( size_size_list_nat @ Ns ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M3 @ N ) @ R ) )
        | ( ( M3 = N )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_737_Cons__lenlex__iff,axiom,
    ! [M3: term_a_b,Ms: list_term_a_b,N: term_a_b,Ns: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ M3 @ Ms ) @ ( cons_term_a_b @ N @ Ns ) ) @ ( lenlex_term_a_b @ R ) )
      = ( ( ord_less_nat @ ( size_s8906293707977694520rm_a_b @ Ms ) @ ( size_s8906293707977694520rm_a_b @ Ns ) )
        | ( ( ( size_s8906293707977694520rm_a_b @ Ms )
            = ( size_s8906293707977694520rm_a_b @ Ns ) )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ M3 @ N ) @ R ) )
        | ( ( M3 = N )
          & ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ms @ Ns ) @ ( lenlex_term_a_b @ R ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_738_Cons__in__lex,axiom,
    ! [X3: nat,Xs: list_nat,Y3: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) @ ( lex_nat @ R ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
          & ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) ) )
        | ( ( X3 = Y3 )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_739_Cons__in__lex,axiom,
    ! [X3: term_a_b,Xs: list_term_a_b,Y3: term_a_b,Ys: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X3 @ Xs ) @ ( cons_term_a_b @ Y3 @ Ys ) ) @ ( lex_term_a_b @ R ) )
      = ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
          & ( ( size_s8906293707977694520rm_a_b @ Xs )
            = ( size_s8906293707977694520rm_a_b @ Ys ) ) )
        | ( ( X3 = Y3 )
          & ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( lex_term_a_b @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_740_Cons__listrel1__Cons,axiom,
    ! [X3: nat,Xs: list_nat,Y3: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel1_nat @ R ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
          & ( Xs = Ys ) )
        | ( ( X3 = Y3 )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_741_Cons__listrel1__Cons,axiom,
    ! [X3: term_a_b,Xs: list_term_a_b,Y3: term_a_b,Ys: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X3 @ Xs ) @ ( cons_term_a_b @ Y3 @ Ys ) ) @ ( listrel1_term_a_b @ R ) )
      = ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
          & ( Xs = Ys ) )
        | ( ( X3 = Y3 )
          & ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listrel1_term_a_b @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_742_psubsetD,axiom,
    ! [A5: set_list_nat,B5: set_list_nat,C: list_nat] :
      ( ( ord_le1190675801316882794st_nat @ A5 @ B5 )
     => ( ( member_list_nat2 @ C @ A5 )
       => ( member_list_nat2 @ C @ B5 ) ) ) ).

% psubsetD
thf(fact_743_psubsetD,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C: produc357393685978478089rm_a_b] :
      ( ( ord_le4883060154188259093rm_a_b @ A5 @ B5 )
     => ( ( member5869715511025134514rm_a_b @ C @ A5 )
       => ( member5869715511025134514rm_a_b @ C @ B5 ) ) ) ).

% psubsetD
thf(fact_744_psubsetD,axiom,
    ! [A5: set_nat,B5: set_nat,C: nat] :
      ( ( ord_less_set_nat @ A5 @ B5 )
     => ( ( member_nat2 @ C @ A5 )
       => ( member_nat2 @ C @ B5 ) ) ) ).

% psubsetD
thf(fact_745_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs2: list_nat,Ws: list_nat,P4: list_nat > list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs2 ) )
       => ( ( ( size_size_list_nat @ Zs2 )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P4 @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
           => ( ! [X: nat,Xs3: list_nat,Y: nat,Ys3: list_nat,Z: nat,Zs: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_nat @ Ys3 ) )
                 => ( ( ( size_size_list_nat @ Ys3 )
                      = ( size_size_list_nat @ Zs ) )
                   => ( ( ( size_size_list_nat @ Zs )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P4 @ Xs3 @ Ys3 @ Zs @ Ws2 )
                       => ( P4 @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys3 ) @ ( cons_nat @ Z @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P4 @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_746_list__induct4,axiom,
    ! [Xs: list_term_a_b,Ys: list_nat,Zs2: list_nat,Ws: list_nat,P4: list_term_a_b > list_nat > list_nat > list_nat > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs2 ) )
       => ( ( ( size_size_list_nat @ Zs2 )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P4 @ nil_term_a_b @ nil_nat @ nil_nat @ nil_nat )
           => ( ! [X: term_a_b,Xs3: list_term_a_b,Y: nat,Ys3: list_nat,Z: nat,Zs: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_s8906293707977694520rm_a_b @ Xs3 )
                    = ( size_size_list_nat @ Ys3 ) )
                 => ( ( ( size_size_list_nat @ Ys3 )
                      = ( size_size_list_nat @ Zs ) )
                   => ( ( ( size_size_list_nat @ Zs )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P4 @ Xs3 @ Ys3 @ Zs @ Ws2 )
                       => ( P4 @ ( cons_term_a_b @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys3 ) @ ( cons_nat @ Z @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P4 @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_747_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_term_a_b,Zs2: list_nat,Ws: list_nat,P4: list_nat > list_term_a_b > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Ys )
          = ( size_size_list_nat @ Zs2 ) )
       => ( ( ( size_size_list_nat @ Zs2 )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P4 @ nil_nat @ nil_term_a_b @ nil_nat @ nil_nat )
           => ( ! [X: nat,Xs3: list_nat,Y: term_a_b,Ys3: list_term_a_b,Z: nat,Zs: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_s8906293707977694520rm_a_b @ Ys3 ) )
                 => ( ( ( size_s8906293707977694520rm_a_b @ Ys3 )
                      = ( size_size_list_nat @ Zs ) )
                   => ( ( ( size_size_list_nat @ Zs )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P4 @ Xs3 @ Ys3 @ Zs @ Ws2 )
                       => ( P4 @ ( cons_nat @ X @ Xs3 ) @ ( cons_term_a_b @ Y @ Ys3 ) @ ( cons_nat @ Z @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P4 @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_748_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs2: list_term_a_b,Ws: list_nat,P4: list_nat > list_nat > list_term_a_b > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s8906293707977694520rm_a_b @ Zs2 ) )
       => ( ( ( size_s8906293707977694520rm_a_b @ Zs2 )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P4 @ nil_nat @ nil_nat @ nil_term_a_b @ nil_nat )
           => ( ! [X: nat,Xs3: list_nat,Y: nat,Ys3: list_nat,Z: term_a_b,Zs: list_term_a_b,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_nat @ Ys3 ) )
                 => ( ( ( size_size_list_nat @ Ys3 )
                      = ( size_s8906293707977694520rm_a_b @ Zs ) )
                   => ( ( ( size_s8906293707977694520rm_a_b @ Zs )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P4 @ Xs3 @ Ys3 @ Zs @ Ws2 )
                       => ( P4 @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys3 ) @ ( cons_term_a_b @ Z @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P4 @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_749_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs2: list_nat,Ws: list_term_a_b,P4: list_nat > list_nat > list_nat > list_term_a_b > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs2 ) )
       => ( ( ( size_size_list_nat @ Zs2 )
            = ( size_s8906293707977694520rm_a_b @ Ws ) )
         => ( ( P4 @ nil_nat @ nil_nat @ nil_nat @ nil_term_a_b )
           => ( ! [X: nat,Xs3: list_nat,Y: nat,Ys3: list_nat,Z: nat,Zs: list_nat,W: term_a_b,Ws2: list_term_a_b] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_nat @ Ys3 ) )
                 => ( ( ( size_size_list_nat @ Ys3 )
                      = ( size_size_list_nat @ Zs ) )
                   => ( ( ( size_size_list_nat @ Zs )
                        = ( size_s8906293707977694520rm_a_b @ Ws2 ) )
                     => ( ( P4 @ Xs3 @ Ys3 @ Zs @ Ws2 )
                       => ( P4 @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys3 ) @ ( cons_nat @ Z @ Zs ) @ ( cons_term_a_b @ W @ Ws2 ) ) ) ) ) )
             => ( P4 @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_750_list__induct4,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b,Zs2: list_nat,Ws: list_nat,P4: list_term_a_b > list_term_a_b > list_nat > list_nat > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Ys )
          = ( size_size_list_nat @ Zs2 ) )
       => ( ( ( size_size_list_nat @ Zs2 )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P4 @ nil_term_a_b @ nil_term_a_b @ nil_nat @ nil_nat )
           => ( ! [X: term_a_b,Xs3: list_term_a_b,Y: term_a_b,Ys3: list_term_a_b,Z: nat,Zs: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_s8906293707977694520rm_a_b @ Xs3 )
                    = ( size_s8906293707977694520rm_a_b @ Ys3 ) )
                 => ( ( ( size_s8906293707977694520rm_a_b @ Ys3 )
                      = ( size_size_list_nat @ Zs ) )
                   => ( ( ( size_size_list_nat @ Zs )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P4 @ Xs3 @ Ys3 @ Zs @ Ws2 )
                       => ( P4 @ ( cons_term_a_b @ X @ Xs3 ) @ ( cons_term_a_b @ Y @ Ys3 ) @ ( cons_nat @ Z @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P4 @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_751_list__induct4,axiom,
    ! [Xs: list_term_a_b,Ys: list_nat,Zs2: list_term_a_b,Ws: list_nat,P4: list_term_a_b > list_nat > list_term_a_b > list_nat > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s8906293707977694520rm_a_b @ Zs2 ) )
       => ( ( ( size_s8906293707977694520rm_a_b @ Zs2 )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P4 @ nil_term_a_b @ nil_nat @ nil_term_a_b @ nil_nat )
           => ( ! [X: term_a_b,Xs3: list_term_a_b,Y: nat,Ys3: list_nat,Z: term_a_b,Zs: list_term_a_b,W: nat,Ws2: list_nat] :
                  ( ( ( size_s8906293707977694520rm_a_b @ Xs3 )
                    = ( size_size_list_nat @ Ys3 ) )
                 => ( ( ( size_size_list_nat @ Ys3 )
                      = ( size_s8906293707977694520rm_a_b @ Zs ) )
                   => ( ( ( size_s8906293707977694520rm_a_b @ Zs )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P4 @ Xs3 @ Ys3 @ Zs @ Ws2 )
                       => ( P4 @ ( cons_term_a_b @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys3 ) @ ( cons_term_a_b @ Z @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P4 @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_752_list__induct4,axiom,
    ! [Xs: list_term_a_b,Ys: list_nat,Zs2: list_nat,Ws: list_term_a_b,P4: list_term_a_b > list_nat > list_nat > list_term_a_b > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs2 ) )
       => ( ( ( size_size_list_nat @ Zs2 )
            = ( size_s8906293707977694520rm_a_b @ Ws ) )
         => ( ( P4 @ nil_term_a_b @ nil_nat @ nil_nat @ nil_term_a_b )
           => ( ! [X: term_a_b,Xs3: list_term_a_b,Y: nat,Ys3: list_nat,Z: nat,Zs: list_nat,W: term_a_b,Ws2: list_term_a_b] :
                  ( ( ( size_s8906293707977694520rm_a_b @ Xs3 )
                    = ( size_size_list_nat @ Ys3 ) )
                 => ( ( ( size_size_list_nat @ Ys3 )
                      = ( size_size_list_nat @ Zs ) )
                   => ( ( ( size_size_list_nat @ Zs )
                        = ( size_s8906293707977694520rm_a_b @ Ws2 ) )
                     => ( ( P4 @ Xs3 @ Ys3 @ Zs @ Ws2 )
                       => ( P4 @ ( cons_term_a_b @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys3 ) @ ( cons_nat @ Z @ Zs ) @ ( cons_term_a_b @ W @ Ws2 ) ) ) ) ) )
             => ( P4 @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_753_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_term_a_b,Zs2: list_term_a_b,Ws: list_nat,P4: list_nat > list_term_a_b > list_term_a_b > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Ys )
          = ( size_s8906293707977694520rm_a_b @ Zs2 ) )
       => ( ( ( size_s8906293707977694520rm_a_b @ Zs2 )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P4 @ nil_nat @ nil_term_a_b @ nil_term_a_b @ nil_nat )
           => ( ! [X: nat,Xs3: list_nat,Y: term_a_b,Ys3: list_term_a_b,Z: term_a_b,Zs: list_term_a_b,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_s8906293707977694520rm_a_b @ Ys3 ) )
                 => ( ( ( size_s8906293707977694520rm_a_b @ Ys3 )
                      = ( size_s8906293707977694520rm_a_b @ Zs ) )
                   => ( ( ( size_s8906293707977694520rm_a_b @ Zs )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P4 @ Xs3 @ Ys3 @ Zs @ Ws2 )
                       => ( P4 @ ( cons_nat @ X @ Xs3 ) @ ( cons_term_a_b @ Y @ Ys3 ) @ ( cons_term_a_b @ Z @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P4 @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_754_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_term_a_b,Zs2: list_nat,Ws: list_term_a_b,P4: list_nat > list_term_a_b > list_nat > list_term_a_b > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Ys )
          = ( size_size_list_nat @ Zs2 ) )
       => ( ( ( size_size_list_nat @ Zs2 )
            = ( size_s8906293707977694520rm_a_b @ Ws ) )
         => ( ( P4 @ nil_nat @ nil_term_a_b @ nil_nat @ nil_term_a_b )
           => ( ! [X: nat,Xs3: list_nat,Y: term_a_b,Ys3: list_term_a_b,Z: nat,Zs: list_nat,W: term_a_b,Ws2: list_term_a_b] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_s8906293707977694520rm_a_b @ Ys3 ) )
                 => ( ( ( size_s8906293707977694520rm_a_b @ Ys3 )
                      = ( size_size_list_nat @ Zs ) )
                   => ( ( ( size_size_list_nat @ Zs )
                        = ( size_s8906293707977694520rm_a_b @ Ws2 ) )
                     => ( ( P4 @ Xs3 @ Ys3 @ Zs @ Ws2 )
                       => ( P4 @ ( cons_nat @ X @ Xs3 ) @ ( cons_term_a_b @ Y @ Ys3 ) @ ( cons_nat @ Z @ Zs ) @ ( cons_term_a_b @ W @ Ws2 ) ) ) ) ) )
             => ( P4 @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_755_list__induct3,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b,Zs2: list_term_a_b,P4: list_term_a_b > list_term_a_b > list_term_a_b > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Ys )
          = ( size_s8906293707977694520rm_a_b @ Zs2 ) )
       => ( ( P4 @ nil_term_a_b @ nil_term_a_b @ nil_term_a_b )
         => ( ! [X: term_a_b,Xs3: list_term_a_b,Y: term_a_b,Ys3: list_term_a_b,Z: term_a_b,Zs: list_term_a_b] :
                ( ( ( size_s8906293707977694520rm_a_b @ Xs3 )
                  = ( size_s8906293707977694520rm_a_b @ Ys3 ) )
               => ( ( ( size_s8906293707977694520rm_a_b @ Ys3 )
                    = ( size_s8906293707977694520rm_a_b @ Zs ) )
                 => ( ( P4 @ Xs3 @ Ys3 @ Zs )
                   => ( P4 @ ( cons_term_a_b @ X @ Xs3 ) @ ( cons_term_a_b @ Y @ Ys3 ) @ ( cons_term_a_b @ Z @ Zs ) ) ) ) )
           => ( P4 @ Xs @ Ys @ Zs2 ) ) ) ) ) ).

% list_induct3
thf(fact_756_list__induct3,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b,Zs2: list_nat,P4: list_term_a_b > list_term_a_b > list_nat > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Ys )
          = ( size_size_list_nat @ Zs2 ) )
       => ( ( P4 @ nil_term_a_b @ nil_term_a_b @ nil_nat )
         => ( ! [X: term_a_b,Xs3: list_term_a_b,Y: term_a_b,Ys3: list_term_a_b,Z: nat,Zs: list_nat] :
                ( ( ( size_s8906293707977694520rm_a_b @ Xs3 )
                  = ( size_s8906293707977694520rm_a_b @ Ys3 ) )
               => ( ( ( size_s8906293707977694520rm_a_b @ Ys3 )
                    = ( size_size_list_nat @ Zs ) )
                 => ( ( P4 @ Xs3 @ Ys3 @ Zs )
                   => ( P4 @ ( cons_term_a_b @ X @ Xs3 ) @ ( cons_term_a_b @ Y @ Ys3 ) @ ( cons_nat @ Z @ Zs ) ) ) ) )
           => ( P4 @ Xs @ Ys @ Zs2 ) ) ) ) ) ).

% list_induct3
thf(fact_757_list__induct3,axiom,
    ! [Xs: list_term_a_b,Ys: list_nat,Zs2: list_term_a_b,P4: list_term_a_b > list_nat > list_term_a_b > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s8906293707977694520rm_a_b @ Zs2 ) )
       => ( ( P4 @ nil_term_a_b @ nil_nat @ nil_term_a_b )
         => ( ! [X: term_a_b,Xs3: list_term_a_b,Y: nat,Ys3: list_nat,Z: term_a_b,Zs: list_term_a_b] :
                ( ( ( size_s8906293707977694520rm_a_b @ Xs3 )
                  = ( size_size_list_nat @ Ys3 ) )
               => ( ( ( size_size_list_nat @ Ys3 )
                    = ( size_s8906293707977694520rm_a_b @ Zs ) )
                 => ( ( P4 @ Xs3 @ Ys3 @ Zs )
                   => ( P4 @ ( cons_term_a_b @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys3 ) @ ( cons_term_a_b @ Z @ Zs ) ) ) ) )
           => ( P4 @ Xs @ Ys @ Zs2 ) ) ) ) ) ).

% list_induct3
thf(fact_758_list__induct3,axiom,
    ! [Xs: list_term_a_b,Ys: list_nat,Zs2: list_nat,P4: list_term_a_b > list_nat > list_nat > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs2 ) )
       => ( ( P4 @ nil_term_a_b @ nil_nat @ nil_nat )
         => ( ! [X: term_a_b,Xs3: list_term_a_b,Y: nat,Ys3: list_nat,Z: nat,Zs: list_nat] :
                ( ( ( size_s8906293707977694520rm_a_b @ Xs3 )
                  = ( size_size_list_nat @ Ys3 ) )
               => ( ( ( size_size_list_nat @ Ys3 )
                    = ( size_size_list_nat @ Zs ) )
                 => ( ( P4 @ Xs3 @ Ys3 @ Zs )
                   => ( P4 @ ( cons_term_a_b @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys3 ) @ ( cons_nat @ Z @ Zs ) ) ) ) )
           => ( P4 @ Xs @ Ys @ Zs2 ) ) ) ) ) ).

% list_induct3
thf(fact_759_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_term_a_b,Zs2: list_term_a_b,P4: list_nat > list_term_a_b > list_term_a_b > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Ys )
          = ( size_s8906293707977694520rm_a_b @ Zs2 ) )
       => ( ( P4 @ nil_nat @ nil_term_a_b @ nil_term_a_b )
         => ( ! [X: nat,Xs3: list_nat,Y: term_a_b,Ys3: list_term_a_b,Z: term_a_b,Zs: list_term_a_b] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_s8906293707977694520rm_a_b @ Ys3 ) )
               => ( ( ( size_s8906293707977694520rm_a_b @ Ys3 )
                    = ( size_s8906293707977694520rm_a_b @ Zs ) )
                 => ( ( P4 @ Xs3 @ Ys3 @ Zs )
                   => ( P4 @ ( cons_nat @ X @ Xs3 ) @ ( cons_term_a_b @ Y @ Ys3 ) @ ( cons_term_a_b @ Z @ Zs ) ) ) ) )
           => ( P4 @ Xs @ Ys @ Zs2 ) ) ) ) ) ).

% list_induct3
thf(fact_760_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_term_a_b,Zs2: list_nat,P4: list_nat > list_term_a_b > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Ys )
          = ( size_size_list_nat @ Zs2 ) )
       => ( ( P4 @ nil_nat @ nil_term_a_b @ nil_nat )
         => ( ! [X: nat,Xs3: list_nat,Y: term_a_b,Ys3: list_term_a_b,Z: nat,Zs: list_nat] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_s8906293707977694520rm_a_b @ Ys3 ) )
               => ( ( ( size_s8906293707977694520rm_a_b @ Ys3 )
                    = ( size_size_list_nat @ Zs ) )
                 => ( ( P4 @ Xs3 @ Ys3 @ Zs )
                   => ( P4 @ ( cons_nat @ X @ Xs3 ) @ ( cons_term_a_b @ Y @ Ys3 ) @ ( cons_nat @ Z @ Zs ) ) ) ) )
           => ( P4 @ Xs @ Ys @ Zs2 ) ) ) ) ) ).

% list_induct3
thf(fact_761_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs2: list_term_a_b,P4: list_nat > list_nat > list_term_a_b > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s8906293707977694520rm_a_b @ Zs2 ) )
       => ( ( P4 @ nil_nat @ nil_nat @ nil_term_a_b )
         => ( ! [X: nat,Xs3: list_nat,Y: nat,Ys3: list_nat,Z: term_a_b,Zs: list_term_a_b] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_size_list_nat @ Ys3 ) )
               => ( ( ( size_size_list_nat @ Ys3 )
                    = ( size_s8906293707977694520rm_a_b @ Zs ) )
                 => ( ( P4 @ Xs3 @ Ys3 @ Zs )
                   => ( P4 @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys3 ) @ ( cons_term_a_b @ Z @ Zs ) ) ) ) )
           => ( P4 @ Xs @ Ys @ Zs2 ) ) ) ) ) ).

% list_induct3
thf(fact_762_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs2: list_nat,P4: list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs2 ) )
       => ( ( P4 @ nil_nat @ nil_nat @ nil_nat )
         => ( ! [X: nat,Xs3: list_nat,Y: nat,Ys3: list_nat,Z: nat,Zs: list_nat] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_size_list_nat @ Ys3 ) )
               => ( ( ( size_size_list_nat @ Ys3 )
                    = ( size_size_list_nat @ Zs ) )
                 => ( ( P4 @ Xs3 @ Ys3 @ Zs )
                   => ( P4 @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys3 ) @ ( cons_nat @ Z @ Zs ) ) ) ) )
           => ( P4 @ Xs @ Ys @ Zs2 ) ) ) ) ) ).

% list_induct3
thf(fact_763_list__induct2,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b,P4: list_term_a_b > list_term_a_b > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( P4 @ nil_term_a_b @ nil_term_a_b )
       => ( ! [X: term_a_b,Xs3: list_term_a_b,Y: term_a_b,Ys3: list_term_a_b] :
              ( ( ( size_s8906293707977694520rm_a_b @ Xs3 )
                = ( size_s8906293707977694520rm_a_b @ Ys3 ) )
             => ( ( P4 @ Xs3 @ Ys3 )
               => ( P4 @ ( cons_term_a_b @ X @ Xs3 ) @ ( cons_term_a_b @ Y @ Ys3 ) ) ) )
         => ( P4 @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_764_list__induct2,axiom,
    ! [Xs: list_term_a_b,Ys: list_nat,P4: list_term_a_b > list_nat > $o] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P4 @ nil_term_a_b @ nil_nat )
       => ( ! [X: term_a_b,Xs3: list_term_a_b,Y: nat,Ys3: list_nat] :
              ( ( ( size_s8906293707977694520rm_a_b @ Xs3 )
                = ( size_size_list_nat @ Ys3 ) )
             => ( ( P4 @ Xs3 @ Ys3 )
               => ( P4 @ ( cons_term_a_b @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys3 ) ) ) )
         => ( P4 @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_765_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_term_a_b,P4: list_nat > list_term_a_b > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( P4 @ nil_nat @ nil_term_a_b )
       => ( ! [X: nat,Xs3: list_nat,Y: term_a_b,Ys3: list_term_a_b] :
              ( ( ( size_size_list_nat @ Xs3 )
                = ( size_s8906293707977694520rm_a_b @ Ys3 ) )
             => ( ( P4 @ Xs3 @ Ys3 )
               => ( P4 @ ( cons_nat @ X @ Xs3 ) @ ( cons_term_a_b @ Y @ Ys3 ) ) ) )
         => ( P4 @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_766_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_nat,P4: list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P4 @ nil_nat @ nil_nat )
       => ( ! [X: nat,Xs3: list_nat,Y: nat,Ys3: list_nat] :
              ( ( ( size_size_list_nat @ Xs3 )
                = ( size_size_list_nat @ Ys3 ) )
             => ( ( P4 @ Xs3 @ Ys3 )
               => ( P4 @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y @ Ys3 ) ) ) )
         => ( P4 @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_767_impossible__Cons,axiom,
    ! [Xs: list_nat,Ys: list_nat,X3: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
     => ( Xs
       != ( cons_nat @ X3 @ Ys ) ) ) ).

% impossible_Cons
thf(fact_768_listrel1I2,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,X3: nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ X3 @ Ys ) ) @ ( listrel1_nat @ R ) ) ) ).

% listrel1I2
thf(fact_769_not__listrel1__Nil,axiom,
    ! [Xs: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ nil_term_a_b ) @ ( listrel1_term_a_b @ R ) ) ).

% not_listrel1_Nil
thf(fact_770_not__listrel1__Nil,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel1_nat @ R ) ) ).

% not_listrel1_Nil
thf(fact_771_not__Nil__listrel1,axiom,
    ! [Xs: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ nil_term_a_b @ Xs ) @ ( listrel1_term_a_b @ R ) ) ).

% not_Nil_listrel1
thf(fact_772_not__Nil__listrel1,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel1_nat @ R ) ) ).

% not_Nil_listrel1
thf(fact_773_same__length__different,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( Xs != Ys )
     => ( ( ( size_s8906293707977694520rm_a_b @ Xs )
          = ( size_s8906293707977694520rm_a_b @ Ys ) )
       => ? [Pre: list_term_a_b,X: term_a_b,Xs4: list_term_a_b,Y: term_a_b,Ys5: list_term_a_b] :
            ( ( X != Y )
            & ( Xs
              = ( append_term_a_b @ Pre @ ( append_term_a_b @ ( cons_term_a_b @ X @ nil_term_a_b ) @ Xs4 ) ) )
            & ( Ys
              = ( append_term_a_b @ Pre @ ( append_term_a_b @ ( cons_term_a_b @ Y @ nil_term_a_b ) @ Ys5 ) ) ) ) ) ) ).

% same_length_different
thf(fact_774_same__length__different,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs != Ys )
     => ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
       => ? [Pre: list_nat,X: nat,Xs4: list_nat,Y: nat,Ys5: list_nat] :
            ( ( X != Y )
            & ( Xs
              = ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X @ nil_nat ) @ Xs4 ) ) )
            & ( Ys
              = ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y @ nil_nat ) @ Ys5 ) ) ) ) ) ) ).

% same_length_different
thf(fact_775_Cons__listrel1E2,axiom,
    ! [Xs: list_nat,Y3: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel1_nat @ R ) )
     => ( ! [X: nat] :
            ( ( Xs
              = ( cons_nat @ X @ Ys ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R ) )
       => ~ ! [Zs: list_nat] :
              ( ( Xs
                = ( cons_nat @ Y3 @ Zs ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_776_Cons__listrel1E2,axiom,
    ! [Xs: list_term_a_b,Y3: term_a_b,Ys: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ ( cons_term_a_b @ Y3 @ Ys ) ) @ ( listrel1_term_a_b @ R ) )
     => ( ! [X: term_a_b] :
            ( ( Xs
              = ( cons_term_a_b @ X @ Ys ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R ) )
       => ~ ! [Zs: list_term_a_b] :
              ( ( Xs
                = ( cons_term_a_b @ Y3 @ Zs ) )
             => ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Zs @ Ys ) @ ( listrel1_term_a_b @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_777_Cons__listrel1E1,axiom,
    ! [X3: nat,Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ Ys ) @ ( listrel1_nat @ R ) )
     => ( ! [Y: nat] :
            ( ( Ys
              = ( cons_nat @ Y @ Xs ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R ) )
       => ~ ! [Zs: list_nat] :
              ( ( Ys
                = ( cons_nat @ X3 @ Zs ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_778_Cons__listrel1E1,axiom,
    ! [X3: term_a_b,Xs: list_term_a_b,Ys: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X3 @ Xs ) @ Ys ) @ ( listrel1_term_a_b @ R ) )
     => ( ! [Y: term_a_b] :
            ( ( Ys
              = ( cons_term_a_b @ Y @ Xs ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ R ) )
       => ~ ! [Zs: list_term_a_b] :
              ( ( Ys
                = ( cons_term_a_b @ X3 @ Zs ) )
             => ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Zs ) @ ( listrel1_term_a_b @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_779_listrel1I1,axiom,
    ! [X3: nat,Y3: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Xs ) ) @ ( listrel1_nat @ R ) ) ) ).

% listrel1I1
thf(fact_780_listrel1I1,axiom,
    ! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
     => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X3 @ Xs ) @ ( cons_term_a_b @ Y3 @ Xs ) ) @ ( listrel1_term_a_b @ R ) ) ) ).

% listrel1I1
thf(fact_781_listrel1E,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
     => ~ ! [X: nat,Y: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
           => ! [Us: list_nat,Vs: list_nat] :
                ( ( Xs
                  = ( append_nat @ Us @ ( cons_nat @ X @ Vs ) ) )
               => ( Ys
                 != ( append_nat @ Us @ ( cons_nat @ Y @ Vs ) ) ) ) ) ) ).

% listrel1E
thf(fact_782_listrel1E,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listrel1_term_a_b @ R ) )
     => ~ ! [X: term_a_b,Y: term_a_b] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
           => ! [Us: list_term_a_b,Vs: list_term_a_b] :
                ( ( Xs
                  = ( append_term_a_b @ Us @ ( cons_term_a_b @ X @ Vs ) ) )
               => ( Ys
                 != ( append_term_a_b @ Us @ ( cons_term_a_b @ Y @ Vs ) ) ) ) ) ) ).

% listrel1E
thf(fact_783_listrel1I,axiom,
    ! [X3: nat,Y3: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat,Us2: list_nat,Vs2: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
     => ( ( Xs
          = ( append_nat @ Us2 @ ( cons_nat @ X3 @ Vs2 ) ) )
       => ( ( Ys
            = ( append_nat @ Us2 @ ( cons_nat @ Y3 @ Vs2 ) ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% listrel1I
thf(fact_784_listrel1I,axiom,
    ! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b,Us2: list_term_a_b,Vs2: list_term_a_b,Ys: list_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
     => ( ( Xs
          = ( append_term_a_b @ Us2 @ ( cons_term_a_b @ X3 @ Vs2 ) ) )
       => ( ( Ys
            = ( append_term_a_b @ Us2 @ ( cons_term_a_b @ Y3 @ Vs2 ) ) )
         => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listrel1_term_a_b @ R ) ) ) ) ) ).

% listrel1I
thf(fact_785_listrel__iff__nth,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listre1194016999521814427rm_a_b @ R ) )
      = ( ( ( size_s8906293707977694520rm_a_b @ Xs )
          = ( size_s8906293707977694520rm_a_b @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
           => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Xs @ N3 ) @ ( nth_term_a_b @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_786_subseqs_Osimps_I1_J,axiom,
    ( ( subseqs_term_a_b @ nil_term_a_b )
    = ( cons_list_term_a_b @ nil_term_a_b @ nil_list_term_a_b ) ) ).

% subseqs.simps(1)
thf(fact_787_subseqs_Osimps_I1_J,axiom,
    ( ( subseqs_nat @ nil_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% subseqs.simps(1)
thf(fact_788_nth__append__length,axiom,
    ! [Xs: list_nat,X3: nat,Ys: list_nat] :
      ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X3 @ Ys ) ) @ ( size_size_list_nat @ Xs ) )
      = X3 ) ).

% nth_append_length
thf(fact_789_refl__onD2,axiom,
    ! [A5: set_list_nat,R: set_Pr3451248702717554689st_nat,X3: list_nat,Y3: list_nat] :
      ( ( refl_on_list_nat @ A5 @ R )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ R )
       => ( member_list_nat2 @ Y3 @ A5 ) ) ) ).

% refl_onD2
thf(fact_790_refl__onD2,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
      ( ( refl_o5344404488616963464rm_a_b @ A5 @ R )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ R )
       => ( member5869715511025134514rm_a_b @ Y3 @ A5 ) ) ) ).

% refl_onD2
thf(fact_791_refl__onD2,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X3: nat,Y3: nat] :
      ( ( refl_on_nat @ A5 @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
       => ( member_nat2 @ Y3 @ A5 ) ) ) ).

% refl_onD2
thf(fact_792_refl__onD2,axiom,
    ! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b] :
      ( ( refl_on_term_a_b @ A5 @ R )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
       => ( member_term_a_b2 @ Y3 @ A5 ) ) ) ).

% refl_onD2
thf(fact_793_refl__onD1,axiom,
    ! [A5: set_list_nat,R: set_Pr3451248702717554689st_nat,X3: list_nat,Y3: list_nat] :
      ( ( refl_on_list_nat @ A5 @ R )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ R )
       => ( member_list_nat2 @ X3 @ A5 ) ) ) ).

% refl_onD1
thf(fact_794_refl__onD1,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
      ( ( refl_o5344404488616963464rm_a_b @ A5 @ R )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ R )
       => ( member5869715511025134514rm_a_b @ X3 @ A5 ) ) ) ).

% refl_onD1
thf(fact_795_refl__onD1,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X3: nat,Y3: nat] :
      ( ( refl_on_nat @ A5 @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
       => ( member_nat2 @ X3 @ A5 ) ) ) ).

% refl_onD1
thf(fact_796_refl__onD1,axiom,
    ! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b] :
      ( ( refl_on_term_a_b @ A5 @ R )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
       => ( member_term_a_b2 @ X3 @ A5 ) ) ) ).

% refl_onD1
thf(fact_797_refl__onD,axiom,
    ! [A5: set_list_nat,R: set_Pr3451248702717554689st_nat,A: list_nat] :
      ( ( refl_on_list_nat @ A5 @ R )
     => ( ( member_list_nat2 @ A @ A5 )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_798_refl__onD,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b] :
      ( ( refl_o5344404488616963464rm_a_b @ A5 @ R )
     => ( ( member5869715511025134514rm_a_b @ A @ A5 )
       => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_799_refl__onD,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( refl_on_nat @ A5 @ R )
     => ( ( member_nat2 @ A @ A5 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_800_refl__onD,axiom,
    ! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
      ( ( refl_on_term_a_b @ A5 @ R )
     => ( ( member_term_a_b2 @ A @ A5 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_801_reflD,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
      ( ( refl_on_term_a_b @ top_top_set_term_a_b @ R )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R ) ) ).

% reflD
thf(fact_802_reflI,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ! [X: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R )
     => ( refl_on_term_a_b @ top_top_set_term_a_b @ R ) ) ).

% reflI
thf(fact_803_poss__of__term__Cons,axiom,
    ! [I: nat,P: list_nat,U: term_a_b,F2: a,Ts: list_term_a_b] :
      ( ( member_list_nat2 @ ( cons_nat @ I @ P ) @ ( terms_7168686267159881682rm_a_b @ U @ ( fun_a_b @ F2 @ Ts ) ) )
     => ( member_list_nat2 @ P @ ( terms_7168686267159881682rm_a_b @ U @ ( nth_term_a_b @ Ts @ I ) ) ) ) ).

% poss_of_term_Cons
thf(fact_804_trans__refl__imp__O__id,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b] :
      ( ( trans_on_term_a_b @ top_top_set_term_a_b @ R )
     => ( ( refl_on_term_a_b @ top_top_set_term_a_b @ R )
       => ( ( relcom370159955682700863rm_a_b @ R @ R )
          = R ) ) ) ).

% trans_refl_imp_O_id
thf(fact_805_refl__on__domain,axiom,
    ! [A5: set_list_nat,R: set_Pr3451248702717554689st_nat,A: list_nat,B: list_nat] :
      ( ( refl_on_list_nat @ A5 @ R )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ R )
       => ( ( member_list_nat2 @ A @ A5 )
          & ( member_list_nat2 @ B @ A5 ) ) ) ) ).

% refl_on_domain
thf(fact_806_refl__on__domain,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
      ( ( refl_o5344404488616963464rm_a_b @ A5 @ R )
     => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R )
       => ( ( member5869715511025134514rm_a_b @ A @ A5 )
          & ( member5869715511025134514rm_a_b @ B @ A5 ) ) ) ) ).

% refl_on_domain
thf(fact_807_refl__on__domain,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( refl_on_nat @ A5 @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
       => ( ( member_nat2 @ A @ A5 )
          & ( member_nat2 @ B @ A5 ) ) ) ) ).

% refl_on_domain
thf(fact_808_refl__on__domain,axiom,
    ! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
      ( ( refl_on_term_a_b @ A5 @ R )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
       => ( ( member_term_a_b2 @ A @ A5 )
          & ( member_term_a_b2 @ B @ A5 ) ) ) ) ).

% refl_on_domain
thf(fact_809_listrel1__iff__update,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listrel1_term_a_b @ R ) )
      = ( ? [Y4: term_a_b,N3: nat] :
            ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Xs @ N3 ) @ Y4 ) @ R )
            & ( ord_less_nat @ N3 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
            & ( Ys
              = ( list_update_term_a_b @ Xs @ N3 @ Y4 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_810_list__update__nonempty,axiom,
    ! [Xs: list_term_a_b,K: nat,X3: term_a_b] :
      ( ( ( list_update_term_a_b @ Xs @ K @ X3 )
        = nil_term_a_b )
      = ( Xs = nil_term_a_b ) ) ).

% list_update_nonempty
thf(fact_811_list__update__nonempty,axiom,
    ! [Xs: list_nat,K: nat,X3: nat] :
      ( ( ( list_update_nat @ Xs @ K @ X3 )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% list_update_nonempty
thf(fact_812_list__update__length,axiom,
    ! [Xs: list_nat,X3: nat,Ys: list_nat,Y3: nat] :
      ( ( list_update_nat @ ( append_nat @ Xs @ ( cons_nat @ X3 @ Ys ) ) @ ( size_size_list_nat @ Xs ) @ Y3 )
      = ( append_nat @ Xs @ ( cons_nat @ Y3 @ Ys ) ) ) ).

% list_update_length
thf(fact_813_list__update_Osimps_I1_J,axiom,
    ! [I: nat,V: term_a_b] :
      ( ( list_update_term_a_b @ nil_term_a_b @ I @ V )
      = nil_term_a_b ) ).

% list_update.simps(1)
thf(fact_814_list__update_Osimps_I1_J,axiom,
    ! [I: nat,V: nat] :
      ( ( list_update_nat @ nil_nat @ I @ V )
      = nil_nat ) ).

% list_update.simps(1)
thf(fact_815_list__update__code_I1_J,axiom,
    ! [I: nat,Y3: term_a_b] :
      ( ( list_update_term_a_b @ nil_term_a_b @ I @ Y3 )
      = nil_term_a_b ) ).

% list_update_code(1)
thf(fact_816_list__update__code_I1_J,axiom,
    ! [I: nat,Y3: nat] :
      ( ( list_update_nat @ nil_nat @ I @ Y3 )
      = nil_nat ) ).

% list_update_code(1)
thf(fact_817_replace__term__at_Osimps_I3_J,axiom,
    ! [I: nat,Ts: list_term_a_b,F2: a,Ps: list_nat,T: term_a_b] :
      ( ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ts ) )
       => ( ( term_r6860082780075436317at_a_b @ ( fun_a_b @ F2 @ Ts ) @ ( cons_nat @ I @ Ps ) @ T )
          = ( fun_a_b @ F2 @ ( list_update_term_a_b @ Ts @ I @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ts @ I ) @ Ps @ T ) ) ) ) )
      & ( ~ ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ts ) )
       => ( ( term_r6860082780075436317at_a_b @ ( fun_a_b @ F2 @ Ts ) @ ( cons_nat @ I @ Ps ) @ T )
          = ( fun_a_b @ F2 @ Ts ) ) ) ) ).

% replace_term_at.simps(3)
thf(fact_818_filter2_Oelims,axiom,
    ! [X3: term_a_b > term_a_b > $o,Xa: list_term_a_b,Xb: list_term_a_b,Y3: produc51424535725745577rm_a_b] :
      ( ( ( missin5478484803207743724rm_a_b @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( ( Xa = nil_term_a_b )
         => ( Y3
           != ( produc4885699992713594593rm_a_b @ nil_term_a_b @ nil_term_a_b ) ) )
       => ( ( ? [V2: term_a_b,Va: list_term_a_b] :
                ( Xa
                = ( cons_term_a_b @ V2 @ Va ) )
           => ( ( Xb = nil_term_a_b )
             => ( Y3
               != ( produc4885699992713594593rm_a_b @ nil_term_a_b @ nil_term_a_b ) ) ) )
         => ~ ! [A4: term_a_b,As: list_term_a_b] :
                ( ( Xa
                  = ( cons_term_a_b @ A4 @ As ) )
               => ! [B3: term_a_b,Bs: list_term_a_b] :
                    ( ( Xb
                      = ( cons_term_a_b @ B3 @ Bs ) )
                   => ~ ( ( ( X3 @ A4 @ B3 )
                         => ( Y3
                            = ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ A4 @ ( produc3174924977638783693rm_a_b @ ( missin5478484803207743724rm_a_b @ X3 @ As @ Bs ) ) ) @ ( cons_term_a_b @ B3 @ ( produc3756843454654926607rm_a_b @ ( missin5478484803207743724rm_a_b @ X3 @ As @ Bs ) ) ) ) ) )
                        & ( ~ ( X3 @ A4 @ B3 )
                         => ( Y3
                            = ( missin5478484803207743724rm_a_b @ X3 @ As @ Bs ) ) ) ) ) ) ) ) ) ).

% filter2.elims
thf(fact_819_filter2_Oelims,axiom,
    ! [X3: term_a_b > nat > $o,Xa: list_term_a_b,Xb: list_nat,Y3: produc895009549562661605st_nat] :
      ( ( ( missin8610218420629654184_b_nat @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( ( Xa = nil_term_a_b )
         => ( Y3
           != ( produc5386604813764302749st_nat @ nil_term_a_b @ nil_nat ) ) )
       => ( ( ? [V2: term_a_b,Va: list_term_a_b] :
                ( Xa
                = ( cons_term_a_b @ V2 @ Va ) )
           => ( ( Xb = nil_nat )
             => ( Y3
               != ( produc5386604813764302749st_nat @ nil_term_a_b @ nil_nat ) ) ) )
         => ~ ! [A4: term_a_b,As: list_term_a_b] :
                ( ( Xa
                  = ( cons_term_a_b @ A4 @ As ) )
               => ! [B3: nat,Bs: list_nat] :
                    ( ( Xb
                      = ( cons_nat @ B3 @ Bs ) )
                   => ~ ( ( ( X3 @ A4 @ B3 )
                         => ( Y3
                            = ( produc5386604813764302749st_nat @ ( cons_term_a_b @ A4 @ ( produc1668564938548490633st_nat @ ( missin8610218420629654184_b_nat @ X3 @ As @ Bs ) ) ) @ ( cons_nat @ B3 @ ( produc6625369319960465099st_nat @ ( missin8610218420629654184_b_nat @ X3 @ As @ Bs ) ) ) ) ) )
                        & ( ~ ( X3 @ A4 @ B3 )
                         => ( Y3
                            = ( missin8610218420629654184_b_nat @ X3 @ As @ Bs ) ) ) ) ) ) ) ) ) ).

% filter2.elims
thf(fact_820_filter2_Oelims,axiom,
    ! [X3: nat > term_a_b > $o,Xa: list_nat,Xb: list_term_a_b,Y3: produc9043357390500214885rm_a_b] :
      ( ( ( missin7648764293464682024rm_a_b @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( ( Xa = nil_nat )
         => ( Y3
           != ( produc4336470666470712605rm_a_b @ nil_nat @ nil_term_a_b ) ) )
       => ( ( ? [V2: nat,Va: list_nat] :
                ( Xa
                = ( cons_nat @ V2 @ Va ) )
           => ( ( Xb = nil_term_a_b )
             => ( Y3
               != ( produc4336470666470712605rm_a_b @ nil_nat @ nil_term_a_b ) ) ) )
         => ~ ! [A4: nat,As: list_nat] :
                ( ( Xa
                  = ( cons_nat @ A4 @ As ) )
               => ! [B3: term_a_b,Bs: list_term_a_b] :
                    ( ( Xb
                      = ( cons_term_a_b @ B3 @ Bs ) )
                   => ~ ( ( ( X3 @ A4 @ B3 )
                         => ( Y3
                            = ( produc4336470666470712605rm_a_b @ ( cons_nat @ A4 @ ( produc618430791254900489rm_a_b @ ( missin7648764293464682024rm_a_b @ X3 @ As @ Bs ) ) ) @ ( cons_term_a_b @ B3 @ ( produc5575235172666874955rm_a_b @ ( missin7648764293464682024rm_a_b @ X3 @ As @ Bs ) ) ) ) ) )
                        & ( ~ ( X3 @ A4 @ B3 )
                         => ( Y3
                            = ( missin7648764293464682024rm_a_b @ X3 @ As @ Bs ) ) ) ) ) ) ) ) ) ).

% filter2.elims
thf(fact_821_filter2_Oelims,axiom,
    ! [X3: nat > nat > $o,Xa: list_nat,Xb: list_nat,Y3: produc1828647624359046049st_nat] :
      ( ( ( missin7656342814955748836at_nat @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( ( Xa = nil_nat )
         => ( Y3
           != ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) ) )
       => ( ( ? [V2: nat,Va: list_nat] :
                ( Xa
                = ( cons_nat @ V2 @ Va ) )
           => ( ( Xb = nil_nat )
             => ( Y3
               != ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) ) ) )
         => ~ ! [A4: nat,As: list_nat] :
                ( ( Xa
                  = ( cons_nat @ A4 @ As ) )
               => ! [B3: nat,Bs: list_nat] :
                    ( ( Xb
                      = ( cons_nat @ B3 @ Bs ) )
                   => ~ ( ( ( X3 @ A4 @ B3 )
                         => ( Y3
                            = ( produc2694037385005941721st_nat @ ( cons_nat @ A4 @ ( produc1382935764643595205st_nat @ ( missin7656342814955748836at_nat @ X3 @ As @ Bs ) ) ) @ ( cons_nat @ B3 @ ( produc5865812112468994567st_nat @ ( missin7656342814955748836at_nat @ X3 @ As @ Bs ) ) ) ) ) )
                        & ( ~ ( X3 @ A4 @ B3 )
                         => ( Y3
                            = ( missin7656342814955748836at_nat @ X3 @ As @ Bs ) ) ) ) ) ) ) ) ) ).

% filter2.elims
thf(fact_822_filter2_Osimps_I3_J,axiom,
    ! [P4: nat > nat > $o,A: nat,B: nat,As2: list_nat,Bs2: list_nat] :
      ( ( ( P4 @ A @ B )
       => ( ( missin7656342814955748836at_nat @ P4 @ ( cons_nat @ A @ As2 ) @ ( cons_nat @ B @ Bs2 ) )
          = ( produc2694037385005941721st_nat @ ( cons_nat @ A @ ( produc1382935764643595205st_nat @ ( missin7656342814955748836at_nat @ P4 @ As2 @ Bs2 ) ) ) @ ( cons_nat @ B @ ( produc5865812112468994567st_nat @ ( missin7656342814955748836at_nat @ P4 @ As2 @ Bs2 ) ) ) ) ) )
      & ( ~ ( P4 @ A @ B )
       => ( ( missin7656342814955748836at_nat @ P4 @ ( cons_nat @ A @ As2 ) @ ( cons_nat @ B @ Bs2 ) )
          = ( missin7656342814955748836at_nat @ P4 @ As2 @ Bs2 ) ) ) ) ).

% filter2.simps(3)
thf(fact_823_filter2_Osimps_I1_J,axiom,
    ! [P4: term_a_b > term_a_b > $o,Uu2: list_term_a_b] :
      ( ( missin5478484803207743724rm_a_b @ P4 @ nil_term_a_b @ Uu2 )
      = ( produc4885699992713594593rm_a_b @ nil_term_a_b @ nil_term_a_b ) ) ).

% filter2.simps(1)
thf(fact_824_filter2_Osimps_I1_J,axiom,
    ! [P4: term_a_b > nat > $o,Uu2: list_nat] :
      ( ( missin8610218420629654184_b_nat @ P4 @ nil_term_a_b @ Uu2 )
      = ( produc5386604813764302749st_nat @ nil_term_a_b @ nil_nat ) ) ).

% filter2.simps(1)
thf(fact_825_filter2_Osimps_I1_J,axiom,
    ! [P4: nat > term_a_b > $o,Uu2: list_term_a_b] :
      ( ( missin7648764293464682024rm_a_b @ P4 @ nil_nat @ Uu2 )
      = ( produc4336470666470712605rm_a_b @ nil_nat @ nil_term_a_b ) ) ).

% filter2.simps(1)
thf(fact_826_filter2_Osimps_I1_J,axiom,
    ! [P4: nat > nat > $o,Uu2: list_nat] :
      ( ( missin7656342814955748836at_nat @ P4 @ nil_nat @ Uu2 )
      = ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) ) ).

% filter2.simps(1)
thf(fact_827_filter2_Osimps_I2_J,axiom,
    ! [P4: term_a_b > term_a_b > $o,V: term_a_b,Va2: list_term_a_b] :
      ( ( missin5478484803207743724rm_a_b @ P4 @ ( cons_term_a_b @ V @ Va2 ) @ nil_term_a_b )
      = ( produc4885699992713594593rm_a_b @ nil_term_a_b @ nil_term_a_b ) ) ).

% filter2.simps(2)
thf(fact_828_filter2_Osimps_I2_J,axiom,
    ! [P4: term_a_b > nat > $o,V: term_a_b,Va2: list_term_a_b] :
      ( ( missin8610218420629654184_b_nat @ P4 @ ( cons_term_a_b @ V @ Va2 ) @ nil_nat )
      = ( produc5386604813764302749st_nat @ nil_term_a_b @ nil_nat ) ) ).

% filter2.simps(2)
thf(fact_829_filter2_Osimps_I2_J,axiom,
    ! [P4: nat > term_a_b > $o,V: nat,Va2: list_nat] :
      ( ( missin7648764293464682024rm_a_b @ P4 @ ( cons_nat @ V @ Va2 ) @ nil_term_a_b )
      = ( produc4336470666470712605rm_a_b @ nil_nat @ nil_term_a_b ) ) ).

% filter2.simps(2)
thf(fact_830_filter2_Osimps_I2_J,axiom,
    ! [P4: nat > nat > $o,V: nat,Va2: list_nat] :
      ( ( missin7656342814955748836at_nat @ P4 @ ( cons_nat @ V @ Va2 ) @ nil_nat )
      = ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) ) ).

% filter2.simps(2)
thf(fact_831_replace__term__at_Oelims,axiom,
    ! [X3: term_a_b,Xa: list_nat,Xb: term_a_b,Y3: term_a_b] :
      ( ( ( term_r6860082780075436317at_a_b @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( ( Xa = nil_nat )
         => ( Y3 != Xb ) )
       => ( ! [X: b] :
              ( ( X3
                = ( var_b_a @ X ) )
             => ( ? [V2: nat,Va: list_nat] :
                    ( Xa
                    = ( cons_nat @ V2 @ Va ) )
               => ( Y3
                 != ( var_b_a @ X ) ) ) )
         => ~ ! [F3: a,Ts2: list_term_a_b] :
                ( ( X3
                  = ( fun_a_b @ F3 @ Ts2 ) )
               => ! [I2: nat,Ps2: list_nat] :
                    ( ( Xa
                      = ( cons_nat @ I2 @ Ps2 ) )
                   => ~ ( ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
                         => ( Y3
                            = ( fun_a_b @ F3 @ ( list_update_term_a_b @ Ts2 @ I2 @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ts2 @ I2 ) @ Ps2 @ Xb ) ) ) ) )
                        & ( ~ ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
                         => ( Y3
                            = ( fun_a_b @ F3 @ Ts2 ) ) ) ) ) ) ) ) ) ).

% replace_term_at.elims
thf(fact_832_funas__term_Ocases,axiom,
    ! [X3: term_a_b] :
      ( ! [X: b] :
          ( X3
         != ( var_b_a @ X ) )
     => ~ ! [F3: a,Ts2: list_term_a_b] :
            ( X3
           != ( fun_a_b @ F3 @ Ts2 ) ) ) ).

% funas_term.cases
thf(fact_833_linear__term_Ocases,axiom,
    ! [X3: term_a_b] :
      ( ! [Uu: b] :
          ( X3
         != ( var_b_a @ Uu ) )
     => ~ ! [Uv: a,Ts2: list_term_a_b] :
            ( X3
           != ( fun_a_b @ Uv @ Ts2 ) ) ) ).

% linear_term.cases
thf(fact_834_Term_Oterm_Oexhaust,axiom,
    ! [Y3: term_a_b] :
      ( ! [X12: b] :
          ( Y3
         != ( var_b_a @ X12 ) )
     => ~ ! [X212: a,X222: list_term_a_b] :
            ( Y3
           != ( fun_a_b @ X212 @ X222 ) ) ) ).

% Term.term.exhaust
thf(fact_835_term_Odistinct_I1_J,axiom,
    ! [X1: b,X21: a,X22: list_term_a_b] :
      ( ( var_b_a @ X1 )
     != ( fun_a_b @ X21 @ X22 ) ) ).

% term.distinct(1)
thf(fact_836_term__to__sig_Ocases,axiom,
    ! [X3: produc5279506192219892694rm_a_b] :
      ( ! [F4: set_Pr4934435412358123699_a_nat,V2: b,X: b] :
          ( X3
         != ( produc8030969961714872974rm_a_b @ F4 @ ( produc1437816968797971900rm_a_b @ V2 @ ( var_b_a @ X ) ) ) )
     => ~ ! [F4: set_Pr4934435412358123699_a_nat,V2: b,F3: a,Ts2: list_term_a_b] :
            ( X3
           != ( produc8030969961714872974rm_a_b @ F4 @ ( produc1437816968797971900rm_a_b @ V2 @ ( fun_a_b @ F3 @ Ts2 ) ) ) ) ) ).

% term_to_sig.cases
thf(fact_837_replace__term__at_Osimps_I2_J,axiom,
    ! [X3: b,V: nat,Va2: list_nat,T: term_a_b] :
      ( ( term_r6860082780075436317at_a_b @ ( var_b_a @ X3 ) @ ( cons_nat @ V @ Va2 ) @ T )
      = ( var_b_a @ X3 ) ) ).

% replace_term_at.simps(2)
thf(fact_838_depth_Ocases,axiom,
    ! [X3: term_a_b] :
      ( ! [X: b] :
          ( X3
         != ( var_b_a @ X ) )
     => ( ! [F3: a] :
            ( X3
           != ( fun_a_b @ F3 @ nil_term_a_b ) )
       => ~ ! [F3: a,V2: term_a_b,Va: list_term_a_b] :
              ( X3
             != ( fun_a_b @ F3 @ ( cons_term_a_b @ V2 @ Va ) ) ) ) ) ).

% depth.cases
thf(fact_839_replace__term__at_Ocases,axiom,
    ! [X3: produc2732850333517536310rm_a_b] :
      ( ! [S2: term_a_b,T2: term_a_b] :
          ( X3
         != ( produc3812856575676843240rm_a_b @ S2 @ ( produc5151171985953862413rm_a_b @ nil_nat @ T2 ) ) )
     => ( ! [X: b,V2: nat,Va: list_nat,T2: term_a_b] :
            ( X3
           != ( produc3812856575676843240rm_a_b @ ( var_b_a @ X ) @ ( produc5151171985953862413rm_a_b @ ( cons_nat @ V2 @ Va ) @ T2 ) ) )
       => ~ ! [F3: a,Ts2: list_term_a_b,I2: nat,Ps2: list_nat,T2: term_a_b] :
              ( X3
             != ( produc3812856575676843240rm_a_b @ ( fun_a_b @ F3 @ Ts2 ) @ ( produc5151171985953862413rm_a_b @ ( cons_nat @ I2 @ Ps2 ) @ T2 ) ) ) ) ) ).

% replace_term_at.cases
thf(fact_840_ctxt__at__pos_Ocases,axiom,
    ! [X3: produc3697673438841856213st_nat] :
      ( ! [S2: term_a_b] :
          ( X3
         != ( produc4563063199488751885st_nat @ S2 @ nil_nat ) )
     => ( ! [F3: a,Ss: list_term_a_b,I2: nat,P2: list_nat] :
            ( X3
           != ( produc4563063199488751885st_nat @ ( fun_a_b @ F3 @ Ss ) @ ( cons_nat @ I2 @ P2 ) ) )
       => ~ ! [X: b,V2: nat,Va: list_nat] :
              ( X3
             != ( produc4563063199488751885st_nat @ ( var_b_a @ X ) @ ( cons_nat @ V2 @ Va ) ) ) ) ) ).

% ctxt_at_pos.cases
thf(fact_841_subt__at_Ocases,axiom,
    ! [X3: produc3697673438841856213st_nat] :
      ( ! [S2: term_a_b] :
          ( X3
         != ( produc4563063199488751885st_nat @ S2 @ nil_nat ) )
     => ( ! [F3: a,Ss: list_term_a_b,I2: nat,P2: list_nat] :
            ( X3
           != ( produc4563063199488751885st_nat @ ( fun_a_b @ F3 @ Ss ) @ ( cons_nat @ I2 @ P2 ) ) )
       => ~ ! [X: b,V2: nat,Va: list_nat] :
              ( X3
             != ( produc4563063199488751885st_nat @ ( var_b_a @ X ) @ ( cons_nat @ V2 @ Va ) ) ) ) ) ).

% subt_at.cases
thf(fact_842_fun__at_Ocases,axiom,
    ! [X3: produc3697673438841856213st_nat] :
      ( ! [X: b] :
          ( X3
         != ( produc4563063199488751885st_nat @ ( var_b_a @ X ) @ nil_nat ) )
     => ( ! [F3: a,Ts2: list_term_a_b] :
            ( X3
           != ( produc4563063199488751885st_nat @ ( fun_a_b @ F3 @ Ts2 ) @ nil_nat ) )
       => ( ! [F3: a,Ts2: list_term_a_b,I2: nat,P2: list_nat] :
              ( X3
             != ( produc4563063199488751885st_nat @ ( fun_a_b @ F3 @ Ts2 ) @ ( cons_nat @ I2 @ P2 ) ) )
         => ~ ! [Vb: b,V2: nat,Va: list_nat] :
                ( X3
               != ( produc4563063199488751885st_nat @ ( var_b_a @ Vb ) @ ( cons_nat @ V2 @ Va ) ) ) ) ) ) ).

% fun_at.cases
thf(fact_843_replace__term__at_Opelims,axiom,
    ! [X3: term_a_b,Xa: list_nat,Xb: term_a_b,Y3: term_a_b] :
      ( ( ( term_r6860082780075436317at_a_b @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ X3 @ ( produc5151171985953862413rm_a_b @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_nat )
           => ( ( Y3 = Xb )
             => ~ ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ X3 @ ( produc5151171985953862413rm_a_b @ nil_nat @ Xb ) ) ) ) )
         => ( ! [X: b] :
                ( ( X3
                  = ( var_b_a @ X ) )
               => ! [V2: nat,Va: list_nat] :
                    ( ( Xa
                      = ( cons_nat @ V2 @ Va ) )
                   => ( ( Y3
                        = ( var_b_a @ X ) )
                     => ~ ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ ( var_b_a @ X ) @ ( produc5151171985953862413rm_a_b @ ( cons_nat @ V2 @ Va ) @ Xb ) ) ) ) ) )
           => ~ ! [F3: a,Ts2: list_term_a_b] :
                  ( ( X3
                    = ( fun_a_b @ F3 @ Ts2 ) )
                 => ! [I2: nat,Ps2: list_nat] :
                      ( ( Xa
                        = ( cons_nat @ I2 @ Ps2 ) )
                     => ( ( ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
                           => ( Y3
                              = ( fun_a_b @ F3 @ ( list_update_term_a_b @ Ts2 @ I2 @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ts2 @ I2 ) @ Ps2 @ Xb ) ) ) ) )
                          & ( ~ ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
                           => ( Y3
                              = ( fun_a_b @ F3 @ Ts2 ) ) ) )
                       => ~ ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ ( fun_a_b @ F3 @ Ts2 ) @ ( produc5151171985953862413rm_a_b @ ( cons_nat @ I2 @ Ps2 ) @ Xb ) ) ) ) ) ) ) ) ) ) ).

% replace_term_at.pelims
thf(fact_844_gterm__of__term_Ocases,axiom,
    ! [X3: term_a_b] :
      ( ! [F3: a,Ts2: list_term_a_b] :
          ( X3
         != ( fun_a_b @ F3 @ Ts2 ) )
     => ~ ! [V2: b] :
            ( X3
           != ( var_b_a @ V2 ) ) ) ).

% gterm_of_term.cases
thf(fact_845_filter2_Opelims,axiom,
    ! [X3: term_a_b > term_a_b > $o,Xa: list_term_a_b,Xb: list_term_a_b,Y3: produc51424535725745577rm_a_b] :
      ( ( ( missin5478484803207743724rm_a_b @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P1470738975870918523rm_a_b @ missin5945364489896066401rm_a_b @ ( produc1278856658843369954rm_a_b @ X3 @ ( produc4885699992713594593rm_a_b @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_term_a_b )
           => ( ( Y3
                = ( produc4885699992713594593rm_a_b @ nil_term_a_b @ nil_term_a_b ) )
             => ~ ( accp_P1470738975870918523rm_a_b @ missin5945364489896066401rm_a_b @ ( produc1278856658843369954rm_a_b @ X3 @ ( produc4885699992713594593rm_a_b @ nil_term_a_b @ Xb ) ) ) ) )
         => ( ! [V2: term_a_b,Va: list_term_a_b] :
                ( ( Xa
                  = ( cons_term_a_b @ V2 @ Va ) )
               => ( ( Xb = nil_term_a_b )
                 => ( ( Y3
                      = ( produc4885699992713594593rm_a_b @ nil_term_a_b @ nil_term_a_b ) )
                   => ~ ( accp_P1470738975870918523rm_a_b @ missin5945364489896066401rm_a_b @ ( produc1278856658843369954rm_a_b @ X3 @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ V2 @ Va ) @ nil_term_a_b ) ) ) ) ) )
           => ~ ! [A4: term_a_b,As: list_term_a_b] :
                  ( ( Xa
                    = ( cons_term_a_b @ A4 @ As ) )
                 => ! [B3: term_a_b,Bs: list_term_a_b] :
                      ( ( Xb
                        = ( cons_term_a_b @ B3 @ Bs ) )
                     => ( ( ( ( X3 @ A4 @ B3 )
                           => ( Y3
                              = ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ A4 @ ( produc3174924977638783693rm_a_b @ ( missin5478484803207743724rm_a_b @ X3 @ As @ Bs ) ) ) @ ( cons_term_a_b @ B3 @ ( produc3756843454654926607rm_a_b @ ( missin5478484803207743724rm_a_b @ X3 @ As @ Bs ) ) ) ) ) )
                          & ( ~ ( X3 @ A4 @ B3 )
                           => ( Y3
                              = ( missin5478484803207743724rm_a_b @ X3 @ As @ Bs ) ) ) )
                       => ~ ( accp_P1470738975870918523rm_a_b @ missin5945364489896066401rm_a_b @ ( produc1278856658843369954rm_a_b @ X3 @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ A4 @ As ) @ ( cons_term_a_b @ B3 @ Bs ) ) ) ) ) ) ) ) ) ) ) ).

% filter2.pelims
thf(fact_846_filter2_Opelims,axiom,
    ! [X3: term_a_b > nat > $o,Xa: list_term_a_b,Xb: list_nat,Y3: produc895009549562661605st_nat] :
      ( ( ( missin8610218420629654184_b_nat @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P3739882250219358835st_nat @ missin3682495897811778973_b_nat @ ( produc7813177934582050522st_nat @ X3 @ ( produc5386604813764302749st_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_term_a_b )
           => ( ( Y3
                = ( produc5386604813764302749st_nat @ nil_term_a_b @ nil_nat ) )
             => ~ ( accp_P3739882250219358835st_nat @ missin3682495897811778973_b_nat @ ( produc7813177934582050522st_nat @ X3 @ ( produc5386604813764302749st_nat @ nil_term_a_b @ Xb ) ) ) ) )
         => ( ! [V2: term_a_b,Va: list_term_a_b] :
                ( ( Xa
                  = ( cons_term_a_b @ V2 @ Va ) )
               => ( ( Xb = nil_nat )
                 => ( ( Y3
                      = ( produc5386604813764302749st_nat @ nil_term_a_b @ nil_nat ) )
                   => ~ ( accp_P3739882250219358835st_nat @ missin3682495897811778973_b_nat @ ( produc7813177934582050522st_nat @ X3 @ ( produc5386604813764302749st_nat @ ( cons_term_a_b @ V2 @ Va ) @ nil_nat ) ) ) ) ) )
           => ~ ! [A4: term_a_b,As: list_term_a_b] :
                  ( ( Xa
                    = ( cons_term_a_b @ A4 @ As ) )
                 => ! [B3: nat,Bs: list_nat] :
                      ( ( Xb
                        = ( cons_nat @ B3 @ Bs ) )
                     => ( ( ( ( X3 @ A4 @ B3 )
                           => ( Y3
                              = ( produc5386604813764302749st_nat @ ( cons_term_a_b @ A4 @ ( produc1668564938548490633st_nat @ ( missin8610218420629654184_b_nat @ X3 @ As @ Bs ) ) ) @ ( cons_nat @ B3 @ ( produc6625369319960465099st_nat @ ( missin8610218420629654184_b_nat @ X3 @ As @ Bs ) ) ) ) ) )
                          & ( ~ ( X3 @ A4 @ B3 )
                           => ( Y3
                              = ( missin8610218420629654184_b_nat @ X3 @ As @ Bs ) ) ) )
                       => ~ ( accp_P3739882250219358835st_nat @ missin3682495897811778973_b_nat @ ( produc7813177934582050522st_nat @ X3 @ ( produc5386604813764302749st_nat @ ( cons_term_a_b @ A4 @ As ) @ ( cons_nat @ B3 @ Bs ) ) ) ) ) ) ) ) ) ) ) ).

% filter2.pelims
thf(fact_847_filter2_Opelims,axiom,
    ! [X3: nat > term_a_b > $o,Xa: list_nat,Xb: list_term_a_b,Y3: produc9043357390500214885rm_a_b] :
      ( ( ( missin7648764293464682024rm_a_b @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P5656397868249209331rm_a_b @ missin2721041770646806813rm_a_b @ ( produc2750092164434624090rm_a_b @ X3 @ ( produc4336470666470712605rm_a_b @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_nat )
           => ( ( Y3
                = ( produc4336470666470712605rm_a_b @ nil_nat @ nil_term_a_b ) )
             => ~ ( accp_P5656397868249209331rm_a_b @ missin2721041770646806813rm_a_b @ ( produc2750092164434624090rm_a_b @ X3 @ ( produc4336470666470712605rm_a_b @ nil_nat @ Xb ) ) ) ) )
         => ( ! [V2: nat,Va: list_nat] :
                ( ( Xa
                  = ( cons_nat @ V2 @ Va ) )
               => ( ( Xb = nil_term_a_b )
                 => ( ( Y3
                      = ( produc4336470666470712605rm_a_b @ nil_nat @ nil_term_a_b ) )
                   => ~ ( accp_P5656397868249209331rm_a_b @ missin2721041770646806813rm_a_b @ ( produc2750092164434624090rm_a_b @ X3 @ ( produc4336470666470712605rm_a_b @ ( cons_nat @ V2 @ Va ) @ nil_term_a_b ) ) ) ) ) )
           => ~ ! [A4: nat,As: list_nat] :
                  ( ( Xa
                    = ( cons_nat @ A4 @ As ) )
                 => ! [B3: term_a_b,Bs: list_term_a_b] :
                      ( ( Xb
                        = ( cons_term_a_b @ B3 @ Bs ) )
                     => ( ( ( ( X3 @ A4 @ B3 )
                           => ( Y3
                              = ( produc4336470666470712605rm_a_b @ ( cons_nat @ A4 @ ( produc618430791254900489rm_a_b @ ( missin7648764293464682024rm_a_b @ X3 @ As @ Bs ) ) ) @ ( cons_term_a_b @ B3 @ ( produc5575235172666874955rm_a_b @ ( missin7648764293464682024rm_a_b @ X3 @ As @ Bs ) ) ) ) ) )
                          & ( ~ ( X3 @ A4 @ B3 )
                           => ( Y3
                              = ( missin7648764293464682024rm_a_b @ X3 @ As @ Bs ) ) ) )
                       => ~ ( accp_P5656397868249209331rm_a_b @ missin2721041770646806813rm_a_b @ ( produc2750092164434624090rm_a_b @ X3 @ ( produc4336470666470712605rm_a_b @ ( cons_nat @ A4 @ As ) @ ( cons_term_a_b @ B3 @ Bs ) ) ) ) ) ) ) ) ) ) ) ).

% filter2.pelims
thf(fact_848_filter2_Opelims,axiom,
    ! [X3: nat > nat > $o,Xa: list_nat,Xb: list_nat,Y3: produc1828647624359046049st_nat] :
      ( ( ( missin7656342814955748836at_nat @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P7766589077260926187st_nat @ missin8708480906560974681at_nat @ ( produc3127733452865184594st_nat @ X3 @ ( produc2694037385005941721st_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_nat )
           => ( ( Y3
                = ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) )
             => ~ ( accp_P7766589077260926187st_nat @ missin8708480906560974681at_nat @ ( produc3127733452865184594st_nat @ X3 @ ( produc2694037385005941721st_nat @ nil_nat @ Xb ) ) ) ) )
         => ( ! [V2: nat,Va: list_nat] :
                ( ( Xa
                  = ( cons_nat @ V2 @ Va ) )
               => ( ( Xb = nil_nat )
                 => ( ( Y3
                      = ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) )
                   => ~ ( accp_P7766589077260926187st_nat @ missin8708480906560974681at_nat @ ( produc3127733452865184594st_nat @ X3 @ ( produc2694037385005941721st_nat @ ( cons_nat @ V2 @ Va ) @ nil_nat ) ) ) ) ) )
           => ~ ! [A4: nat,As: list_nat] :
                  ( ( Xa
                    = ( cons_nat @ A4 @ As ) )
                 => ! [B3: nat,Bs: list_nat] :
                      ( ( Xb
                        = ( cons_nat @ B3 @ Bs ) )
                     => ( ( ( ( X3 @ A4 @ B3 )
                           => ( Y3
                              = ( produc2694037385005941721st_nat @ ( cons_nat @ A4 @ ( produc1382935764643595205st_nat @ ( missin7656342814955748836at_nat @ X3 @ As @ Bs ) ) ) @ ( cons_nat @ B3 @ ( produc5865812112468994567st_nat @ ( missin7656342814955748836at_nat @ X3 @ As @ Bs ) ) ) ) ) )
                          & ( ~ ( X3 @ A4 @ B3 )
                           => ( Y3
                              = ( missin7656342814955748836at_nat @ X3 @ As @ Bs ) ) ) )
                       => ~ ( accp_P7766589077260926187st_nat @ missin8708480906560974681at_nat @ ( produc3127733452865184594st_nat @ X3 @ ( produc2694037385005941721st_nat @ ( cons_nat @ A4 @ As ) @ ( cons_nat @ B3 @ Bs ) ) ) ) ) ) ) ) ) ) ) ).

% filter2.pelims
thf(fact_849_map__tailrec__rev_Opelims,axiom,
    ! [X3: term_a_b > nat,Xa: list_term_a_b,Xb: list_nat,Y3: list_nat] :
      ( ( ( map_ta4339121960748780251_b_nat @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P6552001184412372814st_nat @ map_ta8187459354307304746_b_nat @ ( produc8252734431196571465st_nat @ X3 @ ( produc5386604813764302749st_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_term_a_b )
           => ( ( Y3 = Xb )
             => ~ ( accp_P6552001184412372814st_nat @ map_ta8187459354307304746_b_nat @ ( produc8252734431196571465st_nat @ X3 @ ( produc5386604813764302749st_nat @ nil_term_a_b @ Xb ) ) ) ) )
         => ~ ! [A4: term_a_b,As: list_term_a_b] :
                ( ( Xa
                  = ( cons_term_a_b @ A4 @ As ) )
               => ( ( Y3
                    = ( map_ta4339121960748780251_b_nat @ X3 @ As @ ( cons_nat @ ( X3 @ A4 ) @ Xb ) ) )
                 => ~ ( accp_P6552001184412372814st_nat @ map_ta8187459354307304746_b_nat @ ( produc8252734431196571465st_nat @ X3 @ ( produc5386604813764302749st_nat @ ( cons_term_a_b @ A4 @ As ) @ Xb ) ) ) ) ) ) ) ) ).

% map_tailrec_rev.pelims
thf(fact_850_map__tailrec__rev_Opelims,axiom,
    ! [X3: nat > nat,Xa: list_nat,Xb: list_nat,Y3: list_nat] :
      ( ( ( map_ta7164188454487880599at_nat @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P3924000266242761678st_nat @ map_ta8615873517111064934at_nat @ ( produc4626581765195395529st_nat @ X3 @ ( produc2694037385005941721st_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_nat )
           => ( ( Y3 = Xb )
             => ~ ( accp_P3924000266242761678st_nat @ map_ta8615873517111064934at_nat @ ( produc4626581765195395529st_nat @ X3 @ ( produc2694037385005941721st_nat @ nil_nat @ Xb ) ) ) ) )
         => ~ ! [A4: nat,As: list_nat] :
                ( ( Xa
                  = ( cons_nat @ A4 @ As ) )
               => ( ( Y3
                    = ( map_ta7164188454487880599at_nat @ X3 @ As @ ( cons_nat @ ( X3 @ A4 ) @ Xb ) ) )
                 => ~ ( accp_P3924000266242761678st_nat @ map_ta8615873517111064934at_nat @ ( produc4626581765195395529st_nat @ X3 @ ( produc2694037385005941721st_nat @ ( cons_nat @ A4 @ As ) @ Xb ) ) ) ) ) ) ) ) ).

% map_tailrec_rev.pelims
thf(fact_851_lex__take__index,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( lex_term_a_b @ R ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
           => ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ys ) )
             => ( ( ( take_term_a_b @ I2 @ Xs )
                  = ( take_term_a_b @ I2 @ Ys ) )
               => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Xs @ I2 ) @ ( nth_term_a_b @ Ys @ I2 ) ) @ R ) ) ) ) ) ).

% lex_take_index
thf(fact_852_take__Nil,axiom,
    ! [N: nat] :
      ( ( take_term_a_b @ N @ nil_term_a_b )
      = nil_term_a_b ) ).

% take_Nil
thf(fact_853_take__Nil,axiom,
    ! [N: nat] :
      ( ( take_nat @ N @ nil_nat )
      = nil_nat ) ).

% take_Nil
thf(fact_854_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F2: nat > nat,A: nat,As2: list_nat,Bs2: list_nat] :
      ( ( map_ta7164188454487880599at_nat @ F2 @ ( cons_nat @ A @ As2 ) @ Bs2 )
      = ( map_ta7164188454487880599at_nat @ F2 @ As2 @ ( cons_nat @ ( F2 @ A ) @ Bs2 ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_855_map__tailrec__rev_Oelims,axiom,
    ! [X3: term_a_b > nat,Xa: list_term_a_b,Xb: list_nat,Y3: list_nat] :
      ( ( ( map_ta4339121960748780251_b_nat @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( ( Xa = nil_term_a_b )
         => ( Y3 != Xb ) )
       => ~ ! [A4: term_a_b,As: list_term_a_b] :
              ( ( Xa
                = ( cons_term_a_b @ A4 @ As ) )
             => ( Y3
               != ( map_ta4339121960748780251_b_nat @ X3 @ As @ ( cons_nat @ ( X3 @ A4 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_856_map__tailrec__rev_Oelims,axiom,
    ! [X3: nat > nat,Xa: list_nat,Xb: list_nat,Y3: list_nat] :
      ( ( ( map_ta7164188454487880599at_nat @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( ( Xa = nil_nat )
         => ( Y3 != Xb ) )
       => ~ ! [A4: nat,As: list_nat] :
              ( ( Xa
                = ( cons_nat @ A4 @ As ) )
             => ( Y3
               != ( map_ta7164188454487880599at_nat @ X3 @ As @ ( cons_nat @ ( X3 @ A4 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_857_take__Suc__conv__app__nth,axiom,
    ! [I: nat,Xs: list_term_a_b] :
      ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
     => ( ( take_term_a_b @ ( suc @ I ) @ Xs )
        = ( append_term_a_b @ ( take_term_a_b @ I @ Xs ) @ ( cons_term_a_b @ ( nth_term_a_b @ Xs @ I ) @ nil_term_a_b ) ) ) ) ).

% take_Suc_conv_app_nth
thf(fact_858_take__Suc__conv__app__nth,axiom,
    ! [I: nat,Xs: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( take_nat @ ( suc @ I ) @ Xs )
        = ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I ) @ nil_nat ) ) ) ) ).

% take_Suc_conv_app_nth
thf(fact_859_lexord__take__index__conv,axiom,
    ! [X3: list_term_a_b,Y3: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X3 @ Y3 ) @ ( lexord_term_a_b @ R ) )
      = ( ( ( ord_less_nat @ ( size_s8906293707977694520rm_a_b @ X3 ) @ ( size_s8906293707977694520rm_a_b @ Y3 ) )
          & ( ( take_term_a_b @ ( size_s8906293707977694520rm_a_b @ X3 ) @ Y3 )
            = X3 ) )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( ord_min_nat @ ( size_s8906293707977694520rm_a_b @ X3 ) @ ( size_s8906293707977694520rm_a_b @ Y3 ) ) )
            & ( ( take_term_a_b @ I3 @ X3 )
              = ( take_term_a_b @ I3 @ Y3 ) )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ X3 @ I3 ) @ ( nth_term_a_b @ Y3 @ I3 ) ) @ R ) ) ) ) ).

% lexord_take_index_conv
thf(fact_860_in__listsI,axiom,
    ! [Xs: list_list_nat,A5: set_list_nat] :
      ( ! [X: list_nat] :
          ( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
         => ( member_list_nat2 @ X @ A5 ) )
     => ( member_list_list_nat @ Xs @ ( lists_list_nat @ A5 ) ) ) ).

% in_listsI
thf(fact_861_in__listsI,axiom,
    ! [Xs: list_P8875379029341186191rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ! [X: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
         => ( member5869715511025134514rm_a_b @ X @ A5 ) )
     => ( member7646937072540562872rm_a_b @ Xs @ ( lists_3936190290964281341rm_a_b @ A5 ) ) ) ).

% in_listsI
thf(fact_862_in__listsI,axiom,
    ! [Xs: list_nat,A5: set_nat] :
      ( ! [X: nat] :
          ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
         => ( member_nat2 @ X @ A5 ) )
     => ( member_list_nat2 @ Xs @ ( lists_nat @ A5 ) ) ) ).

% in_listsI
thf(fact_863_nth__Cons__Suc,axiom,
    ! [X3: nat,Xs: list_nat,N: nat] :
      ( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ ( suc @ N ) )
      = ( nth_nat @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_864_take__Suc__Cons,axiom,
    ! [N: nat,X3: nat,Xs: list_nat] :
      ( ( take_nat @ ( suc @ N ) @ ( cons_nat @ X3 @ Xs ) )
      = ( cons_nat @ X3 @ ( take_nat @ N @ Xs ) ) ) ).

% take_Suc_Cons
thf(fact_865_in__set__takeD,axiom,
    ! [X3: list_nat,N: nat,Xs: list_list_nat] :
      ( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ ( take_list_nat @ N @ Xs ) ) )
     => ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) ) ) ).

% in_set_takeD
thf(fact_866_in__set__takeD,axiom,
    ! [X3: produc357393685978478089rm_a_b,N: nat,Xs: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ ( take_P7698341771522494711rm_a_b @ N @ Xs ) ) )
     => ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) ) ) ).

% in_set_takeD
thf(fact_867_in__set__takeD,axiom,
    ! [X3: nat,N: nat,Xs: list_nat] :
      ( ( member_nat2 @ X3 @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) )
     => ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) ) ) ).

% in_set_takeD
thf(fact_868_subset__code_I1_J,axiom,
    ! [Xs: list_list_nat,B5: set_list_nat] :
      ( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ B5 )
      = ( ! [X4: list_nat] :
            ( ( member_list_nat2 @ X4 @ ( set_list_nat2 @ Xs ) )
           => ( member_list_nat2 @ X4 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_869_subset__code_I1_J,axiom,
    ! [Xs: list_P8875379029341186191rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ ( set_Pr2370880052973118494rm_a_b @ Xs ) @ B5 )
      = ( ! [X4: produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ X4 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
           => ( member5869715511025134514rm_a_b @ X4 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_870_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B5 )
      = ( ! [X4: nat] :
            ( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
           => ( member_nat2 @ X4 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_871_in__listsD,axiom,
    ! [Xs: list_list_nat,A5: set_list_nat] :
      ( ( member_list_list_nat @ Xs @ ( lists_list_nat @ A5 ) )
     => ! [X5: list_nat] :
          ( ( member_list_nat2 @ X5 @ ( set_list_nat2 @ Xs ) )
         => ( member_list_nat2 @ X5 @ A5 ) ) ) ).

% in_listsD
thf(fact_872_in__listsD,axiom,
    ! [Xs: list_P8875379029341186191rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member7646937072540562872rm_a_b @ Xs @ ( lists_3936190290964281341rm_a_b @ A5 ) )
     => ! [X5: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X5 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
         => ( member5869715511025134514rm_a_b @ X5 @ A5 ) ) ) ).

% in_listsD
thf(fact_873_in__listsD,axiom,
    ! [Xs: list_nat,A5: set_nat] :
      ( ( member_list_nat2 @ Xs @ ( lists_nat @ A5 ) )
     => ! [X5: nat] :
          ( ( member_nat2 @ X5 @ ( set_nat2 @ Xs ) )
         => ( member_nat2 @ X5 @ A5 ) ) ) ).

% in_listsD
thf(fact_874_in__lists__conv__set,axiom,
    ! [Xs: list_list_nat,A5: set_list_nat] :
      ( ( member_list_list_nat @ Xs @ ( lists_list_nat @ A5 ) )
      = ( ! [X4: list_nat] :
            ( ( member_list_nat2 @ X4 @ ( set_list_nat2 @ Xs ) )
           => ( member_list_nat2 @ X4 @ A5 ) ) ) ) ).

% in_lists_conv_set
thf(fact_875_in__lists__conv__set,axiom,
    ! [Xs: list_P8875379029341186191rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
      ( ( member7646937072540562872rm_a_b @ Xs @ ( lists_3936190290964281341rm_a_b @ A5 ) )
      = ( ! [X4: produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ X4 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
           => ( member5869715511025134514rm_a_b @ X4 @ A5 ) ) ) ) ).

% in_lists_conv_set
thf(fact_876_in__lists__conv__set,axiom,
    ! [Xs: list_nat,A5: set_nat] :
      ( ( member_list_nat2 @ Xs @ ( lists_nat @ A5 ) )
      = ( ! [X4: nat] :
            ( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
           => ( member_nat2 @ X4 @ A5 ) ) ) ) ).

% in_lists_conv_set
thf(fact_877_list__ex1__iff,axiom,
    ( list_ex1_list_nat
    = ( ^ [P7: list_nat > $o,Xs2: list_list_nat] :
        ? [X4: list_nat] :
          ( ( member_list_nat2 @ X4 @ ( set_list_nat2 @ Xs2 ) )
          & ( P7 @ X4 )
          & ! [Y4: list_nat] :
              ( ( ( member_list_nat2 @ Y4 @ ( set_list_nat2 @ Xs2 ) )
                & ( P7 @ Y4 ) )
             => ( Y4 = X4 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_878_list__ex1__iff,axiom,
    ( list_e2893281867963234770rm_a_b
    = ( ^ [P7: produc357393685978478089rm_a_b > $o,Xs2: list_P8875379029341186191rm_a_b] :
        ? [X4: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X4 @ ( set_Pr2370880052973118494rm_a_b @ Xs2 ) )
          & ( P7 @ X4 )
          & ! [Y4: produc357393685978478089rm_a_b] :
              ( ( ( member5869715511025134514rm_a_b @ Y4 @ ( set_Pr2370880052973118494rm_a_b @ Xs2 ) )
                & ( P7 @ Y4 ) )
             => ( Y4 = X4 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_879_list__ex1__iff,axiom,
    ( list_ex1_nat
    = ( ^ [P7: nat > $o,Xs2: list_nat] :
        ? [X4: nat] :
          ( ( member_nat2 @ X4 @ ( set_nat2 @ Xs2 ) )
          & ( P7 @ X4 )
          & ! [Y4: nat] :
              ( ( ( member_nat2 @ Y4 @ ( set_nat2 @ Xs2 ) )
                & ( P7 @ Y4 ) )
             => ( Y4 = X4 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_880_subseqs__refl,axiom,
    ! [Xs: list_nat] : ( member_list_nat2 @ Xs @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ).

% subseqs_refl
thf(fact_881_in__set__member,axiom,
    ! [X3: list_nat,Xs: list_list_nat] :
      ( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
      = ( member_list_nat @ Xs @ X3 ) ) ).

% in_set_member
thf(fact_882_in__set__member,axiom,
    ! [X3: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
      = ( member1202727356028726564rm_a_b @ Xs @ X3 ) ) ).

% in_set_member
thf(fact_883_in__set__member,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
      = ( member_nat @ Xs @ X3 ) ) ).

% in_set_member
thf(fact_884_list_Oset__intros_I2_J,axiom,
    ! [Y3: list_nat,X22: list_list_nat,X21: list_nat] :
      ( ( member_list_nat2 @ Y3 @ ( set_list_nat2 @ X22 ) )
     => ( member_list_nat2 @ Y3 @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_885_list_Oset__intros_I2_J,axiom,
    ! [Y3: produc357393685978478089rm_a_b,X22: list_P8875379029341186191rm_a_b,X21: produc357393685978478089rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ Y3 @ ( set_Pr2370880052973118494rm_a_b @ X22 ) )
     => ( member5869715511025134514rm_a_b @ Y3 @ ( set_Pr2370880052973118494rm_a_b @ ( cons_P3536305108106557631rm_a_b @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_886_list_Oset__intros_I2_J,axiom,
    ! [Y3: nat,X22: list_nat,X21: nat] :
      ( ( member_nat2 @ Y3 @ ( set_nat2 @ X22 ) )
     => ( member_nat2 @ Y3 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_887_list_Oset__intros_I1_J,axiom,
    ! [X21: list_nat,X22: list_list_nat] : ( member_list_nat2 @ X21 @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_888_list_Oset__intros_I1_J,axiom,
    ! [X21: produc357393685978478089rm_a_b,X22: list_P8875379029341186191rm_a_b] : ( member5869715511025134514rm_a_b @ X21 @ ( set_Pr2370880052973118494rm_a_b @ ( cons_P3536305108106557631rm_a_b @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_889_list_Oset__intros_I1_J,axiom,
    ! [X21: nat,X22: list_nat] : ( member_nat2 @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_890_list_Oset__cases,axiom,
    ! [E: list_nat,A: list_list_nat] :
      ( ( member_list_nat2 @ E @ ( set_list_nat2 @ A ) )
     => ( ! [Z22: list_list_nat] :
            ( A
           != ( cons_list_nat @ E @ Z22 ) )
       => ~ ! [Z1: list_nat,Z22: list_list_nat] :
              ( ( A
                = ( cons_list_nat @ Z1 @ Z22 ) )
             => ~ ( member_list_nat2 @ E @ ( set_list_nat2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_891_list_Oset__cases,axiom,
    ! [E: produc357393685978478089rm_a_b,A: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ E @ ( set_Pr2370880052973118494rm_a_b @ A ) )
     => ( ! [Z22: list_P8875379029341186191rm_a_b] :
            ( A
           != ( cons_P3536305108106557631rm_a_b @ E @ Z22 ) )
       => ~ ! [Z1: produc357393685978478089rm_a_b,Z22: list_P8875379029341186191rm_a_b] :
              ( ( A
                = ( cons_P3536305108106557631rm_a_b @ Z1 @ Z22 ) )
             => ~ ( member5869715511025134514rm_a_b @ E @ ( set_Pr2370880052973118494rm_a_b @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_892_list_Oset__cases,axiom,
    ! [E: nat,A: list_nat] :
      ( ( member_nat2 @ E @ ( set_nat2 @ A ) )
     => ( ! [Z22: list_nat] :
            ( A
           != ( cons_nat @ E @ Z22 ) )
       => ~ ! [Z1: nat,Z22: list_nat] :
              ( ( A
                = ( cons_nat @ Z1 @ Z22 ) )
             => ~ ( member_nat2 @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_893_set__ConsD,axiom,
    ! [Y3: list_nat,X3: list_nat,Xs: list_list_nat] :
      ( ( member_list_nat2 @ Y3 @ ( set_list_nat2 @ ( cons_list_nat @ X3 @ Xs ) ) )
     => ( ( Y3 = X3 )
        | ( member_list_nat2 @ Y3 @ ( set_list_nat2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_894_set__ConsD,axiom,
    ! [Y3: produc357393685978478089rm_a_b,X3: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ Y3 @ ( set_Pr2370880052973118494rm_a_b @ ( cons_P3536305108106557631rm_a_b @ X3 @ Xs ) ) )
     => ( ( Y3 = X3 )
        | ( member5869715511025134514rm_a_b @ Y3 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_895_set__ConsD,axiom,
    ! [Y3: nat,X3: nat,Xs: list_nat] :
      ( ( member_nat2 @ Y3 @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) )
     => ( ( Y3 = X3 )
        | ( member_nat2 @ Y3 @ ( set_nat2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_896_set__subset__Cons,axiom,
    ! [Xs: list_nat,X3: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_897_split__list__first__prop__iff,axiom,
    ! [Xs: list_nat,P4: nat > $o] :
      ( ( ? [X4: nat] :
            ( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
            & ( P4 @ X4 ) ) )
      = ( ? [Ys2: list_nat,X4: nat] :
            ( ? [Zs3: list_nat] :
                ( Xs
                = ( append_nat @ Ys2 @ ( cons_nat @ X4 @ Zs3 ) ) )
            & ( P4 @ X4 )
            & ! [Y4: nat] :
                ( ( member_nat2 @ Y4 @ ( set_nat2 @ Ys2 ) )
               => ~ ( P4 @ Y4 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_898_split__list__last__prop__iff,axiom,
    ! [Xs: list_nat,P4: nat > $o] :
      ( ( ? [X4: nat] :
            ( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
            & ( P4 @ X4 ) ) )
      = ( ? [Ys2: list_nat,X4: nat,Zs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ X4 @ Zs3 ) ) )
            & ( P4 @ X4 )
            & ! [Y4: nat] :
                ( ( member_nat2 @ Y4 @ ( set_nat2 @ Zs3 ) )
               => ~ ( P4 @ Y4 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_899_in__set__conv__decomp__first,axiom,
    ! [X3: list_nat,Xs: list_list_nat] :
      ( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
      = ( ? [Ys2: list_list_nat,Zs3: list_list_nat] :
            ( ( Xs
              = ( append_list_nat @ Ys2 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
            & ~ ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_900_in__set__conv__decomp__first,axiom,
    ! [X3: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
      = ( ? [Ys2: list_P8875379029341186191rm_a_b,Zs3: list_P8875379029341186191rm_a_b] :
            ( ( Xs
              = ( append1152621114427393060rm_a_b @ Ys2 @ ( cons_P3536305108106557631rm_a_b @ X3 @ Zs3 ) ) )
            & ~ ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_901_in__set__conv__decomp__first,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
      = ( ? [Ys2: list_nat,Zs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs3 ) ) )
            & ~ ( member_nat2 @ X3 @ ( set_nat2 @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_902_in__set__conv__decomp__last,axiom,
    ! [X3: list_nat,Xs: list_list_nat] :
      ( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
      = ( ? [Ys2: list_list_nat,Zs3: list_list_nat] :
            ( ( Xs
              = ( append_list_nat @ Ys2 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
            & ~ ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_903_in__set__conv__decomp__last,axiom,
    ! [X3: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
      = ( ? [Ys2: list_P8875379029341186191rm_a_b,Zs3: list_P8875379029341186191rm_a_b] :
            ( ( Xs
              = ( append1152621114427393060rm_a_b @ Ys2 @ ( cons_P3536305108106557631rm_a_b @ X3 @ Zs3 ) ) )
            & ~ ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_904_in__set__conv__decomp__last,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
      = ( ? [Ys2: list_nat,Zs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs3 ) ) )
            & ~ ( member_nat2 @ X3 @ ( set_nat2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_905_split__list__first__propE,axiom,
    ! [Xs: list_nat,P4: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat2 @ X5 @ ( set_nat2 @ Xs ) )
          & ( P4 @ X5 ) )
     => ~ ! [Ys3: list_nat,X: nat] :
            ( ? [Zs: list_nat] :
                ( Xs
                = ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs ) ) )
           => ( ( P4 @ X )
             => ~ ! [Xa2: nat] :
                    ( ( member_nat2 @ Xa2 @ ( set_nat2 @ Ys3 ) )
                   => ~ ( P4 @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_906_split__list__last__propE,axiom,
    ! [Xs: list_nat,P4: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat2 @ X5 @ ( set_nat2 @ Xs ) )
          & ( P4 @ X5 ) )
     => ~ ! [Ys3: list_nat,X: nat,Zs: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs ) ) )
           => ( ( P4 @ X )
             => ~ ! [Xa2: nat] :
                    ( ( member_nat2 @ Xa2 @ ( set_nat2 @ Zs ) )
                   => ~ ( P4 @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_907_split__list__first__prop,axiom,
    ! [Xs: list_nat,P4: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat2 @ X5 @ ( set_nat2 @ Xs ) )
          & ( P4 @ X5 ) )
     => ? [Ys3: list_nat,X: nat] :
          ( ? [Zs: list_nat] :
              ( Xs
              = ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs ) ) )
          & ( P4 @ X )
          & ! [Xa2: nat] :
              ( ( member_nat2 @ Xa2 @ ( set_nat2 @ Ys3 ) )
             => ~ ( P4 @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_908_split__list__last__prop,axiom,
    ! [Xs: list_nat,P4: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat2 @ X5 @ ( set_nat2 @ Xs ) )
          & ( P4 @ X5 ) )
     => ? [Ys3: list_nat,X: nat,Zs: list_nat] :
          ( ( Xs
            = ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs ) ) )
          & ( P4 @ X )
          & ! [Xa2: nat] :
              ( ( member_nat2 @ Xa2 @ ( set_nat2 @ Zs ) )
             => ~ ( P4 @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_909_in__set__conv__decomp,axiom,
    ! [X3: list_nat,Xs: list_list_nat] :
      ( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
      = ( ? [Ys2: list_list_nat,Zs3: list_list_nat] :
            ( Xs
            = ( append_list_nat @ Ys2 @ ( cons_list_nat @ X3 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_910_in__set__conv__decomp,axiom,
    ! [X3: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
      = ( ? [Ys2: list_P8875379029341186191rm_a_b,Zs3: list_P8875379029341186191rm_a_b] :
            ( Xs
            = ( append1152621114427393060rm_a_b @ Ys2 @ ( cons_P3536305108106557631rm_a_b @ X3 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_911_in__set__conv__decomp,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
      = ( ? [Ys2: list_nat,Zs3: list_nat] :
            ( Xs
            = ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_912_append__Cons__eq__iff,axiom,
    ! [X3: list_nat,Xs: list_list_nat,Ys: list_list_nat,Xs5: list_list_nat,Ys6: list_list_nat] :
      ( ~ ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
     => ( ~ ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Ys ) )
       => ( ( ( append_list_nat @ Xs @ ( cons_list_nat @ X3 @ Ys ) )
            = ( append_list_nat @ Xs5 @ ( cons_list_nat @ X3 @ Ys6 ) ) )
          = ( ( Xs = Xs5 )
            & ( Ys = Ys6 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_913_append__Cons__eq__iff,axiom,
    ! [X3: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b,Ys: list_P8875379029341186191rm_a_b,Xs5: list_P8875379029341186191rm_a_b,Ys6: list_P8875379029341186191rm_a_b] :
      ( ~ ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
     => ( ~ ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Ys ) )
       => ( ( ( append1152621114427393060rm_a_b @ Xs @ ( cons_P3536305108106557631rm_a_b @ X3 @ Ys ) )
            = ( append1152621114427393060rm_a_b @ Xs5 @ ( cons_P3536305108106557631rm_a_b @ X3 @ Ys6 ) ) )
          = ( ( Xs = Xs5 )
            & ( Ys = Ys6 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_914_append__Cons__eq__iff,axiom,
    ! [X3: nat,Xs: list_nat,Ys: list_nat,Xs5: list_nat,Ys6: list_nat] :
      ( ~ ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
     => ( ~ ( member_nat2 @ X3 @ ( set_nat2 @ Ys ) )
       => ( ( ( append_nat @ Xs @ ( cons_nat @ X3 @ Ys ) )
            = ( append_nat @ Xs5 @ ( cons_nat @ X3 @ Ys6 ) ) )
          = ( ( Xs = Xs5 )
            & ( Ys = Ys6 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_915_split__list__propE,axiom,
    ! [Xs: list_nat,P4: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat2 @ X5 @ ( set_nat2 @ Xs ) )
          & ( P4 @ X5 ) )
     => ~ ! [Ys3: list_nat,X: nat] :
            ( ? [Zs: list_nat] :
                ( Xs
                = ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs ) ) )
           => ~ ( P4 @ X ) ) ) ).

% split_list_propE
thf(fact_916_split__list__first,axiom,
    ! [X3: list_nat,Xs: list_list_nat] :
      ( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
     => ? [Ys3: list_list_nat,Zs: list_list_nat] :
          ( ( Xs
            = ( append_list_nat @ Ys3 @ ( cons_list_nat @ X3 @ Zs ) ) )
          & ~ ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_917_split__list__first,axiom,
    ! [X3: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
     => ? [Ys3: list_P8875379029341186191rm_a_b,Zs: list_P8875379029341186191rm_a_b] :
          ( ( Xs
            = ( append1152621114427393060rm_a_b @ Ys3 @ ( cons_P3536305108106557631rm_a_b @ X3 @ Zs ) ) )
          & ~ ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_918_split__list__first,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
     => ? [Ys3: list_nat,Zs: list_nat] :
          ( ( Xs
            = ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs ) ) )
          & ~ ( member_nat2 @ X3 @ ( set_nat2 @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_919_split__list__prop,axiom,
    ! [Xs: list_nat,P4: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat2 @ X5 @ ( set_nat2 @ Xs ) )
          & ( P4 @ X5 ) )
     => ? [Ys3: list_nat,X: nat] :
          ( ? [Zs: list_nat] :
              ( Xs
              = ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs ) ) )
          & ( P4 @ X ) ) ) ).

% split_list_prop
thf(fact_920_split__list__last,axiom,
    ! [X3: list_nat,Xs: list_list_nat] :
      ( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
     => ? [Ys3: list_list_nat,Zs: list_list_nat] :
          ( ( Xs
            = ( append_list_nat @ Ys3 @ ( cons_list_nat @ X3 @ Zs ) ) )
          & ~ ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Zs ) ) ) ) ).

% split_list_last
thf(fact_921_split__list__last,axiom,
    ! [X3: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
     => ? [Ys3: list_P8875379029341186191rm_a_b,Zs: list_P8875379029341186191rm_a_b] :
          ( ( Xs
            = ( append1152621114427393060rm_a_b @ Ys3 @ ( cons_P3536305108106557631rm_a_b @ X3 @ Zs ) ) )
          & ~ ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Zs ) ) ) ) ).

% split_list_last
thf(fact_922_split__list__last,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
     => ? [Ys3: list_nat,Zs: list_nat] :
          ( ( Xs
            = ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs ) ) )
          & ~ ( member_nat2 @ X3 @ ( set_nat2 @ Zs ) ) ) ) ).

% split_list_last
thf(fact_923_split__list,axiom,
    ! [X3: list_nat,Xs: list_list_nat] :
      ( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
     => ? [Ys3: list_list_nat,Zs: list_list_nat] :
          ( Xs
          = ( append_list_nat @ Ys3 @ ( cons_list_nat @ X3 @ Zs ) ) ) ) ).

% split_list
thf(fact_924_split__list,axiom,
    ! [X3: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
     => ? [Ys3: list_P8875379029341186191rm_a_b,Zs: list_P8875379029341186191rm_a_b] :
          ( Xs
          = ( append1152621114427393060rm_a_b @ Ys3 @ ( cons_P3536305108106557631rm_a_b @ X3 @ Zs ) ) ) ) ).

% split_list
thf(fact_925_split__list,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
     => ? [Ys3: list_nat,Zs: list_nat] :
          ( Xs
          = ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs ) ) ) ) ).

% split_list
thf(fact_926_length__Suc__conv,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( suc @ N ) )
      = ( ? [Y4: nat,Ys2: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y4 @ Ys2 ) )
            & ( ( size_size_list_nat @ Ys2 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_927_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ( suc @ N )
        = ( size_size_list_nat @ Xs ) )
      = ( ? [Y4: nat,Ys2: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y4 @ Ys2 ) )
            & ( ( size_size_list_nat @ Ys2 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_928_Abstract__Rewriting_Ochain__mono,axiom,
    ! [R4: set_Pr4386577575007340137rm_a_b,R2: set_Pr4386577575007340137rm_a_b,Seq: nat > term_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ R4 @ R2 )
     => ( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( Seq @ I2 ) @ ( Seq @ ( suc @ I2 ) ) ) @ R4 )
       => ! [I4: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( Seq @ I4 ) @ ( Seq @ ( suc @ I4 ) ) ) @ R2 ) ) ) ).

% Abstract_Rewriting.chain_mono
thf(fact_929_set__update__subsetI,axiom,
    ! [Xs: list_list_nat,A5: set_list_nat,X3: list_nat,I: nat] :
      ( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A5 )
     => ( ( member_list_nat2 @ X3 @ A5 )
       => ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( list_update_list_nat @ Xs @ I @ X3 ) ) @ A5 ) ) ) ).

% set_update_subsetI
thf(fact_930_set__update__subsetI,axiom,
    ! [Xs: list_P8875379029341186191rm_a_b,A5: set_Pr4386577575007340137rm_a_b,X3: produc357393685978478089rm_a_b,I: nat] :
      ( ( ord_le118470702582115849rm_a_b @ ( set_Pr2370880052973118494rm_a_b @ Xs ) @ A5 )
     => ( ( member5869715511025134514rm_a_b @ X3 @ A5 )
       => ( ord_le118470702582115849rm_a_b @ ( set_Pr2370880052973118494rm_a_b @ ( list_u4211539095652206455rm_a_b @ Xs @ I @ X3 ) ) @ A5 ) ) ) ).

% set_update_subsetI
thf(fact_931_set__update__subsetI,axiom,
    ! [Xs: list_nat,A5: set_nat,X3: nat,I: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A5 )
     => ( ( member_nat2 @ X3 @ A5 )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X3 ) ) @ A5 ) ) ) ).

% set_update_subsetI
thf(fact_932_list__update__code_I3_J,axiom,
    ! [X3: nat,Xs: list_nat,I: nat,Y3: nat] :
      ( ( list_update_nat @ ( cons_nat @ X3 @ Xs ) @ ( suc @ I ) @ Y3 )
      = ( cons_nat @ X3 @ ( list_update_nat @ Xs @ I @ Y3 ) ) ) ).

% list_update_code(3)
thf(fact_933_Cons__in__subseqsD,axiom,
    ! [Y3: nat,Ys: list_nat,Xs: list_nat] :
      ( ( member_list_nat2 @ ( cons_nat @ Y3 @ Ys ) @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
     => ( member_list_nat2 @ Ys @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_934_empty__subseqs,axiom,
    ! [Xs: list_term_a_b] : ( member_list_term_a_b2 @ nil_term_a_b @ ( set_list_term_a_b2 @ ( subseqs_term_a_b @ Xs ) ) ) ).

% empty_subseqs
thf(fact_935_empty__subseqs,axiom,
    ! [Xs: list_nat] : ( member_list_nat2 @ nil_nat @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ).

% empty_subseqs
thf(fact_936_subset__code_I2_J,axiom,
    ! [A5: set_list_nat,Ys: list_list_nat] :
      ( ( ord_le6045566169113846134st_nat @ A5 @ ( coset_list_nat @ Ys ) )
      = ( ! [X4: list_nat] :
            ( ( member_list_nat2 @ X4 @ ( set_list_nat2 @ Ys ) )
           => ~ ( member_list_nat2 @ X4 @ A5 ) ) ) ) ).

% subset_code(2)
thf(fact_937_subset__code_I2_J,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,Ys: list_P8875379029341186191rm_a_b] :
      ( ( ord_le118470702582115849rm_a_b @ A5 @ ( coset_9213161874213406204rm_a_b @ Ys ) )
      = ( ! [X4: produc357393685978478089rm_a_b] :
            ( ( member5869715511025134514rm_a_b @ X4 @ ( set_Pr2370880052973118494rm_a_b @ Ys ) )
           => ~ ( member5869715511025134514rm_a_b @ X4 @ A5 ) ) ) ) ).

% subset_code(2)
thf(fact_938_subset__code_I2_J,axiom,
    ! [A5: set_nat,Ys: list_nat] :
      ( ( ord_less_eq_set_nat @ A5 @ ( coset_nat @ Ys ) )
      = ( ! [X4: nat] :
            ( ( member_nat2 @ X4 @ ( set_nat2 @ Ys ) )
           => ~ ( member_nat2 @ X4 @ A5 ) ) ) ) ).

% subset_code(2)
thf(fact_939_in__set__product__lists__length,axiom,
    ! [Xs: list_nat,Xss2: list_list_nat] :
      ( ( member_list_nat2 @ Xs @ ( set_list_nat2 @ ( product_lists_nat @ Xss2 ) ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s3023201423986296836st_nat @ Xss2 ) ) ) ).

% in_set_product_lists_length
thf(fact_940_gen__length__code_I2_J,axiom,
    ! [N: nat,X3: nat,Xs: list_nat] :
      ( ( gen_length_nat @ N @ ( cons_nat @ X3 @ Xs ) )
      = ( gen_length_nat @ ( suc @ N ) @ Xs ) ) ).

% gen_length_code(2)
thf(fact_941_nth__mem,axiom,
    ! [N: nat,Xs: list_list_nat] :
      ( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
     => ( member_list_nat2 @ ( nth_list_nat @ Xs @ N ) @ ( set_list_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_942_nth__mem,axiom,
    ! [N: nat,Xs: list_P8875379029341186191rm_a_b] :
      ( ( ord_less_nat @ N @ ( size_s8600120110137556219rm_a_b @ Xs ) )
     => ( member5869715511025134514rm_a_b @ ( nth_Pr1098598711982373392rm_a_b @ Xs @ N ) @ ( set_Pr2370880052973118494rm_a_b @ Xs ) ) ) ).

% nth_mem
thf(fact_943_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat2 @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_944_in__set__conv__nth,axiom,
    ! [X3: list_nat,Xs: list_list_nat] :
      ( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
            & ( ( nth_list_nat @ Xs @ I3 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_945_in__set__conv__nth,axiom,
    ! [X3: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s8600120110137556219rm_a_b @ Xs ) )
            & ( ( nth_Pr1098598711982373392rm_a_b @ Xs @ I3 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_946_in__set__conv__nth,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ I3 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_947_all__nth__imp__all__set,axiom,
    ! [Xs: list_list_nat,P4: list_nat > $o,X3: list_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
         => ( P4 @ ( nth_list_nat @ Xs @ I2 ) ) )
     => ( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
       => ( P4 @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_948_all__nth__imp__all__set,axiom,
    ! [Xs: list_P8875379029341186191rm_a_b,P4: produc357393685978478089rm_a_b > $o,X3: produc357393685978478089rm_a_b] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s8600120110137556219rm_a_b @ Xs ) )
         => ( P4 @ ( nth_Pr1098598711982373392rm_a_b @ Xs @ I2 ) ) )
     => ( ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
       => ( P4 @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_949_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P4: nat > $o,X3: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
         => ( P4 @ ( nth_nat @ Xs @ I2 ) ) )
     => ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
       => ( P4 @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_950_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
      = ( ? [X4: nat,Ys2: list_nat] :
            ( ( Xs
              = ( cons_nat @ X4 @ Ys2 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_951_set__update__memI,axiom,
    ! [N: nat,Xs: list_list_nat,X3: list_nat] :
      ( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
     => ( member_list_nat2 @ X3 @ ( set_list_nat2 @ ( list_update_list_nat @ Xs @ N @ X3 ) ) ) ) ).

% set_update_memI
thf(fact_952_set__update__memI,axiom,
    ! [N: nat,Xs: list_P8875379029341186191rm_a_b,X3: produc357393685978478089rm_a_b] :
      ( ( ord_less_nat @ N @ ( size_s8600120110137556219rm_a_b @ Xs ) )
     => ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ ( list_u4211539095652206455rm_a_b @ Xs @ N @ X3 ) ) ) ) ).

% set_update_memI
thf(fact_953_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X3: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat2 @ X3 @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X3 ) ) ) ) ).

% set_update_memI
thf(fact_954_lexord__partial__trans,axiom,
    ! [Xs: list_list_nat,R: set_Pr3451248702717554689st_nat,Ys: list_list_nat,Zs2: list_list_nat] :
      ( ! [X: list_nat,Y: list_nat,Z: list_nat] :
          ( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
         => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R )
           => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y @ Z ) @ R )
             => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Z ) @ R ) ) ) )
     => ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys ) @ ( lexord_list_nat @ R ) )
       => ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Ys @ Zs2 ) @ ( lexord_list_nat @ R ) )
         => ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Zs2 ) @ ( lexord_list_nat @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_955_lexord__partial__trans,axiom,
    ! [Xs: list_P8875379029341186191rm_a_b,R: set_Pr2972776593051762503rm_a_b,Ys: list_P8875379029341186191rm_a_b,Zs2: list_P8875379029341186191rm_a_b] :
      ( ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
         => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R )
           => ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z ) @ R )
             => ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z ) @ R ) ) ) )
     => ( ( member2631635837537579280rm_a_b @ ( produc129289007660123479rm_a_b @ Xs @ Ys ) @ ( lexord1555492263668129660rm_a_b @ R ) )
       => ( ( member2631635837537579280rm_a_b @ ( produc129289007660123479rm_a_b @ Ys @ Zs2 ) @ ( lexord1555492263668129660rm_a_b @ R ) )
         => ( member2631635837537579280rm_a_b @ ( produc129289007660123479rm_a_b @ Xs @ Zs2 ) @ ( lexord1555492263668129660rm_a_b @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_956_lexord__partial__trans,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat,Ys: list_nat,Zs2: list_nat] :
      ( ! [X: nat,Y: nat,Z: nat] :
          ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ R )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R ) ) ) )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lexord_nat @ R ) )
       => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs2 ) @ ( lexord_nat @ R ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs2 ) @ ( lexord_nat @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_957_lexord__partial__trans,axiom,
    ! [Xs: list_term_a_b,R: set_Pr4386577575007340137rm_a_b,Ys: list_term_a_b,Zs2: list_term_a_b] :
      ( ! [X: term_a_b,Y: term_a_b,Z: term_a_b] :
          ( ( member_term_a_b2 @ X @ ( set_term_a_b2 @ Xs ) )
         => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
           => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ R )
             => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ R ) ) ) )
     => ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( lexord_term_a_b @ R ) )
       => ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ys @ Zs2 ) @ ( lexord_term_a_b @ R ) )
         => ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Zs2 ) @ ( lexord_term_a_b @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_958_subset__code_I3_J,axiom,
    ~ ( ord_le2705286416250468010rm_a_b @ ( coset_term_a_b @ nil_term_a_b ) @ ( set_term_a_b2 @ nil_term_a_b ) ) ).

% subset_code(3)
thf(fact_959_subset__code_I3_J,axiom,
    ~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).

% subset_code(3)
thf(fact_960_length__Suc__conv__rev,axiom,
    ! [Xs: list_term_a_b,N: nat] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( suc @ N ) )
      = ( ? [Y4: term_a_b,Ys2: list_term_a_b] :
            ( ( Xs
              = ( append_term_a_b @ Ys2 @ ( cons_term_a_b @ Y4 @ nil_term_a_b ) ) )
            & ( ( size_s8906293707977694520rm_a_b @ Ys2 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_961_length__Suc__conv__rev,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( suc @ N ) )
      = ( ? [Y4: nat,Ys2: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ Y4 @ nil_nat ) ) )
            & ( ( size_size_list_nat @ Ys2 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_962_length__append__singleton,axiom,
    ! [Xs: list_term_a_b,X3: term_a_b] :
      ( ( size_s8906293707977694520rm_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ X3 @ nil_term_a_b ) ) )
      = ( suc @ ( size_s8906293707977694520rm_a_b @ Xs ) ) ) ).

% length_append_singleton
thf(fact_963_length__append__singleton,axiom,
    ! [Xs: list_nat,X3: nat] :
      ( ( size_size_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X3 @ nil_nat ) ) )
      = ( suc @ ( size_size_list_nat @ Xs ) ) ) ).

% length_append_singleton
thf(fact_964_lexord__same__pref__iff,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b,Zs2: list_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
      ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ Xs @ Ys ) @ ( append_term_a_b @ Xs @ Zs2 ) ) @ ( lexord_term_a_b @ R ) )
      = ( ? [X4: term_a_b] :
            ( ( member_term_a_b2 @ X4 @ ( set_term_a_b2 @ Xs ) )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ X4 ) @ R ) )
        | ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ys @ Zs2 ) @ ( lexord_term_a_b @ R ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_965_the__elem__set,axiom,
    ! [X3: term_a_b] :
      ( ( the_elem_term_a_b @ ( set_term_a_b2 @ ( cons_term_a_b @ X3 @ nil_term_a_b ) ) )
      = X3 ) ).

% the_elem_set
thf(fact_966_the__elem__set,axiom,
    ! [X3: nat] :
      ( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X3 @ nil_nat ) ) )
      = X3 ) ).

% the_elem_set
thf(fact_967_can__select__def,axiom,
    ( can_select_list_nat
    = ( ^ [P7: list_nat > $o,A6: set_list_nat] :
        ? [X4: list_nat] :
          ( ( member_list_nat2 @ X4 @ A6 )
          & ( P7 @ X4 )
          & ! [Y4: list_nat] :
              ( ( ( member_list_nat2 @ Y4 @ A6 )
                & ( P7 @ Y4 ) )
             => ( Y4 = X4 ) ) ) ) ) ).

% can_select_def
thf(fact_968_can__select__def,axiom,
    ( can_se4942192940725369094rm_a_b
    = ( ^ [P7: produc357393685978478089rm_a_b > $o,A6: set_Pr4386577575007340137rm_a_b] :
        ? [X4: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X4 @ A6 )
          & ( P7 @ X4 )
          & ! [Y4: produc357393685978478089rm_a_b] :
              ( ( ( member5869715511025134514rm_a_b @ Y4 @ A6 )
                & ( P7 @ Y4 ) )
             => ( Y4 = X4 ) ) ) ) ) ).

% can_select_def
thf(fact_969_can__select__def,axiom,
    ( can_select_nat
    = ( ^ [P7: nat > $o,A6: set_nat] :
        ? [X4: nat] :
          ( ( member_nat2 @ X4 @ A6 )
          & ( P7 @ X4 )
          & ! [Y4: nat] :
              ( ( ( member_nat2 @ Y4 @ A6 )
                & ( P7 @ Y4 ) )
             => ( Y4 = X4 ) ) ) ) ) ).

% can_select_def
thf(fact_970_term__not__arg,axiom,
    ! [F2: a,Ss2: list_term_a_b] :
      ~ ( member_term_a_b2 @ ( fun_a_b @ F2 @ Ss2 ) @ ( set_term_a_b2 @ Ss2 ) ) ).

% term_not_arg
thf(fact_971_upd__conv__take__nth__drop,axiom,
    ! [I: nat,Xs: list_nat,A: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( list_update_nat @ Xs @ I @ A )
        = ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ A @ ( drop_nat @ ( suc @ I ) @ Xs ) ) ) ) ) ).

% upd_conv_take_nth_drop
thf(fact_972_drop__Suc__Cons,axiom,
    ! [N: nat,X3: nat,Xs: list_nat] :
      ( ( drop_nat @ ( suc @ N ) @ ( cons_nat @ X3 @ Xs ) )
      = ( drop_nat @ N @ Xs ) ) ).

% drop_Suc_Cons
thf(fact_973_drop__all,axiom,
    ! [Xs: list_term_a_b,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ N )
     => ( ( drop_term_a_b @ N @ Xs )
        = nil_term_a_b ) ) ).

% drop_all
thf(fact_974_drop__all,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
     => ( ( drop_nat @ N @ Xs )
        = nil_nat ) ) ).

% drop_all
thf(fact_975_drop__eq__Nil,axiom,
    ! [N: nat,Xs: list_term_a_b] :
      ( ( ( drop_term_a_b @ N @ Xs )
        = nil_term_a_b )
      = ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ N ) ) ).

% drop_eq_Nil
thf(fact_976_drop__eq__Nil,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ( drop_nat @ N @ Xs )
        = nil_nat )
      = ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).

% drop_eq_Nil
thf(fact_977_drop__eq__Nil2,axiom,
    ! [N: nat,Xs: list_term_a_b] :
      ( ( nil_term_a_b
        = ( drop_term_a_b @ N @ Xs ) )
      = ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ N ) ) ).

% drop_eq_Nil2
thf(fact_978_drop__eq__Nil2,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( nil_nat
        = ( drop_nat @ N @ Xs ) )
      = ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).

% drop_eq_Nil2
thf(fact_979_in__set__dropD,axiom,
    ! [X3: list_nat,N: nat,Xs: list_list_nat] :
      ( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ ( drop_list_nat @ N @ Xs ) ) )
     => ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) ) ) ).

% in_set_dropD
thf(fact_980_in__set__dropD,axiom,
    ! [X3: produc357393685978478089rm_a_b,N: nat,Xs: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ ( drop_P2146130716461001295rm_a_b @ N @ Xs ) ) )
     => ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) ) ) ).

% in_set_dropD
thf(fact_981_in__set__dropD,axiom,
    ! [X3: nat,N: nat,Xs: list_nat] :
      ( ( member_nat2 @ X3 @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) )
     => ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) ) ) ).

% in_set_dropD
thf(fact_982_drop__Nil,axiom,
    ! [N: nat] :
      ( ( drop_term_a_b @ N @ nil_term_a_b )
      = nil_term_a_b ) ).

% drop_Nil
thf(fact_983_drop__Nil,axiom,
    ! [N: nat] :
      ( ( drop_nat @ N @ nil_nat )
      = nil_nat ) ).

% drop_Nil
thf(fact_984_nth__via__drop,axiom,
    ! [N: nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
      ( ( ( drop_nat @ N @ Xs )
        = ( cons_nat @ Y3 @ Ys ) )
     => ( ( nth_nat @ Xs @ N )
        = Y3 ) ) ).

% nth_via_drop
thf(fact_985_Cons__nth__drop__Suc,axiom,
    ! [I: nat,Xs: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( cons_nat @ ( nth_nat @ Xs @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs ) )
        = ( drop_nat @ I @ Xs ) ) ) ).

% Cons_nth_drop_Suc
thf(fact_986_id__take__nth__drop,axiom,
    ! [I: nat,Xs: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( Xs
        = ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs ) ) ) ) ) ).

% id_take_nth_drop
thf(fact_987_take__hd__drop,axiom,
    ! [N: nat,Xs: list_term_a_b] :
      ( ( ord_less_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) )
     => ( ( append_term_a_b @ ( take_term_a_b @ N @ Xs ) @ ( cons_term_a_b @ ( hd_term_a_b @ ( drop_term_a_b @ N @ Xs ) ) @ nil_term_a_b ) )
        = ( take_term_a_b @ ( suc @ N ) @ Xs ) ) ) ).

% take_hd_drop
thf(fact_988_take__hd__drop,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( append_nat @ ( take_nat @ N @ Xs ) @ ( cons_nat @ ( hd_nat @ ( drop_nat @ N @ Xs ) ) @ nil_nat ) )
        = ( take_nat @ ( suc @ N ) @ Xs ) ) ) ).

% take_hd_drop
thf(fact_989_hd__append2,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( Xs != nil_term_a_b )
     => ( ( hd_term_a_b @ ( append_term_a_b @ Xs @ Ys ) )
        = ( hd_term_a_b @ Xs ) ) ) ).

% hd_append2
thf(fact_990_hd__append2,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs != nil_nat )
     => ( ( hd_nat @ ( append_nat @ Xs @ Ys ) )
        = ( hd_nat @ Xs ) ) ) ).

% hd_append2
thf(fact_991_list_Osel_I1_J,axiom,
    ! [X21: nat,X22: list_nat] :
      ( ( hd_nat @ ( cons_nat @ X21 @ X22 ) )
      = X21 ) ).

% list.sel(1)
thf(fact_992_longest__common__prefix,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b] :
    ? [Ps2: list_term_a_b,Xs4: list_term_a_b,Ys5: list_term_a_b] :
      ( ( Xs
        = ( append_term_a_b @ Ps2 @ Xs4 ) )
      & ( Ys
        = ( append_term_a_b @ Ps2 @ Ys5 ) )
      & ( ( Xs4 = nil_term_a_b )
        | ( Ys5 = nil_term_a_b )
        | ( ( hd_term_a_b @ Xs4 )
         != ( hd_term_a_b @ Ys5 ) ) ) ) ).

% longest_common_prefix
thf(fact_993_longest__common__prefix,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
    ? [Ps2: list_nat,Xs4: list_nat,Ys5: list_nat] :
      ( ( Xs
        = ( append_nat @ Ps2 @ Xs4 ) )
      & ( Ys
        = ( append_nat @ Ps2 @ Ys5 ) )
      & ( ( Xs4 = nil_nat )
        | ( Ys5 = nil_nat )
        | ( ( hd_nat @ Xs4 )
         != ( hd_nat @ Ys5 ) ) ) ) ).

% longest_common_prefix
thf(fact_994_hd__append,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( ( Xs = nil_term_a_b )
       => ( ( hd_term_a_b @ ( append_term_a_b @ Xs @ Ys ) )
          = ( hd_term_a_b @ Ys ) ) )
      & ( ( Xs != nil_term_a_b )
       => ( ( hd_term_a_b @ ( append_term_a_b @ Xs @ Ys ) )
          = ( hd_term_a_b @ Xs ) ) ) ) ).

% hd_append
thf(fact_995_hd__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( Xs = nil_nat )
       => ( ( hd_nat @ ( append_nat @ Xs @ Ys ) )
          = ( hd_nat @ Ys ) ) )
      & ( ( Xs != nil_nat )
       => ( ( hd_nat @ ( append_nat @ Xs @ Ys ) )
          = ( hd_nat @ Xs ) ) ) ) ).

% hd_append
thf(fact_996_list_Oset__sel_I1_J,axiom,
    ! [A: list_list_nat] :
      ( ( A != nil_list_nat )
     => ( member_list_nat2 @ ( hd_list_nat @ A ) @ ( set_list_nat2 @ A ) ) ) ).

% list.set_sel(1)
thf(fact_997_list_Oset__sel_I1_J,axiom,
    ! [A: list_P8875379029341186191rm_a_b] :
      ( ( A != nil_Pr6942174756412032271rm_a_b )
     => ( member5869715511025134514rm_a_b @ ( hd_Pro6001814652670619252rm_a_b @ A ) @ ( set_Pr2370880052973118494rm_a_b @ A ) ) ) ).

% list.set_sel(1)
thf(fact_998_list_Oset__sel_I1_J,axiom,
    ! [A: list_term_a_b] :
      ( ( A != nil_term_a_b )
     => ( member_term_a_b2 @ ( hd_term_a_b @ A ) @ ( set_term_a_b2 @ A ) ) ) ).

% list.set_sel(1)
thf(fact_999_list_Oset__sel_I1_J,axiom,
    ! [A: list_nat] :
      ( ( A != nil_nat )
     => ( member_nat2 @ ( hd_nat @ A ) @ ( set_nat2 @ A ) ) ) ).

% list.set_sel(1)
thf(fact_1000_hd__in__set,axiom,
    ! [Xs: list_list_nat] :
      ( ( Xs != nil_list_nat )
     => ( member_list_nat2 @ ( hd_list_nat @ Xs ) @ ( set_list_nat2 @ Xs ) ) ) ).

% hd_in_set
thf(fact_1001_hd__in__set,axiom,
    ! [Xs: list_P8875379029341186191rm_a_b] :
      ( ( Xs != nil_Pr6942174756412032271rm_a_b )
     => ( member5869715511025134514rm_a_b @ ( hd_Pro6001814652670619252rm_a_b @ Xs ) @ ( set_Pr2370880052973118494rm_a_b @ Xs ) ) ) ).

% hd_in_set
thf(fact_1002_hd__in__set,axiom,
    ! [Xs: list_term_a_b] :
      ( ( Xs != nil_term_a_b )
     => ( member_term_a_b2 @ ( hd_term_a_b @ Xs ) @ ( set_term_a_b2 @ Xs ) ) ) ).

% hd_in_set
thf(fact_1003_hd__in__set,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ( member_nat2 @ ( hd_nat @ Xs ) @ ( set_nat2 @ Xs ) ) ) ).

% hd_in_set
thf(fact_1004_supteqp_Osimps,axiom,
    ( subter3057829548224121575qp_a_b
    = ( ^ [A12: term_a_b,A23: term_a_b] :
          ( ? [T3: term_a_b] :
              ( ( A12 = T3 )
              & ( A23 = T3 ) )
          | ? [U5: term_a_b,Ss3: list_term_a_b,T3: term_a_b,F: a] :
              ( ( A12
                = ( fun_a_b @ F @ Ss3 ) )
              & ( A23 = T3 )
              & ( member_term_a_b2 @ U5 @ ( set_term_a_b2 @ Ss3 ) )
              & ( subter3057829548224121575qp_a_b @ U5 @ T3 ) ) ) ) ) ).

% supteqp.simps
thf(fact_1005_suptp_Osimps,axiom,
    ( subter8709468037939584051tp_a_b
    = ( ^ [A12: term_a_b,A23: term_a_b] :
          ( ? [S3: term_a_b,Ss3: list_term_a_b,F: a] :
              ( ( A12
                = ( fun_a_b @ F @ Ss3 ) )
              & ( A23 = S3 )
              & ( member_term_a_b2 @ S3 @ ( set_term_a_b2 @ Ss3 ) ) )
          | ? [S3: term_a_b,Ss3: list_term_a_b,T3: term_a_b,F: a] :
              ( ( A12
                = ( fun_a_b @ F @ Ss3 ) )
              & ( A23 = T3 )
              & ( member_term_a_b2 @ S3 @ ( set_term_a_b2 @ Ss3 ) )
              & ( subter8709468037939584051tp_a_b @ S3 @ T3 ) ) ) ) ) ).

% suptp.simps
thf(fact_1006_nth__equal__first__eq,axiom,
    ! [X3: list_nat,Xs: list_list_nat,N: nat] :
      ( ~ ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
       => ( ( ( nth_list_nat @ ( cons_list_nat @ X3 @ Xs ) @ N )
            = X3 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_1007_nth__equal__first__eq,axiom,
    ! [X3: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b,N: nat] :
      ( ~ ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s8600120110137556219rm_a_b @ Xs ) )
       => ( ( ( nth_Pr1098598711982373392rm_a_b @ ( cons_P3536305108106557631rm_a_b @ X3 @ Xs ) @ N )
            = X3 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_1008_nth__equal__first__eq,axiom,
    ! [X3: nat,Xs: list_nat,N: nat] :
      ( ~ ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ N )
            = X3 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_1009_length__0__conv,axiom,
    ! [Xs: list_term_a_b] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_term_a_b ) ) ).

% length_0_conv
thf(fact_1010_length__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_nat ) ) ).

% length_0_conv
thf(fact_1011_nth__Cons__0,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ zero_zero_nat )
      = X3 ) ).

% nth_Cons_0
thf(fact_1012_take0,axiom,
    ( ( take_term_a_b @ zero_zero_nat )
    = ( ^ [Xs2: list_term_a_b] : nil_term_a_b ) ) ).

% take0
thf(fact_1013_take0,axiom,
    ( ( take_nat @ zero_zero_nat )
    = ( ^ [Xs2: list_nat] : nil_nat ) ) ).

% take0
thf(fact_1014_take__eq__Nil,axiom,
    ! [N: nat,Xs: list_term_a_b] :
      ( ( ( take_term_a_b @ N @ Xs )
        = nil_term_a_b )
      = ( ( N = zero_zero_nat )
        | ( Xs = nil_term_a_b ) ) ) ).

% take_eq_Nil
thf(fact_1015_take__eq__Nil,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ( take_nat @ N @ Xs )
        = nil_nat )
      = ( ( N = zero_zero_nat )
        | ( Xs = nil_nat ) ) ) ).

% take_eq_Nil
thf(fact_1016_take__eq__Nil2,axiom,
    ! [N: nat,Xs: list_term_a_b] :
      ( ( nil_term_a_b
        = ( take_term_a_b @ N @ Xs ) )
      = ( ( N = zero_zero_nat )
        | ( Xs = nil_term_a_b ) ) ) ).

% take_eq_Nil2
thf(fact_1017_take__eq__Nil2,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( nil_nat
        = ( take_nat @ N @ Xs ) )
      = ( ( N = zero_zero_nat )
        | ( Xs = nil_nat ) ) ) ).

% take_eq_Nil2
thf(fact_1018_length__greater__0__conv,axiom,
    ! [Xs: list_term_a_b] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) )
      = ( Xs != nil_term_a_b ) ) ).

% length_greater_0_conv
thf(fact_1019_length__greater__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
      = ( Xs != nil_nat ) ) ).

% length_greater_0_conv
thf(fact_1020_list__update__code_I2_J,axiom,
    ! [X3: nat,Xs: list_nat,Y3: nat] :
      ( ( list_update_nat @ ( cons_nat @ X3 @ Xs ) @ zero_zero_nat @ Y3 )
      = ( cons_nat @ Y3 @ Xs ) ) ).

% list_update_code(2)
thf(fact_1021_list_Osize_I3_J,axiom,
    ( ( size_s8906293707977694520rm_a_b @ nil_term_a_b )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_1022_list_Osize_I3_J,axiom,
    ( ( size_size_list_nat @ nil_nat )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_1023_image__Pow__mono,axiom,
    ! [F2: nat > nat,A5: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A5 ) @ B5 )
     => ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F2 ) @ ( pow_nat @ A5 ) ) @ ( pow_nat @ B5 ) ) ) ).

% image_Pow_mono
thf(fact_1024_image__Pow__surj,axiom,
    ! [F2: nat > nat,A5: set_nat,B5: set_nat] :
      ( ( ( image_nat_nat @ F2 @ A5 )
        = B5 )
     => ( ( image_7916887816326733075et_nat @ ( image_nat_nat @ F2 ) @ ( pow_nat @ A5 ) )
        = ( pow_nat @ B5 ) ) ) ).

% image_Pow_surj
thf(fact_1025_take__0,axiom,
    ! [Xs: list_term_a_b] :
      ( ( take_term_a_b @ zero_zero_nat @ Xs )
      = nil_term_a_b ) ).

% take_0
thf(fact_1026_take__0,axiom,
    ! [Xs: list_nat] :
      ( ( take_nat @ zero_zero_nat @ Xs )
      = nil_nat ) ).

% take_0
thf(fact_1027_length__pos__if__in__set,axiom,
    ! [X3: list_nat,Xs: list_list_nat] :
      ( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1028_length__pos__if__in__set,axiom,
    ! [X3: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
      ( ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s8600120110137556219rm_a_b @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1029_length__pos__if__in__set,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1030_hd__conv__nth,axiom,
    ! [Xs: list_term_a_b] :
      ( ( Xs != nil_term_a_b )
     => ( ( hd_term_a_b @ Xs )
        = ( nth_term_a_b @ Xs @ zero_zero_nat ) ) ) ).

% hd_conv_nth
thf(fact_1031_hd__conv__nth,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ( ( hd_nat @ Xs )
        = ( nth_nat @ Xs @ zero_zero_nat ) ) ) ).

% hd_conv_nth
thf(fact_1032_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_lists_term_a_b @ N @ nil_term_a_b )
          = ( cons_list_term_a_b @ nil_term_a_b @ nil_list_term_a_b ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_lists_term_a_b @ N @ nil_term_a_b )
          = nil_list_term_a_b ) ) ) ).

% n_lists_Nil
thf(fact_1033_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_lists_nat @ N @ nil_nat )
          = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_lists_nat @ N @ nil_nat )
          = nil_list_nat ) ) ) ).

% n_lists_Nil
thf(fact_1034_list__of__permutation__element__n_Osimps_I1_J,axiom,
    ! [X3: term_a_b,L4: list_term_a_b] :
      ( ( basic_7447718476515137801rm_a_b @ X3 @ zero_zero_nat @ L4 )
      = ( cons_list_term_a_b @ nil_term_a_b @ nil_list_term_a_b ) ) ).

% list_of_permutation_element_n.simps(1)
thf(fact_1035_list__of__permutation__element__n_Osimps_I1_J,axiom,
    ! [X3: nat,L4: list_nat] :
      ( ( basic_7079635023375748421_n_nat @ X3 @ zero_zero_nat @ L4 )
      = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% list_of_permutation_element_n.simps(1)
thf(fact_1036_length__n__lists__elem,axiom,
    ! [Ys: list_nat,N: nat,Xs: list_nat] :
      ( ( member_list_nat2 @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) ) )
     => ( ( size_size_list_nat @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_1037_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_term_a_b] :
      ( ( n_lists_term_a_b @ zero_zero_nat @ Xs )
      = ( cons_list_term_a_b @ nil_term_a_b @ nil_list_term_a_b ) ) ).

% n_lists.simps(1)
thf(fact_1038_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_nat] :
      ( ( n_lists_nat @ zero_zero_nat @ Xs )
      = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% n_lists.simps(1)
thf(fact_1039_image__Fpow__mono,axiom,
    ! [F2: nat > nat,A5: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A5 ) @ B5 )
     => ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F2 ) @ ( finite_Fpow_nat @ A5 ) ) @ ( finite_Fpow_nat @ B5 ) ) ) ).

% image_Fpow_mono
thf(fact_1040_zero__notin__Suc__image,axiom,
    ! [A5: set_nat] :
      ~ ( member_nat2 @ zero_zero_nat @ ( image_nat_nat @ suc @ A5 ) ) ).

% zero_notin_Suc_image
thf(fact_1041_nth__image,axiom,
    ! [L: nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ L @ ( size_size_list_nat @ Xs ) )
     => ( ( image_nat_nat @ ( nth_nat @ Xs ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ L ) )
        = ( set_nat2 @ ( take_nat @ L @ Xs ) ) ) ) ).

% nth_image
thf(fact_1042_depth_Osimps_I2_J,axiom,
    ! [F2: a] :
      ( ( term_depth_a_b @ ( fun_a_b @ F2 @ nil_term_a_b ) )
      = zero_zero_nat ) ).

% depth.simps(2)
thf(fact_1043_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
      = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastLessThan
thf(fact_1044_all__nat__less__eq,axiom,
    ! [N: nat,P4: nat > $o] :
      ( ( ! [M2: nat] :
            ( ( ord_less_nat @ M2 @ N )
           => ( P4 @ M2 ) ) )
      = ( ! [X4: nat] :
            ( ( member_nat2 @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P4 @ X4 ) ) ) ) ).

% all_nat_less_eq
thf(fact_1045_ex__nat__less__eq,axiom,
    ! [N: nat,P4: nat > $o] :
      ( ( ? [M2: nat] :
            ( ( ord_less_nat @ M2 @ N )
            & ( P4 @ M2 ) ) )
      = ( ? [X4: nat] :
            ( ( member_nat2 @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P4 @ X4 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_1046_depth_Oelims,axiom,
    ! [X3: term_a_b,Y3: nat] :
      ( ( ( term_depth_a_b @ X3 )
        = Y3 )
     => ( ( ? [X: b] :
              ( X3
              = ( var_b_a @ X ) )
         => ( Y3 != zero_zero_nat ) )
       => ( ( ? [F3: a] :
                ( X3
                = ( fun_a_b @ F3 @ nil_term_a_b ) )
           => ( Y3 != zero_zero_nat ) )
         => ~ ! [F3: a,V2: term_a_b,Va: list_term_a_b] :
                ( ( X3
                  = ( fun_a_b @ F3 @ ( cons_term_a_b @ V2 @ Va ) ) )
               => ( Y3
                 != ( suc @ ( lattic8265883725875713057ax_nat @ ( image_term_a_b_nat @ term_depth_a_b @ ( set_term_a_b2 @ ( cons_term_a_b @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ).

% depth.elims
thf(fact_1047_depth_Osimps_I3_J,axiom,
    ! [F2: a,V: term_a_b,Va2: list_term_a_b] :
      ( ( term_depth_a_b @ ( fun_a_b @ F2 @ ( cons_term_a_b @ V @ Va2 ) ) )
      = ( suc @ ( lattic8265883725875713057ax_nat @ ( image_term_a_b_nat @ term_depth_a_b @ ( set_term_a_b2 @ ( cons_term_a_b @ V @ Va2 ) ) ) ) ) ) ).

% depth.simps(3)
thf(fact_1048_depth_Opelims,axiom,
    ! [X3: term_a_b,Y3: nat] :
      ( ( ( term_depth_a_b @ X3 )
        = Y3 )
     => ( ( accp_term_a_b @ term_depth_rel_a_b @ X3 )
       => ( ! [X: b] :
              ( ( X3
                = ( var_b_a @ X ) )
             => ( ( Y3 = zero_zero_nat )
               => ~ ( accp_term_a_b @ term_depth_rel_a_b @ ( var_b_a @ X ) ) ) )
         => ( ! [F3: a] :
                ( ( X3
                  = ( fun_a_b @ F3 @ nil_term_a_b ) )
               => ( ( Y3 = zero_zero_nat )
                 => ~ ( accp_term_a_b @ term_depth_rel_a_b @ ( fun_a_b @ F3 @ nil_term_a_b ) ) ) )
           => ~ ! [F3: a,V2: term_a_b,Va: list_term_a_b] :
                  ( ( X3
                    = ( fun_a_b @ F3 @ ( cons_term_a_b @ V2 @ Va ) ) )
                 => ( ( Y3
                      = ( suc @ ( lattic8265883725875713057ax_nat @ ( image_term_a_b_nat @ term_depth_a_b @ ( set_term_a_b2 @ ( cons_term_a_b @ V2 @ Va ) ) ) ) ) )
                   => ~ ( accp_term_a_b @ term_depth_rel_a_b @ ( fun_a_b @ F3 @ ( cons_term_a_b @ V2 @ Va ) ) ) ) ) ) ) ) ) ).

% depth.pelims
thf(fact_1049_list__inter_Oelims,axiom,
    ! [X3: list_list_nat,Xa: list_list_nat,Y3: list_list_nat] :
      ( ( ( missin6532874241183986279st_nat @ X3 @ Xa )
        = Y3 )
     => ( ( ( X3 = nil_list_nat )
         => ( Y3 != nil_list_nat ) )
       => ~ ! [A4: list_nat,As: list_list_nat] :
              ( ( X3
                = ( cons_list_nat @ A4 @ As ) )
             => ~ ( ( ( member_list_nat2 @ A4 @ ( set_list_nat2 @ Xa ) )
                   => ( Y3
                      = ( cons_list_nat @ A4 @ ( missin6532874241183986279st_nat @ As @ Xa ) ) ) )
                  & ( ~ ( member_list_nat2 @ A4 @ ( set_list_nat2 @ Xa ) )
                   => ( Y3
                      = ( missin6532874241183986279st_nat @ As @ Xa ) ) ) ) ) ) ) ).

% list_inter.elims
thf(fact_1050_list__inter_Oelims,axiom,
    ! [X3: list_P8875379029341186191rm_a_b,Xa: list_P8875379029341186191rm_a_b,Y3: list_P8875379029341186191rm_a_b] :
      ( ( ( missin3381965633867963714rm_a_b @ X3 @ Xa )
        = Y3 )
     => ( ( ( X3 = nil_Pr6942174756412032271rm_a_b )
         => ( Y3 != nil_Pr6942174756412032271rm_a_b ) )
       => ~ ! [A4: produc357393685978478089rm_a_b,As: list_P8875379029341186191rm_a_b] :
              ( ( X3
                = ( cons_P3536305108106557631rm_a_b @ A4 @ As ) )
             => ~ ( ( ( member5869715511025134514rm_a_b @ A4 @ ( set_Pr2370880052973118494rm_a_b @ Xa ) )
                   => ( Y3
                      = ( cons_P3536305108106557631rm_a_b @ A4 @ ( missin3381965633867963714rm_a_b @ As @ Xa ) ) ) )
                  & ( ~ ( member5869715511025134514rm_a_b @ A4 @ ( set_Pr2370880052973118494rm_a_b @ Xa ) )
                   => ( Y3
                      = ( missin3381965633867963714rm_a_b @ As @ Xa ) ) ) ) ) ) ) ).

% list_inter.elims
thf(fact_1051_list__inter_Oelims,axiom,
    ! [X3: list_term_a_b,Xa: list_term_a_b,Y3: list_term_a_b] :
      ( ( ( missin8990008842131906971rm_a_b @ X3 @ Xa )
        = Y3 )
     => ( ( ( X3 = nil_term_a_b )
         => ( Y3 != nil_term_a_b ) )
       => ~ ! [A4: term_a_b,As: list_term_a_b] :
              ( ( X3
                = ( cons_term_a_b @ A4 @ As ) )
             => ~ ( ( ( member_term_a_b2 @ A4 @ ( set_term_a_b2 @ Xa ) )
                   => ( Y3
                      = ( cons_term_a_b @ A4 @ ( missin8990008842131906971rm_a_b @ As @ Xa ) ) ) )
                  & ( ~ ( member_term_a_b2 @ A4 @ ( set_term_a_b2 @ Xa ) )
                   => ( Y3
                      = ( missin8990008842131906971rm_a_b @ As @ Xa ) ) ) ) ) ) ) ).

% list_inter.elims
thf(fact_1052_list__inter_Oelims,axiom,
    ! [X3: list_nat,Xa: list_nat,Y3: list_nat] :
      ( ( ( missin6377695591745783511er_nat @ X3 @ Xa )
        = Y3 )
     => ( ( ( X3 = nil_nat )
         => ( Y3 != nil_nat ) )
       => ~ ! [A4: nat,As: list_nat] :
              ( ( X3
                = ( cons_nat @ A4 @ As ) )
             => ~ ( ( ( member_nat2 @ A4 @ ( set_nat2 @ Xa ) )
                   => ( Y3
                      = ( cons_nat @ A4 @ ( missin6377695591745783511er_nat @ As @ Xa ) ) ) )
                  & ( ~ ( member_nat2 @ A4 @ ( set_nat2 @ Xa ) )
                   => ( Y3
                      = ( missin6377695591745783511er_nat @ As @ Xa ) ) ) ) ) ) ) ).

% list_inter.elims
thf(fact_1053_list__inter_Osimps_I1_J,axiom,
    ! [Bs2: list_term_a_b] :
      ( ( missin8990008842131906971rm_a_b @ nil_term_a_b @ Bs2 )
      = nil_term_a_b ) ).

% list_inter.simps(1)
thf(fact_1054_list__inter_Osimps_I1_J,axiom,
    ! [Bs2: list_nat] :
      ( ( missin6377695591745783511er_nat @ nil_nat @ Bs2 )
      = nil_nat ) ).

% list_inter.simps(1)
thf(fact_1055_take__Suc,axiom,
    ! [Xs: list_term_a_b,N: nat] :
      ( ( Xs != nil_term_a_b )
     => ( ( take_term_a_b @ ( suc @ N ) @ Xs )
        = ( cons_term_a_b @ ( hd_term_a_b @ Xs ) @ ( take_term_a_b @ N @ ( tl_term_a_b @ Xs ) ) ) ) ) ).

% take_Suc
thf(fact_1056_take__Suc,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( Xs != nil_nat )
     => ( ( take_nat @ ( suc @ N ) @ Xs )
        = ( cons_nat @ ( hd_nat @ Xs ) @ ( take_nat @ N @ ( tl_nat @ Xs ) ) ) ) ) ).

% take_Suc
thf(fact_1057_image__add__0,axiom,
    ! [S4: set_nat] :
      ( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S4 )
      = S4 ) ).

% image_add_0
thf(fact_1058_image__add__atLeastLessThan,axiom,
    ! [K: nat,I: nat,J: nat] :
      ( ( image_nat_nat @ ( plus_plus_nat @ K ) @ ( set_or4665077453230672383an_nat @ I @ J ) )
      = ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% image_add_atLeastLessThan
thf(fact_1059_tl__append2,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( Xs != nil_term_a_b )
     => ( ( tl_term_a_b @ ( append_term_a_b @ Xs @ Ys ) )
        = ( append_term_a_b @ ( tl_term_a_b @ Xs ) @ Ys ) ) ) ).

% tl_append2
thf(fact_1060_tl__append2,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs != nil_nat )
     => ( ( tl_nat @ ( append_nat @ Xs @ Ys ) )
        = ( append_nat @ ( tl_nat @ Xs ) @ Ys ) ) ) ).

% tl_append2
thf(fact_1061_list_Ocollapse,axiom,
    ! [List: list_term_a_b] :
      ( ( List != nil_term_a_b )
     => ( ( cons_term_a_b @ ( hd_term_a_b @ List ) @ ( tl_term_a_b @ List ) )
        = List ) ) ).

% list.collapse
thf(fact_1062_list_Ocollapse,axiom,
    ! [List: list_nat] :
      ( ( List != nil_nat )
     => ( ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) )
        = List ) ) ).

% list.collapse
thf(fact_1063_hd__Cons__tl,axiom,
    ! [Xs: list_term_a_b] :
      ( ( Xs != nil_term_a_b )
     => ( ( cons_term_a_b @ ( hd_term_a_b @ Xs ) @ ( tl_term_a_b @ Xs ) )
        = Xs ) ) ).

% hd_Cons_tl
thf(fact_1064_hd__Cons__tl,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ( ( cons_nat @ ( hd_nat @ Xs ) @ ( tl_nat @ Xs ) )
        = Xs ) ) ).

% hd_Cons_tl
thf(fact_1065_list_Osel_I3_J,axiom,
    ! [X21: nat,X22: list_nat] :
      ( ( tl_nat @ ( cons_nat @ X21 @ X22 ) )
      = X22 ) ).

% list.sel(3)
thf(fact_1066_list_Osel_I2_J,axiom,
    ( ( tl_term_a_b @ nil_term_a_b )
    = nil_term_a_b ) ).

% list.sel(2)
thf(fact_1067_list_Osel_I2_J,axiom,
    ( ( tl_nat @ nil_nat )
    = nil_nat ) ).

% list.sel(2)
thf(fact_1068_tl__append__if,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( ( Xs = nil_term_a_b )
       => ( ( tl_term_a_b @ ( append_term_a_b @ Xs @ Ys ) )
          = ( tl_term_a_b @ Ys ) ) )
      & ( ( Xs != nil_term_a_b )
       => ( ( tl_term_a_b @ ( append_term_a_b @ Xs @ Ys ) )
          = ( append_term_a_b @ ( tl_term_a_b @ Xs ) @ Ys ) ) ) ) ).

% tl_append_if
thf(fact_1069_tl__append__if,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( Xs = nil_nat )
       => ( ( tl_nat @ ( append_nat @ Xs @ Ys ) )
          = ( tl_nat @ Ys ) ) )
      & ( ( Xs != nil_nat )
       => ( ( tl_nat @ ( append_nat @ Xs @ Ys ) )
          = ( append_nat @ ( tl_nat @ Xs ) @ Ys ) ) ) ) ).

% tl_append_if
thf(fact_1070_list_Oset__sel_I2_J,axiom,
    ! [A: list_list_nat,X3: list_nat] :
      ( ( A != nil_list_nat )
     => ( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ ( tl_list_nat @ A ) ) )
       => ( member_list_nat2 @ X3 @ ( set_list_nat2 @ A ) ) ) ) ).

% list.set_sel(2)
thf(fact_1071_list_Oset__sel_I2_J,axiom,
    ! [A: list_P8875379029341186191rm_a_b,X3: produc357393685978478089rm_a_b] :
      ( ( A != nil_Pr6942174756412032271rm_a_b )
     => ( ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ ( tl_Pro9011950709736066168rm_a_b @ A ) ) )
       => ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ A ) ) ) ) ).

% list.set_sel(2)
thf(fact_1072_list_Oset__sel_I2_J,axiom,
    ! [A: list_term_a_b,X3: term_a_b] :
      ( ( A != nil_term_a_b )
     => ( ( member_term_a_b2 @ X3 @ ( set_term_a_b2 @ ( tl_term_a_b @ A ) ) )
       => ( member_term_a_b2 @ X3 @ ( set_term_a_b2 @ A ) ) ) ) ).

% list.set_sel(2)
thf(fact_1073_list_Oset__sel_I2_J,axiom,
    ! [A: list_nat,X3: nat] :
      ( ( A != nil_nat )
     => ( ( member_nat2 @ X3 @ ( set_nat2 @ ( tl_nat @ A ) ) )
       => ( member_nat2 @ X3 @ ( set_nat2 @ A ) ) ) ) ).

% list.set_sel(2)
thf(fact_1074_Nil__tl,axiom,
    ! [Xs: list_term_a_b] :
      ( ( nil_term_a_b
        = ( tl_term_a_b @ Xs ) )
      = ( ( Xs = nil_term_a_b )
        | ? [X4: term_a_b] :
            ( Xs
            = ( cons_term_a_b @ X4 @ nil_term_a_b ) ) ) ) ).

% Nil_tl
thf(fact_1075_Nil__tl,axiom,
    ! [Xs: list_nat] :
      ( ( nil_nat
        = ( tl_nat @ Xs ) )
      = ( ( Xs = nil_nat )
        | ? [X4: nat] :
            ( Xs
            = ( cons_nat @ X4 @ nil_nat ) ) ) ) ).

% Nil_tl
thf(fact_1076_tl__Nil,axiom,
    ! [Xs: list_term_a_b] :
      ( ( ( tl_term_a_b @ Xs )
        = nil_term_a_b )
      = ( ( Xs = nil_term_a_b )
        | ? [X4: term_a_b] :
            ( Xs
            = ( cons_term_a_b @ X4 @ nil_term_a_b ) ) ) ) ).

% tl_Nil
thf(fact_1077_tl__Nil,axiom,
    ! [Xs: list_nat] :
      ( ( ( tl_nat @ Xs )
        = nil_nat )
      = ( ( Xs = nil_nat )
        | ? [X4: nat] :
            ( Xs
            = ( cons_nat @ X4 @ nil_nat ) ) ) ) ).

% tl_Nil
thf(fact_1078_list_Oexpand,axiom,
    ! [List: list_term_a_b,List2: list_term_a_b] :
      ( ( ( List = nil_term_a_b )
        = ( List2 = nil_term_a_b ) )
     => ( ( ( List != nil_term_a_b )
         => ( ( List2 != nil_term_a_b )
           => ( ( ( hd_term_a_b @ List )
                = ( hd_term_a_b @ List2 ) )
              & ( ( tl_term_a_b @ List )
                = ( tl_term_a_b @ List2 ) ) ) ) )
       => ( List = List2 ) ) ) ).

% list.expand
thf(fact_1079_list_Oexpand,axiom,
    ! [List: list_nat,List2: list_nat] :
      ( ( ( List = nil_nat )
        = ( List2 = nil_nat ) )
     => ( ( ( List != nil_nat )
         => ( ( List2 != nil_nat )
           => ( ( ( hd_nat @ List )
                = ( hd_nat @ List2 ) )
              & ( ( tl_nat @ List )
                = ( tl_nat @ List2 ) ) ) ) )
       => ( List = List2 ) ) ) ).

% list.expand
thf(fact_1080_list_Oexhaust__sel,axiom,
    ! [List: list_term_a_b] :
      ( ( List != nil_term_a_b )
     => ( List
        = ( cons_term_a_b @ ( hd_term_a_b @ List ) @ ( tl_term_a_b @ List ) ) ) ) ).

% list.exhaust_sel
thf(fact_1081_list_Oexhaust__sel,axiom,
    ! [List: list_nat] :
      ( ( List != nil_nat )
     => ( List
        = ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) ) ) ) ).

% list.exhaust_sel
thf(fact_1082_list_Osize_I4_J,axiom,
    ! [X21: nat,X22: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_1083_Nitpick_Osize__list__simp_I2_J,axiom,
    ( size_s8906293707977694520rm_a_b
    = ( ^ [Xs2: list_term_a_b] : ( if_nat @ ( Xs2 = nil_term_a_b ) @ zero_zero_nat @ ( suc @ ( size_s8906293707977694520rm_a_b @ ( tl_term_a_b @ Xs2 ) ) ) ) ) ) ).

% Nitpick.size_list_simp(2)
thf(fact_1084_Nitpick_Osize__list__simp_I2_J,axiom,
    ( size_size_list_nat
    = ( ^ [Xs2: list_nat] : ( if_nat @ ( Xs2 = nil_nat ) @ zero_zero_nat @ ( suc @ ( size_size_list_nat @ ( tl_nat @ Xs2 ) ) ) ) ) ) ).

% Nitpick.size_list_simp(2)
thf(fact_1085_Nitpick_Osize__list__simp_I1_J,axiom,
    ( size_list_term_a_b
    = ( ^ [F: term_a_b > nat,Xs2: list_term_a_b] : ( if_nat @ ( Xs2 = nil_term_a_b ) @ zero_zero_nat @ ( suc @ ( plus_plus_nat @ ( F @ ( hd_term_a_b @ Xs2 ) ) @ ( size_list_term_a_b @ F @ ( tl_term_a_b @ Xs2 ) ) ) ) ) ) ) ).

% Nitpick.size_list_simp(1)
thf(fact_1086_Nitpick_Osize__list__simp_I1_J,axiom,
    ( size_list_nat
    = ( ^ [F: nat > nat,Xs2: list_nat] : ( if_nat @ ( Xs2 = nil_nat ) @ zero_zero_nat @ ( suc @ ( plus_plus_nat @ ( F @ ( hd_nat @ Xs2 ) ) @ ( size_list_nat @ F @ ( tl_nat @ Xs2 ) ) ) ) ) ) ) ).

% Nitpick.size_list_simp(1)
thf(fact_1087_list_Osize__gen_I1_J,axiom,
    ! [X3: term_a_b > nat] :
      ( ( size_list_term_a_b @ X3 @ nil_term_a_b )
      = zero_zero_nat ) ).

% list.size_gen(1)
thf(fact_1088_list_Osize__gen_I1_J,axiom,
    ! [X3: nat > nat] :
      ( ( size_list_nat @ X3 @ nil_nat )
      = zero_zero_nat ) ).

% list.size_gen(1)
thf(fact_1089_size__list__estimation,axiom,
    ! [X3: list_nat,Xs: list_list_nat,Y3: nat,F2: list_nat > nat] :
      ( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
     => ( ( ord_less_nat @ Y3 @ ( F2 @ X3 ) )
       => ( ord_less_nat @ Y3 @ ( size_list_list_nat @ F2 @ Xs ) ) ) ) ).

% size_list_estimation
thf(fact_1090_size__list__estimation,axiom,
    ! [X3: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b,Y3: nat,F2: produc357393685978478089rm_a_b > nat] :
      ( ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
     => ( ( ord_less_nat @ Y3 @ ( F2 @ X3 ) )
       => ( ord_less_nat @ Y3 @ ( size_l3466988539524548625rm_a_b @ F2 @ Xs ) ) ) ) ).

% size_list_estimation
thf(fact_1091_size__list__estimation,axiom,
    ! [X3: nat,Xs: list_nat,Y3: nat,F2: nat > nat] :
      ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
     => ( ( ord_less_nat @ Y3 @ ( F2 @ X3 ) )
       => ( ord_less_nat @ Y3 @ ( size_list_nat @ F2 @ Xs ) ) ) ) ).

% size_list_estimation
thf(fact_1092_size__list__pointwise,axiom,
    ! [Xs: list_list_nat,F2: list_nat > nat,G: list_nat > nat] :
      ( ! [X: list_nat] :
          ( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
         => ( ord_less_eq_nat @ ( F2 @ X ) @ ( G @ X ) ) )
     => ( ord_less_eq_nat @ ( size_list_list_nat @ F2 @ Xs ) @ ( size_list_list_nat @ G @ Xs ) ) ) ).

% size_list_pointwise
thf(fact_1093_size__list__pointwise,axiom,
    ! [Xs: list_P8875379029341186191rm_a_b,F2: produc357393685978478089rm_a_b > nat,G: produc357393685978478089rm_a_b > nat] :
      ( ! [X: produc357393685978478089rm_a_b] :
          ( ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
         => ( ord_less_eq_nat @ ( F2 @ X ) @ ( G @ X ) ) )
     => ( ord_less_eq_nat @ ( size_l3466988539524548625rm_a_b @ F2 @ Xs ) @ ( size_l3466988539524548625rm_a_b @ G @ Xs ) ) ) ).

% size_list_pointwise
thf(fact_1094_size__list__pointwise,axiom,
    ! [Xs: list_nat,F2: nat > nat,G: nat > nat] :
      ( ! [X: nat] :
          ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
         => ( ord_less_eq_nat @ ( F2 @ X ) @ ( G @ X ) ) )
     => ( ord_less_eq_nat @ ( size_list_nat @ F2 @ Xs ) @ ( size_list_nat @ G @ Xs ) ) ) ).

% size_list_pointwise
thf(fact_1095_size__list__estimation_H,axiom,
    ! [X3: list_nat,Xs: list_list_nat,Y3: nat,F2: list_nat > nat] :
      ( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
     => ( ( ord_less_eq_nat @ Y3 @ ( F2 @ X3 ) )
       => ( ord_less_eq_nat @ Y3 @ ( size_list_list_nat @ F2 @ Xs ) ) ) ) ).

% size_list_estimation'
thf(fact_1096_size__list__estimation_H,axiom,
    ! [X3: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b,Y3: nat,F2: produc357393685978478089rm_a_b > nat] :
      ( ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
     => ( ( ord_less_eq_nat @ Y3 @ ( F2 @ X3 ) )
       => ( ord_less_eq_nat @ Y3 @ ( size_l3466988539524548625rm_a_b @ F2 @ Xs ) ) ) ) ).

% size_list_estimation'
thf(fact_1097_size__list__estimation_H,axiom,
    ! [X3: nat,Xs: list_nat,Y3: nat,F2: nat > nat] :
      ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
     => ( ( ord_less_eq_nat @ Y3 @ ( F2 @ X3 ) )
       => ( ord_less_eq_nat @ Y3 @ ( size_list_nat @ F2 @ Xs ) ) ) ) ).

% size_list_estimation'
thf(fact_1098_rotate1__hd__tl,axiom,
    ! [Xs: list_term_a_b] :
      ( ( Xs != nil_term_a_b )
     => ( ( rotate1_term_a_b @ Xs )
        = ( append_term_a_b @ ( tl_term_a_b @ Xs ) @ ( cons_term_a_b @ ( hd_term_a_b @ Xs ) @ nil_term_a_b ) ) ) ) ).

% rotate1_hd_tl
thf(fact_1099_rotate1__hd__tl,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ( ( rotate1_nat @ Xs )
        = ( append_nat @ ( tl_nat @ Xs ) @ ( cons_nat @ ( hd_nat @ Xs ) @ nil_nat ) ) ) ) ).

% rotate1_hd_tl
thf(fact_1100_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_term_a_b] :
      ( ( ( rotate1_term_a_b @ Xs )
        = nil_term_a_b )
      = ( Xs = nil_term_a_b ) ) ).

% rotate1_is_Nil_conv
thf(fact_1101_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ( rotate1_nat @ Xs )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% rotate1_is_Nil_conv
thf(fact_1102_rotate1_Osimps_I1_J,axiom,
    ( ( rotate1_term_a_b @ nil_term_a_b )
    = nil_term_a_b ) ).

% rotate1.simps(1)
thf(fact_1103_rotate1_Osimps_I1_J,axiom,
    ( ( rotate1_nat @ nil_nat )
    = nil_nat ) ).

% rotate1.simps(1)
thf(fact_1104_rotate1_Osimps_I2_J,axiom,
    ! [X3: term_a_b,Xs: list_term_a_b] :
      ( ( rotate1_term_a_b @ ( cons_term_a_b @ X3 @ Xs ) )
      = ( append_term_a_b @ Xs @ ( cons_term_a_b @ X3 @ nil_term_a_b ) ) ) ).

% rotate1.simps(2)
thf(fact_1105_rotate1_Osimps_I2_J,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( rotate1_nat @ ( cons_nat @ X3 @ Xs ) )
      = ( append_nat @ Xs @ ( cons_nat @ X3 @ nil_nat ) ) ) ).

% rotate1.simps(2)
thf(fact_1106_zip__eq__Nil__iff,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( ( zip_te2260252107881041761rm_a_b @ Xs @ Ys )
        = nil_Pr6942174756412032271rm_a_b )
      = ( ( Xs = nil_term_a_b )
        | ( Ys = nil_term_a_b ) ) ) ).

% zip_eq_Nil_iff
thf(fact_1107_zip__eq__Nil__iff,axiom,
    ! [Xs: list_term_a_b,Ys: list_nat] :
      ( ( ( zip_term_a_b_nat @ Xs @ Ys )
        = nil_Pr5682614035376309451_b_nat )
      = ( ( Xs = nil_term_a_b )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_1108_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_term_a_b] :
      ( ( ( zip_nat_term_a_b @ Xs @ Ys )
        = nil_Pr4197086119080101963rm_a_b )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_term_a_b ) ) ) ).

% zip_eq_Nil_iff
thf(fact_1109_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( zip_nat_nat @ Xs @ Ys )
        = nil_Pr5478986624290739719at_nat )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_1110_Nil__eq__zip__iff,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( nil_Pr6942174756412032271rm_a_b
        = ( zip_te2260252107881041761rm_a_b @ Xs @ Ys ) )
      = ( ( Xs = nil_term_a_b )
        | ( Ys = nil_term_a_b ) ) ) ).

% Nil_eq_zip_iff
thf(fact_1111_Nil__eq__zip__iff,axiom,
    ! [Xs: list_term_a_b,Ys: list_nat] :
      ( ( nil_Pr5682614035376309451_b_nat
        = ( zip_term_a_b_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_term_a_b )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_1112_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_term_a_b] :
      ( ( nil_Pr4197086119080101963rm_a_b
        = ( zip_nat_term_a_b @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_term_a_b ) ) ) ).

% Nil_eq_zip_iff
thf(fact_1113_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( nil_Pr5478986624290739719at_nat
        = ( zip_nat_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_1114_enumerate__simps_I1_J,axiom,
    ! [N: nat] :
      ( ( enumerate_term_a_b @ N @ nil_term_a_b )
      = nil_Pr4197086119080101963rm_a_b ) ).

% enumerate_simps(1)
thf(fact_1115_enumerate__simps_I1_J,axiom,
    ! [N: nat] :
      ( ( enumerate_nat @ N @ nil_nat )
      = nil_Pr5478986624290739719at_nat ) ).

% enumerate_simps(1)
thf(fact_1116_zip__Cons__Cons,axiom,
    ! [X3: nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
      ( ( zip_nat_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1117_zip__Cons__Cons,axiom,
    ! [X3: term_a_b,Xs: list_term_a_b,Y3: term_a_b,Ys: list_term_a_b] :
      ( ( zip_te2260252107881041761rm_a_b @ ( cons_term_a_b @ X3 @ Xs ) @ ( cons_term_a_b @ Y3 @ Ys ) )
      = ( cons_P3536305108106557631rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( zip_te2260252107881041761rm_a_b @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1118_enumerate__simps_I2_J,axiom,
    ! [N: nat,X3: nat,Xs: list_nat] :
      ( ( enumerate_nat @ N @ ( cons_nat @ X3 @ Xs ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ X3 ) @ ( enumerate_nat @ ( suc @ N ) @ Xs ) ) ) ).

% enumerate_simps(2)
thf(fact_1119_nth__zip,axiom,
    ! [I: nat,Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ys ) )
       => ( ( nth_Pr1098598711982373392rm_a_b @ ( zip_te2260252107881041761rm_a_b @ Xs @ Ys ) @ I )
          = ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Xs @ I ) @ ( nth_term_a_b @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1120_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b,X3: term_a_b] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( member_term_a_b2 @ X3 @ ( set_term_a_b2 @ Xs ) )
       => ~ ! [Y: term_a_b] :
              ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( set_Pr2370880052973118494rm_a_b @ ( zip_te2260252107881041761rm_a_b @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1121_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b,Y3: term_a_b] :
      ( ( ( size_s8906293707977694520rm_a_b @ Xs )
        = ( size_s8906293707977694520rm_a_b @ Ys ) )
     => ( ( member_term_a_b2 @ Y3 @ ( set_term_a_b2 @ Ys ) )
       => ~ ! [X: term_a_b] :
              ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ ( set_Pr2370880052973118494rm_a_b @ ( zip_te2260252107881041761rm_a_b @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1122_zip__update,axiom,
    ! [Xs: list_term_a_b,I: nat,X3: term_a_b,Ys: list_term_a_b,Y3: term_a_b] :
      ( ( zip_te2260252107881041761rm_a_b @ ( list_update_term_a_b @ Xs @ I @ X3 ) @ ( list_update_term_a_b @ Ys @ I @ Y3 ) )
      = ( list_u4211539095652206455rm_a_b @ ( zip_te2260252107881041761rm_a_b @ Xs @ Ys ) @ I @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) ) ) ).

% zip_update
thf(fact_1123_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_nat,Xy: product_prod_nat_nat,Xys: list_P6011104703257516679at_nat] :
      ( ( ( zip_nat_nat @ Xs @ Ys )
        = ( cons_P6512896166579812791at_nat @ Xy @ Xys ) )
     => ~ ! [X: nat,Xs4: list_nat] :
            ( ( Xs
              = ( cons_nat @ X @ Xs4 ) )
           => ! [Y: nat,Ys5: list_nat] :
                ( ( Ys
                  = ( cons_nat @ Y @ Ys5 ) )
               => ( ( Xy
                    = ( product_Pair_nat_nat @ X @ Y ) )
                 => ( Xys
                   != ( zip_nat_nat @ Xs4 @ Ys5 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_1124_zip__eq__ConsE,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b,Xy: produc357393685978478089rm_a_b,Xys: list_P8875379029341186191rm_a_b] :
      ( ( ( zip_te2260252107881041761rm_a_b @ Xs @ Ys )
        = ( cons_P3536305108106557631rm_a_b @ Xy @ Xys ) )
     => ~ ! [X: term_a_b,Xs4: list_term_a_b] :
            ( ( Xs
              = ( cons_term_a_b @ X @ Xs4 ) )
           => ! [Y: term_a_b,Ys5: list_term_a_b] :
                ( ( Ys
                  = ( cons_term_a_b @ Y @ Ys5 ) )
               => ( ( Xy
                    = ( produc7020197800436672577rm_a_b @ X @ Y ) )
                 => ( Xys
                   != ( zip_te2260252107881041761rm_a_b @ Xs4 @ Ys5 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_1125_set__zip__rightD,axiom,
    ! [X3: term_a_b,Y3: term_a_b,Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( set_Pr2370880052973118494rm_a_b @ ( zip_te2260252107881041761rm_a_b @ Xs @ Ys ) ) )
     => ( member_term_a_b2 @ Y3 @ ( set_term_a_b2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1126_set__zip__leftD,axiom,
    ! [X3: term_a_b,Y3: term_a_b,Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( set_Pr2370880052973118494rm_a_b @ ( zip_te2260252107881041761rm_a_b @ Xs @ Ys ) ) )
     => ( member_term_a_b2 @ X3 @ ( set_term_a_b2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1127_in__set__zipE,axiom,
    ! [X3: list_nat,Y3: list_nat,Xs: list_list_nat,Ys: list_list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ ( set_Pr3842133991353686454st_nat @ ( zip_li7157463729305086713st_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
         => ~ ( member_list_nat2 @ Y3 @ ( set_list_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1128_in__set__zipE,axiom,
    ! [X3: list_nat,Y3: produc357393685978478089rm_a_b,Xs: list_list_nat,Ys: list_P8875379029341186191rm_a_b] :
      ( ( member4140859439665155893rm_a_b @ ( produc3015861401532503632rm_a_b @ X3 @ Y3 ) @ ( set_Pr7273799732993181129rm_a_b @ ( zip_li1978623282951132976rm_a_b @ Xs @ Ys ) ) )
     => ~ ( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
         => ~ ( member5869715511025134514rm_a_b @ Y3 @ ( set_Pr2370880052973118494rm_a_b @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1129_in__set__zipE,axiom,
    ! [X3: list_nat,Y3: nat,Xs: list_list_nat,Ys: list_nat] :
      ( ( member7820552544826472890at_nat @ ( produc1298395424260782409at_nat @ X3 @ Y3 ) @ ( set_Pr7715406932454250534at_nat @ ( zip_list_nat_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
         => ~ ( member_nat2 @ Y3 @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1130_in__set__zipE,axiom,
    ! [X3: produc357393685978478089rm_a_b,Y3: list_nat,Xs: list_P8875379029341186191rm_a_b,Ys: list_list_nat] :
      ( ( member190277321972437807st_nat @ ( produc5056965718246025362st_nat @ X3 @ Y3 ) @ ( set_Pr3323217615300463043st_nat @ ( zip_Pr4019727599664654706st_nat @ Xs @ Ys ) ) )
     => ~ ( ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
         => ~ ( member_list_nat2 @ Y3 @ ( set_list_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1131_in__set__zipE,axiom,
    ! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b,Ys: list_P8875379029341186191rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ ( set_Pr1141850553043368956rm_a_b @ ( zip_Pr5674138466097917431rm_a_b @ Xs @ Ys ) ) )
     => ~ ( ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
         => ~ ( member5869715511025134514rm_a_b @ Y3 @ ( set_Pr2370880052973118494rm_a_b @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1132_in__set__zipE,axiom,
    ! [X3: produc357393685978478089rm_a_b,Y3: nat,Xs: list_P8875379029341186191rm_a_b,Ys: list_nat] :
      ( ( member5878619692762604063_b_nat @ ( produc6844290133701299842_b_nat @ X3 @ Y3 ) @ ( set_Pr225064623739089587_b_nat @ ( zip_Pr6666957785173753698_b_nat @ Xs @ Ys ) ) )
     => ~ ( ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
         => ~ ( member_nat2 @ Y3 @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1133_in__set__zipE,axiom,
    ! [X3: nat,Y3: list_nat,Xs: list_nat,Ys: list_list_nat] :
      ( ( member1631564025489475386st_nat @ ( produc8282810413953273033st_nat @ X3 @ Y3 ) @ ( set_Pr1526418413117253030st_nat @ ( zip_nat_list_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
         => ~ ( member_list_nat2 @ Y3 @ ( set_list_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1134_in__set__zipE,axiom,
    ! [X3: nat,Y3: produc357393685978478089rm_a_b,Xs: list_nat,Ys: list_P8875379029341186191rm_a_b] :
      ( ( member755457025140340037rm_a_b @ ( produc8902844540702284128rm_a_b @ X3 @ Y3 ) @ ( set_Pr4325273992971601369rm_a_b @ ( zip_na8725512192174737984rm_a_b @ Xs @ Ys ) ) )
     => ~ ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
         => ~ ( member5869715511025134514rm_a_b @ Y3 @ ( set_Pr2370880052973118494rm_a_b @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1135_in__set__zipE,axiom,
    ! [X3: nat,Y3: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
         => ~ ( member_nat2 @ Y3 @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1136_in__set__zipE,axiom,
    ! [X3: term_a_b,Y3: term_a_b,Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( set_Pr2370880052973118494rm_a_b @ ( zip_te2260252107881041761rm_a_b @ Xs @ Ys ) ) )
     => ~ ( ( member_term_a_b2 @ X3 @ ( set_term_a_b2 @ Xs ) )
         => ~ ( member_term_a_b2 @ Y3 @ ( set_term_a_b2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1137_zip__same,axiom,
    ! [A: list_nat,B: list_nat,Xs: list_list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( set_Pr3842133991353686454st_nat @ ( zip_li7157463729305086713st_nat @ Xs @ Xs ) ) )
      = ( ( member_list_nat2 @ A @ ( set_list_nat2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_1138_zip__same,axiom,
    ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
      ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( set_Pr1141850553043368956rm_a_b @ ( zip_Pr5674138466097917431rm_a_b @ Xs @ Xs ) ) )
      = ( ( member5869715511025134514rm_a_b @ A @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_1139_zip__same,axiom,
    ! [A: nat,B: nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Xs ) ) )
      = ( ( member_nat2 @ A @ ( set_nat2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_1140_zip__same,axiom,
    ! [A: term_a_b,B: term_a_b,Xs: list_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( set_Pr2370880052973118494rm_a_b @ ( zip_te2260252107881041761rm_a_b @ Xs @ Xs ) ) )
      = ( ( member_term_a_b2 @ A @ ( set_term_a_b2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_1141_zip__fst,axiom,
    ! [P: produc357393685978478089rm_a_b,As2: list_term_a_b,Bs2: list_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ P @ ( set_Pr2370880052973118494rm_a_b @ ( zip_te2260252107881041761rm_a_b @ As2 @ Bs2 ) ) )
     => ( member_term_a_b2 @ ( produc5709096180074326061rm_a_b @ P ) @ ( set_term_a_b2 @ As2 ) ) ) ).

% zip_fst
thf(fact_1142_zip__snd,axiom,
    ! [P: produc357393685978478089rm_a_b,As2: list_term_a_b,Bs2: list_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ P @ ( set_Pr2370880052973118494rm_a_b @ ( zip_te2260252107881041761rm_a_b @ As2 @ Bs2 ) ) )
     => ( member_term_a_b2 @ ( produc968600491044949615rm_a_b @ P ) @ ( set_term_a_b2 @ Bs2 ) ) ) ).

% zip_snd
thf(fact_1143_hd__zip,axiom,
    ! [Xs: list_term_a_b,Ys: list_nat] :
      ( ( Xs != nil_term_a_b )
     => ( ( Ys != nil_nat )
       => ( ( hd_Pro5563056963574031536_b_nat @ ( zip_term_a_b_nat @ Xs @ Ys ) )
          = ( produc2478027105211390077_b_nat @ ( hd_term_a_b @ Xs ) @ ( hd_nat @ Ys ) ) ) ) ) ).

% hd_zip
thf(fact_1144_hd__zip,axiom,
    ! [Xs: list_nat,Ys: list_term_a_b] :
      ( ( Xs != nil_nat )
     => ( ( Ys != nil_term_a_b )
       => ( ( hd_Pro4077529047277824048rm_a_b @ ( zip_nat_term_a_b @ Xs @ Ys ) )
          = ( produc1516572978046417917rm_a_b @ ( hd_nat @ Xs ) @ ( hd_term_a_b @ Ys ) ) ) ) ) ).

% hd_zip
thf(fact_1145_hd__zip,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs != nil_nat )
     => ( ( Ys != nil_nat )
       => ( ( hd_Pro3460610213475200108at_nat @ ( zip_nat_nat @ Xs @ Ys ) )
          = ( product_Pair_nat_nat @ ( hd_nat @ Xs ) @ ( hd_nat @ Ys ) ) ) ) ) ).

% hd_zip
thf(fact_1146_hd__zip,axiom,
    ! [Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( Xs != nil_term_a_b )
     => ( ( Ys != nil_term_a_b )
       => ( ( hd_Pro6001814652670619252rm_a_b @ ( zip_te2260252107881041761rm_a_b @ Xs @ Ys ) )
          = ( produc7020197800436672577rm_a_b @ ( hd_term_a_b @ Xs ) @ ( hd_term_a_b @ Ys ) ) ) ) ) ).

% hd_zip
thf(fact_1147_in__set__zip,axiom,
    ! [P: produc357393685978478089rm_a_b,Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ P @ ( set_Pr2370880052973118494rm_a_b @ ( zip_te2260252107881041761rm_a_b @ Xs @ Ys ) ) )
      = ( ? [N3: nat] :
            ( ( ( nth_term_a_b @ Xs @ N3 )
              = ( produc5709096180074326061rm_a_b @ P ) )
            & ( ( nth_term_a_b @ Ys @ N3 )
              = ( produc968600491044949615rm_a_b @ P ) )
            & ( ord_less_nat @ N3 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
            & ( ord_less_nat @ N3 @ ( size_s8906293707977694520rm_a_b @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_1148_size__simp4,axiom,
    ! [X3: term_a_b,Y3: term_a_b,Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( set_Pr2370880052973118494rm_a_b @ ( zip_te2260252107881041761rm_a_b @ Xs @ Ys ) ) )
     => ( ord_less_nat @ ( size_size_term_a_b @ Y3 ) @ ( suc @ ( size_list_term_a_b @ size_size_term_a_b @ Ys ) ) ) ) ).

% size_simp4
thf(fact_1149_size__simp3,axiom,
    ! [X3: term_a_b,Y3: term_a_b,Xs: list_term_a_b,Ys: list_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( set_Pr2370880052973118494rm_a_b @ ( zip_te2260252107881041761rm_a_b @ Xs @ Ys ) ) )
     => ( ord_less_nat @ ( size_size_term_a_b @ X3 ) @ ( suc @ ( size_list_term_a_b @ size_size_term_a_b @ Xs ) ) ) ) ).

% size_simp3
thf(fact_1150_term_Osize_I4_J,axiom,
    ! [X21: a,X22: list_term_a_b] :
      ( ( size_size_term_a_b @ ( fun_a_b @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_list_term_a_b @ size_size_term_a_b @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% term.size(4)
thf(fact_1151_size__simp6,axiom,
    ! [S: term_a_b,Ss2: list_term_a_b,T: term_a_b] :
      ( ( member_term_a_b2 @ S @ ( set_term_a_b2 @ Ss2 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
       => ( ( ord_less_eq_nat @ ( size_size_term_a_b @ T ) @ ( size_size_term_a_b @ S ) )
         => ( ord_less_eq_nat @ ( size_size_term_a_b @ T ) @ ( suc @ ( size_list_term_a_b @ size_size_term_a_b @ Ss2 ) ) ) ) ) ) ).

% size_simp6
thf(fact_1152_size__simp5,axiom,
    ! [S: term_a_b,Ss2: list_term_a_b,T: term_a_b] :
      ( ( member_term_a_b2 @ S @ ( set_term_a_b2 @ Ss2 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
       => ( ( ord_less_nat @ ( size_size_term_a_b @ T ) @ ( size_size_term_a_b @ S ) )
         => ( ord_less_nat @ ( size_size_term_a_b @ T ) @ ( suc @ ( size_list_term_a_b @ size_size_term_a_b @ Ss2 ) ) ) ) ) ) ).

% size_simp5
thf(fact_1153_suptI,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( A != B )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b ) ) ) ).

% suptI
thf(fact_1154_Fun__supt,axiom,
    ! [F2: a,Ts: list_term_a_b,S: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( fun_a_b @ F2 @ Ts ) @ S ) @ subterm_and_supt_a_b )
     => ~ ! [T2: term_a_b] :
            ( ( member_term_a_b2 @ T2 @ ( set_term_a_b2 @ Ts ) )
           => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ S ) @ subter523971068842742411eq_a_b ) ) ) ).

% Fun_supt
thf(fact_1155_set__supteq__into__supt,axiom,
    ! [T: term_a_b,Ts: list_term_a_b,S: term_a_b,F2: a] :
      ( ( member_term_a_b2 @ T @ ( set_term_a_b2 @ Ts ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( fun_a_b @ F2 @ Ts ) @ S ) @ subterm_and_supt_a_b ) ) ) ).

% set_supteq_into_supt
thf(fact_1156_image__diff__subset,axiom,
    ! [F2: nat > nat,A5: set_nat,B5: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F2 @ A5 ) @ ( image_nat_nat @ F2 @ B5 ) ) @ ( image_nat_nat @ F2 @ ( minus_minus_set_nat @ A5 @ B5 ) ) ) ).

% image_diff_subset
thf(fact_1157_psubset__imp__ex__mem,axiom,
    ! [A5: set_list_nat,B5: set_list_nat] :
      ( ( ord_le1190675801316882794st_nat @ A5 @ B5 )
     => ? [B3: list_nat] : ( member_list_nat2 @ B3 @ ( minus_7954133019191499631st_nat @ B5 @ A5 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1158_psubset__imp__ex__mem,axiom,
    ! [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
      ( ( ord_le4883060154188259093rm_a_b @ A5 @ B5 )
     => ? [B3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ B3 @ ( minus_5192120951422937424rm_a_b @ B5 @ A5 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1159_psubset__imp__ex__mem,axiom,
    ! [A5: set_nat,B5: set_nat] :
      ( ( ord_less_set_nat @ A5 @ B5 )
     => ? [B3: nat] : ( member_nat2 @ B3 @ ( minus_minus_set_nat @ B5 @ A5 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1160_subterm_OleD,axiom,
    ! [X3: term_a_b,Y3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ subter523971068842742411eq_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subterm_and_supt_a_b ) ) ).

% subterm.leD
thf(fact_1161_subterm_Ole__less,axiom,
    ! [Y3: term_a_b,X3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subter523971068842742411eq_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subterm_and_supt_a_b )
        | ( X3 = Y3 ) ) ) ).

% subterm.le_less
thf(fact_1162_subterm_Oless__le,axiom,
    ! [Y3: term_a_b,X3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subter523971068842742411eq_a_b )
        & ( X3 != Y3 ) ) ) ).

% subterm.less_le
thf(fact_1163_subterm_Onless__le,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b ) )
      = ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
        | ( A = B ) ) ) ).

% subterm.nless_le
thf(fact_1164_subterm_Oless__imp__le,axiom,
    ! [Y3: term_a_b,X3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subterm_and_supt_a_b )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subter523971068842742411eq_a_b ) ) ).

% subterm.less_imp_le
thf(fact_1165_subterm_Ole__neq__trans,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
     => ( ( A != B )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.le_neq_trans
thf(fact_1166_subterm_Oantisym__conv1,axiom,
    ! [Y3: term_a_b,X3: term_a_b] :
      ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subter523971068842742411eq_a_b )
        = ( X3 = Y3 ) ) ) ).

% subterm.antisym_conv1
thf(fact_1167_subterm_Oantisym__conv2,axiom,
    ! [Y3: term_a_b,X3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subter523971068842742411eq_a_b )
     => ( ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subterm_and_supt_a_b ) )
        = ( X3 = Y3 ) ) ) ).

% subterm.antisym_conv2
thf(fact_1168_subterm_Ole__less__trans,axiom,
    ! [Y3: term_a_b,X3: term_a_b,Z2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ Y3 ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ X3 ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.le_less_trans
thf(fact_1169_subterm_Oless__le__trans,axiom,
    ! [Y3: term_a_b,X3: term_a_b,Z2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ Y3 ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ X3 ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.less_le_trans
thf(fact_1170_subterm_Oless__le__not__le,axiom,
    ! [Y3: term_a_b,X3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subter523971068842742411eq_a_b )
        & ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.less_le_not_le
thf(fact_1171_subterm_Ole__imp__less__or__eq,axiom,
    ! [Y3: term_a_b,X3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subterm_and_supt_a_b )
        | ( X3 = Y3 ) ) ) ).

% subterm.le_imp_less_or_eq
thf(fact_1172_subterm_Oorder_Ostrict__trans1,axiom,
    ! [B: term_a_b,A: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.order.strict_trans1
thf(fact_1173_subterm_Oorder_Ostrict__trans2,axiom,
    ! [B: term_a_b,A: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.order.strict_trans2
thf(fact_1174_subterm_Oorder_Ostrict__iff__not,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
        & ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.order.strict_iff_not
thf(fact_1175_subterm_Oorder_Oorder__iff__strict,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
        | ( A = B ) ) ) ).

% subterm.order.order_iff_strict
thf(fact_1176_subterm_Oorder_Ostrict__iff__order,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
        & ( A != B ) ) ) ).

% subterm.order.strict_iff_order
thf(fact_1177_subterm_Odual__order_Ostrict__trans1,axiom,
    ! [A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.dual_order.strict_trans1
thf(fact_1178_subterm_Odual__order_Ostrict__trans2,axiom,
    ! [A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.dual_order.strict_trans2
thf(fact_1179_subterm_Odual__order_Ostrict__iff__not,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
        & ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.dual_order.strict_iff_not
thf(fact_1180_subterm_Oorder_Ostrict__implies__order,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b ) ) ).

% subterm.order.strict_implies_order
thf(fact_1181_subterm_Odual__order_Oorder__iff__strict,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
        | ( A = B ) ) ) ).

% subterm.dual_order.order_iff_strict
thf(fact_1182_subterm_Odual__order_Ostrict__iff__order,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
        & ( A != B ) ) ) ).

% subterm.dual_order.strict_iff_order
thf(fact_1183_subterm_Odual__order_Ostrict__implies__order,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b ) ) ).

% subterm.dual_order.strict_implies_order
thf(fact_1184_subterm_Oorder_Onot__eq__order__implies__strict,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( A != B )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.order.not_eq_order_implies_strict
thf(fact_1185_suptE,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
       => ( S != T ) ) ) ).

% suptE
thf(fact_1186_supt__imp__supteq,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b ) ) ).

% supt_imp_supteq
thf(fact_1187_supteq__not__supt,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
     => ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
       => ( S = T ) ) ) ).

% supteq_not_supt
thf(fact_1188_supt__supteq__conv,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
        & ( S != T ) ) ) ).

% supt_supteq_conv
thf(fact_1189_supteq__supt__conv,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
        | ( S = T ) ) ) ).

% supteq_supt_conv
thf(fact_1190_supt__supteq__trans,axiom,
    ! [S: term_a_b,T: term_a_b,U: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ U ) @ subterm_and_supt_a_b ) ) ) ).

% supt_supteq_trans
thf(fact_1191_supteq__supt__trans,axiom,
    ! [S: term_a_b,T: term_a_b,U: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ U ) @ subterm_and_supt_a_b ) ) ) ).

% supteq_supt_trans
thf(fact_1192_supt__supteq__not__supteq,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
      = ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
        & ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ subter523971068842742411eq_a_b ) ) ) ).

% supt_supteq_not_supteq
thf(fact_1193_supteq_Orefl,axiom,
    ! [T: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ T ) @ subter523971068842742411eq_a_b ) ).

% supteq.refl
thf(fact_1194_subterm_Oeq__refl,axiom,
    ! [X3: term_a_b,Y3: term_a_b] :
      ( ( X3 = Y3 )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subter523971068842742411eq_a_b ) ) ).

% subterm.eq_refl
thf(fact_1195_subterm_Oorder__refl,axiom,
    ! [X3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ subter523971068842742411eq_a_b ) ).

% subterm.order_refl
thf(fact_1196_subterm_Oorder_Otrans,axiom,
    ! [B: term_a_b,A: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.order.trans
thf(fact_1197_subterm_Oorder__trans,axiom,
    ! [Y3: term_a_b,X3: term_a_b,Z2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ Y3 ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ X3 ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.order_trans
thf(fact_1198_subterm_Oantisym__conv,axiom,
    ! [X3: term_a_b,Y3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subter523971068842742411eq_a_b )
        = ( X3 = Y3 ) ) ) ).

% subterm.antisym_conv
thf(fact_1199_subterm_Oorder_Oeq__iff,axiom,
    ( ( ^ [Y6: term_a_b,Z4: term_a_b] : ( Y6 = Z4 ) )
    = ( ^ [A3: term_a_b,B4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ A3 ) @ subter523971068842742411eq_a_b )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B4 ) @ subter523971068842742411eq_a_b ) ) ) ) ).

% subterm.order.eq_iff
thf(fact_1200_subterm_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: term_a_b,Z4: term_a_b] : ( Y6 = Z4 ) )
    = ( ^ [X4: term_a_b,Y4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ X4 ) @ subter523971068842742411eq_a_b )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y4 ) @ subter523971068842742411eq_a_b ) ) ) ) ).

% subterm.order_eq_iff
thf(fact_1201_subterm_Oorder_Oantisym,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
       => ( A = B ) ) ) ).

% subterm.order.antisym
thf(fact_1202_subterm_Oorder__antisym,axiom,
    ! [Y3: term_a_b,X3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ subter523971068842742411eq_a_b )
       => ( X3 = Y3 ) ) ) ).

% subterm.order_antisym
thf(fact_1203_subterm_Odual__order_Orefl,axiom,
    ! [A: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ subter523971068842742411eq_a_b ) ).

% subterm.dual_order.refl
thf(fact_1204_subterm_Oord__eq__le__trans,axiom,
    ! [A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( A = B )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.ord_eq_le_trans
thf(fact_1205_subterm_Oord__le__eq__trans,axiom,
    ! [B: term_a_b,A: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
     => ( ( B = C )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.ord_le_eq_trans
thf(fact_1206_subterm_Odual__order_Otrans,axiom,
    ! [A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.dual_order.trans
thf(fact_1207_subterm_Odual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: term_a_b,Z4: term_a_b] : ( Y6 = Z4 ) )
    = ( ^ [A3: term_a_b,B4: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B4 ) @ subter523971068842742411eq_a_b )
          & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ A3 ) @ subter523971068842742411eq_a_b ) ) ) ) ).

% subterm.dual_order.eq_iff
thf(fact_1208_subterm_Odual__order_Oantisym,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
       => ( A = B ) ) ) ).

% subterm.dual_order.antisym
thf(fact_1209_eq__supteq,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( S = T )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b ) ) ).

% eq_supteq
thf(fact_1210_supteq__trans,axiom,
    ! [S: term_a_b,T: term_a_b,U: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ subter523971068842742411eq_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ U ) @ subter523971068842742411eq_a_b ) ) ) ).

% supteq_trans
thf(fact_1211_supteq__antisym,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ subter523971068842742411eq_a_b )
       => ( S = T ) ) ) ).

% supteq_antisym
thf(fact_1212_subterm_Oless__asym,axiom,
    ! [Y3: term_a_b,X3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ subterm_and_supt_a_b ) ) ).

% subterm.less_asym
thf(fact_1213_subterm_Oless__asym_H,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b ) ) ).

% subterm.less_asym'
thf(fact_1214_subterm_Oless__trans,axiom,
    ! [Y3: term_a_b,X3: term_a_b,Z2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ Y3 ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ X3 ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.less_trans
thf(fact_1215_subterm_Oorder_Oasym,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b ) ) ).

% subterm.order.asym
thf(fact_1216_subterm_Oless__irrefl,axiom,
    ! [X3: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ subterm_and_supt_a_b ) ).

% subterm.less_irrefl
thf(fact_1217_subterm_Oless__imp__neq,axiom,
    ! [Y3: term_a_b,X3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subterm_and_supt_a_b )
     => ( X3 != Y3 ) ) ).

% subterm.less_imp_neq
thf(fact_1218_subterm_Oless__not__sym,axiom,
    ! [Y3: term_a_b,X3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ subterm_and_supt_a_b ) ) ).

% subterm.less_not_sym
thf(fact_1219_subterm_Oless__imp__triv,axiom,
    ! [Y3: term_a_b,X3: term_a_b,P4: $o] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ subterm_and_supt_a_b )
       => P4 ) ) ).

% subterm.less_imp_triv
thf(fact_1220_subterm_Odual__order_Oasym,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b ) ) ).

% subterm.dual_order.asym
thf(fact_1221_subterm_Oless__imp__not__eq,axiom,
    ! [Y3: term_a_b,X3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subterm_and_supt_a_b )
     => ( X3 != Y3 ) ) ).

% subterm.less_imp_not_eq
thf(fact_1222_subterm_Oless__imp__not__eq2,axiom,
    ! [Y3: term_a_b,X3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subterm_and_supt_a_b )
     => ( Y3 != X3 ) ) ).

% subterm.less_imp_not_eq2
thf(fact_1223_subterm_Odual__order_Oirrefl,axiom,
    ! [A: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ subterm_and_supt_a_b ) ).

% subterm.dual_order.irrefl
thf(fact_1224_subterm_Oless__imp__not__less,axiom,
    ! [Y3: term_a_b,X3: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ subterm_and_supt_a_b ) ) ).

% subterm.less_imp_not_less
thf(fact_1225_subterm_Oord__eq__less__trans,axiom,
    ! [A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( A = B )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.ord_eq_less_trans
thf(fact_1226_subterm_Oord__less__eq__trans,axiom,
    ! [B: term_a_b,A: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ( ( B = C )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.ord_less_eq_trans
thf(fact_1227_subterm_Oorder_Ostrict__trans,axiom,
    ! [B: term_a_b,A: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.order.strict_trans
thf(fact_1228_subterm_Odual__order_Ostrict__trans,axiom,
    ! [A: term_a_b,B: term_a_b,C: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.dual_order.strict_trans
thf(fact_1229_subterm_Oorder_Ostrict__implies__not__eq,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ( A != B ) ) ).

% subterm.order.strict_implies_not_eq
thf(fact_1230_subterm_Odual__order_Ostrict__implies__not__eq,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
     => ( A != B ) ) ).

% subterm.dual_order.strict_implies_not_eq
thf(fact_1231_supt__neqD,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ( S != T ) ) ).

% supt_neqD
thf(fact_1232_supt__trans,axiom,
    ! [S: term_a_b,T: term_a_b,U: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ U ) @ subterm_and_supt_a_b ) ) ) ).

% supt_trans
thf(fact_1233_supt__irrefl,axiom,
    ! [T: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ T ) @ subterm_and_supt_a_b ) ).

% supt_irrefl
thf(fact_1234_supt__not__sym,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ subterm_and_supt_a_b ) ) ).

% supt_not_sym
thf(fact_1235_supt__not__refl,axiom,
    ! [T: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ T ) @ subterm_and_supt_a_b ) ).

% supt_not_refl
thf(fact_1236_subterm__induct,axiom,
    ! [P4: term_a_b > $o,T: term_a_b] :
      ( ! [T2: term_a_b] :
          ( ! [S6: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ S6 ) @ subterm_and_supt_a_b )
             => ( P4 @ S6 ) )
         => ( P4 @ T2 ) )
     => ( P4 @ T ) ) ).

% subterm_induct
thf(fact_1237_supteq__var__imp__eq,axiom,
    ! [X3: b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( var_b_a @ X3 ) @ T ) @ subter523971068842742411eq_a_b )
      = ( T
        = ( var_b_a @ X3 ) ) ) ).

% supteq_var_imp_eq
thf(fact_1238_supt__var,axiom,
    ! [X3: b,U: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( var_b_a @ X3 ) @ U ) @ subterm_and_supt_a_b ) ).

% supt_var
thf(fact_1239_Var__supt,axiom,
    ! [X3: b,T: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( var_b_a @ X3 ) @ T ) @ subterm_and_supt_a_b ) ).

% Var_supt
thf(fact_1240_subterm_Olift__Suc__mono__less,axiom,
    ! [F2: nat > term_a_b,N: nat,N4: nat] :
      ( ! [N5: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ ( suc @ N5 ) ) @ ( F2 @ N5 ) ) @ subterm_and_supt_a_b )
     => ( ( ord_less_nat @ N @ N4 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ N4 ) @ ( F2 @ N ) ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.lift_Suc_mono_less
thf(fact_1241_subterm_Olift__Suc__mono__less__iff,axiom,
    ! [F2: nat > term_a_b,M3: nat,N: nat] :
      ( ! [N5: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ ( suc @ N5 ) ) @ ( F2 @ N5 ) ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ M3 ) @ ( F2 @ N ) ) @ subterm_and_supt_a_b )
        = ( ord_less_nat @ N @ M3 ) ) ) ).

% subterm.lift_Suc_mono_less_iff
thf(fact_1242_subterm_Olift__Suc__antimono__le,axiom,
    ! [F2: nat > term_a_b,N: nat,N4: nat] :
      ( ! [N5: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ N5 ) @ ( F2 @ ( suc @ N5 ) ) ) @ subter523971068842742411eq_a_b )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ N ) @ ( F2 @ N4 ) ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.lift_Suc_antimono_le
thf(fact_1243_subterm_Olift__Suc__mono__le,axiom,
    ! [F2: nat > term_a_b,N: nat,N4: nat] :
      ( ! [N5: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ ( suc @ N5 ) ) @ ( F2 @ N5 ) ) @ subter523971068842742411eq_a_b )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ N4 ) @ ( F2 @ N ) ) @ subter523971068842742411eq_a_b ) ) ) ).

% subterm.lift_Suc_mono_le
thf(fact_1244_supt_Oarg,axiom,
    ! [S: term_a_b,Ss2: list_term_a_b,F2: a] :
      ( ( member_term_a_b2 @ S @ ( set_term_a_b2 @ Ss2 ) )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( fun_a_b @ F2 @ Ss2 ) @ S ) @ subterm_and_supt_a_b ) ) ).

% supt.arg
thf(fact_1245_supt_Osubt,axiom,
    ! [S: term_a_b,Ss2: list_term_a_b,T: term_a_b,F2: a] :
      ( ( member_term_a_b2 @ S @ ( set_term_a_b2 @ Ss2 ) )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( fun_a_b @ F2 @ Ss2 ) @ T ) @ subterm_and_supt_a_b ) ) ) ).

% supt.subt
thf(fact_1246_supt_Ocases,axiom,
    ! [A1: term_a_b,A22: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ subterm_and_supt_a_b )
     => ( ! [S2: term_a_b,Ss: list_term_a_b] :
            ( ? [F3: a] :
                ( A1
                = ( fun_a_b @ F3 @ Ss ) )
           => ( ( A22 = S2 )
             => ~ ( member_term_a_b2 @ S2 @ ( set_term_a_b2 @ Ss ) ) ) )
       => ~ ! [S2: term_a_b,Ss: list_term_a_b] :
              ( ? [F3: a] :
                  ( A1
                  = ( fun_a_b @ F3 @ Ss ) )
             => ( ( member_term_a_b2 @ S2 @ ( set_term_a_b2 @ Ss ) )
               => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S2 @ A22 ) @ subterm_and_supt_a_b ) ) ) ) ) ).

% supt.cases
thf(fact_1247_supt_Osimps,axiom,
    ! [A1: term_a_b,A22: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ subterm_and_supt_a_b )
      = ( ? [S3: term_a_b,Ss3: list_term_a_b,F: a] :
            ( ( A1
              = ( fun_a_b @ F @ Ss3 ) )
            & ( A22 = S3 )
            & ( member_term_a_b2 @ S3 @ ( set_term_a_b2 @ Ss3 ) ) )
        | ? [S3: term_a_b,Ss3: list_term_a_b,T3: term_a_b,F: a] :
            ( ( A1
              = ( fun_a_b @ F @ Ss3 ) )
            & ( A22 = T3 )
            & ( member_term_a_b2 @ S3 @ ( set_term_a_b2 @ Ss3 ) )
            & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S3 @ T3 ) @ subterm_and_supt_a_b ) ) ) ) ).

% supt.simps
thf(fact_1248_supt__Fun,axiom,
    ! [S: term_a_b,F2: a,Ss2: list_term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ ( fun_a_b @ F2 @ Ss2 ) ) @ subterm_and_supt_a_b )
     => ~ ( member_term_a_b2 @ S @ ( set_term_a_b2 @ Ss2 ) ) ) ).

% supt_Fun

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
    ! [P4: $o] :
      ( ( P4 = $true )
      | ( P4 = $false ) ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( if_nat @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( if_nat @ $true @ X3 @ Y3 )
      = X3 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ? [Q2: list_nat] :
        ( ( member_list_nat2 @ Q2 @ ( terms_7168686267159881682rm_a_b @ ( fun_a_b @ c @ nil_term_a_b ) @ t ) )
        & ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( term_r6860082780075436317at_a_b @ s @ p @ u2 ) @ ( term_r6860082780075436317at_a_b @ t @ Q2 @ u2 ) ) @ ( relcom370159955682700863rm_a_b @ u @ v ) ) )
    | ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( term_r6860082780075436317at_a_b @ s @ p @ u2 ) @ t ) @ ( relcom370159955682700863rm_a_b @ u @ v ) ) ) ).

%------------------------------------------------------------------------------