%------------------------------------------------------------------------------
% 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    : FO_Theory_Rewriting/0055_NF/prob_00221_009334__18427790_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1731 ( 491 unt; 483 typ;   0 def)
%            Number of atoms       : 3908 (1508 equ;   0 cnn)
%            Maximal formula atoms :   20 (   3 avg)
%            Number of connectives : 15181 ( 584   ~; 104   |; 332   &;12189   @)
%                                         (   0 <=>;1972  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   8 avg)
%            Number of types       :  118 ( 117 usr)
%            Number of type conns  : 1004 (1004   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  369 ( 366 usr;  35 con; 0-3 aty)
%            Number of variables   : 4387 ( 121   ^;4014   !; 252   ?;4387   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 12:01:17.624
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    set_Pr4719305120195102542st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    set_Pr9093778441882193744at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J,type,
    produc8027700073345061287rm_f_v: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc3902171991451677348st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__a_J_Mt__Term__Oterm_Itf__f_Mtf__a_J_J,type,
    produc1754468282888495997rm_f_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J,type,
    produc5954175706252429988rm_f_v: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc1828647624359046049st_nat: $tType ).

thf(ty_n_t__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__List__Olist_It__Term__Oterm_Itf__f_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    list_t5008422188120450138term_f: $tType ).

thf(ty_n_t__FSet__Ofset_It__Term__Oterm_Itf__f_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    fset_t2965824081267281850term_f: $tType ).

thf(ty_n_t__Set__Oset_It__Term__Oterm_Itf__f_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    set_te5056978688659945524term_f: $tType ).

thf(ty_n_t__List__Olist_I_062_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Nat__Onat_J_J,type,
    list_B8305986481258897165_f_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    produc8199716216217303280at_nat: $tType ).

thf(ty_n_t__Tree____Automata__Ota_It__Bot____Terms__Obot____term_Itf__f_J_Mtf__f_J,type,
    tree_t2044664379607036063rm_f_f: $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__FSet__Ofset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fset_P5670320511379867111at_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__f_Mtf__v_J_Mt__Nat__Onat_J_J,type,
    list_term_f_v_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J,type,
    list_P3903862279629787026_f_nat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
    list_list_term_f_v: $tType ).

thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J,type,
    fset_P6228066233360383026_f_nat: $tType ).

thf(ty_n_t__Tree____Automata__Ota_It__Term__Oterm_Itf__f_Mtf__a_J_Mtf__f_J,type,
    tree_ta_term_f_a_f: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J,type,
    set_Pr5245412377734449720_f_nat: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
    set_list_term_f_v: $tType ).

thf(ty_n_t__Term__Oterm_Itf__f_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    term_f7969170442531588692term_f: $tType ).

thf(ty_n_t__List__Olist_It__Bot____Terms__Obot____term_Itf__f_J_J,type,
    list_Bot_bot_term_f: $tType ).

thf(ty_n_t__FSet__Ofset_It__Bot____Terms__Obot____term_Itf__f_J_J,type,
    fset_Bot_bot_term_f: $tType ).

thf(ty_n_t__Term__Oterm_Itf__f_Mt__Term__Oterm_Itf__f_Mtf__a_J_J,type,
    term_f_term_f_a: $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__Set__Oset_It__Bot____Terms__Obot____term_Itf__f_J_J,type,
    set_Bot_bot_term_f: $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__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J,type,
    product_prod_f_nat: $tType ).

thf(ty_n_t__List__Olist_It__Term__Oterm_Itf__f_Mtf__v_J_J,type,
    list_term_f_v: $tType ).

thf(ty_n_t__List__Olist_It__Term__Oterm_Itf__f_Mtf__a_J_J,type,
    list_term_f_a: $tType ).

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

thf(ty_n_t__FSet__Ofset_It__Term__Oterm_Itf__f_Mtf__v_J_J,type,
    fset_term_f_v: $tType ).

thf(ty_n_t__FSet__Ofset_It__Term__Oterm_Itf__f_Mtf__a_J_J,type,
    fset_term_f_a: $tType ).

thf(ty_n_t__Set__Oset_It__Term__Oterm_Itf__f_Mtf__v_J_J,type,
    set_term_f_v: $tType ).

thf(ty_n_t__Set__Oset_It__Term__Oterm_Itf__f_Mtf__a_J_J,type,
    set_term_f_a: $tType ).

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

thf(ty_n_t__Tree____Automata__Ota_Itf__v_Mtf__f_J,type,
    tree_ta_v_f: $tType ).

thf(ty_n_t__Tree____Automata__Ota_Itf__a_Mtf__f_J,type,
    tree_ta_a_f: $tType ).

thf(ty_n_t__Bot____Terms__Obot____term_Itf__f_J,type,
    bot_bot_term_f: $tType ).

thf(ty_n_t__Term__Oterm_Itf__f_Mtf__v_J,type,
    term_f_v: $tType ).

thf(ty_n_t__Term__Oterm_Itf__f_Mtf__a_J,type,
    term_f_a: $tType ).

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

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

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

thf(ty_n_t__FSet__Ofset_Itf__v_J,type,
    fset_v: $tType ).

thf(ty_n_t__FSet__Ofset_Itf__a_J,type,
    fset_a: $tType ).

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

thf(ty_n_tf__v,type,
    v: $tType ).

thf(ty_n_tf__f,type,
    f: $tType ).

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

% Explicit typings (366)
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__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    basic_2716954541200321574term_f: produc250624644672406269term_f > list_P1780566427699531651term_f > list_l9126709995455877001term_f ).

thf(sy_c_Basic__Utils_Oadd__elem__list__lists_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J,type,
    basic_9152604556761636560rm_f_v: produc8027700073345061287rm_f_v > list_P4093298276913796397rm_f_v > list_l4945967923587861555rm_f_v ).

thf(sy_c_Basic__Utils_Oadd__elem__list__lists_001t__Term__Oterm_Itf__f_Mtf__v_J,type,
    basic_6102686972066033914rm_f_v: term_f_v > list_term_f_v > list_list_term_f_v ).

thf(sy_c_Bot__Terms_Obless__eq_001tf__f,type,
    bot_bless_eq_f: set_Pr6841673932529763677term_f ).

thf(sy_c_Bot__Terms_Oterm__to__bot__term_001tf__f_001tf__a,type,
    bot_te7903553565439629403rm_f_a: term_f_a > bot_bot_term_f ).

thf(sy_c_Bot__Terms_Oterm__to__bot__term_001tf__f_001tf__v,type,
    bot_te7903553565439629424rm_f_v: term_f_v > bot_bot_term_f ).

thf(sy_c_FSet_Ofmember_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    fmembe1418995490368054919term_f: bot_bot_term_f > fset_Bot_bot_term_f > $o ).

thf(sy_c_FSet_Ofmember_001t__Nat__Onat,type,
    fmember_nat: nat > fset_nat > $o ).

thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    fmembe6950792022577883716term_f: produc250624644672406269term_f > fset_P2882900458636368611term_f > $o ).

thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    fmembe1449479052724974408at_nat: product_prod_nat_nat > fset_P5670320511379867111at_nat > $o ).

thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__a_J_Mt__Term__Oterm_Itf__f_Mtf__a_J_J,type,
    fmembe3725132237878994628rm_f_a: produc1754468282888495997rm_f_a > fset_P1463011978278004579rm_f_a > $o ).

thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J,type,
    fmembe774991991480784110rm_f_v: produc8027700073345061287rm_f_v > fset_P8018961893305114765rm_f_v > $o ).

thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J,type,
    fmembe6795308708001954811_f_nat: product_prod_f_nat > fset_P6228066233360383026_f_nat > $o ).

thf(sy_c_FSet_Ofmember_001t__Term__Oterm_Itf__f_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    fmembe7587999265257414299term_f: term_f7969170442531588692term_f > fset_t2965824081267281850term_f > $o ).

thf(sy_c_FSet_Ofmember_001t__Term__Oterm_Itf__f_Mtf__a_J,type,
    fmember_term_f_a: term_f_a > fset_term_f_a > $o ).

thf(sy_c_FSet_Ofmember_001t__Term__Oterm_Itf__f_Mtf__v_J,type,
    fmember_term_f_v: term_f_v > fset_term_f_v > $o ).

thf(sy_c_FSet_Ofmember_001tf__a,type,
    fmember_a: a > fset_a > $o ).

thf(sy_c_FSet_Ofmember_001tf__v,type,
    fmember_v: v > fset_v > $o ).

thf(sy_c_FSet_Ofset_Ofset_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    fset_Bot_bot_term_f2: fset_Bot_bot_term_f > set_Bot_bot_term_f ).

thf(sy_c_FSet_Ofset_Ofset_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    fset_P8925168969356593900term_f: fset_P2882900458636368611term_f > set_Pr6841673932529763677term_f ).

thf(sy_c_FSet_Ofset_Ofset_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J,type,
    fset_P4617584883882644886rm_f_v: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v ).

thf(sy_c_FSet_Ofset_Ofset_001t__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J,type,
    fset_P3576968334923099475_f_nat: fset_P6228066233360383026_f_nat > set_Pr5245412377734449720_f_nat ).

thf(sy_c_FSet_Ofset_Ofset_001t__Term__Oterm_Itf__f_Mtf__a_J,type,
    fset_term_f_a2: fset_term_f_a > set_term_f_a ).

thf(sy_c_FSet_Ofset_Ofset_001t__Term__Oterm_Itf__f_Mtf__v_J,type,
    fset_term_f_v2: fset_term_f_v > set_term_f_v ).

thf(sy_c_FSet_Ofset__of__list_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    fset_o7715858803885244232term_f: list_Bot_bot_term_f > fset_Bot_bot_term_f ).

thf(sy_c_FSet_Ofset__of__list_001t__Nat__Onat,type,
    fset_of_list_nat: list_nat > fset_nat ).

thf(sy_c_FSet_Ofset__of__list_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    fset_o6125718882019406531term_f: list_P1780566427699531651term_f > fset_P2882900458636368611term_f ).

thf(sy_c_FSet_Ofset__of__list_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J,type,
    fset_o4970844032613833069rm_f_v: list_P4093298276913796397rm_f_v > fset_P8018961893305114765rm_f_v ).

thf(sy_c_FSet_Ofset__of__list_001t__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J,type,
    fset_o8009517685352940092_f_nat: list_P3903862279629787026_f_nat > fset_P6228066233360383026_f_nat ).

thf(sy_c_FSet_Ofset__of__list_001t__Term__Oterm_Itf__f_Mtf__a_J,type,
    fset_o6574051009341269640rm_f_a: list_term_f_a > fset_term_f_a ).

thf(sy_c_FSet_Ofset__of__list_001t__Term__Oterm_Itf__f_Mtf__v_J,type,
    fset_o6574051099709074461rm_f_v: list_term_f_v > fset_term_f_v ).

thf(sy_c_FSet__Utils_OfId__on_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    fSet_f5231733434221412358term_f: fset_Bot_bot_term_f > fset_P2882900458636368611term_f ).

thf(sy_c_FSet__Utils_OfId__on_001t__Nat__Onat,type,
    fSet_fId_on_nat: fset_nat > fset_P5670320511379867111at_nat ).

thf(sy_c_FSet__Utils_OfId__on_001t__Term__Oterm_Itf__f_Mtf__a_J,type,
    fSet_fId_on_term_f_a: fset_term_f_a > fset_P1463011978278004579rm_f_a ).

thf(sy_c_FSet__Utils_OfId__on_001t__Term__Oterm_Itf__f_Mtf__v_J,type,
    fSet_fId_on_term_f_v: fset_term_f_v > fset_P8018961893305114765rm_f_v ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: nat > nat > nat ).

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__Bot____Terms__Obot____term_Itf__f_J,type,
    if_Bot_bot_term_f: $o > bot_bot_term_f > bot_bot_term_f > bot_bot_term_f ).

thf(sy_c_If_001t__FSet__Ofset_It__Bot____Terms__Obot____term_Itf__f_J_J,type,
    if_fse3524882727628502648term_f: $o > fset_Bot_bot_term_f > fset_Bot_bot_term_f > fset_Bot_bot_term_f ).

thf(sy_c_If_001t__FSet__Ofset_It__Term__Oterm_Itf__f_Mtf__a_J_J,type,
    if_fset_term_f_a: $o > fset_term_f_a > fset_term_f_a > fset_term_f_a ).

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

thf(sy_c_If_001t__Term__Oterm_Itf__f_Mtf__a_J,type,
    if_term_f_a: $o > term_f_a > term_f_a > term_f_a ).

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__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    append2162745965627483416term_f: list_P1780566427699531651term_f > list_P1780566427699531651term_f > list_P1780566427699531651term_f ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J,type,
    append8822927501793976258rm_f_v: list_P4093298276913796397rm_f_v > list_P4093298276913796397rm_f_v > list_P4093298276913796397rm_f_v ).

thf(sy_c_List_Oappend_001t__Term__Oterm_Itf__f_Mtf__v_J,type,
    append_term_f_v: list_term_f_v > list_term_f_v > list_term_f_v ).

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

thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    distin6927496147406123510term_f: list_P1780566427699531651term_f > $o ).

thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J,type,
    distin6546576869490926752rm_f_v: list_P4093298276913796397rm_f_v > $o ).

thf(sy_c_List_Odistinct_001t__Term__Oterm_Itf__f_Mtf__v_J,type,
    distinct_term_f_v: list_term_f_v > $o ).

thf(sy_c_List_Olenlex_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    lenlex569208748157492291term_f: set_Pr6841673932529763677term_f > set_Pr488239067965912317term_f ).

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

thf(sy_c_List_Olenlex_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    lenlex9017302659811057544term_f: set_Pr4896805198020829255term_f > set_Pr6162250758287926727term_f ).

thf(sy_c_List_Olenlex_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J,type,
    lenlex5726706343053400626rm_f_v: set_Pr7897041799657265351rm_f_v > set_Pr3656563536019104583rm_f_v ).

thf(sy_c_List_Olenlex_001t__Term__Oterm_Itf__f_Mtf__v_J,type,
    lenlex_term_f_v: set_Pr8827868859434726151rm_f_v > set_Pr1526384100762628903rm_f_v ).

thf(sy_c_List_Olex_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    lex_Bot_bot_term_f: set_Pr6841673932529763677term_f > set_Pr488239067965912317term_f ).

thf(sy_c_List_Olex_001t__Nat__Onat,type,
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thf(sy_c_List_Olex_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
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thf(sy_c_Missing__List_Oadjust__idx,type,
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thf(sy_c_Missing__List_Oadjust__idx__rev,type,
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    tree_ta_der_a_f2: tree_ta_a_f > term_f_a > fset_term_f_a ).

thf(sy_c_Tree__Automata_Ota__der_H_001tf__v_001tf__f,type,
    tree_ta_der_v_f2: tree_ta_v_f > term_f_v > fset_term_f_v ).

thf(sy_c_Tree__Automata_Ota__lang_001t__Bot____Terms__Obot____term_Itf__f_J_001tf__f_001tf__v,type,
    tree_t1514651371760659444_f_f_v: fset_Bot_bot_term_f > tree_t2044664379607036063rm_f_f > set_term_f_v ).

thf(sy_c_Tree__Automata_Ota__lang_001t__Term__Oterm_Itf__f_Mtf__a_J_001tf__f_001tf__v,type,
    tree_t6230644817623509812_a_f_v: fset_term_f_a > tree_ta_term_f_a_f > set_term_f_v ).

thf(sy_c_Tree__Automata_Ota__lang_001tf__v_001tf__f_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    tree_t3074183349113346900term_f: fset_v > tree_ta_v_f > set_te5056978688659945524term_f ).

thf(sy_c_Tree__Automata_Ota__lang_001tf__v_001tf__f_001tf__v,type,
    tree_ta_lang_v_f_v: fset_v > tree_ta_v_f > set_term_f_v ).

thf(sy_c_Utils_Otrancl__list_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    trancl6444742785402773766term_f: set_Pr6841673932529763677term_f > set_Pr488239067965912317term_f ).

thf(sy_c_Utils_Otrancl__list_001t__Nat__Onat,type,
    trancl_list_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_Utils_Otrancl__list_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    trancl1147503139693895301term_f: set_Pr4896805198020829255term_f > set_Pr6162250758287926727term_f ).

thf(sy_c_Utils_Otrancl__list_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J,type,
    trancl7964251992849616175rm_f_v: set_Pr7897041799657265351rm_f_v > set_Pr3656563536019104583rm_f_v ).

thf(sy_c_Utils_Otrancl__list_001t__Term__Oterm_Itf__f_Mtf__v_J,type,
    trancl_list_term_f_v: set_Pr8827868859434726151rm_f_v > set_Pr1526384100762628903rm_f_v ).

thf(sy_c_Utils_Otrancl__listp_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    trancl2235149414931352451term_f: ( produc250624644672406269term_f > produc250624644672406269term_f > $o ) > list_P1780566427699531651term_f > list_P1780566427699531651term_f > $o ).

thf(sy_c_Utils_Otrancl__listp_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J,type,
    trancl8258233303137563181rm_f_v: ( produc8027700073345061287rm_f_v > produc8027700073345061287rm_f_v > $o ) > list_P4093298276913796397rm_f_v > list_P4093298276913796397rm_f_v > $o ).

thf(sy_c_Utils_Otrancl__listp_001t__Term__Oterm_Itf__f_Mtf__v_J,type,
    trancl7233808882382938973rm_f_v: ( term_f_v > term_f_v > $o ) > list_term_f_v > list_term_f_v > $o ).

thf(sy_c_Utils_Ovarposs__list_001tf__f_001tf__v,type,
    varposs_list_f_v: term_f_v > list_list_nat ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
    accp_P6476584661967079469rm_f_v: ( produc6479857609258389878rm_f_v > produc6479857609258389878rm_f_v > $o ) > produc6479857609258389878rm_f_v > $o ).

thf(sy_c_member_001t__Bot____Terms__Obot____term_Itf__f_J,type,
    member2723211850833494437term_f: bot_bot_term_f > set_Bot_bot_term_f > $o ).

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

thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    member6232163599979755692term_f: list_P1780566427699531651term_f > set_li2099592054954659939term_f > $o ).

thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
    member2864856320113173078rm_f_v: list_P4093298276913796397rm_f_v > set_li1204841918454117645rm_f_v > $o ).

thf(sy_c_member_001t__List__Olist_It__Term__Oterm_Itf__f_Mtf__v_J_J,type,
    member_list_term_f_v: list_term_f_v > set_list_term_f_v > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    member4687689947140789414term_f: produc250624644672406269term_f > set_Pr6841673932529763677term_f > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member6987746275253522745st_nat: produc4155638208356863248st_nat > set_Pr5771542735269860976st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Bot____Terms__Obot____term_Itf__f_J_J_Mt__List__Olist_It__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    member878789856639434822term_f: produc7885403226499153821term_f > set_Pr488239067965912317term_f > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J_J,type,
    member2822224418761970704term_f: produc6721322714573097447term_f > set_Pr6162250758287926727term_f > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J,type,
    member3469830491719462074rm_f_v: produc1780116556837289617rm_f_v > set_Pr2493473585119558001rm_f_v > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J_Mt__List__Olist_It__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
    member8709465672751168444rm_f_v: produc8443351192948919077rm_f_v > set_Pr192651762572823643rm_f_v > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J_J,type,
    member7203770481494966502term_f: produc5514056546612794045term_f > set_Pr5407177960468871069term_f > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J,type,
    member4621649979522838928rm_f_v: produc785048979587161959rm_f_v > set_Pr3656563536019104583rm_f_v > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_Mt__List__Olist_It__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
    member2857318640343734118rm_f_v: produc5612110391004957391rm_f_v > set_Pr3027792620959023877rm_f_v > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Term__Oterm_Itf__f_Mtf__v_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J_J,type,
    member6476877439930291548term_f: produc6210762960128042181term_f > set_Pr5427172866150716923term_f > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Term__Oterm_Itf__f_Mtf__v_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J,type,
    member253218050144787974rm_f_v: produc3008009800806011247rm_f_v > set_Pr3935753902997436837rm_f_v > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Term__Oterm_Itf__f_Mtf__v_J_J_Mt__List__Olist_It__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
    member2355812552444855152rm_f_v: produc7225343704568979143rm_f_v > set_Pr1526384100762628903rm_f_v > $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__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    member3795392313164512464at_nat: produc492020038885415591at_nat > set_Pr7047708467701517959at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_Mt__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    member4806591834699997840term_f: produc6853483567480260455term_f > set_Pr4896805198020829255term_f > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
    member6153102960928152634rm_f_v: produc474063087423118609rm_f_v > set_Pr4527924768817157361rm_f_v > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J,type,
    member9108189709895347494rm_f_v: produc8919046586786123151rm_f_v > set_Pr8897580718372814021rm_f_v > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
    member7300054593328134928rm_f_v: produc3406425829894461415rm_f_v > set_Pr7897041799657265351rm_f_v > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J,type,
    member6884080165642858192rm_f_v: produc9116948288474477369rm_f_v > set_Pr4552519640582707567rm_f_v > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__a_J_Mt__Term__Oterm_Itf__f_Mtf__a_J_J,type,
    member7266790107935152422rm_f_a: produc1754468282888495997rm_f_a > set_Pr2271918944407615965rm_f_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__f_J_Mt__Bot____Terms__Obot____term_Itf__f_J_J_J,type,
    member1278244878481305830term_f: produc1089101755372081487term_f > set_Pr6532846849355489413term_f > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
    member6048657017474400912rm_f_v: produc8281525140306020089rm_f_v > set_Pr4810970677413006639rm_f_v > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J,type,
    member4316649861536941904rm_f_v: produc8027700073345061287rm_f_v > set_Pr8827868859434726151rm_f_v > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J,type,
    member2678038196313615385_f_nat: product_prod_f_nat > set_Pr5245412377734449720_f_nat > $o ).

thf(sy_c_member_001t__Term__Oterm_Itf__f_Mt__Bot____Terms__Obot____term_Itf__f_J_J,type,
    member1719022467595452157term_f: term_f7969170442531588692term_f > set_te5056978688659945524term_f > $o ).

thf(sy_c_member_001t__Term__Oterm_Itf__f_Mtf__a_J,type,
    member_term_f_a: term_f_a > set_term_f_a > $o ).

thf(sy_c_member_001t__Term__Oterm_Itf__f_Mtf__v_J,type,
    member_term_f_v: term_f_v > set_term_f_v > $o ).

thf(sy_v_R,type,
    r: fset_term_f_a ).

thf(sy_v__092_060F_062,type,
    f2: fset_P6228066233360383026_f_nat ).

thf(sy_v_f____,type,
    f3: f ).

thf(sy_v_i,type,
    i: nat ).

thf(sy_v_l,type,
    l: term_f_a ).

thf(sy_v_s,type,
    s: term_f_v ).

thf(sy_v_t,type,
    t: term_f_v ).

thf(sy_v_ts____,type,
    ts: list_term_f_v ).

% Relevant facts (1233)
thf(fact_0_nth__mem,axiom,
    ! [N: nat,Xs: list_P4093298276913796397rm_f_v] :
      ( ( ord_less_nat @ N @ ( size_s3818039357710166425rm_f_v @ Xs ) )
     => ( member4316649861536941904rm_f_v @ ( nth_Pr8768905099348956590rm_f_v @ Xs @ N ) @ ( set_Pr817814403484925884rm_f_v @ Xs ) ) ) ).

% nth_mem
thf(fact_1_nth__mem,axiom,
    ! [N: nat,Xs: list_P1780566427699531651term_f] :
      ( ( ord_less_nat @ N @ ( size_s9037951226665808879term_f @ Xs ) )
     => ( member4687689947140789414term_f @ ( nth_Pr7080171896067517188term_f @ Xs @ N ) @ ( set_Pr1368550810757897490term_f @ Xs ) ) ) ).

% nth_mem
thf(fact_2_nth__mem,axiom,
    ! [N: nat,Xs: list_term_f_v] :
      ( ( ord_less_nat @ N @ ( size_s7148729505879680775rm_f_v @ Xs ) )
     => ( member_term_f_v @ ( nth_term_f_v @ Xs @ N ) @ ( set_term_f_v2 @ Xs ) ) ) ).

% nth_mem
thf(fact_3_Fun_Oprems_I3_J,axiom,
    ! [L: term_f_a,S: term_f_v] :
      ( ( fmember_term_f_a @ L @ r )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( fun_f_v @ f3 @ ts ) @ S ) @ subter5033437232688913882eq_f_v )
       => ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_te7903553565439629403rm_f_a @ L ) @ ( bot_te7903553565439629424rm_f_v @ S ) ) @ bot_bless_eq_f ) ) ) ).

% Fun.prems(3)
thf(fact_4_NF,axiom,
    ! [L: term_f_a,S: term_f_v] :
      ( ( fmember_term_f_a @ L @ r )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ t @ S ) @ subter5033437232688913882eq_f_v )
       => ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_te7903553565439629403rm_f_a @ L ) @ ( bot_te7903553565439629424rm_f_v @ S ) ) @ bot_bless_eq_f ) ) ) ).

% NF
thf(fact_5_lift__total_Otrans,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,Z: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ bot_bless_eq_f )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y @ Z ) @ bot_bless_eq_f )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Z ) @ bot_bless_eq_f ) ) ) ).

% lift_total.trans
thf(fact_6_lift__total_Oanti__sym,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ bot_bless_eq_f )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y @ X ) @ bot_bless_eq_f )
       => ( X = Y ) ) ) ).

% lift_total.anti_sym
thf(fact_7_nth__equalityI,axiom,
    ! [Xs: list_P1780566427699531651term_f,Ys: list_P1780566427699531651term_f] :
      ( ( ( size_s9037951226665808879term_f @ Xs )
        = ( size_s9037951226665808879term_f @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s9037951226665808879term_f @ Xs ) )
           => ( ( nth_Pr7080171896067517188term_f @ Xs @ I )
              = ( nth_Pr7080171896067517188term_f @ Ys @ I ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_8_nth__equalityI,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,Ys: list_P4093298276913796397rm_f_v] :
      ( ( ( size_s3818039357710166425rm_f_v @ Xs )
        = ( size_s3818039357710166425rm_f_v @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3818039357710166425rm_f_v @ Xs ) )
           => ( ( nth_Pr8768905099348956590rm_f_v @ Xs @ I )
              = ( nth_Pr8768905099348956590rm_f_v @ Ys @ I ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_9_nth__equalityI,axiom,
    ! [Xs: list_term_f_v,Ys: list_term_f_v] :
      ( ( ( size_s7148729505879680775rm_f_v @ Xs )
        = ( size_s7148729505879680775rm_f_v @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s7148729505879680775rm_f_v @ Xs ) )
           => ( ( nth_term_f_v @ Xs @ I )
              = ( nth_term_f_v @ Ys @ I ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_10_nth__equalityE,axiom,
    ! [Xs: list_P1780566427699531651term_f,Ys: list_P1780566427699531651term_f] :
      ( ( Xs = Ys )
     => ~ ( ( ( size_s9037951226665808879term_f @ Xs )
            = ( size_s9037951226665808879term_f @ Ys ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s9037951226665808879term_f @ Xs ) )
               => ( ( nth_Pr7080171896067517188term_f @ Xs @ I2 )
                  = ( nth_Pr7080171896067517188term_f @ Ys @ I2 ) ) ) ) ) ).

% nth_equalityE
thf(fact_11_nth__equalityE,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,Ys: list_P4093298276913796397rm_f_v] :
      ( ( Xs = Ys )
     => ~ ( ( ( size_s3818039357710166425rm_f_v @ Xs )
            = ( size_s3818039357710166425rm_f_v @ Ys ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
               => ( ( nth_Pr8768905099348956590rm_f_v @ Xs @ I2 )
                  = ( nth_Pr8768905099348956590rm_f_v @ Ys @ I2 ) ) ) ) ) ).

% nth_equalityE
thf(fact_12_nth__equalityE,axiom,
    ! [Xs: list_term_f_v,Ys: list_term_f_v] :
      ( ( Xs = Ys )
     => ~ ( ( ( size_s7148729505879680775rm_f_v @ Xs )
            = ( size_s7148729505879680775rm_f_v @ Ys ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
               => ( ( nth_term_f_v @ Xs @ I2 )
                  = ( nth_term_f_v @ Ys @ I2 ) ) ) ) ) ).

% nth_equalityE
thf(fact_13_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > produc250624644672406269term_f > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X2: produc250624644672406269term_f] : ( P @ I3 @ X2 ) ) )
      = ( ? [Xs2: list_P1780566427699531651term_f] :
            ( ( ( size_s9037951226665808879term_f @ Xs2 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_Pr7080171896067517188term_f @ Xs2 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_14_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > produc8027700073345061287rm_f_v > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X2: produc8027700073345061287rm_f_v] : ( P @ I3 @ X2 ) ) )
      = ( ? [Xs2: list_P4093298276913796397rm_f_v] :
            ( ( ( size_s3818039357710166425rm_f_v @ Xs2 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_Pr8768905099348956590rm_f_v @ Xs2 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_15_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > term_f_v > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X2: term_f_v] : ( P @ I3 @ X2 ) ) )
      = ( ? [Xs2: list_term_f_v] :
            ( ( ( size_s7148729505879680775rm_f_v @ Xs2 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_term_f_v @ Xs2 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_16_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y2: list_P1780566427699531651term_f,Z2: list_P1780566427699531651term_f] : ( Y2 = Z2 ) )
    = ( ^ [Xs2: list_P1780566427699531651term_f,Ys2: list_P1780566427699531651term_f] :
          ( ( ( size_s9037951226665808879term_f @ Xs2 )
            = ( size_s9037951226665808879term_f @ Ys2 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s9037951226665808879term_f @ Xs2 ) )
             => ( ( nth_Pr7080171896067517188term_f @ Xs2 @ I3 )
                = ( nth_Pr7080171896067517188term_f @ Ys2 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_17_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y2: list_P4093298276913796397rm_f_v,Z2: list_P4093298276913796397rm_f_v] : ( Y2 = Z2 ) )
    = ( ^ [Xs2: list_P4093298276913796397rm_f_v,Ys2: list_P4093298276913796397rm_f_v] :
          ( ( ( size_s3818039357710166425rm_f_v @ Xs2 )
            = ( size_s3818039357710166425rm_f_v @ Ys2 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s3818039357710166425rm_f_v @ Xs2 ) )
             => ( ( nth_Pr8768905099348956590rm_f_v @ Xs2 @ I3 )
                = ( nth_Pr8768905099348956590rm_f_v @ Ys2 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_18_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y2: list_term_f_v,Z2: list_term_f_v] : ( Y2 = Z2 ) )
    = ( ^ [Xs2: list_term_f_v,Ys2: list_term_f_v] :
          ( ( ( size_s7148729505879680775rm_f_v @ Xs2 )
            = ( size_s7148729505879680775rm_f_v @ Ys2 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s7148729505879680775rm_f_v @ Xs2 ) )
             => ( ( nth_term_f_v @ Xs2 @ I3 )
                = ( nth_term_f_v @ Ys2 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_19_Ex__list__of__length__P,axiom,
    ! [N: nat,P: produc250624644672406269term_f > nat > $o] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ N )
         => ? [X3: produc250624644672406269term_f] : ( P @ X3 @ I ) )
     => ? [Xs3: list_P1780566427699531651term_f] :
          ( ( ( size_s9037951226665808879term_f @ Xs3 )
            = N )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ N )
             => ( P @ ( nth_Pr7080171896067517188term_f @ Xs3 @ I2 ) @ I2 ) ) ) ) ).

% Ex_list_of_length_P
thf(fact_20_Ex__list__of__length__P,axiom,
    ! [N: nat,P: produc8027700073345061287rm_f_v > nat > $o] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ N )
         => ? [X3: produc8027700073345061287rm_f_v] : ( P @ X3 @ I ) )
     => ? [Xs3: list_P4093298276913796397rm_f_v] :
          ( ( ( size_s3818039357710166425rm_f_v @ Xs3 )
            = N )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ N )
             => ( P @ ( nth_Pr8768905099348956590rm_f_v @ Xs3 @ I2 ) @ I2 ) ) ) ) ).

% Ex_list_of_length_P
thf(fact_21_Ex__list__of__length__P,axiom,
    ! [N: nat,P: term_f_v > nat > $o] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ N )
         => ? [X3: term_f_v] : ( P @ X3 @ I ) )
     => ? [Xs3: list_term_f_v] :
          ( ( ( size_s7148729505879680775rm_f_v @ Xs3 )
            = N )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ N )
             => ( P @ ( nth_term_f_v @ Xs3 @ I2 ) @ I2 ) ) ) ) ).

% Ex_list_of_length_P
thf(fact_22_Fun_Oprems_I1_J,axiom,
    term_ground_f_v @ ( fun_f_v @ f3 @ ts ) ).

% Fun.prems(1)
thf(fact_23_prod_Oinject,axiom,
    ! [X1: term_f_v,X22: term_f_v,Y1: term_f_v,Y22: term_f_v] :
      ( ( ( produc2510790566102465375rm_f_v @ X1 @ X22 )
        = ( produc2510790566102465375rm_f_v @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_24_prod_Oinject,axiom,
    ! [X1: bot_bot_term_f,X22: bot_bot_term_f,Y1: bot_bot_term_f,Y22: bot_bot_term_f] :
      ( ( ( produc2865564700782054965term_f @ X1 @ X22 )
        = ( produc2865564700782054965term_f @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_25_prod_Oinject,axiom,
    ! [X1: nat > nat,X22: nat,Y1: nat > nat,Y22: nat] :
      ( ( ( produc72220940542539688at_nat @ X1 @ X22 )
        = ( produc72220940542539688at_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_26_prod_Oinject,axiom,
    ! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X22 )
        = ( product_Pair_nat_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_27_old_Oprod_Oinject,axiom,
    ! [A: term_f_v,B: term_f_v,A2: term_f_v,B2: term_f_v] :
      ( ( ( produc2510790566102465375rm_f_v @ A @ B )
        = ( produc2510790566102465375rm_f_v @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_28_old_Oprod_Oinject,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,A2: bot_bot_term_f,B2: bot_bot_term_f] :
      ( ( ( produc2865564700782054965term_f @ A @ B )
        = ( produc2865564700782054965term_f @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_29_old_Oprod_Oinject,axiom,
    ! [A: nat > nat,B: nat,A2: nat > nat,B2: nat] :
      ( ( ( produc72220940542539688at_nat @ A @ B )
        = ( produc72220940542539688at_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_30_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_31_bless__eq__refl,axiom,
    ! [S: bot_bot_term_f] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S @ S ) @ bot_bless_eq_f ) ).

% bless_eq_refl
thf(fact_32_bless__eq__trans,axiom,
    ! [S: bot_bot_term_f,T: bot_bot_term_f,U: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S @ T ) @ bot_bless_eq_f )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ T @ U ) @ bot_bless_eq_f )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S @ U ) @ bot_bless_eq_f ) ) ) ).

% bless_eq_trans
thf(fact_33_bless__eq__anti__sym,axiom,
    ! [S: bot_bot_term_f,T: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ S @ T ) @ bot_bless_eq_f )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ T @ S ) @ bot_bless_eq_f )
       => ( S = T ) ) ) ).

% bless_eq_anti_sym
thf(fact_34_supteq_Orefl,axiom,
    ! [T: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ T ) @ subter5033437232688913882eq_f_v ) ).

% supteq.refl
thf(fact_35_assms_I2_J,axiom,
    term_ground_f_v @ t ).

% assms(2)
thf(fact_36_term__not__arg,axiom,
    ! [F: f,Ss: list_term_f_v] :
      ~ ( member_term_f_v @ ( fun_f_v @ F @ Ss ) @ ( set_term_f_v2 @ Ss ) ) ).

% term_not_arg
thf(fact_37_arg__subteq,axiom,
    ! [T: term_f_v,Ts: list_term_f_v,F: f] :
      ( ( member_term_f_v @ T @ ( set_term_f_v2 @ Ts ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( fun_f_v @ F @ Ts ) @ T ) @ subter5033437232688913882eq_f_v ) ) ).

% arg_subteq
thf(fact_38_supteq_Osimps,axiom,
    ! [A1: term_f_v,A22: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ subter5033437232688913882eq_f_v )
      = ( ? [T2: term_f_v] :
            ( ( A1 = T2 )
            & ( A22 = T2 ) )
        | ? [U2: term_f_v,Ss2: list_term_f_v,T2: term_f_v,F2: f] :
            ( ( A1
              = ( fun_f_v @ F2 @ Ss2 ) )
            & ( A22 = T2 )
            & ( member_term_f_v @ U2 @ ( set_term_f_v2 @ Ss2 ) )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ U2 @ T2 ) @ subter5033437232688913882eq_f_v ) ) ) ) ).

% supteq.simps
thf(fact_39_supteq_Ocases,axiom,
    ! [A1: term_f_v,A22: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ subter5033437232688913882eq_f_v )
     => ( ( A22 != A1 )
       => ~ ! [U3: term_f_v,Ss3: list_term_f_v] :
              ( ? [F3: f] :
                  ( A1
                  = ( fun_f_v @ F3 @ Ss3 ) )
             => ( ( member_term_f_v @ U3 @ ( set_term_f_v2 @ Ss3 ) )
               => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ U3 @ A22 ) @ subter5033437232688913882eq_f_v ) ) ) ) ) ).

% supteq.cases
thf(fact_40_supteq_Osubt,axiom,
    ! [U: term_f_v,Ss: list_term_f_v,T: term_f_v,F: f] :
      ( ( member_term_f_v @ U @ ( set_term_f_v2 @ Ss ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ U @ T ) @ subter5033437232688913882eq_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( fun_f_v @ F @ Ss ) @ T ) @ subter5033437232688913882eq_f_v ) ) ) ).

% supteq.subt
thf(fact_41_Pair__inject,axiom,
    ! [A: term_f_v,B: term_f_v,A2: term_f_v,B2: term_f_v] :
      ( ( ( produc2510790566102465375rm_f_v @ A @ B )
        = ( produc2510790566102465375rm_f_v @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_42_Pair__inject,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,A2: bot_bot_term_f,B2: bot_bot_term_f] :
      ( ( ( produc2865564700782054965term_f @ A @ B )
        = ( produc2865564700782054965term_f @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_43_Pair__inject,axiom,
    ! [A: nat > nat,B: nat,A2: nat > nat,B2: nat] :
      ( ( ( produc72220940542539688at_nat @ A @ B )
        = ( produc72220940542539688at_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_44_Pair__inject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_45_prod__cases,axiom,
    ! [P: produc8027700073345061287rm_f_v > $o,P2: produc8027700073345061287rm_f_v] :
      ( ! [A3: term_f_v,B3: term_f_v] : ( P @ ( produc2510790566102465375rm_f_v @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_46_prod__cases,axiom,
    ! [P: produc250624644672406269term_f > $o,P2: produc250624644672406269term_f] :
      ( ! [A3: bot_bot_term_f,B3: bot_bot_term_f] : ( P @ ( produc2865564700782054965term_f @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_47_prod__cases,axiom,
    ! [P: produc8199716216217303280at_nat > $o,P2: produc8199716216217303280at_nat] :
      ( ! [A3: nat > nat,B3: nat] : ( P @ ( produc72220940542539688at_nat @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_48_prod__cases,axiom,
    ! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_49_surj__pair,axiom,
    ! [P2: produc8027700073345061287rm_f_v] :
    ? [X4: term_f_v,Y3: term_f_v] :
      ( P2
      = ( produc2510790566102465375rm_f_v @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_50_surj__pair,axiom,
    ! [P2: produc250624644672406269term_f] :
    ? [X4: bot_bot_term_f,Y3: bot_bot_term_f] :
      ( P2
      = ( produc2865564700782054965term_f @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_51_surj__pair,axiom,
    ! [P2: produc8199716216217303280at_nat] :
    ? [X4: nat > nat,Y3: nat] :
      ( P2
      = ( produc72220940542539688at_nat @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_52_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X4: nat,Y3: nat] :
      ( P2
      = ( product_Pair_nat_nat @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_53_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8027700073345061287rm_f_v] :
      ~ ! [A3: term_f_v,B3: term_f_v] :
          ( Y
         != ( produc2510790566102465375rm_f_v @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_54_old_Oprod_Oexhaust,axiom,
    ! [Y: produc250624644672406269term_f] :
      ~ ! [A3: bot_bot_term_f,B3: bot_bot_term_f] :
          ( Y
         != ( produc2865564700782054965term_f @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_55_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8199716216217303280at_nat] :
      ~ ! [A3: nat > nat,B3: nat] :
          ( Y
         != ( produc72220940542539688at_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_56_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A3: nat,B3: nat] :
          ( Y
         != ( product_Pair_nat_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_57_neq__if__length__neq,axiom,
    ! [Xs: list_term_f_v,Ys: list_term_f_v] :
      ( ( ( size_s7148729505879680775rm_f_v @ Xs )
       != ( size_s7148729505879680775rm_f_v @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_58_neq__if__length__neq,axiom,
    ! [Xs: list_P1780566427699531651term_f,Ys: list_P1780566427699531651term_f] :
      ( ( ( size_s9037951226665808879term_f @ Xs )
       != ( size_s9037951226665808879term_f @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_59_neq__if__length__neq,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,Ys: list_P4093298276913796397rm_f_v] :
      ( ( ( size_s3818039357710166425rm_f_v @ Xs )
       != ( size_s3818039357710166425rm_f_v @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_60_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_term_f_v] :
      ( ( size_s7148729505879680775rm_f_v @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_61_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_P1780566427699531651term_f] :
      ( ( size_s9037951226665808879term_f @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_62_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_P4093298276913796397rm_f_v] :
      ( ( size_s3818039357710166425rm_f_v @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_63_ex__set__conv__ex__nth,axiom,
    ! [Xs: list_term_f_v,P: term_f_v > $o] :
      ( ( ? [X5: term_f_v] :
            ( ( member_term_f_v @ X5 @ ( set_term_f_v2 @ Xs ) )
            & ( P @ X5 ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
            & ( P @ ( nth_term_f_v @ Xs @ I3 ) ) ) ) ) ).

% ex_set_conv_ex_nth
thf(fact_64_ex__set__conv__ex__nth,axiom,
    ! [Xs: list_P1780566427699531651term_f,P: produc250624644672406269term_f > $o] :
      ( ( ? [X5: produc250624644672406269term_f] :
            ( ( member4687689947140789414term_f @ X5 @ ( set_Pr1368550810757897490term_f @ Xs ) )
            & ( P @ X5 ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s9037951226665808879term_f @ Xs ) )
            & ( P @ ( nth_Pr7080171896067517188term_f @ Xs @ I3 ) ) ) ) ) ).

% ex_set_conv_ex_nth
thf(fact_65_ex__set__conv__ex__nth,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,P: produc8027700073345061287rm_f_v > $o] :
      ( ( ? [X5: produc8027700073345061287rm_f_v] :
            ( ( member4316649861536941904rm_f_v @ X5 @ ( set_Pr817814403484925884rm_f_v @ Xs ) )
            & ( P @ X5 ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
            & ( P @ ( nth_Pr8768905099348956590rm_f_v @ Xs @ I3 ) ) ) ) ) ).

% ex_set_conv_ex_nth
thf(fact_66_all__set__conv__all__nth,axiom,
    ! [Xs: list_term_f_v,P: term_f_v > $o] :
      ( ( ! [X5: term_f_v] :
            ( ( member_term_f_v @ X5 @ ( set_term_f_v2 @ Xs ) )
           => ( P @ X5 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
           => ( P @ ( nth_term_f_v @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_67_all__set__conv__all__nth,axiom,
    ! [Xs: list_P1780566427699531651term_f,P: produc250624644672406269term_f > $o] :
      ( ( ! [X5: produc250624644672406269term_f] :
            ( ( member4687689947140789414term_f @ X5 @ ( set_Pr1368550810757897490term_f @ Xs ) )
           => ( P @ X5 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s9037951226665808879term_f @ Xs ) )
           => ( P @ ( nth_Pr7080171896067517188term_f @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_68_all__set__conv__all__nth,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,P: produc8027700073345061287rm_f_v > $o] :
      ( ( ! [X5: produc8027700073345061287rm_f_v] :
            ( ( member4316649861536941904rm_f_v @ X5 @ ( set_Pr817814403484925884rm_f_v @ Xs ) )
           => ( P @ X5 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
           => ( P @ ( nth_Pr8768905099348956590rm_f_v @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_69_all__nth__imp__all__set,axiom,
    ! [Xs: list_term_f_v,P: term_f_v > $o,X: term_f_v] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s7148729505879680775rm_f_v @ Xs ) )
         => ( P @ ( nth_term_f_v @ Xs @ I ) ) )
     => ( ( member_term_f_v @ X @ ( set_term_f_v2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_70_all__nth__imp__all__set,axiom,
    ! [Xs: list_P1780566427699531651term_f,P: produc250624644672406269term_f > $o,X: produc250624644672406269term_f] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s9037951226665808879term_f @ Xs ) )
         => ( P @ ( nth_Pr7080171896067517188term_f @ Xs @ I ) ) )
     => ( ( member4687689947140789414term_f @ X @ ( set_Pr1368550810757897490term_f @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_71_all__nth__imp__all__set,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,P: produc8027700073345061287rm_f_v > $o,X: produc8027700073345061287rm_f_v] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s3818039357710166425rm_f_v @ Xs ) )
         => ( P @ ( nth_Pr8768905099348956590rm_f_v @ Xs @ I ) ) )
     => ( ( member4316649861536941904rm_f_v @ X @ ( set_Pr817814403484925884rm_f_v @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_72_mem__Collect__eq,axiom,
    ! [A: produc8027700073345061287rm_f_v,P: produc8027700073345061287rm_f_v > $o] :
      ( ( member4316649861536941904rm_f_v @ A @ ( collec7769483783292507282rm_f_v @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_73_mem__Collect__eq,axiom,
    ! [A: produc250624644672406269term_f,P: produc250624644672406269term_f > $o] :
      ( ( member4687689947140789414term_f @ A @ ( collec753421845813909992term_f @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_74_mem__Collect__eq,axiom,
    ! [A: term_f_v,P: term_f_v > $o] :
      ( ( member_term_f_v @ A @ ( collect_term_f_v @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_75_Collect__mem__eq,axiom,
    ! [A4: set_Pr8827868859434726151rm_f_v] :
      ( ( collec7769483783292507282rm_f_v
        @ ^ [X5: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X5 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_76_Collect__mem__eq,axiom,
    ! [A4: set_Pr6841673932529763677term_f] :
      ( ( collec753421845813909992term_f
        @ ^ [X5: produc250624644672406269term_f] : ( member4687689947140789414term_f @ X5 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_77_Collect__mem__eq,axiom,
    ! [A4: set_term_f_v] :
      ( ( collect_term_f_v
        @ ^ [X5: term_f_v] : ( member_term_f_v @ X5 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_78_in__set__conv__nth,axiom,
    ! [X: term_f_v,Xs: list_term_f_v] :
      ( ( member_term_f_v @ X @ ( set_term_f_v2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
            & ( ( nth_term_f_v @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_79_in__set__conv__nth,axiom,
    ! [X: produc250624644672406269term_f,Xs: list_P1780566427699531651term_f] :
      ( ( member4687689947140789414term_f @ X @ ( set_Pr1368550810757897490term_f @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s9037951226665808879term_f @ Xs ) )
            & ( ( nth_Pr7080171896067517188term_f @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_80_in__set__conv__nth,axiom,
    ! [X: produc8027700073345061287rm_f_v,Xs: list_P4093298276913796397rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ X @ ( set_Pr817814403484925884rm_f_v @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
            & ( ( nth_Pr8768905099348956590rm_f_v @ Xs @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_81_list__ball__nth,axiom,
    ! [N: nat,Xs: list_term_f_v,P: term_f_v > $o] :
      ( ( ord_less_nat @ N @ ( size_s7148729505879680775rm_f_v @ Xs ) )
     => ( ! [X4: term_f_v] :
            ( ( member_term_f_v @ X4 @ ( set_term_f_v2 @ Xs ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_term_f_v @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_82_list__ball__nth,axiom,
    ! [N: nat,Xs: list_P1780566427699531651term_f,P: produc250624644672406269term_f > $o] :
      ( ( ord_less_nat @ N @ ( size_s9037951226665808879term_f @ Xs ) )
     => ( ! [X4: produc250624644672406269term_f] :
            ( ( member4687689947140789414term_f @ X4 @ ( set_Pr1368550810757897490term_f @ Xs ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_Pr7080171896067517188term_f @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_83_list__ball__nth,axiom,
    ! [N: nat,Xs: list_P4093298276913796397rm_f_v,P: produc8027700073345061287rm_f_v > $o] :
      ( ( ord_less_nat @ N @ ( size_s3818039357710166425rm_f_v @ Xs ) )
     => ( ! [X4: produc8027700073345061287rm_f_v] :
            ( ( member4316649861536941904rm_f_v @ X4 @ ( set_Pr817814403484925884rm_f_v @ Xs ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_Pr8768905099348956590rm_f_v @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_84_length__induct,axiom,
    ! [P: list_term_f_v > $o,Xs: list_term_f_v] :
      ( ! [Xs3: list_term_f_v] :
          ( ! [Ys3: list_term_f_v] :
              ( ( ord_less_nat @ ( size_s7148729505879680775rm_f_v @ Ys3 ) @ ( size_s7148729505879680775rm_f_v @ Xs3 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_85_length__induct,axiom,
    ! [P: list_P1780566427699531651term_f > $o,Xs: list_P1780566427699531651term_f] :
      ( ! [Xs3: list_P1780566427699531651term_f] :
          ( ! [Ys3: list_P1780566427699531651term_f] :
              ( ( ord_less_nat @ ( size_s9037951226665808879term_f @ Ys3 ) @ ( size_s9037951226665808879term_f @ Xs3 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_86_length__induct,axiom,
    ! [P: list_P4093298276913796397rm_f_v > $o,Xs: list_P4093298276913796397rm_f_v] :
      ( ! [Xs3: list_P4093298276913796397rm_f_v] :
          ( ! [Ys3: list_P4093298276913796397rm_f_v] :
              ( ( ord_less_nat @ ( size_s3818039357710166425rm_f_v @ Ys3 ) @ ( size_s3818039357710166425rm_f_v @ Xs3 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_87_supteq__antisym,axiom,
    ! [S: term_f_v,T: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subter5033437232688913882eq_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ S ) @ subter5033437232688913882eq_f_v )
       => ( S = T ) ) ) ).

% supteq_antisym
thf(fact_88_supteq__trans,axiom,
    ! [S: term_f_v,T: term_f_v,U: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subter5033437232688913882eq_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ U ) @ subter5033437232688913882eq_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ U ) @ subter5033437232688913882eq_f_v ) ) ) ).

% supteq_trans
thf(fact_89_eq__supteq,axiom,
    ! [S: term_f_v,T: term_f_v] :
      ( ( S = T )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subter5033437232688913882eq_f_v ) ) ).

% eq_supteq
thf(fact_90_subterm_Odual__order_Oantisym,axiom,
    ! [A: term_f_v,B: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subter5033437232688913882eq_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subter5033437232688913882eq_f_v )
       => ( A = B ) ) ) ).

% subterm.dual_order.antisym
thf(fact_91_subterm_Odual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: term_f_v,Z2: term_f_v] : ( Y2 = Z2 ) )
    = ( ^ [A5: term_f_v,B4: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A5 @ B4 ) @ subter5033437232688913882eq_f_v )
          & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B4 @ A5 ) @ subter5033437232688913882eq_f_v ) ) ) ) ).

% subterm.dual_order.eq_iff
thf(fact_92_subterm_Odual__order_Otrans,axiom,
    ! [A: term_f_v,B: term_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subter5033437232688913882eq_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ subter5033437232688913882eq_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ subter5033437232688913882eq_f_v ) ) ) ).

% subterm.dual_order.trans
thf(fact_93_subterm_Oord__le__eq__trans,axiom,
    ! [B: term_f_v,A: term_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subter5033437232688913882eq_f_v )
     => ( ( B = C )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C @ A ) @ subter5033437232688913882eq_f_v ) ) ) ).

% subterm.ord_le_eq_trans
thf(fact_94_subterm_Oord__eq__le__trans,axiom,
    ! [A: term_f_v,B: term_f_v,C: term_f_v] :
      ( ( A = B )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C @ B ) @ subter5033437232688913882eq_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C @ A ) @ subter5033437232688913882eq_f_v ) ) ) ).

% subterm.ord_eq_le_trans
thf(fact_95_subterm_Odual__order_Orefl,axiom,
    ! [A: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ A ) @ subter5033437232688913882eq_f_v ) ).

% subterm.dual_order.refl
thf(fact_96_subterm_Oorder__antisym,axiom,
    ! [Y: term_f_v,X: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subter5033437232688913882eq_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ subter5033437232688913882eq_f_v )
       => ( X = Y ) ) ) ).

% subterm.order_antisym
thf(fact_97_subterm_Oorder_Oantisym,axiom,
    ! [B: term_f_v,A: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subter5033437232688913882eq_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subter5033437232688913882eq_f_v )
       => ( A = B ) ) ) ).

% subterm.order.antisym
thf(fact_98_subterm_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: term_f_v,Z2: term_f_v] : ( Y2 = Z2 ) )
    = ( ^ [X5: term_f_v,Y4: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y4 @ X5 ) @ subter5033437232688913882eq_f_v )
          & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X5 @ Y4 ) @ subter5033437232688913882eq_f_v ) ) ) ) ).

% subterm.order_eq_iff
thf(fact_99_subterm_Oorder_Oeq__iff,axiom,
    ( ( ^ [Y2: term_f_v,Z2: term_f_v] : ( Y2 = Z2 ) )
    = ( ^ [A5: term_f_v,B4: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B4 @ A5 ) @ subter5033437232688913882eq_f_v )
          & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A5 @ B4 ) @ subter5033437232688913882eq_f_v ) ) ) ) ).

% subterm.order.eq_iff
thf(fact_100_subterm_Oantisym__conv,axiom,
    ! [X: term_f_v,Y: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ subter5033437232688913882eq_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subter5033437232688913882eq_f_v )
        = ( X = Y ) ) ) ).

% subterm.antisym_conv
thf(fact_101_subterm_Oorder__trans,axiom,
    ! [Y: term_f_v,X: term_f_v,Z: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subter5033437232688913882eq_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z @ Y ) @ subter5033437232688913882eq_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z @ X ) @ subter5033437232688913882eq_f_v ) ) ) ).

% subterm.order_trans
thf(fact_102_subterm_Oorder_Otrans,axiom,
    ! [B: term_f_v,A: term_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subter5033437232688913882eq_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C @ B ) @ subter5033437232688913882eq_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C @ A ) @ subter5033437232688913882eq_f_v ) ) ) ).

% subterm.order.trans
thf(fact_103_subterm_Oorder__refl,axiom,
    ! [X: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ X ) @ subter5033437232688913882eq_f_v ) ).

% subterm.order_refl
thf(fact_104_subterm_Oeq__refl,axiom,
    ! [X: term_f_v,Y: term_f_v] :
      ( ( X = Y )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subter5033437232688913882eq_f_v ) ) ).

% subterm.eq_refl
thf(fact_105_in__set__idx,axiom,
    ! [X: term_f_v,Xs: list_term_f_v] :
      ( ( member_term_f_v @ X @ ( set_term_f_v2 @ Xs ) )
     => ? [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s7148729505879680775rm_f_v @ Xs ) )
          & ( ( nth_term_f_v @ Xs @ I )
            = X ) ) ) ).

% in_set_idx
thf(fact_106_in__set__idx,axiom,
    ! [X: produc250624644672406269term_f,Xs: list_P1780566427699531651term_f] :
      ( ( member4687689947140789414term_f @ X @ ( set_Pr1368550810757897490term_f @ Xs ) )
     => ? [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s9037951226665808879term_f @ Xs ) )
          & ( ( nth_Pr7080171896067517188term_f @ Xs @ I )
            = X ) ) ) ).

% in_set_idx
thf(fact_107_in__set__idx,axiom,
    ! [X: produc8027700073345061287rm_f_v,Xs: list_P4093298276913796397rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ X @ ( set_Pr817814403484925884rm_f_v @ Xs ) )
     => ? [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s3818039357710166425rm_f_v @ Xs ) )
          & ( ( nth_Pr8768905099348956590rm_f_v @ Xs @ I )
            = X ) ) ) ).

% in_set_idx
thf(fact_108_subterm__eq__pres__ground,axiom,
    ! [S: term_f_v,T: term_f_v] :
      ( ( term_ground_f_v @ S )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subter5033437232688913882eq_f_v )
       => ( term_ground_f_v @ T ) ) ) ).

% subterm_eq_pres_ground
thf(fact_109_permut__sound,axiom,
    ! [I4: nat,As: list_term_f_v,F: nat > nat] :
      ( ( ord_less_nat @ I4 @ ( size_s7148729505879680775rm_f_v @ As ) )
     => ( ( nth_term_f_v @ ( missin703555983608898443rm_f_v @ As @ F ) @ I4 )
        = ( nth_term_f_v @ As @ ( F @ I4 ) ) ) ) ).

% permut_sound
thf(fact_110_permut__sound,axiom,
    ! [I4: nat,As: list_P1780566427699531651term_f,F: nat > nat] :
      ( ( ord_less_nat @ I4 @ ( size_s9037951226665808879term_f @ As ) )
     => ( ( nth_Pr7080171896067517188term_f @ ( missin8994855842826764501term_f @ As @ F ) @ I4 )
        = ( nth_Pr7080171896067517188term_f @ As @ ( F @ I4 ) ) ) ) ).

% permut_sound
thf(fact_111_permut__sound,axiom,
    ! [I4: nat,As: list_P4093298276913796397rm_f_v,F: nat > nat] :
      ( ( ord_less_nat @ I4 @ ( size_s3818039357710166425rm_f_v @ As ) )
     => ( ( nth_Pr8768905099348956590rm_f_v @ ( missin1444235793242550143rm_f_v @ As @ F ) @ I4 )
        = ( nth_Pr8768905099348956590rm_f_v @ As @ ( F @ I4 ) ) ) ) ).

% permut_sound
thf(fact_112_term_Oinject_I2_J,axiom,
    ! [X21: f,X222: list_term_f_v,Y21: f,Y222: list_term_f_v] :
      ( ( ( fun_f_v @ X21 @ X222 )
        = ( fun_f_v @ Y21 @ Y222 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% term.inject(2)
thf(fact_113_permut__aux__sound,axiom,
    ! [I4: nat,As: list_term_f_v,F: nat > nat,Bs: list_term_f_v] :
      ( ( ord_less_nat @ I4 @ ( size_s7148729505879680775rm_f_v @ As ) )
     => ( ( nth_term_f_v @ ( missin947829892312348577rm_f_v @ As @ F @ Bs ) @ I4 )
        = ( nth_term_f_v @ Bs @ ( F @ I4 ) ) ) ) ).

% permut_aux_sound
thf(fact_114_permut__aux__sound,axiom,
    ! [I4: nat,As: list_P1780566427699531651term_f,F: nat > nat,Bs: list_P1780566427699531651term_f] :
      ( ( ord_less_nat @ I4 @ ( size_s9037951226665808879term_f @ As ) )
     => ( ( nth_Pr7080171896067517188term_f @ ( missin5869388257194169407term_f @ As @ F @ Bs ) @ I4 )
        = ( nth_Pr7080171896067517188term_f @ Bs @ ( F @ I4 ) ) ) ) ).

% permut_aux_sound
thf(fact_115_permut__aux__sound,axiom,
    ! [I4: nat,As: list_P4093298276913796397rm_f_v,F: nat > nat,Bs: list_P4093298276913796397rm_f_v] :
      ( ( ord_less_nat @ I4 @ ( size_s3818039357710166425rm_f_v @ As ) )
     => ( ( nth_Pr8768905099348956590rm_f_v @ ( missin615867081890605801rm_f_v @ As @ F @ Bs ) @ I4 )
        = ( nth_Pr8768905099348956590rm_f_v @ Bs @ ( F @ I4 ) ) ) ) ).

% permut_aux_sound
thf(fact_116_ta__der_H__Fun2,axiom,
    ! [F: f,Ps: list_term_f_a,A6: tree_ta_a_f,G: f,Ts: list_term_f_a] :
      ( ( fmember_term_f_a @ ( fun_f_a @ F @ Ps ) @ ( tree_ta_der_a_f2 @ A6 @ ( fun_f_a @ G @ Ts ) ) )
      = ( ( F = G )
        & ( ( size_s5657014664906415090rm_f_a @ Ps )
          = ( size_s5657014664906415090rm_f_a @ Ts ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5657014664906415090rm_f_a @ Ts ) )
           => ( fmember_term_f_a @ ( nth_term_f_a @ Ps @ I3 ) @ ( tree_ta_der_a_f2 @ A6 @ ( nth_term_f_a @ Ts @ I3 ) ) ) ) ) ) ).

% ta_der'_Fun2
thf(fact_117_ta__der_H__Fun2,axiom,
    ! [F: f,Ps: list_term_f_v,A6: tree_ta_v_f,G: f,Ts: list_term_f_v] :
      ( ( fmember_term_f_v @ ( fun_f_v @ F @ Ps ) @ ( tree_ta_der_v_f2 @ A6 @ ( fun_f_v @ G @ Ts ) ) )
      = ( ( F = G )
        & ( ( size_s7148729505879680775rm_f_v @ Ps )
          = ( size_s7148729505879680775rm_f_v @ Ts ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7148729505879680775rm_f_v @ Ts ) )
           => ( fmember_term_f_v @ ( nth_term_f_v @ Ps @ I3 ) @ ( tree_ta_der_v_f2 @ A6 @ ( nth_term_f_v @ Ts @ I3 ) ) ) ) ) ) ).

% ta_der'_Fun2
thf(fact_118_trancl__list_Obase,axiom,
    ! [Xs: list_Bot_bot_term_f,Ys: list_Bot_bot_term_f,R: set_Pr6841673932529763677term_f] :
      ( ( ( size_s1458857753309042354term_f @ Xs )
        = ( size_s1458857753309042354term_f @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s1458857753309042354term_f @ Ys ) )
           => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( nth_Bot_bot_term_f @ Xs @ I ) @ ( nth_Bot_bot_term_f @ Ys @ I ) ) @ R ) )
       => ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Xs @ Ys ) @ ( trancl6444742785402773766term_f @ R ) ) ) ) ).

% trancl_list.base
thf(fact_119_trancl__list_Obase,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) @ R ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( trancl_list_nat @ R ) ) ) ) ).

% trancl_list.base
thf(fact_120_trancl__list_Obase,axiom,
    ! [Xs: list_term_f_v,Ys: list_term_f_v,R: set_Pr8827868859434726151rm_f_v] :
      ( ( ( size_s7148729505879680775rm_f_v @ Xs )
        = ( size_s7148729505879680775rm_f_v @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s7148729505879680775rm_f_v @ Ys ) )
           => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( nth_term_f_v @ Xs @ I ) @ ( nth_term_f_v @ Ys @ I ) ) @ R ) )
       => ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Xs @ Ys ) @ ( trancl_list_term_f_v @ R ) ) ) ) ).

% trancl_list.base
thf(fact_121_trancl__list_Obase,axiom,
    ! [Xs: list_P1780566427699531651term_f,Ys: list_P1780566427699531651term_f,R: set_Pr4896805198020829255term_f] :
      ( ( ( size_s9037951226665808879term_f @ Xs )
        = ( size_s9037951226665808879term_f @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s9037951226665808879term_f @ Ys ) )
           => ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( nth_Pr7080171896067517188term_f @ Xs @ I ) @ ( nth_Pr7080171896067517188term_f @ Ys @ I ) ) @ R ) )
       => ( member2822224418761970704term_f @ ( produc1033786750570529367term_f @ Xs @ Ys ) @ ( trancl1147503139693895301term_f @ R ) ) ) ) ).

% trancl_list.base
thf(fact_122_trancl__list_Obase,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,Ys: list_P4093298276913796397rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
      ( ( ( size_s3818039357710166425rm_f_v @ Xs )
        = ( size_s3818039357710166425rm_f_v @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3818039357710166425rm_f_v @ Ys ) )
           => ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( nth_Pr8768905099348956590rm_f_v @ Xs @ I ) @ ( nth_Pr8768905099348956590rm_f_v @ Ys @ I ) ) @ R ) )
       => ( member4621649979522838928rm_f_v @ ( produc3444105967875339479rm_f_v @ Xs @ Ys ) @ ( trancl7964251992849616175rm_f_v @ R ) ) ) ) ).

% trancl_list.base
thf(fact_123_trancl__listp_Obase,axiom,
    ! [Xs: list_term_f_v,Ys: list_term_f_v,R: term_f_v > term_f_v > $o] :
      ( ( ( size_s7148729505879680775rm_f_v @ Xs )
        = ( size_s7148729505879680775rm_f_v @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s7148729505879680775rm_f_v @ Ys ) )
           => ( R @ ( nth_term_f_v @ Xs @ I ) @ ( nth_term_f_v @ Ys @ I ) ) )
       => ( trancl7233808882382938973rm_f_v @ R @ Xs @ Ys ) ) ) ).

% trancl_listp.base
thf(fact_124_trancl__listp_Obase,axiom,
    ! [Xs: list_P1780566427699531651term_f,Ys: list_P1780566427699531651term_f,R: produc250624644672406269term_f > produc250624644672406269term_f > $o] :
      ( ( ( size_s9037951226665808879term_f @ Xs )
        = ( size_s9037951226665808879term_f @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s9037951226665808879term_f @ Ys ) )
           => ( R @ ( nth_Pr7080171896067517188term_f @ Xs @ I ) @ ( nth_Pr7080171896067517188term_f @ Ys @ I ) ) )
       => ( trancl2235149414931352451term_f @ R @ Xs @ Ys ) ) ) ).

% trancl_listp.base
thf(fact_125_trancl__listp_Obase,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,Ys: list_P4093298276913796397rm_f_v,R: produc8027700073345061287rm_f_v > produc8027700073345061287rm_f_v > $o] :
      ( ( ( size_s3818039357710166425rm_f_v @ Xs )
        = ( size_s3818039357710166425rm_f_v @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3818039357710166425rm_f_v @ Ys ) )
           => ( R @ ( nth_Pr8768905099348956590rm_f_v @ Xs @ I ) @ ( nth_Pr8768905099348956590rm_f_v @ Ys @ I ) ) )
       => ( trancl8258233303137563181rm_f_v @ R @ Xs @ Ys ) ) ) ).

% trancl_listp.base
thf(fact_126_remove__nth__P__compat,axiom,
    ! [As: list_term_f_v,Bs: list_term_f_v,P: term_f_v > term_f_v > $o,P2: nat] :
      ( ( ( size_s7148729505879680775rm_f_v @ As )
        = ( size_s7148729505879680775rm_f_v @ Bs ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s7148729505879680775rm_f_v @ As ) )
           => ( P @ ( nth_term_f_v @ As @ I ) @ ( nth_term_f_v @ Bs @ I ) ) )
       => ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s7148729505879680775rm_f_v @ ( missin2877041621942143146rm_f_v @ P2 @ As ) ) )
           => ( P @ ( nth_term_f_v @ ( missin2877041621942143146rm_f_v @ P2 @ As ) @ I2 ) @ ( nth_term_f_v @ ( missin2877041621942143146rm_f_v @ P2 @ Bs ) @ I2 ) ) ) ) ) ).

% remove_nth_P_compat
thf(fact_127_remove__nth__P__compat,axiom,
    ! [As: list_term_f_v,Bs: list_P1780566427699531651term_f,P: term_f_v > produc250624644672406269term_f > $o,P2: nat] :
      ( ( ( size_s7148729505879680775rm_f_v @ As )
        = ( size_s9037951226665808879term_f @ Bs ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s7148729505879680775rm_f_v @ As ) )
           => ( P @ ( nth_term_f_v @ As @ I ) @ ( nth_Pr7080171896067517188term_f @ Bs @ I ) ) )
       => ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s7148729505879680775rm_f_v @ ( missin2877041621942143146rm_f_v @ P2 @ As ) ) )
           => ( P @ ( nth_term_f_v @ ( missin2877041621942143146rm_f_v @ P2 @ As ) @ I2 ) @ ( nth_Pr7080171896067517188term_f @ ( missin5571439781501833334term_f @ P2 @ Bs ) @ I2 ) ) ) ) ) ).

% remove_nth_P_compat
thf(fact_128_remove__nth__P__compat,axiom,
    ! [As: list_term_f_v,Bs: list_P4093298276913796397rm_f_v,P: term_f_v > produc8027700073345061287rm_f_v > $o,P2: nat] :
      ( ( ( size_s7148729505879680775rm_f_v @ As )
        = ( size_s3818039357710166425rm_f_v @ Bs ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s7148729505879680775rm_f_v @ As ) )
           => ( P @ ( nth_term_f_v @ As @ I ) @ ( nth_Pr8768905099348956590rm_f_v @ Bs @ I ) ) )
       => ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s7148729505879680775rm_f_v @ ( missin2877041621942143146rm_f_v @ P2 @ As ) ) )
           => ( P @ ( nth_term_f_v @ ( missin2877041621942143146rm_f_v @ P2 @ As ) @ I2 ) @ ( nth_Pr8768905099348956590rm_f_v @ ( missin5548135175598359328rm_f_v @ P2 @ Bs ) @ I2 ) ) ) ) ) ).

% remove_nth_P_compat
thf(fact_129_remove__nth__P__compat,axiom,
    ! [As: list_P1780566427699531651term_f,Bs: list_term_f_v,P: produc250624644672406269term_f > term_f_v > $o,P2: nat] :
      ( ( ( size_s9037951226665808879term_f @ As )
        = ( size_s7148729505879680775rm_f_v @ Bs ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s9037951226665808879term_f @ As ) )
           => ( P @ ( nth_Pr7080171896067517188term_f @ As @ I ) @ ( nth_term_f_v @ Bs @ I ) ) )
       => ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s9037951226665808879term_f @ ( missin5571439781501833334term_f @ P2 @ As ) ) )
           => ( P @ ( nth_Pr7080171896067517188term_f @ ( missin5571439781501833334term_f @ P2 @ As ) @ I2 ) @ ( nth_term_f_v @ ( missin2877041621942143146rm_f_v @ P2 @ Bs ) @ I2 ) ) ) ) ) ).

% remove_nth_P_compat
thf(fact_130_remove__nth__P__compat,axiom,
    ! [As: list_P1780566427699531651term_f,Bs: list_P1780566427699531651term_f,P: produc250624644672406269term_f > produc250624644672406269term_f > $o,P2: nat] :
      ( ( ( size_s9037951226665808879term_f @ As )
        = ( size_s9037951226665808879term_f @ Bs ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s9037951226665808879term_f @ As ) )
           => ( P @ ( nth_Pr7080171896067517188term_f @ As @ I ) @ ( nth_Pr7080171896067517188term_f @ Bs @ I ) ) )
       => ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s9037951226665808879term_f @ ( missin5571439781501833334term_f @ P2 @ As ) ) )
           => ( P @ ( nth_Pr7080171896067517188term_f @ ( missin5571439781501833334term_f @ P2 @ As ) @ I2 ) @ ( nth_Pr7080171896067517188term_f @ ( missin5571439781501833334term_f @ P2 @ Bs ) @ I2 ) ) ) ) ) ).

% remove_nth_P_compat
thf(fact_131_remove__nth__P__compat,axiom,
    ! [As: list_P1780566427699531651term_f,Bs: list_P4093298276913796397rm_f_v,P: produc250624644672406269term_f > produc8027700073345061287rm_f_v > $o,P2: nat] :
      ( ( ( size_s9037951226665808879term_f @ As )
        = ( size_s3818039357710166425rm_f_v @ Bs ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s9037951226665808879term_f @ As ) )
           => ( P @ ( nth_Pr7080171896067517188term_f @ As @ I ) @ ( nth_Pr8768905099348956590rm_f_v @ Bs @ I ) ) )
       => ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s9037951226665808879term_f @ ( missin5571439781501833334term_f @ P2 @ As ) ) )
           => ( P @ ( nth_Pr7080171896067517188term_f @ ( missin5571439781501833334term_f @ P2 @ As ) @ I2 ) @ ( nth_Pr8768905099348956590rm_f_v @ ( missin5548135175598359328rm_f_v @ P2 @ Bs ) @ I2 ) ) ) ) ) ).

% remove_nth_P_compat
thf(fact_132_remove__nth__P__compat,axiom,
    ! [As: list_P4093298276913796397rm_f_v,Bs: list_term_f_v,P: produc8027700073345061287rm_f_v > term_f_v > $o,P2: nat] :
      ( ( ( size_s3818039357710166425rm_f_v @ As )
        = ( size_s7148729505879680775rm_f_v @ Bs ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3818039357710166425rm_f_v @ As ) )
           => ( P @ ( nth_Pr8768905099348956590rm_f_v @ As @ I ) @ ( nth_term_f_v @ Bs @ I ) ) )
       => ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3818039357710166425rm_f_v @ ( missin5548135175598359328rm_f_v @ P2 @ As ) ) )
           => ( P @ ( nth_Pr8768905099348956590rm_f_v @ ( missin5548135175598359328rm_f_v @ P2 @ As ) @ I2 ) @ ( nth_term_f_v @ ( missin2877041621942143146rm_f_v @ P2 @ Bs ) @ I2 ) ) ) ) ) ).

% remove_nth_P_compat
thf(fact_133_remove__nth__P__compat,axiom,
    ! [As: list_P4093298276913796397rm_f_v,Bs: list_P1780566427699531651term_f,P: produc8027700073345061287rm_f_v > produc250624644672406269term_f > $o,P2: nat] :
      ( ( ( size_s3818039357710166425rm_f_v @ As )
        = ( size_s9037951226665808879term_f @ Bs ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3818039357710166425rm_f_v @ As ) )
           => ( P @ ( nth_Pr8768905099348956590rm_f_v @ As @ I ) @ ( nth_Pr7080171896067517188term_f @ Bs @ I ) ) )
       => ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3818039357710166425rm_f_v @ ( missin5548135175598359328rm_f_v @ P2 @ As ) ) )
           => ( P @ ( nth_Pr8768905099348956590rm_f_v @ ( missin5548135175598359328rm_f_v @ P2 @ As ) @ I2 ) @ ( nth_Pr7080171896067517188term_f @ ( missin5571439781501833334term_f @ P2 @ Bs ) @ I2 ) ) ) ) ) ).

% remove_nth_P_compat
thf(fact_134_remove__nth__P__compat,axiom,
    ! [As: list_P4093298276913796397rm_f_v,Bs: list_P4093298276913796397rm_f_v,P: produc8027700073345061287rm_f_v > produc8027700073345061287rm_f_v > $o,P2: nat] :
      ( ( ( size_s3818039357710166425rm_f_v @ As )
        = ( size_s3818039357710166425rm_f_v @ Bs ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3818039357710166425rm_f_v @ As ) )
           => ( P @ ( nth_Pr8768905099348956590rm_f_v @ As @ I ) @ ( nth_Pr8768905099348956590rm_f_v @ Bs @ I ) ) )
       => ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3818039357710166425rm_f_v @ ( missin5548135175598359328rm_f_v @ P2 @ As ) ) )
           => ( P @ ( nth_Pr8768905099348956590rm_f_v @ ( missin5548135175598359328rm_f_v @ P2 @ As ) @ I2 ) @ ( nth_Pr8768905099348956590rm_f_v @ ( missin5548135175598359328rm_f_v @ P2 @ Bs ) @ I2 ) ) ) ) ) ).

% remove_nth_P_compat
thf(fact_135_set__swap,axiom,
    ! [I4: nat,Xs: list_term_f_v,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s7148729505879680775rm_f_v @ Xs ) )
       => ( ( set_term_f_v2 @ ( list_update_term_f_v @ ( list_update_term_f_v @ Xs @ I4 @ ( nth_term_f_v @ Xs @ J ) ) @ J @ ( nth_term_f_v @ Xs @ I4 ) ) )
          = ( set_term_f_v2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_136_set__swap,axiom,
    ! [I4: nat,Xs: list_P1780566427699531651term_f,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s9037951226665808879term_f @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s9037951226665808879term_f @ Xs ) )
       => ( ( set_Pr1368550810757897490term_f @ ( list_u2231791752812534891term_f @ ( list_u2231791752812534891term_f @ Xs @ I4 @ ( nth_Pr7080171896067517188term_f @ Xs @ J ) ) @ J @ ( nth_Pr7080171896067517188term_f @ Xs @ I4 ) ) )
          = ( set_Pr1368550810757897490term_f @ Xs ) ) ) ) ).

% set_swap
thf(fact_137_set__swap,axiom,
    ! [I4: nat,Xs: list_P4093298276913796397rm_f_v,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s3818039357710166425rm_f_v @ Xs ) )
       => ( ( set_Pr817814403484925884rm_f_v @ ( list_u2658473446164013845rm_f_v @ ( list_u2658473446164013845rm_f_v @ Xs @ I4 @ ( nth_Pr8768905099348956590rm_f_v @ Xs @ J ) ) @ J @ ( nth_Pr8768905099348956590rm_f_v @ Xs @ I4 ) ) )
          = ( set_Pr817814403484925884rm_f_v @ Xs ) ) ) ) ).

% set_swap
thf(fact_138_length__list__update,axiom,
    ! [Xs: list_term_f_v,I4: nat,X: term_f_v] :
      ( ( size_s7148729505879680775rm_f_v @ ( list_update_term_f_v @ Xs @ I4 @ X ) )
      = ( size_s7148729505879680775rm_f_v @ Xs ) ) ).

% length_list_update
thf(fact_139_length__list__update,axiom,
    ! [Xs: list_P1780566427699531651term_f,I4: nat,X: produc250624644672406269term_f] :
      ( ( size_s9037951226665808879term_f @ ( list_u2231791752812534891term_f @ Xs @ I4 @ X ) )
      = ( size_s9037951226665808879term_f @ Xs ) ) ).

% length_list_update
thf(fact_140_length__list__update,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,I4: nat,X: produc8027700073345061287rm_f_v] :
      ( ( size_s3818039357710166425rm_f_v @ ( list_u2658473446164013845rm_f_v @ Xs @ I4 @ X ) )
      = ( size_s3818039357710166425rm_f_v @ Xs ) ) ).

% length_list_update
thf(fact_141_list__update__id,axiom,
    ! [Xs: list_term_f_v,I4: nat] :
      ( ( list_update_term_f_v @ Xs @ I4 @ ( nth_term_f_v @ Xs @ I4 ) )
      = Xs ) ).

% list_update_id
thf(fact_142_list__update__id,axiom,
    ! [Xs: list_P1780566427699531651term_f,I4: nat] :
      ( ( list_u2231791752812534891term_f @ Xs @ I4 @ ( nth_Pr7080171896067517188term_f @ Xs @ I4 ) )
      = Xs ) ).

% list_update_id
thf(fact_143_list__update__id,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,I4: nat] :
      ( ( list_u2658473446164013845rm_f_v @ Xs @ I4 @ ( nth_Pr8768905099348956590rm_f_v @ Xs @ I4 ) )
      = Xs ) ).

% list_update_id
thf(fact_144_nth__list__update__neq,axiom,
    ! [I4: nat,J: nat,Xs: list_term_f_v,X: term_f_v] :
      ( ( I4 != J )
     => ( ( nth_term_f_v @ ( list_update_term_f_v @ Xs @ I4 @ X ) @ J )
        = ( nth_term_f_v @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_145_nth__list__update__neq,axiom,
    ! [I4: nat,J: nat,Xs: list_P1780566427699531651term_f,X: produc250624644672406269term_f] :
      ( ( I4 != J )
     => ( ( nth_Pr7080171896067517188term_f @ ( list_u2231791752812534891term_f @ Xs @ I4 @ X ) @ J )
        = ( nth_Pr7080171896067517188term_f @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_146_nth__list__update__neq,axiom,
    ! [I4: nat,J: nat,Xs: list_P4093298276913796397rm_f_v,X: produc8027700073345061287rm_f_v] :
      ( ( I4 != J )
     => ( ( nth_Pr8768905099348956590rm_f_v @ ( list_u2658473446164013845rm_f_v @ Xs @ I4 @ X ) @ J )
        = ( nth_Pr8768905099348956590rm_f_v @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_147_nth__list__update__eq,axiom,
    ! [I4: nat,Xs: list_term_f_v,X: term_f_v] :
      ( ( ord_less_nat @ I4 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
     => ( ( nth_term_f_v @ ( list_update_term_f_v @ Xs @ I4 @ X ) @ I4 )
        = X ) ) ).

% nth_list_update_eq
thf(fact_148_nth__list__update__eq,axiom,
    ! [I4: nat,Xs: list_P1780566427699531651term_f,X: produc250624644672406269term_f] :
      ( ( ord_less_nat @ I4 @ ( size_s9037951226665808879term_f @ Xs ) )
     => ( ( nth_Pr7080171896067517188term_f @ ( list_u2231791752812534891term_f @ Xs @ I4 @ X ) @ I4 )
        = X ) ) ).

% nth_list_update_eq
thf(fact_149_nth__list__update__eq,axiom,
    ! [I4: nat,Xs: list_P4093298276913796397rm_f_v,X: produc8027700073345061287rm_f_v] :
      ( ( ord_less_nat @ I4 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
     => ( ( nth_Pr8768905099348956590rm_f_v @ ( list_u2658473446164013845rm_f_v @ Xs @ I4 @ X ) @ I4 )
        = X ) ) ).

% nth_list_update_eq
thf(fact_150_ta__der_H__refl,axiom,
    ! [T: term_f_a,A6: tree_ta_a_f] : ( fmember_term_f_a @ T @ ( tree_ta_der_a_f2 @ A6 @ T ) ) ).

% ta_der'_refl
thf(fact_151_ta__der_H__trans,axiom,
    ! [T: term_f_a,A6: tree_ta_a_f,S: term_f_a,U: term_f_a] :
      ( ( fmember_term_f_a @ T @ ( tree_ta_der_a_f2 @ A6 @ S ) )
     => ( ( fmember_term_f_a @ U @ ( tree_ta_der_a_f2 @ A6 @ T ) )
       => ( fmember_term_f_a @ U @ ( tree_ta_der_a_f2 @ A6 @ S ) ) ) ) ).

% ta_der'_trans
thf(fact_152_trancl__listp_Ocases,axiom,
    ! [R: term_f_v > term_f_v > $o,A1: list_term_f_v,A22: list_term_f_v] :
      ( ( trancl7233808882382938973rm_f_v @ R @ A1 @ A22 )
     => ( ( ( ( size_s7148729505879680775rm_f_v @ A1 )
            = ( size_s7148729505879680775rm_f_v @ A22 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s7148729505879680775rm_f_v @ A22 ) )
               => ( R @ ( nth_term_f_v @ A1 @ I2 ) @ ( nth_term_f_v @ A22 @ I2 ) ) ) )
       => ~ ! [Ys4: list_term_f_v,I: nat,Z3: term_f_v] :
              ( ( A22
                = ( list_update_term_f_v @ Ys4 @ I @ Z3 ) )
             => ( ( trancl7233808882382938973rm_f_v @ R @ A1 @ Ys4 )
               => ( ( ord_less_nat @ I @ ( size_s7148729505879680775rm_f_v @ Ys4 ) )
                 => ~ ( R @ ( nth_term_f_v @ Ys4 @ I ) @ Z3 ) ) ) ) ) ) ).

% trancl_listp.cases
thf(fact_153_trancl__listp_Ocases,axiom,
    ! [R: produc250624644672406269term_f > produc250624644672406269term_f > $o,A1: list_P1780566427699531651term_f,A22: list_P1780566427699531651term_f] :
      ( ( trancl2235149414931352451term_f @ R @ A1 @ A22 )
     => ( ( ( ( size_s9037951226665808879term_f @ A1 )
            = ( size_s9037951226665808879term_f @ A22 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s9037951226665808879term_f @ A22 ) )
               => ( R @ ( nth_Pr7080171896067517188term_f @ A1 @ I2 ) @ ( nth_Pr7080171896067517188term_f @ A22 @ I2 ) ) ) )
       => ~ ! [Ys4: list_P1780566427699531651term_f,I: nat,Z3: produc250624644672406269term_f] :
              ( ( A22
                = ( list_u2231791752812534891term_f @ Ys4 @ I @ Z3 ) )
             => ( ( trancl2235149414931352451term_f @ R @ A1 @ Ys4 )
               => ( ( ord_less_nat @ I @ ( size_s9037951226665808879term_f @ Ys4 ) )
                 => ~ ( R @ ( nth_Pr7080171896067517188term_f @ Ys4 @ I ) @ Z3 ) ) ) ) ) ) ).

% trancl_listp.cases
thf(fact_154_trancl__listp_Ocases,axiom,
    ! [R: produc8027700073345061287rm_f_v > produc8027700073345061287rm_f_v > $o,A1: list_P4093298276913796397rm_f_v,A22: list_P4093298276913796397rm_f_v] :
      ( ( trancl8258233303137563181rm_f_v @ R @ A1 @ A22 )
     => ( ( ( ( size_s3818039357710166425rm_f_v @ A1 )
            = ( size_s3818039357710166425rm_f_v @ A22 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s3818039357710166425rm_f_v @ A22 ) )
               => ( R @ ( nth_Pr8768905099348956590rm_f_v @ A1 @ I2 ) @ ( nth_Pr8768905099348956590rm_f_v @ A22 @ I2 ) ) ) )
       => ~ ! [Ys4: list_P4093298276913796397rm_f_v,I: nat,Z3: produc8027700073345061287rm_f_v] :
              ( ( A22
                = ( list_u2658473446164013845rm_f_v @ Ys4 @ I @ Z3 ) )
             => ( ( trancl8258233303137563181rm_f_v @ R @ A1 @ Ys4 )
               => ( ( ord_less_nat @ I @ ( size_s3818039357710166425rm_f_v @ Ys4 ) )
                 => ~ ( R @ ( nth_Pr8768905099348956590rm_f_v @ Ys4 @ I ) @ Z3 ) ) ) ) ) ) ).

% trancl_listp.cases
thf(fact_155_trancl__listp_Osimps,axiom,
    ( trancl7233808882382938973rm_f_v
    = ( ^ [R2: term_f_v > term_f_v > $o,A12: list_term_f_v,A23: list_term_f_v] :
          ( ? [Xs2: list_term_f_v,Ys2: list_term_f_v] :
              ( ( A12 = Xs2 )
              & ( A23 = Ys2 )
              & ( ( size_s7148729505879680775rm_f_v @ Xs2 )
                = ( size_s7148729505879680775rm_f_v @ Ys2 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_s7148729505879680775rm_f_v @ Ys2 ) )
                 => ( R2 @ ( nth_term_f_v @ Xs2 @ I3 ) @ ( nth_term_f_v @ Ys2 @ I3 ) ) ) )
          | ? [Xs2: list_term_f_v,Ys2: list_term_f_v,I3: nat,Z4: term_f_v] :
              ( ( A12 = Xs2 )
              & ( A23
                = ( list_update_term_f_v @ Ys2 @ I3 @ Z4 ) )
              & ( trancl7233808882382938973rm_f_v @ R2 @ Xs2 @ Ys2 )
              & ( ord_less_nat @ I3 @ ( size_s7148729505879680775rm_f_v @ Ys2 ) )
              & ( R2 @ ( nth_term_f_v @ Ys2 @ I3 ) @ Z4 ) ) ) ) ) ).

% trancl_listp.simps
thf(fact_156_trancl__listp_Osimps,axiom,
    ( trancl2235149414931352451term_f
    = ( ^ [R2: produc250624644672406269term_f > produc250624644672406269term_f > $o,A12: list_P1780566427699531651term_f,A23: list_P1780566427699531651term_f] :
          ( ? [Xs2: list_P1780566427699531651term_f,Ys2: list_P1780566427699531651term_f] :
              ( ( A12 = Xs2 )
              & ( A23 = Ys2 )
              & ( ( size_s9037951226665808879term_f @ Xs2 )
                = ( size_s9037951226665808879term_f @ Ys2 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_s9037951226665808879term_f @ Ys2 ) )
                 => ( R2 @ ( nth_Pr7080171896067517188term_f @ Xs2 @ I3 ) @ ( nth_Pr7080171896067517188term_f @ Ys2 @ I3 ) ) ) )
          | ? [Xs2: list_P1780566427699531651term_f,Ys2: list_P1780566427699531651term_f,I3: nat,Z4: produc250624644672406269term_f] :
              ( ( A12 = Xs2 )
              & ( A23
                = ( list_u2231791752812534891term_f @ Ys2 @ I3 @ Z4 ) )
              & ( trancl2235149414931352451term_f @ R2 @ Xs2 @ Ys2 )
              & ( ord_less_nat @ I3 @ ( size_s9037951226665808879term_f @ Ys2 ) )
              & ( R2 @ ( nth_Pr7080171896067517188term_f @ Ys2 @ I3 ) @ Z4 ) ) ) ) ) ).

% trancl_listp.simps
thf(fact_157_trancl__listp_Osimps,axiom,
    ( trancl8258233303137563181rm_f_v
    = ( ^ [R2: produc8027700073345061287rm_f_v > produc8027700073345061287rm_f_v > $o,A12: list_P4093298276913796397rm_f_v,A23: list_P4093298276913796397rm_f_v] :
          ( ? [Xs2: list_P4093298276913796397rm_f_v,Ys2: list_P4093298276913796397rm_f_v] :
              ( ( A12 = Xs2 )
              & ( A23 = Ys2 )
              & ( ( size_s3818039357710166425rm_f_v @ Xs2 )
                = ( size_s3818039357710166425rm_f_v @ Ys2 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_s3818039357710166425rm_f_v @ Ys2 ) )
                 => ( R2 @ ( nth_Pr8768905099348956590rm_f_v @ Xs2 @ I3 ) @ ( nth_Pr8768905099348956590rm_f_v @ Ys2 @ I3 ) ) ) )
          | ? [Xs2: list_P4093298276913796397rm_f_v,Ys2: list_P4093298276913796397rm_f_v,I3: nat,Z4: produc8027700073345061287rm_f_v] :
              ( ( A12 = Xs2 )
              & ( A23
                = ( list_u2658473446164013845rm_f_v @ Ys2 @ I3 @ Z4 ) )
              & ( trancl8258233303137563181rm_f_v @ R2 @ Xs2 @ Ys2 )
              & ( ord_less_nat @ I3 @ ( size_s3818039357710166425rm_f_v @ Ys2 ) )
              & ( R2 @ ( nth_Pr8768905099348956590rm_f_v @ Ys2 @ I3 ) @ Z4 ) ) ) ) ) ).

% trancl_listp.simps
thf(fact_158_trancl__listp_Olist__trancl,axiom,
    ! [R: term_f_v > term_f_v > $o,Xs: list_term_f_v,Ys: list_term_f_v,I4: nat,Z: term_f_v] :
      ( ( trancl7233808882382938973rm_f_v @ R @ Xs @ Ys )
     => ( ( ord_less_nat @ I4 @ ( size_s7148729505879680775rm_f_v @ Ys ) )
       => ( ( R @ ( nth_term_f_v @ Ys @ I4 ) @ Z )
         => ( trancl7233808882382938973rm_f_v @ R @ Xs @ ( list_update_term_f_v @ Ys @ I4 @ Z ) ) ) ) ) ).

% trancl_listp.list_trancl
thf(fact_159_trancl__listp_Olist__trancl,axiom,
    ! [R: produc250624644672406269term_f > produc250624644672406269term_f > $o,Xs: list_P1780566427699531651term_f,Ys: list_P1780566427699531651term_f,I4: nat,Z: produc250624644672406269term_f] :
      ( ( trancl2235149414931352451term_f @ R @ Xs @ Ys )
     => ( ( ord_less_nat @ I4 @ ( size_s9037951226665808879term_f @ Ys ) )
       => ( ( R @ ( nth_Pr7080171896067517188term_f @ Ys @ I4 ) @ Z )
         => ( trancl2235149414931352451term_f @ R @ Xs @ ( list_u2231791752812534891term_f @ Ys @ I4 @ Z ) ) ) ) ) ).

% trancl_listp.list_trancl
thf(fact_160_trancl__listp_Olist__trancl,axiom,
    ! [R: produc8027700073345061287rm_f_v > produc8027700073345061287rm_f_v > $o,Xs: list_P4093298276913796397rm_f_v,Ys: list_P4093298276913796397rm_f_v,I4: nat,Z: produc8027700073345061287rm_f_v] :
      ( ( trancl8258233303137563181rm_f_v @ R @ Xs @ Ys )
     => ( ( ord_less_nat @ I4 @ ( size_s3818039357710166425rm_f_v @ Ys ) )
       => ( ( R @ ( nth_Pr8768905099348956590rm_f_v @ Ys @ I4 ) @ Z )
         => ( trancl8258233303137563181rm_f_v @ R @ Xs @ ( list_u2658473446164013845rm_f_v @ Ys @ I4 @ Z ) ) ) ) ) ).

% trancl_listp.list_trancl
thf(fact_161_trancl__list_Ocases,axiom,
    ! [A1: list_Bot_bot_term_f,A22: list_Bot_bot_term_f,R: set_Pr6841673932529763677term_f] :
      ( ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ A1 @ A22 ) @ ( trancl6444742785402773766term_f @ R ) )
     => ( ( ( ( size_s1458857753309042354term_f @ A1 )
            = ( size_s1458857753309042354term_f @ A22 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s1458857753309042354term_f @ A22 ) )
               => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( nth_Bot_bot_term_f @ A1 @ I2 ) @ ( nth_Bot_bot_term_f @ A22 @ I2 ) ) @ R ) ) )
       => ~ ! [Ys4: list_Bot_bot_term_f,I: nat,Z3: bot_bot_term_f] :
              ( ( A22
                = ( list_u5503441994469622688term_f @ Ys4 @ I @ Z3 ) )
             => ( ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ A1 @ Ys4 ) @ ( trancl6444742785402773766term_f @ R ) )
               => ( ( ord_less_nat @ I @ ( size_s1458857753309042354term_f @ Ys4 ) )
                 => ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( nth_Bot_bot_term_f @ Ys4 @ I ) @ Z3 ) @ R ) ) ) ) ) ) ).

% trancl_list.cases
thf(fact_162_trancl__list_Ocases,axiom,
    ! [A1: list_nat,A22: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( trancl_list_nat @ R ) )
     => ( ( ( ( size_size_list_nat @ A1 )
            = ( size_size_list_nat @ A22 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ A22 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ A1 @ I2 ) @ ( nth_nat @ A22 @ I2 ) ) @ R ) ) )
       => ~ ! [Ys4: list_nat,I: nat,Z3: nat] :
              ( ( A22
                = ( list_update_nat @ Ys4 @ I @ Z3 ) )
             => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ Ys4 ) @ ( trancl_list_nat @ R ) )
               => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys4 ) )
                 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys4 @ I ) @ Z3 ) @ R ) ) ) ) ) ) ).

% trancl_list.cases
thf(fact_163_trancl__list_Ocases,axiom,
    ! [A1: list_term_f_v,A22: list_term_f_v,R: set_Pr8827868859434726151rm_f_v] :
      ( ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ A1 @ A22 ) @ ( trancl_list_term_f_v @ R ) )
     => ( ( ( ( size_s7148729505879680775rm_f_v @ A1 )
            = ( size_s7148729505879680775rm_f_v @ A22 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s7148729505879680775rm_f_v @ A22 ) )
               => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( nth_term_f_v @ A1 @ I2 ) @ ( nth_term_f_v @ A22 @ I2 ) ) @ R ) ) )
       => ~ ! [Ys4: list_term_f_v,I: nat,Z3: term_f_v] :
              ( ( A22
                = ( list_update_term_f_v @ Ys4 @ I @ Z3 ) )
             => ( ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ A1 @ Ys4 ) @ ( trancl_list_term_f_v @ R ) )
               => ( ( ord_less_nat @ I @ ( size_s7148729505879680775rm_f_v @ Ys4 ) )
                 => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( nth_term_f_v @ Ys4 @ I ) @ Z3 ) @ R ) ) ) ) ) ) ).

% trancl_list.cases
thf(fact_164_trancl__list_Ocases,axiom,
    ! [A1: list_P1780566427699531651term_f,A22: list_P1780566427699531651term_f,R: set_Pr4896805198020829255term_f] :
      ( ( member2822224418761970704term_f @ ( produc1033786750570529367term_f @ A1 @ A22 ) @ ( trancl1147503139693895301term_f @ R ) )
     => ( ( ( ( size_s9037951226665808879term_f @ A1 )
            = ( size_s9037951226665808879term_f @ A22 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s9037951226665808879term_f @ A22 ) )
               => ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( nth_Pr7080171896067517188term_f @ A1 @ I2 ) @ ( nth_Pr7080171896067517188term_f @ A22 @ I2 ) ) @ R ) ) )
       => ~ ! [Ys4: list_P1780566427699531651term_f,I: nat,Z3: produc250624644672406269term_f] :
              ( ( A22
                = ( list_u2231791752812534891term_f @ Ys4 @ I @ Z3 ) )
             => ( ( member2822224418761970704term_f @ ( produc1033786750570529367term_f @ A1 @ Ys4 ) @ ( trancl1147503139693895301term_f @ R ) )
               => ( ( ord_less_nat @ I @ ( size_s9037951226665808879term_f @ Ys4 ) )
                 => ~ ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( nth_Pr7080171896067517188term_f @ Ys4 @ I ) @ Z3 ) @ R ) ) ) ) ) ) ).

% trancl_list.cases
thf(fact_165_trancl__list_Ocases,axiom,
    ! [A1: list_P4093298276913796397rm_f_v,A22: list_P4093298276913796397rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
      ( ( member4621649979522838928rm_f_v @ ( produc3444105967875339479rm_f_v @ A1 @ A22 ) @ ( trancl7964251992849616175rm_f_v @ R ) )
     => ( ( ( ( size_s3818039357710166425rm_f_v @ A1 )
            = ( size_s3818039357710166425rm_f_v @ A22 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s3818039357710166425rm_f_v @ A22 ) )
               => ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( nth_Pr8768905099348956590rm_f_v @ A1 @ I2 ) @ ( nth_Pr8768905099348956590rm_f_v @ A22 @ I2 ) ) @ R ) ) )
       => ~ ! [Ys4: list_P4093298276913796397rm_f_v,I: nat,Z3: produc8027700073345061287rm_f_v] :
              ( ( A22
                = ( list_u2658473446164013845rm_f_v @ Ys4 @ I @ Z3 ) )
             => ( ( member4621649979522838928rm_f_v @ ( produc3444105967875339479rm_f_v @ A1 @ Ys4 ) @ ( trancl7964251992849616175rm_f_v @ R ) )
               => ( ( ord_less_nat @ I @ ( size_s3818039357710166425rm_f_v @ Ys4 ) )
                 => ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( nth_Pr8768905099348956590rm_f_v @ Ys4 @ I ) @ Z3 ) @ R ) ) ) ) ) ) ).

% trancl_list.cases
thf(fact_166_trancl__list_Osimps,axiom,
    ! [A1: list_Bot_bot_term_f,A22: list_Bot_bot_term_f,R: set_Pr6841673932529763677term_f] :
      ( ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ A1 @ A22 ) @ ( trancl6444742785402773766term_f @ R ) )
      = ( ? [Xs2: list_Bot_bot_term_f,Ys2: list_Bot_bot_term_f] :
            ( ( A1 = Xs2 )
            & ( A22 = Ys2 )
            & ( ( size_s1458857753309042354term_f @ Xs2 )
              = ( size_s1458857753309042354term_f @ Ys2 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s1458857753309042354term_f @ Ys2 ) )
               => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( nth_Bot_bot_term_f @ Xs2 @ I3 ) @ ( nth_Bot_bot_term_f @ Ys2 @ I3 ) ) @ R ) ) )
        | ? [Xs2: list_Bot_bot_term_f,Ys2: list_Bot_bot_term_f,I3: nat,Z4: bot_bot_term_f] :
            ( ( A1 = Xs2 )
            & ( A22
              = ( list_u5503441994469622688term_f @ Ys2 @ I3 @ Z4 ) )
            & ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Xs2 @ Ys2 ) @ ( trancl6444742785402773766term_f @ R ) )
            & ( ord_less_nat @ I3 @ ( size_s1458857753309042354term_f @ Ys2 ) )
            & ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( nth_Bot_bot_term_f @ Ys2 @ I3 ) @ Z4 ) @ R ) ) ) ) ).

% trancl_list.simps
thf(fact_167_trancl__list_Osimps,axiom,
    ! [A1: list_nat,A22: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( trancl_list_nat @ R ) )
      = ( ? [Xs2: list_nat,Ys2: list_nat] :
            ( ( A1 = Xs2 )
            & ( A22 = Ys2 )
            & ( ( size_size_list_nat @ Xs2 )
              = ( size_size_list_nat @ Ys2 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys2 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs2 @ I3 ) @ ( nth_nat @ Ys2 @ I3 ) ) @ R ) ) )
        | ? [Xs2: list_nat,Ys2: list_nat,I3: nat,Z4: nat] :
            ( ( A1 = Xs2 )
            & ( A22
              = ( list_update_nat @ Ys2 @ I3 @ Z4 ) )
            & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( trancl_list_nat @ R ) )
            & ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys2 ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys2 @ I3 ) @ Z4 ) @ R ) ) ) ) ).

% trancl_list.simps
thf(fact_168_trancl__list_Osimps,axiom,
    ! [A1: list_term_f_v,A22: list_term_f_v,R: set_Pr8827868859434726151rm_f_v] :
      ( ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ A1 @ A22 ) @ ( trancl_list_term_f_v @ R ) )
      = ( ? [Xs2: list_term_f_v,Ys2: list_term_f_v] :
            ( ( A1 = Xs2 )
            & ( A22 = Ys2 )
            & ( ( size_s7148729505879680775rm_f_v @ Xs2 )
              = ( size_s7148729505879680775rm_f_v @ Ys2 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s7148729505879680775rm_f_v @ Ys2 ) )
               => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( nth_term_f_v @ Xs2 @ I3 ) @ ( nth_term_f_v @ Ys2 @ I3 ) ) @ R ) ) )
        | ? [Xs2: list_term_f_v,Ys2: list_term_f_v,I3: nat,Z4: term_f_v] :
            ( ( A1 = Xs2 )
            & ( A22
              = ( list_update_term_f_v @ Ys2 @ I3 @ Z4 ) )
            & ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Xs2 @ Ys2 ) @ ( trancl_list_term_f_v @ R ) )
            & ( ord_less_nat @ I3 @ ( size_s7148729505879680775rm_f_v @ Ys2 ) )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( nth_term_f_v @ Ys2 @ I3 ) @ Z4 ) @ R ) ) ) ) ).

% trancl_list.simps
thf(fact_169_trancl__list_Osimps,axiom,
    ! [A1: list_P1780566427699531651term_f,A22: list_P1780566427699531651term_f,R: set_Pr4896805198020829255term_f] :
      ( ( member2822224418761970704term_f @ ( produc1033786750570529367term_f @ A1 @ A22 ) @ ( trancl1147503139693895301term_f @ R ) )
      = ( ? [Xs2: list_P1780566427699531651term_f,Ys2: list_P1780566427699531651term_f] :
            ( ( A1 = Xs2 )
            & ( A22 = Ys2 )
            & ( ( size_s9037951226665808879term_f @ Xs2 )
              = ( size_s9037951226665808879term_f @ Ys2 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s9037951226665808879term_f @ Ys2 ) )
               => ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( nth_Pr7080171896067517188term_f @ Xs2 @ I3 ) @ ( nth_Pr7080171896067517188term_f @ Ys2 @ I3 ) ) @ R ) ) )
        | ? [Xs2: list_P1780566427699531651term_f,Ys2: list_P1780566427699531651term_f,I3: nat,Z4: produc250624644672406269term_f] :
            ( ( A1 = Xs2 )
            & ( A22
              = ( list_u2231791752812534891term_f @ Ys2 @ I3 @ Z4 ) )
            & ( member2822224418761970704term_f @ ( produc1033786750570529367term_f @ Xs2 @ Ys2 ) @ ( trancl1147503139693895301term_f @ R ) )
            & ( ord_less_nat @ I3 @ ( size_s9037951226665808879term_f @ Ys2 ) )
            & ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( nth_Pr7080171896067517188term_f @ Ys2 @ I3 ) @ Z4 ) @ R ) ) ) ) ).

% trancl_list.simps
thf(fact_170_trancl__list_Osimps,axiom,
    ! [A1: list_P4093298276913796397rm_f_v,A22: list_P4093298276913796397rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
      ( ( member4621649979522838928rm_f_v @ ( produc3444105967875339479rm_f_v @ A1 @ A22 ) @ ( trancl7964251992849616175rm_f_v @ R ) )
      = ( ? [Xs2: list_P4093298276913796397rm_f_v,Ys2: list_P4093298276913796397rm_f_v] :
            ( ( A1 = Xs2 )
            & ( A22 = Ys2 )
            & ( ( size_s3818039357710166425rm_f_v @ Xs2 )
              = ( size_s3818039357710166425rm_f_v @ Ys2 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s3818039357710166425rm_f_v @ Ys2 ) )
               => ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( nth_Pr8768905099348956590rm_f_v @ Xs2 @ I3 ) @ ( nth_Pr8768905099348956590rm_f_v @ Ys2 @ I3 ) ) @ R ) ) )
        | ? [Xs2: list_P4093298276913796397rm_f_v,Ys2: list_P4093298276913796397rm_f_v,I3: nat,Z4: produc8027700073345061287rm_f_v] :
            ( ( A1 = Xs2 )
            & ( A22
              = ( list_u2658473446164013845rm_f_v @ Ys2 @ I3 @ Z4 ) )
            & ( member4621649979522838928rm_f_v @ ( produc3444105967875339479rm_f_v @ Xs2 @ Ys2 ) @ ( trancl7964251992849616175rm_f_v @ R ) )
            & ( ord_less_nat @ I3 @ ( size_s3818039357710166425rm_f_v @ Ys2 ) )
            & ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( nth_Pr8768905099348956590rm_f_v @ Ys2 @ I3 ) @ Z4 ) @ R ) ) ) ) ).

% trancl_list.simps
thf(fact_171_trancl__list_Olist__trancl,axiom,
    ! [Xs: list_Bot_bot_term_f,Ys: list_Bot_bot_term_f,R: set_Pr6841673932529763677term_f,I4: nat,Z: bot_bot_term_f] :
      ( ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Xs @ Ys ) @ ( trancl6444742785402773766term_f @ R ) )
     => ( ( ord_less_nat @ I4 @ ( size_s1458857753309042354term_f @ Ys ) )
       => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( nth_Bot_bot_term_f @ Ys @ I4 ) @ Z ) @ R )
         => ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Xs @ ( list_u5503441994469622688term_f @ Ys @ I4 @ Z ) ) @ ( trancl6444742785402773766term_f @ R ) ) ) ) ) ).

% trancl_list.list_trancl
thf(fact_172_trancl__list_Olist__trancl,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,I4: nat,Z: nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( trancl_list_nat @ R ) )
     => ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys @ I4 ) @ Z ) @ R )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( list_update_nat @ Ys @ I4 @ Z ) ) @ ( trancl_list_nat @ R ) ) ) ) ) ).

% trancl_list.list_trancl
thf(fact_173_trancl__list_Olist__trancl,axiom,
    ! [Xs: list_term_f_v,Ys: list_term_f_v,R: set_Pr8827868859434726151rm_f_v,I4: nat,Z: term_f_v] :
      ( ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Xs @ Ys ) @ ( trancl_list_term_f_v @ R ) )
     => ( ( ord_less_nat @ I4 @ ( size_s7148729505879680775rm_f_v @ Ys ) )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( nth_term_f_v @ Ys @ I4 ) @ Z ) @ R )
         => ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Xs @ ( list_update_term_f_v @ Ys @ I4 @ Z ) ) @ ( trancl_list_term_f_v @ R ) ) ) ) ) ).

% trancl_list.list_trancl
thf(fact_174_trancl__list_Olist__trancl,axiom,
    ! [Xs: list_P1780566427699531651term_f,Ys: list_P1780566427699531651term_f,R: set_Pr4896805198020829255term_f,I4: nat,Z: produc250624644672406269term_f] :
      ( ( member2822224418761970704term_f @ ( produc1033786750570529367term_f @ Xs @ Ys ) @ ( trancl1147503139693895301term_f @ R ) )
     => ( ( ord_less_nat @ I4 @ ( size_s9037951226665808879term_f @ Ys ) )
       => ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( nth_Pr7080171896067517188term_f @ Ys @ I4 ) @ Z ) @ R )
         => ( member2822224418761970704term_f @ ( produc1033786750570529367term_f @ Xs @ ( list_u2231791752812534891term_f @ Ys @ I4 @ Z ) ) @ ( trancl1147503139693895301term_f @ R ) ) ) ) ) ).

% trancl_list.list_trancl
thf(fact_175_trancl__list_Olist__trancl,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,Ys: list_P4093298276913796397rm_f_v,R: set_Pr7897041799657265351rm_f_v,I4: nat,Z: produc8027700073345061287rm_f_v] :
      ( ( member4621649979522838928rm_f_v @ ( produc3444105967875339479rm_f_v @ Xs @ Ys ) @ ( trancl7964251992849616175rm_f_v @ R ) )
     => ( ( ord_less_nat @ I4 @ ( size_s3818039357710166425rm_f_v @ Ys ) )
       => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( nth_Pr8768905099348956590rm_f_v @ Ys @ I4 ) @ Z ) @ R )
         => ( member4621649979522838928rm_f_v @ ( produc3444105967875339479rm_f_v @ Xs @ ( list_u2658473446164013845rm_f_v @ Ys @ I4 @ Z ) ) @ ( trancl7964251992849616175rm_f_v @ R ) ) ) ) ) ).

% trancl_list.list_trancl
thf(fact_176_set__update__memI,axiom,
    ! [N: nat,Xs: list_term_f_v,X: term_f_v] :
      ( ( ord_less_nat @ N @ ( size_s7148729505879680775rm_f_v @ Xs ) )
     => ( member_term_f_v @ X @ ( set_term_f_v2 @ ( list_update_term_f_v @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_177_set__update__memI,axiom,
    ! [N: nat,Xs: list_P1780566427699531651term_f,X: produc250624644672406269term_f] :
      ( ( ord_less_nat @ N @ ( size_s9037951226665808879term_f @ Xs ) )
     => ( member4687689947140789414term_f @ X @ ( set_Pr1368550810757897490term_f @ ( list_u2231791752812534891term_f @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_178_set__update__memI,axiom,
    ! [N: nat,Xs: list_P4093298276913796397rm_f_v,X: produc8027700073345061287rm_f_v] :
      ( ( ord_less_nat @ N @ ( size_s3818039357710166425rm_f_v @ Xs ) )
     => ( member4316649861536941904rm_f_v @ X @ ( set_Pr817814403484925884rm_f_v @ ( list_u2658473446164013845rm_f_v @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_179_nth__list__update,axiom,
    ! [I4: nat,Xs: list_term_f_v,J: nat,X: term_f_v] :
      ( ( ord_less_nat @ I4 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
     => ( ( ( I4 = J )
         => ( ( nth_term_f_v @ ( list_update_term_f_v @ Xs @ I4 @ X ) @ J )
            = X ) )
        & ( ( I4 != J )
         => ( ( nth_term_f_v @ ( list_update_term_f_v @ Xs @ I4 @ X ) @ J )
            = ( nth_term_f_v @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_180_nth__list__update,axiom,
    ! [I4: nat,Xs: list_P1780566427699531651term_f,J: nat,X: produc250624644672406269term_f] :
      ( ( ord_less_nat @ I4 @ ( size_s9037951226665808879term_f @ Xs ) )
     => ( ( ( I4 = J )
         => ( ( nth_Pr7080171896067517188term_f @ ( list_u2231791752812534891term_f @ Xs @ I4 @ X ) @ J )
            = X ) )
        & ( ( I4 != J )
         => ( ( nth_Pr7080171896067517188term_f @ ( list_u2231791752812534891term_f @ Xs @ I4 @ X ) @ J )
            = ( nth_Pr7080171896067517188term_f @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_181_nth__list__update,axiom,
    ! [I4: nat,Xs: list_P4093298276913796397rm_f_v,J: nat,X: produc8027700073345061287rm_f_v] :
      ( ( ord_less_nat @ I4 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
     => ( ( ( I4 = J )
         => ( ( nth_Pr8768905099348956590rm_f_v @ ( list_u2658473446164013845rm_f_v @ Xs @ I4 @ X ) @ J )
            = X ) )
        & ( ( I4 != J )
         => ( ( nth_Pr8768905099348956590rm_f_v @ ( list_u2658473446164013845rm_f_v @ Xs @ I4 @ X ) @ J )
            = ( nth_Pr8768905099348956590rm_f_v @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_182_list__update__same__conv,axiom,
    ! [I4: nat,Xs: list_term_f_v,X: term_f_v] :
      ( ( ord_less_nat @ I4 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
     => ( ( ( list_update_term_f_v @ Xs @ I4 @ X )
          = Xs )
        = ( ( nth_term_f_v @ Xs @ I4 )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_183_list__update__same__conv,axiom,
    ! [I4: nat,Xs: list_P1780566427699531651term_f,X: produc250624644672406269term_f] :
      ( ( ord_less_nat @ I4 @ ( size_s9037951226665808879term_f @ Xs ) )
     => ( ( ( list_u2231791752812534891term_f @ Xs @ I4 @ X )
          = Xs )
        = ( ( nth_Pr7080171896067517188term_f @ Xs @ I4 )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_184_list__update__same__conv,axiom,
    ! [I4: nat,Xs: list_P4093298276913796397rm_f_v,X: produc8027700073345061287rm_f_v] :
      ( ( ord_less_nat @ I4 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
     => ( ( ( list_u2658473446164013845rm_f_v @ Xs @ I4 @ X )
          = Xs )
        = ( ( nth_Pr8768905099348956590rm_f_v @ Xs @ I4 )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_185_parallel__list__update,axiom,
    ! [N: nat,R3: term_f_v > term_f_v > $o,P2: list_term_f_v > $o,Xs: list_term_f_v,Ys: list_term_f_v] :
      ( ! [Xs3: list_term_f_v,I: nat,Y3: term_f_v] :
          ( ( ( size_s7148729505879680775rm_f_v @ Xs3 )
            = N )
         => ( ( ord_less_nat @ I @ N )
           => ( ( R3 @ ( nth_term_f_v @ Xs3 @ I ) @ Y3 )
             => ( ( P2 @ Xs3 )
               => ( P2 @ ( list_update_term_f_v @ Xs3 @ I @ Y3 ) ) ) ) ) )
     => ( ( ( size_s7148729505879680775rm_f_v @ Xs )
          = N )
       => ( ( P2 @ Xs )
         => ( ( ( size_s7148729505879680775rm_f_v @ Ys )
              = N )
           => ( ! [I: nat] :
                  ( ( ord_less_nat @ I @ N )
                 => ( R3 @ ( nth_term_f_v @ Xs @ I ) @ ( nth_term_f_v @ Ys @ I ) ) )
             => ( P2 @ Ys ) ) ) ) ) ) ).

% parallel_list_update
thf(fact_186_parallel__list__update,axiom,
    ! [N: nat,R3: produc250624644672406269term_f > produc250624644672406269term_f > $o,P2: list_P1780566427699531651term_f > $o,Xs: list_P1780566427699531651term_f,Ys: list_P1780566427699531651term_f] :
      ( ! [Xs3: list_P1780566427699531651term_f,I: nat,Y3: produc250624644672406269term_f] :
          ( ( ( size_s9037951226665808879term_f @ Xs3 )
            = N )
         => ( ( ord_less_nat @ I @ N )
           => ( ( R3 @ ( nth_Pr7080171896067517188term_f @ Xs3 @ I ) @ Y3 )
             => ( ( P2 @ Xs3 )
               => ( P2 @ ( list_u2231791752812534891term_f @ Xs3 @ I @ Y3 ) ) ) ) ) )
     => ( ( ( size_s9037951226665808879term_f @ Xs )
          = N )
       => ( ( P2 @ Xs )
         => ( ( ( size_s9037951226665808879term_f @ Ys )
              = N )
           => ( ! [I: nat] :
                  ( ( ord_less_nat @ I @ N )
                 => ( R3 @ ( nth_Pr7080171896067517188term_f @ Xs @ I ) @ ( nth_Pr7080171896067517188term_f @ Ys @ I ) ) )
             => ( P2 @ Ys ) ) ) ) ) ) ).

% parallel_list_update
thf(fact_187_parallel__list__update,axiom,
    ! [N: nat,R3: produc8027700073345061287rm_f_v > produc8027700073345061287rm_f_v > $o,P2: list_P4093298276913796397rm_f_v > $o,Xs: list_P4093298276913796397rm_f_v,Ys: list_P4093298276913796397rm_f_v] :
      ( ! [Xs3: list_P4093298276913796397rm_f_v,I: nat,Y3: produc8027700073345061287rm_f_v] :
          ( ( ( size_s3818039357710166425rm_f_v @ Xs3 )
            = N )
         => ( ( ord_less_nat @ I @ N )
           => ( ( R3 @ ( nth_Pr8768905099348956590rm_f_v @ Xs3 @ I ) @ Y3 )
             => ( ( P2 @ Xs3 )
               => ( P2 @ ( list_u2658473446164013845rm_f_v @ Xs3 @ I @ Y3 ) ) ) ) ) )
     => ( ( ( size_s3818039357710166425rm_f_v @ Xs )
          = N )
       => ( ( P2 @ Xs )
         => ( ( ( size_s3818039357710166425rm_f_v @ Ys )
              = N )
           => ( ! [I: nat] :
                  ( ( ord_less_nat @ I @ N )
                 => ( R3 @ ( nth_Pr8768905099348956590rm_f_v @ Xs @ I ) @ ( nth_Pr8768905099348956590rm_f_v @ Ys @ I ) ) )
             => ( P2 @ Ys ) ) ) ) ) ) ).

% parallel_list_update
thf(fact_188_remove__nth__sound__l,axiom,
    ! [P2: nat,N: nat,Xs: list_term_f_v] :
      ( ( ord_less_nat @ P2 @ N )
     => ( ( nth_term_f_v @ ( missin2877041621942143146rm_f_v @ N @ Xs ) @ P2 )
        = ( nth_term_f_v @ Xs @ P2 ) ) ) ).

% remove_nth_sound_l
thf(fact_189_remove__nth__sound__l,axiom,
    ! [P2: nat,N: nat,Xs: list_P1780566427699531651term_f] :
      ( ( ord_less_nat @ P2 @ N )
     => ( ( nth_Pr7080171896067517188term_f @ ( missin5571439781501833334term_f @ N @ Xs ) @ P2 )
        = ( nth_Pr7080171896067517188term_f @ Xs @ P2 ) ) ) ).

% remove_nth_sound_l
thf(fact_190_remove__nth__sound__l,axiom,
    ! [P2: nat,N: nat,Xs: list_P4093298276913796397rm_f_v] :
      ( ( ord_less_nat @ P2 @ N )
     => ( ( nth_Pr8768905099348956590rm_f_v @ ( missin5548135175598359328rm_f_v @ N @ Xs ) @ P2 )
        = ( nth_Pr8768905099348956590rm_f_v @ Xs @ P2 ) ) ) ).

% remove_nth_sound_l
thf(fact_191_adjust__idx__rev__nth,axiom,
    ! [I4: nat,Xs: list_term_f_v,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
     => ( ( J != I4 )
       => ( ( nth_term_f_v @ ( missin2877041621942143146rm_f_v @ I4 @ Xs ) @ ( missin3815256168798769645dx_rev @ I4 @ J ) )
          = ( nth_term_f_v @ Xs @ J ) ) ) ) ).

% adjust_idx_rev_nth
thf(fact_192_adjust__idx__rev__nth,axiom,
    ! [I4: nat,Xs: list_P1780566427699531651term_f,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s9037951226665808879term_f @ Xs ) )
     => ( ( J != I4 )
       => ( ( nth_Pr7080171896067517188term_f @ ( missin5571439781501833334term_f @ I4 @ Xs ) @ ( missin3815256168798769645dx_rev @ I4 @ J ) )
          = ( nth_Pr7080171896067517188term_f @ Xs @ J ) ) ) ) ).

% adjust_idx_rev_nth
thf(fact_193_adjust__idx__rev__nth,axiom,
    ! [I4: nat,Xs: list_P4093298276913796397rm_f_v,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
     => ( ( J != I4 )
       => ( ( nth_Pr8768905099348956590rm_f_v @ ( missin5548135175598359328rm_f_v @ I4 @ Xs ) @ ( missin3815256168798769645dx_rev @ I4 @ J ) )
          = ( nth_Pr8768905099348956590rm_f_v @ Xs @ J ) ) ) ) ).

% adjust_idx_rev_nth
thf(fact_194_adjust__idx__nth,axiom,
    ! [I4: nat,Xs: list_term_f_v,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
     => ( ( nth_term_f_v @ ( missin2877041621942143146rm_f_v @ I4 @ Xs ) @ J )
        = ( nth_term_f_v @ Xs @ ( missing_adjust_idx @ I4 @ J ) ) ) ) ).

% adjust_idx_nth
thf(fact_195_adjust__idx__nth,axiom,
    ! [I4: nat,Xs: list_P1780566427699531651term_f,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s9037951226665808879term_f @ Xs ) )
     => ( ( nth_Pr7080171896067517188term_f @ ( missin5571439781501833334term_f @ I4 @ Xs ) @ J )
        = ( nth_Pr7080171896067517188term_f @ Xs @ ( missing_adjust_idx @ I4 @ J ) ) ) ) ).

% adjust_idx_nth
thf(fact_196_adjust__idx__nth,axiom,
    ! [I4: nat,Xs: list_P4093298276913796397rm_f_v,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
     => ( ( nth_Pr8768905099348956590rm_f_v @ ( missin5548135175598359328rm_f_v @ I4 @ Xs ) @ J )
        = ( nth_Pr8768905099348956590rm_f_v @ Xs @ ( missing_adjust_idx @ I4 @ J ) ) ) ) ).

% adjust_idx_nth
thf(fact_197_adjust__idx__rev__length,axiom,
    ! [I4: nat,Xs: list_term_f_v,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s7148729505879680775rm_f_v @ Xs ) )
       => ( ( J != I4 )
         => ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I4 @ J ) @ ( size_s7148729505879680775rm_f_v @ ( missin2877041621942143146rm_f_v @ I4 @ Xs ) ) ) ) ) ) ).

% adjust_idx_rev_length
thf(fact_198_adjust__idx__rev__length,axiom,
    ! [I4: nat,Xs: list_P1780566427699531651term_f,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s9037951226665808879term_f @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s9037951226665808879term_f @ Xs ) )
       => ( ( J != I4 )
         => ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I4 @ J ) @ ( size_s9037951226665808879term_f @ ( missin5571439781501833334term_f @ I4 @ Xs ) ) ) ) ) ) ).

% adjust_idx_rev_length
thf(fact_199_adjust__idx__rev__length,axiom,
    ! [I4: nat,Xs: list_P4093298276913796397rm_f_v,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s3818039357710166425rm_f_v @ Xs ) )
       => ( ( J != I4 )
         => ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I4 @ J ) @ ( size_s3818039357710166425rm_f_v @ ( missin5548135175598359328rm_f_v @ I4 @ Xs ) ) ) ) ) ) ).

% adjust_idx_rev_length
thf(fact_200_adjust__idx__length,axiom,
    ! [I4: nat,Xs: list_term_f_v,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s7148729505879680775rm_f_v @ ( missin2877041621942143146rm_f_v @ I4 @ Xs ) ) )
       => ( ord_less_nat @ ( missing_adjust_idx @ I4 @ J ) @ ( size_s7148729505879680775rm_f_v @ Xs ) ) ) ) ).

% adjust_idx_length
thf(fact_201_adjust__idx__length,axiom,
    ! [I4: nat,Xs: list_P1780566427699531651term_f,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s9037951226665808879term_f @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s9037951226665808879term_f @ ( missin5571439781501833334term_f @ I4 @ Xs ) ) )
       => ( ord_less_nat @ ( missing_adjust_idx @ I4 @ J ) @ ( size_s9037951226665808879term_f @ Xs ) ) ) ) ).

% adjust_idx_length
thf(fact_202_adjust__idx__length,axiom,
    ! [I4: nat,Xs: list_P4093298276913796397rm_f_v,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s3818039357710166425rm_f_v @ ( missin5548135175598359328rm_f_v @ I4 @ Xs ) ) )
       => ( ord_less_nat @ ( missing_adjust_idx @ I4 @ J ) @ ( size_s3818039357710166425rm_f_v @ Xs ) ) ) ) ).

% adjust_idx_length
thf(fact_203_listrel1__iff__update,axiom,
    ! [Xs: list_Bot_bot_term_f,Ys: list_Bot_bot_term_f,R3: set_Pr6841673932529763677term_f] :
      ( ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Xs @ Ys ) @ ( listre4162385684681681519term_f @ R3 ) )
      = ( ? [Y4: bot_bot_term_f,N2: nat] :
            ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( nth_Bot_bot_term_f @ Xs @ N2 ) @ Y4 ) @ R3 )
            & ( ord_less_nat @ N2 @ ( size_s1458857753309042354term_f @ Xs ) )
            & ( Ys
              = ( list_u5503441994469622688term_f @ Xs @ N2 @ Y4 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_204_listrel1__iff__update,axiom,
    ! [Xs: list_nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R3 ) )
      = ( ? [Y4: nat,N2: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ Y4 ) @ R3 )
            & ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
            & ( Ys
              = ( list_update_nat @ Xs @ N2 @ Y4 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_205_listrel1__iff__update,axiom,
    ! [Xs: list_term_f_v,Ys: list_term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Xs @ Ys ) @ ( listrel1_term_f_v @ R3 ) )
      = ( ? [Y4: term_f_v,N2: nat] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( nth_term_f_v @ Xs @ N2 ) @ Y4 ) @ R3 )
            & ( ord_less_nat @ N2 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
            & ( Ys
              = ( list_update_term_f_v @ Xs @ N2 @ Y4 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_206_listrel1__iff__update,axiom,
    ! [Xs: list_P1780566427699531651term_f,Ys: list_P1780566427699531651term_f,R3: set_Pr4896805198020829255term_f] :
      ( ( member2822224418761970704term_f @ ( produc1033786750570529367term_f @ Xs @ Ys ) @ ( listre5302397532715845724term_f @ R3 ) )
      = ( ? [Y4: produc250624644672406269term_f,N2: nat] :
            ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( nth_Pr7080171896067517188term_f @ Xs @ N2 ) @ Y4 ) @ R3 )
            & ( ord_less_nat @ N2 @ ( size_s9037951226665808879term_f @ Xs ) )
            & ( Ys
              = ( list_u2231791752812534891term_f @ Xs @ N2 @ Y4 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_207_listrel1__iff__update,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,Ys: list_P4093298276913796397rm_f_v,R3: set_Pr7897041799657265351rm_f_v] :
      ( ( member4621649979522838928rm_f_v @ ( produc3444105967875339479rm_f_v @ Xs @ Ys ) @ ( listre7866737091979268870rm_f_v @ R3 ) )
      = ( ? [Y4: produc8027700073345061287rm_f_v,N2: nat] :
            ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( nth_Pr8768905099348956590rm_f_v @ Xs @ N2 ) @ Y4 ) @ R3 )
            & ( ord_less_nat @ N2 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
            & ( Ys
              = ( list_u2658473446164013845rm_f_v @ Xs @ N2 @ Y4 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_208_trancl__list__conv,axiom,
    ! [Xs: list_Bot_bot_term_f,Ys: list_Bot_bot_term_f,R: set_Pr6841673932529763677term_f] :
      ( ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Xs @ Ys ) @ ( trancl6444742785402773766term_f @ R ) )
      = ( ( ( size_s1458857753309042354term_f @ Xs )
          = ( size_s1458857753309042354term_f @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1458857753309042354term_f @ Ys ) )
           => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( nth_Bot_bot_term_f @ Xs @ I3 ) @ ( nth_Bot_bot_term_f @ Ys @ I3 ) ) @ ( transi2774326432670905901term_f @ R ) ) ) ) ) ).

% trancl_list_conv
thf(fact_209_trancl__list__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( trancl_list_nat @ R ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ) ).

% trancl_list_conv
thf(fact_210_trancl__list__conv,axiom,
    ! [Xs: list_term_f_v,Ys: list_term_f_v,R: set_Pr8827868859434726151rm_f_v] :
      ( ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Xs @ Ys ) @ ( trancl_list_term_f_v @ R ) )
      = ( ( ( size_s7148729505879680775rm_f_v @ Xs )
          = ( size_s7148729505879680775rm_f_v @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7148729505879680775rm_f_v @ Ys ) )
           => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( nth_term_f_v @ Xs @ I3 ) @ ( nth_term_f_v @ Ys @ I3 ) ) @ ( transi3208867851621559554rm_f_v @ R ) ) ) ) ) ).

% trancl_list_conv
thf(fact_211_trancl__list__conv,axiom,
    ! [Xs: list_P1780566427699531651term_f,Ys: list_P1780566427699531651term_f,R: set_Pr4896805198020829255term_f] :
      ( ( member2822224418761970704term_f @ ( produc1033786750570529367term_f @ Xs @ Ys ) @ ( trancl1147503139693895301term_f @ R ) )
      = ( ( ( size_s9037951226665808879term_f @ Xs )
          = ( size_s9037951226665808879term_f @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s9037951226665808879term_f @ Ys ) )
           => ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( nth_Pr7080171896067517188term_f @ Xs @ I3 ) @ ( nth_Pr7080171896067517188term_f @ Ys @ I3 ) ) @ ( transi2747239000429906462term_f @ R ) ) ) ) ) ).

% trancl_list_conv
thf(fact_212_trancl__list__conv,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,Ys: list_P4093298276913796397rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
      ( ( member4621649979522838928rm_f_v @ ( produc3444105967875339479rm_f_v @ Xs @ Ys ) @ ( trancl7964251992849616175rm_f_v @ R ) )
      = ( ( ( size_s3818039357710166425rm_f_v @ Xs )
          = ( size_s3818039357710166425rm_f_v @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3818039357710166425rm_f_v @ Ys ) )
           => ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( nth_Pr8768905099348956590rm_f_v @ Xs @ I3 ) @ ( nth_Pr8768905099348956590rm_f_v @ Ys @ I3 ) ) @ ( transi7658437189833989320rm_f_v @ R ) ) ) ) ) ).

% trancl_list_conv
thf(fact_213_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R3 ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R3 ) ) ) ) ).

% listrel_iff_nth
thf(fact_214_listrel__iff__nth,axiom,
    ! [Xs: list_Bot_bot_term_f,Ys: list_Bot_bot_term_f,R3: set_Pr6841673932529763677term_f] :
      ( ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Xs @ Ys ) @ ( listre686400052008961423term_f @ R3 ) )
      = ( ( ( size_s1458857753309042354term_f @ Xs )
          = ( size_s1458857753309042354term_f @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s1458857753309042354term_f @ Xs ) )
           => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( nth_Bot_bot_term_f @ Xs @ N2 ) @ ( nth_Bot_bot_term_f @ Ys @ N2 ) ) @ R3 ) ) ) ) ).

% listrel_iff_nth
thf(fact_215_listrel__iff__nth,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat,R3: set_Pr9093778441882193744at_nat] :
      ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs @ Ys ) @ ( listrel_nat_nat_nat @ R3 ) )
      = ( ( ( size_s8208510060688613859at_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s8208510060688613859at_nat @ Xs ) )
           => ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ ( nth_nat_nat @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R3 ) ) ) ) ).

% listrel_iff_nth
thf(fact_216_listrel__iff__nth,axiom,
    ! [Xs: list_term_f_v,Ys: list_term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Xs @ Ys ) @ ( listre5907981802042383033rm_f_v @ R3 ) )
      = ( ( ( size_s7148729505879680775rm_f_v @ Xs )
          = ( size_s7148729505879680775rm_f_v @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
           => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( nth_term_f_v @ Xs @ N2 ) @ ( nth_term_f_v @ Ys @ N2 ) ) @ R3 ) ) ) ) ).

% listrel_iff_nth
thf(fact_217_listrel__iff__nth,axiom,
    ! [Xs: list_term_f_v,Ys: list_P1780566427699531651term_f,R3: set_Pr6532846849355489413term_f] :
      ( ( member6476877439930291548term_f @ ( produc1235233103486118263term_f @ Xs @ Ys ) @ ( listre3919867748209901351term_f @ R3 ) )
      = ( ( ( size_s7148729505879680775rm_f_v @ Xs )
          = ( size_s9037951226665808879term_f @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
           => ( member1278244878481305830term_f @ ( produc6247301185409243649term_f @ ( nth_term_f_v @ Xs @ N2 ) @ ( nth_Pr7080171896067517188term_f @ Ys @ N2 ) ) @ R3 ) ) ) ) ).

% listrel_iff_nth
thf(fact_218_listrel__iff__nth,axiom,
    ! [Xs: list_P1780566427699531651term_f,Ys: list_term_f_v,R3: set_Pr8897580718372814021rm_f_v] :
      ( ( member8709465672751168444rm_f_v @ ( produc7783624727839950495rm_f_v @ Xs @ Ys ) @ ( listre1924844212442861871rm_f_v @ R3 ) )
      = ( ( ( size_s9037951226665808879term_f @ Xs )
          = ( size_s7148729505879680775rm_f_v @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s9037951226665808879term_f @ Xs ) )
           => ( member9108189709895347494rm_f_v @ ( produc4252277649642204169rm_f_v @ ( nth_Pr7080171896067517188term_f @ Xs @ N2 ) @ ( nth_term_f_v @ Ys @ N2 ) ) @ R3 ) ) ) ) ).

% listrel_iff_nth
thf(fact_219_listrel__iff__nth,axiom,
    ! [Xs: list_term_f_v,Ys: list_P4093298276913796397rm_f_v,R3: set_Pr4810970677413006639rm_f_v] :
      ( ( member253218050144787974rm_f_v @ ( produc5335960931383885601rm_f_v @ Xs @ Ys ) @ ( listre8977894228209776081rm_f_v @ R3 ) )
      = ( ( ( size_s7148729505879680775rm_f_v @ Xs )
          = ( size_s3818039357710166425rm_f_v @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
           => ( member6048657017474400912rm_f_v @ ( produc8152674082280945835rm_f_v @ ( nth_term_f_v @ Xs @ N2 ) @ ( nth_Pr8768905099348956590rm_f_v @ Ys @ N2 ) ) @ R3 ) ) ) ) ).

% listrel_iff_nth
thf(fact_220_listrel__iff__nth,axiom,
    ! [Xs: list_P1780566427699531651term_f,Ys: list_P1780566427699531651term_f,R3: set_Pr4896805198020829255term_f] :
      ( ( member2822224418761970704term_f @ ( produc1033786750570529367term_f @ Xs @ Ys ) @ ( listre8631483066960881713term_f @ R3 ) )
      = ( ( ( size_s9037951226665808879term_f @ Xs )
          = ( size_s9037951226665808879term_f @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s9037951226665808879term_f @ Xs ) )
           => ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( nth_Pr7080171896067517188term_f @ Xs @ N2 ) @ ( nth_Pr7080171896067517188term_f @ Ys @ N2 ) ) @ R3 ) ) ) ) ).

% listrel_iff_nth
thf(fact_221_listrel__iff__nth,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,Ys: list_term_f_v,R3: set_Pr4552519640582707567rm_f_v] :
      ( ( member2857318640343734118rm_f_v @ ( produc6435190340566256201rm_f_v @ Xs @ Ys ) @ ( listre6193689439955784409rm_f_v @ R3 ) )
      = ( ( ( size_s3818039357710166425rm_f_v @ Xs )
          = ( size_s7148729505879680775rm_f_v @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
           => ( member6884080165642858192rm_f_v @ ( produc5368469294026954163rm_f_v @ ( nth_Pr8768905099348956590rm_f_v @ Xs @ N2 ) @ ( nth_term_f_v @ Ys @ N2 ) ) @ R3 ) ) ) ) ).

% listrel_iff_nth
thf(fact_222_listrel__iff__nth,axiom,
    ! [Xs: list_P1780566427699531651term_f,Ys: list_P4093298276913796397rm_f_v,R3: set_Pr4527924768817157361rm_f_v] :
      ( ( member3469830491719462074rm_f_v @ ( produc1826272848763574273rm_f_v @ Xs @ Ys ) @ ( listre6899962708045809371rm_f_v @ R3 ) )
      = ( ( ( size_s9037951226665808879term_f @ Xs )
          = ( size_s3818039357710166425rm_f_v @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s9037951226665808879term_f @ Xs ) )
           => ( member6153102960928152634rm_f_v @ ( produc2290582403270764673rm_f_v @ ( nth_Pr7080171896067517188term_f @ Xs @ N2 ) @ ( nth_Pr8768905099348956590rm_f_v @ Ys @ N2 ) ) @ R3 ) ) ) ) ).

% listrel_iff_nth
thf(fact_223_supteqp_Osimps,axiom,
    ( subter7567295712070293046qp_f_v
    = ( ^ [A12: term_f_v,A23: term_f_v] :
          ( ? [T2: term_f_v] :
              ( ( A12 = T2 )
              & ( A23 = T2 ) )
          | ? [U2: term_f_v,Ss2: list_term_f_v,T2: term_f_v,F2: f] :
              ( ( A12
                = ( fun_f_v @ F2 @ Ss2 ) )
              & ( A23 = T2 )
              & ( member_term_f_v @ U2 @ ( set_term_f_v2 @ Ss2 ) )
              & ( subter7567295712070293046qp_f_v @ U2 @ T2 ) ) ) ) ) ).

% supteqp.simps
thf(fact_224_trancl__supteq,axiom,
    ( ( transi3208867851621559554rm_f_v @ subter5033437232688913882eq_f_v )
    = subter5033437232688913882eq_f_v ) ).

% trancl_supteq
thf(fact_225_adjust__idx__rev2,axiom,
    ! [J: nat,I4: nat] :
      ( ( J != I4 )
     => ( ( missing_adjust_idx @ I4 @ ( missin3815256168798769645dx_rev @ I4 @ J ) )
        = J ) ) ).

% adjust_idx_rev2
thf(fact_226_adjust__idx__rev1,axiom,
    ! [I4: nat,J: nat] :
      ( ( missin3815256168798769645dx_rev @ I4 @ ( missing_adjust_idx @ I4 @ J ) )
      = J ) ).

% adjust_idx_rev1
thf(fact_227_adjust__idx__i,axiom,
    ! [I4: nat,J: nat] :
      ( ( missing_adjust_idx @ I4 @ J )
     != I4 ) ).

% adjust_idx_i
thf(fact_228_trancl__map,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,F: term_f_v > term_f_v,S: set_Pr8827868859434726151rm_f_v,X: term_f_v,Y: term_f_v] :
      ( ! [X4: term_f_v,Y3: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y3 ) @ R3 )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F @ X4 ) @ ( F @ Y3 ) ) @ S ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F @ X ) @ ( F @ Y ) ) @ ( transi3208867851621559554rm_f_v @ S ) ) ) ) ).

% trancl_map
thf(fact_229_trancl__map,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,F: term_f_v > bot_bot_term_f,S: set_Pr6841673932529763677term_f,X: term_f_v,Y: term_f_v] :
      ( ! [X4: term_f_v,Y3: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y3 ) @ R3 )
         => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( F @ X4 ) @ ( F @ Y3 ) ) @ S ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( F @ X ) @ ( F @ Y ) ) @ ( transi2774326432670905901term_f @ S ) ) ) ) ).

% trancl_map
thf(fact_230_trancl__map,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,F: term_f_v > nat,S: set_Pr1261947904930325089at_nat,X: term_f_v,Y: term_f_v] :
      ( ! [X4: term_f_v,Y3: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y3 ) @ R3 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) @ S ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y ) ) @ ( transi6264000038957366511cl_nat @ S ) ) ) ) ).

% trancl_map
thf(fact_231_trancl__map,axiom,
    ! [R3: set_Pr6841673932529763677term_f,F: bot_bot_term_f > term_f_v,S: set_Pr8827868859434726151rm_f_v,X: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ! [X4: bot_bot_term_f,Y3: bot_bot_term_f] :
          ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X4 @ Y3 ) @ R3 )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F @ X4 ) @ ( F @ Y3 ) ) @ S ) )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( transi2774326432670905901term_f @ R3 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F @ X ) @ ( F @ Y ) ) @ ( transi3208867851621559554rm_f_v @ S ) ) ) ) ).

% trancl_map
thf(fact_232_trancl__map,axiom,
    ! [R3: set_Pr6841673932529763677term_f,F: bot_bot_term_f > bot_bot_term_f,S: set_Pr6841673932529763677term_f,X: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ! [X4: bot_bot_term_f,Y3: bot_bot_term_f] :
          ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X4 @ Y3 ) @ R3 )
         => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( F @ X4 ) @ ( F @ Y3 ) ) @ S ) )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( transi2774326432670905901term_f @ R3 ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( F @ X ) @ ( F @ Y ) ) @ ( transi2774326432670905901term_f @ S ) ) ) ) ).

% trancl_map
thf(fact_233_trancl__map,axiom,
    ! [R3: set_Pr6841673932529763677term_f,F: bot_bot_term_f > nat,S: set_Pr1261947904930325089at_nat,X: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ! [X4: bot_bot_term_f,Y3: bot_bot_term_f] :
          ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X4 @ Y3 ) @ R3 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) @ S ) )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( transi2774326432670905901term_f @ R3 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y ) ) @ ( transi6264000038957366511cl_nat @ S ) ) ) ) ).

% trancl_map
thf(fact_234_trancl__map,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,F: nat > term_f_v,S: set_Pr8827868859434726151rm_f_v,X: nat,Y: nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R3 )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F @ X4 ) @ ( F @ Y3 ) ) @ S ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F @ X ) @ ( F @ Y ) ) @ ( transi3208867851621559554rm_f_v @ S ) ) ) ) ).

% trancl_map
thf(fact_235_trancl__map,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,F: nat > bot_bot_term_f,S: set_Pr6841673932529763677term_f,X: nat,Y: nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R3 )
         => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( F @ X4 ) @ ( F @ Y3 ) ) @ S ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( F @ X ) @ ( F @ Y ) ) @ ( transi2774326432670905901term_f @ S ) ) ) ) ).

% trancl_map
thf(fact_236_trancl__map,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,F: nat > nat,S: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R3 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) @ S ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y ) ) @ ( transi6264000038957366511cl_nat @ S ) ) ) ) ).

% trancl_map
thf(fact_237_listrel1__eq__len,axiom,
    ! [Xs: list_term_f_v,Ys: list_term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Xs @ Ys ) @ ( listrel1_term_f_v @ R3 ) )
     => ( ( size_s7148729505879680775rm_f_v @ Xs )
        = ( size_s7148729505879680775rm_f_v @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_238_listrel1__eq__len,axiom,
    ! [Xs: list_P1780566427699531651term_f,Ys: list_P1780566427699531651term_f,R3: set_Pr4896805198020829255term_f] :
      ( ( member2822224418761970704term_f @ ( produc1033786750570529367term_f @ Xs @ Ys ) @ ( listre5302397532715845724term_f @ R3 ) )
     => ( ( size_s9037951226665808879term_f @ Xs )
        = ( size_s9037951226665808879term_f @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_239_listrel1__eq__len,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,Ys: list_P4093298276913796397rm_f_v,R3: set_Pr7897041799657265351rm_f_v] :
      ( ( member4621649979522838928rm_f_v @ ( produc3444105967875339479rm_f_v @ Xs @ Ys ) @ ( listre7866737091979268870rm_f_v @ R3 ) )
     => ( ( size_s3818039357710166425rm_f_v @ Xs )
        = ( size_s3818039357710166425rm_f_v @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_240_listrel__eq__len,axiom,
    ! [Xs: list_term_f_v,Ys: list_term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Xs @ Ys ) @ ( listre5907981802042383033rm_f_v @ R3 ) )
     => ( ( size_s7148729505879680775rm_f_v @ Xs )
        = ( size_s7148729505879680775rm_f_v @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_241_listrel__eq__len,axiom,
    ! [Xs: list_term_f_v,Ys: list_P1780566427699531651term_f,R3: set_Pr6532846849355489413term_f] :
      ( ( member6476877439930291548term_f @ ( produc1235233103486118263term_f @ Xs @ Ys ) @ ( listre3919867748209901351term_f @ R3 ) )
     => ( ( size_s7148729505879680775rm_f_v @ Xs )
        = ( size_s9037951226665808879term_f @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_242_listrel__eq__len,axiom,
    ! [Xs: list_term_f_v,Ys: list_P4093298276913796397rm_f_v,R3: set_Pr4810970677413006639rm_f_v] :
      ( ( member253218050144787974rm_f_v @ ( produc5335960931383885601rm_f_v @ Xs @ Ys ) @ ( listre8977894228209776081rm_f_v @ R3 ) )
     => ( ( size_s7148729505879680775rm_f_v @ Xs )
        = ( size_s3818039357710166425rm_f_v @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_243_listrel__eq__len,axiom,
    ! [Xs: list_P1780566427699531651term_f,Ys: list_term_f_v,R3: set_Pr8897580718372814021rm_f_v] :
      ( ( member8709465672751168444rm_f_v @ ( produc7783624727839950495rm_f_v @ Xs @ Ys ) @ ( listre1924844212442861871rm_f_v @ R3 ) )
     => ( ( size_s9037951226665808879term_f @ Xs )
        = ( size_s7148729505879680775rm_f_v @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_244_listrel__eq__len,axiom,
    ! [Xs: list_P1780566427699531651term_f,Ys: list_P1780566427699531651term_f,R3: set_Pr4896805198020829255term_f] :
      ( ( member2822224418761970704term_f @ ( produc1033786750570529367term_f @ Xs @ Ys ) @ ( listre8631483066960881713term_f @ R3 ) )
     => ( ( size_s9037951226665808879term_f @ Xs )
        = ( size_s9037951226665808879term_f @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_245_listrel__eq__len,axiom,
    ! [Xs: list_P1780566427699531651term_f,Ys: list_P4093298276913796397rm_f_v,R3: set_Pr4527924768817157361rm_f_v] :
      ( ( member3469830491719462074rm_f_v @ ( produc1826272848763574273rm_f_v @ Xs @ Ys ) @ ( listre6899962708045809371rm_f_v @ R3 ) )
     => ( ( size_s9037951226665808879term_f @ Xs )
        = ( size_s3818039357710166425rm_f_v @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_246_listrel__eq__len,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,Ys: list_term_f_v,R3: set_Pr4552519640582707567rm_f_v] :
      ( ( member2857318640343734118rm_f_v @ ( produc6435190340566256201rm_f_v @ Xs @ Ys ) @ ( listre6193689439955784409rm_f_v @ R3 ) )
     => ( ( size_s3818039357710166425rm_f_v @ Xs )
        = ( size_s7148729505879680775rm_f_v @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_247_listrel__eq__len,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,Ys: list_P1780566427699531651term_f,R3: set_Pr7007807420363213853term_f] :
      ( ( member7203770481494966502term_f @ ( produc8962910302844323629term_f @ Xs @ Ys ) @ ( listre7107951199197490951term_f @ R3 ) )
     => ( ( size_s3818039357710166425rm_f_v @ Xs )
        = ( size_s9037951226665808879term_f @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_248_listrel__eq__len,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,Ys: list_P4093298276913796397rm_f_v,R3: set_Pr7897041799657265351rm_f_v] :
      ( ( member4621649979522838928rm_f_v @ ( produc3444105967875339479rm_f_v @ Xs @ Ys ) @ ( listre6309640189867419057rm_f_v @ R3 ) )
     => ( ( size_s3818039357710166425rm_f_v @ Xs )
        = ( size_s3818039357710166425rm_f_v @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_249_trancl__list__induct,axiom,
    ! [Ss: list_Bot_bot_term_f,Ts: list_Bot_bot_term_f,R: set_Pr6841673932529763677term_f,P: list_Bot_bot_term_f > list_Bot_bot_term_f > $o] :
      ( ( ( size_s1458857753309042354term_f @ Ss )
        = ( size_s1458857753309042354term_f @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s1458857753309042354term_f @ Ts ) )
           => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( nth_Bot_bot_term_f @ Ss @ I ) @ ( nth_Bot_bot_term_f @ Ts @ I ) ) @ ( transi2774326432670905901term_f @ R ) ) )
       => ( ! [Xs3: list_Bot_bot_term_f,Ys4: list_Bot_bot_term_f] :
              ( ( ( size_s1458857753309042354term_f @ Xs3 )
                = ( size_s1458857753309042354term_f @ Ys4 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( size_s1458857753309042354term_f @ Ys4 ) )
                   => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( nth_Bot_bot_term_f @ Xs3 @ I2 ) @ ( nth_Bot_bot_term_f @ Ys4 @ I2 ) ) @ R ) )
               => ( P @ Xs3 @ Ys4 ) ) )
         => ( ! [Xs3: list_Bot_bot_term_f,Ys4: list_Bot_bot_term_f,I: nat,Z3: bot_bot_term_f] :
                ( ( ( size_s1458857753309042354term_f @ Xs3 )
                  = ( size_s1458857753309042354term_f @ Ys4 ) )
               => ( ! [Ia: nat] :
                      ( ( ord_less_nat @ Ia @ ( size_s1458857753309042354term_f @ Ys4 ) )
                     => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( nth_Bot_bot_term_f @ Xs3 @ Ia ) @ ( nth_Bot_bot_term_f @ Ys4 @ Ia ) ) @ ( transi2774326432670905901term_f @ R ) ) )
                 => ( ( P @ Xs3 @ Ys4 )
                   => ( ( ord_less_nat @ I @ ( size_s1458857753309042354term_f @ Ys4 ) )
                     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( nth_Bot_bot_term_f @ Ys4 @ I ) @ Z3 ) @ R )
                       => ( P @ Xs3 @ ( list_u5503441994469622688term_f @ Ys4 @ I @ Z3 ) ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% trancl_list_induct
thf(fact_250_trancl__list__induct,axiom,
    ! [Ss: list_nat,Ts: list_nat,R: set_Pr1261947904930325089at_nat,P: list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Ss )
        = ( size_size_list_nat @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ts ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ss @ I ) @ ( nth_nat @ Ts @ I ) ) @ ( transi6264000038957366511cl_nat @ R ) ) )
       => ( ! [Xs3: list_nat,Ys4: list_nat] :
              ( ( ( size_size_list_nat @ Xs3 )
                = ( size_size_list_nat @ Ys4 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys4 ) )
                   => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs3 @ I2 ) @ ( nth_nat @ Ys4 @ I2 ) ) @ R ) )
               => ( P @ Xs3 @ Ys4 ) ) )
         => ( ! [Xs3: list_nat,Ys4: list_nat,I: nat,Z3: nat] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ! [Ia: nat] :
                      ( ( ord_less_nat @ Ia @ ( size_size_list_nat @ Ys4 ) )
                     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs3 @ Ia ) @ ( nth_nat @ Ys4 @ Ia ) ) @ ( transi6264000038957366511cl_nat @ R ) ) )
                 => ( ( P @ Xs3 @ Ys4 )
                   => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys4 ) )
                     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys4 @ I ) @ Z3 ) @ R )
                       => ( P @ Xs3 @ ( list_update_nat @ Ys4 @ I @ Z3 ) ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% trancl_list_induct
thf(fact_251_trancl__list__induct,axiom,
    ! [Ss: list_term_f_v,Ts: list_term_f_v,R: set_Pr8827868859434726151rm_f_v,P: list_term_f_v > list_term_f_v > $o] :
      ( ( ( size_s7148729505879680775rm_f_v @ Ss )
        = ( size_s7148729505879680775rm_f_v @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s7148729505879680775rm_f_v @ Ts ) )
           => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( nth_term_f_v @ Ss @ I ) @ ( nth_term_f_v @ Ts @ I ) ) @ ( transi3208867851621559554rm_f_v @ R ) ) )
       => ( ! [Xs3: list_term_f_v,Ys4: list_term_f_v] :
              ( ( ( size_s7148729505879680775rm_f_v @ Xs3 )
                = ( size_s7148729505879680775rm_f_v @ Ys4 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( size_s7148729505879680775rm_f_v @ Ys4 ) )
                   => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( nth_term_f_v @ Xs3 @ I2 ) @ ( nth_term_f_v @ Ys4 @ I2 ) ) @ R ) )
               => ( P @ Xs3 @ Ys4 ) ) )
         => ( ! [Xs3: list_term_f_v,Ys4: list_term_f_v,I: nat,Z3: term_f_v] :
                ( ( ( size_s7148729505879680775rm_f_v @ Xs3 )
                  = ( size_s7148729505879680775rm_f_v @ Ys4 ) )
               => ( ! [Ia: nat] :
                      ( ( ord_less_nat @ Ia @ ( size_s7148729505879680775rm_f_v @ Ys4 ) )
                     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( nth_term_f_v @ Xs3 @ Ia ) @ ( nth_term_f_v @ Ys4 @ Ia ) ) @ ( transi3208867851621559554rm_f_v @ R ) ) )
                 => ( ( P @ Xs3 @ Ys4 )
                   => ( ( ord_less_nat @ I @ ( size_s7148729505879680775rm_f_v @ Ys4 ) )
                     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( nth_term_f_v @ Ys4 @ I ) @ Z3 ) @ R )
                       => ( P @ Xs3 @ ( list_update_term_f_v @ Ys4 @ I @ Z3 ) ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% trancl_list_induct
thf(fact_252_trancl__list__induct,axiom,
    ! [Ss: list_P1780566427699531651term_f,Ts: list_P1780566427699531651term_f,R: set_Pr4896805198020829255term_f,P: list_P1780566427699531651term_f > list_P1780566427699531651term_f > $o] :
      ( ( ( size_s9037951226665808879term_f @ Ss )
        = ( size_s9037951226665808879term_f @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s9037951226665808879term_f @ Ts ) )
           => ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( nth_Pr7080171896067517188term_f @ Ss @ I ) @ ( nth_Pr7080171896067517188term_f @ Ts @ I ) ) @ ( transi2747239000429906462term_f @ R ) ) )
       => ( ! [Xs3: list_P1780566427699531651term_f,Ys4: list_P1780566427699531651term_f] :
              ( ( ( size_s9037951226665808879term_f @ Xs3 )
                = ( size_s9037951226665808879term_f @ Ys4 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( size_s9037951226665808879term_f @ Ys4 ) )
                   => ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( nth_Pr7080171896067517188term_f @ Xs3 @ I2 ) @ ( nth_Pr7080171896067517188term_f @ Ys4 @ I2 ) ) @ R ) )
               => ( P @ Xs3 @ Ys4 ) ) )
         => ( ! [Xs3: list_P1780566427699531651term_f,Ys4: list_P1780566427699531651term_f,I: nat,Z3: produc250624644672406269term_f] :
                ( ( ( size_s9037951226665808879term_f @ Xs3 )
                  = ( size_s9037951226665808879term_f @ Ys4 ) )
               => ( ! [Ia: nat] :
                      ( ( ord_less_nat @ Ia @ ( size_s9037951226665808879term_f @ Ys4 ) )
                     => ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( nth_Pr7080171896067517188term_f @ Xs3 @ Ia ) @ ( nth_Pr7080171896067517188term_f @ Ys4 @ Ia ) ) @ ( transi2747239000429906462term_f @ R ) ) )
                 => ( ( P @ Xs3 @ Ys4 )
                   => ( ( ord_less_nat @ I @ ( size_s9037951226665808879term_f @ Ys4 ) )
                     => ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( nth_Pr7080171896067517188term_f @ Ys4 @ I ) @ Z3 ) @ R )
                       => ( P @ Xs3 @ ( list_u2231791752812534891term_f @ Ys4 @ I @ Z3 ) ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% trancl_list_induct
thf(fact_253_trancl__list__induct,axiom,
    ! [Ss: list_P4093298276913796397rm_f_v,Ts: list_P4093298276913796397rm_f_v,R: set_Pr7897041799657265351rm_f_v,P: list_P4093298276913796397rm_f_v > list_P4093298276913796397rm_f_v > $o] :
      ( ( ( size_s3818039357710166425rm_f_v @ Ss )
        = ( size_s3818039357710166425rm_f_v @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3818039357710166425rm_f_v @ Ts ) )
           => ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( nth_Pr8768905099348956590rm_f_v @ Ss @ I ) @ ( nth_Pr8768905099348956590rm_f_v @ Ts @ I ) ) @ ( transi7658437189833989320rm_f_v @ R ) ) )
       => ( ! [Xs3: list_P4093298276913796397rm_f_v,Ys4: list_P4093298276913796397rm_f_v] :
              ( ( ( size_s3818039357710166425rm_f_v @ Xs3 )
                = ( size_s3818039357710166425rm_f_v @ Ys4 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( size_s3818039357710166425rm_f_v @ Ys4 ) )
                   => ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( nth_Pr8768905099348956590rm_f_v @ Xs3 @ I2 ) @ ( nth_Pr8768905099348956590rm_f_v @ Ys4 @ I2 ) ) @ R ) )
               => ( P @ Xs3 @ Ys4 ) ) )
         => ( ! [Xs3: list_P4093298276913796397rm_f_v,Ys4: list_P4093298276913796397rm_f_v,I: nat,Z3: produc8027700073345061287rm_f_v] :
                ( ( ( size_s3818039357710166425rm_f_v @ Xs3 )
                  = ( size_s3818039357710166425rm_f_v @ Ys4 ) )
               => ( ! [Ia: nat] :
                      ( ( ord_less_nat @ Ia @ ( size_s3818039357710166425rm_f_v @ Ys4 ) )
                     => ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( nth_Pr8768905099348956590rm_f_v @ Xs3 @ Ia ) @ ( nth_Pr8768905099348956590rm_f_v @ Ys4 @ Ia ) ) @ ( transi7658437189833989320rm_f_v @ R ) ) )
                 => ( ( P @ Xs3 @ Ys4 )
                   => ( ( ord_less_nat @ I @ ( size_s3818039357710166425rm_f_v @ Ys4 ) )
                     => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( nth_Pr8768905099348956590rm_f_v @ Ys4 @ I ) @ Z3 ) @ R )
                       => ( P @ Xs3 @ ( list_u2658473446164013845rm_f_v @ Ys4 @ I @ Z3 ) ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% trancl_list_induct
thf(fact_254_suptp_Osimps,axiom,
    ( subter3995562164930979714tp_f_v
    = ( ^ [A12: term_f_v,A23: term_f_v] :
          ( ? [S2: term_f_v,Ss2: list_term_f_v,F2: f] :
              ( ( A12
                = ( fun_f_v @ F2 @ Ss2 ) )
              & ( A23 = S2 )
              & ( member_term_f_v @ S2 @ ( set_term_f_v2 @ Ss2 ) ) )
          | ? [S2: term_f_v,Ss2: list_term_f_v,T2: term_f_v,F2: f] :
              ( ( A12
                = ( fun_f_v @ F2 @ Ss2 ) )
              & ( A23 = T2 )
              & ( member_term_f_v @ S2 @ ( set_term_f_v2 @ Ss2 ) )
              & ( subter3995562164930979714tp_f_v @ S2 @ T2 ) ) ) ) ) ).

% suptp.simps
thf(fact_255_converse__trancl__induct,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v,P: term_f_v > $o] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
     => ( ! [Y3: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ B ) @ R3 )
           => ( P @ Y3 ) )
       => ( ! [Y3: term_f_v,Z3: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z3 ) @ R3 )
             => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z3 @ B ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
               => ( ( P @ Z3 )
                 => ( P @ Y3 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_256_converse__trancl__induct,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R3: set_Pr6841673932529763677term_f,P: bot_bot_term_f > $o] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi2774326432670905901term_f @ R3 ) )
     => ( ! [Y3: bot_bot_term_f] :
            ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y3 @ B ) @ R3 )
           => ( P @ Y3 ) )
       => ( ! [Y3: bot_bot_term_f,Z3: bot_bot_term_f] :
              ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y3 @ Z3 ) @ R3 )
             => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Z3 @ B ) @ ( transi2774326432670905901term_f @ R3 ) )
               => ( ( P @ Z3 )
                 => ( P @ Y3 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_257_converse__trancl__induct,axiom,
    ! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,P: nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R3 ) )
     => ( ! [Y3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ B ) @ R3 )
           => ( P @ Y3 ) )
       => ( ! [Y3: nat,Z3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z3 ) @ R3 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z3 @ B ) @ ( transi6264000038957366511cl_nat @ R3 ) )
               => ( ( P @ Z3 )
                 => ( P @ Y3 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_258_trancl__trans__induct,axiom,
    ! [X: term_f_v,Y: term_f_v,R3: set_Pr8827868859434726151rm_f_v,P: term_f_v > term_f_v > $o] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
     => ( ! [X4: term_f_v,Y3: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y3 ) @ R3 )
           => ( P @ X4 @ Y3 ) )
       => ( ! [X4: term_f_v,Y3: term_f_v,Z3: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y3 ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
             => ( ( P @ X4 @ Y3 )
               => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z3 ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
                 => ( ( P @ Y3 @ Z3 )
                   => ( P @ X4 @ Z3 ) ) ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_259_trancl__trans__induct,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,R3: set_Pr6841673932529763677term_f,P: bot_bot_term_f > bot_bot_term_f > $o] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( transi2774326432670905901term_f @ R3 ) )
     => ( ! [X4: bot_bot_term_f,Y3: bot_bot_term_f] :
            ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X4 @ Y3 ) @ R3 )
           => ( P @ X4 @ Y3 ) )
       => ( ! [X4: bot_bot_term_f,Y3: bot_bot_term_f,Z3: bot_bot_term_f] :
              ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X4 @ Y3 ) @ ( transi2774326432670905901term_f @ R3 ) )
             => ( ( P @ X4 @ Y3 )
               => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y3 @ Z3 ) @ ( transi2774326432670905901term_f @ R3 ) )
                 => ( ( P @ Y3 @ Z3 )
                   => ( P @ X4 @ Z3 ) ) ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_260_trancl__trans__induct,axiom,
    ! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat,P: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) )
     => ( ! [X4: nat,Y3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R3 )
           => ( P @ X4 @ Y3 ) )
       => ( ! [X4: nat,Y3: nat,Z3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ ( transi6264000038957366511cl_nat @ R3 ) )
             => ( ( P @ X4 @ Y3 )
               => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z3 ) @ ( transi6264000038957366511cl_nat @ R3 ) )
                 => ( ( P @ Y3 @ Z3 )
                   => ( P @ X4 @ Z3 ) ) ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_261_trancl__into__trancl2,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R3 )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3208867851621559554rm_f_v @ R3 ) ) ) ) ).

% trancl_into_trancl2
thf(fact_262_trancl__into__trancl2,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R3: set_Pr6841673932529763677term_f,C: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ R3 )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B @ C ) @ ( transi2774326432670905901term_f @ R3 ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ C ) @ ( transi2774326432670905901term_f @ R3 ) ) ) ) ).

% trancl_into_trancl2
thf(fact_263_trancl__into__trancl2,axiom,
    ! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,C: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( transi6264000038957366511cl_nat @ R3 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ).

% trancl_into_trancl2
thf(fact_264_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ R3 )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3208867851621559554rm_f_v @ R3 ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_265_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R3: set_Pr6841673932529763677term_f,C: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi2774326432670905901term_f @ R3 ) )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B @ C ) @ R3 )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ C ) @ ( transi2774326432670905901term_f @ R3 ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_266_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,C: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R3 ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ R3 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_267_irrefl__trancl__rD,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,X: term_f_v,Y: term_f_v] :
      ( ! [X4: term_f_v] :
          ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ X4 ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ R3 )
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_268_irrefl__trancl__rD,axiom,
    ! [R3: set_Pr6841673932529763677term_f,X: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ! [X4: bot_bot_term_f] :
          ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X4 @ X4 ) @ ( transi2774326432670905901term_f @ R3 ) )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ R3 )
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_269_irrefl__trancl__rD,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ! [X4: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ ( transi6264000038957366511cl_nat @ R3 ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 )
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_270_converse__tranclE,axiom,
    ! [X: term_f_v,Z: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Z ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
     => ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Z ) @ R3 )
       => ~ ! [Y3: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R3 )
             => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z ) @ ( transi3208867851621559554rm_f_v @ R3 ) ) ) ) ) ).

% converse_tranclE
thf(fact_271_converse__tranclE,axiom,
    ! [X: bot_bot_term_f,Z: bot_bot_term_f,R3: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Z ) @ ( transi2774326432670905901term_f @ R3 ) )
     => ( ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Z ) @ R3 )
       => ~ ! [Y3: bot_bot_term_f] :
              ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y3 ) @ R3 )
             => ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y3 @ Z ) @ ( transi2774326432670905901term_f @ R3 ) ) ) ) ) ).

% converse_tranclE
thf(fact_272_converse__tranclE,axiom,
    ! [X: nat,Z: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi6264000038957366511cl_nat @ R3 ) )
     => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R3 )
       => ~ ! [Y3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R3 )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ) ).

% converse_tranclE
thf(fact_273_r__r__into__trancl,axiom,
    ! [A: term_f_v,B: term_f_v,R4: set_Pr8827868859434726151rm_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R4 )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ R4 )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3208867851621559554rm_f_v @ R4 ) ) ) ) ).

% r_r_into_trancl
thf(fact_274_r__r__into__trancl,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R4: set_Pr6841673932529763677term_f,C: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ R4 )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B @ C ) @ R4 )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ C ) @ ( transi2774326432670905901term_f @ R4 ) ) ) ) ).

% r_r_into_trancl
thf(fact_275_r__r__into__trancl,axiom,
    ! [A: nat,B: nat,R4: set_Pr1261947904930325089at_nat,C: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ R4 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C ) @ ( transi6264000038957366511cl_nat @ R4 ) ) ) ) ).

% r_r_into_trancl
thf(fact_276_trancl__induct,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v,P: term_f_v > $o] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
     => ( ! [Y3: term_f_v] :
            ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ Y3 ) @ R3 )
           => ( P @ Y3 ) )
       => ( ! [Y3: term_f_v,Z3: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ Y3 ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
             => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z3 ) @ R3 )
               => ( ( P @ Y3 )
                 => ( P @ Z3 ) ) ) )
         => ( P @ B ) ) ) ) ).

% trancl_induct
thf(fact_277_trancl__induct,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R3: set_Pr6841673932529763677term_f,P: bot_bot_term_f > $o] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi2774326432670905901term_f @ R3 ) )
     => ( ! [Y3: bot_bot_term_f] :
            ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ Y3 ) @ R3 )
           => ( P @ Y3 ) )
       => ( ! [Y3: bot_bot_term_f,Z3: bot_bot_term_f] :
              ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ Y3 ) @ ( transi2774326432670905901term_f @ R3 ) )
             => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y3 @ Z3 ) @ R3 )
               => ( ( P @ Y3 )
                 => ( P @ Z3 ) ) ) )
         => ( P @ B ) ) ) ) ).

% trancl_induct
thf(fact_278_trancl__induct,axiom,
    ! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,P: nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R3 ) )
     => ( ! [Y3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ R3 )
           => ( P @ Y3 ) )
       => ( ! [Y3: nat,Z3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ ( transi6264000038957366511cl_nat @ R3 ) )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z3 ) @ R3 )
               => ( ( P @ Y3 )
                 => ( P @ Z3 ) ) ) )
         => ( P @ B ) ) ) ) ).

% trancl_induct
thf(fact_279_trancl__trans,axiom,
    ! [X: term_f_v,Y: term_f_v,R3: set_Pr8827868859434726151rm_f_v,Z: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Z ) @ ( transi3208867851621559554rm_f_v @ R3 ) ) ) ) ).

% trancl_trans
thf(fact_280_trancl__trans,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,R3: set_Pr6841673932529763677term_f,Z: bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( transi2774326432670905901term_f @ R3 ) )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y @ Z ) @ ( transi2774326432670905901term_f @ R3 ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Z ) @ ( transi2774326432670905901term_f @ R3 ) ) ) ) ).

% trancl_trans
thf(fact_281_trancl__trans,axiom,
    ! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat,Z: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ ( transi6264000038957366511cl_nat @ R3 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ).

% trancl_trans
thf(fact_282_trancl__induct2,axiom,
    ! [Ax: term_f_v,Ay: term_f_v,Bx: term_f_v,By: term_f_v,R3: set_Pr7897041799657265351rm_f_v,P: term_f_v > term_f_v > $o] :
      ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Ax @ Ay ) @ ( produc2510790566102465375rm_f_v @ Bx @ By ) ) @ ( transi7658437189833989320rm_f_v @ R3 ) )
     => ( ! [A3: term_f_v,B3: term_f_v] :
            ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Ax @ Ay ) @ ( produc2510790566102465375rm_f_v @ A3 @ B3 ) ) @ R3 )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: term_f_v,B3: term_f_v,Aa: term_f_v,Ba: term_f_v] :
              ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Ax @ Ay ) @ ( produc2510790566102465375rm_f_v @ A3 @ B3 ) ) @ ( transi7658437189833989320rm_f_v @ R3 ) )
             => ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ A3 @ B3 ) @ ( produc2510790566102465375rm_f_v @ Aa @ Ba ) ) @ R3 )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_283_trancl__induct2,axiom,
    ! [Ax: bot_bot_term_f,Ay: bot_bot_term_f,Bx: bot_bot_term_f,By: bot_bot_term_f,R3: set_Pr4896805198020829255term_f,P: bot_bot_term_f > bot_bot_term_f > $o] :
      ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( produc2865564700782054965term_f @ Ax @ Ay ) @ ( produc2865564700782054965term_f @ Bx @ By ) ) @ ( transi2747239000429906462term_f @ R3 ) )
     => ( ! [A3: bot_bot_term_f,B3: bot_bot_term_f] :
            ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( produc2865564700782054965term_f @ Ax @ Ay ) @ ( produc2865564700782054965term_f @ A3 @ B3 ) ) @ R3 )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: bot_bot_term_f,B3: bot_bot_term_f,Aa: bot_bot_term_f,Ba: bot_bot_term_f] :
              ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( produc2865564700782054965term_f @ Ax @ Ay ) @ ( produc2865564700782054965term_f @ A3 @ B3 ) ) @ ( transi2747239000429906462term_f @ R3 ) )
             => ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( produc2865564700782054965term_f @ A3 @ B3 ) @ ( produc2865564700782054965term_f @ Aa @ Ba ) ) @ R3 )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_284_trancl__induct2,axiom,
    ! [Ax: nat > nat,Ay: nat,Bx: nat > nat,By: nat,R3: set_Pr7047708467701517959at_nat,P: ( nat > nat ) > nat > $o] :
      ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ Bx @ By ) ) @ ( transi3146596849628481425at_nat @ R3 ) )
     => ( ! [A3: nat > nat,B3: nat] :
            ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ A3 @ B3 ) ) @ R3 )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: nat > nat,B3: nat,Aa: nat > nat,Ba: nat] :
              ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ A3 @ B3 ) ) @ ( transi3146596849628481425at_nat @ R3 ) )
             => ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ A3 @ B3 ) @ ( produc72220940542539688at_nat @ Aa @ Ba ) ) @ R3 )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_285_trancl__induct2,axiom,
    ! [Ax: nat,Ay: nat,Bx: nat,By: nat,R3: set_Pr8693737435421807431at_nat,P: nat > nat > $o] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi243908449541399842at_nat @ R3 ) )
     => ( ! [A3: nat,B3: nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A3 @ B3 ) ) @ R3 )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: nat,B3: nat,Aa: nat,Ba: nat] :
              ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A3 @ B3 ) ) @ ( transi243908449541399842at_nat @ R3 ) )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R3 )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_286_r__into__trancl_H,axiom,
    ! [P2: produc8027700073345061287rm_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ P2 @ R3 )
     => ( member4316649861536941904rm_f_v @ P2 @ ( transi3208867851621559554rm_f_v @ R3 ) ) ) ).

% r_into_trancl'
thf(fact_287_r__into__trancl_H,axiom,
    ! [P2: produc250624644672406269term_f,R3: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ P2 @ R3 )
     => ( member4687689947140789414term_f @ P2 @ ( transi2774326432670905901term_f @ R3 ) ) ) ).

% r_into_trancl'
thf(fact_288_trancl_Ocases,axiom,
    ! [A1: term_f_v,A22: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
     => ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ R3 )
       => ~ ! [B3: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ B3 ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
             => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B3 @ A22 ) @ R3 ) ) ) ) ).

% trancl.cases
thf(fact_289_trancl_Ocases,axiom,
    ! [A1: bot_bot_term_f,A22: bot_bot_term_f,R3: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A1 @ A22 ) @ ( transi2774326432670905901term_f @ R3 ) )
     => ( ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A1 @ A22 ) @ R3 )
       => ~ ! [B3: bot_bot_term_f] :
              ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A1 @ B3 ) @ ( transi2774326432670905901term_f @ R3 ) )
             => ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B3 @ A22 ) @ R3 ) ) ) ) ).

% trancl.cases
thf(fact_290_trancl_Ocases,axiom,
    ! [A1: nat,A22: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi6264000038957366511cl_nat @ R3 ) )
     => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ R3 )
       => ~ ! [B3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B3 ) @ ( transi6264000038957366511cl_nat @ R3 ) )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ R3 ) ) ) ) ).

% trancl.cases
thf(fact_291_trancl_Osimps,axiom,
    ! [A1: term_f_v,A22: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
      = ( ? [A5: term_f_v,B4: term_f_v] :
            ( ( A1 = A5 )
            & ( A22 = B4 )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A5 @ B4 ) @ R3 ) )
        | ? [A5: term_f_v,B4: term_f_v,C2: term_f_v] :
            ( ( A1 = A5 )
            & ( A22 = C2 )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A5 @ B4 ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B4 @ C2 ) @ R3 ) ) ) ) ).

% trancl.simps
thf(fact_292_trancl_Osimps,axiom,
    ! [A1: bot_bot_term_f,A22: bot_bot_term_f,R3: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A1 @ A22 ) @ ( transi2774326432670905901term_f @ R3 ) )
      = ( ? [A5: bot_bot_term_f,B4: bot_bot_term_f] :
            ( ( A1 = A5 )
            & ( A22 = B4 )
            & ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A5 @ B4 ) @ R3 ) )
        | ? [A5: bot_bot_term_f,B4: bot_bot_term_f,C2: bot_bot_term_f] :
            ( ( A1 = A5 )
            & ( A22 = C2 )
            & ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A5 @ B4 ) @ ( transi2774326432670905901term_f @ R3 ) )
            & ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ B4 @ C2 ) @ R3 ) ) ) ) ).

% trancl.simps
thf(fact_293_trancl_Osimps,axiom,
    ! [A1: nat,A22: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi6264000038957366511cl_nat @ R3 ) )
      = ( ? [A5: nat,B4: nat] :
            ( ( A1 = A5 )
            & ( A22 = B4 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B4 ) @ R3 ) )
        | ? [A5: nat,B4: nat,C2: nat] :
            ( ( A1 = A5 )
            & ( A22 = C2 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B4 ) @ ( transi6264000038957366511cl_nat @ R3 ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C2 ) @ R3 ) ) ) ) ).

% trancl.simps
thf(fact_294_trancl_Or__into__trancl,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R3 )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R3 ) ) ) ).

% trancl.r_into_trancl
thf(fact_295_trancl_Or__into__trancl,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R3: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ R3 )
     => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi2774326432670905901term_f @ R3 ) ) ) ).

% trancl.r_into_trancl
thf(fact_296_trancl_Or__into__trancl,axiom,
    ! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ).

% trancl.r_into_trancl
thf(fact_297_tranclE,axiom,
    ! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
     => ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R3 )
       => ~ ! [C3: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
             => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C3 @ B ) @ R3 ) ) ) ) ).

% tranclE
thf(fact_298_tranclE,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,R3: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( transi2774326432670905901term_f @ R3 ) )
     => ( ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ B ) @ R3 )
       => ~ ! [C3: bot_bot_term_f] :
              ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ A @ C3 ) @ ( transi2774326432670905901term_f @ R3 ) )
             => ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ C3 @ B ) @ R3 ) ) ) ) ).

% tranclE
thf(fact_299_tranclE,axiom,
    ! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R3 ) )
     => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
       => ~ ! [C3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi6264000038957366511cl_nat @ R3 ) )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ C3 @ B ) @ R3 ) ) ) ) ).

% tranclE
thf(fact_300_trancl__list__append__tranclI,axiom,
    ! [X: term_f_v,Y: term_f_v,R: set_Pr8827868859434726151rm_f_v,Xs: list_term_f_v,Ys: list_term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( transi3208867851621559554rm_f_v @ R ) )
     => ( ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Xs @ Ys ) @ ( trancl_list_term_f_v @ R ) )
       => ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ ( cons_term_f_v @ X @ Xs ) @ ( cons_term_f_v @ Y @ Ys ) ) @ ( trancl_list_term_f_v @ R ) ) ) ) ).

% trancl_list_append_tranclI
thf(fact_301_trancl__list__append__tranclI,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,R: set_Pr6841673932529763677term_f,Xs: list_Bot_bot_term_f,Ys: list_Bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( transi2774326432670905901term_f @ R ) )
     => ( ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Xs @ Ys ) @ ( trancl6444742785402773766term_f @ R ) )
       => ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ ( cons_Bot_bot_term_f @ X @ Xs ) @ ( cons_Bot_bot_term_f @ Y @ Ys ) ) @ ( trancl6444742785402773766term_f @ R ) ) ) ) ).

% trancl_list_append_tranclI
thf(fact_302_trancl__list__append__tranclI,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R ) )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( trancl_list_nat @ R ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( trancl_list_nat @ R ) ) ) ) ).

% trancl_list_append_tranclI
thf(fact_303_list__ex__length,axiom,
    ( list_ex_term_f_v
    = ( ^ [P3: term_f_v > $o,Xs2: list_term_f_v] :
        ? [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( size_s7148729505879680775rm_f_v @ Xs2 ) )
          & ( P3 @ ( nth_term_f_v @ Xs2 @ N2 ) ) ) ) ) ).

% list_ex_length
thf(fact_304_list__ex__length,axiom,
    ( list_e1994534584214639169term_f
    = ( ^ [P3: produc250624644672406269term_f > $o,Xs2: list_P1780566427699531651term_f] :
        ? [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( size_s9037951226665808879term_f @ Xs2 ) )
          & ( P3 @ ( nth_Pr7080171896067517188term_f @ Xs2 @ N2 ) ) ) ) ) ).

% list_ex_length
thf(fact_305_list__ex__length,axiom,
    ( list_e327092010371199211rm_f_v
    = ( ^ [P3: produc8027700073345061287rm_f_v > $o,Xs2: list_P4093298276913796397rm_f_v] :
        ? [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( size_s3818039357710166425rm_f_v @ Xs2 ) )
          & ( P3 @ ( nth_Pr8768905099348956590rm_f_v @ Xs2 @ N2 ) ) ) ) ) ).

% list_ex_length
thf(fact_306_distinct__swap,axiom,
    ! [I4: nat,Xs: list_term_f_v,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s7148729505879680775rm_f_v @ Xs ) )
       => ( ( distinct_term_f_v @ ( list_update_term_f_v @ ( list_update_term_f_v @ Xs @ I4 @ ( nth_term_f_v @ Xs @ J ) ) @ J @ ( nth_term_f_v @ Xs @ I4 ) ) )
          = ( distinct_term_f_v @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_307_distinct__swap,axiom,
    ! [I4: nat,Xs: list_P1780566427699531651term_f,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s9037951226665808879term_f @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s9037951226665808879term_f @ Xs ) )
       => ( ( distin6927496147406123510term_f @ ( list_u2231791752812534891term_f @ ( list_u2231791752812534891term_f @ Xs @ I4 @ ( nth_Pr7080171896067517188term_f @ Xs @ J ) ) @ J @ ( nth_Pr7080171896067517188term_f @ Xs @ I4 ) ) )
          = ( distin6927496147406123510term_f @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_308_distinct__swap,axiom,
    ! [I4: nat,Xs: list_P4093298276913796397rm_f_v,J: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s3818039357710166425rm_f_v @ Xs ) )
       => ( ( distin6546576869490926752rm_f_v @ ( list_u2658473446164013845rm_f_v @ ( list_u2658473446164013845rm_f_v @ Xs @ I4 @ ( nth_Pr8768905099348956590rm_f_v @ Xs @ J ) ) @ J @ ( nth_Pr8768905099348956590rm_f_v @ Xs @ I4 ) ) )
          = ( distin6546576869490926752rm_f_v @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_309_in__fset__idx,axiom,
    ! [X: term_f_a,Xs: list_term_f_a] :
      ( ( fmember_term_f_a @ X @ ( fset_o6574051009341269640rm_f_a @ Xs ) )
     => ? [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s5657014664906415090rm_f_a @ Xs ) )
          & ( ( nth_term_f_a @ Xs @ I )
            = X ) ) ) ).

% in_fset_idx
thf(fact_310_in__fset__idx,axiom,
    ! [X: bot_bot_term_f,Xs: list_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ X @ ( fset_o7715858803885244232term_f @ Xs ) )
     => ? [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s1458857753309042354term_f @ Xs ) )
          & ( ( nth_Bot_bot_term_f @ Xs @ I )
            = X ) ) ) ).

% in_fset_idx
thf(fact_311_in__fset__idx,axiom,
    ! [X: term_f_v,Xs: list_term_f_v] :
      ( ( fmember_term_f_v @ X @ ( fset_o6574051099709074461rm_f_v @ Xs ) )
     => ? [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s7148729505879680775rm_f_v @ Xs ) )
          & ( ( nth_term_f_v @ Xs @ I )
            = X ) ) ) ).

% in_fset_idx
thf(fact_312_in__fset__idx,axiom,
    ! [X: produc250624644672406269term_f,Xs: list_P1780566427699531651term_f] :
      ( ( fmembe6950792022577883716term_f @ X @ ( fset_o6125718882019406531term_f @ Xs ) )
     => ? [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s9037951226665808879term_f @ Xs ) )
          & ( ( nth_Pr7080171896067517188term_f @ Xs @ I )
            = X ) ) ) ).

% in_fset_idx
thf(fact_313_in__fset__idx,axiom,
    ! [X: produc8027700073345061287rm_f_v,Xs: list_P4093298276913796397rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ X @ ( fset_o4970844032613833069rm_f_v @ Xs ) )
     => ? [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s3818039357710166425rm_f_v @ Xs ) )
          & ( ( nth_Pr8768905099348956590rm_f_v @ Xs @ I )
            = X ) ) ) ).

% in_fset_idx
thf(fact_314_eps__statesI_I2_J,axiom,
    ! [P2: term_f_a,Q: term_f_a,Delta: fset_P1463011978278004579rm_f_a] :
      ( ( fmembe3725132237878994628rm_f_a @ ( produc6952645563106136757rm_f_a @ P2 @ Q ) @ Delta )
     => ( fmember_term_f_a @ Q @ ( tree_e8707155189387792095rm_f_a @ Delta ) ) ) ).

% eps_statesI(2)
thf(fact_315_eps__statesI_I2_J,axiom,
    ! [P2: term_f_v,Q: term_f_v,Delta: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ P2 @ Q ) @ Delta )
     => ( fmember_term_f_v @ Q @ ( tree_e8707155279755596916rm_f_v @ Delta ) ) ) ).

% eps_statesI(2)
thf(fact_316_eps__statesI_I2_J,axiom,
    ! [P2: bot_bot_term_f,Q: bot_bot_term_f,Delta: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ P2 @ Q ) @ Delta )
     => ( fmembe1418995490368054919term_f @ Q @ ( tree_e551949757491771807term_f @ Delta ) ) ) ).

% eps_statesI(2)
thf(fact_317_eps__statesI_I2_J,axiom,
    ! [P2: nat,Q: nat,Delta: fset_P5670320511379867111at_nat] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ P2 @ Q ) @ Delta )
     => ( fmember_nat @ Q @ ( tree_eps_states_nat @ Delta ) ) ) ).

% eps_statesI(2)
thf(fact_318_eps__statesI_I1_J,axiom,
    ! [P2: term_f_a,Q: term_f_a,Delta: fset_P1463011978278004579rm_f_a] :
      ( ( fmembe3725132237878994628rm_f_a @ ( produc6952645563106136757rm_f_a @ P2 @ Q ) @ Delta )
     => ( fmember_term_f_a @ P2 @ ( tree_e8707155189387792095rm_f_a @ Delta ) ) ) ).

% eps_statesI(1)
thf(fact_319_eps__statesI_I1_J,axiom,
    ! [P2: term_f_v,Q: term_f_v,Delta: fset_P8018961893305114765rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ P2 @ Q ) @ Delta )
     => ( fmember_term_f_v @ P2 @ ( tree_e8707155279755596916rm_f_v @ Delta ) ) ) ).

% eps_statesI(1)
thf(fact_320_eps__statesI_I1_J,axiom,
    ! [P2: bot_bot_term_f,Q: bot_bot_term_f,Delta: fset_P2882900458636368611term_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ P2 @ Q ) @ Delta )
     => ( fmembe1418995490368054919term_f @ P2 @ ( tree_e551949757491771807term_f @ Delta ) ) ) ).

% eps_statesI(1)
thf(fact_321_eps__statesI_I1_J,axiom,
    ! [P2: nat,Q: nat,Delta: fset_P5670320511379867111at_nat] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ P2 @ Q ) @ Delta )
     => ( fmember_nat @ P2 @ ( tree_eps_states_nat @ Delta ) ) ) ).

% eps_statesI(1)
thf(fact_322_nth__remove__nth__conv,axiom,
    ! [I4: nat,N: nat,Xs: list_term_f_v] :
      ( ( ord_less_nat @ I4 @ ( size_s7148729505879680775rm_f_v @ ( missin2877041621942143146rm_f_v @ N @ Xs ) ) )
     => ( ( nth_term_f_v @ ( missin2877041621942143146rm_f_v @ N @ Xs ) @ I4 )
        = ( nth_term_f_v @ Xs @ ( if_nat @ ( ord_less_nat @ I4 @ N ) @ I4 @ ( suc @ I4 ) ) ) ) ) ).

% nth_remove_nth_conv
thf(fact_323_nth__remove__nth__conv,axiom,
    ! [I4: nat,N: nat,Xs: list_P1780566427699531651term_f] :
      ( ( ord_less_nat @ I4 @ ( size_s9037951226665808879term_f @ ( missin5571439781501833334term_f @ N @ Xs ) ) )
     => ( ( nth_Pr7080171896067517188term_f @ ( missin5571439781501833334term_f @ N @ Xs ) @ I4 )
        = ( nth_Pr7080171896067517188term_f @ Xs @ ( if_nat @ ( ord_less_nat @ I4 @ N ) @ I4 @ ( suc @ I4 ) ) ) ) ) ).

% nth_remove_nth_conv
thf(fact_324_nth__remove__nth__conv,axiom,
    ! [I4: nat,N: nat,Xs: list_P4093298276913796397rm_f_v] :
      ( ( ord_less_nat @ I4 @ ( size_s3818039357710166425rm_f_v @ ( missin5548135175598359328rm_f_v @ N @ Xs ) ) )
     => ( ( nth_Pr8768905099348956590rm_f_v @ ( missin5548135175598359328rm_f_v @ N @ Xs ) @ I4 )
        = ( nth_Pr8768905099348956590rm_f_v @ Xs @ ( if_nat @ ( ord_less_nat @ I4 @ N ) @ I4 @ ( suc @ I4 ) ) ) ) ) ).

% nth_remove_nth_conv
thf(fact_325_Fun__supt,axiom,
    ! [F: f,Ts: list_term_f_v,S: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( fun_f_v @ F @ Ts ) @ S ) @ subterm_and_supt_f_v )
     => ~ ! [T3: term_f_v] :
            ( ( member_term_f_v @ T3 @ ( set_term_f_v2 @ Ts ) )
           => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T3 @ S ) @ subter5033437232688913882eq_f_v ) ) ) ).

% Fun_supt
thf(fact_326_list_Oinject,axiom,
    ! [X21: nat,X222: list_nat,Y21: nat,Y222: list_nat] :
      ( ( ( cons_nat @ X21 @ X222 )
        = ( cons_nat @ Y21 @ Y222 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% list.inject
thf(fact_327_list__ex__simps_I1_J,axiom,
    ! [P: nat > $o,X: nat,Xs: list_nat] :
      ( ( list_ex_nat @ P @ ( cons_nat @ X @ Xs ) )
      = ( ( P @ X )
        | ( list_ex_nat @ P @ Xs ) ) ) ).

% list_ex_simps(1)
thf(fact_328_nth__Cons__Suc,axiom,
    ! [X: term_f_v,Xs: list_term_f_v,N: nat] :
      ( ( nth_term_f_v @ ( cons_term_f_v @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_term_f_v @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_329_nth__Cons__Suc,axiom,
    ! [X: produc250624644672406269term_f,Xs: list_P1780566427699531651term_f,N: nat] :
      ( ( nth_Pr7080171896067517188term_f @ ( cons_P3711653291972799923term_f @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_Pr7080171896067517188term_f @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_330_nth__Cons__Suc,axiom,
    ! [X: produc8027700073345061287rm_f_v,Xs: list_P4093298276913796397rm_f_v,N: nat] :
      ( ( nth_Pr8768905099348956590rm_f_v @ ( cons_P1983239458618365021rm_f_v @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_Pr8768905099348956590rm_f_v @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_331_nth__Cons__Suc,axiom,
    ! [X: nat,Xs: list_nat,N: nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_nat @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_332_suptI,axiom,
    ! [A: term_f_v,B: term_f_v] :
      ( ( A != B )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subter5033437232688913882eq_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subterm_and_supt_f_v ) ) ) ).

% suptI
thf(fact_333_Cons__listrel1__Cons,axiom,
    ! [X: term_f_v,Xs: list_term_f_v,Y: term_f_v,Ys: list_term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ ( cons_term_f_v @ X @ Xs ) @ ( cons_term_f_v @ Y @ Ys ) ) @ ( listrel1_term_f_v @ R3 ) )
      = ( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ R3 )
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Xs @ Ys ) @ ( listrel1_term_f_v @ R3 ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_334_Cons__listrel1__Cons,axiom,
    ! [X: bot_bot_term_f,Xs: list_Bot_bot_term_f,Y: bot_bot_term_f,Ys: list_Bot_bot_term_f,R3: set_Pr6841673932529763677term_f] :
      ( ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ ( cons_Bot_bot_term_f @ X @ Xs ) @ ( cons_Bot_bot_term_f @ Y @ Ys ) ) @ ( listre4162385684681681519term_f @ R3 ) )
      = ( ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ R3 )
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Xs @ Ys ) @ ( listre4162385684681681519term_f @ R3 ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_335_Cons__listrel1__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R3 ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 )
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R3 ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_336_trancl__list__appendI,axiom,
    ! [Xs: list_term_f_v,Ys: list_term_f_v,R: set_Pr8827868859434726151rm_f_v,X: term_f_v,Y: term_f_v] :
      ( ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Xs @ Ys ) @ ( trancl_list_term_f_v @ R ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ R )
       => ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ ( cons_term_f_v @ X @ Xs ) @ ( cons_term_f_v @ Y @ Ys ) ) @ ( trancl_list_term_f_v @ R ) ) ) ) ).

% trancl_list_appendI
thf(fact_337_trancl__list__appendI,axiom,
    ! [Xs: list_Bot_bot_term_f,Ys: list_Bot_bot_term_f,R: set_Pr6841673932529763677term_f,X: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Xs @ Ys ) @ ( trancl6444742785402773766term_f @ R ) )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ R )
       => ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ ( cons_Bot_bot_term_f @ X @ Xs ) @ ( cons_Bot_bot_term_f @ Y @ Ys ) ) @ ( trancl6444742785402773766term_f @ R ) ) ) ) ).

% trancl_list_appendI
thf(fact_338_trancl__list__appendI,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( trancl_list_nat @ R ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( trancl_list_nat @ R ) ) ) ) ).

% trancl_list_appendI
thf(fact_339_list__update__code_I3_J,axiom,
    ! [X: nat,Xs: list_nat,I4: nat,Y: nat] :
      ( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ I4 ) @ Y )
      = ( cons_nat @ X @ ( list_update_nat @ Xs @ I4 @ Y ) ) ) ).

% list_update_code(3)
thf(fact_340_distinct_Osimps_I2_J,axiom,
    ! [X: term_f_v,Xs: list_term_f_v] :
      ( ( distinct_term_f_v @ ( cons_term_f_v @ X @ Xs ) )
      = ( ~ ( member_term_f_v @ X @ ( set_term_f_v2 @ Xs ) )
        & ( distinct_term_f_v @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_341_distinct_Osimps_I2_J,axiom,
    ! [X: produc250624644672406269term_f,Xs: list_P1780566427699531651term_f] :
      ( ( distin6927496147406123510term_f @ ( cons_P3711653291972799923term_f @ X @ Xs ) )
      = ( ~ ( member4687689947140789414term_f @ X @ ( set_Pr1368550810757897490term_f @ Xs ) )
        & ( distin6927496147406123510term_f @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_342_distinct_Osimps_I2_J,axiom,
    ! [X: produc8027700073345061287rm_f_v,Xs: list_P4093298276913796397rm_f_v] :
      ( ( distin6546576869490926752rm_f_v @ ( cons_P1983239458618365021rm_f_v @ X @ Xs ) )
      = ( ~ ( member4316649861536941904rm_f_v @ X @ ( set_Pr817814403484925884rm_f_v @ Xs ) )
        & ( distin6546576869490926752rm_f_v @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_343_distinct_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( distinct_nat @ ( cons_nat @ X @ Xs ) )
      = ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
        & ( distinct_nat @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_344_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_345_length__Suc__conv,axiom,
    ! [Xs: list_term_f_v,N: nat] :
      ( ( ( size_s7148729505879680775rm_f_v @ Xs )
        = ( suc @ N ) )
      = ( ? [Y4: term_f_v,Ys2: list_term_f_v] :
            ( ( Xs
              = ( cons_term_f_v @ Y4 @ Ys2 ) )
            & ( ( size_s7148729505879680775rm_f_v @ Ys2 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_346_length__Suc__conv,axiom,
    ! [Xs: list_P1780566427699531651term_f,N: nat] :
      ( ( ( size_s9037951226665808879term_f @ Xs )
        = ( suc @ N ) )
      = ( ? [Y4: produc250624644672406269term_f,Ys2: list_P1780566427699531651term_f] :
            ( ( Xs
              = ( cons_P3711653291972799923term_f @ Y4 @ Ys2 ) )
            & ( ( size_s9037951226665808879term_f @ Ys2 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_347_length__Suc__conv,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,N: nat] :
      ( ( ( size_s3818039357710166425rm_f_v @ Xs )
        = ( suc @ N ) )
      = ( ? [Y4: produc8027700073345061287rm_f_v,Ys2: list_P4093298276913796397rm_f_v] :
            ( ( Xs
              = ( cons_P1983239458618365021rm_f_v @ Y4 @ Ys2 ) )
            & ( ( size_s3818039357710166425rm_f_v @ Ys2 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_348_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_349_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_term_f_v] :
      ( ( ( suc @ N )
        = ( size_s7148729505879680775rm_f_v @ Xs ) )
      = ( ? [Y4: term_f_v,Ys2: list_term_f_v] :
            ( ( Xs
              = ( cons_term_f_v @ Y4 @ Ys2 ) )
            & ( ( size_s7148729505879680775rm_f_v @ Ys2 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_350_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_P1780566427699531651term_f] :
      ( ( ( suc @ N )
        = ( size_s9037951226665808879term_f @ Xs ) )
      = ( ? [Y4: produc250624644672406269term_f,Ys2: list_P1780566427699531651term_f] :
            ( ( Xs
              = ( cons_P3711653291972799923term_f @ Y4 @ Ys2 ) )
            & ( ( size_s9037951226665808879term_f @ Ys2 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_351_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_P4093298276913796397rm_f_v] :
      ( ( ( suc @ N )
        = ( size_s3818039357710166425rm_f_v @ Xs ) )
      = ( ? [Y4: produc8027700073345061287rm_f_v,Ys2: list_P4093298276913796397rm_f_v] :
            ( ( Xs
              = ( cons_P1983239458618365021rm_f_v @ Y4 @ Ys2 ) )
            & ( ( size_s3818039357710166425rm_f_v @ Ys2 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_352_distinct__length__2__or__more,axiom,
    ! [A: nat,B: nat,Xs: list_nat] :
      ( ( distinct_nat @ ( cons_nat @ A @ ( cons_nat @ B @ Xs ) ) )
      = ( ( A != B )
        & ( distinct_nat @ ( cons_nat @ A @ Xs ) )
        & ( distinct_nat @ ( cons_nat @ B @ Xs ) ) ) ) ).

% distinct_length_2_or_more
thf(fact_353_not__Cons__self2,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( cons_nat @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_354_subterm_Olift__Suc__mono__less__iff,axiom,
    ! [F: nat > term_f_v,M: nat,N: nat] :
      ( ! [N3: nat] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) ) @ subterm_and_supt_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F @ M ) @ ( F @ N ) ) @ subterm_and_supt_f_v )
        = ( ord_less_nat @ N @ M ) ) ) ).

% subterm.lift_Suc_mono_less_iff
thf(fact_355_subterm_Olift__Suc__mono__less,axiom,
    ! [F: nat > term_f_v,N: nat,N4: nat] :
      ( ! [N3: nat] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) ) @ subterm_and_supt_f_v )
     => ( ( ord_less_nat @ N @ N4 )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F @ N4 ) @ ( F @ N ) ) @ subterm_and_supt_f_v ) ) ) ).

% subterm.lift_Suc_mono_less
thf(fact_356_list_Oset__intros_I2_J,axiom,
    ! [Y: term_f_v,X222: list_term_f_v,X21: term_f_v] :
      ( ( member_term_f_v @ Y @ ( set_term_f_v2 @ X222 ) )
     => ( member_term_f_v @ Y @ ( set_term_f_v2 @ ( cons_term_f_v @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_357_list_Oset__intros_I2_J,axiom,
    ! [Y: produc250624644672406269term_f,X222: list_P1780566427699531651term_f,X21: produc250624644672406269term_f] :
      ( ( member4687689947140789414term_f @ Y @ ( set_Pr1368550810757897490term_f @ X222 ) )
     => ( member4687689947140789414term_f @ Y @ ( set_Pr1368550810757897490term_f @ ( cons_P3711653291972799923term_f @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_358_list_Oset__intros_I2_J,axiom,
    ! [Y: produc8027700073345061287rm_f_v,X222: list_P4093298276913796397rm_f_v,X21: produc8027700073345061287rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ Y @ ( set_Pr817814403484925884rm_f_v @ X222 ) )
     => ( member4316649861536941904rm_f_v @ Y @ ( set_Pr817814403484925884rm_f_v @ ( cons_P1983239458618365021rm_f_v @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_359_list_Oset__intros_I2_J,axiom,
    ! [Y: nat,X222: list_nat,X21: nat] :
      ( ( member_nat @ Y @ ( set_nat2 @ X222 ) )
     => ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_360_list_Oset__intros_I1_J,axiom,
    ! [X21: term_f_v,X222: list_term_f_v] : ( member_term_f_v @ X21 @ ( set_term_f_v2 @ ( cons_term_f_v @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_361_list_Oset__intros_I1_J,axiom,
    ! [X21: produc250624644672406269term_f,X222: list_P1780566427699531651term_f] : ( member4687689947140789414term_f @ X21 @ ( set_Pr1368550810757897490term_f @ ( cons_P3711653291972799923term_f @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_362_list_Oset__intros_I1_J,axiom,
    ! [X21: produc8027700073345061287rm_f_v,X222: list_P4093298276913796397rm_f_v] : ( member4316649861536941904rm_f_v @ X21 @ ( set_Pr817814403484925884rm_f_v @ ( cons_P1983239458618365021rm_f_v @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_363_list_Oset__intros_I1_J,axiom,
    ! [X21: nat,X222: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_364_list_Oset__cases,axiom,
    ! [E: term_f_v,A: list_term_f_v] :
      ( ( member_term_f_v @ E @ ( set_term_f_v2 @ A ) )
     => ( ! [Z22: list_term_f_v] :
            ( A
           != ( cons_term_f_v @ E @ Z22 ) )
       => ~ ! [Z1: term_f_v,Z22: list_term_f_v] :
              ( ( A
                = ( cons_term_f_v @ Z1 @ Z22 ) )
             => ~ ( member_term_f_v @ E @ ( set_term_f_v2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_365_list_Oset__cases,axiom,
    ! [E: produc250624644672406269term_f,A: list_P1780566427699531651term_f] :
      ( ( member4687689947140789414term_f @ E @ ( set_Pr1368550810757897490term_f @ A ) )
     => ( ! [Z22: list_P1780566427699531651term_f] :
            ( A
           != ( cons_P3711653291972799923term_f @ E @ Z22 ) )
       => ~ ! [Z1: produc250624644672406269term_f,Z22: list_P1780566427699531651term_f] :
              ( ( A
                = ( cons_P3711653291972799923term_f @ Z1 @ Z22 ) )
             => ~ ( member4687689947140789414term_f @ E @ ( set_Pr1368550810757897490term_f @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_366_list_Oset__cases,axiom,
    ! [E: produc8027700073345061287rm_f_v,A: list_P4093298276913796397rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ E @ ( set_Pr817814403484925884rm_f_v @ A ) )
     => ( ! [Z22: list_P4093298276913796397rm_f_v] :
            ( A
           != ( cons_P1983239458618365021rm_f_v @ E @ Z22 ) )
       => ~ ! [Z1: produc8027700073345061287rm_f_v,Z22: list_P4093298276913796397rm_f_v] :
              ( ( A
                = ( cons_P1983239458618365021rm_f_v @ Z1 @ Z22 ) )
             => ~ ( member4316649861536941904rm_f_v @ E @ ( set_Pr817814403484925884rm_f_v @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_367_list_Oset__cases,axiom,
    ! [E: nat,A: list_nat] :
      ( ( member_nat @ E @ ( set_nat2 @ A ) )
     => ( ! [Z22: list_nat] :
            ( A
           != ( cons_nat @ E @ Z22 ) )
       => ~ ! [Z1: nat,Z22: list_nat] :
              ( ( A
                = ( cons_nat @ Z1 @ Z22 ) )
             => ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_368_set__ConsD,axiom,
    ! [Y: term_f_v,X: term_f_v,Xs: list_term_f_v] :
      ( ( member_term_f_v @ Y @ ( set_term_f_v2 @ ( cons_term_f_v @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member_term_f_v @ Y @ ( set_term_f_v2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_369_set__ConsD,axiom,
    ! [Y: produc250624644672406269term_f,X: produc250624644672406269term_f,Xs: list_P1780566427699531651term_f] :
      ( ( member4687689947140789414term_f @ Y @ ( set_Pr1368550810757897490term_f @ ( cons_P3711653291972799923term_f @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member4687689947140789414term_f @ Y @ ( set_Pr1368550810757897490term_f @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_370_set__ConsD,axiom,
    ! [Y: produc8027700073345061287rm_f_v,X: produc8027700073345061287rm_f_v,Xs: list_P4093298276913796397rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ Y @ ( set_Pr817814403484925884rm_f_v @ ( cons_P1983239458618365021rm_f_v @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member4316649861536941904rm_f_v @ Y @ ( set_Pr817814403484925884rm_f_v @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_371_set__ConsD,axiom,
    ! [Y: nat,X: nat,Xs: list_nat] :
      ( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member_nat @ Y @ ( set_nat2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_372_subterm__induct,axiom,
    ! [P: term_f_v > $o,T: term_f_v] :
      ( ! [T3: term_f_v] :
          ( ! [S3: term_f_v] :
              ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T3 @ S3 ) @ subterm_and_supt_f_v )
             => ( P @ S3 ) )
         => ( P @ T3 ) )
     => ( P @ T ) ) ).

% subterm_induct
thf(fact_373_supt__not__refl,axiom,
    ! [T: term_f_v] :
      ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ T ) @ subterm_and_supt_f_v ) ).

% supt_not_refl
thf(fact_374_supt__not__sym,axiom,
    ! [S: term_f_v,T: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subterm_and_supt_f_v )
     => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ S ) @ subterm_and_supt_f_v ) ) ).

% supt_not_sym
thf(fact_375_supt__irrefl,axiom,
    ! [T: term_f_v] :
      ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ T ) @ subterm_and_supt_f_v ) ).

% supt_irrefl
thf(fact_376_supt__trans,axiom,
    ! [S: term_f_v,T: term_f_v,U: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subterm_and_supt_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ U ) @ subterm_and_supt_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ U ) @ subterm_and_supt_f_v ) ) ) ).

% supt_trans
thf(fact_377_supt__neqD,axiom,
    ! [S: term_f_v,T: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subterm_and_supt_f_v )
     => ( S != T ) ) ).

% supt_neqD
thf(fact_378_subterm_Odual__order_Ostrict__implies__not__eq,axiom,
    ! [A: term_f_v,B: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subterm_and_supt_f_v )
     => ( A != B ) ) ).

% subterm.dual_order.strict_implies_not_eq
thf(fact_379_subterm_Oorder_Ostrict__implies__not__eq,axiom,
    ! [B: term_f_v,A: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subterm_and_supt_f_v )
     => ( A != B ) ) ).

% subterm.order.strict_implies_not_eq
thf(fact_380_subterm_Odual__order_Ostrict__trans,axiom,
    ! [A: term_f_v,B: term_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subterm_and_supt_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ subterm_and_supt_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ subterm_and_supt_f_v ) ) ) ).

% subterm.dual_order.strict_trans
thf(fact_381_subterm_Oorder_Ostrict__trans,axiom,
    ! [B: term_f_v,A: term_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subterm_and_supt_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C @ B ) @ subterm_and_supt_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C @ A ) @ subterm_and_supt_f_v ) ) ) ).

% subterm.order.strict_trans
thf(fact_382_subterm_Oord__less__eq__trans,axiom,
    ! [B: term_f_v,A: term_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subterm_and_supt_f_v )
     => ( ( B = C )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C @ A ) @ subterm_and_supt_f_v ) ) ) ).

% subterm.ord_less_eq_trans
thf(fact_383_subterm_Oord__eq__less__trans,axiom,
    ! [A: term_f_v,B: term_f_v,C: term_f_v] :
      ( ( A = B )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C @ B ) @ subterm_and_supt_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C @ A ) @ subterm_and_supt_f_v ) ) ) ).

% subterm.ord_eq_less_trans
thf(fact_384_subterm_Oless__imp__not__less,axiom,
    ! [Y: term_f_v,X: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subterm_and_supt_f_v )
     => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ subterm_and_supt_f_v ) ) ).

% subterm.less_imp_not_less
thf(fact_385_subterm_Odual__order_Oirrefl,axiom,
    ! [A: term_f_v] :
      ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ A ) @ subterm_and_supt_f_v ) ).

% subterm.dual_order.irrefl
thf(fact_386_subterm_Oless__imp__not__eq2,axiom,
    ! [Y: term_f_v,X: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subterm_and_supt_f_v )
     => ( Y != X ) ) ).

% subterm.less_imp_not_eq2
thf(fact_387_subterm_Oless__imp__not__eq,axiom,
    ! [Y: term_f_v,X: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subterm_and_supt_f_v )
     => ( X != Y ) ) ).

% subterm.less_imp_not_eq
thf(fact_388_subterm_Odual__order_Oasym,axiom,
    ! [A: term_f_v,B: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subterm_and_supt_f_v )
     => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subterm_and_supt_f_v ) ) ).

% subterm.dual_order.asym
thf(fact_389_subterm_Oless__imp__triv,axiom,
    ! [Y: term_f_v,X: term_f_v,P: $o] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subterm_and_supt_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ subterm_and_supt_f_v )
       => P ) ) ).

% subterm.less_imp_triv
thf(fact_390_subterm_Oless__not__sym,axiom,
    ! [Y: term_f_v,X: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subterm_and_supt_f_v )
     => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ subterm_and_supt_f_v ) ) ).

% subterm.less_not_sym
thf(fact_391_subterm_Oless__imp__neq,axiom,
    ! [Y: term_f_v,X: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subterm_and_supt_f_v )
     => ( X != Y ) ) ).

% subterm.less_imp_neq
thf(fact_392_subterm_Oless__irrefl,axiom,
    ! [X: term_f_v] :
      ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ X ) @ subterm_and_supt_f_v ) ).

% subterm.less_irrefl
thf(fact_393_subterm_Oorder_Oasym,axiom,
    ! [B: term_f_v,A: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subterm_and_supt_f_v )
     => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subterm_and_supt_f_v ) ) ).

% subterm.order.asym
thf(fact_394_subterm_Oless__trans,axiom,
    ! [Y: term_f_v,X: term_f_v,Z: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subterm_and_supt_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z @ Y ) @ subterm_and_supt_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z @ X ) @ subterm_and_supt_f_v ) ) ) ).

% subterm.less_trans
thf(fact_395_subterm_Oless__asym_H,axiom,
    ! [B: term_f_v,A: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subterm_and_supt_f_v )
     => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subterm_and_supt_f_v ) ) ).

% subterm.less_asym'
thf(fact_396_subterm_Oless__asym,axiom,
    ! [Y: term_f_v,X: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subterm_and_supt_f_v )
     => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ subterm_and_supt_f_v ) ) ).

% subterm.less_asym
thf(fact_397_list__ex__cong,axiom,
    ! [Xs: list_term_f_v,Ys: list_term_f_v,F: term_f_v > $o,G: term_f_v > $o] :
      ( ( Xs = Ys )
     => ( ! [X4: term_f_v] :
            ( ( member_term_f_v @ X4 @ ( set_term_f_v2 @ Ys ) )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) )
       => ( ( list_ex_term_f_v @ F @ Xs )
          = ( list_ex_term_f_v @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_398_list__ex__cong,axiom,
    ! [Xs: list_P1780566427699531651term_f,Ys: list_P1780566427699531651term_f,F: produc250624644672406269term_f > $o,G: produc250624644672406269term_f > $o] :
      ( ( Xs = Ys )
     => ( ! [X4: produc250624644672406269term_f] :
            ( ( member4687689947140789414term_f @ X4 @ ( set_Pr1368550810757897490term_f @ Ys ) )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) )
       => ( ( list_e1994534584214639169term_f @ F @ Xs )
          = ( list_e1994534584214639169term_f @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_399_list__ex__cong,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,Ys: list_P4093298276913796397rm_f_v,F: produc8027700073345061287rm_f_v > $o,G: produc8027700073345061287rm_f_v > $o] :
      ( ( Xs = Ys )
     => ( ! [X4: produc8027700073345061287rm_f_v] :
            ( ( member4316649861536941904rm_f_v @ X4 @ ( set_Pr817814403484925884rm_f_v @ Ys ) )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) )
       => ( ( list_e327092010371199211rm_f_v @ F @ Xs )
          = ( list_e327092010371199211rm_f_v @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_400_supt__supteq__not__supteq,axiom,
    ! [S: term_f_v,T: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subterm_and_supt_f_v )
      = ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subter5033437232688913882eq_f_v )
        & ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ S ) @ subter5033437232688913882eq_f_v ) ) ) ).

% supt_supteq_not_supteq
thf(fact_401_supteq__supt__trans,axiom,
    ! [S: term_f_v,T: term_f_v,U: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subter5033437232688913882eq_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ U ) @ subterm_and_supt_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ U ) @ subterm_and_supt_f_v ) ) ) ).

% supteq_supt_trans
thf(fact_402_supt__supteq__trans,axiom,
    ! [S: term_f_v,T: term_f_v,U: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subterm_and_supt_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ U ) @ subter5033437232688913882eq_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ U ) @ subterm_and_supt_f_v ) ) ) ).

% supt_supteq_trans
thf(fact_403_supteq__supt__conv,axiom,
    ! [S: term_f_v,T: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subter5033437232688913882eq_f_v )
      = ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subterm_and_supt_f_v )
        | ( S = T ) ) ) ).

% supteq_supt_conv
thf(fact_404_supt__supteq__conv,axiom,
    ! [S: term_f_v,T: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subterm_and_supt_f_v )
      = ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subter5033437232688913882eq_f_v )
        & ( S != T ) ) ) ).

% supt_supteq_conv
thf(fact_405_supteq__not__supt,axiom,
    ! [S: term_f_v,T: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subter5033437232688913882eq_f_v )
     => ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subterm_and_supt_f_v )
       => ( S = T ) ) ) ).

% supteq_not_supt
thf(fact_406_supt__imp__supteq,axiom,
    ! [S: term_f_v,T: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subterm_and_supt_f_v )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subter5033437232688913882eq_f_v ) ) ).

% supt_imp_supteq
thf(fact_407_suptE,axiom,
    ! [S: term_f_v,T: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subterm_and_supt_f_v )
     => ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subter5033437232688913882eq_f_v )
       => ( S != T ) ) ) ).

% suptE
thf(fact_408_subterm_Oorder_Onot__eq__order__implies__strict,axiom,
    ! [A: term_f_v,B: term_f_v] :
      ( ( A != B )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subter5033437232688913882eq_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subterm_and_supt_f_v ) ) ) ).

% subterm.order.not_eq_order_implies_strict
thf(fact_409_subterm_Odual__order_Ostrict__implies__order,axiom,
    ! [A: term_f_v,B: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subterm_and_supt_f_v )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subter5033437232688913882eq_f_v ) ) ).

% subterm.dual_order.strict_implies_order
thf(fact_410_subterm_Odual__order_Ostrict__iff__order,axiom,
    ! [A: term_f_v,B: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subterm_and_supt_f_v )
      = ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subter5033437232688913882eq_f_v )
        & ( A != B ) ) ) ).

% subterm.dual_order.strict_iff_order
thf(fact_411_subterm_Odual__order_Oorder__iff__strict,axiom,
    ! [A: term_f_v,B: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subter5033437232688913882eq_f_v )
      = ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subterm_and_supt_f_v )
        | ( A = B ) ) ) ).

% subterm.dual_order.order_iff_strict
thf(fact_412_subterm_Oorder_Ostrict__implies__order,axiom,
    ! [B: term_f_v,A: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subterm_and_supt_f_v )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subter5033437232688913882eq_f_v ) ) ).

% subterm.order.strict_implies_order
thf(fact_413_subterm_Odual__order_Ostrict__iff__not,axiom,
    ! [A: term_f_v,B: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subterm_and_supt_f_v )
      = ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subter5033437232688913882eq_f_v )
        & ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subter5033437232688913882eq_f_v ) ) ) ).

% subterm.dual_order.strict_iff_not
thf(fact_414_subterm_Odual__order_Ostrict__trans2,axiom,
    ! [A: term_f_v,B: term_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subterm_and_supt_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ subter5033437232688913882eq_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ subterm_and_supt_f_v ) ) ) ).

% subterm.dual_order.strict_trans2
thf(fact_415_subterm_Odual__order_Ostrict__trans1,axiom,
    ! [A: term_f_v,B: term_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subter5033437232688913882eq_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ subterm_and_supt_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ subterm_and_supt_f_v ) ) ) ).

% subterm.dual_order.strict_trans1
thf(fact_416_subterm_Oorder_Ostrict__iff__order,axiom,
    ! [B: term_f_v,A: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subterm_and_supt_f_v )
      = ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subter5033437232688913882eq_f_v )
        & ( A != B ) ) ) ).

% subterm.order.strict_iff_order
thf(fact_417_subterm_Oorder_Oorder__iff__strict,axiom,
    ! [B: term_f_v,A: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subter5033437232688913882eq_f_v )
      = ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subterm_and_supt_f_v )
        | ( A = B ) ) ) ).

% subterm.order.order_iff_strict
thf(fact_418_subterm_Oorder_Ostrict__iff__not,axiom,
    ! [B: term_f_v,A: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subterm_and_supt_f_v )
      = ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subter5033437232688913882eq_f_v )
        & ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ subter5033437232688913882eq_f_v ) ) ) ).

% subterm.order.strict_iff_not
thf(fact_419_subterm_Oorder_Ostrict__trans2,axiom,
    ! [B: term_f_v,A: term_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subterm_and_supt_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C @ B ) @ subter5033437232688913882eq_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C @ A ) @ subterm_and_supt_f_v ) ) ) ).

% subterm.order.strict_trans2
thf(fact_420_subterm_Oorder_Ostrict__trans1,axiom,
    ! [B: term_f_v,A: term_f_v,C: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subter5033437232688913882eq_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C @ B ) @ subterm_and_supt_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C @ A ) @ subterm_and_supt_f_v ) ) ) ).

% subterm.order.strict_trans1
thf(fact_421_subterm_Ole__imp__less__or__eq,axiom,
    ! [Y: term_f_v,X: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subter5033437232688913882eq_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subterm_and_supt_f_v )
        | ( X = Y ) ) ) ).

% subterm.le_imp_less_or_eq
thf(fact_422_subterm_Oless__le__not__le,axiom,
    ! [Y: term_f_v,X: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subterm_and_supt_f_v )
      = ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subter5033437232688913882eq_f_v )
        & ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ subter5033437232688913882eq_f_v ) ) ) ).

% subterm.less_le_not_le
thf(fact_423_subterm_Oless__le__trans,axiom,
    ! [Y: term_f_v,X: term_f_v,Z: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subterm_and_supt_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z @ Y ) @ subter5033437232688913882eq_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z @ X ) @ subterm_and_supt_f_v ) ) ) ).

% subterm.less_le_trans
thf(fact_424_subterm_Ole__less__trans,axiom,
    ! [Y: term_f_v,X: term_f_v,Z: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subter5033437232688913882eq_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z @ Y ) @ subterm_and_supt_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z @ X ) @ subterm_and_supt_f_v ) ) ) ).

% subterm.le_less_trans
thf(fact_425_subterm_Oantisym__conv2,axiom,
    ! [Y: term_f_v,X: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subter5033437232688913882eq_f_v )
     => ( ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subterm_and_supt_f_v ) )
        = ( X = Y ) ) ) ).

% subterm.antisym_conv2
thf(fact_426_subterm_Oantisym__conv1,axiom,
    ! [Y: term_f_v,X: term_f_v] :
      ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subterm_and_supt_f_v )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subter5033437232688913882eq_f_v )
        = ( X = Y ) ) ) ).

% subterm.antisym_conv1
thf(fact_427_subterm_Ole__neq__trans,axiom,
    ! [B: term_f_v,A: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subter5033437232688913882eq_f_v )
     => ( ( A != B )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subterm_and_supt_f_v ) ) ) ).

% subterm.le_neq_trans
thf(fact_428_subterm_Oless__imp__le,axiom,
    ! [Y: term_f_v,X: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subterm_and_supt_f_v )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subter5033437232688913882eq_f_v ) ) ).

% subterm.less_imp_le
thf(fact_429_subterm_Onless__le,axiom,
    ! [B: term_f_v,A: term_f_v] :
      ( ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subterm_and_supt_f_v ) )
      = ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ subter5033437232688913882eq_f_v )
        | ( A = B ) ) ) ).

% subterm.nless_le
thf(fact_430_subterm_Oless__le,axiom,
    ! [Y: term_f_v,X: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subterm_and_supt_f_v )
      = ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subter5033437232688913882eq_f_v )
        & ( X != Y ) ) ) ).

% subterm.less_le
thf(fact_431_subterm_Ole__less,axiom,
    ! [Y: term_f_v,X: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subter5033437232688913882eq_f_v )
      = ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subterm_and_supt_f_v )
        | ( X = Y ) ) ) ).

% subterm.le_less
thf(fact_432_subterm_OleD,axiom,
    ! [X: term_f_v,Y: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ subter5033437232688913882eq_f_v )
     => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X ) @ subterm_and_supt_f_v ) ) ).

% subterm.leD
thf(fact_433_listrel1I2,axiom,
    ! [Xs: list_nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat,X: nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R3 ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ X @ Ys ) ) @ ( listrel1_nat @ R3 ) ) ) ).

% listrel1I2
thf(fact_434_adjust__idx__def,axiom,
    ( missing_adjust_idx
    = ( ^ [I3: nat,J2: nat] : ( if_nat @ ( ord_less_nat @ J2 @ I3 ) @ J2 @ ( suc @ J2 ) ) ) ) ).

% adjust_idx_def
thf(fact_435_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_term_f_v,I4: nat,J: nat] :
      ( ( distinct_term_f_v @ Xs )
     => ( ( ord_less_nat @ I4 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_s7148729505879680775rm_f_v @ Xs ) )
         => ( ( ( nth_term_f_v @ Xs @ I4 )
              = ( nth_term_f_v @ Xs @ J ) )
            = ( I4 = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_436_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_P1780566427699531651term_f,I4: nat,J: nat] :
      ( ( distin6927496147406123510term_f @ Xs )
     => ( ( ord_less_nat @ I4 @ ( size_s9037951226665808879term_f @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_s9037951226665808879term_f @ Xs ) )
         => ( ( ( nth_Pr7080171896067517188term_f @ Xs @ I4 )
              = ( nth_Pr7080171896067517188term_f @ Xs @ J ) )
            = ( I4 = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_437_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,I4: nat,J: nat] :
      ( ( distin6546576869490926752rm_f_v @ Xs )
     => ( ( ord_less_nat @ I4 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_s3818039357710166425rm_f_v @ Xs ) )
         => ( ( ( nth_Pr8768905099348956590rm_f_v @ Xs @ I4 )
              = ( nth_Pr8768905099348956590rm_f_v @ Xs @ J ) )
            = ( I4 = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_438_distinct__conv__nth,axiom,
    ( distinct_term_f_v
    = ( ^ [Xs2: list_term_f_v] :
        ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s7148729505879680775rm_f_v @ Xs2 ) )
         => ! [J2: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s7148729505879680775rm_f_v @ Xs2 ) )
             => ( ( I3 != J2 )
               => ( ( nth_term_f_v @ Xs2 @ I3 )
                 != ( nth_term_f_v @ Xs2 @ J2 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_439_distinct__conv__nth,axiom,
    ( distin6927496147406123510term_f
    = ( ^ [Xs2: list_P1780566427699531651term_f] :
        ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s9037951226665808879term_f @ Xs2 ) )
         => ! [J2: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s9037951226665808879term_f @ Xs2 ) )
             => ( ( I3 != J2 )
               => ( ( nth_Pr7080171896067517188term_f @ Xs2 @ I3 )
                 != ( nth_Pr7080171896067517188term_f @ Xs2 @ J2 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_440_distinct__conv__nth,axiom,
    ( distin6546576869490926752rm_f_v
    = ( ^ [Xs2: list_P4093298276913796397rm_f_v] :
        ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s3818039357710166425rm_f_v @ Xs2 ) )
         => ! [J2: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s3818039357710166425rm_f_v @ Xs2 ) )
             => ( ( I3 != J2 )
               => ( ( nth_Pr8768905099348956590rm_f_v @ Xs2 @ I3 )
                 != ( nth_Pr8768905099348956590rm_f_v @ Xs2 @ J2 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_441_supt_Oarg,axiom,
    ! [S: term_f_v,Ss: list_term_f_v,F: f] :
      ( ( member_term_f_v @ S @ ( set_term_f_v2 @ Ss ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( fun_f_v @ F @ Ss ) @ S ) @ subterm_and_supt_f_v ) ) ).

% supt.arg
thf(fact_442_supt_Osubt,axiom,
    ! [S: term_f_v,Ss: list_term_f_v,T: term_f_v,F: f] :
      ( ( member_term_f_v @ S @ ( set_term_f_v2 @ Ss ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subterm_and_supt_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( fun_f_v @ F @ Ss ) @ T ) @ subterm_and_supt_f_v ) ) ) ).

% supt.subt
thf(fact_443_supt_Ocases,axiom,
    ! [A1: term_f_v,A22: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ subterm_and_supt_f_v )
     => ( ! [S4: term_f_v,Ss3: list_term_f_v] :
            ( ? [F3: f] :
                ( A1
                = ( fun_f_v @ F3 @ Ss3 ) )
           => ( ( A22 = S4 )
             => ~ ( member_term_f_v @ S4 @ ( set_term_f_v2 @ Ss3 ) ) ) )
       => ~ ! [S4: term_f_v,Ss3: list_term_f_v] :
              ( ? [F3: f] :
                  ( A1
                  = ( fun_f_v @ F3 @ Ss3 ) )
             => ( ( member_term_f_v @ S4 @ ( set_term_f_v2 @ Ss3 ) )
               => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S4 @ A22 ) @ subterm_and_supt_f_v ) ) ) ) ) ).

% supt.cases
thf(fact_444_supt_Osimps,axiom,
    ! [A1: term_f_v,A22: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ subterm_and_supt_f_v )
      = ( ? [S2: term_f_v,Ss2: list_term_f_v,F2: f] :
            ( ( A1
              = ( fun_f_v @ F2 @ Ss2 ) )
            & ( A22 = S2 )
            & ( member_term_f_v @ S2 @ ( set_term_f_v2 @ Ss2 ) ) )
        | ? [S2: term_f_v,Ss2: list_term_f_v,T2: term_f_v,F2: f] :
            ( ( A1
              = ( fun_f_v @ F2 @ Ss2 ) )
            & ( A22 = T2 )
            & ( member_term_f_v @ S2 @ ( set_term_f_v2 @ Ss2 ) )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S2 @ T2 ) @ subterm_and_supt_f_v ) ) ) ) ).

% supt.simps
thf(fact_445_supt__Fun,axiom,
    ! [S: term_f_v,F: f,Ss: list_term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ ( fun_f_v @ F @ Ss ) ) @ subterm_and_supt_f_v )
     => ~ ( member_term_f_v @ S @ ( set_term_f_v2 @ Ss ) ) ) ).

% supt_Fun
thf(fact_446_listrel1I1,axiom,
    ! [X: term_f_v,Y: term_f_v,R3: set_Pr8827868859434726151rm_f_v,Xs: list_term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ R3 )
     => ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ ( cons_term_f_v @ X @ Xs ) @ ( cons_term_f_v @ Y @ Xs ) ) @ ( listrel1_term_f_v @ R3 ) ) ) ).

% listrel1I1
thf(fact_447_listrel1I1,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,R3: set_Pr6841673932529763677term_f,Xs: list_Bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ R3 )
     => ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ ( cons_Bot_bot_term_f @ X @ Xs ) @ ( cons_Bot_bot_term_f @ Y @ Xs ) ) @ ( listre4162385684681681519term_f @ R3 ) ) ) ).

% listrel1I1
thf(fact_448_listrel1I1,axiom,
    ! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Xs ) ) @ ( listrel1_nat @ R3 ) ) ) ).

% listrel1I1
thf(fact_449_Cons__listrel1E1,axiom,
    ! [X: term_f_v,Xs: list_term_f_v,Ys: list_term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ ( cons_term_f_v @ X @ Xs ) @ Ys ) @ ( listrel1_term_f_v @ R3 ) )
     => ( ! [Y3: term_f_v] :
            ( ( Ys
              = ( cons_term_f_v @ Y3 @ Xs ) )
           => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R3 ) )
       => ~ ! [Zs: list_term_f_v] :
              ( ( Ys
                = ( cons_term_f_v @ X @ Zs ) )
             => ~ ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Xs @ Zs ) @ ( listrel1_term_f_v @ R3 ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_450_Cons__listrel1E1,axiom,
    ! [X: bot_bot_term_f,Xs: list_Bot_bot_term_f,Ys: list_Bot_bot_term_f,R3: set_Pr6841673932529763677term_f] :
      ( ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ ( cons_Bot_bot_term_f @ X @ Xs ) @ Ys ) @ ( listre4162385684681681519term_f @ R3 ) )
     => ( ! [Y3: bot_bot_term_f] :
            ( ( Ys
              = ( cons_Bot_bot_term_f @ Y3 @ Xs ) )
           => ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y3 ) @ R3 ) )
       => ~ ! [Zs: list_Bot_bot_term_f] :
              ( ( Ys
                = ( cons_Bot_bot_term_f @ X @ Zs ) )
             => ~ ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Xs @ Zs ) @ ( listre4162385684681681519term_f @ R3 ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_451_Cons__listrel1E1,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ Ys ) @ ( listrel1_nat @ R3 ) )
     => ( ! [Y3: nat] :
            ( ( Ys
              = ( cons_nat @ Y3 @ Xs ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R3 ) )
       => ~ ! [Zs: list_nat] :
              ( ( Ys
                = ( cons_nat @ X @ Zs ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs ) @ ( listrel1_nat @ R3 ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_452_Cons__listrel1E2,axiom,
    ! [Xs: list_term_f_v,Y: term_f_v,Ys: list_term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Xs @ ( cons_term_f_v @ Y @ Ys ) ) @ ( listrel1_term_f_v @ R3 ) )
     => ( ! [X4: term_f_v] :
            ( ( Xs
              = ( cons_term_f_v @ X4 @ Ys ) )
           => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y ) @ R3 ) )
       => ~ ! [Zs: list_term_f_v] :
              ( ( Xs
                = ( cons_term_f_v @ Y @ Zs ) )
             => ~ ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Zs @ Ys ) @ ( listrel1_term_f_v @ R3 ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_453_Cons__listrel1E2,axiom,
    ! [Xs: list_Bot_bot_term_f,Y: bot_bot_term_f,Ys: list_Bot_bot_term_f,R3: set_Pr6841673932529763677term_f] :
      ( ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Xs @ ( cons_Bot_bot_term_f @ Y @ Ys ) ) @ ( listre4162385684681681519term_f @ R3 ) )
     => ( ! [X4: bot_bot_term_f] :
            ( ( Xs
              = ( cons_Bot_bot_term_f @ X4 @ Ys ) )
           => ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X4 @ Y ) @ R3 ) )
       => ~ ! [Zs: list_Bot_bot_term_f] :
              ( ( Xs
                = ( cons_Bot_bot_term_f @ Y @ Zs ) )
             => ~ ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Zs @ Ys ) @ ( listre4162385684681681519term_f @ R3 ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_454_Cons__listrel1E2,axiom,
    ! [Xs: list_nat,Y: nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R3 ) )
     => ( ! [X4: nat] :
            ( ( Xs
              = ( cons_nat @ X4 @ Ys ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y ) @ R3 ) )
       => ~ ! [Zs: list_nat] :
              ( ( Xs
                = ( cons_nat @ Y @ Zs ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs @ Ys ) @ ( listrel1_nat @ R3 ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_455_eps__statesE,axiom,
    ! [P2: term_f_a,Delta: fset_P1463011978278004579rm_f_a] :
      ( ( fmember_term_f_a @ P2 @ ( tree_e8707155189387792095rm_f_a @ Delta ) )
     => ~ ! [Q2: term_f_a] :
            ~ ( ( fmembe3725132237878994628rm_f_a @ ( produc6952645563106136757rm_f_a @ P2 @ Q2 ) @ Delta )
              | ( fmembe3725132237878994628rm_f_a @ ( produc6952645563106136757rm_f_a @ Q2 @ P2 ) @ Delta ) ) ) ).

% eps_statesE
thf(fact_456_eps__statesE,axiom,
    ! [P2: term_f_v,Delta: fset_P8018961893305114765rm_f_v] :
      ( ( fmember_term_f_v @ P2 @ ( tree_e8707155279755596916rm_f_v @ Delta ) )
     => ~ ! [Q2: term_f_v] :
            ~ ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ P2 @ Q2 ) @ Delta )
              | ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ Q2 @ P2 ) @ Delta ) ) ) ).

% eps_statesE
thf(fact_457_eps__statesE,axiom,
    ! [P2: bot_bot_term_f,Delta: fset_P2882900458636368611term_f] :
      ( ( fmembe1418995490368054919term_f @ P2 @ ( tree_e551949757491771807term_f @ Delta ) )
     => ~ ! [Q2: bot_bot_term_f] :
            ~ ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ P2 @ Q2 ) @ Delta )
              | ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ Q2 @ P2 ) @ Delta ) ) ) ).

% eps_statesE
thf(fact_458_eps__statesE,axiom,
    ! [P2: nat,Delta: fset_P5670320511379867111at_nat] :
      ( ( fmember_nat @ P2 @ ( tree_eps_states_nat @ Delta ) )
     => ~ ! [Q2: nat] :
            ~ ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ P2 @ Q2 ) @ Delta )
              | ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ Q2 @ P2 ) @ Delta ) ) ) ).

% eps_statesE
thf(fact_459_listrel_OCons,axiom,
    ! [X: term_f_v,Y: term_f_v,R3: set_Pr8827868859434726151rm_f_v,Xs: list_term_f_v,Ys: list_term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ R3 )
     => ( ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Xs @ Ys ) @ ( listre5907981802042383033rm_f_v @ R3 ) )
       => ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ ( cons_term_f_v @ X @ Xs ) @ ( cons_term_f_v @ Y @ Ys ) ) @ ( listre5907981802042383033rm_f_v @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_460_listrel_OCons,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,R3: set_Pr6841673932529763677term_f,Xs: list_Bot_bot_term_f,Ys: list_Bot_bot_term_f] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ R3 )
     => ( ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Xs @ Ys ) @ ( listre686400052008961423term_f @ R3 ) )
       => ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ ( cons_Bot_bot_term_f @ X @ Xs ) @ ( cons_Bot_bot_term_f @ Y @ Ys ) ) @ ( listre686400052008961423term_f @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_461_listrel_OCons,axiom,
    ! [X: nat > nat,Y: nat,R3: set_Pr9093778441882193744at_nat,Xs: list_nat_nat,Ys: list_nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y ) @ R3 )
     => ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs @ Ys ) @ ( listrel_nat_nat_nat @ R3 ) )
       => ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ ( cons_nat_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat_nat @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_462_listrel_OCons,axiom,
    ! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R3 ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_463_listrel__Cons1,axiom,
    ! [Y: term_f_v,Ys: list_term_f_v,Xs: list_term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ ( cons_term_f_v @ Y @ Ys ) @ Xs ) @ ( listre5907981802042383033rm_f_v @ R3 ) )
     => ~ ! [Y3: term_f_v,Ys4: list_term_f_v] :
            ( ( Xs
              = ( cons_term_f_v @ Y3 @ Ys4 ) )
           => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Y3 ) @ R3 )
             => ~ ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Ys @ Ys4 ) @ ( listre5907981802042383033rm_f_v @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_464_listrel__Cons1,axiom,
    ! [Y: bot_bot_term_f,Ys: list_Bot_bot_term_f,Xs: list_Bot_bot_term_f,R3: set_Pr6841673932529763677term_f] :
      ( ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ ( cons_Bot_bot_term_f @ Y @ Ys ) @ Xs ) @ ( listre686400052008961423term_f @ R3 ) )
     => ~ ! [Y3: bot_bot_term_f,Ys4: list_Bot_bot_term_f] :
            ( ( Xs
              = ( cons_Bot_bot_term_f @ Y3 @ Ys4 ) )
           => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Y @ Y3 ) @ R3 )
             => ~ ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Ys @ Ys4 ) @ ( listre686400052008961423term_f @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_465_listrel__Cons1,axiom,
    ! [Y: nat > nat,Ys: list_nat_nat,Xs: list_nat,R3: set_Pr9093778441882193744at_nat] :
      ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ ( cons_nat_nat @ Y @ Ys ) @ Xs ) @ ( listrel_nat_nat_nat @ R3 ) )
     => ~ ! [Y3: nat,Ys4: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y3 @ Ys4 ) )
           => ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ Y @ Y3 ) @ R3 )
             => ~ ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Ys @ Ys4 ) @ ( listrel_nat_nat_nat @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_466_listrel__Cons1,axiom,
    ! [Y: nat,Ys: list_nat,Xs: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y @ Ys ) @ Xs ) @ ( listrel_nat_nat @ R3 ) )
     => ~ ! [Y3: nat,Ys4: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y3 @ Ys4 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Y3 ) @ R3 )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Ys4 ) @ ( listrel_nat_nat @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_467_listrel__Cons2,axiom,
    ! [Xs: list_term_f_v,Y: term_f_v,Ys: list_term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Xs @ ( cons_term_f_v @ Y @ Ys ) ) @ ( listre5907981802042383033rm_f_v @ R3 ) )
     => ~ ! [X4: term_f_v,Xs3: list_term_f_v] :
            ( ( Xs
              = ( cons_term_f_v @ X4 @ Xs3 ) )
           => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y ) @ R3 )
             => ~ ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Xs3 @ Ys ) @ ( listre5907981802042383033rm_f_v @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_468_listrel__Cons2,axiom,
    ! [Xs: list_Bot_bot_term_f,Y: bot_bot_term_f,Ys: list_Bot_bot_term_f,R3: set_Pr6841673932529763677term_f] :
      ( ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Xs @ ( cons_Bot_bot_term_f @ Y @ Ys ) ) @ ( listre686400052008961423term_f @ R3 ) )
     => ~ ! [X4: bot_bot_term_f,Xs3: list_Bot_bot_term_f] :
            ( ( Xs
              = ( cons_Bot_bot_term_f @ X4 @ Xs3 ) )
           => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X4 @ Y ) @ R3 )
             => ~ ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Xs3 @ Ys ) @ ( listre686400052008961423term_f @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_469_listrel__Cons2,axiom,
    ! [Xs: list_nat_nat,Y: nat,Ys: list_nat,R3: set_Pr9093778441882193744at_nat] :
      ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat_nat @ R3 ) )
     => ~ ! [X4: nat > nat,Xs3: list_nat_nat] :
            ( ( Xs
              = ( cons_nat_nat @ X4 @ Xs3 ) )
           => ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X4 @ Y ) @ R3 )
             => ~ ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs3 @ Ys ) @ ( listrel_nat_nat_nat @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_470_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y: nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat @ R3 ) )
     => ~ ! [X4: nat,Xs3: list_nat] :
            ( ( Xs
              = ( cons_nat @ X4 @ Xs3 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y ) @ R3 )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys ) @ ( listrel_nat_nat @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_471_remove__nth__len,axiom,
    ! [I4: nat,Xs: list_term_f_v] :
      ( ( ord_less_nat @ I4 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
     => ( ( size_s7148729505879680775rm_f_v @ Xs )
        = ( suc @ ( size_s7148729505879680775rm_f_v @ ( missin2877041621942143146rm_f_v @ I4 @ Xs ) ) ) ) ) ).

% remove_nth_len
thf(fact_472_remove__nth__len,axiom,
    ! [I4: nat,Xs: list_P1780566427699531651term_f] :
      ( ( ord_less_nat @ I4 @ ( size_s9037951226665808879term_f @ Xs ) )
     => ( ( size_s9037951226665808879term_f @ Xs )
        = ( suc @ ( size_s9037951226665808879term_f @ ( missin5571439781501833334term_f @ I4 @ Xs ) ) ) ) ) ).

% remove_nth_len
thf(fact_473_remove__nth__len,axiom,
    ! [I4: nat,Xs: list_P4093298276913796397rm_f_v] :
      ( ( ord_less_nat @ I4 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
     => ( ( size_s3818039357710166425rm_f_v @ Xs )
        = ( suc @ ( size_s3818039357710166425rm_f_v @ ( missin5548135175598359328rm_f_v @ I4 @ Xs ) ) ) ) ) ).

% remove_nth_len
thf(fact_474_distinct__Ex1,axiom,
    ! [Xs: list_term_f_v,X: term_f_v] :
      ( ( distinct_term_f_v @ Xs )
     => ( ( member_term_f_v @ X @ ( set_term_f_v2 @ Xs ) )
       => ? [X4: nat] :
            ( ( ord_less_nat @ X4 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
            & ( ( nth_term_f_v @ Xs @ X4 )
              = X )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
                  & ( ( nth_term_f_v @ Xs @ Y5 )
                    = X ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_475_distinct__Ex1,axiom,
    ! [Xs: list_P1780566427699531651term_f,X: produc250624644672406269term_f] :
      ( ( distin6927496147406123510term_f @ Xs )
     => ( ( member4687689947140789414term_f @ X @ ( set_Pr1368550810757897490term_f @ Xs ) )
       => ? [X4: nat] :
            ( ( ord_less_nat @ X4 @ ( size_s9037951226665808879term_f @ Xs ) )
            & ( ( nth_Pr7080171896067517188term_f @ Xs @ X4 )
              = X )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_s9037951226665808879term_f @ Xs ) )
                  & ( ( nth_Pr7080171896067517188term_f @ Xs @ Y5 )
                    = X ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_476_distinct__Ex1,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,X: produc8027700073345061287rm_f_v] :
      ( ( distin6546576869490926752rm_f_v @ Xs )
     => ( ( member4316649861536941904rm_f_v @ X @ ( set_Pr817814403484925884rm_f_v @ Xs ) )
       => ? [X4: nat] :
            ( ( ord_less_nat @ X4 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
            & ( ( nth_Pr8768905099348956590rm_f_v @ Xs @ X4 )
              = X )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
                  & ( ( nth_Pr8768905099348956590rm_f_v @ Xs @ Y5 )
                    = X ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_477_set__supteq__into__supt,axiom,
    ! [T: term_f_v,Ts: list_term_f_v,S: term_f_v,F: f] :
      ( ( member_term_f_v @ T @ ( set_term_f_v2 @ Ts ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ S ) @ subter5033437232688913882eq_f_v )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( fun_f_v @ F @ Ts ) @ S ) @ subterm_and_supt_f_v ) ) ) ).

% set_supteq_into_supt
thf(fact_478_in__fset__conv__nth,axiom,
    ! [X: term_f_a,Xs: list_term_f_a] :
      ( ( fmember_term_f_a @ X @ ( fset_o6574051009341269640rm_f_a @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5657014664906415090rm_f_a @ Xs ) )
            & ( ( nth_term_f_a @ Xs @ I3 )
              = X ) ) ) ) ).

% in_fset_conv_nth
thf(fact_479_in__fset__conv__nth,axiom,
    ! [X: bot_bot_term_f,Xs: list_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ X @ ( fset_o7715858803885244232term_f @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1458857753309042354term_f @ Xs ) )
            & ( ( nth_Bot_bot_term_f @ Xs @ I3 )
              = X ) ) ) ) ).

% in_fset_conv_nth
thf(fact_480_in__fset__conv__nth,axiom,
    ! [X: term_f_v,Xs: list_term_f_v] :
      ( ( fmember_term_f_v @ X @ ( fset_o6574051099709074461rm_f_v @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
            & ( ( nth_term_f_v @ Xs @ I3 )
              = X ) ) ) ) ).

% in_fset_conv_nth
thf(fact_481_in__fset__conv__nth,axiom,
    ! [X: produc250624644672406269term_f,Xs: list_P1780566427699531651term_f] :
      ( ( fmembe6950792022577883716term_f @ X @ ( fset_o6125718882019406531term_f @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s9037951226665808879term_f @ Xs ) )
            & ( ( nth_Pr7080171896067517188term_f @ Xs @ I3 )
              = X ) ) ) ) ).

% in_fset_conv_nth
thf(fact_482_in__fset__conv__nth,axiom,
    ! [X: produc8027700073345061287rm_f_v,Xs: list_P4093298276913796397rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ X @ ( fset_o4970844032613833069rm_f_v @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
            & ( ( nth_Pr8768905099348956590rm_f_v @ Xs @ I3 )
              = X ) ) ) ) ).

% in_fset_conv_nth
thf(fact_483_fnth__mem,axiom,
    ! [N: nat,Xs: list_term_f_a] :
      ( ( ord_less_nat @ N @ ( size_s5657014664906415090rm_f_a @ Xs ) )
     => ( fmember_term_f_a @ ( nth_term_f_a @ Xs @ N ) @ ( fset_o6574051009341269640rm_f_a @ Xs ) ) ) ).

% fnth_mem
thf(fact_484_fnth__mem,axiom,
    ! [N: nat,Xs: list_Bot_bot_term_f] :
      ( ( ord_less_nat @ N @ ( size_s1458857753309042354term_f @ Xs ) )
     => ( fmembe1418995490368054919term_f @ ( nth_Bot_bot_term_f @ Xs @ N ) @ ( fset_o7715858803885244232term_f @ Xs ) ) ) ).

% fnth_mem
thf(fact_485_fnth__mem,axiom,
    ! [N: nat,Xs: list_term_f_v] :
      ( ( ord_less_nat @ N @ ( size_s7148729505879680775rm_f_v @ Xs ) )
     => ( fmember_term_f_v @ ( nth_term_f_v @ Xs @ N ) @ ( fset_o6574051099709074461rm_f_v @ Xs ) ) ) ).

% fnth_mem
thf(fact_486_fnth__mem,axiom,
    ! [N: nat,Xs: list_P1780566427699531651term_f] :
      ( ( ord_less_nat @ N @ ( size_s9037951226665808879term_f @ Xs ) )
     => ( fmembe6950792022577883716term_f @ ( nth_Pr7080171896067517188term_f @ Xs @ N ) @ ( fset_o6125718882019406531term_f @ Xs ) ) ) ).

% fnth_mem
thf(fact_487_fnth__mem,axiom,
    ! [N: nat,Xs: list_P4093298276913796397rm_f_v] :
      ( ( ord_less_nat @ N @ ( size_s3818039357710166425rm_f_v @ Xs ) )
     => ( fmembe774991991480784110rm_f_v @ ( nth_Pr8768905099348956590rm_f_v @ Xs @ N ) @ ( fset_o4970844032613833069rm_f_v @ Xs ) ) ) ).

% fnth_mem
thf(fact_488_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_less_eq
thf(fact_489_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_490_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_491_Cons__less__Cons,axiom,
    ! [A: nat,X: list_nat,B: nat,Y: list_nat] :
      ( ( ord_less_list_nat @ ( cons_nat @ A @ X ) @ ( cons_nat @ B @ Y ) )
      = ( ( ord_less_nat @ A @ B )
        | ( ( A = B )
          & ( ord_less_list_nat @ X @ Y ) ) ) ) ).

% Cons_less_Cons
thf(fact_492_chain__imp__trancl,axiom,
    ! [S5: nat > term_f_v,R3: set_Pr8827868859434726151rm_f_v,I4: nat,J: nat] :
      ( ! [I: nat] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( S5 @ I ) @ ( S5 @ ( suc @ I ) ) ) @ R3 )
     => ( ( ord_less_nat @ I4 @ J )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( S5 @ I4 ) @ ( S5 @ J ) ) @ ( transi3208867851621559554rm_f_v @ R3 ) ) ) ) ).

% chain_imp_trancl
thf(fact_493_chain__imp__trancl,axiom,
    ! [S5: nat > bot_bot_term_f,R3: set_Pr6841673932529763677term_f,I4: nat,J: nat] :
      ( ! [I: nat] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( S5 @ I ) @ ( S5 @ ( suc @ I ) ) ) @ R3 )
     => ( ( ord_less_nat @ I4 @ J )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( S5 @ I4 ) @ ( S5 @ J ) ) @ ( transi2774326432670905901term_f @ R3 ) ) ) ) ).

% chain_imp_trancl
thf(fact_494_chain__imp__trancl,axiom,
    ! [S5: nat > nat,R3: set_Pr1261947904930325089at_nat,I4: nat,J: nat] :
      ( ! [I: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( S5 @ I ) @ ( S5 @ ( suc @ I ) ) ) @ R3 )
     => ( ( ord_less_nat @ I4 @ J )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( S5 @ I4 ) @ ( S5 @ J ) ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ).

% chain_imp_trancl
thf(fact_495_fset__of__list__elem,axiom,
    ! [X: term_f_a,Xs: list_term_f_a] :
      ( ( fmember_term_f_a @ X @ ( fset_o6574051009341269640rm_f_a @ Xs ) )
      = ( member_term_f_a @ X @ ( set_term_f_a2 @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_496_fset__of__list__elem,axiom,
    ! [X: bot_bot_term_f,Xs: list_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ X @ ( fset_o7715858803885244232term_f @ Xs ) )
      = ( member2723211850833494437term_f @ X @ ( set_Bot_bot_term_f2 @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_497_fset__of__list__elem,axiom,
    ! [X: term_f_v,Xs: list_term_f_v] :
      ( ( fmember_term_f_v @ X @ ( fset_o6574051099709074461rm_f_v @ Xs ) )
      = ( member_term_f_v @ X @ ( set_term_f_v2 @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_498_fset__of__list__elem,axiom,
    ! [X: produc250624644672406269term_f,Xs: list_P1780566427699531651term_f] :
      ( ( fmembe6950792022577883716term_f @ X @ ( fset_o6125718882019406531term_f @ Xs ) )
      = ( member4687689947140789414term_f @ X @ ( set_Pr1368550810757897490term_f @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_499_fset__of__list__elem,axiom,
    ! [X: produc8027700073345061287rm_f_v,Xs: list_P4093298276913796397rm_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ X @ ( fset_o4970844032613833069rm_f_v @ Xs ) )
      = ( member4316649861536941904rm_f_v @ X @ ( set_Pr817814403484925884rm_f_v @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_500_length__Cons,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X @ Xs ) )
      = ( suc @ ( size_size_list_nat @ Xs ) ) ) ).

% length_Cons
thf(fact_501_length__Cons,axiom,
    ! [X: term_f_v,Xs: list_term_f_v] :
      ( ( size_s7148729505879680775rm_f_v @ ( cons_term_f_v @ X @ Xs ) )
      = ( suc @ ( size_s7148729505879680775rm_f_v @ Xs ) ) ) ).

% length_Cons
thf(fact_502_length__Cons,axiom,
    ! [X: produc250624644672406269term_f,Xs: list_P1780566427699531651term_f] :
      ( ( size_s9037951226665808879term_f @ ( cons_P3711653291972799923term_f @ X @ Xs ) )
      = ( suc @ ( size_s9037951226665808879term_f @ Xs ) ) ) ).

% length_Cons
thf(fact_503_length__Cons,axiom,
    ! [X: produc8027700073345061287rm_f_v,Xs: list_P4093298276913796397rm_f_v] :
      ( ( size_s3818039357710166425rm_f_v @ ( cons_P1983239458618365021rm_f_v @ X @ Xs ) )
      = ( suc @ ( size_s3818039357710166425rm_f_v @ Xs ) ) ) ).

% length_Cons
thf(fact_504_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_505_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% nat.inject
thf(fact_506_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_507_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_508_size__neq__size__imp__neq,axiom,
    ! [X: list_term_f_v,Y: list_term_f_v] :
      ( ( ( size_s7148729505879680775rm_f_v @ X )
       != ( size_s7148729505879680775rm_f_v @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_509_size__neq__size__imp__neq,axiom,
    ! [X: list_P1780566427699531651term_f,Y: list_P1780566427699531651term_f] :
      ( ( ( size_s9037951226665808879term_f @ X )
       != ( size_s9037951226665808879term_f @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_510_size__neq__size__imp__neq,axiom,
    ! [X: list_P4093298276913796397rm_f_v,Y: list_P4093298276913796397rm_f_v] :
      ( ( ( size_s3818039357710166425rm_f_v @ X )
       != ( size_s3818039357710166425rm_f_v @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_511_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
      = ( ( ord_less_nat @ M @ N )
        | ( ord_less_nat @ N @ M ) ) ) ).

% nat_neq_iff
thf(fact_512_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_513_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_514_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_515_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_516_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N3 )
             => ( P @ M2 ) )
         => ( P @ N3 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_517_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ~ ( P @ N3 )
         => ? [M2: nat] :
              ( ( ord_less_nat @ M2 @ N3 )
              & ~ ( P @ M2 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_518_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE_nat
thf(fact_519_pfsubsetD,axiom,
    ! [A4: fset_term_f_a,B5: fset_term_f_a,C: term_f_a] :
      ( ( ord_le672024502089505362rm_f_a @ A4 @ B5 )
     => ( ( fmember_term_f_a @ C @ A4 )
       => ( fmember_term_f_a @ C @ B5 ) ) ) ).

% pfsubsetD
thf(fact_520_pfsubsetD,axiom,
    ! [A4: fset_Bot_bot_term_f,B5: fset_Bot_bot_term_f,C: bot_bot_term_f] :
      ( ( ord_le1453809042743219474term_f @ A4 @ B5 )
     => ( ( fmembe1418995490368054919term_f @ C @ A4 )
       => ( fmembe1418995490368054919term_f @ C @ B5 ) ) ) ).

% pfsubsetD
thf(fact_521_fset__eqI,axiom,
    ! [A4: fset_term_f_a,B5: fset_term_f_a] :
      ( ! [X4: term_f_a] :
          ( ( fmember_term_f_a @ X4 @ A4 )
          = ( fmember_term_f_a @ X4 @ B5 ) )
     => ( A4 = B5 ) ) ).

% fset_eqI
thf(fact_522_fset__eqI,axiom,
    ! [A4: fset_Bot_bot_term_f,B5: fset_Bot_bot_term_f] :
      ( ! [X4: bot_bot_term_f] :
          ( ( fmembe1418995490368054919term_f @ X4 @ A4 )
          = ( fmembe1418995490368054919term_f @ X4 @ B5 ) )
     => ( A4 = B5 ) ) ).

% fset_eqI
thf(fact_523_fequalityCE,axiom,
    ! [A4: fset_term_f_a,B5: fset_term_f_a,C: term_f_a] :
      ( ( A4 = B5 )
     => ( ( ( fmember_term_f_a @ C @ A4 )
         => ~ ( fmember_term_f_a @ C @ B5 ) )
       => ~ ( ~ ( fmember_term_f_a @ C @ A4 )
           => ( fmember_term_f_a @ C @ B5 ) ) ) ) ).

% fequalityCE
thf(fact_524_fequalityCE,axiom,
    ! [A4: fset_Bot_bot_term_f,B5: fset_Bot_bot_term_f,C: bot_bot_term_f] :
      ( ( A4 = B5 )
     => ( ( ( fmembe1418995490368054919term_f @ C @ A4 )
         => ~ ( fmembe1418995490368054919term_f @ C @ B5 ) )
       => ~ ( ~ ( fmembe1418995490368054919term_f @ C @ A4 )
           => ( fmembe1418995490368054919term_f @ C @ B5 ) ) ) ) ).

% fequalityCE
thf(fact_525_eq__fmem__trans,axiom,
    ! [A: term_f_a,B: term_f_a,A4: fset_term_f_a] :
      ( ( A = B )
     => ( ( fmember_term_f_a @ B @ A4 )
       => ( fmember_term_f_a @ A @ A4 ) ) ) ).

% eq_fmem_trans
thf(fact_526_eq__fmem__trans,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,A4: fset_Bot_bot_term_f] :
      ( ( A = B )
     => ( ( fmembe1418995490368054919term_f @ B @ A4 )
       => ( fmembe1418995490368054919term_f @ A @ A4 ) ) ) ).

% eq_fmem_trans
thf(fact_527_eqfset__imp__iff,axiom,
    ! [A4: fset_term_f_a,B5: fset_term_f_a,X: term_f_a] :
      ( ( A4 = B5 )
     => ( ( fmember_term_f_a @ X @ A4 )
        = ( fmember_term_f_a @ X @ B5 ) ) ) ).

% eqfset_imp_iff
thf(fact_528_eqfset__imp__iff,axiom,
    ! [A4: fset_Bot_bot_term_f,B5: fset_Bot_bot_term_f,X: bot_bot_term_f] :
      ( ( A4 = B5 )
     => ( ( fmembe1418995490368054919term_f @ X @ A4 )
        = ( fmembe1418995490368054919term_f @ X @ B5 ) ) ) ).

% eqfset_imp_iff
thf(fact_529_if__split__fmem1,axiom,
    ! [Q3: $o,X: term_f_a,Y: term_f_a,B: fset_term_f_a] :
      ( ( fmember_term_f_a @ ( if_term_f_a @ Q3 @ X @ Y ) @ B )
      = ( ( Q3
         => ( fmember_term_f_a @ X @ B ) )
        & ( ~ Q3
         => ( fmember_term_f_a @ Y @ B ) ) ) ) ).

% if_split_fmem1
thf(fact_530_if__split__fmem1,axiom,
    ! [Q3: $o,X: bot_bot_term_f,Y: bot_bot_term_f,B: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ ( if_Bot_bot_term_f @ Q3 @ X @ Y ) @ B )
      = ( ( Q3
         => ( fmembe1418995490368054919term_f @ X @ B ) )
        & ( ~ Q3
         => ( fmembe1418995490368054919term_f @ Y @ B ) ) ) ) ).

% if_split_fmem1
thf(fact_531_if__split__fmem2,axiom,
    ! [A: term_f_a,Q3: $o,X: fset_term_f_a,Y: fset_term_f_a] :
      ( ( fmember_term_f_a @ A @ ( if_fset_term_f_a @ Q3 @ X @ Y ) )
      = ( ( Q3
         => ( fmember_term_f_a @ A @ X ) )
        & ( ~ Q3
         => ( fmember_term_f_a @ A @ Y ) ) ) ) ).

% if_split_fmem2
thf(fact_532_if__split__fmem2,axiom,
    ! [A: bot_bot_term_f,Q3: $o,X: fset_Bot_bot_term_f,Y: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ A @ ( if_fse3524882727628502648term_f @ Q3 @ X @ Y ) )
      = ( ( Q3
         => ( fmembe1418995490368054919term_f @ A @ X ) )
        & ( ~ Q3
         => ( fmembe1418995490368054919term_f @ A @ Y ) ) ) ) ).

% if_split_fmem2
thf(fact_533_eqfelem__imp__iff,axiom,
    ! [X: term_f_a,Y: term_f_a,A4: fset_term_f_a] :
      ( ( X = Y )
     => ( ( fmember_term_f_a @ X @ A4 )
        = ( fmember_term_f_a @ Y @ A4 ) ) ) ).

% eqfelem_imp_iff
thf(fact_534_eqfelem__imp__iff,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,A4: fset_Bot_bot_term_f] :
      ( ( X = Y )
     => ( ( fmembe1418995490368054919term_f @ X @ A4 )
        = ( fmembe1418995490368054919term_f @ Y @ A4 ) ) ) ).

% eqfelem_imp_iff
thf(fact_535_less__list__code_I3_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( ord_less_list_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
      = ( ( ord_less_nat @ X @ Y )
        | ( ( X = Y )
          & ( ord_less_list_nat @ Xs @ Ys ) ) ) ) ).

% less_list_code(3)
thf(fact_536_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_537_strict__inc__induct,axiom,
    ! [I4: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I4 @ J )
     => ( ! [I: nat] :
            ( ( J
              = ( suc @ I ) )
           => ( P @ I ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ J )
             => ( ( P @ ( suc @ I ) )
               => ( P @ I ) ) )
         => ( P @ I4 ) ) ) ) ).

% strict_inc_induct
thf(fact_538_less__Suc__induct,axiom,
    ! [I4: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I4 @ J )
     => ( ! [I: nat] : ( P @ I @ ( suc @ I ) )
       => ( ! [I: nat,J3: nat,K2: nat] :
              ( ( ord_less_nat @ I @ J3 )
             => ( ( ord_less_nat @ J3 @ K2 )
               => ( ( P @ I @ J3 )
                 => ( ( P @ J3 @ K2 )
                   => ( P @ I @ K2 ) ) ) ) )
         => ( P @ I4 @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_539_less__trans__Suc,axiom,
    ! [I4: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I4 @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I4 ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_540_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_less_SucD
thf(fact_541_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_542_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M )
      = ( ? [M3: nat] :
            ( ( M
              = ( suc @ M3 ) )
            & ( ord_less_nat @ N @ M3 ) ) ) ) ).

% Suc_less_eq2
thf(fact_543_All__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
           => ( P @ I3 ) ) )
      = ( ( P @ N )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
           => ( P @ I3 ) ) ) ) ).

% All_less_Suc
thf(fact_544_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_545_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( M = N ) ) ) ).

% less_Suc_eq
thf(fact_546_Ex__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
            & ( P @ I3 ) ) )
      = ( ( P @ N )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
            & ( P @ I3 ) ) ) ) ).

% Ex_less_Suc
thf(fact_547_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_548_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_549_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ( suc @ M )
         != N )
       => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).

% Suc_lessI
thf(fact_550_Suc__lessE,axiom,
    ! [I4: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I4 ) @ K )
     => ~ ! [J3: nat] :
            ( ( ord_less_nat @ I4 @ J3 )
           => ( K
             != ( suc @ J3 ) ) ) ) ).

% Suc_lessE
thf(fact_551_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_552_Nat_OlessE,axiom,
    ! [I4: nat,K: nat] :
      ( ( ord_less_nat @ I4 @ K )
     => ( ( K
         != ( suc @ I4 ) )
       => ~ ! [J3: nat] :
              ( ( ord_less_nat @ I4 @ J3 )
             => ( K
               != ( suc @ J3 ) ) ) ) ) ).

% Nat.lessE
thf(fact_553_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_554_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_555_Cons__lenlex__iff,axiom,
    ! [M: bot_bot_term_f,Ms: list_Bot_bot_term_f,N: bot_bot_term_f,Ns: list_Bot_bot_term_f,R3: set_Pr6841673932529763677term_f] :
      ( ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ ( cons_Bot_bot_term_f @ M @ Ms ) @ ( cons_Bot_bot_term_f @ N @ Ns ) ) @ ( lenlex569208748157492291term_f @ R3 ) )
      = ( ( ord_less_nat @ ( size_s1458857753309042354term_f @ Ms ) @ ( size_s1458857753309042354term_f @ Ns ) )
        | ( ( ( size_s1458857753309042354term_f @ Ms )
            = ( size_s1458857753309042354term_f @ Ns ) )
          & ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ M @ N ) @ R3 ) )
        | ( ( M = N )
          & ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Ms @ Ns ) @ ( lenlex569208748157492291term_f @ R3 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_556_Cons__lenlex__iff,axiom,
    ! [M: nat,Ms: list_nat,N: nat,Ns: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R3 ) )
      = ( ( ord_less_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) )
        | ( ( ( size_size_list_nat @ Ms )
            = ( size_size_list_nat @ Ns ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ R3 ) )
        | ( ( M = N )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R3 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_557_Cons__lenlex__iff,axiom,
    ! [M: term_f_v,Ms: list_term_f_v,N: term_f_v,Ns: list_term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ ( cons_term_f_v @ M @ Ms ) @ ( cons_term_f_v @ N @ Ns ) ) @ ( lenlex_term_f_v @ R3 ) )
      = ( ( ord_less_nat @ ( size_s7148729505879680775rm_f_v @ Ms ) @ ( size_s7148729505879680775rm_f_v @ Ns ) )
        | ( ( ( size_s7148729505879680775rm_f_v @ Ms )
            = ( size_s7148729505879680775rm_f_v @ Ns ) )
          & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ M @ N ) @ R3 ) )
        | ( ( M = N )
          & ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Ms @ Ns ) @ ( lenlex_term_f_v @ R3 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_558_Cons__lenlex__iff,axiom,
    ! [M: produc250624644672406269term_f,Ms: list_P1780566427699531651term_f,N: produc250624644672406269term_f,Ns: list_P1780566427699531651term_f,R3: set_Pr4896805198020829255term_f] :
      ( ( member2822224418761970704term_f @ ( produc1033786750570529367term_f @ ( cons_P3711653291972799923term_f @ M @ Ms ) @ ( cons_P3711653291972799923term_f @ N @ Ns ) ) @ ( lenlex9017302659811057544term_f @ R3 ) )
      = ( ( ord_less_nat @ ( size_s9037951226665808879term_f @ Ms ) @ ( size_s9037951226665808879term_f @ Ns ) )
        | ( ( ( size_s9037951226665808879term_f @ Ms )
            = ( size_s9037951226665808879term_f @ Ns ) )
          & ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ M @ N ) @ R3 ) )
        | ( ( M = N )
          & ( member2822224418761970704term_f @ ( produc1033786750570529367term_f @ Ms @ Ns ) @ ( lenlex9017302659811057544term_f @ R3 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_559_Cons__lenlex__iff,axiom,
    ! [M: produc8027700073345061287rm_f_v,Ms: list_P4093298276913796397rm_f_v,N: produc8027700073345061287rm_f_v,Ns: list_P4093298276913796397rm_f_v,R3: set_Pr7897041799657265351rm_f_v] :
      ( ( member4621649979522838928rm_f_v @ ( produc3444105967875339479rm_f_v @ ( cons_P1983239458618365021rm_f_v @ M @ Ms ) @ ( cons_P1983239458618365021rm_f_v @ N @ Ns ) ) @ ( lenlex5726706343053400626rm_f_v @ R3 ) )
      = ( ( ord_less_nat @ ( size_s3818039357710166425rm_f_v @ Ms ) @ ( size_s3818039357710166425rm_f_v @ Ns ) )
        | ( ( ( size_s3818039357710166425rm_f_v @ Ms )
            = ( size_s3818039357710166425rm_f_v @ Ns ) )
          & ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ M @ N ) @ R3 ) )
        | ( ( M = N )
          & ( member4621649979522838928rm_f_v @ ( produc3444105967875339479rm_f_v @ Ms @ Ns ) @ ( lenlex5726706343053400626rm_f_v @ R3 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_560_Cons__in__lex,axiom,
    ! [X: bot_bot_term_f,Xs: list_Bot_bot_term_f,Y: bot_bot_term_f,Ys: list_Bot_bot_term_f,R3: set_Pr6841673932529763677term_f] :
      ( ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ ( cons_Bot_bot_term_f @ X @ Xs ) @ ( cons_Bot_bot_term_f @ Y @ Ys ) ) @ ( lex_Bot_bot_term_f @ R3 ) )
      = ( ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ R3 )
          & ( ( size_s1458857753309042354term_f @ Xs )
            = ( size_s1458857753309042354term_f @ Ys ) ) )
        | ( ( X = Y )
          & ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Xs @ Ys ) @ ( lex_Bot_bot_term_f @ R3 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_561_Cons__in__lex,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( lex_nat @ R3 ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 )
          & ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) ) )
        | ( ( X = Y )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R3 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_562_Cons__in__lex,axiom,
    ! [X: term_f_v,Xs: list_term_f_v,Y: term_f_v,Ys: list_term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ ( cons_term_f_v @ X @ Xs ) @ ( cons_term_f_v @ Y @ Ys ) ) @ ( lex_term_f_v @ R3 ) )
      = ( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ R3 )
          & ( ( size_s7148729505879680775rm_f_v @ Xs )
            = ( size_s7148729505879680775rm_f_v @ Ys ) ) )
        | ( ( X = Y )
          & ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Xs @ Ys ) @ ( lex_term_f_v @ R3 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_563_Cons__in__lex,axiom,
    ! [X: produc250624644672406269term_f,Xs: list_P1780566427699531651term_f,Y: produc250624644672406269term_f,Ys: list_P1780566427699531651term_f,R3: set_Pr4896805198020829255term_f] :
      ( ( member2822224418761970704term_f @ ( produc1033786750570529367term_f @ ( cons_P3711653291972799923term_f @ X @ Xs ) @ ( cons_P3711653291972799923term_f @ Y @ Ys ) ) @ ( lex_Pr7099805488739160679term_f @ R3 ) )
      = ( ( ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ X @ Y ) @ R3 )
          & ( ( size_s9037951226665808879term_f @ Xs )
            = ( size_s9037951226665808879term_f @ Ys ) ) )
        | ( ( X = Y )
          & ( member2822224418761970704term_f @ ( produc1033786750570529367term_f @ Xs @ Ys ) @ ( lex_Pr7099805488739160679term_f @ R3 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_564_Cons__in__lex,axiom,
    ! [X: produc8027700073345061287rm_f_v,Xs: list_P4093298276913796397rm_f_v,Y: produc8027700073345061287rm_f_v,Ys: list_P4093298276913796397rm_f_v,R3: set_Pr7897041799657265351rm_f_v] :
      ( ( member4621649979522838928rm_f_v @ ( produc3444105967875339479rm_f_v @ ( cons_P1983239458618365021rm_f_v @ X @ Xs ) @ ( cons_P1983239458618365021rm_f_v @ Y @ Ys ) ) @ ( lex_Pr8333588729306351889rm_f_v @ R3 ) )
      = ( ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X @ Y ) @ R3 )
          & ( ( size_s3818039357710166425rm_f_v @ Xs )
            = ( size_s3818039357710166425rm_f_v @ Ys ) ) )
        | ( ( X = Y )
          & ( member4621649979522838928rm_f_v @ ( produc3444105967875339479rm_f_v @ Xs @ Ys ) @ ( lex_Pr8333588729306351889rm_f_v @ R3 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_565_fId__onI,axiom,
    ! [A: term_f_a,A4: fset_term_f_a] :
      ( ( fmember_term_f_a @ A @ A4 )
     => ( fmembe3725132237878994628rm_f_a @ ( produc6952645563106136757rm_f_a @ A @ A ) @ ( fSet_fId_on_term_f_a @ A4 ) ) ) ).

% fId_onI
thf(fact_566_fId__onI,axiom,
    ! [A: term_f_v,A4: fset_term_f_v] :
      ( ( fmember_term_f_v @ A @ A4 )
     => ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ A ) @ ( fSet_fId_on_term_f_v @ A4 ) ) ) ).

% fId_onI
thf(fact_567_fId__onI,axiom,
    ! [A: bot_bot_term_f,A4: fset_Bot_bot_term_f] :
      ( ( fmembe1418995490368054919term_f @ A @ A4 )
     => ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ A ) @ ( fSet_f5231733434221412358term_f @ A4 ) ) ) ).

% fId_onI
thf(fact_568_fId__onI,axiom,
    ! [A: nat,A4: fset_nat] :
      ( ( fmember_nat @ A @ A4 )
     => ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( fSet_fId_on_nat @ A4 ) ) ) ).

% fId_onI
thf(fact_569_position__diff__Cons,axiom,
    ! [I4: nat,Ps: list_nat,Qs: list_nat] :
      ( ( term_pos_diff_nat @ ( cons_nat @ I4 @ Ps ) @ ( cons_nat @ I4 @ Qs ) )
      = ( term_pos_diff_nat @ Ps @ Qs ) ) ).

% position_diff_Cons
thf(fact_570_replace__term__at_Osimps_I3_J,axiom,
    ! [I4: nat,Ts: list_term_f_v,F: f,Ps: list_nat,T: term_f_v] :
      ( ( ( ord_less_nat @ I4 @ ( size_s7148729505879680775rm_f_v @ Ts ) )
       => ( ( term_r2146176907066831980at_f_v @ ( fun_f_v @ F @ Ts ) @ ( cons_nat @ I4 @ Ps ) @ T )
          = ( fun_f_v @ F @ ( list_update_term_f_v @ Ts @ I4 @ ( term_r2146176907066831980at_f_v @ ( nth_term_f_v @ Ts @ I4 ) @ Ps @ T ) ) ) ) )
      & ( ~ ( ord_less_nat @ I4 @ ( size_s7148729505879680775rm_f_v @ Ts ) )
       => ( ( term_r2146176907066831980at_f_v @ ( fun_f_v @ F @ Ts ) @ ( cons_nat @ I4 @ Ps ) @ T )
          = ( fun_f_v @ F @ Ts ) ) ) ) ).

% replace_term_at.simps(3)
thf(fact_571_lenlex__irreflexive,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,Xs: list_term_f_v] :
      ( ! [X4: term_f_v] :
          ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ X4 ) @ R3 )
     => ~ ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Xs @ Xs ) @ ( lenlex_term_f_v @ R3 ) ) ) ).

% lenlex_irreflexive
thf(fact_572_lenlex__irreflexive,axiom,
    ! [R3: set_Pr6841673932529763677term_f,Xs: list_Bot_bot_term_f] :
      ( ! [X4: bot_bot_term_f] :
          ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X4 @ X4 ) @ R3 )
     => ~ ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Xs @ Xs ) @ ( lenlex569208748157492291term_f @ R3 ) ) ) ).

% lenlex_irreflexive
thf(fact_573_lenlex__irreflexive,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ! [X4: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R3 )
     => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lenlex_nat @ R3 ) ) ) ).

% lenlex_irreflexive
thf(fact_574_supt__size,axiom,
    ! [S: term_f_v,T: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subterm_and_supt_f_v )
     => ( ord_less_nat @ ( size_size_term_f_v @ T ) @ ( size_size_term_f_v @ S ) ) ) ).

% supt_size
thf(fact_575_fId__on__iff,axiom,
    ! [X: term_f_a,Y: term_f_a,A4: fset_term_f_a] :
      ( ( fmembe3725132237878994628rm_f_a @ ( produc6952645563106136757rm_f_a @ X @ Y ) @ ( fSet_fId_on_term_f_a @ A4 ) )
      = ( ( X = Y )
        & ( fmember_term_f_a @ X @ A4 ) ) ) ).

% fId_on_iff
thf(fact_576_fId__on__iff,axiom,
    ! [X: term_f_v,Y: term_f_v,A4: fset_term_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( fSet_fId_on_term_f_v @ A4 ) )
      = ( ( X = Y )
        & ( fmember_term_f_v @ X @ A4 ) ) ) ).

% fId_on_iff
thf(fact_577_fId__on__iff,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,A4: fset_Bot_bot_term_f] :
      ( ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( fSet_f5231733434221412358term_f @ A4 ) )
      = ( ( X = Y )
        & ( fmembe1418995490368054919term_f @ X @ A4 ) ) ) ).

% fId_on_iff
thf(fact_578_fId__on__iff,axiom,
    ! [X: nat,Y: nat,A4: fset_nat] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( fSet_fId_on_nat @ A4 ) )
      = ( ( X = Y )
        & ( fmember_nat @ X @ A4 ) ) ) ).

% fId_on_iff
thf(fact_579_fId__on__eqI,axiom,
    ! [A: term_f_a,B: term_f_a,A4: fset_term_f_a] :
      ( ( A = B )
     => ( ( fmember_term_f_a @ A @ A4 )
       => ( fmembe3725132237878994628rm_f_a @ ( produc6952645563106136757rm_f_a @ A @ B ) @ ( fSet_fId_on_term_f_a @ A4 ) ) ) ) ).

% fId_on_eqI
thf(fact_580_fId__on__eqI,axiom,
    ! [A: term_f_v,B: term_f_v,A4: fset_term_f_v] :
      ( ( A = B )
     => ( ( fmember_term_f_v @ A @ A4 )
       => ( fmembe774991991480784110rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( fSet_fId_on_term_f_v @ A4 ) ) ) ) ).

% fId_on_eqI
thf(fact_581_fId__on__eqI,axiom,
    ! [A: bot_bot_term_f,B: bot_bot_term_f,A4: fset_Bot_bot_term_f] :
      ( ( A = B )
     => ( ( fmembe1418995490368054919term_f @ A @ A4 )
       => ( fmembe6950792022577883716term_f @ ( produc2865564700782054965term_f @ A @ B ) @ ( fSet_f5231733434221412358term_f @ A4 ) ) ) ) ).

% fId_on_eqI
thf(fact_582_fId__on__eqI,axiom,
    ! [A: nat,B: nat,A4: fset_nat] :
      ( ( A = B )
     => ( ( fmember_nat @ A @ A4 )
       => ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( fSet_fId_on_nat @ A4 ) ) ) ) ).

% fId_on_eqI
thf(fact_583_fId__onE,axiom,
    ! [C: produc1754468282888495997rm_f_a,A4: fset_term_f_a] :
      ( ( fmembe3725132237878994628rm_f_a @ C @ ( fSet_fId_on_term_f_a @ A4 ) )
     => ~ ! [X4: term_f_a] :
            ( ( fmember_term_f_a @ X4 @ A4 )
           => ( C
             != ( produc6952645563106136757rm_f_a @ X4 @ X4 ) ) ) ) ).

% fId_onE
thf(fact_584_fId__onE,axiom,
    ! [C: produc8027700073345061287rm_f_v,A4: fset_term_f_v] :
      ( ( fmembe774991991480784110rm_f_v @ C @ ( fSet_fId_on_term_f_v @ A4 ) )
     => ~ ! [X4: term_f_v] :
            ( ( fmember_term_f_v @ X4 @ A4 )
           => ( C
             != ( produc2510790566102465375rm_f_v @ X4 @ X4 ) ) ) ) ).

% fId_onE
thf(fact_585_fId__onE,axiom,
    ! [C: produc250624644672406269term_f,A4: fset_Bot_bot_term_f] :
      ( ( fmembe6950792022577883716term_f @ C @ ( fSet_f5231733434221412358term_f @ A4 ) )
     => ~ ! [X4: bot_bot_term_f] :
            ( ( fmembe1418995490368054919term_f @ X4 @ A4 )
           => ( C
             != ( produc2865564700782054965term_f @ X4 @ X4 ) ) ) ) ).

% fId_onE
thf(fact_586_fId__onE,axiom,
    ! [C: product_prod_nat_nat,A4: fset_nat] :
      ( ( fmembe1449479052724974408at_nat @ C @ ( fSet_fId_on_nat @ A4 ) )
     => ~ ! [X4: nat] :
            ( ( fmember_nat @ X4 @ A4 )
           => ( C
             != ( product_Pair_nat_nat @ X4 @ X4 ) ) ) ) ).

% fId_onE
thf(fact_587_vaposs__list__fun,axiom,
    ! [P2: list_nat,F: f,Ts: list_term_f_v] :
      ( ( member_list_nat @ P2 @ ( set_list_nat2 @ ( varposs_list_f_v @ ( fun_f_v @ F @ Ts ) ) ) )
     => ~ ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s7148729505879680775rm_f_v @ Ts ) )
           => ! [Ps2: list_nat] :
                ( P2
               != ( cons_nat @ I @ Ps2 ) ) ) ) ).

% vaposs_list_fun
thf(fact_588_size__simp5,axiom,
    ! [S: term_f_v,Ss: list_term_f_v,T: term_f_v] :
      ( ( member_term_f_v @ S @ ( set_term_f_v2 @ Ss ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subterm_and_supt_f_v )
       => ( ( ord_less_nat @ ( size_size_term_f_v @ T ) @ ( size_size_term_f_v @ S ) )
         => ( ord_less_nat @ ( size_size_term_f_v @ T ) @ ( suc @ ( size_list_term_f_v @ size_size_term_f_v @ Ss ) ) ) ) ) ) ).

% size_simp5
thf(fact_589_lex__take__index,axiom,
    ! [Xs: list_Bot_bot_term_f,Ys: list_Bot_bot_term_f,R3: set_Pr6841673932529763677term_f] :
      ( ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Xs @ Ys ) @ ( lex_Bot_bot_term_f @ R3 ) )
     => ~ ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s1458857753309042354term_f @ Xs ) )
           => ( ( ord_less_nat @ I @ ( size_s1458857753309042354term_f @ Ys ) )
             => ( ( ( take_Bot_bot_term_f @ I @ Xs )
                  = ( take_Bot_bot_term_f @ I @ Ys ) )
               => ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( nth_Bot_bot_term_f @ Xs @ I ) @ ( nth_Bot_bot_term_f @ Ys @ I ) ) @ R3 ) ) ) ) ) ).

% lex_take_index
thf(fact_590_lex__take__index,axiom,
    ! [Xs: list_nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R3 ) )
     => ~ ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
           => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
             => ( ( ( take_nat @ I @ Xs )
                  = ( take_nat @ I @ Ys ) )
               => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) @ R3 ) ) ) ) ) ).

% lex_take_index
thf(fact_591_lex__take__index,axiom,
    ! [Xs: list_term_f_v,Ys: list_term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Xs @ Ys ) @ ( lex_term_f_v @ R3 ) )
     => ~ ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s7148729505879680775rm_f_v @ Xs ) )
           => ( ( ord_less_nat @ I @ ( size_s7148729505879680775rm_f_v @ Ys ) )
             => ( ( ( take_term_f_v @ I @ Xs )
                  = ( take_term_f_v @ I @ Ys ) )
               => ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( nth_term_f_v @ Xs @ I ) @ ( nth_term_f_v @ Ys @ I ) ) @ R3 ) ) ) ) ) ).

% lex_take_index
thf(fact_592_lex__take__index,axiom,
    ! [Xs: list_P1780566427699531651term_f,Ys: list_P1780566427699531651term_f,R3: set_Pr4896805198020829255term_f] :
      ( ( member2822224418761970704term_f @ ( produc1033786750570529367term_f @ Xs @ Ys ) @ ( lex_Pr7099805488739160679term_f @ R3 ) )
     => ~ ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s9037951226665808879term_f @ Xs ) )
           => ( ( ord_less_nat @ I @ ( size_s9037951226665808879term_f @ Ys ) )
             => ( ( ( take_P4256413432045430251term_f @ I @ Xs )
                  = ( take_P4256413432045430251term_f @ I @ Ys ) )
               => ~ ( member4806591834699997840term_f @ ( produc7121717381804241367term_f @ ( nth_Pr7080171896067517188term_f @ Xs @ I ) @ ( nth_Pr7080171896067517188term_f @ Ys @ I ) ) @ R3 ) ) ) ) ) ).

% lex_take_index
thf(fact_593_lex__take__index,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,Ys: list_P4093298276913796397rm_f_v,R3: set_Pr7897041799657265351rm_f_v] :
      ( ( member4621649979522838928rm_f_v @ ( produc3444105967875339479rm_f_v @ Xs @ Ys ) @ ( lex_Pr8333588729306351889rm_f_v @ R3 ) )
     => ~ ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3818039357710166425rm_f_v @ Xs ) )
           => ( ( ord_less_nat @ I @ ( size_s3818039357710166425rm_f_v @ Ys ) )
             => ( ( ( take_P6145276122034302101rm_f_v @ I @ Xs )
                  = ( take_P6145276122034302101rm_f_v @ I @ Ys ) )
               => ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( nth_Pr8768905099348956590rm_f_v @ Xs @ I ) @ ( nth_Pr8768905099348956590rm_f_v @ Ys @ I ) ) @ R3 ) ) ) ) ) ).

% lex_take_index
thf(fact_594_in__measures_I2_J,axiom,
    ! [X: term_f_v,Y: term_f_v,F: term_f_v > nat,Fs: list_term_f_v_nat] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( measures_term_f_v @ ( cons_term_f_v_nat @ F @ Fs ) ) )
      = ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
        | ( ( ( F @ X )
            = ( F @ Y ) )
          & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( measures_term_f_v @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_595_in__measures_I2_J,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f,F: bot_bot_term_f > nat,Fs: list_B8305986481258897165_f_nat] :
      ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( measur619265046736277902term_f @ ( cons_B6385786799753249095_f_nat @ F @ Fs ) ) )
      = ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
        | ( ( ( F @ X )
            = ( F @ Y ) )
          & ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( measur619265046736277902term_f @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_596_in__measures_I2_J,axiom,
    ! [X: nat,Y: nat,F: nat > nat,Fs: list_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) )
      = ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
        | ( ( ( F @ X )
            = ( F @ Y ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_597_remove__nth__sound__r,axiom,
    ! [N: nat,P2: nat,Xs: list_term_f_v] :
      ( ( ord_less_eq_nat @ N @ P2 )
     => ( ( ord_less_nat @ P2 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
       => ( ( nth_term_f_v @ ( missin2877041621942143146rm_f_v @ N @ Xs ) @ P2 )
          = ( nth_term_f_v @ Xs @ ( suc @ P2 ) ) ) ) ) ).

% remove_nth_sound_r
thf(fact_598_remove__nth__sound__r,axiom,
    ! [N: nat,P2: nat,Xs: list_P1780566427699531651term_f] :
      ( ( ord_less_eq_nat @ N @ P2 )
     => ( ( ord_less_nat @ P2 @ ( size_s9037951226665808879term_f @ Xs ) )
       => ( ( nth_Pr7080171896067517188term_f @ ( missin5571439781501833334term_f @ N @ Xs ) @ P2 )
          = ( nth_Pr7080171896067517188term_f @ Xs @ ( suc @ P2 ) ) ) ) ) ).

% remove_nth_sound_r
thf(fact_599_remove__nth__sound__r,axiom,
    ! [N: nat,P2: nat,Xs: list_P4093298276913796397rm_f_v] :
      ( ( ord_less_eq_nat @ N @ P2 )
     => ( ( ord_less_nat @ P2 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
       => ( ( nth_Pr8768905099348956590rm_f_v @ ( missin5548135175598359328rm_f_v @ N @ Xs ) @ P2 )
          = ( nth_Pr8768905099348956590rm_f_v @ Xs @ ( suc @ P2 ) ) ) ) ) ).

% remove_nth_sound_r
thf(fact_600_fsubseteq__fset__conv__nth,axiom,
    ! [Ss: list_term_f_a,T4: fset_term_f_a] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5657014664906415090rm_f_a @ Ss ) )
           => ( fmember_term_f_a @ ( nth_term_f_a @ Ss @ I3 ) @ T4 ) ) )
      = ( ord_le5168046524967222366rm_f_a @ ( fset_o6574051009341269640rm_f_a @ Ss ) @ T4 ) ) ).

% fsubseteq_fset_conv_nth
thf(fact_601_fsubseteq__fset__conv__nth,axiom,
    ! [Ss: list_Bot_bot_term_f,T4: fset_Bot_bot_term_f] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1458857753309042354term_f @ Ss ) )
           => ( fmembe1418995490368054919term_f @ ( nth_Bot_bot_term_f @ Ss @ I3 ) @ T4 ) ) )
      = ( ord_le7572167314378612510term_f @ ( fset_o7715858803885244232term_f @ Ss ) @ T4 ) ) ).

% fsubseteq_fset_conv_nth
thf(fact_602_fsubseteq__fset__conv__nth,axiom,
    ! [Ss: list_term_f_v,T4: fset_term_f_v] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7148729505879680775rm_f_v @ Ss ) )
           => ( fmember_term_f_v @ ( nth_term_f_v @ Ss @ I3 ) @ T4 ) ) )
      = ( ord_le6659761365940488051rm_f_v @ ( fset_o6574051099709074461rm_f_v @ Ss ) @ T4 ) ) ).

% fsubseteq_fset_conv_nth
thf(fact_603_fsubseteq__fset__conv__nth,axiom,
    ! [Ss: list_P1780566427699531651term_f,T4: fset_P2882900458636368611term_f] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s9037951226665808879term_f @ Ss ) )
           => ( fmembe6950792022577883716term_f @ ( nth_Pr7080171896067517188term_f @ Ss @ I3 ) @ T4 ) ) )
      = ( ord_le7092741666294821507term_f @ ( fset_o6125718882019406531term_f @ Ss ) @ T4 ) ) ).

% fsubseteq_fset_conv_nth
thf(fact_604_fsubseteq__fset__conv__nth,axiom,
    ! [Ss: list_P4093298276913796397rm_f_v,T4: fset_P8018961893305114765rm_f_v] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3818039357710166425rm_f_v @ Ss ) )
           => ( fmembe774991991480784110rm_f_v @ ( nth_Pr8768905099348956590rm_f_v @ Ss @ I3 ) @ T4 ) ) )
      = ( ord_le4587745213494032429rm_f_v @ ( fset_o4970844032613833069rm_f_v @ Ss ) @ T4 ) ) ).

% fsubseteq_fset_conv_nth
thf(fact_605_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N @ M ) ) ).

% Suc_le_mono
thf(fact_606_fsubsetI,axiom,
    ! [A4: fset_term_f_a,B5: fset_term_f_a] :
      ( ! [X4: term_f_a] :
          ( ( fmember_term_f_a @ X4 @ A4 )
         => ( fmember_term_f_a @ X4 @ B5 ) )
     => ( ord_le5168046524967222366rm_f_a @ A4 @ B5 ) ) ).

% fsubsetI
thf(fact_607_fsubsetI,axiom,
    ! [A4: fset_Bot_bot_term_f,B5: fset_Bot_bot_term_f] :
      ( ! [X4: bot_bot_term_f] :
          ( ( fmembe1418995490368054919term_f @ X4 @ A4 )
         => ( fmembe1418995490368054919term_f @ X4 @ B5 ) )
     => ( ord_le7572167314378612510term_f @ A4 @ B5 ) ) ).

% fsubsetI
thf(fact_608_take__Suc__Cons,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( ( take_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs ) )
      = ( cons_nat @ X @ ( take_nat @ N @ Xs ) ) ) ).

% take_Suc_Cons
thf(fact_609_take__all__iff,axiom,
    ! [N: nat,Xs: list_term_f_v] :
      ( ( ( take_term_f_v @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_s7148729505879680775rm_f_v @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_610_take__all__iff,axiom,
    ! [N: nat,Xs: list_P1780566427699531651term_f] :
      ( ( ( take_P4256413432045430251term_f @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_s9037951226665808879term_f @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_611_take__all__iff,axiom,
    ! [N: nat,Xs: list_P4093298276913796397rm_f_v] :
      ( ( ( take_P6145276122034302101rm_f_v @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_s3818039357710166425rm_f_v @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_612_take__all,axiom,
    ! [Xs: list_term_f_v,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s7148729505879680775rm_f_v @ Xs ) @ N )
     => ( ( take_term_f_v @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_613_take__all,axiom,
    ! [Xs: list_P1780566427699531651term_f,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s9037951226665808879term_f @ Xs ) @ N )
     => ( ( take_P4256413432045430251term_f @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_614_take__all,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s3818039357710166425rm_f_v @ Xs ) @ N )
     => ( ( take_P6145276122034302101rm_f_v @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_615_nth__take,axiom,
    ! [I4: nat,N: nat,Xs: list_term_f_v] :
      ( ( ord_less_nat @ I4 @ N )
     => ( ( nth_term_f_v @ ( take_term_f_v @ N @ Xs ) @ I4 )
        = ( nth_term_f_v @ Xs @ I4 ) ) ) ).

% nth_take
thf(fact_616_nth__take,axiom,
    ! [I4: nat,N: nat,Xs: list_P1780566427699531651term_f] :
      ( ( ord_less_nat @ I4 @ N )
     => ( ( nth_Pr7080171896067517188term_f @ ( take_P4256413432045430251term_f @ N @ Xs ) @ I4 )
        = ( nth_Pr7080171896067517188term_f @ Xs @ I4 ) ) ) ).

% nth_take
thf(fact_617_nth__take,axiom,
    ! [I4: nat,N: nat,Xs: list_P4093298276913796397rm_f_v] :
      ( ( ord_less_nat @ I4 @ N )
     => ( ( nth_Pr8768905099348956590rm_f_v @ ( take_P6145276122034302101rm_f_v @ N @ Xs ) @ I4 )
        = ( nth_Pr8768905099348956590rm_f_v @ Xs @ I4 ) ) ) ).

% nth_take
thf(fact_618_list__update__beyond,axiom,
    ! [Xs: list_term_f_v,I4: nat,X: term_f_v] :
      ( ( ord_less_eq_nat @ ( size_s7148729505879680775rm_f_v @ Xs ) @ I4 )
     => ( ( list_update_term_f_v @ Xs @ I4 @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_619_list__update__beyond,axiom,
    ! [Xs: list_P1780566427699531651term_f,I4: nat,X: produc250624644672406269term_f] :
      ( ( ord_less_eq_nat @ ( size_s9037951226665808879term_f @ Xs ) @ I4 )
     => ( ( list_u2231791752812534891term_f @ Xs @ I4 @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_620_list__update__beyond,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,I4: nat,X: produc8027700073345061287rm_f_v] :
      ( ( ord_less_eq_nat @ ( size_s3818039357710166425rm_f_v @ Xs ) @ I4 )
     => ( ( list_u2658473446164013845rm_f_v @ Xs @ I4 @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_621_measures__lesseq,axiom,
    ! [F: term_f_v > nat,X: term_f_v,Y: term_f_v,Fs: list_term_f_v_nat] :
      ( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( measures_term_f_v @ Fs ) )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( measures_term_f_v @ ( cons_term_f_v_nat @ F @ Fs ) ) ) ) ) ).

% measures_lesseq
thf(fact_622_measures__lesseq,axiom,
    ! [F: bot_bot_term_f > nat,X: bot_bot_term_f,Y: bot_bot_term_f,Fs: list_B8305986481258897165_f_nat] :
      ( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( measur619265046736277902term_f @ Fs ) )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( measur619265046736277902term_f @ ( cons_B6385786799753249095_f_nat @ F @ Fs ) ) ) ) ) ).

% measures_lesseq
thf(fact_623_measures__lesseq,axiom,
    ! [F: nat > nat,X: nat,Y: nat,Fs: list_nat_nat] :
      ( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ Fs ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) ) ) ) ).

% measures_lesseq
thf(fact_624_size__list__pointwise,axiom,
    ! [Xs: list_term_f_v,F: term_f_v > nat,G: term_f_v > nat] :
      ( ! [X4: term_f_v] :
          ( ( member_term_f_v @ X4 @ ( set_term_f_v2 @ Xs ) )
         => ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
     => ( ord_less_eq_nat @ ( size_list_term_f_v @ F @ Xs ) @ ( size_list_term_f_v @ G @ Xs ) ) ) ).

% size_list_pointwise
thf(fact_625_size__list__pointwise,axiom,
    ! [Xs: list_P1780566427699531651term_f,F: produc250624644672406269term_f > nat,G: produc250624644672406269term_f > nat] :
      ( ! [X4: produc250624644672406269term_f] :
          ( ( member4687689947140789414term_f @ X4 @ ( set_Pr1368550810757897490term_f @ Xs ) )
         => ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
     => ( ord_less_eq_nat @ ( size_l6230091532269729029term_f @ F @ Xs ) @ ( size_l6230091532269729029term_f @ G @ Xs ) ) ) ).

% size_list_pointwise
thf(fact_626_size__list__pointwise,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,F: produc8027700073345061287rm_f_v > nat,G: produc8027700073345061287rm_f_v > nat] :
      ( ! [X4: produc8027700073345061287rm_f_v] :
          ( ( member4316649861536941904rm_f_v @ X4 @ ( set_Pr817814403484925884rm_f_v @ Xs ) )
         => ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
     => ( ord_less_eq_nat @ ( size_l1913922890036356015rm_f_v @ F @ Xs ) @ ( size_l1913922890036356015rm_f_v @ G @ Xs ) ) ) ).

% size_list_pointwise
thf(fact_627_size__list__estimation_H,axiom,
    ! [X: term_f_v,Xs: list_term_f_v,Y: nat,F: term_f_v > nat] :
      ( ( member_term_f_v @ X @ ( set_term_f_v2 @ Xs ) )
     => ( ( ord_less_eq_nat @ Y @ ( F @ X ) )
       => ( ord_less_eq_nat @ Y @ ( size_list_term_f_v @ F @ Xs ) ) ) ) ).

% size_list_estimation'
thf(fact_628_size__list__estimation_H,axiom,
    ! [X: produc250624644672406269term_f,Xs: list_P1780566427699531651term_f,Y: nat,F: produc250624644672406269term_f > nat] :
      ( ( member4687689947140789414term_f @ X @ ( set_Pr1368550810757897490term_f @ Xs ) )
     => ( ( ord_less_eq_nat @ Y @ ( F @ X ) )
       => ( ord_less_eq_nat @ Y @ ( size_l6230091532269729029term_f @ F @ Xs ) ) ) ) ).

% size_list_estimation'
thf(fact_629_size__list__estimation_H,axiom,
    ! [X: produc8027700073345061287rm_f_v,Xs: list_P4093298276913796397rm_f_v,Y: nat,F: produc8027700073345061287rm_f_v > nat] :
      ( ( member4316649861536941904rm_f_v @ X @ ( set_Pr817814403484925884rm_f_v @ Xs ) )
     => ( ( ord_less_eq_nat @ Y @ ( F @ X ) )
       => ( ord_less_eq_nat @ Y @ ( size_l1913922890036356015rm_f_v @ F @ Xs ) ) ) ) ).

% size_list_estimation'
thf(fact_630_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_631_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_Pr5245412377734449720_f_nat,N: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_le8976984241387448984_f_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_le8976984241387448984_f_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_632_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_633_lift__Suc__mono__le,axiom,
    ! [F: nat > set_Pr5245412377734449720_f_nat,N: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_le8976984241387448984_f_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_le8976984241387448984_f_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_634_in__set__takeD,axiom,
    ! [X: term_f_v,N: nat,Xs: list_term_f_v] :
      ( ( member_term_f_v @ X @ ( set_term_f_v2 @ ( take_term_f_v @ N @ Xs ) ) )
     => ( member_term_f_v @ X @ ( set_term_f_v2 @ Xs ) ) ) ).

% in_set_takeD
thf(fact_635_in__set__takeD,axiom,
    ! [X: produc250624644672406269term_f,N: nat,Xs: list_P1780566427699531651term_f] :
      ( ( member4687689947140789414term_f @ X @ ( set_Pr1368550810757897490term_f @ ( take_P4256413432045430251term_f @ N @ Xs ) ) )
     => ( member4687689947140789414term_f @ X @ ( set_Pr1368550810757897490term_f @ Xs ) ) ) ).

% in_set_takeD
thf(fact_636_in__set__takeD,axiom,
    ! [X: produc8027700073345061287rm_f_v,N: nat,Xs: list_P4093298276913796397rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ X @ ( set_Pr817814403484925884rm_f_v @ ( take_P6145276122034302101rm_f_v @ N @ Xs ) ) )
     => ( member4316649861536941904rm_f_v @ X @ ( set_Pr817814403484925884rm_f_v @ Xs ) ) ) ).

% in_set_takeD
thf(fact_637_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_638_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M @ N )
       => ( M
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_639_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_640_Suc__le__D,axiom,
    ! [N: nat,M4: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
     => ? [M5: nat] :
          ( M4
          = ( suc @ M5 ) ) ) ).

% Suc_le_D
thf(fact_641_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M @ N )
        | ( M
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_642_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_643_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).

% not_less_eq_eq
thf(fact_644_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
             => ( P @ M2 ) )
         => ( P @ N3 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_645_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P @ M )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ M @ N3 )
             => ( ( P @ N3 )
               => ( P @ ( suc @ N3 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_646_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R4: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X4: nat] : ( R4 @ X4 @ X4 )
       => ( ! [X4: nat,Y3: nat,Z3: nat] :
              ( ( R4 @ X4 @ Y3 )
             => ( ( R4 @ Y3 @ Z3 )
               => ( R4 @ X4 @ Z3 ) ) )
         => ( ! [N3: nat] : ( R4 @ N3 @ ( suc @ N3 ) )
           => ( R4 @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_647_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I4: nat,J: nat] :
      ( ! [I: nat,J3: nat] :
          ( ( ord_less_nat @ I @ J3 )
         => ( ord_less_nat @ ( F @ I ) @ ( F @ J3 ) ) )
     => ( ( ord_less_eq_nat @ I4 @ J )
       => ( ord_less_eq_nat @ ( F @ I4 ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_648_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( M != N )
       => ( ord_less_nat @ M @ N ) ) ) ).

% le_neq_implies_less
thf(fact_649_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_nat @ M @ N )
        | ( M = N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_or_eq_imp_le
thf(fact_650_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M6: nat,N2: nat] :
          ( ( ord_less_nat @ M6 @ N2 )
          | ( M6 = N2 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_651_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_imp_le_nat
thf(fact_652_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M6: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M6 @ N2 )
          & ( M6 != N2 ) ) ) ) ).

% nat_less_le
thf(fact_653_fsubsetD,axiom,
    ! [A4: fset_term_f_a,B5: fset_term_f_a,C: term_f_a] :
      ( ( ord_le5168046524967222366rm_f_a @ A4 @ B5 )
     => ( ( fmember_term_f_a @ C @ A4 )
       => ( fmember_term_f_a @ C @ B5 ) ) ) ).

% fsubsetD
thf(fact_654_fsubsetD,axiom,
    ! [A4: fset_Bot_bot_term_f,B5: fset_Bot_bot_term_f,C: bot_bot_term_f] :
      ( ( ord_le7572167314378612510term_f @ A4 @ B5 )
     => ( ( fmembe1418995490368054919term_f @ C @ A4 )
       => ( fmembe1418995490368054919term_f @ C @ B5 ) ) ) ).

% fsubsetD
thf(fact_655_fin__mono,axiom,
    ! [A4: fset_term_f_a,B5: fset_term_f_a,X: term_f_a] :
      ( ( ord_le5168046524967222366rm_f_a @ A4 @ B5 )
     => ( ( fmember_term_f_a @ X @ A4 )
       => ( fmember_term_f_a @ X @ B5 ) ) ) ).

% fin_mono
thf(fact_656_fin__mono,axiom,
    ! [A4: fset_Bot_bot_term_f,B5: fset_Bot_bot_term_f,X: bot_bot_term_f] :
      ( ( ord_le7572167314378612510term_f @ A4 @ B5 )
     => ( ( fmembe1418995490368054919term_f @ X @ A4 )
       => ( fmembe1418995490368054919term_f @ X @ B5 ) ) ) ).

% fin_mono
thf(fact_657_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_term_f_v,Ys: list_term_f_v] :
      ( ( ord_less_eq_nat @ K @ ( size_s7148729505879680775rm_f_v @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s7148729505879680775rm_f_v @ Ys ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ K )
             => ( ( nth_term_f_v @ Xs @ I )
                = ( nth_term_f_v @ Ys @ I ) ) )
         => ( ( take_term_f_v @ K @ Xs )
            = ( take_term_f_v @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_658_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_P1780566427699531651term_f,Ys: list_P1780566427699531651term_f] :
      ( ( ord_less_eq_nat @ K @ ( size_s9037951226665808879term_f @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s9037951226665808879term_f @ Ys ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ K )
             => ( ( nth_Pr7080171896067517188term_f @ Xs @ I )
                = ( nth_Pr7080171896067517188term_f @ Ys @ I ) ) )
         => ( ( take_P4256413432045430251term_f @ K @ Xs )
            = ( take_P4256413432045430251term_f @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_659_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_P4093298276913796397rm_f_v,Ys: list_P4093298276913796397rm_f_v] :
      ( ( ord_less_eq_nat @ K @ ( size_s3818039357710166425rm_f_v @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s3818039357710166425rm_f_v @ Ys ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ K )
             => ( ( nth_Pr8768905099348956590rm_f_v @ Xs @ I )
                = ( nth_Pr8768905099348956590rm_f_v @ Ys @ I ) ) )
         => ( ( take_P6145276122034302101rm_f_v @ K @ Xs )
            = ( take_P6145276122034302101rm_f_v @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_660_nth__take__prefix,axiom,
    ! [Ys: list_term_f_v,Xs: list_term_f_v] :
      ( ( ord_less_eq_nat @ ( size_s7148729505879680775rm_f_v @ Ys ) @ ( size_s7148729505879680775rm_f_v @ Xs ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s7148729505879680775rm_f_v @ Ys ) )
           => ( ( nth_term_f_v @ Xs @ I )
              = ( nth_term_f_v @ Ys @ I ) ) )
       => ( ( take_term_f_v @ ( size_s7148729505879680775rm_f_v @ Ys ) @ Xs )
          = Ys ) ) ) ).

% nth_take_prefix
thf(fact_661_nth__take__prefix,axiom,
    ! [Ys: list_P1780566427699531651term_f,Xs: list_P1780566427699531651term_f] :
      ( ( ord_less_eq_nat @ ( size_s9037951226665808879term_f @ Ys ) @ ( size_s9037951226665808879term_f @ Xs ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s9037951226665808879term_f @ Ys ) )
           => ( ( nth_Pr7080171896067517188term_f @ Xs @ I )
              = ( nth_Pr7080171896067517188term_f @ Ys @ I ) ) )
       => ( ( take_P4256413432045430251term_f @ ( size_s9037951226665808879term_f @ Ys ) @ Xs )
          = Ys ) ) ) ).

% nth_take_prefix
thf(fact_662_nth__take__prefix,axiom,
    ! [Ys: list_P4093298276913796397rm_f_v,Xs: list_P4093298276913796397rm_f_v] :
      ( ( ord_less_eq_nat @ ( size_s3818039357710166425rm_f_v @ Ys ) @ ( size_s3818039357710166425rm_f_v @ Xs ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3818039357710166425rm_f_v @ Ys ) )
           => ( ( nth_Pr8768905099348956590rm_f_v @ Xs @ I )
              = ( nth_Pr8768905099348956590rm_f_v @ Ys @ I ) ) )
       => ( ( take_P6145276122034302101rm_f_v @ ( size_s3818039357710166425rm_f_v @ Ys ) @ Xs )
          = Ys ) ) ) ).

% nth_take_prefix
thf(fact_663_impossible__Cons,axiom,
    ! [Xs: list_nat,Ys: list_nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
     => ( Xs
       != ( cons_nat @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_664_impossible__Cons,axiom,
    ! [Xs: list_term_f_v,Ys: list_term_f_v,X: term_f_v] :
      ( ( ord_less_eq_nat @ ( size_s7148729505879680775rm_f_v @ Xs ) @ ( size_s7148729505879680775rm_f_v @ Ys ) )
     => ( Xs
       != ( cons_term_f_v @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_665_impossible__Cons,axiom,
    ! [Xs: list_P1780566427699531651term_f,Ys: list_P1780566427699531651term_f,X: produc250624644672406269term_f] :
      ( ( ord_less_eq_nat @ ( size_s9037951226665808879term_f @ Xs ) @ ( size_s9037951226665808879term_f @ Ys ) )
     => ( Xs
       != ( cons_P3711653291972799923term_f @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_666_impossible__Cons,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,Ys: list_P4093298276913796397rm_f_v,X: produc8027700073345061287rm_f_v] :
      ( ( ord_less_eq_nat @ ( size_s3818039357710166425rm_f_v @ Xs ) @ ( size_s3818039357710166425rm_f_v @ Ys ) )
     => ( Xs
       != ( cons_P1983239458618365021rm_f_v @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_667_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_668_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_le_eq
thf(fact_669_dec__induct,axiom,
    ! [I4: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I4 @ J )
     => ( ( P @ I4 )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I4 @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P @ N3 )
                 => ( P @ ( suc @ N3 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_670_inc__induct,axiom,
    ! [I4: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I4 @ J )
     => ( ( P @ J )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I4 @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P @ ( suc @ N3 ) )
                 => ( P @ N3 ) ) ) )
         => ( P @ I4 ) ) ) ) ).

% inc_induct
thf(fact_671_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_le_lessD
thf(fact_672_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% le_less_Suc_eq
thf(fact_673_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% less_Suc_eq_le
thf(fact_674_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_675_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_676_size__simp6,axiom,
    ! [S: term_f_v,Ss: list_term_f_v,T: term_f_v] :
      ( ( member_term_f_v @ S @ ( set_term_f_v2 @ Ss ) )
     => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subter5033437232688913882eq_f_v )
       => ( ( ord_less_eq_nat @ ( size_size_term_f_v @ T ) @ ( size_size_term_f_v @ S ) )
         => ( ord_less_eq_nat @ ( size_size_term_f_v @ T ) @ ( suc @ ( size_list_term_f_v @ size_size_term_f_v @ Ss ) ) ) ) ) ) ).

% size_simp6
thf(fact_677_supteq__size,axiom,
    ! [S: term_f_v,T: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ subter5033437232688913882eq_f_v )
     => ( ord_less_eq_nat @ ( size_size_term_f_v @ T ) @ ( size_size_term_f_v @ S ) ) ) ).

% supteq_size
thf(fact_678_remove__nth__id,axiom,
    ! [Xs: list_term_f_v,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s7148729505879680775rm_f_v @ Xs ) @ N )
     => ( ( missin2877041621942143146rm_f_v @ N @ Xs )
        = Xs ) ) ).

% remove_nth_id
thf(fact_679_remove__nth__id,axiom,
    ! [Xs: list_P1780566427699531651term_f,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s9037951226665808879term_f @ Xs ) @ N )
     => ( ( missin5571439781501833334term_f @ N @ Xs )
        = Xs ) ) ).

% remove_nth_id
thf(fact_680_remove__nth__id,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s3818039357710166425rm_f_v @ Xs ) @ N )
     => ( ( missin5548135175598359328rm_f_v @ N @ Xs )
        = Xs ) ) ).

% remove_nth_id
thf(fact_681_size__list__estimation,axiom,
    ! [X: term_f_v,Xs: list_term_f_v,Y: nat,F: term_f_v > nat] :
      ( ( member_term_f_v @ X @ ( set_term_f_v2 @ Xs ) )
     => ( ( ord_less_nat @ Y @ ( F @ X ) )
       => ( ord_less_nat @ Y @ ( size_list_term_f_v @ F @ Xs ) ) ) ) ).

% size_list_estimation
thf(fact_682_size__list__estimation,axiom,
    ! [X: produc250624644672406269term_f,Xs: list_P1780566427699531651term_f,Y: nat,F: produc250624644672406269term_f > nat] :
      ( ( member4687689947140789414term_f @ X @ ( set_Pr1368550810757897490term_f @ Xs ) )
     => ( ( ord_less_nat @ Y @ ( F @ X ) )
       => ( ord_less_nat @ Y @ ( size_l6230091532269729029term_f @ F @ Xs ) ) ) ) ).

% size_list_estimation
thf(fact_683_size__list__estimation,axiom,
    ! [X: produc8027700073345061287rm_f_v,Xs: list_P4093298276913796397rm_f_v,Y: nat,F: produc8027700073345061287rm_f_v > nat] :
      ( ( member4316649861536941904rm_f_v @ X @ ( set_Pr817814403484925884rm_f_v @ Xs ) )
     => ( ( ord_less_nat @ Y @ ( F @ X ) )
       => ( ord_less_nat @ Y @ ( size_l1913922890036356015rm_f_v @ F @ Xs ) ) ) ) ).

% size_list_estimation
thf(fact_684_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
      = ( ? [X5: nat,Ys2: list_nat] :
            ( ( Xs
              = ( cons_nat @ X5 @ Ys2 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_685_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_term_f_v] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s7148729505879680775rm_f_v @ Xs ) )
      = ( ? [X5: term_f_v,Ys2: list_term_f_v] :
            ( ( Xs
              = ( cons_term_f_v @ X5 @ Ys2 ) )
            & ( ord_less_eq_nat @ N @ ( size_s7148729505879680775rm_f_v @ Ys2 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_686_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_P1780566427699531651term_f] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s9037951226665808879term_f @ Xs ) )
      = ( ? [X5: produc250624644672406269term_f,Ys2: list_P1780566427699531651term_f] :
            ( ( Xs
              = ( cons_P3711653291972799923term_f @ X5 @ Ys2 ) )
            & ( ord_less_eq_nat @ N @ ( size_s9037951226665808879term_f @ Ys2 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_687_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_P4093298276913796397rm_f_v] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s3818039357710166425rm_f_v @ Xs ) )
      = ( ? [X5: produc8027700073345061287rm_f_v,Ys2: list_P4093298276913796397rm_f_v] :
            ( ( Xs
              = ( cons_P1983239458618365021rm_f_v @ X5 @ Ys2 ) )
            & ( ord_less_eq_nat @ N @ ( size_s3818039357710166425rm_f_v @ Ys2 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_688_subterm_Olift__Suc__mono__le,axiom,
    ! [F: nat > term_f_v,N: nat,N4: nat] :
      ( ! [N3: nat] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) ) @ subter5033437232688913882eq_f_v )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F @ N4 ) @ ( F @ N ) ) @ subter5033437232688913882eq_f_v ) ) ) ).

% subterm.lift_Suc_mono_le
thf(fact_689_subterm_Olift__Suc__antimono__le,axiom,
    ! [F: nat > term_f_v,N: nat,N4: nat] :
      ( ! [N3: nat] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) ) @ subter5033437232688913882eq_f_v )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F @ N ) @ ( F @ N4 ) ) @ subter5033437232688913882eq_f_v ) ) ) ).

% subterm.lift_Suc_antimono_le
thf(fact_690_lenlex__length,axiom,
    ! [Ms: list_term_f_v,Ns: list_term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Ms @ Ns ) @ ( lenlex_term_f_v @ R3 ) )
     => ( ord_less_eq_nat @ ( size_s7148729505879680775rm_f_v @ Ms ) @ ( size_s7148729505879680775rm_f_v @ Ns ) ) ) ).

% lenlex_length
thf(fact_691_lenlex__length,axiom,
    ! [Ms: list_P1780566427699531651term_f,Ns: list_P1780566427699531651term_f,R3: set_Pr4896805198020829255term_f] :
      ( ( member2822224418761970704term_f @ ( produc1033786750570529367term_f @ Ms @ Ns ) @ ( lenlex9017302659811057544term_f @ R3 ) )
     => ( ord_less_eq_nat @ ( size_s9037951226665808879term_f @ Ms ) @ ( size_s9037951226665808879term_f @ Ns ) ) ) ).

% lenlex_length
thf(fact_692_lenlex__length,axiom,
    ! [Ms: list_P4093298276913796397rm_f_v,Ns: list_P4093298276913796397rm_f_v,R3: set_Pr7897041799657265351rm_f_v] :
      ( ( member4621649979522838928rm_f_v @ ( produc3444105967875339479rm_f_v @ Ms @ Ns ) @ ( lenlex5726706343053400626rm_f_v @ R3 ) )
     => ( ord_less_eq_nat @ ( size_s3818039357710166425rm_f_v @ Ms ) @ ( size_s3818039357710166425rm_f_v @ Ns ) ) ) ).

% lenlex_length
thf(fact_693_set__take__nth,axiom,
    ! [X: term_f_v,I4: nat,Xs: list_term_f_v] :
      ( ( member_term_f_v @ X @ ( set_term_f_v2 @ ( take_term_f_v @ I4 @ Xs ) ) )
     => ? [J3: nat] :
          ( ( ord_less_nat @ J3 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
          & ( ord_less_nat @ J3 @ I4 )
          & ( ( nth_term_f_v @ Xs @ J3 )
            = X ) ) ) ).

% set_take_nth
thf(fact_694_set__take__nth,axiom,
    ! [X: produc250624644672406269term_f,I4: nat,Xs: list_P1780566427699531651term_f] :
      ( ( member4687689947140789414term_f @ X @ ( set_Pr1368550810757897490term_f @ ( take_P4256413432045430251term_f @ I4 @ Xs ) ) )
     => ? [J3: nat] :
          ( ( ord_less_nat @ J3 @ ( size_s9037951226665808879term_f @ Xs ) )
          & ( ord_less_nat @ J3 @ I4 )
          & ( ( nth_Pr7080171896067517188term_f @ Xs @ J3 )
            = X ) ) ) ).

% set_take_nth
thf(fact_695_set__take__nth,axiom,
    ! [X: produc8027700073345061287rm_f_v,I4: nat,Xs: list_P4093298276913796397rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ X @ ( set_Pr817814403484925884rm_f_v @ ( take_P6145276122034302101rm_f_v @ I4 @ Xs ) ) )
     => ? [J3: nat] :
          ( ( ord_less_nat @ J3 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
          & ( ord_less_nat @ J3 @ I4 )
          & ( ( nth_Pr8768905099348956590rm_f_v @ Xs @ J3 )
            = X ) ) ) ).

% set_take_nth
thf(fact_696_measures__less,axiom,
    ! [F: term_f_v > nat,X: term_f_v,Y: term_f_v,Fs: list_term_f_v_nat] :
      ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( measures_term_f_v @ ( cons_term_f_v_nat @ F @ Fs ) ) ) ) ).

% measures_less
thf(fact_697_measures__less,axiom,
    ! [F: bot_bot_term_f > nat,X: bot_bot_term_f,Y: bot_bot_term_f,Fs: list_B8305986481258897165_f_nat] :
      ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( measur619265046736277902term_f @ ( cons_B6385786799753249095_f_nat @ F @ Fs ) ) ) ) ).

% measures_less
thf(fact_698_measures__less,axiom,
    ! [F: nat > nat,X: nat,Y: nat,Fs: list_nat_nat] :
      ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) ) ) ).

% measures_less
thf(fact_699_size__simp1,axiom,
    ! [T: term_f_v,Ts: list_term_f_v] :
      ( ( member_term_f_v @ T @ ( set_term_f_v2 @ Ts ) )
     => ( ord_less_nat @ ( size_size_term_f_v @ T ) @ ( suc @ ( size_list_term_f_v @ size_size_term_f_v @ Ts ) ) ) ) ).

% size_simp1
thf(fact_700_size__simp1,axiom,
    ! [T: produc250624644672406269term_f,Ts: list_P1780566427699531651term_f] :
      ( ( member4687689947140789414term_f @ T @ ( set_Pr1368550810757897490term_f @ Ts ) )
     => ( ord_less_nat @ ( size_s304001196500897129term_f @ T ) @ ( suc @ ( size_l6230091532269729029term_f @ size_s304001196500897129term_f @ Ts ) ) ) ) ).

% size_simp1
thf(fact_701_size__simp1,axiom,
    ! [T: produc8027700073345061287rm_f_v,Ts: list_P4093298276913796397rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ T @ ( set_Pr817814403484925884rm_f_v @ Ts ) )
     => ( ord_less_nat @ ( size_s103389704072612883rm_f_v @ T ) @ ( suc @ ( size_l1913922890036356015rm_f_v @ size_s103389704072612883rm_f_v @ Ts ) ) ) ) ).

% size_simp1
thf(fact_702_size__simp1,axiom,
    ! [T: list_term_f_v,Ts: list_list_term_f_v] :
      ( ( member_list_term_f_v @ T @ ( set_list_term_f_v2 @ Ts ) )
     => ( ord_less_nat @ ( size_s7148729505879680775rm_f_v @ T ) @ ( suc @ ( size_l5830554566379815531rm_f_v @ size_s7148729505879680775rm_f_v @ Ts ) ) ) ) ).

% size_simp1
thf(fact_703_size__simp1,axiom,
    ! [T: list_P1780566427699531651term_f,Ts: list_l9126709995455877001term_f] :
      ( ( member6232163599979755692term_f @ T @ ( set_li3839205001427786520term_f @ Ts ) )
     => ( ord_less_nat @ ( size_s9037951226665808879term_f @ T ) @ ( suc @ ( size_l1335942268763413899term_f @ size_s9037951226665808879term_f @ Ts ) ) ) ) ).

% size_simp1
thf(fact_704_size__simp1,axiom,
    ! [T: list_P4093298276913796397rm_f_v,Ts: list_l4945967923587861555rm_f_v] :
      ( ( member2864856320113173078rm_f_v @ T @ ( set_li3345156661183018178rm_f_v @ Ts ) )
     => ( ord_less_nat @ ( size_s3818039357710166425rm_f_v @ T ) @ ( suc @ ( size_l8652417004037522229rm_f_v @ size_s3818039357710166425rm_f_v @ Ts ) ) ) ) ).

% size_simp1
thf(fact_705_fset__list__fsubset__eq__nth__conv,axiom,
    ! [Xs: list_term_f_a,A4: fset_term_f_a] :
      ( ( ord_le5168046524967222366rm_f_a @ ( fset_o6574051009341269640rm_f_a @ Xs ) @ A4 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5657014664906415090rm_f_a @ Xs ) )
           => ( fmember_term_f_a @ ( nth_term_f_a @ Xs @ I3 ) @ A4 ) ) ) ) ).

% fset_list_fsubset_eq_nth_conv
thf(fact_706_fset__list__fsubset__eq__nth__conv,axiom,
    ! [Xs: list_Bot_bot_term_f,A4: fset_Bot_bot_term_f] :
      ( ( ord_le7572167314378612510term_f @ ( fset_o7715858803885244232term_f @ Xs ) @ A4 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1458857753309042354term_f @ Xs ) )
           => ( fmembe1418995490368054919term_f @ ( nth_Bot_bot_term_f @ Xs @ I3 ) @ A4 ) ) ) ) ).

% fset_list_fsubset_eq_nth_conv
thf(fact_707_fset__list__fsubset__eq__nth__conv,axiom,
    ! [Xs: list_term_f_v,A4: fset_term_f_v] :
      ( ( ord_le6659761365940488051rm_f_v @ ( fset_o6574051099709074461rm_f_v @ Xs ) @ A4 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
           => ( fmember_term_f_v @ ( nth_term_f_v @ Xs @ I3 ) @ A4 ) ) ) ) ).

% fset_list_fsubset_eq_nth_conv
thf(fact_708_fset__list__fsubset__eq__nth__conv,axiom,
    ! [Xs: list_P1780566427699531651term_f,A4: fset_P2882900458636368611term_f] :
      ( ( ord_le7092741666294821507term_f @ ( fset_o6125718882019406531term_f @ Xs ) @ A4 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s9037951226665808879term_f @ Xs ) )
           => ( fmembe6950792022577883716term_f @ ( nth_Pr7080171896067517188term_f @ Xs @ I3 ) @ A4 ) ) ) ) ).

% fset_list_fsubset_eq_nth_conv
thf(fact_709_fset__list__fsubset__eq__nth__conv,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,A4: fset_P8018961893305114765rm_f_v] :
      ( ( ord_le4587745213494032429rm_f_v @ ( fset_o4970844032613833069rm_f_v @ Xs ) @ A4 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
           => ( fmembe774991991480784110rm_f_v @ ( nth_Pr8768905099348956590rm_f_v @ Xs @ I3 ) @ A4 ) ) ) ) ).

% fset_list_fsubset_eq_nth_conv
thf(fact_710_less__eq__prod__simp,axiom,
    ! [X1: nat > nat,Y1: nat,X22: nat > nat,Y22: nat] :
      ( ( ord_le2819838839419867280at_nat @ ( produc72220940542539688at_nat @ X1 @ Y1 ) @ ( produc72220940542539688at_nat @ X22 @ Y22 ) )
      = ( ( ord_less_nat_nat @ X1 @ X22 )
        | ( ( ord_less_eq_nat_nat @ X1 @ X22 )
          & ( ord_less_eq_nat @ Y1 @ Y22 ) ) ) ) ).

% less_eq_prod_simp
thf(fact_711_less__eq__prod__simp,axiom,
    ! [X1: nat,Y1: nat,X22: nat,Y22: nat] :
      ( ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ X1 @ Y1 ) @ ( product_Pair_nat_nat @ X22 @ Y22 ) )
      = ( ( ord_less_nat @ X1 @ X22 )
        | ( ( ord_less_eq_nat @ X1 @ X22 )
          & ( ord_less_eq_nat @ Y1 @ Y22 ) ) ) ) ).

% less_eq_prod_simp
thf(fact_712_less__eq__prod__simp,axiom,
    ! [X1: nat,Y1: set_Pr5245412377734449720_f_nat,X22: nat,Y22: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le36920181279911177_f_nat @ ( produc2676473986085469089_f_nat @ X1 @ Y1 ) @ ( produc2676473986085469089_f_nat @ X22 @ Y22 ) )
      = ( ( ord_less_nat @ X1 @ X22 )
        | ( ( ord_less_eq_nat @ X1 @ X22 )
          & ( ord_le8976984241387448984_f_nat @ Y1 @ Y22 ) ) ) ) ).

% less_eq_prod_simp
thf(fact_713_less__eq__prod__simp,axiom,
    ! [X1: set_Pr5245412377734449720_f_nat,Y1: nat,X22: set_Pr5245412377734449720_f_nat,Y22: nat] :
      ( ( ord_le7494303434561702409at_nat @ ( produc3509043581404444833at_nat @ X1 @ Y1 ) @ ( produc3509043581404444833at_nat @ X22 @ Y22 ) )
      = ( ( ord_le2793513434575419532_f_nat @ X1 @ X22 )
        | ( ( ord_le8976984241387448984_f_nat @ X1 @ X22 )
          & ( ord_less_eq_nat @ Y1 @ Y22 ) ) ) ) ).

% less_eq_prod_simp
thf(fact_714_less__eq__prod__simp,axiom,
    ! [X1: set_Pr5245412377734449720_f_nat,Y1: set_Pr5245412377734449720_f_nat,X22: set_Pr5245412377734449720_f_nat,Y22: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le5034997304165176945_f_nat @ ( produc5809933371186360841_f_nat @ X1 @ Y1 ) @ ( produc5809933371186360841_f_nat @ X22 @ Y22 ) )
      = ( ( ord_le2793513434575419532_f_nat @ X1 @ X22 )
        | ( ( ord_le8976984241387448984_f_nat @ X1 @ X22 )
          & ( ord_le8976984241387448984_f_nat @ Y1 @ Y22 ) ) ) ) ).

% less_eq_prod_simp
thf(fact_715_less__prod__simp,axiom,
    ! [X1: nat > nat,Y1: nat,X22: nat > nat,Y22: nat] :
      ( ( ord_le6964573409913854876at_nat @ ( produc72220940542539688at_nat @ X1 @ Y1 ) @ ( produc72220940542539688at_nat @ X22 @ Y22 ) )
      = ( ( ord_less_nat_nat @ X1 @ X22 )
        | ( ( ord_less_eq_nat_nat @ X1 @ X22 )
          & ( ord_less_nat @ Y1 @ Y22 ) ) ) ) ).

% less_prod_simp
thf(fact_716_less__prod__simp,axiom,
    ! [X1: nat,Y1: nat,X22: nat,Y22: nat] :
      ( ( ord_le1203424502768444845at_nat @ ( product_Pair_nat_nat @ X1 @ Y1 ) @ ( product_Pair_nat_nat @ X22 @ Y22 ) )
      = ( ( ord_less_nat @ X1 @ X22 )
        | ( ( ord_less_eq_nat @ X1 @ X22 )
          & ( ord_less_nat @ Y1 @ Y22 ) ) ) ) ).

% less_prod_simp
thf(fact_717_less__prod__simp,axiom,
    ! [X1: set_Pr5245412377734449720_f_nat,Y1: nat,X22: set_Pr5245412377734449720_f_nat,Y22: nat] :
      ( ( ord_le3035520849313069845at_nat @ ( produc3509043581404444833at_nat @ X1 @ Y1 ) @ ( produc3509043581404444833at_nat @ X22 @ Y22 ) )
      = ( ( ord_le2793513434575419532_f_nat @ X1 @ X22 )
        | ( ( ord_le8976984241387448984_f_nat @ X1 @ X22 )
          & ( ord_less_nat @ Y1 @ Y22 ) ) ) ) ).

% less_prod_simp
thf(fact_718_inf__pigeonhole__principle,axiom,
    ! [N: nat,F: nat > nat > $o] :
      ( ! [K2: nat] :
        ? [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
          & ( F @ K2 @ I2 ) )
     => ? [I: nat] :
          ( ( ord_less_nat @ I @ N )
          & ! [K3: nat] :
            ? [K4: nat] :
              ( ( ord_less_eq_nat @ K3 @ K4 )
              & ( F @ K4 @ I ) ) ) ) ).

% inf_pigeonhole_principle
thf(fact_719_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N @ K2 )
         => ( P @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
           => ( ! [I2: nat] :
                  ( ( ord_less_nat @ K2 @ I2 )
                 => ( P @ I2 ) )
             => ( P @ K2 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_720_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_721_order__le__imp__less__or__eq,axiom,
    ! [X: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ X @ Y )
     => ( ( ord_le2793513434575419532_f_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_722_Cons__le__Cons,axiom,
    ! [A: nat,X: list_nat,B: nat,Y: list_nat] :
      ( ( ord_less_eq_list_nat @ ( cons_nat @ A @ X ) @ ( cons_nat @ B @ Y ) )
      = ( ( ord_less_nat @ A @ B )
        | ( ( A = B )
          & ( ord_less_eq_list_nat @ X @ Y ) ) ) ) ).

% Cons_le_Cons
thf(fact_723_fset__of__list__subset,axiom,
    ! [Xs: list_term_f_v,Ys: list_term_f_v] :
      ( ( ord_le947722214152454265rm_f_v @ ( set_term_f_v2 @ Xs ) @ ( set_term_f_v2 @ Ys ) )
     => ( ord_le6659761365940488051rm_f_v @ ( fset_o6574051099709074461rm_f_v @ Xs ) @ ( fset_o6574051099709074461rm_f_v @ Ys ) ) ) ).

% fset_of_list_subset
thf(fact_724_fset__of__list__subset,axiom,
    ! [Xs: list_P1780566427699531651term_f,Ys: list_P1780566427699531651term_f] :
      ( ( ord_le5176047027162639613term_f @ ( set_Pr1368550810757897490term_f @ Xs ) @ ( set_Pr1368550810757897490term_f @ Ys ) )
     => ( ord_le7092741666294821507term_f @ ( fset_o6125718882019406531term_f @ Xs ) @ ( fset_o6125718882019406531term_f @ Ys ) ) ) ).

% fset_of_list_subset
thf(fact_725_fset__of__list__subset,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,Ys: list_P4093298276913796397rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( set_Pr817814403484925884rm_f_v @ Xs ) @ ( set_Pr817814403484925884rm_f_v @ Ys ) )
     => ( ord_le4587745213494032429rm_f_v @ ( fset_o4970844032613833069rm_f_v @ Xs ) @ ( fset_o4970844032613833069rm_f_v @ Ys ) ) ) ).

% fset_of_list_subset
thf(fact_726_fset__of__list__subset,axiom,
    ! [Xs: list_P3903862279629787026_f_nat,Ys: list_P3903862279629787026_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ ( set_Pr7102205019285007021_f_nat @ Xs ) @ ( set_Pr7102205019285007021_f_nat @ Ys ) )
     => ( ord_le1552505484586773650_f_nat @ ( fset_o8009517685352940092_f_nat @ Xs ) @ ( fset_o8009517685352940092_f_nat @ Ys ) ) ) ).

% fset_of_list_subset
thf(fact_727_listrel__mono,axiom,
    ! [R3: set_Pr5245412377734449720_f_nat,S: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ R3 @ S )
     => ( ord_le1585512986379487150st_nat @ ( listrel_f_nat @ R3 ) @ ( listrel_f_nat @ S ) ) ) ).

% listrel_mono
thf(fact_728_subset__code_I1_J,axiom,
    ! [Xs: list_term_f_v,B5: set_term_f_v] :
      ( ( ord_le947722214152454265rm_f_v @ ( set_term_f_v2 @ Xs ) @ B5 )
      = ( ! [X5: term_f_v] :
            ( ( member_term_f_v @ X5 @ ( set_term_f_v2 @ Xs ) )
           => ( member_term_f_v @ X5 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_729_subset__code_I1_J,axiom,
    ! [Xs: list_P1780566427699531651term_f,B5: set_Pr6841673932529763677term_f] :
      ( ( ord_le5176047027162639613term_f @ ( set_Pr1368550810757897490term_f @ Xs ) @ B5 )
      = ( ! [X5: produc250624644672406269term_f] :
            ( ( member4687689947140789414term_f @ X5 @ ( set_Pr1368550810757897490term_f @ Xs ) )
           => ( member4687689947140789414term_f @ X5 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_730_subset__code_I1_J,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( set_Pr817814403484925884rm_f_v @ Xs ) @ B5 )
      = ( ! [X5: produc8027700073345061287rm_f_v] :
            ( ( member4316649861536941904rm_f_v @ X5 @ ( set_Pr817814403484925884rm_f_v @ Xs ) )
           => ( member4316649861536941904rm_f_v @ X5 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_731_subset__code_I1_J,axiom,
    ! [Xs: list_P3903862279629787026_f_nat,B5: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ ( set_Pr7102205019285007021_f_nat @ Xs ) @ B5 )
      = ( ! [X5: product_prod_f_nat] :
            ( ( member2678038196313615385_f_nat @ X5 @ ( set_Pr7102205019285007021_f_nat @ Xs ) )
           => ( member2678038196313615385_f_nat @ X5 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_732_trancl__mono,axiom,
    ! [P2: produc8027700073345061287rm_f_v,R3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ P2 @ ( transi3208867851621559554rm_f_v @ R3 ) )
     => ( ( ord_le4559761987009501863rm_f_v @ R3 @ S )
       => ( member4316649861536941904rm_f_v @ P2 @ ( transi3208867851621559554rm_f_v @ S ) ) ) ) ).

% trancl_mono
thf(fact_733_trancl__mono,axiom,
    ! [P2: produc250624644672406269term_f,R3: set_Pr6841673932529763677term_f,S: set_Pr6841673932529763677term_f] :
      ( ( member4687689947140789414term_f @ P2 @ ( transi2774326432670905901term_f @ R3 ) )
     => ( ( ord_le5176047027162639613term_f @ R3 @ S )
       => ( member4687689947140789414term_f @ P2 @ ( transi2774326432670905901term_f @ S ) ) ) ) ).

% trancl_mono
thf(fact_734_less__eq__list__code_I3_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( ord_less_eq_list_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
      = ( ( ord_less_nat @ X @ Y )
        | ( ( X = Y )
          & ( ord_less_eq_list_nat @ Xs @ Ys ) ) ) ) ).

% less_eq_list_code(3)
thf(fact_735_set__subset__Cons,axiom,
    ! [Xs: list_term_f_v,X: term_f_v] : ( ord_le947722214152454265rm_f_v @ ( set_term_f_v2 @ Xs ) @ ( set_term_f_v2 @ ( cons_term_f_v @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_736_set__subset__Cons,axiom,
    ! [Xs: list_P1780566427699531651term_f,X: produc250624644672406269term_f] : ( ord_le5176047027162639613term_f @ ( set_Pr1368550810757897490term_f @ Xs ) @ ( set_Pr1368550810757897490term_f @ ( cons_P3711653291972799923term_f @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_737_set__subset__Cons,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,X: produc8027700073345061287rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( set_Pr817814403484925884rm_f_v @ Xs ) @ ( set_Pr817814403484925884rm_f_v @ ( cons_P1983239458618365021rm_f_v @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_738_set__subset__Cons,axiom,
    ! [Xs: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_739_set__subset__Cons,axiom,
    ! [Xs: list_P3903862279629787026_f_nat,X: product_prod_f_nat] : ( ord_le8976984241387448984_f_nat @ ( set_Pr7102205019285007021_f_nat @ Xs ) @ ( set_Pr7102205019285007021_f_nat @ ( cons_P2159016411613705036_f_nat @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_740_set__take__subset,axiom,
    ! [N: nat,Xs: list_term_f_v] : ( ord_le947722214152454265rm_f_v @ ( set_term_f_v2 @ ( take_term_f_v @ N @ Xs ) ) @ ( set_term_f_v2 @ Xs ) ) ).

% set_take_subset
thf(fact_741_set__take__subset,axiom,
    ! [N: nat,Xs: list_P1780566427699531651term_f] : ( ord_le5176047027162639613term_f @ ( set_Pr1368550810757897490term_f @ ( take_P4256413432045430251term_f @ N @ Xs ) ) @ ( set_Pr1368550810757897490term_f @ Xs ) ) ).

% set_take_subset
thf(fact_742_set__take__subset,axiom,
    ! [N: nat,Xs: list_P4093298276913796397rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( set_Pr817814403484925884rm_f_v @ ( take_P6145276122034302101rm_f_v @ N @ Xs ) ) @ ( set_Pr817814403484925884rm_f_v @ Xs ) ) ).

% set_take_subset
thf(fact_743_set__take__subset,axiom,
    ! [N: nat,Xs: list_P3903862279629787026_f_nat] : ( ord_le8976984241387448984_f_nat @ ( set_Pr7102205019285007021_f_nat @ ( take_P4203571807830065428_f_nat @ N @ Xs ) ) @ ( set_Pr7102205019285007021_f_nat @ Xs ) ) ).

% set_take_subset
thf(fact_744_set__update__subsetI,axiom,
    ! [Xs: list_term_f_v,A4: set_term_f_v,X: term_f_v,I4: nat] :
      ( ( ord_le947722214152454265rm_f_v @ ( set_term_f_v2 @ Xs ) @ A4 )
     => ( ( member_term_f_v @ X @ A4 )
       => ( ord_le947722214152454265rm_f_v @ ( set_term_f_v2 @ ( list_update_term_f_v @ Xs @ I4 @ X ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_745_set__update__subsetI,axiom,
    ! [Xs: list_P1780566427699531651term_f,A4: set_Pr6841673932529763677term_f,X: produc250624644672406269term_f,I4: nat] :
      ( ( ord_le5176047027162639613term_f @ ( set_Pr1368550810757897490term_f @ Xs ) @ A4 )
     => ( ( member4687689947140789414term_f @ X @ A4 )
       => ( ord_le5176047027162639613term_f @ ( set_Pr1368550810757897490term_f @ ( list_u2231791752812534891term_f @ Xs @ I4 @ X ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_746_set__update__subsetI,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,A4: set_Pr8827868859434726151rm_f_v,X: produc8027700073345061287rm_f_v,I4: nat] :
      ( ( ord_le4559761987009501863rm_f_v @ ( set_Pr817814403484925884rm_f_v @ Xs ) @ A4 )
     => ( ( member4316649861536941904rm_f_v @ X @ A4 )
       => ( ord_le4559761987009501863rm_f_v @ ( set_Pr817814403484925884rm_f_v @ ( list_u2658473446164013845rm_f_v @ Xs @ I4 @ X ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_747_set__update__subsetI,axiom,
    ! [Xs: list_P3903862279629787026_f_nat,A4: set_Pr5245412377734449720_f_nat,X: product_prod_f_nat,I4: nat] :
      ( ( ord_le8976984241387448984_f_nat @ ( set_Pr7102205019285007021_f_nat @ Xs ) @ A4 )
     => ( ( member2678038196313615385_f_nat @ X @ A4 )
       => ( ord_le8976984241387448984_f_nat @ ( set_Pr7102205019285007021_f_nat @ ( list_u1272160344759216788_f_nat @ Xs @ I4 @ X ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_748_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_749_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_750_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_751_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
      | ( X = Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_752_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_753_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_754_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_755_order__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_756_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_757_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_758_ord__eq__less__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_759_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_760_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_761_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      = ( ( ord_less_nat @ X @ Y )
        | ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_762_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_763_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_764_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_765_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_766_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_767_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_768_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_769_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_nat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: nat] : ( P @ A3 @ A3 )
       => ( ! [A3: nat,B3: nat] :
              ( ( P @ B3 @ A3 )
             => ( P @ A3 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_770_exists__least__iff,axiom,
    ( ( ^ [P4: nat > $o] :
        ? [X6: nat] : ( P4 @ X6 ) )
    = ( ^ [P3: nat > $o] :
        ? [N2: nat] :
          ( ( P3 @ N2 )
          & ! [M6: nat] :
              ( ( ord_less_nat @ M6 @ N2 )
             => ~ ( P3 @ M6 ) ) ) ) ) ).

% exists_least_iff
thf(fact_771_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_772_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_773_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_774_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_775_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X4: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X4 )
             => ( P @ Y5 ) )
         => ( P @ X4 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_776_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_777_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_778_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_779_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_780_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).

% gt_ex
thf(fact_781_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_term_f_v] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_le947722214152454265rm_f_v @ ( set_term_f_v2 @ ( take_term_f_v @ M @ Xs ) ) @ ( set_term_f_v2 @ ( take_term_f_v @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_782_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_P1780566427699531651term_f] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_le5176047027162639613term_f @ ( set_Pr1368550810757897490term_f @ ( take_P4256413432045430251term_f @ M @ Xs ) ) @ ( set_Pr1368550810757897490term_f @ ( take_P4256413432045430251term_f @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_783_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_P4093298276913796397rm_f_v] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_le4559761987009501863rm_f_v @ ( set_Pr817814403484925884rm_f_v @ ( take_P6145276122034302101rm_f_v @ M @ Xs ) ) @ ( set_Pr817814403484925884rm_f_v @ ( take_P6145276122034302101rm_f_v @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_784_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_P3903862279629787026_f_nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_le8976984241387448984_f_nat @ ( set_Pr7102205019285007021_f_nat @ ( take_P4203571807830065428_f_nat @ M @ Xs ) ) @ ( set_Pr7102205019285007021_f_nat @ ( take_P4203571807830065428_f_nat @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_785_subseteq__set__conv__nth,axiom,
    ! [Ss: list_term_f_v,T4: set_term_f_v] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7148729505879680775rm_f_v @ Ss ) )
           => ( member_term_f_v @ ( nth_term_f_v @ Ss @ I3 ) @ T4 ) ) )
      = ( ord_le947722214152454265rm_f_v @ ( set_term_f_v2 @ Ss ) @ T4 ) ) ).

% subseteq_set_conv_nth
thf(fact_786_subseteq__set__conv__nth,axiom,
    ! [Ss: list_P1780566427699531651term_f,T4: set_Pr6841673932529763677term_f] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s9037951226665808879term_f @ Ss ) )
           => ( member4687689947140789414term_f @ ( nth_Pr7080171896067517188term_f @ Ss @ I3 ) @ T4 ) ) )
      = ( ord_le5176047027162639613term_f @ ( set_Pr1368550810757897490term_f @ Ss ) @ T4 ) ) ).

% subseteq_set_conv_nth
thf(fact_787_subseteq__set__conv__nth,axiom,
    ! [Ss: list_P4093298276913796397rm_f_v,T4: set_Pr8827868859434726151rm_f_v] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3818039357710166425rm_f_v @ Ss ) )
           => ( member4316649861536941904rm_f_v @ ( nth_Pr8768905099348956590rm_f_v @ Ss @ I3 ) @ T4 ) ) )
      = ( ord_le4559761987009501863rm_f_v @ ( set_Pr817814403484925884rm_f_v @ Ss ) @ T4 ) ) ).

% subseteq_set_conv_nth
thf(fact_788_subseteq__set__conv__nth,axiom,
    ! [Ss: list_P3903862279629787026_f_nat,T4: set_Pr5245412377734449720_f_nat] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1295974592580694566_f_nat @ Ss ) )
           => ( member2678038196313615385_f_nat @ ( nth_Pr5415315262447610683_f_nat @ Ss @ I3 ) @ T4 ) ) )
      = ( ord_le8976984241387448984_f_nat @ ( set_Pr7102205019285007021_f_nat @ Ss ) @ T4 ) ) ).

% subseteq_set_conv_nth
thf(fact_789_set__list__subset__eq__nth__conv,axiom,
    ! [Xs: list_term_f_v,A4: set_term_f_v] :
      ( ( ord_le947722214152454265rm_f_v @ ( set_term_f_v2 @ Xs ) @ A4 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
           => ( member_term_f_v @ ( nth_term_f_v @ Xs @ I3 ) @ A4 ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_790_set__list__subset__eq__nth__conv,axiom,
    ! [Xs: list_P1780566427699531651term_f,A4: set_Pr6841673932529763677term_f] :
      ( ( ord_le5176047027162639613term_f @ ( set_Pr1368550810757897490term_f @ Xs ) @ A4 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s9037951226665808879term_f @ Xs ) )
           => ( member4687689947140789414term_f @ ( nth_Pr7080171896067517188term_f @ Xs @ I3 ) @ A4 ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_791_set__list__subset__eq__nth__conv,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,A4: set_Pr8827868859434726151rm_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ ( set_Pr817814403484925884rm_f_v @ Xs ) @ A4 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
           => ( member4316649861536941904rm_f_v @ ( nth_Pr8768905099348956590rm_f_v @ Xs @ I3 ) @ A4 ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_792_set__list__subset__eq__nth__conv,axiom,
    ! [Xs: list_P3903862279629787026_f_nat,A4: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ ( set_Pr7102205019285007021_f_nat @ Xs ) @ A4 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1295974592580694566_f_nat @ Xs ) )
           => ( member2678038196313615385_f_nat @ ( nth_Pr5415315262447610683_f_nat @ Xs @ I3 ) @ A4 ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_793_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_794_leD,axiom,
    ! [Y: set_Pr5245412377734449720_f_nat,X: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ Y @ X )
     => ~ ( ord_le2793513434575419532_f_nat @ X @ Y ) ) ).

% leD
thf(fact_795_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_796_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_797_nless__le,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
      ( ( ~ ( ord_le2793513434575419532_f_nat @ A @ B ) )
      = ( ~ ( ord_le8976984241387448984_f_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_798_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_799_antisym__conv1,axiom,
    ! [X: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
      ( ~ ( ord_le2793513434575419532_f_nat @ X @ Y )
     => ( ( ord_le8976984241387448984_f_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_800_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_801_antisym__conv2,axiom,
    ! [X: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ X @ Y )
     => ( ( ~ ( ord_le2793513434575419532_f_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_802_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X5: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X5 @ Y4 )
          & ~ ( ord_less_eq_nat @ Y4 @ X5 ) ) ) ) ).

% less_le_not_le
thf(fact_803_less__le__not__le,axiom,
    ( ord_le2793513434575419532_f_nat
    = ( ^ [X5: set_Pr5245412377734449720_f_nat,Y4: set_Pr5245412377734449720_f_nat] :
          ( ( ord_le8976984241387448984_f_nat @ X5 @ Y4 )
          & ~ ( ord_le8976984241387448984_f_nat @ Y4 @ X5 ) ) ) ) ).

% less_le_not_le
thf(fact_804_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X )
     => ( ord_less_nat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_805_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_nat @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_806_order_Oorder__iff__strict,axiom,
    ( ord_le8976984241387448984_f_nat
    = ( ^ [A5: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
          ( ( ord_le2793513434575419532_f_nat @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_807_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_808_order_Ostrict__iff__order,axiom,
    ( ord_le2793513434575419532_f_nat
    = ( ^ [A5: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
          ( ( ord_le8976984241387448984_f_nat @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_809_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_810_order_Ostrict__trans1,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B )
     => ( ( ord_le2793513434575419532_f_nat @ B @ C )
       => ( ord_le2793513434575419532_f_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_811_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_812_order_Ostrict__trans2,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le2793513434575419532_f_nat @ A @ B )
     => ( ( ord_le8976984241387448984_f_nat @ B @ C )
       => ( ord_le2793513434575419532_f_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_813_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A5 @ B4 )
          & ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_814_order_Ostrict__iff__not,axiom,
    ( ord_le2793513434575419532_f_nat
    = ( ^ [A5: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
          ( ( ord_le8976984241387448984_f_nat @ A5 @ B4 )
          & ~ ( ord_le8976984241387448984_f_nat @ B4 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_815_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( ord_less_nat @ B4 @ A5 )
          | ( A5 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_816_dual__order_Oorder__iff__strict,axiom,
    ( ord_le8976984241387448984_f_nat
    = ( ^ [B4: set_Pr5245412377734449720_f_nat,A5: set_Pr5245412377734449720_f_nat] :
          ( ( ord_le2793513434575419532_f_nat @ B4 @ A5 )
          | ( A5 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_817_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( ord_less_eq_nat @ B4 @ A5 )
          & ( A5 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_818_dual__order_Ostrict__iff__order,axiom,
    ( ord_le2793513434575419532_f_nat
    = ( ^ [B4: set_Pr5245412377734449720_f_nat,A5: set_Pr5245412377734449720_f_nat] :
          ( ( ord_le8976984241387448984_f_nat @ B4 @ A5 )
          & ( A5 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_819_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_820_dual__order_Ostrict__trans1,axiom,
    ! [B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ B @ A )
     => ( ( ord_le2793513434575419532_f_nat @ C @ B )
       => ( ord_le2793513434575419532_f_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_821_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_822_dual__order_Ostrict__trans2,axiom,
    ! [B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le2793513434575419532_f_nat @ B @ A )
     => ( ( ord_le8976984241387448984_f_nat @ C @ B )
       => ( ord_le2793513434575419532_f_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_823_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( ord_less_eq_nat @ B4 @ A5 )
          & ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_824_dual__order_Ostrict__iff__not,axiom,
    ( ord_le2793513434575419532_f_nat
    = ( ^ [B4: set_Pr5245412377734449720_f_nat,A5: set_Pr5245412377734449720_f_nat] :
          ( ( ord_le8976984241387448984_f_nat @ B4 @ A5 )
          & ~ ( ord_le8976984241387448984_f_nat @ A5 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_825_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_826_order_Ostrict__implies__order,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le2793513434575419532_f_nat @ A @ B )
     => ( ord_le8976984241387448984_f_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_827_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_828_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le2793513434575419532_f_nat @ B @ A )
     => ( ord_le8976984241387448984_f_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_829_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X5: nat,Y4: nat] :
          ( ( ord_less_nat @ X5 @ Y4 )
          | ( X5 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_830_order__le__less,axiom,
    ( ord_le8976984241387448984_f_nat
    = ( ^ [X5: set_Pr5245412377734449720_f_nat,Y4: set_Pr5245412377734449720_f_nat] :
          ( ( ord_le2793513434575419532_f_nat @ X5 @ Y4 )
          | ( X5 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_831_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X5: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X5 @ Y4 )
          & ( X5 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_832_order__less__le,axiom,
    ( ord_le2793513434575419532_f_nat
    = ( ^ [X5: set_Pr5245412377734449720_f_nat,Y4: set_Pr5245412377734449720_f_nat] :
          ( ( ord_le8976984241387448984_f_nat @ X5 @ Y4 )
          & ( X5 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_833_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_834_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_835_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_836_order__less__imp__le,axiom,
    ! [X: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le2793513434575419532_f_nat @ X @ Y )
     => ( ord_le8976984241387448984_f_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_837_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_838_order__le__neq__trans,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B )
     => ( ( A != B )
       => ( ord_le2793513434575419532_f_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_839_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_840_order__neq__le__trans,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
      ( ( A != B )
     => ( ( ord_le8976984241387448984_f_nat @ A @ B )
       => ( ord_le2793513434575419532_f_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_841_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_842_order__le__less__trans,axiom,
    ! [X: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat,Z: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ X @ Y )
     => ( ( ord_le2793513434575419532_f_nat @ Y @ Z )
       => ( ord_le2793513434575419532_f_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_843_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_844_order__less__le__trans,axiom,
    ! [X: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat,Z: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le2793513434575419532_f_nat @ X @ Y )
     => ( ( ord_le8976984241387448984_f_nat @ Y @ Z )
       => ( ord_le2793513434575419532_f_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_845_order__le__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_846_order__le__less__subst1,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,F: nat > set_Pr5245412377734449720_f_nat,B: nat,C: nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_le2793513434575419532_f_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le2793513434575419532_f_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_847_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_848_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_le2793513434575419532_f_nat @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_le8976984241387448984_f_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le2793513434575419532_f_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_849_order__le__less__subst2,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,F: set_Pr5245412377734449720_f_nat > nat,C: nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X4: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] :
              ( ( ord_le8976984241387448984_f_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_850_order__le__less__subst2,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,F: set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le8976984241387448984_f_nat @ A @ B )
     => ( ( ord_le2793513434575419532_f_nat @ ( F @ B ) @ C )
       => ( ! [X4: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] :
              ( ( ord_le8976984241387448984_f_nat @ X4 @ Y3 )
             => ( ord_le8976984241387448984_f_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le2793513434575419532_f_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_851_order__less__le__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_852_order__less__le__subst1,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,F: nat > set_Pr5245412377734449720_f_nat,B: nat,C: nat] :
      ( ( ord_le2793513434575419532_f_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_le8976984241387448984_f_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le2793513434575419532_f_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_853_order__less__le__subst1,axiom,
    ! [A: nat,F: set_Pr5245412377734449720_f_nat > nat,B: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_le8976984241387448984_f_nat @ B @ C )
       => ( ! [X4: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] :
              ( ( ord_le8976984241387448984_f_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_854_order__less__le__subst1,axiom,
    ! [A: set_Pr5245412377734449720_f_nat,F: set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( ord_le2793513434575419532_f_nat @ A @ ( F @ B ) )
     => ( ( ord_le8976984241387448984_f_nat @ B @ C )
       => ( ! [X4: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] :
              ( ( ord_le8976984241387448984_f_nat @ X4 @ Y3 )
             => ( ord_le8976984241387448984_f_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le2793513434575419532_f_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_855_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_856_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > set_Pr5245412377734449720_f_nat,C: set_Pr5245412377734449720_f_nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_le8976984241387448984_f_nat @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_le2793513434575419532_f_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le2793513434575419532_f_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_857_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_858_inv__to__set,axiom,
    ! [Ss: list_term_f_v,S5: set_term_f_v] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7148729505879680775rm_f_v @ Ss ) )
           => ( member_term_f_v @ ( nth_term_f_v @ Ss @ I3 ) @ S5 ) ) )
      = ( ord_le947722214152454265rm_f_v @ ( set_term_f_v2 @ Ss ) @ S5 ) ) ).

% inv_to_set
thf(fact_859_inv__to__set,axiom,
    ! [Ss: list_P1780566427699531651term_f,S5: set_Pr6841673932529763677term_f] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s9037951226665808879term_f @ Ss ) )
           => ( member4687689947140789414term_f @ ( nth_Pr7080171896067517188term_f @ Ss @ I3 ) @ S5 ) ) )
      = ( ord_le5176047027162639613term_f @ ( set_Pr1368550810757897490term_f @ Ss ) @ S5 ) ) ).

% inv_to_set
thf(fact_860_inv__to__set,axiom,
    ! [Ss: list_P4093298276913796397rm_f_v,S5: set_Pr8827868859434726151rm_f_v] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3818039357710166425rm_f_v @ Ss ) )
           => ( member4316649861536941904rm_f_v @ ( nth_Pr8768905099348956590rm_f_v @ Ss @ I3 ) @ S5 ) ) )
      = ( ord_le4559761987009501863rm_f_v @ ( set_Pr817814403484925884rm_f_v @ Ss ) @ S5 ) ) ).

% inv_to_set
thf(fact_861_inv__to__set,axiom,
    ! [Ss: list_P3903862279629787026_f_nat,S5: set_Pr5245412377734449720_f_nat] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1295974592580694566_f_nat @ Ss ) )
           => ( member2678038196313615385_f_nat @ ( nth_Pr5415315262447610683_f_nat @ Ss @ I3 ) @ S5 ) ) )
      = ( ord_le8976984241387448984_f_nat @ ( set_Pr7102205019285007021_f_nat @ Ss ) @ S5 ) ) ).

% inv_to_set
thf(fact_862_Abstract__Rewriting_Ochain__mono,axiom,
    ! [R5: set_Pr8827868859434726151rm_f_v,R4: set_Pr8827868859434726151rm_f_v,Seq: nat > term_f_v] :
      ( ( ord_le4559761987009501863rm_f_v @ R5 @ R4 )
     => ( ! [I: nat] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( Seq @ I ) @ ( Seq @ ( suc @ I ) ) ) @ R5 )
       => ! [I2: nat] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( Seq @ I2 ) @ ( Seq @ ( suc @ I2 ) ) ) @ R4 ) ) ) ).

% Abstract_Rewriting.chain_mono
thf(fact_863_Abstract__Rewriting_Ochain__mono,axiom,
    ! [R5: set_Pr6841673932529763677term_f,R4: set_Pr6841673932529763677term_f,Seq: nat > bot_bot_term_f] :
      ( ( ord_le5176047027162639613term_f @ R5 @ R4 )
     => ( ! [I: nat] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( Seq @ I ) @ ( Seq @ ( suc @ I ) ) ) @ R5 )
       => ! [I2: nat] : ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( Seq @ I2 ) @ ( Seq @ ( suc @ I2 ) ) ) @ R4 ) ) ) ).

% Abstract_Rewriting.chain_mono
thf(fact_864_Abstract__Rewriting_Ochain__mono,axiom,
    ! [R5: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat,Seq: nat > nat] :
      ( ( ord_le3146513528884898305at_nat @ R5 @ R4 )
     => ( ! [I: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( Seq @ I ) @ ( Seq @ ( suc @ I ) ) ) @ R5 )
       => ! [I2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( Seq @ I2 ) @ ( Seq @ ( suc @ I2 ) ) ) @ R4 ) ) ) ).

% Abstract_Rewriting.chain_mono
thf(fact_865_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z3 )
     => ~ ( ord_less_eq_nat @ T @ X3 ) ) ).

% minf(8)
thf(fact_866_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z3 )
     => ( ord_less_eq_nat @ X3 @ T ) ) ).

% minf(6)
thf(fact_867_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z3 )
     => ~ ( ord_less_nat @ T @ X3 ) ) ).

% minf(7)
thf(fact_868_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z3 )
     => ( ord_less_nat @ X3 @ T ) ) ).

% minf(5)
thf(fact_869_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z3 )
     => ( X3 != T ) ) ).

% minf(4)
thf(fact_870_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z3 )
     => ( X3 != T ) ) ).

% minf(3)
thf(fact_871_minf_I2_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q3: nat > $o,Q4: nat > $o] :
      ( ? [Z5: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ X4 @ Z5 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z5 )
           => ( ( Q3 @ X4 )
              = ( Q4 @ X4 ) ) )
       => ? [Z3: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z3 )
           => ( ( ( P @ X3 )
                | ( Q3 @ X3 ) )
              = ( ( P5 @ X3 )
                | ( Q4 @ X3 ) ) ) ) ) ) ).

% minf(2)
thf(fact_872_minf_I1_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q3: nat > $o,Q4: nat > $o] :
      ( ? [Z5: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ X4 @ Z5 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z5 )
           => ( ( Q3 @ X4 )
              = ( Q4 @ X4 ) ) )
       => ? [Z3: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z3 )
           => ( ( ( P @ X3 )
                & ( Q3 @ X3 ) )
              = ( ( P5 @ X3 )
                & ( Q4 @ X3 ) ) ) ) ) ) ).

% minf(1)
thf(fact_873_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z3 @ X3 )
     => ( ord_less_nat @ T @ X3 ) ) ).

% pinf(7)
thf(fact_874_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z3 @ X3 )
     => ~ ( ord_less_nat @ X3 @ T ) ) ).

% pinf(5)
thf(fact_875_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z3 @ X3 )
     => ( X3 != T ) ) ).

% pinf(4)
thf(fact_876_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z3 @ X3 )
     => ( X3 != T ) ) ).

% pinf(3)
thf(fact_877_pinf_I2_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q3: nat > $o,Q4: nat > $o] :
      ( ? [Z5: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ Z5 @ X4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z5 @ X4 )
           => ( ( Q3 @ X4 )
              = ( Q4 @ X4 ) ) )
       => ? [Z3: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z3 @ X3 )
           => ( ( ( P @ X3 )
                | ( Q3 @ X3 ) )
              = ( ( P5 @ X3 )
                | ( Q4 @ X3 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_878_pinf_I1_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q3: nat > $o,Q4: nat > $o] :
      ( ? [Z5: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ Z5 @ X4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z5 @ X4 )
           => ( ( Q3 @ X4 )
              = ( Q4 @ X4 ) ) )
       => ? [Z3: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z3 @ X3 )
           => ( ( ( P @ X3 )
                & ( Q3 @ X3 ) )
              = ( ( P5 @ X3 )
                & ( Q4 @ X3 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_879_in__set__product__lists__length,axiom,
    ! [Xs: list_term_f_v,Xss: list_list_term_f_v] :
      ( ( member_list_term_f_v @ Xs @ ( set_list_term_f_v2 @ ( produc4527135265320815924rm_f_v @ Xss ) ) )
     => ( ( size_s7148729505879680775rm_f_v @ Xs )
        = ( size_s600530640984840599rm_f_v @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_880_in__set__product__lists__length,axiom,
    ! [Xs: list_P1780566427699531651term_f,Xss: list_l9126709995455877001term_f] :
      ( ( member6232163599979755692term_f @ Xs @ ( set_li3839205001427786520term_f @ ( produc6629601965052718188term_f @ Xss ) ) )
     => ( ( size_s9037951226665808879term_f @ Xs )
        = ( size_s3597996746158427637term_f @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_881_in__set__product__lists__length,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,Xss: list_l4945967923587861555rm_f_v] :
      ( ( member2864856320113173078rm_f_v @ Xs @ ( set_li3345156661183018178rm_f_v @ ( produc7050385660098377494rm_f_v @ Xss ) ) )
     => ( ( size_s3818039357710166425rm_f_v @ Xs )
        = ( size_s8204897334373333663rm_f_v @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_882_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z3 @ X3 )
     => ~ ( ord_less_eq_nat @ X3 @ T ) ) ).

% pinf(6)
thf(fact_883_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z3 @ X3 )
     => ( ord_less_eq_nat @ T @ X3 ) ) ).

% pinf(8)
thf(fact_884_subrelI,axiom,
    ! [R3: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
      ( ! [X4: term_f_v,Y3: term_f_v] :
          ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y3 ) @ R3 )
         => ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y3 ) @ S ) )
     => ( ord_le4559761987009501863rm_f_v @ R3 @ S ) ) ).

% subrelI
thf(fact_885_subrelI,axiom,
    ! [R3: set_Pr6841673932529763677term_f,S: set_Pr6841673932529763677term_f] :
      ( ! [X4: bot_bot_term_f,Y3: bot_bot_term_f] :
          ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X4 @ Y3 ) @ R3 )
         => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X4 @ Y3 ) @ S ) )
     => ( ord_le5176047027162639613term_f @ R3 @ S ) ) ).

% subrelI
thf(fact_886_subrelI,axiom,
    ! [R3: set_Pr9093778441882193744at_nat,S: set_Pr9093778441882193744at_nat] :
      ( ! [X4: nat > nat,Y3: nat] :
          ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X4 @ Y3 ) @ R3 )
         => ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X4 @ Y3 ) @ S ) )
     => ( ord_le3678578370064672496at_nat @ R3 @ S ) ) ).

% subrelI
thf(fact_887_subrelI,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R3 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ S ) )
     => ( ord_le3146513528884898305at_nat @ R3 @ S ) ) ).

% subrelI
thf(fact_888_subrelI,axiom,
    ! [R3: set_Pr5245412377734449720_f_nat,S: set_Pr5245412377734449720_f_nat] :
      ( ! [X4: f,Y3: nat] :
          ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X4 @ Y3 ) @ R3 )
         => ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X4 @ Y3 ) @ S ) )
     => ( ord_le8976984241387448984_f_nat @ R3 @ S ) ) ).

% subrelI
thf(fact_889_complete__interval,axiom,
    ! [A: nat,B: nat,P: nat > $o] :
      ( ( ord_less_nat @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C3: nat] :
              ( ( ord_less_eq_nat @ A @ C3 )
              & ( ord_less_eq_nat @ C3 @ B )
              & ! [X3: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X3 )
                    & ( ord_less_nat @ X3 @ C3 ) )
                 => ( P @ X3 ) )
              & ! [D: nat] :
                  ( ! [X4: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X4 )
                        & ( ord_less_nat @ X4 @ D ) )
                     => ( P @ X4 ) )
                 => ( ord_less_eq_nat @ D @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_890_verit__comp__simplify1_I3_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
      = ( ord_less_nat @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_891_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_892_length__add__elem__list__lists,axiom,
    ! [Ys: list_term_f_v,X: term_f_v,Xs: list_term_f_v] :
      ( ( member_list_term_f_v @ Ys @ ( set_list_term_f_v2 @ ( basic_6102686972066033914rm_f_v @ X @ Xs ) ) )
     => ( ( size_s7148729505879680775rm_f_v @ Ys )
        = ( suc @ ( size_s7148729505879680775rm_f_v @ Xs ) ) ) ) ).

% length_add_elem_list_lists
thf(fact_893_length__add__elem__list__lists,axiom,
    ! [Ys: list_P1780566427699531651term_f,X: produc250624644672406269term_f,Xs: list_P1780566427699531651term_f] :
      ( ( member6232163599979755692term_f @ Ys @ ( set_li3839205001427786520term_f @ ( basic_2716954541200321574term_f @ X @ Xs ) ) )
     => ( ( size_s9037951226665808879term_f @ Ys )
        = ( suc @ ( size_s9037951226665808879term_f @ Xs ) ) ) ) ).

% length_add_elem_list_lists
thf(fact_894_length__add__elem__list__lists,axiom,
    ! [Ys: list_P4093298276913796397rm_f_v,X: produc8027700073345061287rm_f_v,Xs: list_P4093298276913796397rm_f_v] :
      ( ( member2864856320113173078rm_f_v @ Ys @ ( set_li3345156661183018178rm_f_v @ ( basic_9152604556761636560rm_f_v @ X @ Xs ) ) )
     => ( ( size_s3818039357710166425rm_f_v @ Ys )
        = ( suc @ ( size_s3818039357710166425rm_f_v @ Xs ) ) ) ) ).

% length_add_elem_list_lists
thf(fact_895_min__list__nth,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
           => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) )
       => ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs ) @ ( missing_min_list_nat @ Ys ) ) ) ) ).

% min_list_nth
thf(fact_896_nth__equal__first__eq,axiom,
    ! [X: nat,Xs: list_nat,N: nat] :
      ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_897_nth__equal__first__eq,axiom,
    ! [X: term_f_v,Xs: list_term_f_v,N: nat] :
      ( ~ ( member_term_f_v @ X @ ( set_term_f_v2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s7148729505879680775rm_f_v @ Xs ) )
       => ( ( ( nth_term_f_v @ ( cons_term_f_v @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_898_nth__equal__first__eq,axiom,
    ! [X: produc250624644672406269term_f,Xs: list_P1780566427699531651term_f,N: nat] :
      ( ~ ( member4687689947140789414term_f @ X @ ( set_Pr1368550810757897490term_f @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s9037951226665808879term_f @ Xs ) )
       => ( ( ( nth_Pr7080171896067517188term_f @ ( cons_P3711653291972799923term_f @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_899_nth__equal__first__eq,axiom,
    ! [X: produc8027700073345061287rm_f_v,Xs: list_P4093298276913796397rm_f_v,N: nat] :
      ( ~ ( member4316649861536941904rm_f_v @ X @ ( set_Pr817814403484925884rm_f_v @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s3818039357710166425rm_f_v @ Xs ) )
       => ( ( ( nth_Pr8768905099348956590rm_f_v @ ( cons_P1983239458618365021rm_f_v @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_900_Fun_Oprems_I2_J,axiom,
    ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ ( fun_f_v @ f3 @ ts ) ) @ ( fset_P3576968334923099475_f_nat @ f2 ) ).

% Fun.prems(2)
thf(fact_901_assms_I3_J,axiom,
    ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ t ) @ ( fset_P3576968334923099475_f_nat @ f2 ) ).

% assms(3)
thf(fact_902_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_903_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_904_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_905_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_906_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_907_nth__Cons__0,axiom,
    ! [X: term_f_v,Xs: list_term_f_v] :
      ( ( nth_term_f_v @ ( cons_term_f_v @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_908_nth__Cons__0,axiom,
    ! [X: produc250624644672406269term_f,Xs: list_P1780566427699531651term_f] :
      ( ( nth_Pr7080171896067517188term_f @ ( cons_P3711653291972799923term_f @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_909_nth__Cons__0,axiom,
    ! [X: produc8027700073345061287rm_f_v,Xs: list_P4093298276913796397rm_f_v] :
      ( ( nth_Pr8768905099348956590rm_f_v @ ( cons_P1983239458618365021rm_f_v @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_910_nth__Cons__0,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_911_fmember__iff__member__fset,axiom,
    ( fmembe774991991480784110rm_f_v
    = ( ^ [X5: produc8027700073345061287rm_f_v,A7: fset_P8018961893305114765rm_f_v] : ( member4316649861536941904rm_f_v @ X5 @ ( fset_P4617584883882644886rm_f_v @ A7 ) ) ) ) ).

% fmember_iff_member_fset
thf(fact_912_fmember__iff__member__fset,axiom,
    ( fmembe6950792022577883716term_f
    = ( ^ [X5: produc250624644672406269term_f,A7: fset_P2882900458636368611term_f] : ( member4687689947140789414term_f @ X5 @ ( fset_P8925168969356593900term_f @ A7 ) ) ) ) ).

% fmember_iff_member_fset
thf(fact_913_fmember__iff__member__fset,axiom,
    ( fmember_term_f_v
    = ( ^ [X5: term_f_v,A7: fset_term_f_v] : ( member_term_f_v @ X5 @ ( fset_term_f_v2 @ A7 ) ) ) ) ).

% fmember_iff_member_fset
thf(fact_914_fmember__iff__member__fset,axiom,
    ( fmember_term_f_a
    = ( ^ [X5: term_f_a,A7: fset_term_f_a] : ( member_term_f_a @ X5 @ ( fset_term_f_a2 @ A7 ) ) ) ) ).

% fmember_iff_member_fset
thf(fact_915_fmember__iff__member__fset,axiom,
    ( fmembe1418995490368054919term_f
    = ( ^ [X5: bot_bot_term_f,A7: fset_Bot_bot_term_f] : ( member2723211850833494437term_f @ X5 @ ( fset_Bot_bot_term_f2 @ A7 ) ) ) ) ).

% fmember_iff_member_fset
thf(fact_916_fmember__iff__member__fset,axiom,
    ( fmembe6795308708001954811_f_nat
    = ( ^ [X5: product_prod_f_nat,A7: fset_P6228066233360383026_f_nat] : ( member2678038196313615385_f_nat @ X5 @ ( fset_P3576968334923099475_f_nat @ A7 ) ) ) ) ).

% fmember_iff_member_fset
thf(fact_917_notin__fset,axiom,
    ! [X: produc8027700073345061287rm_f_v,S5: fset_P8018961893305114765rm_f_v] :
      ( ( ~ ( fmembe774991991480784110rm_f_v @ X @ S5 ) )
      = ( ~ ( member4316649861536941904rm_f_v @ X @ ( fset_P4617584883882644886rm_f_v @ S5 ) ) ) ) ).

% notin_fset
thf(fact_918_notin__fset,axiom,
    ! [X: produc250624644672406269term_f,S5: fset_P2882900458636368611term_f] :
      ( ( ~ ( fmembe6950792022577883716term_f @ X @ S5 ) )
      = ( ~ ( member4687689947140789414term_f @ X @ ( fset_P8925168969356593900term_f @ S5 ) ) ) ) ).

% notin_fset
thf(fact_919_notin__fset,axiom,
    ! [X: term_f_v,S5: fset_term_f_v] :
      ( ( ~ ( fmember_term_f_v @ X @ S5 ) )
      = ( ~ ( member_term_f_v @ X @ ( fset_term_f_v2 @ S5 ) ) ) ) ).

% notin_fset
thf(fact_920_notin__fset,axiom,
    ! [X: term_f_a,S5: fset_term_f_a] :
      ( ( ~ ( fmember_term_f_a @ X @ S5 ) )
      = ( ~ ( member_term_f_a @ X @ ( fset_term_f_a2 @ S5 ) ) ) ) ).

% notin_fset
thf(fact_921_notin__fset,axiom,
    ! [X: bot_bot_term_f,S5: fset_Bot_bot_term_f] :
      ( ( ~ ( fmembe1418995490368054919term_f @ X @ S5 ) )
      = ( ~ ( member2723211850833494437term_f @ X @ ( fset_Bot_bot_term_f2 @ S5 ) ) ) ) ).

% notin_fset
thf(fact_922_notin__fset,axiom,
    ! [X: product_prod_f_nat,S5: fset_P6228066233360383026_f_nat] :
      ( ( ~ ( fmembe6795308708001954811_f_nat @ X @ S5 ) )
      = ( ~ ( member2678038196313615385_f_nat @ X @ ( fset_P3576968334923099475_f_nat @ S5 ) ) ) ) ).

% notin_fset
thf(fact_923_fmember_Orep__eq,axiom,
    ( fmembe774991991480784110rm_f_v
    = ( ^ [X5: produc8027700073345061287rm_f_v,Xa: fset_P8018961893305114765rm_f_v] : ( member4316649861536941904rm_f_v @ X5 @ ( fset_P4617584883882644886rm_f_v @ Xa ) ) ) ) ).

% fmember.rep_eq
thf(fact_924_fmember_Orep__eq,axiom,
    ( fmembe6950792022577883716term_f
    = ( ^ [X5: produc250624644672406269term_f,Xa: fset_P2882900458636368611term_f] : ( member4687689947140789414term_f @ X5 @ ( fset_P8925168969356593900term_f @ Xa ) ) ) ) ).

% fmember.rep_eq
thf(fact_925_fmember_Orep__eq,axiom,
    ( fmember_term_f_v
    = ( ^ [X5: term_f_v,Xa: fset_term_f_v] : ( member_term_f_v @ X5 @ ( fset_term_f_v2 @ Xa ) ) ) ) ).

% fmember.rep_eq
thf(fact_926_fmember_Orep__eq,axiom,
    ( fmember_term_f_a
    = ( ^ [X5: term_f_a,Xa: fset_term_f_a] : ( member_term_f_a @ X5 @ ( fset_term_f_a2 @ Xa ) ) ) ) ).

% fmember.rep_eq
thf(fact_927_fmember_Orep__eq,axiom,
    ( fmembe1418995490368054919term_f
    = ( ^ [X5: bot_bot_term_f,Xa: fset_Bot_bot_term_f] : ( member2723211850833494437term_f @ X5 @ ( fset_Bot_bot_term_f2 @ Xa ) ) ) ) ).

% fmember.rep_eq
thf(fact_928_fmember_Orep__eq,axiom,
    ( fmembe6795308708001954811_f_nat
    = ( ^ [X5: product_prod_f_nat,Xa: fset_P6228066233360383026_f_nat] : ( member2678038196313615385_f_nat @ X5 @ ( fset_P3576968334923099475_f_nat @ Xa ) ) ) ) ).

% fmember.rep_eq
thf(fact_929_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ~ ( P @ N3 )
             => ? [M2: nat] :
                  ( ( ord_less_nat @ M2 @ N3 )
                  & ~ ( P @ M2 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_930_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_931_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_932_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_933_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_934_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_935_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_936_inf__concat__simple_Ocases,axiom,
    ! [X: produc8199716216217303280at_nat] :
      ( ! [F3: nat > nat] :
          ( X
         != ( produc72220940542539688at_nat @ F3 @ zero_zero_nat ) )
     => ~ ! [F3: nat > nat,N3: nat] :
            ( X
           != ( produc72220940542539688at_nat @ F3 @ ( suc @ N3 ) ) ) ) ).

% inf_concat_simple.cases
thf(fact_937_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_938_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_939_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_940_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_941_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_942_nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( P @ N3 )
           => ( P @ ( suc @ N3 ) ) )
       => ( P @ N ) ) ) ).

% nat_induct
thf(fact_943_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
     => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
       => ( ! [X4: nat,Y3: nat] :
              ( ( P @ X4 @ Y3 )
             => ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
         => ( P @ M @ N ) ) ) ) ).

% diff_induct
thf(fact_944_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N3: nat] :
            ( ( P @ ( suc @ N3 ) )
           => ( P @ N3 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_945_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_946_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_947_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_948_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M5: nat] :
          ( N
          = ( suc @ M5 ) ) ) ).

% not0_implies_Suc
thf(fact_949_fset__of__list_Orep__eq,axiom,
    ! [X: list_term_f_v] :
      ( ( fset_term_f_v2 @ ( fset_o6574051099709074461rm_f_v @ X ) )
      = ( set_term_f_v2 @ X ) ) ).

% fset_of_list.rep_eq
thf(fact_950_fset__of__list_Orep__eq,axiom,
    ! [X: list_P1780566427699531651term_f] :
      ( ( fset_P8925168969356593900term_f @ ( fset_o6125718882019406531term_f @ X ) )
      = ( set_Pr1368550810757897490term_f @ X ) ) ).

% fset_of_list.rep_eq
thf(fact_951_fset__of__list_Orep__eq,axiom,
    ! [X: list_P4093298276913796397rm_f_v] :
      ( ( fset_P4617584883882644886rm_f_v @ ( fset_o4970844032613833069rm_f_v @ X ) )
      = ( set_Pr817814403484925884rm_f_v @ X ) ) ).

% fset_of_list.rep_eq
thf(fact_952_fset__of__list_Orep__eq,axiom,
    ! [X: list_P3903862279629787026_f_nat] :
      ( ( fset_P3576968334923099475_f_nat @ ( fset_o8009517685352940092_f_nat @ X ) )
      = ( set_Pr7102205019285007021_f_nat @ X ) ) ).

% fset_of_list.rep_eq
thf(fact_953_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M6: nat] :
            ( N
            = ( suc @ M6 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_954_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M5: nat] :
          ( N
          = ( suc @ M5 ) ) ) ).

% gr0_implies_Suc
thf(fact_955_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( M = zero_zero_nat )
        | ? [J2: nat] :
            ( ( M
              = ( suc @ J2 ) )
            & ( ord_less_nat @ J2 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_956_ex__Suc__conv,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
            & ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
            & ( P @ ( suc @ I3 ) ) ) ) ) ).

% ex_Suc_conv
thf(fact_957_all__Suc__conv,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
           => ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
           => ( P @ ( suc @ I3 ) ) ) ) ) ).

% all_Suc_conv
thf(fact_958_all__less__two,axiom,
    ! [P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ ( suc @ zero_zero_nat ) ) )
           => ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        & ( P @ ( suc @ zero_zero_nat ) ) ) ) ).

% all_less_two
thf(fact_959_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K2 )
               => ~ ( P @ I2 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_960_list__update__code_I2_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat] :
      ( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat @ Y )
      = ( cons_nat @ Y @ Xs ) ) ).

% list_update_code(2)
thf(fact_961_min__list,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs ) @ X ) ) ).

% min_list
thf(fact_962_length__pos__if__in__set,axiom,
    ! [X: term_f_v,Xs: list_term_f_v] :
      ( ( member_term_f_v @ X @ ( set_term_f_v2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s7148729505879680775rm_f_v @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_963_length__pos__if__in__set,axiom,
    ! [X: produc250624644672406269term_f,Xs: list_P1780566427699531651term_f] :
      ( ( member4687689947140789414term_f @ X @ ( set_Pr1368550810757897490term_f @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s9037951226665808879term_f @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_964_length__pos__if__in__set,axiom,
    ! [X: produc8027700073345061287rm_f_v,Xs: list_P4093298276913796397rm_f_v] :
      ( ( member4316649861536941904rm_f_v @ X @ ( set_Pr817814403484925884rm_f_v @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3818039357710166425rm_f_v @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_965_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N )
            & ! [I2: nat] :
                ( ( ord_less_eq_nat @ I2 @ K2 )
               => ~ ( P @ I2 ) )
            & ( P @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_966_min__list__Cons,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
       => ( ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs ) @ ( missing_min_list_nat @ Ys ) )
         => ( ord_less_eq_nat @ ( missing_min_list_nat @ ( cons_nat @ X @ Xs ) ) @ ( missing_min_list_nat @ ( cons_nat @ Y @ Ys ) ) ) ) ) ) ).

% min_list_Cons
thf(fact_967_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_968_Fun_Ohyps,axiom,
    ! [X2a: term_f_v] :
      ( ( member_term_f_v @ X2a @ ( set_term_f_v2 @ ts ) )
     => ( ( term_ground_f_v @ X2a )
       => ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ X2a ) @ ( fset_P3576968334923099475_f_nat @ f2 ) )
         => ( ! [L2: term_f_a,S4: term_f_v] :
                ( ( fmember_term_f_a @ L2 @ r )
               => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X2a @ S4 ) @ subter5033437232688913882eq_f_v )
                 => ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_te7903553565439629403rm_f_a @ L2 ) @ ( bot_te7903553565439629424rm_f_v @ S4 ) ) @ bot_bless_eq_f ) ) )
           => ? [Q2: bot_bot_term_f] : ( fmembe1418995490368054919term_f @ Q2 @ ( tree_t4678214921166696585rm_f_f @ ( nf_ta_f_a @ r @ f2 ) @ ( term_a3932776079717421814term_f @ X2a ) ) ) ) ) ) ) ).

% Fun.hyps
thf(fact_969_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N3: nat] :
            ( X
           != ( suc @ N3 ) ) ) ).

% list_decode.cases
thf(fact_970_ground__adapt__vars,axiom,
    ! [T: term_f_v] :
      ( ( term_ground_f_v @ ( term_a2682407027258317995_f_v_v @ T ) )
      = ( term_ground_f_v @ T ) ) ).

% ground_adapt_vars
thf(fact_971_ground__adapt__vars,axiom,
    ! [T: term_f_v] :
      ( ( term_g7881354093776469683term_f @ ( term_a3932776079717421814term_f @ T ) )
      = ( term_ground_f_v @ T ) ) ).

% ground_adapt_vars
thf(fact_972_adapt__vars__adapt__vars,axiom,
    ! [T: term_f_v] :
      ( ( term_ground_f_v @ T )
     => ( ( term_a5606627300349833304rm_f_v @ ( term_a3932776079717421814term_f @ T ) )
        = T ) ) ).

% adapt_vars_adapt_vars
thf(fact_973_adapt__vars__adapt__vars,axiom,
    ! [T: term_f7969170442531588692term_f] :
      ( ( term_g7881354093776469683term_f @ T )
     => ( ( term_a3932776079717421814term_f @ ( term_a5606627300349833304rm_f_v @ T ) )
        = T ) ) ).

% adapt_vars_adapt_vars
thf(fact_974_funas__term__adapt__vars,axiom,
    ! [T: term_f_v] :
      ( ( term_f6993695921694143115term_f @ ( term_a3932776079717421814term_f @ T ) )
      = ( term_funas_term_f_v @ T ) ) ).

% funas_term_adapt_vars
thf(fact_975_funas__term__adapt__vars,axiom,
    ! [T: term_f_v] :
      ( ( term_funas_term_f_v @ ( term_a2682407027258317995_f_v_v @ T ) )
      = ( term_funas_term_f_v @ T ) ) ).

% funas_term_adapt_vars
thf(fact_976_ta__der__adapt__vars__ground,axiom,
    ! [T: term_f_v,A4: tree_ta_v_f] :
      ( ( term_ground_f_v @ T )
     => ( ( tree_ta_der_v_f @ A4 @ ( term_a2682407027258317995_f_v_v @ T ) )
        = ( tree_ta_der_v_f @ A4 @ T ) ) ) ).

% ta_der_adapt_vars_ground
thf(fact_977_ta__der__adapt__vars__ground,axiom,
    ! [T: term_f7969170442531588692term_f,A4: tree_t2044664379607036063rm_f_f] :
      ( ( term_g7881354093776469683term_f @ T )
     => ( ( tree_t4678214921166696585rm_f_f @ A4 @ ( term_a445004366929891977term_f @ T ) )
        = ( tree_t4678214921166696585rm_f_f @ A4 @ T ) ) ) ).

% ta_der_adapt_vars_ground
thf(fact_978_shift_Ocases,axiom,
    ! [X: produc8199716216217303280at_nat] :
      ~ ! [F3: nat > nat,J3: nat] :
          ( X
         != ( produc72220940542539688at_nat @ F3 @ J3 ) ) ).

% shift.cases
thf(fact_979_adapt__vars2,axiom,
    ! [T: term_f_v] :
      ( ( term_a445004366929891977term_f @ ( term_a3932776079717421814term_f @ T ) )
      = ( term_a3932776079717421814term_f @ T ) ) ).

% adapt_vars2
thf(fact_980_adapt__vars2,axiom,
    ! [T: term_f_v] :
      ( ( term_a3932776079717421814term_f @ ( term_a2682407027258317995_f_v_v @ T ) )
      = ( term_a3932776079717421814term_f @ T ) ) ).

% adapt_vars2
thf(fact_981_adapt__vars__inj,axiom,
    ! [X: term_f_v,Y: term_f_v] :
      ( ( ( term_a3932776079717421814term_f @ X )
        = ( term_a3932776079717421814term_f @ Y ) )
     => ( ( term_ground_f_v @ X )
       => ( ( term_ground_f_v @ Y )
         => ( X = Y ) ) ) ) ).

% adapt_vars_inj
thf(fact_982_adapt__vars__reverse,axiom,
    ! [T: term_f_v,T5: term_f7969170442531588692term_f] :
      ( ( term_ground_f_v @ T )
     => ( ( ( term_a5606627300349833304rm_f_v @ T5 )
          = T )
       => ( ( term_a3932776079717421814term_f @ T )
          = T5 ) ) ) ).

% adapt_vars_reverse
thf(fact_983_adapt__vars__reverse,axiom,
    ! [T: term_f7969170442531588692term_f,T5: term_f_v] :
      ( ( term_g7881354093776469683term_f @ T )
     => ( ( ( term_a3932776079717421814term_f @ T5 )
          = T )
       => ( ( term_a5606627300349833304rm_f_v @ T )
          = T5 ) ) ) ).

% adapt_vars_reverse
thf(fact_984_term__instance__of__reach__state,axiom,
    ! [Q: bot_bot_term_f,R4: fset_term_f_a,F4: fset_P6228066233360383026_f_nat,T: term_f_a] :
      ( ( fmembe1418995490368054919term_f @ Q @ ( tree_t4678214921166696585rm_f_f @ ( nf_ta_f_a @ R4 @ F4 ) @ ( term_a4260376763808259937term_f @ T ) ) )
     => ( ( term_ground_f_a @ T )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Q @ ( bot_te7903553565439629403rm_f_a @ T ) ) @ bot_bless_eq_f ) ) ) ).

% term_instance_of_reach_state
thf(fact_985_term__instance__of__reach__state,axiom,
    ! [Q: bot_bot_term_f,R4: fset_term_f_a,F4: fset_P6228066233360383026_f_nat,T: term_f_v] :
      ( ( fmembe1418995490368054919term_f @ Q @ ( tree_t4678214921166696585rm_f_f @ ( nf_ta_f_a @ R4 @ F4 ) @ ( term_a3932776079717421814term_f @ T ) ) )
     => ( ( term_ground_f_v @ T )
       => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ Q @ ( bot_te7903553565439629424rm_f_v @ T ) ) @ bot_bless_eq_f ) ) ) ).

% term_instance_of_reach_state
thf(fact_986_subt__less__eq__res__less__eq,axiom,
    ! [T: term_f_a,L: term_f_a,R4: fset_term_f_a,S: term_f_a,Q: bot_bot_term_f,F4: fset_P6228066233360383026_f_nat] :
      ( ( term_ground_f_a @ T )
     => ( ( fmember_term_f_a @ L @ R4 )
       => ( ( member7266790107935152422rm_f_a @ ( produc6952645563106136757rm_f_a @ L @ S ) @ subterm_and_supt_f_a )
         => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_te7903553565439629403rm_f_a @ S ) @ ( bot_te7903553565439629403rm_f_a @ T ) ) @ bot_bless_eq_f )
           => ( ( fmembe1418995490368054919term_f @ Q @ ( tree_t4678214921166696585rm_f_f @ ( nf_ta_f_a @ R4 @ F4 ) @ ( term_a4260376763808259937term_f @ T ) ) )
             => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_te7903553565439629403rm_f_a @ S ) @ Q ) @ bot_bless_eq_f ) ) ) ) ) ) ).

% subt_less_eq_res_less_eq
thf(fact_987_subt__less__eq__res__less__eq,axiom,
    ! [T: term_f_v,L: term_f_a,R4: fset_term_f_a,S: term_f_a,Q: bot_bot_term_f,F4: fset_P6228066233360383026_f_nat] :
      ( ( term_ground_f_v @ T )
     => ( ( fmember_term_f_a @ L @ R4 )
       => ( ( member7266790107935152422rm_f_a @ ( produc6952645563106136757rm_f_a @ L @ S ) @ subterm_and_supt_f_a )
         => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_te7903553565439629403rm_f_a @ S ) @ ( bot_te7903553565439629424rm_f_v @ T ) ) @ bot_bless_eq_f )
           => ( ( fmembe1418995490368054919term_f @ Q @ ( tree_t4678214921166696585rm_f_f @ ( nf_ta_f_a @ R4 @ F4 ) @ ( term_a3932776079717421814term_f @ T ) ) )
             => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_te7903553565439629403rm_f_a @ S ) @ Q ) @ bot_bless_eq_f ) ) ) ) ) ) ).

% subt_less_eq_res_less_eq
thf(fact_988_subt__less__eq__res__less__eq,axiom,
    ! [T: term_f_a,L: term_f_v,R4: fset_term_f_v,S: term_f_v,Q: bot_bot_term_f,F4: fset_P6228066233360383026_f_nat] :
      ( ( term_ground_f_a @ T )
     => ( ( fmember_term_f_v @ L @ R4 )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ L @ S ) @ subterm_and_supt_f_v )
         => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_te7903553565439629424rm_f_v @ S ) @ ( bot_te7903553565439629403rm_f_a @ T ) ) @ bot_bless_eq_f )
           => ( ( fmembe1418995490368054919term_f @ Q @ ( tree_t4678214921166696585rm_f_f @ ( nf_ta_f_v @ R4 @ F4 ) @ ( term_a4260376763808259937term_f @ T ) ) )
             => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_te7903553565439629424rm_f_v @ S ) @ Q ) @ bot_bless_eq_f ) ) ) ) ) ) ).

% subt_less_eq_res_less_eq
thf(fact_989_subt__less__eq__res__less__eq,axiom,
    ! [T: term_f_v,L: term_f_v,R4: fset_term_f_v,S: term_f_v,Q: bot_bot_term_f,F4: fset_P6228066233360383026_f_nat] :
      ( ( term_ground_f_v @ T )
     => ( ( fmember_term_f_v @ L @ R4 )
       => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ L @ S ) @ subterm_and_supt_f_v )
         => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_te7903553565439629424rm_f_v @ S ) @ ( bot_te7903553565439629424rm_f_v @ T ) ) @ bot_bless_eq_f )
           => ( ( fmembe1418995490368054919term_f @ Q @ ( tree_t4678214921166696585rm_f_f @ ( nf_ta_f_v @ R4 @ F4 ) @ ( term_a3932776079717421814term_f @ T ) ) )
             => ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_te7903553565439629424rm_f_v @ S ) @ Q ) @ bot_bless_eq_f ) ) ) ) ) ) ).

% subt_less_eq_res_less_eq
thf(fact_990_ta__der_H__ta__der,axiom,
    ! [T: term_f_a,A6: tree_ta_a_f,S: term_f_a,P2: a] :
      ( ( fmember_term_f_a @ T @ ( tree_ta_der_a_f2 @ A6 @ S ) )
     => ( ( fmember_a @ P2 @ ( tree_ta_der_a_f @ A6 @ T ) )
       => ( fmember_a @ P2 @ ( tree_ta_der_a_f @ A6 @ S ) ) ) ) ).

% ta_der'_ta_der
thf(fact_991_ta__der_H__ta__der,axiom,
    ! [T: term_f7969170442531588692term_f,A6: tree_t2044664379607036063rm_f_f,S: term_f7969170442531588692term_f,P2: bot_bot_term_f] :
      ( ( fmembe7587999265257414299term_f @ T @ ( tree_t2575918229523990688rm_f_f @ A6 @ S ) )
     => ( ( fmembe1418995490368054919term_f @ P2 @ ( tree_t4678214921166696585rm_f_f @ A6 @ T ) )
       => ( fmembe1418995490368054919term_f @ P2 @ ( tree_t4678214921166696585rm_f_f @ A6 @ S ) ) ) ) ).

% ta_der'_ta_der
thf(fact_992_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_993_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_994_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_995_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_996_ta__nf__tr__to__state,axiom,
    ! [T: term_f_v,Q: bot_bot_term_f,R4: fset_term_f_a,F4: fset_P6228066233360383026_f_nat] :
      ( ( term_ground_f_v @ T )
     => ( ( fmembe1418995490368054919term_f @ Q @ ( tree_t4678214921166696585rm_f_f @ ( nf_ta_f_a @ R4 @ F4 ) @ ( term_a3932776079717421814term_f @ T ) ) )
       => ( fmembe1418995490368054919term_f @ Q @ ( fstates_f_a @ R4 ) ) ) ) ).

% ta_nf_tr_to_state
thf(fact_997_ta__nf__sound1,axiom,
    ! [T: term_f_a,L: term_f_a,R4: fset_term_f_a,F4: fset_P6228066233360383026_f_nat] :
      ( ( term_ground_f_a @ T )
     => ( ( fmember_term_f_a @ L @ R4 )
       => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_te7903553565439629403rm_f_a @ L ) @ ( bot_te7903553565439629403rm_f_a @ T ) ) @ bot_bless_eq_f )
         => ( ( tree_t4678214921166696585rm_f_f @ ( nf_ta_f_a @ R4 @ F4 ) @ ( term_a4260376763808259937term_f @ T ) )
            = bot_bo4668628460741225810term_f ) ) ) ) ).

% ta_nf_sound1
thf(fact_998_ta__nf__sound1,axiom,
    ! [T: term_f_v,L: term_f_a,R4: fset_term_f_a,F4: fset_P6228066233360383026_f_nat] :
      ( ( term_ground_f_v @ T )
     => ( ( fmember_term_f_a @ L @ R4 )
       => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_te7903553565439629403rm_f_a @ L ) @ ( bot_te7903553565439629424rm_f_v @ T ) ) @ bot_bless_eq_f )
         => ( ( tree_t4678214921166696585rm_f_f @ ( nf_ta_f_a @ R4 @ F4 ) @ ( term_a3932776079717421814term_f @ T ) )
            = bot_bo4668628460741225810term_f ) ) ) ) ).

% ta_nf_sound1
thf(fact_999_ta__nf__sound1,axiom,
    ! [T: term_f_a,L: term_f_v,R4: fset_term_f_v,F4: fset_P6228066233360383026_f_nat] :
      ( ( term_ground_f_a @ T )
     => ( ( fmember_term_f_v @ L @ R4 )
       => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_te7903553565439629424rm_f_v @ L ) @ ( bot_te7903553565439629403rm_f_a @ T ) ) @ bot_bless_eq_f )
         => ( ( tree_t4678214921166696585rm_f_f @ ( nf_ta_f_v @ R4 @ F4 ) @ ( term_a4260376763808259937term_f @ T ) )
            = bot_bo4668628460741225810term_f ) ) ) ) ).

% ta_nf_sound1
thf(fact_1000_ta__nf__sound1,axiom,
    ! [T: term_f_v,L: term_f_v,R4: fset_term_f_v,F4: fset_P6228066233360383026_f_nat] :
      ( ( term_ground_f_v @ T )
     => ( ( fmember_term_f_v @ L @ R4 )
       => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_te7903553565439629424rm_f_v @ L ) @ ( bot_te7903553565439629424rm_f_v @ T ) ) @ bot_bless_eq_f )
         => ( ( tree_t4678214921166696585rm_f_f @ ( nf_ta_f_v @ R4 @ F4 ) @ ( term_a3932776079717421814term_f @ T ) )
            = bot_bo4668628460741225810term_f ) ) ) ) ).

% ta_nf_sound1
thf(fact_1001_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_12: nat] : ( P @ X_12 )
       => ? [N3: nat] :
            ( ~ ( P @ N3 )
            & ( P @ ( suc @ N3 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_1002_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_1003_all__not__fin__conv,axiom,
    ! [A4: fset_term_f_a] :
      ( ( ! [X5: term_f_a] :
            ~ ( fmember_term_f_a @ X5 @ A4 ) )
      = ( A4 = bot_bo7557743857269764754rm_f_a ) ) ).

% all_not_fin_conv
thf(fact_1004_all__not__fin__conv,axiom,
    ! [A4: fset_Bot_bot_term_f] :
      ( ( ! [X5: bot_bot_term_f] :
            ~ ( fmembe1418995490368054919term_f @ X5 @ A4 ) )
      = ( A4 = bot_bo4668628460741225810term_f ) ) ).

% all_not_fin_conv
thf(fact_1005_fempty__iff,axiom,
    ! [C: term_f_a] :
      ~ ( fmember_term_f_a @ C @ bot_bo7557743857269764754rm_f_a ) ).

% fempty_iff
thf(fact_1006_fempty__iff,axiom,
    ! [C: bot_bot_term_f] :
      ~ ( fmembe1418995490368054919term_f @ C @ bot_bo4668628460741225810term_f ) ).

% fempty_iff
thf(fact_1007_equalsffemptyI,axiom,
    ! [A4: fset_term_f_a] :
      ( ! [Y3: term_f_a] :
          ~ ( fmember_term_f_a @ Y3 @ A4 )
     => ( A4 = bot_bo7557743857269764754rm_f_a ) ) ).

% equalsffemptyI
thf(fact_1008_equalsffemptyI,axiom,
    ! [A4: fset_Bot_bot_term_f] :
      ( ! [Y3: bot_bot_term_f] :
          ~ ( fmembe1418995490368054919term_f @ Y3 @ A4 )
     => ( A4 = bot_bo4668628460741225810term_f ) ) ).

% equalsffemptyI
thf(fact_1009_equalsffemptyD,axiom,
    ! [A4: fset_term_f_a,A: term_f_a] :
      ( ( A4 = bot_bo7557743857269764754rm_f_a )
     => ~ ( fmember_term_f_a @ A @ A4 ) ) ).

% equalsffemptyD
thf(fact_1010_equalsffemptyD,axiom,
    ! [A4: fset_Bot_bot_term_f,A: bot_bot_term_f] :
      ( ( A4 = bot_bo4668628460741225810term_f )
     => ~ ( fmembe1418995490368054919term_f @ A @ A4 ) ) ).

% equalsffemptyD
thf(fact_1011_ex__fin__conv,axiom,
    ! [A4: fset_term_f_a] :
      ( ( ? [X5: term_f_a] : ( fmember_term_f_a @ X5 @ A4 ) )
      = ( A4 != bot_bo7557743857269764754rm_f_a ) ) ).

% ex_fin_conv
thf(fact_1012_ex__fin__conv,axiom,
    ! [A4: fset_Bot_bot_term_f] :
      ( ( ? [X5: bot_bot_term_f] : ( fmembe1418995490368054919term_f @ X5 @ A4 ) )
      = ( A4 != bot_bo4668628460741225810term_f ) ) ).

% ex_fin_conv
thf(fact_1013_femptyE,axiom,
    ! [A: term_f_a] :
      ~ ( fmember_term_f_a @ A @ bot_bo7557743857269764754rm_f_a ) ).

% femptyE
thf(fact_1014_femptyE,axiom,
    ! [A: bot_bot_term_f] :
      ~ ( fmembe1418995490368054919term_f @ A @ bot_bo4668628460741225810term_f ) ).

% femptyE
thf(fact_1015_bot__prod__def,axiom,
    ( bot_bo2705896066424148828at_nat
    = ( produc72220940542539688at_nat @ bot_bot_nat_nat @ bot_bot_nat ) ) ).

% bot_prod_def
thf(fact_1016_bot__prod__def,axiom,
    ( bot_bo2769642828321324397at_nat
    = ( product_Pair_nat_nat @ bot_bot_nat @ bot_bot_nat ) ) ).

% bot_prod_def
thf(fact_1017_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_1018_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1019_ta__langE,axiom,
    ! [T: term_f_v,Q3: fset_term_f_a,A6: tree_ta_term_f_a_f] :
      ( ( member_term_f_v @ T @ ( tree_t6230644817623509812_a_f_v @ Q3 @ A6 ) )
     => ~ ! [T6: term_f_term_f_a] :
            ( ( term_g3921090622373805043rm_f_a @ T6 )
           => ! [Q2: term_f_a] :
                ( ( fmember_term_f_a @ Q2 @ Q3 )
               => ( ( fmember_term_f_a @ Q2 @ ( tree_t114237181795497801_f_a_f @ A6 @ T6 ) )
                 => ( T
                   != ( term_a258605488875140376_f_a_v @ T6 ) ) ) ) ) ) ).

% ta_langE
thf(fact_1020_ta__langE,axiom,
    ! [T: term_f_v,Q3: fset_v,A6: tree_ta_v_f] :
      ( ( member_term_f_v @ T @ ( tree_ta_lang_v_f_v @ Q3 @ A6 ) )
     => ~ ! [T6: term_f_v] :
            ( ( term_ground_f_v @ T6 )
           => ! [Q2: v] :
                ( ( fmember_v @ Q2 @ Q3 )
               => ( ( fmember_v @ Q2 @ ( tree_ta_der_v_f @ A6 @ T6 ) )
                 => ( T
                   != ( term_a2682407027258317995_f_v_v @ T6 ) ) ) ) ) ) ).

% ta_langE
thf(fact_1021_ta__langE,axiom,
    ! [T: term_f_v,Q3: fset_Bot_bot_term_f,A6: tree_t2044664379607036063rm_f_f] :
      ( ( member_term_f_v @ T @ ( tree_t1514651371760659444_f_f_v @ Q3 @ A6 ) )
     => ~ ! [T6: term_f7969170442531588692term_f] :
            ( ( term_g7881354093776469683term_f @ T6 )
           => ! [Q2: bot_bot_term_f] :
                ( ( fmembe1418995490368054919term_f @ Q2 @ Q3 )
               => ( ( fmembe1418995490368054919term_f @ Q2 @ ( tree_t4678214921166696585rm_f_f @ A6 @ T6 ) )
                 => ( T
                   != ( term_a5606627300349833304rm_f_v @ T6 ) ) ) ) ) ) ).

% ta_langE
thf(fact_1022_ta__langE,axiom,
    ! [T: term_f7969170442531588692term_f,Q3: fset_v,A6: tree_ta_v_f] :
      ( ( member1719022467595452157term_f @ T @ ( tree_t3074183349113346900term_f @ Q3 @ A6 ) )
     => ~ ! [T6: term_f_v] :
            ( ( term_ground_f_v @ T6 )
           => ! [Q2: v] :
                ( ( fmember_v @ Q2 @ Q3 )
               => ( ( fmember_v @ Q2 @ ( tree_ta_der_v_f @ A6 @ T6 ) )
                 => ( T
                   != ( term_a3932776079717421814term_f @ T6 ) ) ) ) ) ) ).

% ta_langE
thf(fact_1023_ta__langI,axiom,
    ! [T5: term_f_term_f_a,Q: term_f_a,Q3: fset_term_f_a,A6: tree_ta_term_f_a_f,T: term_f_v] :
      ( ( term_g3921090622373805043rm_f_a @ T5 )
     => ( ( fmember_term_f_a @ Q @ Q3 )
       => ( ( fmember_term_f_a @ Q @ ( tree_t114237181795497801_f_a_f @ A6 @ T5 ) )
         => ( ( T
              = ( term_a258605488875140376_f_a_v @ T5 ) )
           => ( member_term_f_v @ T @ ( tree_t6230644817623509812_a_f_v @ Q3 @ A6 ) ) ) ) ) ) ).

% ta_langI
thf(fact_1024_ta__langI,axiom,
    ! [T5: term_f_v,Q: v,Q3: fset_v,A6: tree_ta_v_f,T: term_f_v] :
      ( ( term_ground_f_v @ T5 )
     => ( ( fmember_v @ Q @ Q3 )
       => ( ( fmember_v @ Q @ ( tree_ta_der_v_f @ A6 @ T5 ) )
         => ( ( T
              = ( term_a2682407027258317995_f_v_v @ T5 ) )
           => ( member_term_f_v @ T @ ( tree_ta_lang_v_f_v @ Q3 @ A6 ) ) ) ) ) ) ).

% ta_langI
thf(fact_1025_ta__langI,axiom,
    ! [T5: term_f7969170442531588692term_f,Q: bot_bot_term_f,Q3: fset_Bot_bot_term_f,A6: tree_t2044664379607036063rm_f_f,T: term_f_v] :
      ( ( term_g7881354093776469683term_f @ T5 )
     => ( ( fmembe1418995490368054919term_f @ Q @ Q3 )
       => ( ( fmembe1418995490368054919term_f @ Q @ ( tree_t4678214921166696585rm_f_f @ A6 @ T5 ) )
         => ( ( T
              = ( term_a5606627300349833304rm_f_v @ T5 ) )
           => ( member_term_f_v @ T @ ( tree_t1514651371760659444_f_f_v @ Q3 @ A6 ) ) ) ) ) ) ).

% ta_langI
thf(fact_1026_ta__langI,axiom,
    ! [T5: term_f_v,Q: v,Q3: fset_v,A6: tree_ta_v_f,T: term_f7969170442531588692term_f] :
      ( ( term_ground_f_v @ T5 )
     => ( ( fmember_v @ Q @ Q3 )
       => ( ( fmember_v @ Q @ ( tree_ta_der_v_f @ A6 @ T5 ) )
         => ( ( T
              = ( term_a3932776079717421814term_f @ T5 ) )
           => ( member1719022467595452157term_f @ T @ ( tree_t3074183349113346900term_f @ Q3 @ A6 ) ) ) ) ) ) ).

% ta_langI
thf(fact_1027_ta__der_H__Fun,axiom,
    ! [Q: a,A6: tree_ta_a_f,F: f,Ts: list_term_f_a] :
      ( ( fmember_term_f_a @ ( var_a_f @ Q ) @ ( tree_ta_der_a_f2 @ A6 @ ( fun_f_a @ F @ Ts ) ) )
      = ( fmember_a @ Q @ ( tree_ta_der_a_f @ A6 @ ( fun_f_a @ F @ Ts ) ) ) ) ).

% ta_der'_Fun
thf(fact_1028_ta__der_H__Fun,axiom,
    ! [Q: v,A6: tree_ta_v_f,F: f,Ts: list_term_f_v] :
      ( ( fmember_term_f_v @ ( var_v_f @ Q ) @ ( tree_ta_der_v_f2 @ A6 @ ( fun_f_v @ F @ Ts ) ) )
      = ( fmember_v @ Q @ ( tree_ta_der_v_f @ A6 @ ( fun_f_v @ F @ Ts ) ) ) ) ).

% ta_der'_Fun
thf(fact_1029_ta__der_H__Fun,axiom,
    ! [Q: bot_bot_term_f,A6: tree_t2044664379607036063rm_f_f,F: f,Ts: list_t5008422188120450138term_f] :
      ( ( fmembe7587999265257414299term_f @ ( var_Bot_bot_term_f_f @ Q ) @ ( tree_t2575918229523990688rm_f_f @ A6 @ ( fun_f_Bot_bot_term_f @ F @ Ts ) ) )
      = ( fmembe1418995490368054919term_f @ Q @ ( tree_t4678214921166696585rm_f_f @ A6 @ ( fun_f_Bot_bot_term_f @ F @ Ts ) ) ) ) ).

% ta_der'_Fun
thf(fact_1030_ta__der_H_Ocases,axiom,
    ! [X: produc5998591847903407270rm_f_v] :
      ( ! [A8: tree_ta_v_f,P6: v] :
          ( X
         != ( produc5293873249180174558rm_f_v @ A8 @ ( var_v_f @ P6 ) ) )
     => ~ ! [A8: tree_ta_v_f,F3: f,Ts2: list_term_f_v] :
            ( X
           != ( produc5293873249180174558rm_f_v @ A8 @ ( fun_f_v @ F3 @ Ts2 ) ) ) ) ).

% ta_der'.cases
thf(fact_1031_funas__term_Osimps_I1_J,axiom,
    ! [X: v] :
      ( ( term_funas_term_f_v @ ( var_v_f @ X ) )
      = bot_bo136713897782694092_f_nat ) ).

% funas_term.simps(1)
thf(fact_1032_funas__term_Ocases,axiom,
    ! [X: term_f_v] :
      ( ! [X4: v] :
          ( X
         != ( var_v_f @ X4 ) )
     => ~ ! [F3: f,Ts2: list_term_f_v] :
            ( X
           != ( fun_f_v @ F3 @ Ts2 ) ) ) ).

% funas_term.cases
thf(fact_1033_Term_Oterm_Oexhaust,axiom,
    ! [Y: term_f_v] :
      ( ! [X12: v] :
          ( Y
         != ( var_v_f @ X12 ) )
     => ~ ! [X212: f,X223: list_term_f_v] :
            ( Y
           != ( fun_f_v @ X212 @ X223 ) ) ) ).

% Term.term.exhaust
thf(fact_1034_term_Odistinct_I1_J,axiom,
    ! [X1: v,X21: f,X222: list_term_f_v] :
      ( ( var_v_f @ X1 )
     != ( fun_f_v @ X21 @ X222 ) ) ).

% term.distinct(1)
thf(fact_1035_term__to__bot__term_Ocases,axiom,
    ! [X: term_f_v] :
      ( ! [Uu: v] :
          ( X
         != ( var_v_f @ Uu ) )
     => ~ ! [F3: f,Ts2: list_term_f_v] :
            ( X
           != ( fun_f_v @ F3 @ Ts2 ) ) ) ).

% term_to_bot_term.cases
thf(fact_1036_ground_Osimps_I1_J,axiom,
    ! [X: v] :
      ~ ( term_ground_f_v @ ( var_v_f @ X ) ) ).

% ground.simps(1)
thf(fact_1037_supteq__var__imp__eq,axiom,
    ! [X: v,T: term_f_v] :
      ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( var_v_f @ X ) @ T ) @ subter5033437232688913882eq_f_v )
      = ( T
        = ( var_v_f @ X ) ) ) ).

% supteq_var_imp_eq
thf(fact_1038_supt__var,axiom,
    ! [X: v,U: term_f_v] :
      ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( var_v_f @ X ) @ U ) @ subterm_and_supt_f_v ) ).

% supt_var
thf(fact_1039_Var__supt,axiom,
    ! [X: v,T: term_f_v] :
      ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( var_v_f @ X ) @ T ) @ subterm_and_supt_f_v ) ).

% Var_supt
thf(fact_1040_term__fset__induct,axiom,
    ! [P: term_f_a > $o,Term: term_f_a] :
      ( ! [X4: a] : ( P @ ( var_a_f @ X4 ) )
     => ( ! [X1a: f,X23: list_term_f_a] :
            ( ! [X2a2: term_f_a] :
                ( ( fmember_term_f_a @ X2a2 @ ( fset_o6574051009341269640rm_f_a @ X23 ) )
               => ( P @ X2a2 ) )
           => ( P @ ( fun_f_a @ X1a @ X23 ) ) )
       => ( P @ Term ) ) ) ).

% term_fset_induct
thf(fact_1041_term__fset__induct,axiom,
    ! [P: term_f_v > $o,Term: term_f_v] :
      ( ! [X4: v] : ( P @ ( var_v_f @ X4 ) )
     => ( ! [X1a: f,X23: list_term_f_v] :
            ( ! [X2a2: term_f_v] :
                ( ( fmember_term_f_v @ X2a2 @ ( fset_o6574051099709074461rm_f_v @ X23 ) )
               => ( P @ X2a2 ) )
           => ( P @ ( fun_f_v @ X1a @ X23 ) ) )
       => ( P @ Term ) ) ) ).

% term_fset_induct
thf(fact_1042_ta__der__to__ta__der_H,axiom,
    ! [Q: a,A6: tree_ta_a_f,S: term_f_a] :
      ( ( fmember_a @ Q @ ( tree_ta_der_a_f @ A6 @ S ) )
      = ( fmember_term_f_a @ ( var_a_f @ Q ) @ ( tree_ta_der_a_f2 @ A6 @ S ) ) ) ).

% ta_der_to_ta_der'
thf(fact_1043_ta__der__to__ta__der_H,axiom,
    ! [Q: bot_bot_term_f,A6: tree_t2044664379607036063rm_f_f,S: term_f7969170442531588692term_f] :
      ( ( fmembe1418995490368054919term_f @ Q @ ( tree_t4678214921166696585rm_f_f @ A6 @ S ) )
      = ( fmembe7587999265257414299term_f @ ( var_Bot_bot_term_f_f @ Q ) @ ( tree_t2575918229523990688rm_f_f @ A6 @ S ) ) ) ).

% ta_der_to_ta_der'
thf(fact_1044_ta__der_H__to__ta__der,axiom,
    ! [Q: a,A6: tree_ta_a_f,S: term_f_a] :
      ( ( fmember_term_f_a @ ( var_a_f @ Q ) @ ( tree_ta_der_a_f2 @ A6 @ S ) )
     => ( fmember_a @ Q @ ( tree_ta_der_a_f @ A6 @ S ) ) ) ).

% ta_der'_to_ta_der
thf(fact_1045_ta__der_H__to__ta__der,axiom,
    ! [Q: bot_bot_term_f,A6: tree_t2044664379607036063rm_f_f,S: term_f7969170442531588692term_f] :
      ( ( fmembe7587999265257414299term_f @ ( var_Bot_bot_term_f_f @ Q ) @ ( tree_t2575918229523990688rm_f_f @ A6 @ S ) )
     => ( fmembe1418995490368054919term_f @ Q @ ( tree_t4678214921166696585rm_f_f @ A6 @ S ) ) ) ).

% ta_der'_to_ta_der
thf(fact_1046_replace__term__at_Oelims,axiom,
    ! [X: term_f_v,Xa2: list_nat,Xb: term_f_v,Y: term_f_v] :
      ( ( ( term_r2146176907066831980at_f_v @ X @ Xa2 @ Xb )
        = Y )
     => ( ( ( Xa2 = nil_nat )
         => ( Y != Xb ) )
       => ( ! [X4: v] :
              ( ( X
                = ( var_v_f @ X4 ) )
             => ( ? [V: nat,Va: list_nat] :
                    ( Xa2
                    = ( cons_nat @ V @ Va ) )
               => ( Y
                 != ( var_v_f @ X4 ) ) ) )
         => ~ ! [F3: f,Ts2: list_term_f_v] :
                ( ( X
                  = ( fun_f_v @ F3 @ Ts2 ) )
               => ! [I: nat,Ps2: list_nat] :
                    ( ( Xa2
                      = ( cons_nat @ I @ Ps2 ) )
                   => ~ ( ( ( ord_less_nat @ I @ ( size_s7148729505879680775rm_f_v @ Ts2 ) )
                         => ( Y
                            = ( fun_f_v @ F3 @ ( list_update_term_f_v @ Ts2 @ I @ ( term_r2146176907066831980at_f_v @ ( nth_term_f_v @ Ts2 @ I ) @ Ps2 @ Xb ) ) ) ) )
                        & ( ~ ( ord_less_nat @ I @ ( size_s7148729505879680775rm_f_v @ Ts2 ) )
                         => ( Y
                            = ( fun_f_v @ F3 @ Ts2 ) ) ) ) ) ) ) ) ) ).

% replace_term_at.elims
thf(fact_1047_gterm__of__term_Ocases,axiom,
    ! [X: term_f_v] :
      ( ! [F3: f,Ts2: list_term_f_v] :
          ( X
         != ( fun_f_v @ F3 @ Ts2 ) )
     => ~ ! [V: v] :
            ( X
           != ( var_v_f @ V ) ) ) ).

% gterm_of_term.cases
thf(fact_1048_list__update__nonempty,axiom,
    ! [Xs: list_nat,K: nat,X: nat] :
      ( ( ( list_update_nat @ Xs @ K @ X )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% list_update_nonempty
thf(fact_1049_list__ex__simps_I2_J,axiom,
    ! [P: nat > $o] :
      ~ ( list_ex_nat @ P @ nil_nat ) ).

% list_ex_simps(2)
thf(fact_1050_position__diff__Nil,axiom,
    ! [Q: list_nat] :
      ( ( term_pos_diff_nat @ Q @ nil_nat )
      = Q ) ).

% position_diff_Nil
thf(fact_1051_set__empty2,axiom,
    ! [Xs: list_nat] :
      ( ( bot_bot_set_nat
        = ( set_nat2 @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% set_empty2
thf(fact_1052_set__empty2,axiom,
    ! [Xs: list_term_f_v] :
      ( ( bot_bot_set_term_f_v
        = ( set_term_f_v2 @ Xs ) )
      = ( Xs = nil_term_f_v ) ) ).

% set_empty2
thf(fact_1053_set__empty2,axiom,
    ! [Xs: list_P1780566427699531651term_f] :
      ( ( bot_bo5323853609601926601term_f
        = ( set_Pr1368550810757897490term_f @ Xs ) )
      = ( Xs = nil_Pr3198911942259489283term_f ) ) ).

% set_empty2
thf(fact_1054_set__empty2,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v] :
      ( ( bot_bo4638812505780724595rm_f_v
        = ( set_Pr817814403484925884rm_f_v @ Xs ) )
      = ( Xs = nil_Pr5389109106923839661rm_f_v ) ) ).

% set_empty2
thf(fact_1055_set__empty,axiom,
    ! [Xs: list_nat] :
      ( ( ( set_nat2 @ Xs )
        = bot_bot_set_nat )
      = ( Xs = nil_nat ) ) ).

% set_empty
thf(fact_1056_set__empty,axiom,
    ! [Xs: list_term_f_v] :
      ( ( ( set_term_f_v2 @ Xs )
        = bot_bot_set_term_f_v )
      = ( Xs = nil_term_f_v ) ) ).

% set_empty
thf(fact_1057_set__empty,axiom,
    ! [Xs: list_P1780566427699531651term_f] :
      ( ( ( set_Pr1368550810757897490term_f @ Xs )
        = bot_bo5323853609601926601term_f )
      = ( Xs = nil_Pr3198911942259489283term_f ) ) ).

% set_empty
thf(fact_1058_set__empty,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v] :
      ( ( ( set_Pr817814403484925884rm_f_v @ Xs )
        = bot_bo4638812505780724595rm_f_v )
      = ( Xs = nil_Pr5389109106923839661rm_f_v ) ) ).

% set_empty
thf(fact_1059_length__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_nat ) ) ).

% length_0_conv
thf(fact_1060_length__0__conv,axiom,
    ! [Xs: list_term_f_v] :
      ( ( ( size_s7148729505879680775rm_f_v @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_term_f_v ) ) ).

% length_0_conv
thf(fact_1061_length__0__conv,axiom,
    ! [Xs: list_P1780566427699531651term_f] :
      ( ( ( size_s9037951226665808879term_f @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Pr3198911942259489283term_f ) ) ).

% length_0_conv
thf(fact_1062_length__0__conv,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v] :
      ( ( ( size_s3818039357710166425rm_f_v @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Pr5389109106923839661rm_f_v ) ) ).

% length_0_conv
thf(fact_1063_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_1064_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_1065_take0,axiom,
    ( ( take_nat @ zero_zero_nat )
    = ( ^ [Xs2: list_nat] : nil_nat ) ) ).

% take0
thf(fact_1066_fset__of__list__simps_I1_J,axiom,
    ( ( fset_of_list_nat @ nil_nat )
    = bot_bot_fset_nat ) ).

% fset_of_list_simps(1)
thf(fact_1067_Nil__lenlex__iff1,axiom,
    ! [Ns: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ns ) @ ( lenlex_nat @ R3 ) )
      = ( Ns != nil_nat ) ) ).

% Nil_lenlex_iff1
thf(fact_1068_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_1069_length__greater__0__conv,axiom,
    ! [Xs: list_term_f_v] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s7148729505879680775rm_f_v @ Xs ) )
      = ( Xs != nil_term_f_v ) ) ).

% length_greater_0_conv
thf(fact_1070_length__greater__0__conv,axiom,
    ! [Xs: list_P1780566427699531651term_f] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s9037951226665808879term_f @ Xs ) )
      = ( Xs != nil_Pr3198911942259489283term_f ) ) ).

% length_greater_0_conv
thf(fact_1071_length__greater__0__conv,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s3818039357710166425rm_f_v @ Xs ) )
      = ( Xs != nil_Pr5389109106923839661rm_f_v ) ) ).

% length_greater_0_conv
thf(fact_1072_empty__set,axiom,
    ( bot_bot_set_nat
    = ( set_nat2 @ nil_nat ) ) ).

% empty_set
thf(fact_1073_empty__set,axiom,
    ( bot_bot_set_term_f_v
    = ( set_term_f_v2 @ nil_term_f_v ) ) ).

% empty_set
thf(fact_1074_empty__set,axiom,
    ( bot_bo5323853609601926601term_f
    = ( set_Pr1368550810757897490term_f @ nil_Pr3198911942259489283term_f ) ) ).

% empty_set
thf(fact_1075_empty__set,axiom,
    ( bot_bo4638812505780724595rm_f_v
    = ( set_Pr817814403484925884rm_f_v @ nil_Pr5389109106923839661rm_f_v ) ) ).

% empty_set
thf(fact_1076_list__encode_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ~ ! [X4: nat,Xs3: list_nat] :
            ( X
           != ( cons_nat @ X4 @ Xs3 ) ) ) ).

% list_encode.cases
thf(fact_1077_transpose_Ocases,axiom,
    ! [X: list_list_nat] :
      ( ( X != nil_list_nat )
     => ( ! [Xss2: list_list_nat] :
            ( X
           != ( cons_list_nat @ nil_nat @ Xss2 ) )
       => ~ ! [X4: nat,Xs3: list_nat,Xss2: list_list_nat] :
              ( X
             != ( cons_list_nat @ ( cons_nat @ X4 @ Xs3 ) @ Xss2 ) ) ) ) ).

% transpose.cases
thf(fact_1078_list__update_Osimps_I1_J,axiom,
    ! [I4: nat,V2: nat] :
      ( ( list_update_nat @ nil_nat @ I4 @ V2 )
      = nil_nat ) ).

% list_update.simps(1)
thf(fact_1079_list__update__code_I1_J,axiom,
    ! [I4: nat,Y: nat] :
      ( ( list_update_nat @ nil_nat @ I4 @ Y )
      = nil_nat ) ).

% list_update_code(1)
thf(fact_1080_take__Nil,axiom,
    ! [N: nat] :
      ( ( take_nat @ N @ nil_nat )
      = nil_nat ) ).

% take_Nil
thf(fact_1081_distinct_Osimps_I1_J,axiom,
    distinct_nat @ nil_nat ).

% distinct.simps(1)
thf(fact_1082_list__4__cases,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ( ! [X4: nat] :
            ( Xs
           != ( cons_nat @ X4 @ nil_nat ) )
       => ( ! [X4: nat,Y3: nat] :
              ( Xs
             != ( cons_nat @ X4 @ ( cons_nat @ Y3 @ nil_nat ) ) )
         => ~ ! [X4: nat,Y3: nat,Z3: nat,Zs: list_nat] :
                ( Xs
               != ( cons_nat @ X4 @ ( cons_nat @ Y3 @ ( cons_nat @ Z3 @ Zs ) ) ) ) ) ) ) ).

% list_4_cases
thf(fact_1083_list__3__cases,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ( ! [X4: nat] :
            ( Xs
           != ( cons_nat @ X4 @ nil_nat ) )
       => ~ ! [X4: nat,Y3: nat,Ys4: list_nat] :
              ( Xs
             != ( cons_nat @ X4 @ ( cons_nat @ Y3 @ Ys4 ) ) ) ) ) ).

% list_3_cases
thf(fact_1084_list__nonempty__induct,axiom,
    ! [Xs: list_nat,P: list_nat > $o] :
      ( ( Xs != nil_nat )
     => ( ! [X4: nat] : ( P @ ( cons_nat @ X4 @ nil_nat ) )
       => ( ! [X4: nat,Xs3: list_nat] :
              ( ( Xs3 != nil_nat )
             => ( ( P @ Xs3 )
               => ( P @ ( cons_nat @ X4 @ Xs3 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_1085_list__induct2_H,axiom,
    ! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
      ( ( P @ nil_nat @ nil_nat )
     => ( ! [X4: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X4 @ Xs3 ) @ nil_nat )
       => ( ! [Y3: nat,Ys4: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y3 @ Ys4 ) )
         => ( ! [X4: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat] :
                ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_1086_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_1087_Missing__List_Omin__list_Ocases,axiom,
    ! [X: list_nat] :
      ( ! [X4: nat] :
          ( X
         != ( cons_nat @ X4 @ nil_nat ) )
     => ( ! [X4: nat,V: nat,Va: list_nat] :
            ( X
           != ( cons_nat @ X4 @ ( cons_nat @ V @ Va ) ) )
       => ( X = nil_nat ) ) ) ).

% Missing_List.min_list.cases
thf(fact_1088_remdups__adj_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ( ! [X4: nat] :
            ( X
           != ( cons_nat @ X4 @ nil_nat ) )
       => ~ ! [X4: nat,Y3: nat,Xs3: list_nat] :
              ( X
             != ( cons_nat @ X4 @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_1089_List_Omin__list_Ocases,axiom,
    ! [X: list_nat] :
      ( ! [X4: nat,Xs3: list_nat] :
          ( X
         != ( cons_nat @ X4 @ Xs3 ) )
     => ( X = nil_nat ) ) ).

% List.min_list.cases
thf(fact_1090_list_Oexhaust,axiom,
    ! [Y: list_nat] :
      ( ( Y != nil_nat )
     => ~ ! [X212: nat,X223: list_nat] :
            ( Y
           != ( cons_nat @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_1091_list_OdiscI,axiom,
    ! [List: list_nat,X21: nat,X222: list_nat] :
      ( ( List
        = ( cons_nat @ X21 @ X222 ) )
     => ( List != nil_nat ) ) ).

% list.discI
thf(fact_1092_list_Odistinct_I1_J,axiom,
    ! [X21: nat,X222: list_nat] :
      ( nil_nat
     != ( cons_nat @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_1093_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_1094_add__elem__list__lists_Osimps_I1_J,axiom,
    ! [X: nat] :
      ( ( basic_4874698711677410535ts_nat @ X @ nil_nat )
      = ( cons_list_nat @ ( cons_nat @ X @ nil_nat ) @ nil_list_nat ) ) ).

% add_elem_list_lists.simps(1)
thf(fact_1095_successively_Ocases,axiom,
    ! [X: produc254973753779126261st_nat] :
      ( ! [P7: nat > nat > $o] :
          ( X
         != ( produc4727192421694094319st_nat @ P7 @ nil_nat ) )
     => ( ! [P7: nat > nat > $o,X4: nat] :
            ( X
           != ( produc4727192421694094319st_nat @ P7 @ ( cons_nat @ X4 @ nil_nat ) ) )
       => ~ ! [P7: nat > nat > $o,X4: nat,Y3: nat,Xs3: list_nat] :
              ( X
             != ( produc4727192421694094319st_nat @ P7 @ ( cons_nat @ X4 @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_1096_span_Ocases,axiom,
    ! [X: produc4226810134323546766st_nat] :
      ( ! [P7: nat > $o,X4: nat,Xs3: list_nat] :
          ( X
         != ( produc8587622027977423880st_nat @ P7 @ ( cons_nat @ X4 @ Xs3 ) ) )
     => ~ ! [Uu: nat > $o] :
            ( X
           != ( produc8587622027977423880st_nat @ Uu @ nil_nat ) ) ) ).

% span.cases
thf(fact_1097_distinct__eq_Ocases,axiom,
    ! [X: produc254973753779126261st_nat] :
      ( ! [Uu: nat > nat > $o] :
          ( X
         != ( produc4727192421694094319st_nat @ Uu @ nil_nat ) )
     => ~ ! [Eq: nat > nat > $o,X4: nat,Xs3: list_nat] :
            ( X
           != ( produc4727192421694094319st_nat @ Eq @ ( cons_nat @ X4 @ Xs3 ) ) ) ) ).

% distinct_eq.cases
thf(fact_1098_closure__impl_Ocases,axiom,
    ! [X: produc4261437277557208175st_nat] :
      ( ! [F3: nat > nat > nat] :
          ( X
         != ( produc5006845494823863079st_nat @ F3 @ nil_nat ) )
     => ~ ! [F3: nat > nat > nat,X4: nat,S6: list_nat] :
            ( X
           != ( produc5006845494823863079st_nat @ F3 @ ( cons_nat @ X4 @ S6 ) ) ) ) ).

% closure_impl.cases
thf(fact_1099_permut__aux_Osimps_I1_J,axiom,
    ! [Uu2: nat > nat,Uv: list_nat] :
      ( ( missin1888654203714970382ux_nat @ nil_nat @ Uu2 @ Uv )
      = nil_nat ) ).

% permut_aux.simps(1)
thf(fact_1100_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_nat @ nil_nat )
       => ( ! [X4: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat] :
              ( ( ( size_size_list_nat @ Xs3 )
                = ( size_size_list_nat @ Ys4 ) )
             => ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_1101_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_term_f_v,P: list_nat > list_term_f_v > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s7148729505879680775rm_f_v @ Ys ) )
     => ( ( P @ nil_nat @ nil_term_f_v )
       => ( ! [X4: nat,Xs3: list_nat,Y3: term_f_v,Ys4: list_term_f_v] :
              ( ( ( size_size_list_nat @ Xs3 )
                = ( size_s7148729505879680775rm_f_v @ Ys4 ) )
             => ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_term_f_v @ Y3 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_1102_list__induct2,axiom,
    ! [Xs: list_term_f_v,Ys: list_nat,P: list_term_f_v > list_nat > $o] :
      ( ( ( size_s7148729505879680775rm_f_v @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_term_f_v @ nil_nat )
       => ( ! [X4: term_f_v,Xs3: list_term_f_v,Y3: nat,Ys4: list_nat] :
              ( ( ( size_s7148729505879680775rm_f_v @ Xs3 )
                = ( size_size_list_nat @ Ys4 ) )
             => ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_term_f_v @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_1103_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_P1780566427699531651term_f,P: list_nat > list_P1780566427699531651term_f > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s9037951226665808879term_f @ Ys ) )
     => ( ( P @ nil_nat @ nil_Pr3198911942259489283term_f )
       => ( ! [X4: nat,Xs3: list_nat,Y3: produc250624644672406269term_f,Ys4: list_P1780566427699531651term_f] :
              ( ( ( size_size_list_nat @ Xs3 )
                = ( size_s9037951226665808879term_f @ Ys4 ) )
             => ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_P3711653291972799923term_f @ Y3 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_1104_list__induct2,axiom,
    ! [Xs: list_term_f_v,Ys: list_term_f_v,P: list_term_f_v > list_term_f_v > $o] :
      ( ( ( size_s7148729505879680775rm_f_v @ Xs )
        = ( size_s7148729505879680775rm_f_v @ Ys ) )
     => ( ( P @ nil_term_f_v @ nil_term_f_v )
       => ( ! [X4: term_f_v,Xs3: list_term_f_v,Y3: term_f_v,Ys4: list_term_f_v] :
              ( ( ( size_s7148729505879680775rm_f_v @ Xs3 )
                = ( size_s7148729505879680775rm_f_v @ Ys4 ) )
             => ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_term_f_v @ X4 @ Xs3 ) @ ( cons_term_f_v @ Y3 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_1105_list__induct2,axiom,
    ! [Xs: list_P1780566427699531651term_f,Ys: list_nat,P: list_P1780566427699531651term_f > list_nat > $o] :
      ( ( ( size_s9037951226665808879term_f @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_Pr3198911942259489283term_f @ nil_nat )
       => ( ! [X4: produc250624644672406269term_f,Xs3: list_P1780566427699531651term_f,Y3: nat,Ys4: list_nat] :
              ( ( ( size_s9037951226665808879term_f @ Xs3 )
                = ( size_size_list_nat @ Ys4 ) )
             => ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_P3711653291972799923term_f @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_1106_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_P4093298276913796397rm_f_v,P: list_nat > list_P4093298276913796397rm_f_v > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s3818039357710166425rm_f_v @ Ys ) )
     => ( ( P @ nil_nat @ nil_Pr5389109106923839661rm_f_v )
       => ( ! [X4: nat,Xs3: list_nat,Y3: produc8027700073345061287rm_f_v,Ys4: list_P4093298276913796397rm_f_v] :
              ( ( ( size_size_list_nat @ Xs3 )
                = ( size_s3818039357710166425rm_f_v @ Ys4 ) )
             => ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_P1983239458618365021rm_f_v @ Y3 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_1107_list__induct2,axiom,
    ! [Xs: list_term_f_v,Ys: list_P1780566427699531651term_f,P: list_term_f_v > list_P1780566427699531651term_f > $o] :
      ( ( ( size_s7148729505879680775rm_f_v @ Xs )
        = ( size_s9037951226665808879term_f @ Ys ) )
     => ( ( P @ nil_term_f_v @ nil_Pr3198911942259489283term_f )
       => ( ! [X4: term_f_v,Xs3: list_term_f_v,Y3: produc250624644672406269term_f,Ys4: list_P1780566427699531651term_f] :
              ( ( ( size_s7148729505879680775rm_f_v @ Xs3 )
                = ( size_s9037951226665808879term_f @ Ys4 ) )
             => ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_term_f_v @ X4 @ Xs3 ) @ ( cons_P3711653291972799923term_f @ Y3 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_1108_list__induct2,axiom,
    ! [Xs: list_P1780566427699531651term_f,Ys: list_term_f_v,P: list_P1780566427699531651term_f > list_term_f_v > $o] :
      ( ( ( size_s9037951226665808879term_f @ Xs )
        = ( size_s7148729505879680775rm_f_v @ Ys ) )
     => ( ( P @ nil_Pr3198911942259489283term_f @ nil_term_f_v )
       => ( ! [X4: produc250624644672406269term_f,Xs3: list_P1780566427699531651term_f,Y3: term_f_v,Ys4: list_term_f_v] :
              ( ( ( size_s9037951226665808879term_f @ Xs3 )
                = ( size_s7148729505879680775rm_f_v @ Ys4 ) )
             => ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_P3711653291972799923term_f @ X4 @ Xs3 ) @ ( cons_term_f_v @ Y3 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_1109_list__induct2,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,Ys: list_nat,P: list_P4093298276913796397rm_f_v > list_nat > $o] :
      ( ( ( size_s3818039357710166425rm_f_v @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_Pr5389109106923839661rm_f_v @ nil_nat )
       => ( ! [X4: produc8027700073345061287rm_f_v,Xs3: list_P4093298276913796397rm_f_v,Y3: nat,Ys4: list_nat] :
              ( ( ( size_s3818039357710166425rm_f_v @ Xs3 )
                = ( size_size_list_nat @ Ys4 ) )
             => ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_P1983239458618365021rm_f_v @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_1110_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs2: list_nat,P: list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs2 ) )
       => ( ( P @ nil_nat @ nil_nat @ nil_nat )
         => ( ! [X4: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat,Z3: nat,Zs: list_nat] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ( ( size_size_list_nat @ Ys4 )
                    = ( size_size_list_nat @ Zs ) )
                 => ( ( P @ Xs3 @ Ys4 @ Zs )
                   => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs2 ) ) ) ) ) ).

% list_induct3
thf(fact_1111_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs2: list_term_f_v,P: list_nat > list_nat > list_term_f_v > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s7148729505879680775rm_f_v @ Zs2 ) )
       => ( ( P @ nil_nat @ nil_nat @ nil_term_f_v )
         => ( ! [X4: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat,Z3: term_f_v,Zs: list_term_f_v] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ( ( size_size_list_nat @ Ys4 )
                    = ( size_s7148729505879680775rm_f_v @ Zs ) )
                 => ( ( P @ Xs3 @ Ys4 @ Zs )
                   => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_term_f_v @ Z3 @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs2 ) ) ) ) ) ).

% list_induct3
thf(fact_1112_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_term_f_v,Zs2: list_nat,P: list_nat > list_term_f_v > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s7148729505879680775rm_f_v @ Ys ) )
     => ( ( ( size_s7148729505879680775rm_f_v @ Ys )
          = ( size_size_list_nat @ Zs2 ) )
       => ( ( P @ nil_nat @ nil_term_f_v @ nil_nat )
         => ( ! [X4: nat,Xs3: list_nat,Y3: term_f_v,Ys4: list_term_f_v,Z3: nat,Zs: list_nat] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_s7148729505879680775rm_f_v @ Ys4 ) )
               => ( ( ( size_s7148729505879680775rm_f_v @ Ys4 )
                    = ( size_size_list_nat @ Zs ) )
                 => ( ( P @ Xs3 @ Ys4 @ Zs )
                   => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_term_f_v @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs2 ) ) ) ) ) ).

% list_induct3
thf(fact_1113_list__induct3,axiom,
    ! [Xs: list_term_f_v,Ys: list_nat,Zs2: list_nat,P: list_term_f_v > list_nat > list_nat > $o] :
      ( ( ( size_s7148729505879680775rm_f_v @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs2 ) )
       => ( ( P @ nil_term_f_v @ nil_nat @ nil_nat )
         => ( ! [X4: term_f_v,Xs3: list_term_f_v,Y3: nat,Ys4: list_nat,Z3: nat,Zs: list_nat] :
                ( ( ( size_s7148729505879680775rm_f_v @ Xs3 )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ( ( size_size_list_nat @ Ys4 )
                    = ( size_size_list_nat @ Zs ) )
                 => ( ( P @ Xs3 @ Ys4 @ Zs )
                   => ( P @ ( cons_term_f_v @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs2 ) ) ) ) ) ).

% list_induct3
thf(fact_1114_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs2: list_P1780566427699531651term_f,P: list_nat > list_nat > list_P1780566427699531651term_f > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s9037951226665808879term_f @ Zs2 ) )
       => ( ( P @ nil_nat @ nil_nat @ nil_Pr3198911942259489283term_f )
         => ( ! [X4: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat,Z3: produc250624644672406269term_f,Zs: list_P1780566427699531651term_f] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ( ( size_size_list_nat @ Ys4 )
                    = ( size_s9037951226665808879term_f @ Zs ) )
                 => ( ( P @ Xs3 @ Ys4 @ Zs )
                   => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_P3711653291972799923term_f @ Z3 @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs2 ) ) ) ) ) ).

% list_induct3
thf(fact_1115_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_term_f_v,Zs2: list_term_f_v,P: list_nat > list_term_f_v > list_term_f_v > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s7148729505879680775rm_f_v @ Ys ) )
     => ( ( ( size_s7148729505879680775rm_f_v @ Ys )
          = ( size_s7148729505879680775rm_f_v @ Zs2 ) )
       => ( ( P @ nil_nat @ nil_term_f_v @ nil_term_f_v )
         => ( ! [X4: nat,Xs3: list_nat,Y3: term_f_v,Ys4: list_term_f_v,Z3: term_f_v,Zs: list_term_f_v] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_s7148729505879680775rm_f_v @ Ys4 ) )
               => ( ( ( size_s7148729505879680775rm_f_v @ Ys4 )
                    = ( size_s7148729505879680775rm_f_v @ Zs ) )
                 => ( ( P @ Xs3 @ Ys4 @ Zs )
                   => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_term_f_v @ Y3 @ Ys4 ) @ ( cons_term_f_v @ Z3 @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs2 ) ) ) ) ) ).

% list_induct3
thf(fact_1116_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_P1780566427699531651term_f,Zs2: list_nat,P: list_nat > list_P1780566427699531651term_f > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s9037951226665808879term_f @ Ys ) )
     => ( ( ( size_s9037951226665808879term_f @ Ys )
          = ( size_size_list_nat @ Zs2 ) )
       => ( ( P @ nil_nat @ nil_Pr3198911942259489283term_f @ nil_nat )
         => ( ! [X4: nat,Xs3: list_nat,Y3: produc250624644672406269term_f,Ys4: list_P1780566427699531651term_f,Z3: nat,Zs: list_nat] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_s9037951226665808879term_f @ Ys4 ) )
               => ( ( ( size_s9037951226665808879term_f @ Ys4 )
                    = ( size_size_list_nat @ Zs ) )
                 => ( ( P @ Xs3 @ Ys4 @ Zs )
                   => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_P3711653291972799923term_f @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs2 ) ) ) ) ) ).

% list_induct3
thf(fact_1117_list__induct3,axiom,
    ! [Xs: list_term_f_v,Ys: list_nat,Zs2: list_term_f_v,P: list_term_f_v > list_nat > list_term_f_v > $o] :
      ( ( ( size_s7148729505879680775rm_f_v @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s7148729505879680775rm_f_v @ Zs2 ) )
       => ( ( P @ nil_term_f_v @ nil_nat @ nil_term_f_v )
         => ( ! [X4: term_f_v,Xs3: list_term_f_v,Y3: nat,Ys4: list_nat,Z3: term_f_v,Zs: list_term_f_v] :
                ( ( ( size_s7148729505879680775rm_f_v @ Xs3 )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ( ( size_size_list_nat @ Ys4 )
                    = ( size_s7148729505879680775rm_f_v @ Zs ) )
                 => ( ( P @ Xs3 @ Ys4 @ Zs )
                   => ( P @ ( cons_term_f_v @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_term_f_v @ Z3 @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs2 ) ) ) ) ) ).

% list_induct3
thf(fact_1118_list__induct3,axiom,
    ! [Xs: list_term_f_v,Ys: list_term_f_v,Zs2: list_nat,P: list_term_f_v > list_term_f_v > list_nat > $o] :
      ( ( ( size_s7148729505879680775rm_f_v @ Xs )
        = ( size_s7148729505879680775rm_f_v @ Ys ) )
     => ( ( ( size_s7148729505879680775rm_f_v @ Ys )
          = ( size_size_list_nat @ Zs2 ) )
       => ( ( P @ nil_term_f_v @ nil_term_f_v @ nil_nat )
         => ( ! [X4: term_f_v,Xs3: list_term_f_v,Y3: term_f_v,Ys4: list_term_f_v,Z3: nat,Zs: list_nat] :
                ( ( ( size_s7148729505879680775rm_f_v @ Xs3 )
                  = ( size_s7148729505879680775rm_f_v @ Ys4 ) )
               => ( ( ( size_s7148729505879680775rm_f_v @ Ys4 )
                    = ( size_size_list_nat @ Zs ) )
                 => ( ( P @ Xs3 @ Ys4 @ Zs )
                   => ( P @ ( cons_term_f_v @ X4 @ Xs3 ) @ ( cons_term_f_v @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs2 ) ) ) ) ) ).

% list_induct3
thf(fact_1119_list__induct3,axiom,
    ! [Xs: list_P1780566427699531651term_f,Ys: list_nat,Zs2: list_nat,P: list_P1780566427699531651term_f > list_nat > list_nat > $o] :
      ( ( ( size_s9037951226665808879term_f @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs2 ) )
       => ( ( P @ nil_Pr3198911942259489283term_f @ nil_nat @ nil_nat )
         => ( ! [X4: produc250624644672406269term_f,Xs3: list_P1780566427699531651term_f,Y3: nat,Ys4: list_nat,Z3: nat,Zs: list_nat] :
                ( ( ( size_s9037951226665808879term_f @ Xs3 )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ( ( size_size_list_nat @ Ys4 )
                    = ( size_size_list_nat @ Zs ) )
                 => ( ( P @ Xs3 @ Ys4 @ Zs )
                   => ( P @ ( cons_P3711653291972799923term_f @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs2 ) ) ) ) ) ).

% list_induct3
thf(fact_1120_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs2: list_nat,Ws: list_nat,P: list_nat > list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs2 ) )
       => ( ( ( size_size_list_nat @ Zs2 )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
           => ( ! [X4: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat,Z3: nat,Zs: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_size_list_nat @ Zs ) )
                   => ( ( ( size_size_list_nat @ Zs )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys4 @ Zs @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1121_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs2: list_nat,Ws: list_term_f_v,P: list_nat > list_nat > list_nat > list_term_f_v > $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_s7148729505879680775rm_f_v @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_term_f_v )
           => ( ! [X4: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat,Z3: nat,Zs: list_nat,W: term_f_v,Ws2: list_term_f_v] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_size_list_nat @ Zs ) )
                   => ( ( ( size_size_list_nat @ Zs )
                        = ( size_s7148729505879680775rm_f_v @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys4 @ Zs @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs ) @ ( cons_term_f_v @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1122_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs2: list_term_f_v,Ws: list_nat,P: list_nat > list_nat > list_term_f_v > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s7148729505879680775rm_f_v @ Zs2 ) )
       => ( ( ( size_s7148729505879680775rm_f_v @ Zs2 )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_term_f_v @ nil_nat )
           => ( ! [X4: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat,Z3: term_f_v,Zs: list_term_f_v,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_s7148729505879680775rm_f_v @ Zs ) )
                   => ( ( ( size_s7148729505879680775rm_f_v @ Zs )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys4 @ Zs @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_term_f_v @ Z3 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1123_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_term_f_v,Zs2: list_nat,Ws: list_nat,P: list_nat > list_term_f_v > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s7148729505879680775rm_f_v @ Ys ) )
     => ( ( ( size_s7148729505879680775rm_f_v @ Ys )
          = ( size_size_list_nat @ Zs2 ) )
       => ( ( ( size_size_list_nat @ Zs2 )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_term_f_v @ nil_nat @ nil_nat )
           => ( ! [X4: nat,Xs3: list_nat,Y3: term_f_v,Ys4: list_term_f_v,Z3: nat,Zs: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_s7148729505879680775rm_f_v @ Ys4 ) )
                 => ( ( ( size_s7148729505879680775rm_f_v @ Ys4 )
                      = ( size_size_list_nat @ Zs ) )
                   => ( ( ( size_size_list_nat @ Zs )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys4 @ Zs @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_term_f_v @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1124_list__induct4,axiom,
    ! [Xs: list_term_f_v,Ys: list_nat,Zs2: list_nat,Ws: list_nat,P: list_term_f_v > list_nat > list_nat > list_nat > $o] :
      ( ( ( size_s7148729505879680775rm_f_v @ 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 ) )
         => ( ( P @ nil_term_f_v @ nil_nat @ nil_nat @ nil_nat )
           => ( ! [X4: term_f_v,Xs3: list_term_f_v,Y3: nat,Ys4: list_nat,Z3: nat,Zs: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_s7148729505879680775rm_f_v @ Xs3 )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_size_list_nat @ Zs ) )
                   => ( ( ( size_size_list_nat @ Zs )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys4 @ Zs @ Ws2 )
                       => ( P @ ( cons_term_f_v @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1125_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs2: list_nat,Ws: list_P1780566427699531651term_f,P: list_nat > list_nat > list_nat > list_P1780566427699531651term_f > $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_s9037951226665808879term_f @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_Pr3198911942259489283term_f )
           => ( ! [X4: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat,Z3: nat,Zs: list_nat,W: produc250624644672406269term_f,Ws2: list_P1780566427699531651term_f] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_size_list_nat @ Zs ) )
                   => ( ( ( size_size_list_nat @ Zs )
                        = ( size_s9037951226665808879term_f @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys4 @ Zs @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs ) @ ( cons_P3711653291972799923term_f @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1126_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs2: list_term_f_v,Ws: list_term_f_v,P: list_nat > list_nat > list_term_f_v > list_term_f_v > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s7148729505879680775rm_f_v @ Zs2 ) )
       => ( ( ( size_s7148729505879680775rm_f_v @ Zs2 )
            = ( size_s7148729505879680775rm_f_v @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_term_f_v @ nil_term_f_v )
           => ( ! [X4: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat,Z3: term_f_v,Zs: list_term_f_v,W: term_f_v,Ws2: list_term_f_v] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_s7148729505879680775rm_f_v @ Zs ) )
                   => ( ( ( size_s7148729505879680775rm_f_v @ Zs )
                        = ( size_s7148729505879680775rm_f_v @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys4 @ Zs @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_term_f_v @ Z3 @ Zs ) @ ( cons_term_f_v @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1127_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs2: list_P1780566427699531651term_f,Ws: list_nat,P: list_nat > list_nat > list_P1780566427699531651term_f > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s9037951226665808879term_f @ Zs2 ) )
       => ( ( ( size_s9037951226665808879term_f @ Zs2 )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_Pr3198911942259489283term_f @ nil_nat )
           => ( ! [X4: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat,Z3: produc250624644672406269term_f,Zs: list_P1780566427699531651term_f,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_s9037951226665808879term_f @ Zs ) )
                   => ( ( ( size_s9037951226665808879term_f @ Zs )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys4 @ Zs @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_P3711653291972799923term_f @ Z3 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1128_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_term_f_v,Zs2: list_nat,Ws: list_term_f_v,P: list_nat > list_term_f_v > list_nat > list_term_f_v > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s7148729505879680775rm_f_v @ Ys ) )
     => ( ( ( size_s7148729505879680775rm_f_v @ Ys )
          = ( size_size_list_nat @ Zs2 ) )
       => ( ( ( size_size_list_nat @ Zs2 )
            = ( size_s7148729505879680775rm_f_v @ Ws ) )
         => ( ( P @ nil_nat @ nil_term_f_v @ nil_nat @ nil_term_f_v )
           => ( ! [X4: nat,Xs3: list_nat,Y3: term_f_v,Ys4: list_term_f_v,Z3: nat,Zs: list_nat,W: term_f_v,Ws2: list_term_f_v] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_s7148729505879680775rm_f_v @ Ys4 ) )
                 => ( ( ( size_s7148729505879680775rm_f_v @ Ys4 )
                      = ( size_size_list_nat @ Zs ) )
                   => ( ( ( size_size_list_nat @ Zs )
                        = ( size_s7148729505879680775rm_f_v @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys4 @ Zs @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_term_f_v @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs ) @ ( cons_term_f_v @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1129_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_term_f_v,Zs2: list_term_f_v,Ws: list_nat,P: list_nat > list_term_f_v > list_term_f_v > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s7148729505879680775rm_f_v @ Ys ) )
     => ( ( ( size_s7148729505879680775rm_f_v @ Ys )
          = ( size_s7148729505879680775rm_f_v @ Zs2 ) )
       => ( ( ( size_s7148729505879680775rm_f_v @ Zs2 )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_term_f_v @ nil_term_f_v @ nil_nat )
           => ( ! [X4: nat,Xs3: list_nat,Y3: term_f_v,Ys4: list_term_f_v,Z3: term_f_v,Zs: list_term_f_v,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_s7148729505879680775rm_f_v @ Ys4 ) )
                 => ( ( ( size_s7148729505879680775rm_f_v @ Ys4 )
                      = ( size_s7148729505879680775rm_f_v @ Zs ) )
                   => ( ( ( size_s7148729505879680775rm_f_v @ Zs )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys4 @ Zs @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_term_f_v @ Y3 @ Ys4 ) @ ( cons_term_f_v @ Z3 @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1130_list_Osize_I3_J,axiom,
    ( ( size_size_list_nat @ nil_nat )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_1131_list_Osize_I3_J,axiom,
    ( ( size_s7148729505879680775rm_f_v @ nil_term_f_v )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_1132_list_Osize_I3_J,axiom,
    ( ( size_s9037951226665808879term_f @ nil_Pr3198911942259489283term_f )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_1133_list_Osize_I3_J,axiom,
    ( ( size_s3818039357710166425rm_f_v @ nil_Pr5389109106923839661rm_f_v )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_1134_mem__idx_Ocases,axiom,
    ! [X: produc4575160907756185873st_nat] :
      ( ! [Uu: nat] :
          ( X
         != ( produc8282810413953273033st_nat @ Uu @ nil_nat ) )
     => ~ ! [X4: nat,A3: nat,As2: list_nat] :
            ( X
           != ( produc8282810413953273033st_nat @ X4 @ ( cons_nat @ A3 @ As2 ) ) ) ) ).

% mem_idx.cases
thf(fact_1135_subtract__list__sorted_Ocases,axiom,
    ! [X: produc1828647624359046049st_nat] :
      ( ! [X4: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat] :
          ( X
         != ( produc2694037385005941721st_nat @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
     => ( ! [Ys4: list_nat] :
            ( X
           != ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
       => ~ ! [V: nat,Va: list_nat] :
              ( X
             != ( produc2694037385005941721st_nat @ ( cons_nat @ V @ Va ) @ nil_nat ) ) ) ) ).

% subtract_list_sorted.cases
thf(fact_1136_union__list__sorted_Ocases,axiom,
    ! [X: produc1828647624359046049st_nat] :
      ( ! [X4: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat] :
          ( X
         != ( produc2694037385005941721st_nat @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
     => ( ! [Ys4: list_nat] :
            ( X
           != ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
       => ~ ! [V: nat,Va: list_nat] :
              ( X
             != ( produc2694037385005941721st_nat @ ( cons_nat @ V @ Va ) @ nil_nat ) ) ) ) ).

% union_list_sorted.cases
thf(fact_1137_remove__prefix_Ocases,axiom,
    ! [X: produc1828647624359046049st_nat] :
      ( ! [X4: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat] :
          ( X
         != ( produc2694037385005941721st_nat @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
     => ( ! [Ys4: list_nat] :
            ( X
           != ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
       => ~ ! [V: nat,Va: list_nat] :
              ( X
             != ( produc2694037385005941721st_nat @ ( cons_nat @ V @ Va ) @ nil_nat ) ) ) ) ).

% remove_prefix.cases
thf(fact_1138_list__inter_Ocases,axiom,
    ! [X: produc1828647624359046049st_nat] :
      ( ! [Bs2: list_nat] :
          ( X
         != ( produc2694037385005941721st_nat @ nil_nat @ Bs2 ) )
     => ~ ! [A3: nat,As2: list_nat,Bs2: list_nat] :
            ( X
           != ( produc2694037385005941721st_nat @ ( cons_nat @ A3 @ As2 ) @ Bs2 ) ) ) ).

% list_inter.cases
thf(fact_1139_filter2_Ocases,axiom,
    ! [X: produc4787317212837456354st_nat] :
      ( ! [P7: nat > nat > $o,Uu: list_nat] :
          ( X
         != ( produc3127733452865184594st_nat @ P7 @ ( produc2694037385005941721st_nat @ nil_nat @ Uu ) ) )
     => ( ! [P7: nat > nat > $o,V: nat,Va: list_nat] :
            ( X
           != ( produc3127733452865184594st_nat @ P7 @ ( produc2694037385005941721st_nat @ ( cons_nat @ V @ Va ) @ nil_nat ) ) )
       => ~ ! [P7: nat > nat > $o,A3: nat,As2: list_nat,B3: nat,Bs2: list_nat] :
              ( X
             != ( produc3127733452865184594st_nat @ P7 @ ( produc2694037385005941721st_nat @ ( cons_nat @ A3 @ As2 ) @ ( cons_nat @ B3 @ Bs2 ) ) ) ) ) ) ).

% filter2.cases
thf(fact_1140_shuffles_Ocases,axiom,
    ! [X: produc1828647624359046049st_nat] :
      ( ! [Ys4: list_nat] :
          ( X
         != ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
     => ( ! [Xs3: list_nat] :
            ( X
           != ( produc2694037385005941721st_nat @ Xs3 @ nil_nat ) )
       => ~ ! [X4: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat] :
              ( X
             != ( produc2694037385005941721st_nat @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) ) ) ).

% shuffles.cases
thf(fact_1141_distinct__singleton,axiom,
    ! [X: nat] : ( distinct_nat @ ( cons_nat @ X @ nil_nat ) ) ).

% distinct_singleton
thf(fact_1142_take__0,axiom,
    ! [Xs: list_nat] :
      ( ( take_nat @ zero_zero_nat @ Xs )
      = nil_nat ) ).

% take_0
thf(fact_1143_list_Osize__gen_I1_J,axiom,
    ! [X: nat > nat] :
      ( ( size_list_nat @ X @ nil_nat )
      = zero_zero_nat ) ).

% list.size_gen(1)
thf(fact_1144_not__Nil__listrel1,axiom,
    ! [Xs: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel1_nat @ R3 ) ) ).

% not_Nil_listrel1
thf(fact_1145_not__listrel1__Nil,axiom,
    ! [Xs: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel1_nat @ R3 ) ) ).

% not_listrel1_Nil
thf(fact_1146_listrel_ONil,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) @ ( listrel_nat_nat @ R3 ) ) ).

% listrel.Nil
thf(fact_1147_listrel__Nil1,axiom,
    ! [Xs: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel_nat_nat @ R3 ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil1
thf(fact_1148_listrel__Nil2,axiom,
    ! [Xs: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel_nat_nat @ R3 ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil2
thf(fact_1149_Missing__List_Omin__list_Osimps_I1_J,axiom,
    ! [X: nat] :
      ( ( missing_min_list_nat @ ( cons_nat @ X @ nil_nat ) )
      = X ) ).

% Missing_List.min_list.simps(1)
thf(fact_1150_min__list__ex,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ( member_nat @ ( missing_min_list_nat @ Xs ) @ ( set_nat2 @ Xs ) ) ) ).

% min_list_ex
thf(fact_1151_Nil2__notin__lex,axiom,
    ! [Xs: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( lex_nat @ R3 ) ) ).

% Nil2_notin_lex
thf(fact_1152_Nil__notin__lex,axiom,
    ! [Ys: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys ) @ ( lex_nat @ R3 ) ) ).

% Nil_notin_lex
thf(fact_1153_Nil__lenlex__iff2,axiom,
    ! [Ns: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ns @ nil_nat ) @ ( lenlex_nat @ R3 ) ) ).

% Nil_lenlex_iff2
thf(fact_1154_Nil__less__Cons,axiom,
    ! [A: nat,X: list_nat] : ( ord_less_list_nat @ nil_nat @ ( cons_nat @ A @ X ) ) ).

% Nil_less_Cons
thf(fact_1155_less__list__code_I2_J,axiom,
    ! [X: nat,Xs: list_nat] : ( ord_less_list_nat @ nil_nat @ ( cons_nat @ X @ Xs ) ) ).

% less_list_code(2)
thf(fact_1156_less__eq__list__code_I1_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ~ ( ord_less_eq_list_nat @ ( cons_nat @ X @ Xs ) @ nil_nat ) ).

% less_eq_list_code(1)
thf(fact_1157_replace__term__at_Ocases,axiom,
    ! [X: produc6479857609258389878rm_f_v] :
      ( ! [S4: term_f_v,T3: term_f_v] :
          ( X
         != ( produc6079149715188314536rm_f_v @ S4 @ ( produc437266199009834076rm_f_v @ nil_nat @ T3 ) ) )
     => ( ! [X4: v,V: nat,Va: list_nat,T3: term_f_v] :
            ( X
           != ( produc6079149715188314536rm_f_v @ ( var_v_f @ X4 ) @ ( produc437266199009834076rm_f_v @ ( cons_nat @ V @ Va ) @ T3 ) ) )
       => ~ ! [F3: f,Ts2: list_term_f_v,I: nat,Ps2: list_nat,T3: term_f_v] :
              ( X
             != ( produc6079149715188314536rm_f_v @ ( fun_f_v @ F3 @ Ts2 ) @ ( produc437266199009834076rm_f_v @ ( cons_nat @ I @ Ps2 ) @ T3 ) ) ) ) ) ).

% replace_term_at.cases
thf(fact_1158_ctxt__at__pos_Ocases,axiom,
    ! [X: produc3902171991451677348st_nat] :
      ( ! [S4: term_f_v] :
          ( X
         != ( produc4767561752098573020st_nat @ S4 @ nil_nat ) )
     => ( ! [F3: f,Ss3: list_term_f_v,I: nat,P6: list_nat] :
            ( X
           != ( produc4767561752098573020st_nat @ ( fun_f_v @ F3 @ Ss3 ) @ ( cons_nat @ I @ P6 ) ) )
       => ~ ! [X4: v,V: nat,Va: list_nat] :
              ( X
             != ( produc4767561752098573020st_nat @ ( var_v_f @ X4 ) @ ( cons_nat @ V @ Va ) ) ) ) ) ).

% ctxt_at_pos.cases
thf(fact_1159_subt__at_Ocases,axiom,
    ! [X: produc3902171991451677348st_nat] :
      ( ! [S4: term_f_v] :
          ( X
         != ( produc4767561752098573020st_nat @ S4 @ nil_nat ) )
     => ( ! [F3: f,Ss3: list_term_f_v,I: nat,P6: list_nat] :
            ( X
           != ( produc4767561752098573020st_nat @ ( fun_f_v @ F3 @ Ss3 ) @ ( cons_nat @ I @ P6 ) ) )
       => ~ ! [X4: v,V: nat,Va: list_nat] :
              ( X
             != ( produc4767561752098573020st_nat @ ( var_v_f @ X4 ) @ ( cons_nat @ V @ Va ) ) ) ) ) ).

% subt_at.cases
thf(fact_1160_fun__at_Ocases,axiom,
    ! [X: produc3902171991451677348st_nat] :
      ( ! [X4: v] :
          ( X
         != ( produc4767561752098573020st_nat @ ( var_v_f @ X4 ) @ nil_nat ) )
     => ( ! [F3: f,Ts2: list_term_f_v] :
            ( X
           != ( produc4767561752098573020st_nat @ ( fun_f_v @ F3 @ Ts2 ) @ nil_nat ) )
       => ( ! [F3: f,Ts2: list_term_f_v,I: nat,P6: list_nat] :
              ( X
             != ( produc4767561752098573020st_nat @ ( fun_f_v @ F3 @ Ts2 ) @ ( cons_nat @ I @ P6 ) ) )
         => ~ ! [Vb: v,V: nat,Va: list_nat] :
                ( X
               != ( produc4767561752098573020st_nat @ ( var_v_f @ Vb ) @ ( cons_nat @ V @ Va ) ) ) ) ) ) ).

% fun_at.cases
thf(fact_1161_listrel_Ocases,axiom,
    ! [A1: list_term_f_v,A22: list_term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ A1 @ A22 ) @ ( listre5907981802042383033rm_f_v @ R3 ) )
     => ( ( ( A1 = nil_term_f_v )
         => ( A22 != nil_term_f_v ) )
       => ~ ! [X4: term_f_v,Y3: term_f_v,Xs3: list_term_f_v] :
              ( ( A1
                = ( cons_term_f_v @ X4 @ Xs3 ) )
             => ! [Ys4: list_term_f_v] :
                  ( ( A22
                    = ( cons_term_f_v @ Y3 @ Ys4 ) )
                 => ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y3 ) @ R3 )
                   => ~ ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Xs3 @ Ys4 ) @ ( listre5907981802042383033rm_f_v @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_1162_listrel_Ocases,axiom,
    ! [A1: list_Bot_bot_term_f,A22: list_Bot_bot_term_f,R3: set_Pr6841673932529763677term_f] :
      ( ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ A1 @ A22 ) @ ( listre686400052008961423term_f @ R3 ) )
     => ( ( ( A1 = nil_Bot_bot_term_f )
         => ( A22 != nil_Bot_bot_term_f ) )
       => ~ ! [X4: bot_bot_term_f,Y3: bot_bot_term_f,Xs3: list_Bot_bot_term_f] :
              ( ( A1
                = ( cons_Bot_bot_term_f @ X4 @ Xs3 ) )
             => ! [Ys4: list_Bot_bot_term_f] :
                  ( ( A22
                    = ( cons_Bot_bot_term_f @ Y3 @ Ys4 ) )
                 => ( ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X4 @ Y3 ) @ R3 )
                   => ~ ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Xs3 @ Ys4 ) @ ( listre686400052008961423term_f @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_1163_listrel_Ocases,axiom,
    ! [A1: list_nat_nat,A22: list_nat,R3: set_Pr9093778441882193744at_nat] :
      ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ A1 @ A22 ) @ ( listrel_nat_nat_nat @ R3 ) )
     => ( ( ( A1 = nil_nat_nat )
         => ( A22 != nil_nat ) )
       => ~ ! [X4: nat > nat,Y3: nat,Xs3: list_nat_nat] :
              ( ( A1
                = ( cons_nat_nat @ X4 @ Xs3 ) )
             => ! [Ys4: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y3 @ Ys4 ) )
                 => ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X4 @ Y3 ) @ R3 )
                   => ~ ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs3 @ Ys4 ) @ ( listrel_nat_nat_nat @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_1164_listrel_Ocases,axiom,
    ! [A1: list_nat,A22: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R3 ) )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_nat ) )
       => ~ ! [X4: nat,Y3: nat,Xs3: list_nat] :
              ( ( A1
                = ( cons_nat @ X4 @ Xs3 ) )
             => ! [Ys4: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y3 @ Ys4 ) )
                 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R3 )
                   => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys4 ) @ ( listrel_nat_nat @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_1165_listrel_Osimps,axiom,
    ! [A1: list_term_f_v,A22: list_term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
      ( ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ A1 @ A22 ) @ ( listre5907981802042383033rm_f_v @ R3 ) )
      = ( ( ( A1 = nil_term_f_v )
          & ( A22 = nil_term_f_v ) )
        | ? [X5: term_f_v,Y4: term_f_v,Xs2: list_term_f_v,Ys2: list_term_f_v] :
            ( ( A1
              = ( cons_term_f_v @ X5 @ Xs2 ) )
            & ( A22
              = ( cons_term_f_v @ Y4 @ Ys2 ) )
            & ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X5 @ Y4 ) @ R3 )
            & ( member2355812552444855152rm_f_v @ ( produc1355532988472670079rm_f_v @ Xs2 @ Ys2 ) @ ( listre5907981802042383033rm_f_v @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_1166_listrel_Osimps,axiom,
    ! [A1: list_Bot_bot_term_f,A22: list_Bot_bot_term_f,R3: set_Pr6841673932529763677term_f] :
      ( ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ A1 @ A22 ) @ ( listre686400052008961423term_f @ R3 ) )
      = ( ( ( A1 = nil_Bot_bot_term_f )
          & ( A22 = nil_Bot_bot_term_f ) )
        | ? [X5: bot_bot_term_f,Y4: bot_bot_term_f,Xs2: list_Bot_bot_term_f,Ys2: list_Bot_bot_term_f] :
            ( ( A1
              = ( cons_Bot_bot_term_f @ X5 @ Xs2 ) )
            & ( A22
              = ( cons_Bot_bot_term_f @ Y4 @ Ys2 ) )
            & ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X5 @ Y4 ) @ R3 )
            & ( member878789856639434822term_f @ ( produc1421358604722026965term_f @ Xs2 @ Ys2 ) @ ( listre686400052008961423term_f @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_1167_listrel_Osimps,axiom,
    ! [A1: list_nat_nat,A22: list_nat,R3: set_Pr9093778441882193744at_nat] :
      ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ A1 @ A22 ) @ ( listrel_nat_nat_nat @ R3 ) )
      = ( ( ( A1 = nil_nat_nat )
          & ( A22 = nil_nat ) )
        | ? [X5: nat > nat,Y4: nat,Xs2: list_nat_nat,Ys2: list_nat] :
            ( ( A1
              = ( cons_nat_nat @ X5 @ Xs2 ) )
            & ( A22
              = ( cons_nat @ Y4 @ Ys2 ) )
            & ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X5 @ Y4 ) @ R3 )
            & ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat_nat @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_1168_listrel_Osimps,axiom,
    ! [A1: list_nat,A22: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R3 ) )
      = ( ( ( A1 = nil_nat )
          & ( A22 = nil_nat ) )
        | ? [X5: nat,Y4: nat,Xs2: list_nat,Ys2: list_nat] :
            ( ( A1
              = ( cons_nat @ X5 @ Xs2 ) )
            & ( A22
              = ( cons_nat @ Y4 @ Ys2 ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ R3 )
            & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_1169_replace__term__at_Opelims,axiom,
    ! [X: term_f_v,Xa2: list_nat,Xb: term_f_v,Y: term_f_v] :
      ( ( ( term_r2146176907066831980at_f_v @ X @ Xa2 @ Xb )
        = Y )
     => ( ( accp_P6476584661967079469rm_f_v @ term_r5790345193739526189el_f_v @ ( produc6079149715188314536rm_f_v @ X @ ( produc437266199009834076rm_f_v @ Xa2 @ Xb ) ) )
       => ( ( ( Xa2 = nil_nat )
           => ( ( Y = Xb )
             => ~ ( accp_P6476584661967079469rm_f_v @ term_r5790345193739526189el_f_v @ ( produc6079149715188314536rm_f_v @ X @ ( produc437266199009834076rm_f_v @ nil_nat @ Xb ) ) ) ) )
         => ( ! [X4: v] :
                ( ( X
                  = ( var_v_f @ X4 ) )
               => ! [V: nat,Va: list_nat] :
                    ( ( Xa2
                      = ( cons_nat @ V @ Va ) )
                   => ( ( Y
                        = ( var_v_f @ X4 ) )
                     => ~ ( accp_P6476584661967079469rm_f_v @ term_r5790345193739526189el_f_v @ ( produc6079149715188314536rm_f_v @ ( var_v_f @ X4 ) @ ( produc437266199009834076rm_f_v @ ( cons_nat @ V @ Va ) @ Xb ) ) ) ) ) )
           => ~ ! [F3: f,Ts2: list_term_f_v] :
                  ( ( X
                    = ( fun_f_v @ F3 @ Ts2 ) )
                 => ! [I: nat,Ps2: list_nat] :
                      ( ( Xa2
                        = ( cons_nat @ I @ Ps2 ) )
                     => ( ( ( ( ord_less_nat @ I @ ( size_s7148729505879680775rm_f_v @ Ts2 ) )
                           => ( Y
                              = ( fun_f_v @ F3 @ ( list_update_term_f_v @ Ts2 @ I @ ( term_r2146176907066831980at_f_v @ ( nth_term_f_v @ Ts2 @ I ) @ Ps2 @ Xb ) ) ) ) )
                          & ( ~ ( ord_less_nat @ I @ ( size_s7148729505879680775rm_f_v @ Ts2 ) )
                           => ( Y
                              = ( fun_f_v @ F3 @ Ts2 ) ) ) )
                       => ~ ( accp_P6476584661967079469rm_f_v @ term_r5790345193739526189el_f_v @ ( produc6079149715188314536rm_f_v @ ( fun_f_v @ F3 @ Ts2 ) @ ( produc437266199009834076rm_f_v @ ( cons_nat @ I @ Ps2 ) @ Xb ) ) ) ) ) ) ) ) ) ) ).

% replace_term_at.pelims
thf(fact_1170_take__Suc__conv__app__nth,axiom,
    ! [I4: nat,Xs: list_nat] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
     => ( ( take_nat @ ( suc @ I4 ) @ Xs )
        = ( append_nat @ ( take_nat @ I4 @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I4 ) @ nil_nat ) ) ) ) ).

% take_Suc_conv_app_nth
thf(fact_1171_take__Suc__conv__app__nth,axiom,
    ! [I4: nat,Xs: list_term_f_v] :
      ( ( ord_less_nat @ I4 @ ( size_s7148729505879680775rm_f_v @ Xs ) )
     => ( ( take_term_f_v @ ( suc @ I4 ) @ Xs )
        = ( append_term_f_v @ ( take_term_f_v @ I4 @ Xs ) @ ( cons_term_f_v @ ( nth_term_f_v @ Xs @ I4 ) @ nil_term_f_v ) ) ) ) ).

% take_Suc_conv_app_nth
thf(fact_1172_take__Suc__conv__app__nth,axiom,
    ! [I4: nat,Xs: list_P1780566427699531651term_f] :
      ( ( ord_less_nat @ I4 @ ( size_s9037951226665808879term_f @ Xs ) )
     => ( ( take_P4256413432045430251term_f @ ( suc @ I4 ) @ Xs )
        = ( append2162745965627483416term_f @ ( take_P4256413432045430251term_f @ I4 @ Xs ) @ ( cons_P3711653291972799923term_f @ ( nth_Pr7080171896067517188term_f @ Xs @ I4 ) @ nil_Pr3198911942259489283term_f ) ) ) ) ).

% take_Suc_conv_app_nth
thf(fact_1173_take__Suc__conv__app__nth,axiom,
    ! [I4: nat,Xs: list_P4093298276913796397rm_f_v] :
      ( ( ord_less_nat @ I4 @ ( size_s3818039357710166425rm_f_v @ Xs ) )
     => ( ( take_P6145276122034302101rm_f_v @ ( suc @ I4 ) @ Xs )
        = ( append8822927501793976258rm_f_v @ ( take_P6145276122034302101rm_f_v @ I4 @ Xs ) @ ( cons_P1983239458618365021rm_f_v @ ( nth_Pr8768905099348956590rm_f_v @ Xs @ I4 ) @ nil_Pr5389109106923839661rm_f_v ) ) ) ) ).

% take_Suc_conv_app_nth
thf(fact_1174_append_Oright__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ A @ nil_nat )
      = A ) ).

% append.right_neutral
thf(fact_1175_append__Nil2,axiom,
    ! [Xs: list_nat] :
      ( ( append_nat @ Xs @ nil_nat )
      = Xs ) ).

% append_Nil2
thf(fact_1176_append__self__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_nat ) ) ).

% append_self_conv
thf(fact_1177_self__append__conv,axiom,
    ! [Y: list_nat,Ys: list_nat] :
      ( ( Y
        = ( append_nat @ Y @ Ys ) )
      = ( Ys = nil_nat ) ) ).

% self_append_conv
thf(fact_1178_append__self__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_nat ) ) ).

% append_self_conv2
thf(fact_1179_self__append__conv2,axiom,
    ! [Y: list_nat,Xs: list_nat] :
      ( ( Y
        = ( append_nat @ Xs @ Y ) )
      = ( Xs = nil_nat ) ) ).

% self_append_conv2
thf(fact_1180_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_1181_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_1182_append__eq__append__conv,axiom,
    ! [Xs: list_term_f_v,Ys: list_term_f_v,Us: list_term_f_v,Vs: list_term_f_v] :
      ( ( ( ( size_s7148729505879680775rm_f_v @ Xs )
          = ( size_s7148729505879680775rm_f_v @ Ys ) )
        | ( ( size_s7148729505879680775rm_f_v @ Us )
          = ( size_s7148729505879680775rm_f_v @ Vs ) ) )
     => ( ( ( append_term_f_v @ Xs @ Us )
          = ( append_term_f_v @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_1183_append__eq__append__conv,axiom,
    ! [Xs: list_P1780566427699531651term_f,Ys: list_P1780566427699531651term_f,Us: list_P1780566427699531651term_f,Vs: list_P1780566427699531651term_f] :
      ( ( ( ( size_s9037951226665808879term_f @ Xs )
          = ( size_s9037951226665808879term_f @ Ys ) )
        | ( ( size_s9037951226665808879term_f @ Us )
          = ( size_s9037951226665808879term_f @ Vs ) ) )
     => ( ( ( append2162745965627483416term_f @ Xs @ Us )
          = ( append2162745965627483416term_f @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_1184_append__eq__append__conv,axiom,
    ! [Xs: list_P4093298276913796397rm_f_v,Ys: list_P4093298276913796397rm_f_v,Us: list_P4093298276913796397rm_f_v,Vs: list_P4093298276913796397rm_f_v] :
      ( ( ( ( size_s3818039357710166425rm_f_v @ Xs )
          = ( size_s3818039357710166425rm_f_v @ Ys ) )
        | ( ( size_s3818039357710166425rm_f_v @ Us )
          = ( size_s3818039357710166425rm_f_v @ Vs ) ) )
     => ( ( ( append8822927501793976258rm_f_v @ Xs @ Us )
          = ( append8822927501793976258rm_f_v @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_1185_in__measures_I1_J,axiom,
    ! [X: term_f_v,Y: term_f_v] :
      ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y ) @ ( measures_term_f_v @ nil_term_f_v_nat ) ) ).

% in_measures(1)
thf(fact_1186_in__measures_I1_J,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f] :
      ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ X @ Y ) @ ( measur619265046736277902term_f @ nil_Bo63390954156198135_f_nat ) ) ).

% in_measures(1)
thf(fact_1187_in__measures_I1_J,axiom,
    ! [X: nat,Y: nat] :
      ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ nil_nat_nat ) ) ).

% in_measures(1)
thf(fact_1188_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K2: nat,M5: nat] :
          ( X
         != ( product_Pair_nat_nat @ K2 @ M5 ) ) ).

% prod_decode_aux.cases
thf(fact_1189_add__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc_right
thf(fact_1190_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_1191_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_Suc_Suc
thf(fact_1192_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).

% Suc_diff_diff
thf(fact_1193_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_1194_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
      = ( ord_less_nat @ M @ N ) ) ).

% zero_less_diff
thf(fact_1195_Suc__pred,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% Suc_pred
thf(fact_1196_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I4 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I4 @ K ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_1197_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I4 )
        = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I4 ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_1198_diff__Suc__less,axiom,
    ! [N: nat,I4: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) @ N ) ) ).

% diff_Suc_less
thf(fact_1199_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_1200_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_1201_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).

% diff_less
thf(fact_1202_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_1203_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).

% diff_less_mono
thf(fact_1204_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M @ N ) ) ) ) ).

% less_diff_iff
thf(fact_1205_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_1206_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_1207_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ? [K2: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_1208_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M6: nat,N2: nat] :
        ? [K5: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M6 @ K5 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_1209_less__add__Suc2,axiom,
    ! [I4: nat,M: nat] : ( ord_less_nat @ I4 @ ( suc @ ( plus_plus_nat @ M @ I4 ) ) ) ).

% less_add_Suc2
thf(fact_1210_less__add__Suc1,axiom,
    ! [I4: nat,M: nat] : ( ord_less_nat @ I4 @ ( suc @ ( plus_plus_nat @ I4 @ M ) ) ) ).

% less_add_Suc1
thf(fact_1211_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ~ ! [Q2: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).

% less_natE
thf(fact_1212_less__imp__add__positive,axiom,
    ! [I4: nat,J: nat] :
      ( ( ord_less_nat @ I4 @ J )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I4 @ K2 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_1213_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M5: nat,N3: nat] :
          ( ( ord_less_nat @ M5 @ N3 )
         => ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_1214_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D2: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D2 ) )
                & ~ ( P @ D2 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_1215_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P @ zero_zero_nat ) )
        & ! [D2: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D2 ) )
           => ( P @ D2 ) ) ) ) ).

% nat_diff_split
thf(fact_1216_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_1217_less__diff__conv,axiom,
    ! [I4: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I4 @ ( minus_minus_nat @ J @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_1218_add__lessD1,axiom,
    ! [I4: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I4 @ J ) @ K )
     => ( ord_less_nat @ I4 @ K ) ) ).

% add_lessD1
thf(fact_1219_add__less__mono,axiom,
    ! [I4: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I4 @ J )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_1220_not__add__less1,axiom,
    ! [I4: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I4 @ J ) @ I4 ) ).

% not_add_less1
thf(fact_1221_not__add__less2,axiom,
    ! [J: nat,I4: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I4 ) @ I4 ) ).

% not_add_less2
thf(fact_1222_add__less__mono1,axiom,
    ! [I4: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I4 @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_1223_trans__less__add1,axiom,
    ! [I4: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I4 @ J )
     => ( ord_less_nat @ I4 @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_less_add1
thf(fact_1224_trans__less__add2,axiom,
    ! [I4: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I4 @ J )
     => ( ord_less_nat @ I4 @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_less_add2
thf(fact_1225_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% less_add_eq_less
thf(fact_1226_add__Suc__shift,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).

% add_Suc_shift
thf(fact_1227_add__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc
thf(fact_1228_nat__arith_Osuc1,axiom,
    ! [A4: nat,K: nat,A: nat] :
      ( ( A4
        = ( plus_plus_nat @ K @ A ) )
     => ( ( suc @ A4 )
        = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_1229_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ord_less_nat @ M @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_1230_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).

% less_imp_diff_less
thf(fact_1231_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I4: nat] :
      ( ( P @ K )
     => ( ! [N3: nat] :
            ( ( P @ ( suc @ N3 ) )
           => ( P @ N3 ) )
       => ( P @ ( minus_minus_nat @ K @ I4 ) ) ) ) ).

% zero_induct_lemma
thf(fact_1232_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I4 )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I4 @ K ) ) ) ) ).

% less_diff_conv2

% Helper facts (11)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Term__Oterm_Itf__f_Mtf__a_J_T,axiom,
    ! [X: term_f_a,Y: term_f_a] :
      ( ( if_term_f_a @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Term__Oterm_Itf__f_Mtf__a_J_T,axiom,
    ! [X: term_f_a,Y: term_f_a] :
      ( ( if_term_f_a @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Bot____Terms__Obot____term_Itf__f_J_T,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ( if_Bot_bot_term_f @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Bot____Terms__Obot____term_Itf__f_J_T,axiom,
    ! [X: bot_bot_term_f,Y: bot_bot_term_f] :
      ( ( if_Bot_bot_term_f @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__FSet__Ofset_It__Term__Oterm_Itf__f_Mtf__a_J_J_T,axiom,
    ! [X: fset_term_f_a,Y: fset_term_f_a] :
      ( ( if_fset_term_f_a @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_It__Term__Oterm_Itf__f_Mtf__a_J_J_T,axiom,
    ! [X: fset_term_f_a,Y: fset_term_f_a] :
      ( ( if_fset_term_f_a @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__FSet__Ofset_It__Bot____Terms__Obot____term_Itf__f_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__FSet__Ofset_It__Bot____Terms__Obot____term_Itf__f_J_J_T,axiom,
    ! [X: fset_Bot_bot_term_f,Y: fset_Bot_bot_term_f] :
      ( ( if_fse3524882727628502648term_f @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_It__Bot____Terms__Obot____term_Itf__f_J_J_T,axiom,
    ! [X: fset_Bot_bot_term_f,Y: fset_Bot_bot_term_f] :
      ( ( if_fse3524882727628502648term_f @ $true @ X @ Y )
      = X ) ).

% Conjectures (4)
thf(conj_0,hypothesis,
    ord_less_nat @ i @ ( size_s7148729505879680775rm_f_v @ ts ) ).

thf(conj_1,hypothesis,
    fmember_term_f_a @ l @ r ).

thf(conj_2,hypothesis,
    member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( nth_term_f_v @ ts @ i ) @ s ) @ subter5033437232688913882eq_f_v ).

thf(conj_3,conjecture,
    ~ ( member4687689947140789414term_f @ ( produc2865564700782054965term_f @ ( bot_te7903553565439629403rm_f_a @ l ) @ ( bot_te7903553565439629424rm_f_v @ s ) ) @ bot_bless_eq_f ) ).

%------------------------------------------------------------------------------