TPTP Problem File: SLH0833^1.p

%------------------------------------------------------------------------------
% File     : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain   : Archive of Formal Proofs
% Problem  :
% Version  : Especial.
% English  :

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Multiset_Ordering_NPC/0003_Multiset_Ordering_NP_Hard/prob_00166_007750__14095742_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1690 ( 575 unt; 419 typ;   0 def)
%            Number of atoms       : 3621 (1662 equ;   0 cnn)
%            Maximal formula atoms :    8 (   2 avg)
%            Number of connectives : 12798 ( 634   ~;  94   |; 368   &;10065   @)
%                                         (   0 <=>;1637  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   7 avg)
%            Number of types       :  111 ( 110 usr)
%            Number of type conns  :  906 ( 906   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  312 ( 309 usr;  25 con; 0-3 aty)
%            Number of variables   : 4173 ( 133   ^;3654   !; 386   ?;4173   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-18 16:30:34.673
%------------------------------------------------------------------------------
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_M_Eo_J_J,type,
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thf(ty_n_t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__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,
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thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_Mt__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_Mt__Nat__Onat_J,type,
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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,
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thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(ty_n_t__List__Olist_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    list_nat_nat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_Itf__a_M_Eo_J,type,
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thf(ty_n_t__Multiset__Omultiset_It__Nat__Onat_J,type,
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thf(ty_n_t__Multiset__Omultiset_It__Int__Oint_J,type,
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thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
    list_int: $tType ).

thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
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thf(ty_n_t__List__Olist_Itf__a_J,type,
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thf(ty_n_t__String__Ochar,type,
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thf(ty_n_t__Num__Onum,type,
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_t__Int__Oint,type,
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% Explicit typings (309)
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
    abs_abs_int: int > int ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: nat > nat > nat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
    one_one_int: int ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
    uminus_uminus_int: int > int ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_J,type,
    uminus8391578155927437342od_a_o: set_li5011860890973139239od_a_o > set_li5011860890973139239od_a_o ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
    uminus5710092332889474511et_nat: set_nat > set_nat ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    uminus4127886064781087314on_nat: set_Pr2535564044409765851on_nat > set_Pr2535564044409765851on_nat ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    uminus5421032341409327564on_nat: set_Su266273576555847555on_nat > set_Su266273576555847555on_nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
    zero_zero_int: int ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Int__Oint_J,type,
    zero_z3170743180189231877et_int: multiset_int ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
    zero_z7348594199698428585et_nat: multiset_nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_If_001t__Int__Oint,type,
    if_int: $o > int > int > int ).

thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
    if_list_int: $o > list_int > list_int > list_int ).

thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
    if_list_nat: $o > list_nat > list_nat > list_nat ).

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

thf(sy_c_Int_Onat,type,
    nat2: int > nat ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
    ring_1_of_int_int: int > int ).

thf(sy_c_List_Oappend_001t__Int__Oint,type,
    append_int: list_int > list_int > list_int ).

thf(sy_c_List_Oappend_001t__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J,type,
    append2159120467191820514od_a_o: list_l8336264072933094989od_a_o > list_l8336264072933094989od_a_o > list_l8336264072933094989od_a_o ).

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__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    append2865821678206098582on_nat: list_P9127870862658512897on_nat > list_P9127870862658512897on_nat > list_P9127870862658512897on_nat ).

thf(sy_c_List_Oappend_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J,type,
    append3523812819975775986on_nat: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > list_S5253404628753938141on_nat ).

thf(sy_c_List_Obind_001t__Int__Oint_001t__Nat__Onat,type,
    bind_int_nat: list_int > ( int > list_nat ) > list_nat ).

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

thf(sy_c_List_Ocan__select_001t__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J,type,
    can_se5536358244299704004od_a_o: ( list_P3196869577824546119od_a_o > $o ) > set_li5011860890973139239od_a_o > $o ).

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

thf(sy_c_List_Ocan__select_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    can_se7488105254466884472on_nat: ( produc1313295348606329723on_nat > $o ) > set_Pr2535564044409765851on_nat > $o ).

thf(sy_c_List_Ocan__select_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J,type,
    can_se5303938684849109648on_nat: ( sum_su4907927656612468301on_nat > $o ) > set_Su266273576555847555on_nat > $o ).

thf(sy_c_List_Ocoset_001t__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J,type,
    coset_2783394909775257274od_a_o: list_l8336264072933094989od_a_o > set_li5011860890973139239od_a_o ).

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

thf(sy_c_List_Ocoset_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    coset_1816015773736449646on_nat: list_P9127870862658512897on_nat > set_Pr2535564044409765851on_nat ).

thf(sy_c_List_Ocoset_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J,type,
    coset_3487183839294692122on_nat: list_S5253404628753938141on_nat > set_Su266273576555847555on_nat ).

thf(sy_c_List_Ocount__list_001t__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J,type,
    count_5344270436093135555od_a_o: list_l8336264072933094989od_a_o > list_P3196869577824546119od_a_o > nat ).

thf(sy_c_List_Ocount__list_001t__Nat__Onat,type,
    count_list_nat: list_nat > nat > nat ).

thf(sy_c_List_Ocount__list_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    count_9157224493928226551on_nat: list_P9127870862658512897on_nat > produc1313295348606329723on_nat > nat ).

thf(sy_c_List_Ocount__list_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J,type,
    count_3605398483369874897on_nat: list_S5253404628753938141on_nat > sum_su4907927656612468301on_nat > nat ).

thf(sy_c_List_Odistinct_001t__Int__Oint,type,
    distinct_int: list_int > $o ).

thf(sy_c_List_Odistinct_001t__List__Olist_It__Nat__Onat_J,type,
    distinct_list_nat: list_list_nat > $o ).

thf(sy_c_List_Odistinct_001t__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J,type,
    distin690952206374581440od_a_o: list_l8336264072933094989od_a_o > $o ).

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

thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    distin8410642530753306484on_nat: list_P9127870862658512897on_nat > $o ).

thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_Itf__a_M_Eo_J,type,
    distin2380507709364613050od_a_o: list_P3196869577824546119od_a_o > $o ).

thf(sy_c_List_Odistinct_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J,type,
    distin2773651816175405460on_nat: list_S5253404628753938141on_nat > $o ).

thf(sy_c_List_Ogen__length_001t__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J,type,
    gen_le5767993353563561050od_a_o: nat > list_l8336264072933094989od_a_o > nat ).

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

thf(sy_c_List_Ogen__length_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    gen_le5456478678281257998on_nat: nat > list_P9127870862658512897on_nat > nat ).

thf(sy_c_List_Ogen__length_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J,type,
    gen_le4617574567093687674on_nat: nat > list_S5253404628753938141on_nat > nat ).

thf(sy_c_List_Olenlex_001t__Int__Oint,type,
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thf(sy_c_List_Olenlex_001t__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J,type,
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thf(sy_c_List_Onth_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Onth_001t__Int__Oint,type,
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thf(sy_c_List_Onth_001t__Nat__Onat,type,
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thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
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thf(sy_c_List_OremoveAll_001t__Int__Oint,type,
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thf(sy_c_List_OremoveAll_001t__Nat__Onat,type,
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thf(sy_c_List_Osubseqs_001t__Int__Oint,type,
    subseqs_int: list_int > list_list_int ).

thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
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thf(sy_c_List_Otake_001t__Int__Oint,type,
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thf(sy_c_List_Otake_001t__Nat__Onat,type,
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thf(sy_c_List_Ounion_001t__Nat__Onat,type,
    union_nat: list_nat > list_nat > list_nat ).

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

thf(sy_c_List_Oupto__aux,type,
    upto_aux: int > int > list_int > list_int ).

thf(sy_c_List__Order_Olist__order__extension_001t__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J,type,
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thf(sy_c_List__Order_Olist__order__extension_001t__Nat__Onat,type,
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thf(sy_c_List__Order_Olist__order__extension_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J,type,
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thf(sy_c_Multiset_Oadd__mset_001t__Int__Oint,type,
    add_mset_int: int > multiset_int > multiset_int ).

thf(sy_c_Multiset_Oadd__mset_001t__Nat__Onat,type,
    add_mset_nat: nat > multiset_nat > multiset_nat ).

thf(sy_c_Multiset_Olinorder__class_Osorted__list__of__multiset_001t__Int__Oint,type,
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thf(sy_c_Multiset_Olinorder__class_Osorted__list__of__multiset_001t__Nat__Onat,type,
    linord3047872887403683810et_nat: multiset_nat > list_nat ).

thf(sy_c_Multiset__Ordering__NP__Hard_Ovars__of__cnf_001tf__a,type,
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thf(sy_c_Nat_OSuc,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_J_Mt__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    member4418620086355165936on_nat: produc125269551341514585on_nat > set_Pr8643848459224841615on_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Int__Oint_J_J,type,
    member4850886304473975718st_int: produc8561936516282095101st_int > set_Pr7861072320784411741st_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_J_J,type,
    member548352139293716857od_a_o: produc8477798921965902306od_a_o > set_Pr8285060063824045976od_a_o > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    member9199622813682680237on_nat: produc1900388947750868630on_nat > set_Pr764678315123685324on_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    member452587771129616071on_nat: produc4988977186861604126on_nat > set_Pr1225116743179249406on_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    member4629988367768146576on_nat: produc1714117091004681831on_nat > set_Pr8586234676504894535on_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_J,type,
    member5440785652130980624od_a_o: produc8520174026553075431od_a_o > set_Pr7420778282784443591od_a_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_Mt__Nat__Onat_J,type,
    member7885718254550376541_o_nat: produc408724640507517382_o_nat > set_Pr8850484174313502844_o_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    member1833810847379340378on_nat: produc731851057114331587on_nat > set_Pr6144483798281041401on_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_J_J,type,
    member2603097825888147004od_a_o: produc7533119327729271461od_a_o > set_Pr9068572523612285659od_a_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member380504172838623943st_nat: produc4916893588570611998st_nat > set_Pr6075247984620006142st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    member4221009242412147780on_nat: produc5580075136493545115on_nat > set_Pr7828957961435280891on_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
    member4262671552274231302at_int: product_prod_nat_int > set_Pr7995236796853374141at_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_J,type,
    member8910825520198479363od_a_o: produc1433831906155620204od_a_o > set_Pr94232683564258082od_a_o > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    member7352503068387395495on_nat: produc7088870822282640638on_nat > set_Pr7552745763645690846on_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    member1022098016950487056on_nat: produc4634397314152313575on_nat > set_Pr8202701763255893447on_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_J,type,
    member7595723914438784454od_a_o: produc6493764124173775663od_a_o > set_Pr9140199768791955813od_a_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    member8820252441731870247at_nat: produc8556620195627115390at_nat > set_Pr3345771464143190622at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    member5906418179908956708on_nat: produc1313295348606329723on_nat > set_Pr2535564044409765851on_nat > $o ).

thf(sy_c_member_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J,type,
    member8476201367316484836on_nat: sum_su4907927656612468301on_nat > set_Su266273576555847555on_nat > $o ).

thf(sy_v_NS____,type,
    ns: list_P9127870862658512897on_nat ).

thf(sy_v_c____,type,
    c: list_P3196869577824546119od_a_o ).

thf(sy_v_cnf,type,
    cnf: list_l8336264072933094989od_a_o ).

thf(sy_v_cs____,type,
    cs: list_nat ).

thf(sy_v_ns__left____,type,
    ns_left: list_S5253404628753938141on_nat ).

thf(sy_v_ns__right____,type,
    ns_right: list_S5253404628753938141on_nat ).

thf(sy_v_thesis____,type,
    thesis: $o ).

thf(sy_v_xs____,type,
    xs: list_a ).

% Relevant facts (1260)
thf(fact_0_ccnf,axiom,
    member5261019209154752368od_a_o @ c @ ( set_li2629784761229999580od_a_o @ cnf ) ).

% ccnf
thf(fact_1_nth__mem,axiom,
    ! [N: nat,Xs: list_P9127870862658512897on_nat] :
      ( ( ord_less_nat @ N @ ( size_s7490768528479217773on_nat @ Xs ) )
     => ( member5906418179908956708on_nat @ ( nth_Pr3336893728888831618on_nat @ Xs @ N ) @ ( set_Pr937530902923597456on_nat @ Xs ) ) ) ).

% nth_mem
thf(fact_2_nth__mem,axiom,
    ! [N: nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( ord_less_nat @ N @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( member5261019209154752368od_a_o @ ( nth_li3248255784598043854od_a_o @ Xs @ N ) @ ( set_li2629784761229999580od_a_o @ Xs ) ) ) ).

% nth_mem
thf(fact_3_nth__mem,axiom,
    ! [N: nat,Xs: list_S5253404628753938141on_nat] :
      ( ( ord_less_nat @ N @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( member8476201367316484836on_nat @ ( nth_Su8471564416988646406on_nat @ Xs @ N ) @ ( set_Su4848624856859743608on_nat @ Xs ) ) ) ).

% nth_mem
thf(fact_4_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_5_list__ball__nth,axiom,
    ! [N: nat,Xs: list_P9127870862658512897on_nat,P: produc1313295348606329723on_nat > $o] :
      ( ( ord_less_nat @ N @ ( size_s7490768528479217773on_nat @ Xs ) )
     => ( ! [X: produc1313295348606329723on_nat] :
            ( ( member5906418179908956708on_nat @ X @ ( set_Pr937530902923597456on_nat @ Xs ) )
           => ( P @ X ) )
       => ( P @ ( nth_Pr3336893728888831618on_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_6_list__ball__nth,axiom,
    ! [N: nat,Xs: list_l8336264072933094989od_a_o,P: list_P3196869577824546119od_a_o > $o] :
      ( ( ord_less_nat @ N @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ! [X: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X @ ( set_li2629784761229999580od_a_o @ Xs ) )
           => ( P @ X ) )
       => ( P @ ( nth_li3248255784598043854od_a_o @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_7_list__ball__nth,axiom,
    ! [N: nat,Xs: list_S5253404628753938141on_nat,P: sum_su4907927656612468301on_nat > $o] :
      ( ( ord_less_nat @ N @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ! [X: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X @ ( set_Su4848624856859743608on_nat @ Xs ) )
           => ( P @ X ) )
       => ( P @ ( nth_Su8471564416988646406on_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_8_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ( P @ X ) )
       => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_9_in__set__conv__nth,axiom,
    ! [X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Xs ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s7490768528479217773on_nat @ Xs ) )
            & ( ( nth_Pr3336893728888831618on_nat @ Xs @ I )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_10_in__set__conv__nth,axiom,
    ! [X2: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Xs ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Xs ) )
            & ( ( nth_li3248255784598043854od_a_o @ Xs @ I )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_11_in__set__conv__nth,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Xs ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Xs ) )
            & ( ( nth_Su8471564416988646406on_nat @ Xs @ I )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_12_in__set__conv__nth,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ I )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_13_all__nth__imp__all__set,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P: produc1313295348606329723on_nat > $o,X2: produc1313295348606329723on_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s7490768528479217773on_nat @ Xs ) )
         => ( P @ ( nth_Pr3336893728888831618on_nat @ Xs @ I2 ) ) )
     => ( ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Xs ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_14_all__nth__imp__all__set,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,P: list_P3196869577824546119od_a_o > $o,X2: list_P3196869577824546119od_a_o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3379185846251495609od_a_o @ Xs ) )
         => ( P @ ( nth_li3248255784598043854od_a_o @ Xs @ I2 ) ) )
     => ( ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Xs ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_15_all__nth__imp__all__set,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P: sum_su4907927656612468301on_nat > $o,X2: sum_su4907927656612468301on_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s5686234912572693873on_nat @ Xs ) )
         => ( P @ ( nth_Su8471564416988646406on_nat @ Xs @ I2 ) ) )
     => ( ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Xs ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_16_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: nat > $o,X2: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
         => ( P @ ( nth_nat @ Xs @ I2 ) ) )
     => ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_17_all__set__conv__all__nth,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P: produc1313295348606329723on_nat > $o] :
      ( ( ! [X3: produc1313295348606329723on_nat] :
            ( ( member5906418179908956708on_nat @ X3 @ ( set_Pr937530902923597456on_nat @ Xs ) )
           => ( P @ X3 ) ) )
      = ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s7490768528479217773on_nat @ Xs ) )
           => ( P @ ( nth_Pr3336893728888831618on_nat @ Xs @ I ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_18_all__set__conv__all__nth,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,P: list_P3196869577824546119od_a_o > $o] :
      ( ( ! [X3: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X3 @ ( set_li2629784761229999580od_a_o @ Xs ) )
           => ( P @ X3 ) ) )
      = ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Xs ) )
           => ( P @ ( nth_li3248255784598043854od_a_o @ Xs @ I ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_19_all__set__conv__all__nth,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P: sum_su4907927656612468301on_nat > $o] :
      ( ( ! [X3: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X3 @ ( set_Su4848624856859743608on_nat @ Xs ) )
           => ( P @ X3 ) ) )
      = ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Xs ) )
           => ( P @ ( nth_Su8471564416988646406on_nat @ Xs @ I ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_20_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
           => ( P @ X3 ) ) )
      = ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
           => ( P @ ( nth_nat @ Xs @ I ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_21_nth__equalityI,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( ( size_s7490768528479217773on_nat @ Xs )
        = ( size_s7490768528479217773on_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s7490768528479217773on_nat @ Xs ) )
           => ( ( nth_Pr3336893728888831618on_nat @ Xs @ I2 )
              = ( nth_Pr3336893728888831618on_nat @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_22_nth__equalityI,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3379185846251495609od_a_o @ Xs ) )
           => ( ( nth_li3248255784598043854od_a_o @ Xs @ I2 )
              = ( nth_li3248255784598043854od_a_o @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_23_nth__equalityI,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s5686234912572693873on_nat @ Xs ) )
           => ( ( nth_Su8471564416988646406on_nat @ Xs @ I2 )
              = ( nth_Su8471564416988646406on_nat @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_24_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I2 )
              = ( nth_nat @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_25_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > produc1313295348606329723on_nat > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ K )
           => ? [X4: produc1313295348606329723on_nat] : ( P @ I @ X4 ) ) )
      = ( ? [Xs2: list_P9127870862658512897on_nat] :
            ( ( ( size_s7490768528479217773on_nat @ Xs2 )
              = K )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ K )
               => ( P @ I @ ( nth_Pr3336893728888831618on_nat @ Xs2 @ I ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_26_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > list_P3196869577824546119od_a_o > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ K )
           => ? [X4: list_P3196869577824546119od_a_o] : ( P @ I @ X4 ) ) )
      = ( ? [Xs2: list_l8336264072933094989od_a_o] :
            ( ( ( size_s3379185846251495609od_a_o @ Xs2 )
              = K )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ K )
               => ( P @ I @ ( nth_li3248255784598043854od_a_o @ Xs2 @ I ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_27_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > sum_su4907927656612468301on_nat > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ K )
           => ? [X4: sum_su4907927656612468301on_nat] : ( P @ I @ X4 ) ) )
      = ( ? [Xs2: list_S5253404628753938141on_nat] :
            ( ( ( size_s5686234912572693873on_nat @ Xs2 )
              = K )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ K )
               => ( P @ I @ ( nth_Su8471564416988646406on_nat @ Xs2 @ I ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_28_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ K )
           => ? [X4: nat] : ( P @ I @ X4 ) ) )
      = ( ? [Xs2: list_nat] :
            ( ( ( size_size_list_nat @ Xs2 )
              = K )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ K )
               => ( P @ I @ ( nth_nat @ Xs2 @ I ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_29_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y: list_P9127870862658512897on_nat,Z: list_P9127870862658512897on_nat] : ( Y = Z ) )
    = ( ^ [Xs2: list_P9127870862658512897on_nat,Ys2: list_P9127870862658512897on_nat] :
          ( ( ( size_s7490768528479217773on_nat @ Xs2 )
            = ( size_s7490768528479217773on_nat @ Ys2 ) )
          & ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s7490768528479217773on_nat @ Xs2 ) )
             => ( ( nth_Pr3336893728888831618on_nat @ Xs2 @ I )
                = ( nth_Pr3336893728888831618on_nat @ Ys2 @ I ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_30_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y: list_l8336264072933094989od_a_o,Z: list_l8336264072933094989od_a_o] : ( Y = Z ) )
    = ( ^ [Xs2: list_l8336264072933094989od_a_o,Ys2: list_l8336264072933094989od_a_o] :
          ( ( ( size_s3379185846251495609od_a_o @ Xs2 )
            = ( size_s3379185846251495609od_a_o @ Ys2 ) )
          & ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Xs2 ) )
             => ( ( nth_li3248255784598043854od_a_o @ Xs2 @ I )
                = ( nth_li3248255784598043854od_a_o @ Ys2 @ I ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_31_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y: list_S5253404628753938141on_nat,Z: list_S5253404628753938141on_nat] : ( Y = Z ) )
    = ( ^ [Xs2: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat] :
          ( ( ( size_s5686234912572693873on_nat @ Xs2 )
            = ( size_s5686234912572693873on_nat @ Ys2 ) )
          & ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Xs2 ) )
             => ( ( nth_Su8471564416988646406on_nat @ Xs2 @ I )
                = ( nth_Su8471564416988646406on_nat @ Ys2 @ I ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_32_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y: list_nat,Z: list_nat] : ( Y = Z ) )
    = ( ^ [Xs2: list_nat,Ys2: list_nat] :
          ( ( ( size_size_list_nat @ Xs2 )
            = ( size_size_list_nat @ Ys2 ) )
          & ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
             => ( ( nth_nat @ Xs2 @ I )
                = ( nth_nat @ Ys2 @ I ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_33_length__induct,axiom,
    ! [P: list_P9127870862658512897on_nat > $o,Xs: list_P9127870862658512897on_nat] :
      ( ! [Xs3: list_P9127870862658512897on_nat] :
          ( ! [Ys3: list_P9127870862658512897on_nat] :
              ( ( ord_less_nat @ ( size_s7490768528479217773on_nat @ Ys3 ) @ ( size_s7490768528479217773on_nat @ Xs3 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_34_length__induct,axiom,
    ! [P: list_l8336264072933094989od_a_o > $o,Xs: list_l8336264072933094989od_a_o] :
      ( ! [Xs3: list_l8336264072933094989od_a_o] :
          ( ! [Ys3: list_l8336264072933094989od_a_o] :
              ( ( ord_less_nat @ ( size_s3379185846251495609od_a_o @ Ys3 ) @ ( size_s3379185846251495609od_a_o @ Xs3 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_35_length__induct,axiom,
    ! [P: list_S5253404628753938141on_nat > $o,Xs: list_S5253404628753938141on_nat] :
      ( ! [Xs3: list_S5253404628753938141on_nat] :
          ( ! [Ys3: list_S5253404628753938141on_nat] :
              ( ( ord_less_nat @ ( size_s5686234912572693873on_nat @ Ys3 ) @ ( size_s5686234912572693873on_nat @ Xs3 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_36_length__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys3: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_37_cs__def,axiom,
    ( cs
    = ( upt @ zero_zero_nat @ ( size_s3379185846251495609od_a_o @ cnf ) ) ) ).

% cs_def
thf(fact_38_xs__def,axiom,
    ( xs
    = ( multis4741186478440321083_cnf_a @ cnf ) ) ).

% xs_def
thf(fact_39_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_l8336264072933094989od_a_o] :
      ( ( size_s3379185846251495609od_a_o @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_40_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_S5253404628753938141on_nat] :
      ( ( size_s5686234912572693873on_nat @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_41_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_42_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_P9127870862658512897on_nat] :
      ( ( size_s7490768528479217773on_nat @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_43_len,axiom,
    ( ( size_s5686234912572693873on_nat @ ns_left )
    = ( size_s5686234912572693873on_nat @ ns_right ) ) ).

% len
thf(fact_44_length__pos__if__in__set,axiom,
    ! [X2: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3379185846251495609od_a_o @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_45_length__pos__if__in__set,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s5686234912572693873on_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_46_length__pos__if__in__set,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_47_length__pos__if__in__set,axiom,
    ! [X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s7490768528479217773on_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_48_neq__if__length__neq,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
       != ( size_s3379185846251495609od_a_o @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_49_neq__if__length__neq,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
       != ( size_s5686234912572693873on_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_50_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_51_neq__if__length__neq,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( ( size_s7490768528479217773on_nat @ Xs )
       != ( size_s7490768528479217773on_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_52_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_53_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_54_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_55_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_56_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ~ ( P @ N2 )
             => ? [M: nat] :
                  ( ( ord_less_nat @ M @ N2 )
                  & ~ ( P @ M ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_57_gr__implies__not0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_58_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_59_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_60_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_61_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_62_zero__reorient,axiom,
    ! [X2: nat] :
      ( ( zero_zero_nat = X2 )
      = ( X2 = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_63_zero__reorient,axiom,
    ! [X2: int] :
      ( ( zero_zero_int = X2 )
      = ( X2 = zero_zero_int ) ) ).

% zero_reorient
thf(fact_64_nat__neq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 != N )
      = ( ( ord_less_nat @ M2 @ N )
        | ( ord_less_nat @ N @ M2 ) ) ) ).

% nat_neq_iff
thf(fact_65_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_66_less__not__refl2,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ N @ M2 )
     => ( M2 != N ) ) ).

% less_not_refl2
thf(fact_67_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_68_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_69_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M: nat] :
              ( ( ord_less_nat @ M @ N2 )
             => ( P @ M ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_70_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ~ ( P @ N2 )
         => ? [M: nat] :
              ( ( ord_less_nat @ M @ N2 )
              & ~ ( P @ M ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_71_linorder__neqE__nat,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( X2 != Y2 )
     => ( ~ ( ord_less_nat @ X2 @ Y2 )
       => ( ord_less_nat @ Y2 @ X2 ) ) ) ).

% linorder_neqE_nat
thf(fact_72_size__neq__size__imp__neq,axiom,
    ! [X2: list_l8336264072933094989od_a_o,Y2: list_l8336264072933094989od_a_o] :
      ( ( ( size_s3379185846251495609od_a_o @ X2 )
       != ( size_s3379185846251495609od_a_o @ Y2 ) )
     => ( X2 != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_73_size__neq__size__imp__neq,axiom,
    ! [X2: list_S5253404628753938141on_nat,Y2: list_S5253404628753938141on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ X2 )
       != ( size_s5686234912572693873on_nat @ Y2 ) )
     => ( X2 != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_74_size__neq__size__imp__neq,axiom,
    ! [X2: char,Y2: char] :
      ( ( ( size_size_char @ X2 )
       != ( size_size_char @ Y2 ) )
     => ( X2 != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_75_size__neq__size__imp__neq,axiom,
    ! [X2: list_nat,Y2: list_nat] :
      ( ( ( size_size_list_nat @ X2 )
       != ( size_size_list_nat @ Y2 ) )
     => ( X2 != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_76_size__neq__size__imp__neq,axiom,
    ! [X2: list_P9127870862658512897on_nat,Y2: list_P9127870862658512897on_nat] :
      ( ( ( size_s7490768528479217773on_nat @ X2 )
       != ( size_s7490768528479217773on_nat @ Y2 ) )
     => ( X2 != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_77_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_78_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_79_gr__implies__not__zero,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_80_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_81_mem__Collect__eq,axiom,
    ! [A: list_P3196869577824546119od_a_o,P: list_P3196869577824546119od_a_o > $o] :
      ( ( member5261019209154752368od_a_o @ A @ ( collec5117750660557753778od_a_o @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_82_mem__Collect__eq,axiom,
    ! [A: produc1313295348606329723on_nat,P: produc1313295348606329723on_nat > $o] :
      ( ( member5906418179908956708on_nat @ A @ ( collec5585564048161825126on_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_83_mem__Collect__eq,axiom,
    ! [A: sum_su4907927656612468301on_nat,P: sum_su4907927656612468301on_nat > $o] :
      ( ( member8476201367316484836on_nat @ A @ ( collec1912426543561893410on_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_84_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_85_Collect__mem__eq,axiom,
    ! [A2: set_li5011860890973139239od_a_o] :
      ( ( collec5117750660557753778od_a_o
        @ ^ [X3: list_P3196869577824546119od_a_o] : ( member5261019209154752368od_a_o @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_86_Collect__mem__eq,axiom,
    ! [A2: set_Pr2535564044409765851on_nat] :
      ( ( collec5585564048161825126on_nat
        @ ^ [X3: produc1313295348606329723on_nat] : ( member5906418179908956708on_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_87_Collect__mem__eq,axiom,
    ! [A2: set_Su266273576555847555on_nat] :
      ( ( collec1912426543561893410on_nat
        @ ^ [X3: sum_su4907927656612468301on_nat] : ( member8476201367316484836on_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_88_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_89_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_90_ns,axiom,
    ! [I3: nat] :
      ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ ns_right ) )
     => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ ns_left @ I3 ) @ ( nth_Su8471564416988646406on_nat @ ns_right @ I3 ) ) @ ( set_Pr937530902923597456on_nat @ ns ) ) ) ).

% ns
thf(fact_91_in__set__product__lists__length,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,Xss: list_l8336264072933094989od_a_o] :
      ( ( member5261019209154752368od_a_o @ Xs @ ( set_li2629784761229999580od_a_o @ ( produc3814538932794474544od_a_o @ Xss ) ) )
     => ( ( size_s1727999245659094451od_a_o @ Xs )
        = ( size_s3379185846251495609od_a_o @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_92_in__set__product__lists__length,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Xss: list_l7431997615882138323od_a_o] :
      ( ( member8885487059802017782od_a_o @ Xs @ ( set_li3106133282984605794od_a_o @ ( produc6266365578930819638od_a_o @ Xss ) ) )
     => ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_s1509257606055081791od_a_o @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_93_in__set__product__lists__length,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Xss: list_l8999537460041914605on_nat] :
      ( ( member86162308477975540on_nat @ Xs @ ( set_li4371160873219908744on_nat @ ( produc1748266815408604318on_nat @ Xss ) ) )
     => ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s650155532264187265on_nat @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_94_in__set__product__lists__length,axiom,
    ! [Xs: list_nat,Xss: list_list_nat] :
      ( ( member_list_nat @ Xs @ ( set_list_nat2 @ ( product_lists_nat @ Xss ) ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s3023201423986296836st_nat @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_95_in__set__product__lists__length,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Xss: list_l1726306699111145479on_nat] :
      ( ( member783415501022416426on_nat @ Xs @ ( set_li8154783781463198870on_nat @ ( produc2357585119171201002on_nat @ Xss ) ) )
     => ( ( size_s7490768528479217773on_nat @ Xs )
        = ( size_s129639936549468787on_nat @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_96_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_97_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_98_length__code,axiom,
    ( size_s3379185846251495609od_a_o
    = ( gen_le5767993353563561050od_a_o @ zero_zero_nat ) ) ).

% length_code
thf(fact_99_length__code,axiom,
    ( size_s5686234912572693873on_nat
    = ( gen_le4617574567093687674on_nat @ zero_zero_nat ) ) ).

% length_code
thf(fact_100_length__code,axiom,
    ( size_size_list_nat
    = ( gen_length_nat @ zero_zero_nat ) ) ).

% length_code
thf(fact_101_length__code,axiom,
    ( size_s7490768528479217773on_nat
    = ( gen_le5456478678281257998on_nat @ zero_zero_nat ) ) ).

% length_code
thf(fact_102_set__swap,axiom,
    ! [I3: nat,Xs: list_l8336264072933094989od_a_o,J: nat] :
      ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s3379185846251495609od_a_o @ Xs ) )
       => ( ( set_li2629784761229999580od_a_o @ ( list_u329291161401623989od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ I3 @ ( nth_li3248255784598043854od_a_o @ Xs @ J ) ) @ J @ ( nth_li3248255784598043854od_a_o @ Xs @ I3 ) ) )
          = ( set_li2629784761229999580od_a_o @ Xs ) ) ) ) ).

% set_swap
thf(fact_103_set__swap,axiom,
    ! [I3: nat,Xs: list_S5253404628753938141on_nat,J: nat] :
      ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s5686234912572693873on_nat @ Xs ) )
       => ( ( set_Su4848624856859743608on_nat @ ( list_u8577817230714425439on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I3 @ ( nth_Su8471564416988646406on_nat @ Xs @ J ) ) @ J @ ( nth_Su8471564416988646406on_nat @ Xs @ I3 ) ) )
          = ( set_Su4848624856859743608on_nat @ Xs ) ) ) ) ).

% set_swap
thf(fact_104_set__swap,axiom,
    ! [I3: nat,Xs: list_nat,J: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
       => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I3 @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I3 ) ) )
          = ( set_nat2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_105_set__swap,axiom,
    ! [I3: nat,Xs: list_P9127870862658512897on_nat,J: nat] :
      ( ( ord_less_nat @ I3 @ ( size_s7490768528479217773on_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s7490768528479217773on_nat @ Xs ) )
       => ( ( set_Pr937530902923597456on_nat @ ( list_u1840724836389618409on_nat @ ( list_u1840724836389618409on_nat @ Xs @ I3 @ ( nth_Pr3336893728888831618on_nat @ Xs @ J ) ) @ J @ ( nth_Pr3336893728888831618on_nat @ Xs @ I3 ) ) )
          = ( set_Pr937530902923597456on_nat @ Xs ) ) ) ) ).

% set_swap
thf(fact_106_list__ex__length,axiom,
    ( list_e5403782194016407179od_a_o
    = ( ^ [P2: list_P3196869577824546119od_a_o > $o,Xs2: list_l8336264072933094989od_a_o] :
        ? [N3: nat] :
          ( ( ord_less_nat @ N3 @ ( size_s3379185846251495609od_a_o @ Xs2 ) )
          & ( P2 @ ( nth_li3248255784598043854od_a_o @ Xs2 @ N3 ) ) ) ) ) ).

% list_ex_length
thf(fact_107_list__ex__length,axiom,
    ( list_e3769318529992910857on_nat
    = ( ^ [P2: sum_su4907927656612468301on_nat > $o,Xs2: list_S5253404628753938141on_nat] :
        ? [N3: nat] :
          ( ( ord_less_nat @ N3 @ ( size_s5686234912572693873on_nat @ Xs2 ) )
          & ( P2 @ ( nth_Su8471564416988646406on_nat @ Xs2 @ N3 ) ) ) ) ) ).

% list_ex_length
thf(fact_108_list__ex__length,axiom,
    ( list_ex_nat
    = ( ^ [P2: nat > $o,Xs2: list_nat] :
        ? [N3: nat] :
          ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs2 ) )
          & ( P2 @ ( nth_nat @ Xs2 @ N3 ) ) ) ) ) ).

% list_ex_length
thf(fact_109_list__ex__length,axiom,
    ( list_e8248363684495001279on_nat
    = ( ^ [P2: produc1313295348606329723on_nat > $o,Xs2: list_P9127870862658512897on_nat] :
        ? [N3: nat] :
          ( ( ord_less_nat @ N3 @ ( size_s7490768528479217773on_nat @ Xs2 ) )
          & ( P2 @ ( nth_Pr3336893728888831618on_nat @ Xs2 @ N3 ) ) ) ) ) ).

% list_ex_length
thf(fact_110_length__n__lists__elem,axiom,
    ! [Ys: list_P3196869577824546119od_a_o,N: nat,Xs: list_P3196869577824546119od_a_o] :
      ( ( member5261019209154752368od_a_o @ Ys @ ( set_li2629784761229999580od_a_o @ ( n_list4838562191945764047od_a_o @ N @ Xs ) ) )
     => ( ( size_s1727999245659094451od_a_o @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_111_length__n__lists__elem,axiom,
    ! [Ys: list_l8336264072933094989od_a_o,N: nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( member8885487059802017782od_a_o @ Ys @ ( set_li3106133282984605794od_a_o @ ( n_list931218472929919829od_a_o @ N @ Xs ) ) )
     => ( ( size_s3379185846251495609od_a_o @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_112_length__n__lists__elem,axiom,
    ! [Ys: list_S5253404628753938141on_nat,N: nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member86162308477975540on_nat @ Ys @ ( set_li4371160873219908744on_nat @ ( n_list6604620681149647551on_nat @ N @ Xs ) ) )
     => ( ( size_s5686234912572693873on_nat @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_113_length__n__lists__elem,axiom,
    ! [Ys: list_nat,N: nat,Xs: list_nat] :
      ( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) ) )
     => ( ( size_size_list_nat @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_114_length__n__lists__elem,axiom,
    ! [Ys: list_P9127870862658512897on_nat,N: nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member783415501022416426on_nat @ Ys @ ( set_li8154783781463198870on_nat @ ( n_list632955875454079113on_nat @ N @ Xs ) ) )
     => ( ( size_s7490768528479217773on_nat @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_115_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_116_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_117_length__removeAll__less,axiom,
    ! [X2: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Xs ) )
     => ( ord_less_nat @ ( size_s3379185846251495609od_a_o @ ( remove3990864149796893523od_a_o @ X2 @ Xs ) ) @ ( size_s3379185846251495609od_a_o @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_118_length__removeAll__less,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Xs ) )
     => ( ord_less_nat @ ( size_s5686234912572693873on_nat @ ( remove433775707702860097on_nat @ X2 @ Xs ) ) @ ( size_s5686234912572693873on_nat @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_119_length__removeAll__less,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ ( size_size_list_nat @ ( removeAll_nat @ X2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_120_length__removeAll__less,axiom,
    ! [X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Xs ) )
     => ( ord_less_nat @ ( size_s7490768528479217773on_nat @ ( remove6084986979817164679on_nat @ X2 @ Xs ) ) @ ( size_s7490768528479217773on_nat @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_121_length__greater__0__conv,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s3379185846251495609od_a_o @ Xs ) )
      = ( Xs != nil_li1372968520647675725od_a_o ) ) ).

% length_greater_0_conv
thf(fact_122_length__greater__0__conv,axiom,
    ! [Xs: list_S5253404628753938141on_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s5686234912572693873on_nat @ Xs ) )
      = ( Xs != nil_Su3078906344068544711on_nat ) ) ).

% length_greater_0_conv
thf(fact_123_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_124_length__greater__0__conv,axiom,
    ! [Xs: list_P9127870862658512897on_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s7490768528479217773on_nat @ Xs ) )
      = ( Xs != nil_Pr8614093721167239297on_nat ) ) ).

% length_greater_0_conv
thf(fact_125_count__notin,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ( ( count_list_nat @ Xs @ X2 )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_126_count__notin,axiom,
    ! [X2: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ~ ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Xs ) )
     => ( ( count_5344270436093135555od_a_o @ Xs @ X2 )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_127_count__notin,axiom,
    ! [X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ~ ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Xs ) )
     => ( ( count_9157224493928226551on_nat @ Xs @ X2 )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_128_count__notin,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ~ ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Xs ) )
     => ( ( count_3605398483369874897on_nat @ Xs @ X2 )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_129_of__nat__eq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M2 )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M2 = N ) ) ).

% of_nat_eq_iff
thf(fact_130_list__update__nonempty,axiom,
    ! [Xs: list_nat,K: nat,X2: nat] :
      ( ( ( list_update_nat @ Xs @ K @ X2 )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% list_update_nonempty
thf(fact_131_length__list__update,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,I3: nat,X2: list_P3196869577824546119od_a_o] :
      ( ( size_s3379185846251495609od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ I3 @ X2 ) )
      = ( size_s3379185846251495609od_a_o @ Xs ) ) ).

% length_list_update
thf(fact_132_length__list__update,axiom,
    ! [Xs: list_S5253404628753938141on_nat,I3: nat,X2: sum_su4907927656612468301on_nat] :
      ( ( size_s5686234912572693873on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I3 @ X2 ) )
      = ( size_s5686234912572693873on_nat @ Xs ) ) ).

% length_list_update
thf(fact_133_length__list__update,axiom,
    ! [Xs: list_nat,I3: nat,X2: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_list_update
thf(fact_134_length__list__update,axiom,
    ! [Xs: list_P9127870862658512897on_nat,I3: nat,X2: produc1313295348606329723on_nat] :
      ( ( size_s7490768528479217773on_nat @ ( list_u1840724836389618409on_nat @ Xs @ I3 @ X2 ) )
      = ( size_s7490768528479217773on_nat @ Xs ) ) ).

% length_list_update
thf(fact_135_nth__list__update__neq,axiom,
    ! [I3: nat,J: nat,Xs: list_l8336264072933094989od_a_o,X2: list_P3196869577824546119od_a_o] :
      ( ( I3 != J )
     => ( ( nth_li3248255784598043854od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ I3 @ X2 ) @ J )
        = ( nth_li3248255784598043854od_a_o @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_136_nth__list__update__neq,axiom,
    ! [I3: nat,J: nat,Xs: list_S5253404628753938141on_nat,X2: sum_su4907927656612468301on_nat] :
      ( ( I3 != J )
     => ( ( nth_Su8471564416988646406on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I3 @ X2 ) @ J )
        = ( nth_Su8471564416988646406on_nat @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_137_nth__list__update__neq,axiom,
    ! [I3: nat,J: nat,Xs: list_nat,X2: nat] :
      ( ( I3 != J )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ J )
        = ( nth_nat @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_138_nth__list__update__neq,axiom,
    ! [I3: nat,J: nat,Xs: list_P9127870862658512897on_nat,X2: produc1313295348606329723on_nat] :
      ( ( I3 != J )
     => ( ( nth_Pr3336893728888831618on_nat @ ( list_u1840724836389618409on_nat @ Xs @ I3 @ X2 ) @ J )
        = ( nth_Pr3336893728888831618on_nat @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_139_list__update__id,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,I3: nat] :
      ( ( list_u329291161401623989od_a_o @ Xs @ I3 @ ( nth_li3248255784598043854od_a_o @ Xs @ I3 ) )
      = Xs ) ).

% list_update_id
thf(fact_140_list__update__id,axiom,
    ! [Xs: list_S5253404628753938141on_nat,I3: nat] :
      ( ( list_u8577817230714425439on_nat @ Xs @ I3 @ ( nth_Su8471564416988646406on_nat @ Xs @ I3 ) )
      = Xs ) ).

% list_update_id
thf(fact_141_list__update__id,axiom,
    ! [Xs: list_nat,I3: nat] :
      ( ( list_update_nat @ Xs @ I3 @ ( nth_nat @ Xs @ I3 ) )
      = Xs ) ).

% list_update_id
thf(fact_142_list__update__id,axiom,
    ! [Xs: list_P9127870862658512897on_nat,I3: nat] :
      ( ( list_u1840724836389618409on_nat @ Xs @ I3 @ ( nth_Pr3336893728888831618on_nat @ Xs @ I3 ) )
      = Xs ) ).

% list_update_id
thf(fact_143_removeAll__id,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ( ( removeAll_nat @ X2 @ Xs )
        = Xs ) ) ).

% removeAll_id
thf(fact_144_removeAll__id,axiom,
    ! [X2: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ~ ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Xs ) )
     => ( ( remove3990864149796893523od_a_o @ X2 @ Xs )
        = Xs ) ) ).

% removeAll_id
thf(fact_145_removeAll__id,axiom,
    ! [X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ~ ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Xs ) )
     => ( ( remove6084986979817164679on_nat @ X2 @ Xs )
        = Xs ) ) ).

% removeAll_id
thf(fact_146_removeAll__id,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ~ ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Xs ) )
     => ( ( remove433775707702860097on_nat @ X2 @ Xs )
        = Xs ) ) ).

% removeAll_id
thf(fact_147_list__ex__simps_I2_J,axiom,
    ! [P: nat > $o] :
      ~ ( list_ex_nat @ P @ nil_nat ) ).

% list_ex_simps(2)
thf(fact_148_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_149_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_150_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_nat
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_151_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_int
        = ( semiri1314217659103216013at_int @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_152_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M2 )
        = zero_zero_nat )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_153_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( ( semiri1314217659103216013at_int @ M2 )
        = zero_zero_int )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_154_of__nat__less__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_iff
thf(fact_155_of__nat__less__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_iff
thf(fact_156_length__0__conv,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% length_0_conv
thf(fact_157_length__0__conv,axiom,
    ! [Xs: list_S5253404628753938141on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% length_0_conv
thf(fact_158_length__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_nat ) ) ).

% length_0_conv
thf(fact_159_length__0__conv,axiom,
    ! [Xs: list_P9127870862658512897on_nat] :
      ( ( ( size_s7490768528479217773on_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Pr8614093721167239297on_nat ) ) ).

% length_0_conv
thf(fact_160_nth__list__update__eq,axiom,
    ! [I3: nat,Xs: list_l8336264072933094989od_a_o,X2: list_P3196869577824546119od_a_o] :
      ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( nth_li3248255784598043854od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ I3 @ X2 ) @ I3 )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_161_nth__list__update__eq,axiom,
    ! [I3: nat,Xs: list_S5253404628753938141on_nat,X2: sum_su4907927656612468301on_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( nth_Su8471564416988646406on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I3 @ X2 ) @ I3 )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_162_nth__list__update__eq,axiom,
    ! [I3: nat,Xs: list_nat,X2: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ I3 )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_163_nth__list__update__eq,axiom,
    ! [I3: nat,Xs: list_P9127870862658512897on_nat,X2: produc1313295348606329723on_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s7490768528479217773on_nat @ Xs ) )
     => ( ( nth_Pr3336893728888831618on_nat @ ( list_u1840724836389618409on_nat @ Xs @ I3 @ X2 ) @ I3 )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_164_list__update_Osimps_I1_J,axiom,
    ! [I3: nat,V: nat] :
      ( ( list_update_nat @ nil_nat @ I3 @ V )
      = nil_nat ) ).

% list_update.simps(1)
thf(fact_165_list__update__code_I1_J,axiom,
    ! [I3: nat,Y2: nat] :
      ( ( list_update_nat @ nil_nat @ I3 @ Y2 )
      = nil_nat ) ).

% list_update_code(1)
thf(fact_166_removeAll_Osimps_I1_J,axiom,
    ! [X2: nat] :
      ( ( removeAll_nat @ X2 @ nil_nat )
      = nil_nat ) ).

% removeAll.simps(1)
thf(fact_167_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_length_nat @ N @ nil_nat )
      = N ) ).

% gen_length_code(1)
thf(fact_168_count__list_Osimps_I1_J,axiom,
    ! [Y2: nat] :
      ( ( count_list_nat @ nil_nat @ Y2 )
      = zero_zero_nat ) ).

% count_list.simps(1)
thf(fact_169_list__ex__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: nat > $o,G: nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Ys ) )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( list_ex_nat @ F @ Xs )
          = ( list_ex_nat @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_170_list__ex__cong,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,F: list_P3196869577824546119od_a_o > $o,G: list_P3196869577824546119od_a_o > $o] :
      ( ( Xs = Ys )
     => ( ! [X: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X @ ( set_li2629784761229999580od_a_o @ Ys ) )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( list_e5403782194016407179od_a_o @ F @ Xs )
          = ( list_e5403782194016407179od_a_o @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_171_list__ex__cong,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,F: produc1313295348606329723on_nat > $o,G: produc1313295348606329723on_nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X: produc1313295348606329723on_nat] :
            ( ( member5906418179908956708on_nat @ X @ ( set_Pr937530902923597456on_nat @ Ys ) )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( list_e8248363684495001279on_nat @ F @ Xs )
          = ( list_e8248363684495001279on_nat @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_172_list__ex__cong,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,F: sum_su4907927656612468301on_nat > $o,G: sum_su4907927656612468301on_nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X @ ( set_Su4848624856859743608on_nat @ Ys ) )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( list_e3769318529992910857on_nat @ F @ Xs )
          = ( list_e3769318529992910857on_nat @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_173_of__nat__less__0__iff,axiom,
    ! [M2: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_174_of__nat__less__0__iff,axiom,
    ! [M2: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_175_of__nat__less__imp__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
     => ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_imp_less
thf(fact_176_of__nat__less__imp__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
     => ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_imp_less
thf(fact_177_less__imp__of__nat__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_178_less__imp__of__nat__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_179_list_Osize_I3_J,axiom,
    ( ( size_s3379185846251495609od_a_o @ nil_li1372968520647675725od_a_o )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_180_list_Osize_I3_J,axiom,
    ( ( size_s5686234912572693873on_nat @ nil_Su3078906344068544711on_nat )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_181_list_Osize_I3_J,axiom,
    ( ( size_size_list_nat @ nil_nat )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_182_list_Osize_I3_J,axiom,
    ( ( size_s7490768528479217773on_nat @ nil_Pr8614093721167239297on_nat )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_183_upt__0,axiom,
    ! [I3: nat] :
      ( ( upt @ I3 @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_184_count__list__0__iff,axiom,
    ! [Xs: list_nat,X2: nat] :
      ( ( ( count_list_nat @ Xs @ X2 )
        = zero_zero_nat )
      = ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) ) ) ) ).

% count_list_0_iff
thf(fact_185_count__list__0__iff,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X2: list_P3196869577824546119od_a_o] :
      ( ( ( count_5344270436093135555od_a_o @ Xs @ X2 )
        = zero_zero_nat )
      = ( ~ ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Xs ) ) ) ) ).

% count_list_0_iff
thf(fact_186_count__list__0__iff,axiom,
    ! [Xs: list_P9127870862658512897on_nat,X2: produc1313295348606329723on_nat] :
      ( ( ( count_9157224493928226551on_nat @ Xs @ X2 )
        = zero_zero_nat )
      = ( ~ ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Xs ) ) ) ) ).

% count_list_0_iff
thf(fact_187_count__list__0__iff,axiom,
    ! [Xs: list_S5253404628753938141on_nat,X2: sum_su4907927656612468301on_nat] :
      ( ( ( count_3605398483369874897on_nat @ Xs @ X2 )
        = zero_zero_nat )
      = ( ~ ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Xs ) ) ) ) ).

% count_list_0_iff
thf(fact_188_set__update__memI,axiom,
    ! [N: nat,Xs: list_l8336264072933094989od_a_o,X2: list_P3196869577824546119od_a_o] :
      ( ( ord_less_nat @ N @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ N @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_189_set__update__memI,axiom,
    ! [N: nat,Xs: list_S5253404628753938141on_nat,X2: sum_su4907927656612468301on_nat] :
      ( ( ord_less_nat @ N @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ ( list_u8577817230714425439on_nat @ Xs @ N @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_190_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X2: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_191_set__update__memI,axiom,
    ! [N: nat,Xs: list_P9127870862658512897on_nat,X2: produc1313295348606329723on_nat] :
      ( ( ord_less_nat @ N @ ( size_s7490768528479217773on_nat @ Xs ) )
     => ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ ( list_u1840724836389618409on_nat @ Xs @ N @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_192_list__update__same__conv,axiom,
    ! [I3: nat,Xs: list_l8336264072933094989od_a_o,X2: list_P3196869577824546119od_a_o] :
      ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( ( list_u329291161401623989od_a_o @ Xs @ I3 @ X2 )
          = Xs )
        = ( ( nth_li3248255784598043854od_a_o @ Xs @ I3 )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_193_list__update__same__conv,axiom,
    ! [I3: nat,Xs: list_S5253404628753938141on_nat,X2: sum_su4907927656612468301on_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( ( list_u8577817230714425439on_nat @ Xs @ I3 @ X2 )
          = Xs )
        = ( ( nth_Su8471564416988646406on_nat @ Xs @ I3 )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_194_list__update__same__conv,axiom,
    ! [I3: nat,Xs: list_nat,X2: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
     => ( ( ( list_update_nat @ Xs @ I3 @ X2 )
          = Xs )
        = ( ( nth_nat @ Xs @ I3 )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_195_list__update__same__conv,axiom,
    ! [I3: nat,Xs: list_P9127870862658512897on_nat,X2: produc1313295348606329723on_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s7490768528479217773on_nat @ Xs ) )
     => ( ( ( list_u1840724836389618409on_nat @ Xs @ I3 @ X2 )
          = Xs )
        = ( ( nth_Pr3336893728888831618on_nat @ Xs @ I3 )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_196_nth__list__update,axiom,
    ! [I3: nat,Xs: list_l8336264072933094989od_a_o,J: nat,X2: list_P3196869577824546119od_a_o] :
      ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( ( I3 = J )
         => ( ( nth_li3248255784598043854od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ I3 @ X2 ) @ J )
            = X2 ) )
        & ( ( I3 != J )
         => ( ( nth_li3248255784598043854od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ I3 @ X2 ) @ J )
            = ( nth_li3248255784598043854od_a_o @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_197_nth__list__update,axiom,
    ! [I3: nat,Xs: list_S5253404628753938141on_nat,J: nat,X2: sum_su4907927656612468301on_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( ( I3 = J )
         => ( ( nth_Su8471564416988646406on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I3 @ X2 ) @ J )
            = X2 ) )
        & ( ( I3 != J )
         => ( ( nth_Su8471564416988646406on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I3 @ X2 ) @ J )
            = ( nth_Su8471564416988646406on_nat @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_198_nth__list__update,axiom,
    ! [I3: nat,Xs: list_nat,J: nat,X2: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
     => ( ( ( I3 = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ J )
            = X2 ) )
        & ( ( I3 != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ J )
            = ( nth_nat @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_199_nth__list__update,axiom,
    ! [I3: nat,Xs: list_P9127870862658512897on_nat,J: nat,X2: produc1313295348606329723on_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s7490768528479217773on_nat @ Xs ) )
     => ( ( ( I3 = J )
         => ( ( nth_Pr3336893728888831618on_nat @ ( list_u1840724836389618409on_nat @ Xs @ I3 @ X2 ) @ J )
            = X2 ) )
        & ( ( I3 != J )
         => ( ( nth_Pr3336893728888831618on_nat @ ( list_u1840724836389618409on_nat @ Xs @ I3 @ X2 ) @ J )
            = ( nth_Pr3336893728888831618on_nat @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_200_size__char__eq__0,axiom,
    ( size_size_char
    = ( ^ [C: char] : zero_zero_nat ) ) ).

% size_char_eq_0
thf(fact_201_prod_Oinject,axiom,
    ! [X1: sum_su4907927656612468301on_nat,X22: sum_su4907927656612468301on_nat,Y1: sum_su4907927656612468301on_nat,Y22: sum_su4907927656612468301on_nat] :
      ( ( ( produc6704844767664046899on_nat @ X1 @ X22 )
        = ( produc6704844767664046899on_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_202_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_203_old_Oprod_Oinject,axiom,
    ! [A: sum_su4907927656612468301on_nat,B: sum_su4907927656612468301on_nat,A3: sum_su4907927656612468301on_nat,B2: sum_su4907927656612468301on_nat] :
      ( ( ( produc6704844767664046899on_nat @ A @ B )
        = ( produc6704844767664046899on_nat @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_204_old_Oprod_Oinject,axiom,
    ! [A: nat > nat,B: nat,A3: nat > nat,B2: nat] :
      ( ( ( produc72220940542539688at_nat @ A @ B )
        = ( produc72220940542539688at_nat @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_205_pos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ~ ! [N2: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% pos_int_cases
thf(fact_206_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ? [N2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N2 )
          & ( K
            = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_207_listrel1__iff__update,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Xs @ Ys ) @ ( listre4186574385322583590od_a_o @ R ) )
      = ( ? [Y3: list_P3196869577824546119od_a_o,N3: nat] :
            ( ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ Xs @ N3 ) @ Y3 ) @ R )
            & ( ord_less_nat @ N3 @ ( size_s3379185846251495609od_a_o @ Xs ) )
            & ( Ys
              = ( list_u329291161401623989od_a_o @ Xs @ N3 @ Y3 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_208_listrel1__iff__update,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ Ys ) @ ( listre1270753918075970734on_nat @ R ) )
      = ( ? [Y3: sum_su4907927656612468301on_nat,N3: nat] :
            ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ Xs @ N3 ) @ Y3 ) @ R )
            & ( ord_less_nat @ N3 @ ( size_s5686234912572693873on_nat @ Xs ) )
            & ( Ys
              = ( list_u8577817230714425439on_nat @ Xs @ N3 @ Y3 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_209_listrel1__iff__update,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
      = ( ? [Y3: nat,N3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N3 ) @ Y3 ) @ R )
            & ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs ) )
            & ( Ys
              = ( list_update_nat @ Xs @ N3 @ Y3 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_210_listrel1__iff__update,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( listre7129750038259667930on_nat @ R ) )
      = ( ? [Y3: produc1313295348606329723on_nat,N3: nat] :
            ( ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ ( nth_Pr3336893728888831618on_nat @ Xs @ N3 ) @ Y3 ) @ R )
            & ( ord_less_nat @ N3 @ ( size_s7490768528479217773on_nat @ Xs ) )
            & ( Ys
              = ( list_u1840724836389618409on_nat @ Xs @ N3 @ Y3 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_211_distinct__swap,axiom,
    ! [I3: nat,Xs: list_l8336264072933094989od_a_o,J: nat] :
      ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s3379185846251495609od_a_o @ Xs ) )
       => ( ( distin690952206374581440od_a_o @ ( list_u329291161401623989od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ I3 @ ( nth_li3248255784598043854od_a_o @ Xs @ J ) ) @ J @ ( nth_li3248255784598043854od_a_o @ Xs @ I3 ) ) )
          = ( distin690952206374581440od_a_o @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_212_distinct__swap,axiom,
    ! [I3: nat,Xs: list_S5253404628753938141on_nat,J: nat] :
      ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s5686234912572693873on_nat @ Xs ) )
       => ( ( distin2773651816175405460on_nat @ ( list_u8577817230714425439on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I3 @ ( nth_Su8471564416988646406on_nat @ Xs @ J ) ) @ J @ ( nth_Su8471564416988646406on_nat @ Xs @ I3 ) ) )
          = ( distin2773651816175405460on_nat @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_213_distinct__swap,axiom,
    ! [I3: nat,Xs: list_nat,J: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
       => ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs @ I3 @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I3 ) ) )
          = ( distinct_nat @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_214_distinct__swap,axiom,
    ! [I3: nat,Xs: list_P9127870862658512897on_nat,J: nat] :
      ( ( ord_less_nat @ I3 @ ( size_s7490768528479217773on_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s7490768528479217773on_nat @ Xs ) )
       => ( ( distin8410642530753306484on_nat @ ( list_u1840724836389618409on_nat @ ( list_u1840724836389618409on_nat @ Xs @ I3 @ ( nth_Pr3336893728888831618on_nat @ Xs @ J ) ) @ J @ ( nth_Pr3336893728888831618on_nat @ Xs @ I3 ) ) )
          = ( distin8410642530753306484on_nat @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_215_remove__code_I1_J,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( remove_nat @ X2 @ ( set_nat2 @ Xs ) )
      = ( set_nat2 @ ( removeAll_nat @ X2 @ Xs ) ) ) ).

% remove_code(1)
thf(fact_216_remove__code_I1_J,axiom,
    ! [X2: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( remove8508835416573194028od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Xs ) )
      = ( set_li2629784761229999580od_a_o @ ( remove3990864149796893523od_a_o @ X2 @ Xs ) ) ) ).

% remove_code(1)
thf(fact_217_remove__code_I1_J,axiom,
    ! [X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( remove4248810732159152096on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Xs ) )
      = ( set_Pr937530902923597456on_nat @ ( remove6084986979817164679on_nat @ X2 @ Xs ) ) ) ).

% remove_code(1)
thf(fact_218_remove__code_I1_J,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( remove3644547252183752744on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Xs ) )
      = ( set_Su4848624856859743608on_nat @ ( remove433775707702860097on_nat @ X2 @ Xs ) ) ) ).

% remove_code(1)
thf(fact_219_int__int__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M2 )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M2 = N ) ) ).

% int_int_eq
thf(fact_220_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [Xs: list_nat] :
      ( ( distinct_nat @ Xs )
     => ( distinct_nat @ Xs ) ) ).

% sorted_list_of_set.distinct_if_distinct_map
thf(fact_221_less__int__code_I1_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_int_code(1)
thf(fact_222_not__Nil__listrel1,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel1_nat @ R ) ) ).

% not_Nil_listrel1
thf(fact_223_not__listrel1__Nil,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel1_nat @ R ) ) ).

% not_listrel1_Nil
thf(fact_224_listrel1__eq__len,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Xs @ Ys ) @ ( listre4186574385322583590od_a_o @ R ) )
     => ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_225_listrel1__eq__len,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ Ys ) @ ( listre1270753918075970734on_nat @ R ) )
     => ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_226_listrel1__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_227_listrel1__eq__len,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( listre7129750038259667930on_nat @ R ) )
     => ( ( size_s7490768528479217773on_nat @ Xs )
        = ( size_s7490768528479217773on_nat @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_228_distinct_Osimps_I1_J,axiom,
    distinct_nat @ nil_nat ).

% distinct.simps(1)
thf(fact_229_distinct__upt,axiom,
    ! [I3: nat,J: nat] : ( distinct_nat @ ( upt @ I3 @ J ) ) ).

% distinct_upt
thf(fact_230_distinct__product__lists,axiom,
    ! [Xss: list_list_nat] :
      ( ! [X: list_nat] :
          ( ( member_list_nat @ X @ ( set_list_nat2 @ Xss ) )
         => ( distinct_nat @ X ) )
     => ( distinct_list_nat @ ( product_lists_nat @ Xss ) ) ) ).

% distinct_product_lists
thf(fact_231_distinct__product__lists,axiom,
    ! [Xss: list_l8336264072933094989od_a_o] :
      ( ! [X: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ X @ ( set_li2629784761229999580od_a_o @ Xss ) )
         => ( distin2380507709364613050od_a_o @ X ) )
     => ( distin690952206374581440od_a_o @ ( produc3814538932794474544od_a_o @ Xss ) ) ) ).

% distinct_product_lists
thf(fact_232_distinct__removeAll,axiom,
    ! [Xs: list_nat,X2: nat] :
      ( ( distinct_nat @ Xs )
     => ( distinct_nat @ ( removeAll_nat @ X2 @ Xs ) ) ) ).

% distinct_removeAll
thf(fact_233_distinct__conv__nth,axiom,
    ( distin690952206374581440od_a_o
    = ( ^ [Xs2: list_l8336264072933094989od_a_o] :
        ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Xs2 ) )
         => ! [J2: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s3379185846251495609od_a_o @ Xs2 ) )
             => ( ( I != J2 )
               => ( ( nth_li3248255784598043854od_a_o @ Xs2 @ I )
                 != ( nth_li3248255784598043854od_a_o @ Xs2 @ J2 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_234_distinct__conv__nth,axiom,
    ( distin2773651816175405460on_nat
    = ( ^ [Xs2: list_S5253404628753938141on_nat] :
        ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Xs2 ) )
         => ! [J2: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s5686234912572693873on_nat @ Xs2 ) )
             => ( ( I != J2 )
               => ( ( nth_Su8471564416988646406on_nat @ Xs2 @ I )
                 != ( nth_Su8471564416988646406on_nat @ Xs2 @ J2 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_235_distinct__conv__nth,axiom,
    ( distinct_nat
    = ( ^ [Xs2: list_nat] :
        ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
         => ! [J2: nat] :
              ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
             => ( ( I != J2 )
               => ( ( nth_nat @ Xs2 @ I )
                 != ( nth_nat @ Xs2 @ J2 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_236_distinct__conv__nth,axiom,
    ( distin8410642530753306484on_nat
    = ( ^ [Xs2: list_P9127870862658512897on_nat] :
        ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s7490768528479217773on_nat @ Xs2 ) )
         => ! [J2: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s7490768528479217773on_nat @ Xs2 ) )
             => ( ( I != J2 )
               => ( ( nth_Pr3336893728888831618on_nat @ Xs2 @ I )
                 != ( nth_Pr3336893728888831618on_nat @ Xs2 @ J2 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_237_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,I3: nat,J: nat] :
      ( ( distin690952206374581440od_a_o @ Xs )
     => ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_s3379185846251495609od_a_o @ Xs ) )
         => ( ( ( nth_li3248255784598043854od_a_o @ Xs @ I3 )
              = ( nth_li3248255784598043854od_a_o @ Xs @ J ) )
            = ( I3 = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_238_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_S5253404628753938141on_nat,I3: nat,J: nat] :
      ( ( distin2773651816175405460on_nat @ Xs )
     => ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_s5686234912572693873on_nat @ Xs ) )
         => ( ( ( nth_Su8471564416988646406on_nat @ Xs @ I3 )
              = ( nth_Su8471564416988646406on_nat @ Xs @ J ) )
            = ( I3 = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_239_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_nat,I3: nat,J: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
         => ( ( ( nth_nat @ Xs @ I3 )
              = ( nth_nat @ Xs @ J ) )
            = ( I3 = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_240_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_P9127870862658512897on_nat,I3: nat,J: nat] :
      ( ( distin8410642530753306484on_nat @ Xs )
     => ( ( ord_less_nat @ I3 @ ( size_s7490768528479217773on_nat @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_s7490768528479217773on_nat @ Xs ) )
         => ( ( ( nth_Pr3336893728888831618on_nat @ Xs @ I3 )
              = ( nth_Pr3336893728888831618on_nat @ Xs @ J ) )
            = ( I3 = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_241_Pair__inject,axiom,
    ! [A: sum_su4907927656612468301on_nat,B: sum_su4907927656612468301on_nat,A3: sum_su4907927656612468301on_nat,B2: sum_su4907927656612468301on_nat] :
      ( ( ( produc6704844767664046899on_nat @ A @ B )
        = ( produc6704844767664046899on_nat @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_242_Pair__inject,axiom,
    ! [A: nat > nat,B: nat,A3: nat > nat,B2: nat] :
      ( ( ( produc72220940542539688at_nat @ A @ B )
        = ( produc72220940542539688at_nat @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_243_prod__cases,axiom,
    ! [P: produc1313295348606329723on_nat > $o,P3: produc1313295348606329723on_nat] :
      ( ! [A4: sum_su4907927656612468301on_nat,B3: sum_su4907927656612468301on_nat] : ( P @ ( produc6704844767664046899on_nat @ A4 @ B3 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_244_prod__cases,axiom,
    ! [P: produc8199716216217303280at_nat > $o,P3: produc8199716216217303280at_nat] :
      ( ! [A4: nat > nat,B3: nat] : ( P @ ( produc72220940542539688at_nat @ A4 @ B3 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_245_surj__pair,axiom,
    ! [P3: produc1313295348606329723on_nat] :
    ? [X: sum_su4907927656612468301on_nat,Y4: sum_su4907927656612468301on_nat] :
      ( P3
      = ( produc6704844767664046899on_nat @ X @ Y4 ) ) ).

% surj_pair
thf(fact_246_surj__pair,axiom,
    ! [P3: produc8199716216217303280at_nat] :
    ? [X: nat > nat,Y4: nat] :
      ( P3
      = ( produc72220940542539688at_nat @ X @ Y4 ) ) ).

% surj_pair
thf(fact_247_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc1313295348606329723on_nat] :
      ~ ! [A4: sum_su4907927656612468301on_nat,B3: sum_su4907927656612468301on_nat] :
          ( Y2
         != ( produc6704844767664046899on_nat @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_248_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc8199716216217303280at_nat] :
      ~ ! [A4: nat > nat,B3: nat] :
          ( Y2
         != ( produc72220940542539688at_nat @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_249_distinct__Ex1,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X2: list_P3196869577824546119od_a_o] :
      ( ( distin690952206374581440od_a_o @ Xs )
     => ( ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Xs ) )
       => ? [X: nat] :
            ( ( ord_less_nat @ X @ ( size_s3379185846251495609od_a_o @ Xs ) )
            & ( ( nth_li3248255784598043854od_a_o @ Xs @ X )
              = X2 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_s3379185846251495609od_a_o @ Xs ) )
                  & ( ( nth_li3248255784598043854od_a_o @ Xs @ Y5 )
                    = X2 ) )
               => ( Y5 = X ) ) ) ) ) ).

% distinct_Ex1
thf(fact_250_distinct__Ex1,axiom,
    ! [Xs: list_S5253404628753938141on_nat,X2: sum_su4907927656612468301on_nat] :
      ( ( distin2773651816175405460on_nat @ Xs )
     => ( ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Xs ) )
       => ? [X: nat] :
            ( ( ord_less_nat @ X @ ( size_s5686234912572693873on_nat @ Xs ) )
            & ( ( nth_Su8471564416988646406on_nat @ Xs @ X )
              = X2 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_s5686234912572693873on_nat @ Xs ) )
                  & ( ( nth_Su8471564416988646406on_nat @ Xs @ Y5 )
                    = X2 ) )
               => ( Y5 = X ) ) ) ) ) ).

% distinct_Ex1
thf(fact_251_distinct__Ex1,axiom,
    ! [Xs: list_nat,X2: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
       => ? [X: nat] :
            ( ( ord_less_nat @ X @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ X )
              = X2 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_size_list_nat @ Xs ) )
                  & ( ( nth_nat @ Xs @ Y5 )
                    = X2 ) )
               => ( Y5 = X ) ) ) ) ) ).

% distinct_Ex1
thf(fact_252_distinct__Ex1,axiom,
    ! [Xs: list_P9127870862658512897on_nat,X2: produc1313295348606329723on_nat] :
      ( ( distin8410642530753306484on_nat @ Xs )
     => ( ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Xs ) )
       => ? [X: nat] :
            ( ( ord_less_nat @ X @ ( size_s7490768528479217773on_nat @ Xs ) )
            & ( ( nth_Pr3336893728888831618on_nat @ Xs @ X )
              = X2 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_s7490768528479217773on_nat @ Xs ) )
                  & ( ( nth_Pr3336893728888831618on_nat @ Xs @ Y5 )
                    = X2 ) )
               => ( Y5 = X ) ) ) ) ) ).

% distinct_Ex1
thf(fact_253_size_H__char__eq__0,axiom,
    ( size_char
    = ( ^ [C: char] : zero_zero_nat ) ) ).

% size'_char_eq_0
thf(fact_254_member__remove,axiom,
    ! [X2: nat,Y2: nat,A2: set_nat] :
      ( ( member_nat @ X2 @ ( remove_nat @ Y2 @ A2 ) )
      = ( ( member_nat @ X2 @ A2 )
        & ( X2 != Y2 ) ) ) ).

% member_remove
thf(fact_255_member__remove,axiom,
    ! [X2: list_P3196869577824546119od_a_o,Y2: list_P3196869577824546119od_a_o,A2: set_li5011860890973139239od_a_o] :
      ( ( member5261019209154752368od_a_o @ X2 @ ( remove8508835416573194028od_a_o @ Y2 @ A2 ) )
      = ( ( member5261019209154752368od_a_o @ X2 @ A2 )
        & ( X2 != Y2 ) ) ) ).

% member_remove
thf(fact_256_member__remove,axiom,
    ! [X2: produc1313295348606329723on_nat,Y2: produc1313295348606329723on_nat,A2: set_Pr2535564044409765851on_nat] :
      ( ( member5906418179908956708on_nat @ X2 @ ( remove4248810732159152096on_nat @ Y2 @ A2 ) )
      = ( ( member5906418179908956708on_nat @ X2 @ A2 )
        & ( X2 != Y2 ) ) ) ).

% member_remove
thf(fact_257_member__remove,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Y2: sum_su4907927656612468301on_nat,A2: set_Su266273576555847555on_nat] :
      ( ( member8476201367316484836on_nat @ X2 @ ( remove3644547252183752744on_nat @ Y2 @ A2 ) )
      = ( ( member8476201367316484836on_nat @ X2 @ A2 )
        & ( X2 != Y2 ) ) ) ).

% member_remove
thf(fact_258_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_259_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_260_distinct__n__lists,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( distinct_nat @ Xs )
     => ( distinct_list_nat @ ( n_lists_nat @ N @ Xs ) ) ) ).

% distinct_n_lists
thf(fact_261_distinct__union,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( distinct_nat @ ( union_nat @ Xs @ Ys ) )
      = ( distinct_nat @ Ys ) ) ).

% distinct_union
thf(fact_262_sorted__list__of__multiset__empty,axiom,
    ( ( linord3047872887403683810et_nat @ zero_z7348594199698428585et_nat )
    = nil_nat ) ).

% sorted_list_of_multiset_empty
thf(fact_263_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N3 ) @ ( nth_nat @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_264_listrel__iff__nth,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat,R: set_Pr9093778441882193744at_nat] :
      ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs @ Ys ) @ ( listrel_nat_nat_nat @ R ) )
      = ( ( ( size_s8208510060688613859at_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s8208510060688613859at_nat @ Xs ) )
           => ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ ( nth_nat_nat @ Xs @ N3 ) @ ( nth_nat @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_265_listrel__iff__nth,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_nat,R: set_Pr8850484174313502844_o_nat] :
      ( ( member1162260694010923379st_nat @ ( produc4688861313923565270st_nat @ Xs @ Ys ) @ ( listre90293961589800166_o_nat @ R ) )
      = ( ( ( size_s3379185846251495609od_a_o @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s3379185846251495609od_a_o @ Xs ) )
           => ( member7885718254550376541_o_nat @ ( produc3563005473579536832_o_nat @ ( nth_li3248255784598043854od_a_o @ Xs @ N3 ) @ ( nth_nat @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_266_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_l8336264072933094989od_a_o,R: set_Pr94232683564258082od_a_o] :
      ( ( member548352139293716857od_a_o @ ( produc8901617039433855636od_a_o @ Xs @ Ys ) @ ( listre473352748755709764od_a_o @ R ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_s3379185846251495609od_a_o @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs ) )
           => ( member8910825520198479363od_a_o @ ( produc3946064260745446430od_a_o @ ( nth_nat @ Xs @ N3 ) @ ( nth_li3248255784598043854od_a_o @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_267_listrel__iff__nth,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_nat,R: set_Pr3345771464143190622at_nat] :
      ( ( member380504172838623943st_nat @ ( produc845777339779355990st_nat @ Xs @ Ys ) @ ( listre6443138976769382928at_nat @ R ) )
      = ( ( ( size_s5686234912572693873on_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s5686234912572693873on_nat @ Xs ) )
           => ( member8820252441731870247at_nat @ ( produc7500644556467579062at_nat @ ( nth_Su8471564416988646406on_nat @ Xs @ N3 ) @ ( nth_nat @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_268_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_S5253404628753938141on_nat,R: set_Pr7552745763645690846on_nat] :
      ( ( member452587771129616071on_nat @ ( produc9138095300642541526on_nat @ Xs @ Ys ) @ ( listre7340406004007350800on_nat @ R ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_s5686234912572693873on_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs ) )
           => ( member7352503068387395495on_nat @ ( produc8397911583705546934on_nat @ ( nth_nat @ Xs @ N3 ) @ ( nth_Su8471564416988646406on_nat @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_269_listrel__iff__nth,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Xs @ Ys ) @ ( listre9037316752283237297od_a_o @ R ) )
      = ( ( ( size_s3379185846251495609od_a_o @ Xs )
          = ( size_s3379185846251495609od_a_o @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s3379185846251495609od_a_o @ Xs ) )
           => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ Xs @ N3 ) @ ( nth_li3248255784598043854od_a_o @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_270_listrel__iff__nth,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_S5253404628753938141on_nat,R: set_Pr6144483798281041401on_nat] :
      ( ( member4418620086355165936on_nat @ ( produc7184742879426471635on_nat @ Xs @ Ys ) @ ( listre7573934803181360995on_nat @ R ) )
      = ( ( ( size_s3379185846251495609od_a_o @ Xs )
          = ( size_s5686234912572693873on_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s3379185846251495609od_a_o @ Xs ) )
           => ( member1833810847379340378on_nat @ ( produc1428451187817174077on_nat @ ( nth_li3248255784598043854od_a_o @ Xs @ N3 ) @ ( nth_Su8471564416988646406on_nat @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_271_listrel__iff__nth,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_l8336264072933094989od_a_o,R: set_Pr9140199768791955813od_a_o] :
      ( ( member2603097825888147004od_a_o @ ( produc1975650536614756183od_a_o @ Xs @ Ys ) @ ( listre1105410954321393415od_a_o @ R ) )
      = ( ( ( size_s5686234912572693873on_nat @ Xs )
          = ( size_s3379185846251495609od_a_o @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s5686234912572693873on_nat @ Xs ) )
           => ( member7595723914438784454od_a_o @ ( produc4183299375811982305od_a_o @ ( nth_Su8471564416988646406on_nat @ Xs @ N3 ) @ ( nth_li3248255784598043854od_a_o @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_272_listrel__iff__nth,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ Ys ) @ ( listre5398654966139722893on_nat @ R ) )
      = ( ( ( size_s5686234912572693873on_nat @ Xs )
          = ( size_s5686234912572693873on_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s5686234912572693873on_nat @ Xs ) )
           => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ Xs @ N3 ) @ ( nth_Su8471564416988646406on_nat @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_273_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Pr7420778282784443591od_a_o > set_Pr7420778282784443591od_a_o > set_Pr6888711208267297863od_a_o,Ns_list: set_Pr7420778282784443591od_a_o > set_Pr7420778282784443591od_a_o > set_Pr6888711208267297863od_a_o,As: list_l8336264072933094989od_a_o,Bs: list_l8336264072933094989od_a_o,NS: set_Pr7420778282784443591od_a_o,S2: set_Pr7420778282784443591od_a_o] :
      ( ( list_l1016356893908098451od_a_o @ S_list @ Ns_list )
     => ( ( ( size_s3379185846251495609od_a_o @ As )
          = ( size_s3379185846251495609od_a_o @ Bs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s3379185846251495609od_a_o @ Bs ) )
             => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ As @ I2 ) @ ( nth_li3248255784598043854od_a_o @ Bs @ I2 ) ) @ NS ) )
         => ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ As @ Bs ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_274_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Pr2535564044409765851on_nat > set_Pr2535564044409765851on_nat > set_Pr7828957961435280891on_nat,Ns_list: set_Pr2535564044409765851on_nat > set_Pr2535564044409765851on_nat > set_Pr7828957961435280891on_nat,As: list_S5253404628753938141on_nat,Bs: list_S5253404628753938141on_nat,NS: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat] :
      ( ( list_l4998409290742832897on_nat @ S_list @ Ns_list )
     => ( ( ( size_s5686234912572693873on_nat @ As )
          = ( size_s5686234912572693873on_nat @ Bs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s5686234912572693873on_nat @ Bs ) )
             => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ As @ I2 ) @ ( nth_Su8471564416988646406on_nat @ Bs @ I2 ) ) @ NS ) )
         => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ As @ Bs ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_275_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,Ns_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,As: list_nat,Bs: list_nat,NS: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( list_l792762466888043652on_nat @ S_list @ Ns_list )
     => ( ( ( size_size_list_nat @ As )
          = ( size_size_list_nat @ Bs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Bs ) )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) @ NS ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As @ Bs ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_276_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Pr8202701763255893447on_nat > set_Pr8202701763255893447on_nat > set_Pr8586234676504894535on_nat,Ns_list: set_Pr8202701763255893447on_nat > set_Pr8202701763255893447on_nat > set_Pr8586234676504894535on_nat,As: list_P9127870862658512897on_nat,Bs: list_P9127870862658512897on_nat,NS: set_Pr8202701763255893447on_nat,S2: set_Pr8202701763255893447on_nat] :
      ( ( list_l5400313399564392903on_nat @ S_list @ Ns_list )
     => ( ( ( size_s7490768528479217773on_nat @ As )
          = ( size_s7490768528479217773on_nat @ Bs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s7490768528479217773on_nat @ Bs ) )
             => ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ ( nth_Pr3336893728888831618on_nat @ As @ I2 ) @ ( nth_Pr3336893728888831618on_nat @ Bs @ I2 ) ) @ NS ) )
         => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ As @ Bs ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_277_listrel__Nil2,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel_nat_nat @ R ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil2
thf(fact_278_listrel__Nil1,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel_nat_nat @ R ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil1
thf(fact_279_listrel_ONil,axiom,
    ! [R: set_Pr1261947904930325089at_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) @ ( listrel_nat_nat @ R ) ) ).

% listrel.Nil
thf(fact_280_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_281_listrel__eq__len,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_nat,R: set_Pr8850484174313502844_o_nat] :
      ( ( member1162260694010923379st_nat @ ( produc4688861313923565270st_nat @ Xs @ Ys ) @ ( listre90293961589800166_o_nat @ R ) )
     => ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_282_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_l8336264072933094989od_a_o,R: set_Pr94232683564258082od_a_o] :
      ( ( member548352139293716857od_a_o @ ( produc8901617039433855636od_a_o @ Xs @ Ys ) @ ( listre473352748755709764od_a_o @ R ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_283_listrel__eq__len,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_nat,R: set_Pr3345771464143190622at_nat] :
      ( ( member380504172838623943st_nat @ ( produc845777339779355990st_nat @ Xs @ Ys ) @ ( listre6443138976769382928at_nat @ R ) )
     => ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_284_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_S5253404628753938141on_nat,R: set_Pr7552745763645690846on_nat] :
      ( ( member452587771129616071on_nat @ ( produc9138095300642541526on_nat @ Xs @ Ys ) @ ( listre7340406004007350800on_nat @ R ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_285_listrel__eq__len,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Xs @ Ys ) @ ( listre9037316752283237297od_a_o @ R ) )
     => ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_286_listrel__eq__len,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_S5253404628753938141on_nat,R: set_Pr6144483798281041401on_nat] :
      ( ( member4418620086355165936on_nat @ ( produc7184742879426471635on_nat @ Xs @ Ys ) @ ( listre7573934803181360995on_nat @ R ) )
     => ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_287_listrel__eq__len,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_l8336264072933094989od_a_o,R: set_Pr9140199768791955813od_a_o] :
      ( ( member2603097825888147004od_a_o @ ( produc1975650536614756183od_a_o @ Xs @ Ys ) @ ( listre1105410954321393415od_a_o @ R ) )
     => ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_288_listrel__eq__len,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ Ys ) @ ( listre5398654966139722893on_nat @ R ) )
     => ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_289_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr2972819075928174550on_nat] :
      ( ( member9199622813682680237on_nat @ ( produc8802776732007976904on_nat @ Xs @ Ys ) @ ( listre7269971596283755512on_nat @ R ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s7490768528479217773on_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_290_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_291_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_292_int__if,axiom,
    ! [P: $o,A: nat,B: nat] :
      ( ( P
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
          = ( semiri1314217659103216013at_int @ A ) ) )
      & ( ~ P
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
          = ( semiri1314217659103216013at_int @ B ) ) ) ) ).

% int_if
thf(fact_293_nat__int__comparison_I1_J,axiom,
    ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
    = ( ^ [A5: nat,B4: nat] :
          ( ( semiri1314217659103216013at_int @ A5 )
          = ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_int_comparison(1)
thf(fact_294_neg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ~ ! [N2: nat] :
            ( ( K
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% neg_int_cases
thf(fact_295_horner__sum__simps_I1_J,axiom,
    ! [F: nat > nat,A: nat] :
      ( ( groups7488368174851004413at_nat @ F @ A @ nil_nat )
      = zero_zero_nat ) ).

% horner_sum_simps(1)
thf(fact_296_horner__sum__simps_I1_J,axiom,
    ! [F: nat > int,A: int] :
      ( ( groups7485877704341954137at_int @ F @ A @ nil_nat )
      = zero_zero_int ) ).

% horner_sum_simps(1)
thf(fact_297_zero__less__nat__eq,axiom,
    ! [Z2: int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
      = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).

% zero_less_nat_eq
thf(fact_298_upt__eq__Nil__conv,axiom,
    ! [I3: nat,J: nat] :
      ( ( ( upt @ I3 @ J )
        = nil_nat )
      = ( ( J = zero_zero_nat )
        | ( ord_less_eq_nat @ J @ I3 ) ) ) ).

% upt_eq_Nil_conv
thf(fact_299_list__ex1__simps_I1_J,axiom,
    ! [P: nat > $o] :
      ~ ( list_ex1_nat @ P @ nil_nat ) ).

% list_ex1_simps(1)
thf(fact_300_verit__minus__simplify_I4_J,axiom,
    ! [B: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
      = B ) ).

% verit_minus_simplify(4)
thf(fact_301_add_Oinverse__inverse,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_302_neg__equal__iff__equal,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = ( uminus_uminus_int @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_303_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_304_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_305_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = A )
      = ( A = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_306_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A
        = ( uminus_uminus_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_307_neg__equal__0__iff__equal,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
thf(fact_308_neg__0__equal__iff__equal,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( uminus_uminus_int @ A ) )
      = ( zero_zero_int = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_309_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_310_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_311_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_312_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_313_nat__int,axiom,
    ! [N: nat] :
      ( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
      = N ) ).

% nat_int
thf(fact_314_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_315_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% less_eq_neg_nonpos
thf(fact_316_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% neg_le_0_iff_le
thf(fact_317_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% neg_0_le_iff_le
thf(fact_318_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% neg_less_0_iff_less
thf(fact_319_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% neg_0_less_iff_less
thf(fact_320_neg__less__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% neg_less_pos
thf(fact_321_less__neg__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% less_neg_neg
thf(fact_322_of__nat__le__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% of_nat_le_iff
thf(fact_323_of__nat__le__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% of_nat_le_iff
thf(fact_324_negative__eq__positive,axiom,
    ! [N: nat,M2: nat] :
      ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
        = ( semiri1314217659103216013at_int @ M2 ) )
      = ( ( N = zero_zero_nat )
        & ( M2 = zero_zero_nat ) ) ) ).

% negative_eq_positive
thf(fact_325_list__update__beyond,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,I3: nat,X2: list_P3196869577824546119od_a_o] :
      ( ( ord_less_eq_nat @ ( size_s3379185846251495609od_a_o @ Xs ) @ I3 )
     => ( ( list_u329291161401623989od_a_o @ Xs @ I3 @ X2 )
        = Xs ) ) ).

% list_update_beyond
thf(fact_326_list__update__beyond,axiom,
    ! [Xs: list_S5253404628753938141on_nat,I3: nat,X2: sum_su4907927656612468301on_nat] :
      ( ( ord_less_eq_nat @ ( size_s5686234912572693873on_nat @ Xs ) @ I3 )
     => ( ( list_u8577817230714425439on_nat @ Xs @ I3 @ X2 )
        = Xs ) ) ).

% list_update_beyond
thf(fact_327_list__update__beyond,axiom,
    ! [Xs: list_nat,I3: nat,X2: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I3 )
     => ( ( list_update_nat @ Xs @ I3 @ X2 )
        = Xs ) ) ).

% list_update_beyond
thf(fact_328_list__update__beyond,axiom,
    ! [Xs: list_P9127870862658512897on_nat,I3: nat,X2: produc1313295348606329723on_nat] :
      ( ( ord_less_eq_nat @ ( size_s7490768528479217773on_nat @ Xs ) @ I3 )
     => ( ( list_u1840724836389618409on_nat @ Xs @ I3 @ X2 )
        = Xs ) ) ).

% list_update_beyond
thf(fact_329_upt__conv__Nil,axiom,
    ! [J: nat,I3: nat] :
      ( ( ord_less_eq_nat @ J @ I3 )
     => ( ( upt @ I3 @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_330_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_331_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_332_zless__nat__conj,axiom,
    ! [W: int,Z2: int] :
      ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
      = ( ( ord_less_int @ zero_zero_int @ Z2 )
        & ( ord_less_int @ W @ Z2 ) ) ) ).

% zless_nat_conj
thf(fact_333_nat__zminus__int,axiom,
    ! [N: nat] :
      ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
      = zero_zero_nat ) ).

% nat_zminus_int
thf(fact_334_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_335_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_336_verit__negate__coefficient_I3_J,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( ( uminus_uminus_int @ A )
        = ( uminus_uminus_int @ B ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_337_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ ( ord_less_eq_nat @ A @ B )
      | ~ ( ord_less_eq_nat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_338_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ ( ord_less_eq_int @ A @ B )
      | ~ ( ord_less_eq_int @ B @ A ) ) ).

% verit_la_disequality
thf(fact_339_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% equation_minus_iff
thf(fact_340_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( uminus_uminus_int @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_341_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
      = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).

% le_minus_iff
thf(fact_342_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_343_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% le_imp_neg_le
thf(fact_344_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_345_le__trans,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I3 @ K ) ) ) ).

% le_trans
thf(fact_346_eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 = N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% eq_imp_le
thf(fact_347_le__antisym,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( M2 = N ) ) ) ).

% le_antisym
thf(fact_348_nat__le__linear,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
      | ( ord_less_eq_nat @ N @ M2 ) ) ).

% nat_le_linear
thf(fact_349_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ? [X: nat] :
            ( ( P @ X )
            & ! [Y5: nat] :
                ( ( P @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_350_of__nat__mono,axiom,
    ! [I3: nat,J: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).

% of_nat_mono
thf(fact_351_of__nat__mono,axiom,
    ! [I3: nat,J: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I3 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).

% of_nat_mono
thf(fact_352_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
      = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).

% less_minus_iff
thf(fact_353_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
      = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_354_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_355_zero__le,axiom,
    ! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).

% zero_le
thf(fact_356_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_357_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_358_verit__comp__simplify1_I3_J,axiom,
    ! [B2: nat,A3: nat] :
      ( ( ~ ( ord_less_eq_nat @ B2 @ A3 ) )
      = ( ord_less_nat @ A3 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_359_verit__comp__simplify1_I3_J,axiom,
    ! [B2: int,A3: int] :
      ( ( ~ ( ord_less_eq_int @ B2 @ A3 ) )
      = ( ord_less_int @ A3 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_360_uminus__int__code_I1_J,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% uminus_int_code(1)
thf(fact_361_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_362_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_363_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_364_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_365_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
          & ( M3 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_366_less__imp__le__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_imp_le_nat
thf(fact_367_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N3: nat] :
          ( ( ord_less_nat @ M3 @ N3 )
          | ( M3 = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_368_less__or__eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ord_less_nat @ M2 @ N )
        | ( M2 = N ) )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_or_eq_imp_le
thf(fact_369_le__neq__implies__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( M2 != N )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% le_neq_implies_less
thf(fact_370_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I3: nat,J: nat] :
      ( ! [I2: nat,J3: nat] :
          ( ( ord_less_nat @ I2 @ J3 )
         => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J3 ) ) )
     => ( ( ord_less_eq_nat @ I3 @ J )
       => ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_371_int__cases2,axiom,
    ! [Z2: int] :
      ( ! [N2: nat] :
          ( Z2
         != ( semiri1314217659103216013at_int @ N2 ) )
     => ~ ! [N2: nat] :
            ( Z2
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% int_cases2
thf(fact_372_nat__zero__as__int,axiom,
    ( zero_zero_nat
    = ( nat2 @ zero_zero_int ) ) ).

% nat_zero_as_int
thf(fact_373_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).

% of_nat_0_le_iff
thf(fact_374_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).

% of_nat_0_le_iff
thf(fact_375_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K2 )
               => ~ ( P @ I4 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_376_not__int__zless__negative,axiom,
    ! [N: nat,M2: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).

% not_int_zless_negative
thf(fact_377_length__removeAll__less__eq,axiom,
    ! [X2: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] : ( ord_less_eq_nat @ ( size_s3379185846251495609od_a_o @ ( remove3990864149796893523od_a_o @ X2 @ Xs ) ) @ ( size_s3379185846251495609od_a_o @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_378_length__removeAll__less__eq,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] : ( ord_less_eq_nat @ ( size_s5686234912572693873on_nat @ ( remove433775707702860097on_nat @ X2 @ Xs ) ) @ ( size_s5686234912572693873on_nat @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_379_length__removeAll__less__eq,axiom,
    ! [X2: nat,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( removeAll_nat @ X2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_380_length__removeAll__less__eq,axiom,
    ! [X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] : ( ord_less_eq_nat @ ( size_s7490768528479217773on_nat @ ( remove6084986979817164679on_nat @ X2 @ Xs ) ) @ ( size_s7490768528479217773on_nat @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_381_count__le__length,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X2: list_P3196869577824546119od_a_o] : ( ord_less_eq_nat @ ( count_5344270436093135555od_a_o @ Xs @ X2 ) @ ( size_s3379185846251495609od_a_o @ Xs ) ) ).

% count_le_length
thf(fact_382_count__le__length,axiom,
    ! [Xs: list_S5253404628753938141on_nat,X2: sum_su4907927656612468301on_nat] : ( ord_less_eq_nat @ ( count_3605398483369874897on_nat @ Xs @ X2 ) @ ( size_s5686234912572693873on_nat @ Xs ) ) ).

% count_le_length
thf(fact_383_count__le__length,axiom,
    ! [Xs: list_nat,X2: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X2 ) @ ( size_size_list_nat @ Xs ) ) ).

% count_le_length
thf(fact_384_count__le__length,axiom,
    ! [Xs: list_P9127870862658512897on_nat,X2: produc1313295348606329723on_nat] : ( ord_less_eq_nat @ ( count_9157224493928226551on_nat @ Xs @ X2 ) @ ( size_s7490768528479217773on_nat @ Xs ) ) ).

% count_le_length
thf(fact_385_list__ex1__iff,axiom,
    ( list_ex1_nat
    = ( ^ [P2: nat > $o,Xs2: list_nat] :
        ? [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
          & ( P2 @ X3 )
          & ! [Y3: nat] :
              ( ( ( member_nat @ Y3 @ ( set_nat2 @ Xs2 ) )
                & ( P2 @ Y3 ) )
             => ( Y3 = X3 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_386_list__ex1__iff,axiom,
    ( list_e8133552692542901648od_a_o
    = ( ^ [P2: list_P3196869577824546119od_a_o > $o,Xs2: list_l8336264072933094989od_a_o] :
        ? [X3: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ X3 @ ( set_li2629784761229999580od_a_o @ Xs2 ) )
          & ( P2 @ X3 )
          & ! [Y3: list_P3196869577824546119od_a_o] :
              ( ( ( member5261019209154752368od_a_o @ Y3 @ ( set_li2629784761229999580od_a_o @ Xs2 ) )
                & ( P2 @ Y3 ) )
             => ( Y3 = X3 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_387_list__ex1__iff,axiom,
    ( list_e3536954311976427588on_nat
    = ( ^ [P2: produc1313295348606329723on_nat > $o,Xs2: list_P9127870862658512897on_nat] :
        ? [X3: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X3 @ ( set_Pr937530902923597456on_nat @ Xs2 ) )
          & ( P2 @ X3 )
          & ! [Y3: produc1313295348606329723on_nat] :
              ( ( ( member5906418179908956708on_nat @ Y3 @ ( set_Pr937530902923597456on_nat @ Xs2 ) )
                & ( P2 @ Y3 ) )
             => ( Y3 = X3 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_388_list__ex1__iff,axiom,
    ( list_e4461624505202298052on_nat
    = ( ^ [P2: sum_su4907927656612468301on_nat > $o,Xs2: list_S5253404628753938141on_nat] :
        ? [X3: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X3 @ ( set_Su4848624856859743608on_nat @ Xs2 ) )
          & ( P2 @ X3 )
          & ! [Y3: sum_su4907927656612468301on_nat] :
              ( ( ( member8476201367316484836on_nat @ Y3 @ ( set_Su4848624856859743608on_nat @ Xs2 ) )
                & ( P2 @ Y3 ) )
             => ( Y3 = X3 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_389_nat__mono__iff,axiom,
    ! [Z2: int,W: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
        = ( ord_less_int @ W @ Z2 ) ) ) ).

% nat_mono_iff
thf(fact_390_zless__nat__eq__int__zless,axiom,
    ! [M2: nat,Z2: int] :
      ( ( ord_less_nat @ M2 @ ( nat2 @ Z2 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z2 ) ) ).

% zless_nat_eq_int_zless
thf(fact_391_int__cases4,axiom,
    ! [M2: int] :
      ( ! [N2: nat] :
          ( M2
         != ( semiri1314217659103216013at_int @ N2 ) )
     => ~ ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( M2
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% int_cases4
thf(fact_392_split__nat,axiom,
    ! [P: nat > $o,I3: int] :
      ( ( P @ ( nat2 @ I3 ) )
      = ( ! [N3: nat] :
            ( ( I3
              = ( semiri1314217659103216013at_int @ N3 ) )
           => ( P @ N3 ) )
        & ( ( ord_less_int @ I3 @ zero_zero_int )
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_nat
thf(fact_393_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N2: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
       => ~ ! [N2: nat] :
              ( ( K
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% int_cases3
thf(fact_394_can__select__set__list__ex1,axiom,
    ! [P: nat > $o,A2: list_nat] :
      ( ( can_select_nat @ P @ ( set_nat2 @ A2 ) )
      = ( list_ex1_nat @ P @ A2 ) ) ).

% can_select_set_list_ex1
thf(fact_395_can__select__set__list__ex1,axiom,
    ! [P: list_P3196869577824546119od_a_o > $o,A2: list_l8336264072933094989od_a_o] :
      ( ( can_se5536358244299704004od_a_o @ P @ ( set_li2629784761229999580od_a_o @ A2 ) )
      = ( list_e8133552692542901648od_a_o @ P @ A2 ) ) ).

% can_select_set_list_ex1
thf(fact_396_can__select__set__list__ex1,axiom,
    ! [P: produc1313295348606329723on_nat > $o,A2: list_P9127870862658512897on_nat] :
      ( ( can_se7488105254466884472on_nat @ P @ ( set_Pr937530902923597456on_nat @ A2 ) )
      = ( list_e3536954311976427588on_nat @ P @ A2 ) ) ).

% can_select_set_list_ex1
thf(fact_397_can__select__set__list__ex1,axiom,
    ! [P: sum_su4907927656612468301on_nat > $o,A2: list_S5253404628753938141on_nat] :
      ( ( can_se5303938684849109648on_nat @ P @ ( set_Su4848624856859743608on_nat @ A2 ) )
      = ( list_e4461624505202298052on_nat @ P @ A2 ) ) ).

% can_select_set_list_ex1
thf(fact_398_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M2: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N @ K2 )
         => ( P @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ K2 @ I4 )
                 => ( P @ I4 ) )
             => ( P @ K2 ) ) )
       => ( P @ M2 ) ) ) ).

% nat_descend_induct
thf(fact_399_order__le__imp__less__or__eq,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y2 )
     => ( ( ord_less_nat @ X2 @ Y2 )
        | ( X2 = Y2 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_400_order__le__imp__less__or__eq,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ X2 @ Y2 )
     => ( ( ord_less_int @ X2 @ Y2 )
        | ( X2 = Y2 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_401_linorder__le__less__linear,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y2 )
      | ( ord_less_nat @ Y2 @ X2 ) ) ).

% linorder_le_less_linear
thf(fact_402_linorder__le__less__linear,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ X2 @ Y2 )
      | ( ord_less_int @ Y2 @ X2 ) ) ).

% linorder_le_less_linear
thf(fact_403_negative__zle,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).

% negative_zle
thf(fact_404_nat__le__0,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ Z2 @ zero_zero_int )
     => ( ( nat2 @ Z2 )
        = zero_zero_nat ) ) ).

% nat_le_0
thf(fact_405_nat__0__iff,axiom,
    ! [I3: int] :
      ( ( ( nat2 @ I3 )
        = zero_zero_nat )
      = ( ord_less_eq_int @ I3 @ zero_zero_int ) ) ).

% nat_0_iff
thf(fact_406_int__nat__eq,axiom,
    ! [Z2: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
          = Z2 ) )
      & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
          = zero_zero_int ) ) ) ).

% int_nat_eq
thf(fact_407_can__select__def,axiom,
    ( can_select_nat
    = ( ^ [P2: nat > $o,A6: set_nat] :
        ? [X3: nat] :
          ( ( member_nat @ X3 @ A6 )
          & ( P2 @ X3 )
          & ! [Y3: nat] :
              ( ( ( member_nat @ Y3 @ A6 )
                & ( P2 @ Y3 ) )
             => ( Y3 = X3 ) ) ) ) ) ).

% can_select_def
thf(fact_408_can__select__def,axiom,
    ( can_se5536358244299704004od_a_o
    = ( ^ [P2: list_P3196869577824546119od_a_o > $o,A6: set_li5011860890973139239od_a_o] :
        ? [X3: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ X3 @ A6 )
          & ( P2 @ X3 )
          & ! [Y3: list_P3196869577824546119od_a_o] :
              ( ( ( member5261019209154752368od_a_o @ Y3 @ A6 )
                & ( P2 @ Y3 ) )
             => ( Y3 = X3 ) ) ) ) ) ).

% can_select_def
thf(fact_409_can__select__def,axiom,
    ( can_se7488105254466884472on_nat
    = ( ^ [P2: produc1313295348606329723on_nat > $o,A6: set_Pr2535564044409765851on_nat] :
        ? [X3: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X3 @ A6 )
          & ( P2 @ X3 )
          & ! [Y3: produc1313295348606329723on_nat] :
              ( ( ( member5906418179908956708on_nat @ Y3 @ A6 )
                & ( P2 @ Y3 ) )
             => ( Y3 = X3 ) ) ) ) ) ).

% can_select_def
thf(fact_410_can__select__def,axiom,
    ( can_se5303938684849109648on_nat
    = ( ^ [P2: sum_su4907927656612468301on_nat > $o,A6: set_Su266273576555847555on_nat] :
        ? [X3: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X3 @ A6 )
          & ( P2 @ X3 )
          & ! [Y3: sum_su4907927656612468301on_nat] :
              ( ( ( member8476201367316484836on_nat @ Y3 @ A6 )
                & ( P2 @ Y3 ) )
             => ( Y3 = X3 ) ) ) ) ) ).

% can_select_def
thf(fact_411_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B5 )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
           => ( member_nat @ X3 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_412_subset__code_I1_J,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,B5: set_li5011860890973139239od_a_o] :
      ( ( ord_le1271121679792220359od_a_o @ ( set_li2629784761229999580od_a_o @ Xs ) @ B5 )
      = ( ! [X3: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X3 @ ( set_li2629784761229999580od_a_o @ Xs ) )
           => ( member5261019209154752368od_a_o @ X3 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_413_subset__code_I1_J,axiom,
    ! [Xs: list_P9127870862658512897on_nat,B5: set_Pr2535564044409765851on_nat] :
      ( ( ord_le6961793561435735931on_nat @ ( set_Pr937530902923597456on_nat @ Xs ) @ B5 )
      = ( ! [X3: produc1313295348606329723on_nat] :
            ( ( member5906418179908956708on_nat @ X3 @ ( set_Pr937530902923597456on_nat @ Xs ) )
           => ( member5906418179908956708on_nat @ X3 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_414_subset__code_I1_J,axiom,
    ! [Xs: list_S5253404628753938141on_nat,B5: set_Su266273576555847555on_nat] :
      ( ( ord_le6595998496543023075on_nat @ ( set_Su4848624856859743608on_nat @ Xs ) @ B5 )
      = ( ! [X3: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X3 @ ( set_Su4848624856859743608on_nat @ Xs ) )
           => ( member8476201367316484836on_nat @ X3 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_415_less__eq__int__code_I1_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% less_eq_int_code(1)
thf(fact_416_zle__int,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% zle_int
thf(fact_417_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_418_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ? [N2: nat] :
          ( K
          = ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% zero_le_imp_eq_int
thf(fact_419_nonneg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ~ ! [N2: nat] :
            ( K
           != ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% nonneg_int_cases
thf(fact_420_nat__mono,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ X2 @ Y2 )
     => ( ord_less_eq_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y2 ) ) ) ).

% nat_mono
thf(fact_421_eq__nat__nat__iff,axiom,
    ! [Z2: int,Z3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
       => ( ( ( nat2 @ Z2 )
            = ( nat2 @ Z3 ) )
          = ( Z2 = Z3 ) ) ) ) ).

% eq_nat_nat_iff
thf(fact_422_all__nat,axiom,
    ( ( ^ [P4: nat > $o] :
        ! [X5: nat] : ( P4 @ X5 ) )
    = ( ^ [P2: nat > $o] :
        ! [X3: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X3 )
         => ( P2 @ ( nat2 @ X3 ) ) ) ) ) ).

% all_nat
thf(fact_423_ex__nat,axiom,
    ( ( ^ [P4: nat > $o] :
        ? [X5: nat] : ( P4 @ X5 ) )
    = ( ^ [P2: nat > $o] :
        ? [X3: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X3 )
          & ( P2 @ ( nat2 @ X3 ) ) ) ) ) ).

% ex_nat
thf(fact_424_set__update__subsetI,axiom,
    ! [Xs: list_nat,A2: set_nat,X2: nat,I3: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
     => ( ( member_nat @ X2 @ A2 )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I3 @ X2 ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_425_set__update__subsetI,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,A2: set_li5011860890973139239od_a_o,X2: list_P3196869577824546119od_a_o,I3: nat] :
      ( ( ord_le1271121679792220359od_a_o @ ( set_li2629784761229999580od_a_o @ Xs ) @ A2 )
     => ( ( member5261019209154752368od_a_o @ X2 @ A2 )
       => ( ord_le1271121679792220359od_a_o @ ( set_li2629784761229999580od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ I3 @ X2 ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_426_set__update__subsetI,axiom,
    ! [Xs: list_P9127870862658512897on_nat,A2: set_Pr2535564044409765851on_nat,X2: produc1313295348606329723on_nat,I3: nat] :
      ( ( ord_le6961793561435735931on_nat @ ( set_Pr937530902923597456on_nat @ Xs ) @ A2 )
     => ( ( member5906418179908956708on_nat @ X2 @ A2 )
       => ( ord_le6961793561435735931on_nat @ ( set_Pr937530902923597456on_nat @ ( list_u1840724836389618409on_nat @ Xs @ I3 @ X2 ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_427_set__update__subsetI,axiom,
    ! [Xs: list_S5253404628753938141on_nat,A2: set_Su266273576555847555on_nat,X2: sum_su4907927656612468301on_nat,I3: nat] :
      ( ( ord_le6595998496543023075on_nat @ ( set_Su4848624856859743608on_nat @ Xs ) @ A2 )
     => ( ( member8476201367316484836on_nat @ X2 @ A2 )
       => ( ord_le6595998496543023075on_nat @ ( set_Su4848624856859743608on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I3 @ X2 ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_428_order__less__imp__not__less,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_nat @ X2 @ Y2 )
     => ~ ( ord_less_nat @ Y2 @ X2 ) ) ).

% order_less_imp_not_less
thf(fact_429_order__less__imp__not__less,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ X2 @ Y2 )
     => ~ ( ord_less_int @ Y2 @ X2 ) ) ).

% order_less_imp_not_less
thf(fact_430_order__less__imp__not__eq2,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_nat @ X2 @ Y2 )
     => ( Y2 != X2 ) ) ).

% order_less_imp_not_eq2
thf(fact_431_order__less__imp__not__eq2,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ X2 @ Y2 )
     => ( Y2 != X2 ) ) ).

% order_less_imp_not_eq2
thf(fact_432_order__less__imp__not__eq,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_nat @ X2 @ Y2 )
     => ( X2 != Y2 ) ) ).

% order_less_imp_not_eq
thf(fact_433_order__less__imp__not__eq,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ X2 @ Y2 )
     => ( X2 != Y2 ) ) ).

% order_less_imp_not_eq
thf(fact_434_linorder__less__linear,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_nat @ X2 @ Y2 )
      | ( X2 = Y2 )
      | ( ord_less_nat @ Y2 @ X2 ) ) ).

% linorder_less_linear
thf(fact_435_linorder__less__linear,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ X2 @ Y2 )
      | ( X2 = Y2 )
      | ( ord_less_int @ Y2 @ X2 ) ) ).

% linorder_less_linear
thf(fact_436_order__less__imp__triv,axiom,
    ! [X2: nat,Y2: nat,P: $o] :
      ( ( ord_less_nat @ X2 @ Y2 )
     => ( ( ord_less_nat @ Y2 @ X2 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_437_order__less__imp__triv,axiom,
    ! [X2: int,Y2: int,P: $o] :
      ( ( ord_less_int @ X2 @ Y2 )
     => ( ( ord_less_int @ Y2 @ X2 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_438_order__less__not__sym,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_nat @ X2 @ Y2 )
     => ~ ( ord_less_nat @ Y2 @ X2 ) ) ).

% order_less_not_sym
thf(fact_439_order__less__not__sym,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ X2 @ Y2 )
     => ~ ( ord_less_int @ Y2 @ X2 ) ) ).

% order_less_not_sym
thf(fact_440_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_441_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > int,C2: int] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C2 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_442_order__less__subst2,axiom,
    ! [A: int,B: int,F: int > nat,C2: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X: int,Y4: int] :
              ( ( ord_less_int @ X @ Y4 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_443_order__less__subst2,axiom,
    ! [A: int,B: int,F: int > int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C2 )
       => ( ! [X: int,Y4: int] :
              ( ( ord_less_int @ X @ Y4 )
             => ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_444_order__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_445_order__less__subst1,axiom,
    ! [A: nat,F: int > nat,B: int,C2: int] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X: int,Y4: int] :
              ( ( ord_less_int @ X @ Y4 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_446_order__less__subst1,axiom,
    ! [A: int,F: nat > int,B: nat,C2: nat] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_447_order__less__subst1,axiom,
    ! [A: int,F: int > int,B: int,C2: int] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X: int,Y4: int] :
              ( ( ord_less_int @ X @ Y4 )
             => ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_448_order__less__irrefl,axiom,
    ! [X2: nat] :
      ~ ( ord_less_nat @ X2 @ X2 ) ).

% order_less_irrefl
thf(fact_449_order__less__irrefl,axiom,
    ! [X2: int] :
      ~ ( ord_less_int @ X2 @ X2 ) ).

% order_less_irrefl
thf(fact_450_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_451_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > int,C2: int] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_452_ord__less__eq__subst,axiom,
    ! [A: int,B: int,F: int > nat,C2: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X: int,Y4: int] :
              ( ( ord_less_int @ X @ Y4 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_453_ord__less__eq__subst,axiom,
    ! [A: int,B: int,F: int > int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X: int,Y4: int] :
              ( ( ord_less_int @ X @ Y4 )
             => ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_454_ord__eq__less__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_455_ord__eq__less__subst,axiom,
    ! [A: int,F: nat > int,B: nat,C2: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_456_ord__eq__less__subst,axiom,
    ! [A: nat,F: int > nat,B: int,C2: int] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X: int,Y4: int] :
              ( ( ord_less_int @ X @ Y4 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_457_ord__eq__less__subst,axiom,
    ! [A: int,F: int > int,B: int,C2: int] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X: int,Y4: int] :
              ( ( ord_less_int @ X @ Y4 )
             => ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_458_order__less__trans,axiom,
    ! [X2: nat,Y2: nat,Z2: nat] :
      ( ( ord_less_nat @ X2 @ Y2 )
     => ( ( ord_less_nat @ Y2 @ Z2 )
       => ( ord_less_nat @ X2 @ Z2 ) ) ) ).

% order_less_trans
thf(fact_459_order__less__trans,axiom,
    ! [X2: int,Y2: int,Z2: int] :
      ( ( ord_less_int @ X2 @ Y2 )
     => ( ( ord_less_int @ Y2 @ Z2 )
       => ( ord_less_int @ X2 @ Z2 ) ) ) ).

% order_less_trans
thf(fact_460_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_461_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order_less_asym'
thf(fact_462_linorder__neq__iff,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( X2 != Y2 )
      = ( ( ord_less_nat @ X2 @ Y2 )
        | ( ord_less_nat @ Y2 @ X2 ) ) ) ).

% linorder_neq_iff
thf(fact_463_linorder__neq__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( X2 != Y2 )
      = ( ( ord_less_int @ X2 @ Y2 )
        | ( ord_less_int @ Y2 @ X2 ) ) ) ).

% linorder_neq_iff
thf(fact_464_order__less__asym,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_nat @ X2 @ Y2 )
     => ~ ( ord_less_nat @ Y2 @ X2 ) ) ).

% order_less_asym
thf(fact_465_order__less__asym,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ X2 @ Y2 )
     => ~ ( ord_less_int @ Y2 @ X2 ) ) ).

% order_less_asym
thf(fact_466_linorder__neqE,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( X2 != Y2 )
     => ( ~ ( ord_less_nat @ X2 @ Y2 )
       => ( ord_less_nat @ Y2 @ X2 ) ) ) ).

% linorder_neqE
thf(fact_467_linorder__neqE,axiom,
    ! [X2: int,Y2: int] :
      ( ( X2 != Y2 )
     => ( ~ ( ord_less_int @ X2 @ Y2 )
       => ( ord_less_int @ Y2 @ X2 ) ) ) ).

% linorder_neqE
thf(fact_468_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_469_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_470_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_471_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_472_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_473_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C2 @ B )
       => ( ord_less_int @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_474_not__less__iff__gr__or__eq,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
      = ( ( ord_less_nat @ Y2 @ X2 )
        | ( X2 = Y2 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_475_not__less__iff__gr__or__eq,axiom,
    ! [X2: int,Y2: int] :
      ( ( ~ ( ord_less_int @ X2 @ Y2 ) )
      = ( ( ord_less_int @ Y2 @ X2 )
        | ( X2 = Y2 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_476_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_477_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_478_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A4: nat,B3: nat] :
          ( ( ord_less_nat @ A4 @ B3 )
         => ( P @ A4 @ B3 ) )
     => ( ! [A4: nat] : ( P @ A4 @ A4 )
       => ( ! [A4: nat,B3: nat] :
              ( ( P @ B3 @ A4 )
             => ( P @ A4 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_479_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A4: int,B3: int] :
          ( ( ord_less_int @ A4 @ B3 )
         => ( P @ A4 @ B3 ) )
     => ( ! [A4: int] : ( P @ A4 @ A4 )
       => ( ! [A4: int,B3: int] :
              ( ( P @ B3 @ A4 )
             => ( P @ A4 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_480_exists__least__iff,axiom,
    ( ( ^ [P4: nat > $o] :
        ? [X5: nat] : ( P4 @ X5 ) )
    = ( ^ [P2: nat > $o] :
        ? [N3: nat] :
          ( ( P2 @ N3 )
          & ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N3 )
             => ~ ( P2 @ M3 ) ) ) ) ) ).

% exists_least_iff
thf(fact_481_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_482_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_483_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_484_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ~ ( ord_less_int @ A @ B ) ) ).

% dual_order.asym
thf(fact_485_linorder__cases,axiom,
    ! [X2: nat,Y2: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y2 )
     => ( ( X2 != Y2 )
       => ( ord_less_nat @ Y2 @ X2 ) ) ) ).

% linorder_cases
thf(fact_486_linorder__cases,axiom,
    ! [X2: int,Y2: int] :
      ( ~ ( ord_less_int @ X2 @ Y2 )
     => ( ( X2 != Y2 )
       => ( ord_less_int @ Y2 @ X2 ) ) ) ).

% linorder_cases
thf(fact_487_antisym__conv3,axiom,
    ! [Y2: nat,X2: nat] :
      ( ~ ( ord_less_nat @ Y2 @ X2 )
     => ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
        = ( X2 = Y2 ) ) ) ).

% antisym_conv3
thf(fact_488_antisym__conv3,axiom,
    ! [Y2: int,X2: int] :
      ( ~ ( ord_less_int @ Y2 @ X2 )
     => ( ( ~ ( ord_less_int @ X2 @ Y2 ) )
        = ( X2 = Y2 ) ) ) ).

% antisym_conv3
thf(fact_489_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X )
             => ( P @ Y5 ) )
         => ( P @ X ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_490_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_491_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_492_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_493_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A = B )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_494_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_495_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order.asym
thf(fact_496_less__imp__neq,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_nat @ X2 @ Y2 )
     => ( X2 != Y2 ) ) ).

% less_imp_neq
thf(fact_497_less__imp__neq,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ X2 @ Y2 )
     => ( X2 != Y2 ) ) ).

% less_imp_neq
thf(fact_498_gt__ex,axiom,
    ! [X2: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).

% gt_ex
thf(fact_499_gt__ex,axiom,
    ! [X2: int] :
    ? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).

% gt_ex
thf(fact_500_lt__ex,axiom,
    ! [X2: int] :
    ? [Y4: int] : ( ord_less_int @ Y4 @ X2 ) ).

% lt_ex
thf(fact_501_int__zle__neg,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) )
      = ( ( N = zero_zero_nat )
        & ( M2 = zero_zero_nat ) ) ) ).

% int_zle_neg
thf(fact_502_nonpos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ~ ! [N2: nat] :
            ( K
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% nonpos_int_cases
thf(fact_503_negative__zle__0,axiom,
    ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_504_nat__le__iff,axiom,
    ! [X2: int,N: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ X2 ) @ N )
      = ( ord_less_eq_int @ X2 @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% nat_le_iff
thf(fact_505_int__eq__iff,axiom,
    ! [M2: nat,Z2: int] :
      ( ( ( semiri1314217659103216013at_int @ M2 )
        = Z2 )
      = ( ( M2
          = ( nat2 @ Z2 ) )
        & ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ) ).

% int_eq_iff
thf(fact_506_nat__0__le,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
        = Z2 ) ) ).

% nat_0_le
thf(fact_507_nat__eq__iff2,axiom,
    ! [M2: nat,W: int] :
      ( ( M2
        = ( nat2 @ W ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
         => ( W
            = ( semiri1314217659103216013at_int @ M2 ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
         => ( M2 = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
thf(fact_508_nat__eq__iff,axiom,
    ! [W: int,M2: nat] :
      ( ( ( nat2 @ W )
        = M2 )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
         => ( W
            = ( semiri1314217659103216013at_int @ M2 ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
         => ( M2 = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
thf(fact_509_nat__less__eq__zless,axiom,
    ! [W: int,Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ W )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
        = ( ord_less_int @ W @ Z2 ) ) ) ).

% nat_less_eq_zless
thf(fact_510_nat__le__eq__zle,axiom,
    ! [W: int,Z2: int] :
      ( ( ( ord_less_int @ zero_zero_int @ W )
        | ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
     => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
        = ( ord_less_eq_int @ W @ Z2 ) ) ) ).

% nat_le_eq_zle
thf(fact_511_le__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
        = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).

% le_nat_iff
thf(fact_512_nat__less__iff,axiom,
    ! [W: int,M2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ W )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ M2 )
        = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).

% nat_less_iff
thf(fact_513_leD,axiom,
    ! [Y2: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y2 @ X2 )
     => ~ ( ord_less_nat @ X2 @ Y2 ) ) ).

% leD
thf(fact_514_leD,axiom,
    ! [Y2: int,X2: int] :
      ( ( ord_less_eq_int @ Y2 @ X2 )
     => ~ ( ord_less_int @ X2 @ Y2 ) ) ).

% leD
thf(fact_515_leI,axiom,
    ! [X2: nat,Y2: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y2 )
     => ( ord_less_eq_nat @ Y2 @ X2 ) ) ).

% leI
thf(fact_516_leI,axiom,
    ! [X2: int,Y2: int] :
      ( ~ ( ord_less_int @ X2 @ Y2 )
     => ( ord_less_eq_int @ Y2 @ X2 ) ) ).

% leI
thf(fact_517_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_518_nless__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_519_antisym__conv1,axiom,
    ! [X2: nat,Y2: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y2 )
     => ( ( ord_less_eq_nat @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% antisym_conv1
thf(fact_520_antisym__conv1,axiom,
    ! [X2: int,Y2: int] :
      ( ~ ( ord_less_int @ X2 @ Y2 )
     => ( ( ord_less_eq_int @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% antisym_conv1
thf(fact_521_antisym__conv2,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y2 )
     => ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
        = ( X2 = Y2 ) ) ) ).

% antisym_conv2
thf(fact_522_antisym__conv2,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ X2 @ Y2 )
     => ( ( ~ ( ord_less_int @ X2 @ Y2 ) )
        = ( X2 = Y2 ) ) ) ).

% antisym_conv2
thf(fact_523_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y3 )
          & ~ ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_524_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X3: int,Y3: int] :
          ( ( ord_less_eq_int @ X3 @ Y3 )
          & ~ ( ord_less_eq_int @ Y3 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_525_not__le__imp__less,axiom,
    ! [Y2: nat,X2: nat] :
      ( ~ ( ord_less_eq_nat @ Y2 @ X2 )
     => ( ord_less_nat @ X2 @ Y2 ) ) ).

% not_le_imp_less
thf(fact_526_not__le__imp__less,axiom,
    ! [Y2: int,X2: int] :
      ( ~ ( ord_less_eq_int @ Y2 @ X2 )
     => ( ord_less_int @ X2 @ Y2 ) ) ).

% not_le_imp_less
thf(fact_527_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_528_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B4: int] :
          ( ( ord_less_int @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_529_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_530_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A5: int,B4: int] :
          ( ( ord_less_eq_int @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_531_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_532_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_533_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_534_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_535_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_536_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A5: int,B4: int] :
          ( ( ord_less_eq_int @ A5 @ B4 )
          & ~ ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_537_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_538_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A5: int] :
          ( ( ord_less_int @ B4 @ A5 )
          | ( A5 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_539_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_540_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A5: int] :
          ( ( ord_less_eq_int @ B4 @ A5 )
          & ( A5 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_541_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_542_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_int @ C2 @ B )
       => ( ord_less_int @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_543_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_544_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ B )
       => ( ord_less_int @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_545_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_546_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A5: int] :
          ( ( ord_less_eq_int @ B4 @ A5 )
          & ~ ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_547_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_548_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_eq_int @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_549_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_550_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_eq_int @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_551_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X3: nat,Y3: nat] :
          ( ( ord_less_nat @ X3 @ Y3 )
          | ( X3 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_552_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X3: int,Y3: int] :
          ( ( ord_less_int @ X3 @ Y3 )
          | ( X3 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_553_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y3 )
          & ( X3 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_554_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X3: int,Y3: int] :
          ( ( ord_less_eq_int @ X3 @ Y3 )
          & ( X3 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_555_linorder__not__le,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ~ ( ord_less_eq_nat @ X2 @ Y2 ) )
      = ( ord_less_nat @ Y2 @ X2 ) ) ).

% linorder_not_le
thf(fact_556_linorder__not__le,axiom,
    ! [X2: int,Y2: int] :
      ( ( ~ ( ord_less_eq_int @ X2 @ Y2 ) )
      = ( ord_less_int @ Y2 @ X2 ) ) ).

% linorder_not_le
thf(fact_557_linorder__not__less,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
      = ( ord_less_eq_nat @ Y2 @ X2 ) ) ).

% linorder_not_less
thf(fact_558_linorder__not__less,axiom,
    ! [X2: int,Y2: int] :
      ( ( ~ ( ord_less_int @ X2 @ Y2 ) )
      = ( ord_less_eq_int @ Y2 @ X2 ) ) ).

% linorder_not_less
thf(fact_559_order__less__imp__le,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_nat @ X2 @ Y2 )
     => ( ord_less_eq_nat @ X2 @ Y2 ) ) ).

% order_less_imp_le
thf(fact_560_order__less__imp__le,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ X2 @ Y2 )
     => ( ord_less_eq_int @ X2 @ Y2 ) ) ).

% order_less_imp_le
thf(fact_561_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_562_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( A != B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_563_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_564_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_565_order__le__less__trans,axiom,
    ! [X2: nat,Y2: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y2 )
     => ( ( ord_less_nat @ Y2 @ Z2 )
       => ( ord_less_nat @ X2 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_566_order__le__less__trans,axiom,
    ! [X2: int,Y2: int,Z2: int] :
      ( ( ord_less_eq_int @ X2 @ Y2 )
     => ( ( ord_less_int @ Y2 @ Z2 )
       => ( ord_less_int @ X2 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_567_order__less__le__trans,axiom,
    ! [X2: nat,Y2: nat,Z2: nat] :
      ( ( ord_less_nat @ X2 @ Y2 )
     => ( ( ord_less_eq_nat @ Y2 @ Z2 )
       => ( ord_less_nat @ X2 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_568_order__less__le__trans,axiom,
    ! [X2: int,Y2: int,Z2: int] :
      ( ( ord_less_int @ X2 @ Y2 )
     => ( ( ord_less_eq_int @ Y2 @ Z2 )
       => ( ord_less_int @ X2 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_569_order__le__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_570_order__le__less__subst1,axiom,
    ! [A: nat,F: int > nat,B: int,C2: int] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X: int,Y4: int] :
              ( ( ord_less_int @ X @ Y4 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_571_order__le__less__subst1,axiom,
    ! [A: int,F: nat > int,B: nat,C2: nat] :
      ( ( ord_less_eq_int @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_572_order__le__less__subst1,axiom,
    ! [A: int,F: int > int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X: int,Y4: int] :
              ( ( ord_less_int @ X @ Y4 )
             => ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_573_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_574_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > int,C2: int] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C2 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X @ Y4 )
             => ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_575_order__le__less__subst2,axiom,
    ! [A: int,B: int,F: int > nat,C2: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X: int,Y4: int] :
              ( ( ord_less_eq_int @ X @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_576_order__le__less__subst2,axiom,
    ! [A: int,B: int,F: int > int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C2 )
       => ( ! [X: int,Y4: int] :
              ( ( ord_less_eq_int @ X @ Y4 )
             => ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_577_order__less__le__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_578_order__less__le__subst1,axiom,
    ! [A: int,F: nat > int,B: nat,C2: nat] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X @ Y4 )
             => ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_579_order__less__le__subst1,axiom,
    ! [A: nat,F: int > nat,B: int,C2: int] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ! [X: int,Y4: int] :
              ( ( ord_less_eq_int @ X @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_580_order__less__le__subst1,axiom,
    ! [A: int,F: int > int,B: int,C2: int] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ! [X: int,Y4: int] :
              ( ( ord_less_eq_int @ X @ Y4 )
             => ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_581_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_582_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > nat,C2: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
       => ( ! [X: int,Y4: int] :
              ( ( ord_less_int @ X @ Y4 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_583_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > int,C2: int] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_584_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
       => ( ! [X: int,Y4: int] :
              ( ( ord_less_int @ X @ Y4 )
             => ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_585_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z4 )
     => ~ ( ord_less_eq_nat @ T @ X6 ) ) ).

% minf(8)
thf(fact_586_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z4 )
     => ~ ( ord_less_eq_int @ T @ X6 ) ) ).

% minf(8)
thf(fact_587_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z4 )
     => ( ord_less_eq_nat @ X6 @ T ) ) ).

% minf(6)
thf(fact_588_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z4 )
     => ( ord_less_eq_int @ X6 @ T ) ) ).

% minf(6)
thf(fact_589_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z4 @ X6 )
     => ( ord_less_eq_nat @ T @ X6 ) ) ).

% pinf(8)
thf(fact_590_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z4 @ X6 )
     => ( ord_less_eq_int @ T @ X6 ) ) ).

% pinf(8)
thf(fact_591_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z4 @ X6 )
     => ~ ( ord_less_eq_nat @ X6 @ T ) ) ).

% pinf(6)
thf(fact_592_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z4 @ X6 )
     => ~ ( ord_less_eq_int @ X6 @ T ) ) ).

% pinf(6)
thf(fact_593_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 )
              & ! [X6: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X6 )
                    & ( ord_less_nat @ X6 @ C3 ) )
                 => ( P @ X6 ) )
              & ! [D: nat] :
                  ( ! [X: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X )
                        & ( ord_less_nat @ X @ D ) )
                     => ( P @ X ) )
                 => ( ord_less_eq_nat @ D @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_594_complete__interval,axiom,
    ! [A: int,B: int,P: int > $o] :
      ( ( ord_less_int @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C3: int] :
              ( ( ord_less_eq_int @ A @ C3 )
              & ( ord_less_eq_int @ C3 @ B )
              & ! [X6: int] :
                  ( ( ( ord_less_eq_int @ A @ X6 )
                    & ( ord_less_int @ X6 @ C3 ) )
                 => ( P @ X6 ) )
              & ! [D: int] :
                  ( ! [X: int] :
                      ( ( ( ord_less_eq_int @ A @ X )
                        & ( ord_less_int @ X @ D ) )
                     => ( P @ X ) )
                 => ( ord_less_eq_int @ D @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_595_of__int__of__nat,axiom,
    ( ring_1_of_int_int
    = ( ^ [K3: int] : ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K3 ) ) ) ) ) ).

% of_int_of_nat
thf(fact_596_nth__equal__first__eq,axiom,
    ! [X2: int,Xs: list_int,N: nat] :
      ( ~ ( member_int @ X2 @ ( set_int2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_int @ Xs ) )
       => ( ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ N )
            = X2 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_597_nth__equal__first__eq,axiom,
    ! [X2: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o,N: nat] :
      ( ~ ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s3379185846251495609od_a_o @ Xs ) )
       => ( ( ( nth_li3248255784598043854od_a_o @ ( cons_l9179691953475787517od_a_o @ X2 @ Xs ) @ N )
            = X2 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_598_nth__equal__first__eq,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,N: nat] :
      ( ~ ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s5686234912572693873on_nat @ Xs ) )
       => ( ( ( nth_Su8471564416988646406on_nat @ ( cons_S6797538523646374167on_nat @ X2 @ Xs ) @ N )
            = X2 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_599_nth__equal__first__eq,axiom,
    ! [X2: nat,Xs: list_nat,N: nat] :
      ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
            = X2 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_600_nth__equal__first__eq,axiom,
    ! [X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat,N: nat] :
      ( ~ ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s7490768528479217773on_nat @ Xs ) )
       => ( ( ( nth_Pr3336893728888831618on_nat @ ( cons_P1582596238967110705on_nat @ X2 @ Xs ) @ N )
            = X2 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_601_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_602_list_Oinject,axiom,
    ! [X21: int,X222: list_int,Y21: int,Y222: list_int] :
      ( ( ( cons_int @ X21 @ X222 )
        = ( cons_int @ Y21 @ Y222 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% list.inject
thf(fact_603_subsetI,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ! [X: nat] :
          ( ( member_nat @ X @ A2 )
         => ( member_nat @ X @ B5 ) )
     => ( ord_less_eq_set_nat @ A2 @ B5 ) ) ).

% subsetI
thf(fact_604_subsetI,axiom,
    ! [A2: set_li5011860890973139239od_a_o,B5: set_li5011860890973139239od_a_o] :
      ( ! [X: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ X @ A2 )
         => ( member5261019209154752368od_a_o @ X @ B5 ) )
     => ( ord_le1271121679792220359od_a_o @ A2 @ B5 ) ) ).

% subsetI
thf(fact_605_subsetI,axiom,
    ! [A2: set_Pr2535564044409765851on_nat,B5: set_Pr2535564044409765851on_nat] :
      ( ! [X: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X @ A2 )
         => ( member5906418179908956708on_nat @ X @ B5 ) )
     => ( ord_le6961793561435735931on_nat @ A2 @ B5 ) ) ).

% subsetI
thf(fact_606_subsetI,axiom,
    ! [A2: set_Su266273576555847555on_nat,B5: set_Su266273576555847555on_nat] :
      ( ! [X: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X @ A2 )
         => ( member8476201367316484836on_nat @ X @ B5 ) )
     => ( ord_le6595998496543023075on_nat @ A2 @ B5 ) ) ).

% subsetI
thf(fact_607_list__ex__simps_I1_J,axiom,
    ! [P: nat > $o,X2: nat,Xs: list_nat] :
      ( ( list_ex_nat @ P @ ( cons_nat @ X2 @ Xs ) )
      = ( ( P @ X2 )
        | ( list_ex_nat @ P @ Xs ) ) ) ).

% list_ex_simps(1)
thf(fact_608_list__ex__simps_I1_J,axiom,
    ! [P: int > $o,X2: int,Xs: list_int] :
      ( ( list_ex_int @ P @ ( cons_int @ X2 @ Xs ) )
      = ( ( P @ X2 )
        | ( list_ex_int @ P @ Xs ) ) ) ).

% list_ex_simps(1)
thf(fact_609_nth__Cons__0,axiom,
    ! [X2: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( nth_li3248255784598043854od_a_o @ ( cons_l9179691953475787517od_a_o @ X2 @ Xs ) @ zero_zero_nat )
      = X2 ) ).

% nth_Cons_0
thf(fact_610_nth__Cons__0,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( nth_Su8471564416988646406on_nat @ ( cons_S6797538523646374167on_nat @ X2 @ Xs ) @ zero_zero_nat )
      = X2 ) ).

% nth_Cons_0
thf(fact_611_nth__Cons__0,axiom,
    ! [X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( nth_Pr3336893728888831618on_nat @ ( cons_P1582596238967110705on_nat @ X2 @ Xs ) @ zero_zero_nat )
      = X2 ) ).

% nth_Cons_0
thf(fact_612_nth__Cons__0,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat )
      = X2 ) ).

% nth_Cons_0
thf(fact_613_nth__Cons__0,axiom,
    ! [X2: int,Xs: list_int] :
      ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ zero_zero_nat )
      = X2 ) ).

% nth_Cons_0
thf(fact_614_of__int__eq__0__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_1_of_int_int @ Z2 )
        = zero_zero_int )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_615_of__int__0__eq__iff,axiom,
    ! [Z2: int] :
      ( ( zero_zero_int
        = ( ring_1_of_int_int @ Z2 ) )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_616_of__int__0,axiom,
    ( ( ring_1_of_int_int @ zero_zero_int )
    = zero_zero_int ) ).

% of_int_0
thf(fact_617_of__int__le__iff,axiom,
    ! [W: int,Z2: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_eq_int @ W @ Z2 ) ) ).

% of_int_le_iff
thf(fact_618_of__int__less__iff,axiom,
    ! [W: int,Z2: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_int @ W @ Z2 ) ) ).

% of_int_less_iff
thf(fact_619_of__int__minus,axiom,
    ! [Z2: int] :
      ( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z2 ) )
      = ( uminus_uminus_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_minus
thf(fact_620_of__int__of__nat__eq,axiom,
    ! [N: nat] :
      ( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% of_int_of_nat_eq
thf(fact_621_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_lists_nat @ N @ nil_nat )
          = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_lists_nat @ N @ nil_nat )
          = nil_list_nat ) ) ) ).

% n_lists_Nil
thf(fact_622_Cons__listrel1__Cons,axiom,
    ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) @ ( listrel1_nat @ R ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R )
          & ( Xs = Ys ) )
        | ( ( X2 = Y2 )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_623_Cons__listrel1__Cons,axiom,
    ! [X2: int,Xs: list_int,Y2: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X2 @ Xs ) @ ( cons_int @ Y2 @ Ys ) ) @ ( listrel1_int @ R ) )
      = ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y2 ) @ R )
          & ( Xs = Ys ) )
        | ( ( X2 = Y2 )
          & ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_624_Cons__listrel1__Cons,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Y2: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( cons_S6797538523646374167on_nat @ X2 @ Xs ) @ ( cons_S6797538523646374167on_nat @ Y2 @ Ys ) ) @ ( listre1270753918075970734on_nat @ R ) )
      = ( ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X2 @ Y2 ) @ R )
          & ( Xs = Ys ) )
        | ( ( X2 = Y2 )
          & ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ Ys ) @ ( listre1270753918075970734on_nat @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_625_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
      = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_626_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_le_iff
thf(fact_627_of__int__less__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
      = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_628_of__int__0__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_less_iff
thf(fact_629_of__nat__nat,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
        = ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_nat_nat
thf(fact_630_in__mono,axiom,
    ! [A2: set_nat,B5: set_nat,X2: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B5 )
     => ( ( member_nat @ X2 @ A2 )
       => ( member_nat @ X2 @ B5 ) ) ) ).

% in_mono
thf(fact_631_in__mono,axiom,
    ! [A2: set_li5011860890973139239od_a_o,B5: set_li5011860890973139239od_a_o,X2: list_P3196869577824546119od_a_o] :
      ( ( ord_le1271121679792220359od_a_o @ A2 @ B5 )
     => ( ( member5261019209154752368od_a_o @ X2 @ A2 )
       => ( member5261019209154752368od_a_o @ X2 @ B5 ) ) ) ).

% in_mono
thf(fact_632_in__mono,axiom,
    ! [A2: set_Pr2535564044409765851on_nat,B5: set_Pr2535564044409765851on_nat,X2: produc1313295348606329723on_nat] :
      ( ( ord_le6961793561435735931on_nat @ A2 @ B5 )
     => ( ( member5906418179908956708on_nat @ X2 @ A2 )
       => ( member5906418179908956708on_nat @ X2 @ B5 ) ) ) ).

% in_mono
thf(fact_633_in__mono,axiom,
    ! [A2: set_Su266273576555847555on_nat,B5: set_Su266273576555847555on_nat,X2: sum_su4907927656612468301on_nat] :
      ( ( ord_le6595998496543023075on_nat @ A2 @ B5 )
     => ( ( member8476201367316484836on_nat @ X2 @ A2 )
       => ( member8476201367316484836on_nat @ X2 @ B5 ) ) ) ).

% in_mono
thf(fact_634_subsetD,axiom,
    ! [A2: set_nat,B5: set_nat,C2: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B5 )
     => ( ( member_nat @ C2 @ A2 )
       => ( member_nat @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_635_subsetD,axiom,
    ! [A2: set_li5011860890973139239od_a_o,B5: set_li5011860890973139239od_a_o,C2: list_P3196869577824546119od_a_o] :
      ( ( ord_le1271121679792220359od_a_o @ A2 @ B5 )
     => ( ( member5261019209154752368od_a_o @ C2 @ A2 )
       => ( member5261019209154752368od_a_o @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_636_subsetD,axiom,
    ! [A2: set_Pr2535564044409765851on_nat,B5: set_Pr2535564044409765851on_nat,C2: produc1313295348606329723on_nat] :
      ( ( ord_le6961793561435735931on_nat @ A2 @ B5 )
     => ( ( member5906418179908956708on_nat @ C2 @ A2 )
       => ( member5906418179908956708on_nat @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_637_subsetD,axiom,
    ! [A2: set_Su266273576555847555on_nat,B5: set_Su266273576555847555on_nat,C2: sum_su4907927656612468301on_nat] :
      ( ( ord_le6595998496543023075on_nat @ A2 @ B5 )
     => ( ( member8476201367316484836on_nat @ C2 @ A2 )
       => ( member8476201367316484836on_nat @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_638_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ A6 )
         => ( member_nat @ X3 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_639_subset__eq,axiom,
    ( ord_le1271121679792220359od_a_o
    = ( ^ [A6: set_li5011860890973139239od_a_o,B6: set_li5011860890973139239od_a_o] :
        ! [X3: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ X3 @ A6 )
         => ( member5261019209154752368od_a_o @ X3 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_640_subset__eq,axiom,
    ( ord_le6961793561435735931on_nat
    = ( ^ [A6: set_Pr2535564044409765851on_nat,B6: set_Pr2535564044409765851on_nat] :
        ! [X3: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X3 @ A6 )
         => ( member5906418179908956708on_nat @ X3 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_641_subset__eq,axiom,
    ( ord_le6595998496543023075on_nat
    = ( ^ [A6: set_Su266273576555847555on_nat,B6: set_Su266273576555847555on_nat] :
        ! [X3: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X3 @ A6 )
         => ( member8476201367316484836on_nat @ X3 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_642_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A6 )
         => ( member_nat @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_643_subset__iff,axiom,
    ( ord_le1271121679792220359od_a_o
    = ( ^ [A6: set_li5011860890973139239od_a_o,B6: set_li5011860890973139239od_a_o] :
        ! [T2: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ T2 @ A6 )
         => ( member5261019209154752368od_a_o @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_644_subset__iff,axiom,
    ( ord_le6961793561435735931on_nat
    = ( ^ [A6: set_Pr2535564044409765851on_nat,B6: set_Pr2535564044409765851on_nat] :
        ! [T2: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ T2 @ A6 )
         => ( member5906418179908956708on_nat @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_645_subset__iff,axiom,
    ( ord_le6595998496543023075on_nat
    = ( ^ [A6: set_Su266273576555847555on_nat,B6: set_Su266273576555847555on_nat] :
        ! [T2: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ T2 @ A6 )
         => ( member8476201367316484836on_nat @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_646_verit__la__generic,axiom,
    ! [A: int,X2: int] :
      ( ( ord_less_eq_int @ A @ X2 )
      | ( A = X2 )
      | ( ord_less_eq_int @ X2 @ A ) ) ).

% verit_la_generic
thf(fact_647_not__Cons__self2,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( cons_nat @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_648_not__Cons__self2,axiom,
    ! [X2: int,Xs: list_int] :
      ( ( cons_int @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_649_transpose_Ocases,axiom,
    ! [X2: list_list_nat] :
      ( ( X2 != nil_list_nat )
     => ( ! [Xss2: list_list_nat] :
            ( X2
           != ( cons_list_nat @ nil_nat @ Xss2 ) )
       => ~ ! [X: nat,Xs3: list_nat,Xss2: list_list_nat] :
              ( X2
             != ( cons_list_nat @ ( cons_nat @ X @ Xs3 ) @ Xss2 ) ) ) ) ).

% transpose.cases
thf(fact_650_transpose_Ocases,axiom,
    ! [X2: list_list_int] :
      ( ( X2 != nil_list_int )
     => ( ! [Xss2: list_list_int] :
            ( X2
           != ( cons_list_int @ nil_int @ Xss2 ) )
       => ~ ! [X: int,Xs3: list_int,Xss2: list_list_int] :
              ( X2
             != ( cons_list_int @ ( cons_int @ X @ Xs3 ) @ Xss2 ) ) ) ) ).

% transpose.cases
thf(fact_651_sorted__wrt_Ocases,axiom,
    ! [X2: produc254973753779126261st_nat] :
      ( ! [P5: nat > nat > $o] :
          ( X2
         != ( produc4727192421694094319st_nat @ P5 @ nil_nat ) )
     => ~ ! [P5: nat > nat > $o,X: nat,Ys4: list_nat] :
            ( X2
           != ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X @ Ys4 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_652_sorted__wrt_Ocases,axiom,
    ! [X2: produc5834231552977413017st_int] :
      ( ! [P5: int > int > $o] :
          ( X2
         != ( produc8618682346314911123st_int @ P5 @ nil_int ) )
     => ~ ! [P5: int > int > $o,X: int,Ys4: list_int] :
            ( X2
           != ( produc8618682346314911123st_int @ P5 @ ( cons_int @ X @ Ys4 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_653_successively_Ocases,axiom,
    ! [X2: produc254973753779126261st_nat] :
      ( ! [P5: nat > nat > $o] :
          ( X2
         != ( produc4727192421694094319st_nat @ P5 @ nil_nat ) )
     => ( ! [P5: nat > nat > $o,X: nat] :
            ( X2
           != ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X @ nil_nat ) ) )
       => ~ ! [P5: nat > nat > $o,X: nat,Y4: nat,Xs3: list_nat] :
              ( X2
             != ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X @ ( cons_nat @ Y4 @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_654_successively_Ocases,axiom,
    ! [X2: produc5834231552977413017st_int] :
      ( ! [P5: int > int > $o] :
          ( X2
         != ( produc8618682346314911123st_int @ P5 @ nil_int ) )
     => ( ! [P5: int > int > $o,X: int] :
            ( X2
           != ( produc8618682346314911123st_int @ P5 @ ( cons_int @ X @ nil_int ) ) )
       => ~ ! [P5: int > int > $o,X: int,Y4: int,Xs3: list_int] :
              ( X2
             != ( produc8618682346314911123st_int @ P5 @ ( cons_int @ X @ ( cons_int @ Y4 @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_655_list_Odistinct_I1_J,axiom,
    ! [X21: nat,X222: list_nat] :
      ( nil_nat
     != ( cons_nat @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_656_list_Odistinct_I1_J,axiom,
    ! [X21: int,X222: list_int] :
      ( nil_int
     != ( cons_int @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_657_list_OdiscI,axiom,
    ! [List: list_nat,X21: nat,X222: list_nat] :
      ( ( List
        = ( cons_nat @ X21 @ X222 ) )
     => ( List != nil_nat ) ) ).

% list.discI
thf(fact_658_list_OdiscI,axiom,
    ! [List: list_int,X21: int,X222: list_int] :
      ( ( List
        = ( cons_int @ X21 @ X222 ) )
     => ( List != nil_int ) ) ).

% list.discI
thf(fact_659_list_Oexhaust,axiom,
    ! [Y2: list_nat] :
      ( ( Y2 != nil_nat )
     => ~ ! [X212: nat,X223: list_nat] :
            ( Y2
           != ( cons_nat @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_660_list_Oexhaust,axiom,
    ! [Y2: list_int] :
      ( ( Y2 != nil_int )
     => ~ ! [X212: int,X223: list_int] :
            ( Y2
           != ( cons_int @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_661_min__list_Ocases,axiom,
    ! [X2: list_nat] :
      ( ! [X: nat,Xs3: list_nat] :
          ( X2
         != ( cons_nat @ X @ Xs3 ) )
     => ( X2 = nil_nat ) ) ).

% min_list.cases
thf(fact_662_min__list_Ocases,axiom,
    ! [X2: list_int] :
      ( ! [X: int,Xs3: list_int] :
          ( X2
         != ( cons_int @ X @ Xs3 ) )
     => ( X2 = nil_int ) ) ).

% min_list.cases
thf(fact_663_remdups__adj_Ocases,axiom,
    ! [X2: list_nat] :
      ( ( X2 != nil_nat )
     => ( ! [X: nat] :
            ( X2
           != ( cons_nat @ X @ nil_nat ) )
       => ~ ! [X: nat,Y4: nat,Xs3: list_nat] :
              ( X2
             != ( cons_nat @ X @ ( cons_nat @ Y4 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_664_remdups__adj_Ocases,axiom,
    ! [X2: list_int] :
      ( ( X2 != nil_int )
     => ( ! [X: int] :
            ( X2
           != ( cons_int @ X @ nil_int ) )
       => ~ ! [X: int,Y4: int,Xs3: list_int] :
              ( X2
             != ( cons_int @ X @ ( cons_int @ Y4 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_665_neq__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
      = ( ? [Y3: nat,Ys2: list_nat] :
            ( Xs
            = ( cons_nat @ Y3 @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_666_neq__Nil__conv,axiom,
    ! [Xs: list_int] :
      ( ( Xs != nil_int )
      = ( ? [Y3: int,Ys2: list_int] :
            ( Xs
            = ( cons_int @ Y3 @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_667_list__induct2_H,axiom,
    ! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
      ( ( P @ nil_nat @ nil_nat )
     => ( ! [X: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X @ Xs3 ) @ nil_nat )
       => ( ! [Y4: nat,Ys4: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y4 @ Ys4 ) )
         => ( ! [X: nat,Xs3: list_nat,Y4: nat,Ys4: list_nat] :
                ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y4 @ Ys4 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_668_list__induct2_H,axiom,
    ! [P: list_nat > list_int > $o,Xs: list_nat,Ys: list_int] :
      ( ( P @ nil_nat @ nil_int )
     => ( ! [X: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X @ Xs3 ) @ nil_int )
       => ( ! [Y4: int,Ys4: list_int] : ( P @ nil_nat @ ( cons_int @ Y4 @ Ys4 ) )
         => ( ! [X: nat,Xs3: list_nat,Y4: int,Ys4: list_int] :
                ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_int @ Y4 @ Ys4 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_669_list__induct2_H,axiom,
    ! [P: list_int > list_nat > $o,Xs: list_int,Ys: list_nat] :
      ( ( P @ nil_int @ nil_nat )
     => ( ! [X: int,Xs3: list_int] : ( P @ ( cons_int @ X @ Xs3 ) @ nil_nat )
       => ( ! [Y4: nat,Ys4: list_nat] : ( P @ nil_int @ ( cons_nat @ Y4 @ Ys4 ) )
         => ( ! [X: int,Xs3: list_int,Y4: nat,Ys4: list_nat] :
                ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_int @ X @ Xs3 ) @ ( cons_nat @ Y4 @ Ys4 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_670_list__induct2_H,axiom,
    ! [P: list_int > list_int > $o,Xs: list_int,Ys: list_int] :
      ( ( P @ nil_int @ nil_int )
     => ( ! [X: int,Xs3: list_int] : ( P @ ( cons_int @ X @ Xs3 ) @ nil_int )
       => ( ! [Y4: int,Ys4: list_int] : ( P @ nil_int @ ( cons_int @ Y4 @ Ys4 ) )
         => ( ! [X: int,Xs3: list_int,Y4: int,Ys4: list_int] :
                ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y4 @ Ys4 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_671_list__nonempty__induct,axiom,
    ! [Xs: list_nat,P: list_nat > $o] :
      ( ( Xs != nil_nat )
     => ( ! [X: nat] : ( P @ ( cons_nat @ X @ nil_nat ) )
       => ( ! [X: nat,Xs3: list_nat] :
              ( ( Xs3 != nil_nat )
             => ( ( P @ Xs3 )
               => ( P @ ( cons_nat @ X @ Xs3 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_672_list__nonempty__induct,axiom,
    ! [Xs: list_int,P: list_int > $o] :
      ( ( Xs != nil_int )
     => ( ! [X: int] : ( P @ ( cons_int @ X @ nil_int ) )
       => ( ! [X: int,Xs3: list_int] :
              ( ( Xs3 != nil_int )
             => ( ( P @ Xs3 )
               => ( P @ ( cons_int @ X @ Xs3 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_673_list_Oset__intros_I2_J,axiom,
    ! [Y2: list_P3196869577824546119od_a_o,X222: list_l8336264072933094989od_a_o,X21: list_P3196869577824546119od_a_o] :
      ( ( member5261019209154752368od_a_o @ Y2 @ ( set_li2629784761229999580od_a_o @ X222 ) )
     => ( member5261019209154752368od_a_o @ Y2 @ ( set_li2629784761229999580od_a_o @ ( cons_l9179691953475787517od_a_o @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_674_list_Oset__intros_I2_J,axiom,
    ! [Y2: produc1313295348606329723on_nat,X222: list_P9127870862658512897on_nat,X21: produc1313295348606329723on_nat] :
      ( ( member5906418179908956708on_nat @ Y2 @ ( set_Pr937530902923597456on_nat @ X222 ) )
     => ( member5906418179908956708on_nat @ Y2 @ ( set_Pr937530902923597456on_nat @ ( cons_P1582596238967110705on_nat @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_675_list_Oset__intros_I2_J,axiom,
    ! [Y2: sum_su4907927656612468301on_nat,X222: list_S5253404628753938141on_nat,X21: sum_su4907927656612468301on_nat] :
      ( ( member8476201367316484836on_nat @ Y2 @ ( set_Su4848624856859743608on_nat @ X222 ) )
     => ( member8476201367316484836on_nat @ Y2 @ ( set_Su4848624856859743608on_nat @ ( cons_S6797538523646374167on_nat @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_676_list_Oset__intros_I2_J,axiom,
    ! [Y2: nat,X222: list_nat,X21: nat] :
      ( ( member_nat @ Y2 @ ( set_nat2 @ X222 ) )
     => ( member_nat @ Y2 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_677_list_Oset__intros_I2_J,axiom,
    ! [Y2: int,X222: list_int,X21: int] :
      ( ( member_int @ Y2 @ ( set_int2 @ X222 ) )
     => ( member_int @ Y2 @ ( set_int2 @ ( cons_int @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_678_list_Oset__intros_I1_J,axiom,
    ! [X21: list_P3196869577824546119od_a_o,X222: list_l8336264072933094989od_a_o] : ( member5261019209154752368od_a_o @ X21 @ ( set_li2629784761229999580od_a_o @ ( cons_l9179691953475787517od_a_o @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_679_list_Oset__intros_I1_J,axiom,
    ! [X21: produc1313295348606329723on_nat,X222: list_P9127870862658512897on_nat] : ( member5906418179908956708on_nat @ X21 @ ( set_Pr937530902923597456on_nat @ ( cons_P1582596238967110705on_nat @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_680_list_Oset__intros_I1_J,axiom,
    ! [X21: sum_su4907927656612468301on_nat,X222: list_S5253404628753938141on_nat] : ( member8476201367316484836on_nat @ X21 @ ( set_Su4848624856859743608on_nat @ ( cons_S6797538523646374167on_nat @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_681_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_682_list_Oset__intros_I1_J,axiom,
    ! [X21: int,X222: list_int] : ( member_int @ X21 @ ( set_int2 @ ( cons_int @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_683_list_Oset__cases,axiom,
    ! [E: list_P3196869577824546119od_a_o,A: list_l8336264072933094989od_a_o] :
      ( ( member5261019209154752368od_a_o @ E @ ( set_li2629784761229999580od_a_o @ A ) )
     => ( ! [Z22: list_l8336264072933094989od_a_o] :
            ( A
           != ( cons_l9179691953475787517od_a_o @ E @ Z22 ) )
       => ~ ! [Z1: list_P3196869577824546119od_a_o,Z22: list_l8336264072933094989od_a_o] :
              ( ( A
                = ( cons_l9179691953475787517od_a_o @ Z1 @ Z22 ) )
             => ~ ( member5261019209154752368od_a_o @ E @ ( set_li2629784761229999580od_a_o @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_684_list_Oset__cases,axiom,
    ! [E: produc1313295348606329723on_nat,A: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ E @ ( set_Pr937530902923597456on_nat @ A ) )
     => ( ! [Z22: list_P9127870862658512897on_nat] :
            ( A
           != ( cons_P1582596238967110705on_nat @ E @ Z22 ) )
       => ~ ! [Z1: produc1313295348606329723on_nat,Z22: list_P9127870862658512897on_nat] :
              ( ( A
                = ( cons_P1582596238967110705on_nat @ Z1 @ Z22 ) )
             => ~ ( member5906418179908956708on_nat @ E @ ( set_Pr937530902923597456on_nat @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_685_list_Oset__cases,axiom,
    ! [E: sum_su4907927656612468301on_nat,A: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ E @ ( set_Su4848624856859743608on_nat @ A ) )
     => ( ! [Z22: list_S5253404628753938141on_nat] :
            ( A
           != ( cons_S6797538523646374167on_nat @ E @ Z22 ) )
       => ~ ! [Z1: sum_su4907927656612468301on_nat,Z22: list_S5253404628753938141on_nat] :
              ( ( A
                = ( cons_S6797538523646374167on_nat @ Z1 @ Z22 ) )
             => ~ ( member8476201367316484836on_nat @ E @ ( set_Su4848624856859743608on_nat @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_686_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_687_list_Oset__cases,axiom,
    ! [E: int,A: list_int] :
      ( ( member_int @ E @ ( set_int2 @ A ) )
     => ( ! [Z22: list_int] :
            ( A
           != ( cons_int @ E @ Z22 ) )
       => ~ ! [Z1: int,Z22: list_int] :
              ( ( A
                = ( cons_int @ Z1 @ Z22 ) )
             => ~ ( member_int @ E @ ( set_int2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_688_set__ConsD,axiom,
    ! [Y2: list_P3196869577824546119od_a_o,X2: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( member5261019209154752368od_a_o @ Y2 @ ( set_li2629784761229999580od_a_o @ ( cons_l9179691953475787517od_a_o @ X2 @ Xs ) ) )
     => ( ( Y2 = X2 )
        | ( member5261019209154752368od_a_o @ Y2 @ ( set_li2629784761229999580od_a_o @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_689_set__ConsD,axiom,
    ! [Y2: produc1313295348606329723on_nat,X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ Y2 @ ( set_Pr937530902923597456on_nat @ ( cons_P1582596238967110705on_nat @ X2 @ Xs ) ) )
     => ( ( Y2 = X2 )
        | ( member5906418179908956708on_nat @ Y2 @ ( set_Pr937530902923597456on_nat @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_690_set__ConsD,axiom,
    ! [Y2: sum_su4907927656612468301on_nat,X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ Y2 @ ( set_Su4848624856859743608on_nat @ ( cons_S6797538523646374167on_nat @ X2 @ Xs ) ) )
     => ( ( Y2 = X2 )
        | ( member8476201367316484836on_nat @ Y2 @ ( set_Su4848624856859743608on_nat @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_691_set__ConsD,axiom,
    ! [Y2: nat,X2: nat,Xs: list_nat] :
      ( ( member_nat @ Y2 @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) )
     => ( ( Y2 = X2 )
        | ( member_nat @ Y2 @ ( set_nat2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_692_set__ConsD,axiom,
    ! [Y2: int,X2: int,Xs: list_int] :
      ( ( member_int @ Y2 @ ( set_int2 @ ( cons_int @ X2 @ Xs ) ) )
     => ( ( Y2 = X2 )
        | ( member_int @ Y2 @ ( set_int2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_693_set__subset__Cons,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X2: list_P3196869577824546119od_a_o] : ( ord_le1271121679792220359od_a_o @ ( set_li2629784761229999580od_a_o @ Xs ) @ ( set_li2629784761229999580od_a_o @ ( cons_l9179691953475787517od_a_o @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_694_set__subset__Cons,axiom,
    ! [Xs: list_P9127870862658512897on_nat,X2: produc1313295348606329723on_nat] : ( ord_le6961793561435735931on_nat @ ( set_Pr937530902923597456on_nat @ Xs ) @ ( set_Pr937530902923597456on_nat @ ( cons_P1582596238967110705on_nat @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_695_set__subset__Cons,axiom,
    ! [Xs: list_S5253404628753938141on_nat,X2: sum_su4907927656612468301on_nat] : ( ord_le6595998496543023075on_nat @ ( set_Su4848624856859743608on_nat @ Xs ) @ ( set_Su4848624856859743608on_nat @ ( cons_S6797538523646374167on_nat @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_696_set__subset__Cons,axiom,
    ! [Xs: list_nat,X2: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_697_set__subset__Cons,axiom,
    ! [Xs: list_int,X2: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ ( cons_int @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_698_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_699_distinct__length__2__or__more,axiom,
    ! [A: int,B: int,Xs: list_int] :
      ( ( distinct_int @ ( cons_int @ A @ ( cons_int @ B @ Xs ) ) )
      = ( ( A != B )
        & ( distinct_int @ ( cons_int @ A @ Xs ) )
        & ( distinct_int @ ( cons_int @ B @ Xs ) ) ) ) ).

% distinct_length_2_or_more
thf(fact_700_removeAll_Osimps_I2_J,axiom,
    ! [X2: nat,Y2: nat,Xs: list_nat] :
      ( ( ( X2 = Y2 )
       => ( ( removeAll_nat @ X2 @ ( cons_nat @ Y2 @ Xs ) )
          = ( removeAll_nat @ X2 @ Xs ) ) )
      & ( ( X2 != Y2 )
       => ( ( removeAll_nat @ X2 @ ( cons_nat @ Y2 @ Xs ) )
          = ( cons_nat @ Y2 @ ( removeAll_nat @ X2 @ Xs ) ) ) ) ) ).

% removeAll.simps(2)
thf(fact_701_removeAll_Osimps_I2_J,axiom,
    ! [X2: int,Y2: int,Xs: list_int] :
      ( ( ( X2 = Y2 )
       => ( ( removeAll_int @ X2 @ ( cons_int @ Y2 @ Xs ) )
          = ( removeAll_int @ X2 @ Xs ) ) )
      & ( ( X2 != Y2 )
       => ( ( removeAll_int @ X2 @ ( cons_int @ Y2 @ Xs ) )
          = ( cons_int @ Y2 @ ( removeAll_int @ X2 @ Xs ) ) ) ) ) ).

% removeAll.simps(2)
thf(fact_702_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_int,Ws: list_int,P: list_int > list_int > list_int > list_int > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( ( size_size_list_int @ Zs )
            = ( size_size_list_int @ Ws ) )
         => ( ( P @ nil_int @ nil_int @ nil_int @ nil_int )
           => ( ! [X: int,Xs3: list_int,Y4: int,Ys4: list_int,Z4: int,Zs2: list_int,W2: int,Ws2: list_int] :
                  ( ( ( size_size_list_int @ Xs3 )
                    = ( size_size_list_int @ Ys4 ) )
                 => ( ( ( size_size_list_int @ Ys4 )
                      = ( size_size_list_int @ Zs2 ) )
                   => ( ( ( size_size_list_int @ Zs2 )
                        = ( size_size_list_int @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y4 @ Ys4 ) @ ( cons_int @ Z4 @ Zs2 ) @ ( cons_int @ W2 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_703_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_int,Ws: list_nat,P: list_int > list_int > list_int > list_nat > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( ( size_size_list_int @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_int @ nil_int @ nil_int @ nil_nat )
           => ( ! [X: int,Xs3: list_int,Y4: int,Ys4: list_int,Z4: int,Zs2: list_int,W2: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_int @ Xs3 )
                    = ( size_size_list_int @ Ys4 ) )
                 => ( ( ( size_size_list_int @ Ys4 )
                      = ( size_size_list_int @ Zs2 ) )
                   => ( ( ( size_size_list_int @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y4 @ Ys4 ) @ ( cons_int @ Z4 @ Zs2 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_704_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_nat,Ws: list_int,P: list_int > list_int > list_nat > list_int > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_int @ Ws ) )
         => ( ( P @ nil_int @ nil_int @ nil_nat @ nil_int )
           => ( ! [X: int,Xs3: list_int,Y4: int,Ys4: list_int,Z4: nat,Zs2: list_nat,W2: int,Ws2: list_int] :
                  ( ( ( size_size_list_int @ Xs3 )
                    = ( size_size_list_int @ Ys4 ) )
                 => ( ( ( size_size_list_int @ Ys4 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_int @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y4 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs2 ) @ ( cons_int @ W2 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_705_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_nat,Ws: list_nat,P: list_int > list_int > list_nat > list_nat > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_int @ nil_int @ nil_nat @ nil_nat )
           => ( ! [X: int,Xs3: list_int,Y4: int,Ys4: list_int,Z4: nat,Zs2: list_nat,W2: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_int @ Xs3 )
                    = ( size_size_list_int @ Ys4 ) )
                 => ( ( ( size_size_list_int @ Ys4 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y4 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs2 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_706_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_nat,Zs: list_int,Ws: list_int,P: list_int > list_nat > list_int > list_int > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( ( size_size_list_int @ Zs )
            = ( size_size_list_int @ Ws ) )
         => ( ( P @ nil_int @ nil_nat @ nil_int @ nil_int )
           => ( ! [X: int,Xs3: list_int,Y4: nat,Ys4: list_nat,Z4: int,Zs2: list_int,W2: int,Ws2: list_int] :
                  ( ( ( size_size_list_int @ Xs3 )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_size_list_int @ Zs2 ) )
                   => ( ( ( size_size_list_int @ Zs2 )
                        = ( size_size_list_int @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_int @ X @ Xs3 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_int @ Z4 @ Zs2 ) @ ( cons_int @ W2 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_707_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_nat,Zs: list_int,Ws: list_nat,P: list_int > list_nat > list_int > list_nat > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( ( size_size_list_int @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_int @ nil_nat @ nil_int @ nil_nat )
           => ( ! [X: int,Xs3: list_int,Y4: nat,Ys4: list_nat,Z4: int,Zs2: list_int,W2: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_int @ Xs3 )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_size_list_int @ Zs2 ) )
                   => ( ( ( size_size_list_int @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_int @ X @ Xs3 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_int @ Z4 @ Zs2 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_708_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_nat,Zs: list_nat,Ws: list_int,P: list_int > list_nat > list_nat > list_int > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_int @ Ws ) )
         => ( ( P @ nil_int @ nil_nat @ nil_nat @ nil_int )
           => ( ! [X: int,Xs3: list_int,Y4: nat,Ys4: list_nat,Z4: nat,Zs2: list_nat,W2: int,Ws2: list_int] :
                  ( ( ( size_size_list_int @ Xs3 )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_int @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_int @ X @ Xs3 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs2 ) @ ( cons_int @ W2 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_709_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_nat,Zs: list_nat,Ws: list_nat,P: list_int > list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_int @ nil_nat @ nil_nat @ nil_nat )
           => ( ! [X: int,Xs3: list_int,Y4: nat,Ys4: list_nat,Z4: nat,Zs2: list_nat,W2: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_int @ Xs3 )
                    = ( size_size_list_nat @ Ys4 ) )
                 => ( ( ( size_size_list_nat @ Ys4 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_int @ X @ Xs3 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs2 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_710_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_int,Ws: list_int,P: list_nat > list_int > list_int > list_int > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( ( size_size_list_int @ Zs )
            = ( size_size_list_int @ Ws ) )
         => ( ( P @ nil_nat @ nil_int @ nil_int @ nil_int )
           => ( ! [X: nat,Xs3: list_nat,Y4: int,Ys4: list_int,Z4: int,Zs2: list_int,W2: int,Ws2: list_int] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_int @ Ys4 ) )
                 => ( ( ( size_size_list_int @ Ys4 )
                      = ( size_size_list_int @ Zs2 ) )
                   => ( ( ( size_size_list_int @ Zs2 )
                        = ( size_size_list_int @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_int @ Y4 @ Ys4 ) @ ( cons_int @ Z4 @ Zs2 ) @ ( cons_int @ W2 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_711_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_int,Ws: list_nat,P: list_nat > list_int > list_int > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( ( size_size_list_int @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_int @ nil_int @ nil_nat )
           => ( ! [X: nat,Xs3: list_nat,Y4: int,Ys4: list_int,Z4: int,Zs2: list_int,W2: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_int @ Ys4 ) )
                 => ( ( ( size_size_list_int @ Ys4 )
                      = ( size_size_list_int @ Zs2 ) )
                   => ( ( ( size_size_list_int @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_int @ Y4 @ Ys4 ) @ ( cons_int @ Z4 @ Zs2 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_712_list__induct3,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_int,P: list_int > list_int > list_int > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( P @ nil_int @ nil_int @ nil_int )
         => ( ! [X: int,Xs3: list_int,Y4: int,Ys4: list_int,Z4: int,Zs2: list_int] :
                ( ( ( size_size_list_int @ Xs3 )
                  = ( size_size_list_int @ Ys4 ) )
               => ( ( ( size_size_list_int @ Ys4 )
                    = ( size_size_list_int @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y4 @ Ys4 ) @ ( cons_int @ Z4 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_713_list__induct3,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_nat,P: list_int > list_int > list_nat > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_int @ nil_int @ nil_nat )
         => ( ! [X: int,Xs3: list_int,Y4: int,Ys4: list_int,Z4: nat,Zs2: list_nat] :
                ( ( ( size_size_list_int @ Xs3 )
                  = ( size_size_list_int @ Ys4 ) )
               => ( ( ( size_size_list_int @ Ys4 )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y4 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_714_list__induct3,axiom,
    ! [Xs: list_int,Ys: list_nat,Zs: list_int,P: list_int > list_nat > list_int > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( P @ nil_int @ nil_nat @ nil_int )
         => ( ! [X: int,Xs3: list_int,Y4: nat,Ys4: list_nat,Z4: int,Zs2: list_int] :
                ( ( ( size_size_list_int @ Xs3 )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ( ( size_size_list_nat @ Ys4 )
                    = ( size_size_list_int @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_int @ X @ Xs3 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_int @ Z4 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_715_list__induct3,axiom,
    ! [Xs: list_int,Ys: list_nat,Zs: list_nat,P: list_int > list_nat > list_nat > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_int @ nil_nat @ nil_nat )
         => ( ! [X: int,Xs3: list_int,Y4: nat,Ys4: list_nat,Z4: nat,Zs2: list_nat] :
                ( ( ( size_size_list_int @ Xs3 )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ( ( size_size_list_nat @ Ys4 )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_int @ X @ Xs3 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_716_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_int,P: list_nat > list_int > list_int > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( P @ nil_nat @ nil_int @ nil_int )
         => ( ! [X: nat,Xs3: list_nat,Y4: int,Ys4: list_int,Z4: int,Zs2: list_int] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_size_list_int @ Ys4 ) )
               => ( ( ( size_size_list_int @ Ys4 )
                    = ( size_size_list_int @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_int @ Y4 @ Ys4 ) @ ( cons_int @ Z4 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_717_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_nat,P: list_nat > list_int > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_nat @ nil_int @ nil_nat )
         => ( ! [X: nat,Xs3: list_nat,Y4: int,Ys4: list_int,Z4: nat,Zs2: list_nat] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_size_list_int @ Ys4 ) )
               => ( ( ( size_size_list_int @ Ys4 )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_int @ Y4 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_718_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_int,P: list_nat > list_nat > list_int > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( P @ nil_nat @ nil_nat @ nil_int )
         => ( ! [X: nat,Xs3: list_nat,Y4: nat,Ys4: list_nat,Z4: int,Zs2: list_int] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ( ( size_size_list_nat @ Ys4 )
                    = ( size_size_list_int @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_int @ Z4 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_719_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: 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 @ Zs ) )
       => ( ( P @ nil_nat @ nil_nat @ nil_nat )
         => ( ! [X: nat,Xs3: list_nat,Y4: nat,Ys4: list_nat,Z4: nat,Zs2: list_nat] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ( ( size_size_list_nat @ Ys4 )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_nat @ Z4 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_720_list__induct3,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_l8336264072933094989od_a_o,P: list_int > list_int > list_l8336264072933094989od_a_o > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_s3379185846251495609od_a_o @ Zs ) )
       => ( ( P @ nil_int @ nil_int @ nil_li1372968520647675725od_a_o )
         => ( ! [X: int,Xs3: list_int,Y4: int,Ys4: list_int,Z4: list_P3196869577824546119od_a_o,Zs2: list_l8336264072933094989od_a_o] :
                ( ( ( size_size_list_int @ Xs3 )
                  = ( size_size_list_int @ Ys4 ) )
               => ( ( ( size_size_list_int @ Ys4 )
                    = ( size_s3379185846251495609od_a_o @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y4 @ Ys4 ) @ ( cons_l9179691953475787517od_a_o @ Z4 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_721_list__induct3,axiom,
    ! [Xs: list_int,Ys: list_l8336264072933094989od_a_o,Zs: list_int,P: list_int > list_l8336264072933094989od_a_o > list_int > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) )
     => ( ( ( size_s3379185846251495609od_a_o @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( P @ nil_int @ nil_li1372968520647675725od_a_o @ nil_int )
         => ( ! [X: int,Xs3: list_int,Y4: list_P3196869577824546119od_a_o,Ys4: list_l8336264072933094989od_a_o,Z4: int,Zs2: list_int] :
                ( ( ( size_size_list_int @ Xs3 )
                  = ( size_s3379185846251495609od_a_o @ Ys4 ) )
               => ( ( ( size_s3379185846251495609od_a_o @ Ys4 )
                    = ( size_size_list_int @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_int @ X @ Xs3 ) @ ( cons_l9179691953475787517od_a_o @ Y4 @ Ys4 ) @ ( cons_int @ Z4 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_722_list__induct2,axiom,
    ! [Xs: list_int,Ys: list_int,P: list_int > list_int > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( P @ nil_int @ nil_int )
       => ( ! [X: int,Xs3: list_int,Y4: int,Ys4: list_int] :
              ( ( ( size_size_list_int @ Xs3 )
                = ( size_size_list_int @ Ys4 ) )
             => ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y4 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_723_list__induct2,axiom,
    ! [Xs: list_int,Ys: list_nat,P: list_int > list_nat > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_int @ nil_nat )
       => ( ! [X: int,Xs3: list_int,Y4: nat,Ys4: list_nat] :
              ( ( ( size_size_list_int @ Xs3 )
                = ( size_size_list_nat @ Ys4 ) )
             => ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_int @ X @ Xs3 ) @ ( cons_nat @ Y4 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_724_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_int,P: list_nat > list_int > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( P @ nil_nat @ nil_int )
       => ( ! [X: nat,Xs3: list_nat,Y4: int,Ys4: list_int] :
              ( ( ( size_size_list_nat @ Xs3 )
                = ( size_size_list_int @ Ys4 ) )
             => ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_int @ Y4 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_725_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 )
       => ( ! [X: nat,Xs3: list_nat,Y4: nat,Ys4: list_nat] :
              ( ( ( size_size_list_nat @ Xs3 )
                = ( size_size_list_nat @ Ys4 ) )
             => ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y4 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_726_list__induct2,axiom,
    ! [Xs: list_int,Ys: list_l8336264072933094989od_a_o,P: list_int > list_l8336264072933094989od_a_o > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) )
     => ( ( P @ nil_int @ nil_li1372968520647675725od_a_o )
       => ( ! [X: int,Xs3: list_int,Y4: list_P3196869577824546119od_a_o,Ys4: list_l8336264072933094989od_a_o] :
              ( ( ( size_size_list_int @ Xs3 )
                = ( size_s3379185846251495609od_a_o @ Ys4 ) )
             => ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_int @ X @ Xs3 ) @ ( cons_l9179691953475787517od_a_o @ Y4 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_727_list__induct2,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_int,P: list_l8336264072933094989od_a_o > list_int > $o] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( P @ nil_li1372968520647675725od_a_o @ nil_int )
       => ( ! [X: list_P3196869577824546119od_a_o,Xs3: list_l8336264072933094989od_a_o,Y4: int,Ys4: list_int] :
              ( ( ( size_s3379185846251495609od_a_o @ Xs3 )
                = ( size_size_list_int @ Ys4 ) )
             => ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_l9179691953475787517od_a_o @ X @ Xs3 ) @ ( cons_int @ Y4 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_728_list__induct2,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_nat,P: list_l8336264072933094989od_a_o > list_nat > $o] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_li1372968520647675725od_a_o @ nil_nat )
       => ( ! [X: list_P3196869577824546119od_a_o,Xs3: list_l8336264072933094989od_a_o,Y4: nat,Ys4: list_nat] :
              ( ( ( size_s3379185846251495609od_a_o @ Xs3 )
                = ( size_size_list_nat @ Ys4 ) )
             => ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_l9179691953475787517od_a_o @ X @ Xs3 ) @ ( cons_nat @ Y4 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_729_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_l8336264072933094989od_a_o,P: list_nat > list_l8336264072933094989od_a_o > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) )
     => ( ( P @ nil_nat @ nil_li1372968520647675725od_a_o )
       => ( ! [X: nat,Xs3: list_nat,Y4: list_P3196869577824546119od_a_o,Ys4: list_l8336264072933094989od_a_o] :
              ( ( ( size_size_list_nat @ Xs3 )
                = ( size_s3379185846251495609od_a_o @ Ys4 ) )
             => ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_l9179691953475787517od_a_o @ Y4 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_730_list__induct2,axiom,
    ! [Xs: list_int,Ys: list_S5253404628753938141on_nat,P: list_int > list_S5253404628753938141on_nat > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) )
     => ( ( P @ nil_int @ nil_Su3078906344068544711on_nat )
       => ( ! [X: int,Xs3: list_int,Y4: sum_su4907927656612468301on_nat,Ys4: list_S5253404628753938141on_nat] :
              ( ( ( size_size_list_int @ Xs3 )
                = ( size_s5686234912572693873on_nat @ Ys4 ) )
             => ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_int @ X @ Xs3 ) @ ( cons_S6797538523646374167on_nat @ Y4 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_731_list__induct2,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_int,P: list_S5253404628753938141on_nat > list_int > $o] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( P @ nil_Su3078906344068544711on_nat @ nil_int )
       => ( ! [X: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat,Y4: int,Ys4: list_int] :
              ( ( ( size_s5686234912572693873on_nat @ Xs3 )
                = ( size_size_list_int @ Ys4 ) )
             => ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_S6797538523646374167on_nat @ X @ Xs3 ) @ ( cons_int @ Y4 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_732_impossible__Cons,axiom,
    ! [Xs: list_int,Ys: list_int,X2: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) )
     => ( Xs
       != ( cons_int @ X2 @ Ys ) ) ) ).

% impossible_Cons
thf(fact_733_impossible__Cons,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,X2: list_P3196869577824546119od_a_o] :
      ( ( ord_less_eq_nat @ ( size_s3379185846251495609od_a_o @ Xs ) @ ( size_s3379185846251495609od_a_o @ Ys ) )
     => ( Xs
       != ( cons_l9179691953475787517od_a_o @ X2 @ Ys ) ) ) ).

% impossible_Cons
thf(fact_734_impossible__Cons,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,X2: sum_su4907927656612468301on_nat] :
      ( ( ord_less_eq_nat @ ( size_s5686234912572693873on_nat @ Xs ) @ ( size_s5686234912572693873on_nat @ Ys ) )
     => ( Xs
       != ( cons_S6797538523646374167on_nat @ X2 @ Ys ) ) ) ).

% impossible_Cons
thf(fact_735_impossible__Cons,axiom,
    ! [Xs: list_nat,Ys: list_nat,X2: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
     => ( Xs
       != ( cons_nat @ X2 @ Ys ) ) ) ).

% impossible_Cons
thf(fact_736_impossible__Cons,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,X2: produc1313295348606329723on_nat] :
      ( ( ord_less_eq_nat @ ( size_s7490768528479217773on_nat @ Xs ) @ ( size_s7490768528479217773on_nat @ Ys ) )
     => ( Xs
       != ( cons_P1582596238967110705on_nat @ X2 @ Ys ) ) ) ).

% impossible_Cons
thf(fact_737_distinct__singleton,axiom,
    ! [X2: nat] : ( distinct_nat @ ( cons_nat @ X2 @ nil_nat ) ) ).

% distinct_singleton
thf(fact_738_distinct__singleton,axiom,
    ! [X2: int] : ( distinct_int @ ( cons_int @ X2 @ nil_int ) ) ).

% distinct_singleton
thf(fact_739_subset__eq__mset__impl_Ocases,axiom,
    ! [X2: produc1828647624359046049st_nat] :
      ( ! [Ys4: list_nat] :
          ( X2
         != ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
     => ~ ! [X: nat,Xs3: list_nat,Ys4: list_nat] :
            ( X2
           != ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs3 ) @ Ys4 ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_740_subset__eq__mset__impl_Ocases,axiom,
    ! [X2: produc1186641810826059865st_int] :
      ( ! [Ys4: list_int] :
          ( X2
         != ( produc364263696895485585st_int @ nil_int @ Ys4 ) )
     => ~ ! [X: int,Xs3: list_int,Ys4: list_int] :
            ( X2
           != ( produc364263696895485585st_int @ ( cons_int @ X @ Xs3 ) @ Ys4 ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_741_shuffles_Ocases,axiom,
    ! [X2: produc1828647624359046049st_nat] :
      ( ! [Ys4: list_nat] :
          ( X2
         != ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
     => ( ! [Xs3: list_nat] :
            ( X2
           != ( produc2694037385005941721st_nat @ Xs3 @ nil_nat ) )
       => ~ ! [X: nat,Xs3: list_nat,Y4: nat,Ys4: list_nat] :
              ( X2
             != ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y4 @ Ys4 ) ) ) ) ) ).

% shuffles.cases
thf(fact_742_shuffles_Ocases,axiom,
    ! [X2: produc1186641810826059865st_int] :
      ( ! [Ys4: list_int] :
          ( X2
         != ( produc364263696895485585st_int @ nil_int @ Ys4 ) )
     => ( ! [Xs3: list_int] :
            ( X2
           != ( produc364263696895485585st_int @ Xs3 @ nil_int ) )
       => ~ ! [X: int,Xs3: list_int,Y4: int,Ys4: list_int] :
              ( X2
             != ( produc364263696895485585st_int @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y4 @ Ys4 ) ) ) ) ) ).

% shuffles.cases
thf(fact_743_distinct_Osimps_I2_J,axiom,
    ! [X2: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( distin690952206374581440od_a_o @ ( cons_l9179691953475787517od_a_o @ X2 @ Xs ) )
      = ( ~ ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Xs ) )
        & ( distin690952206374581440od_a_o @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_744_distinct_Osimps_I2_J,axiom,
    ! [X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( distin8410642530753306484on_nat @ ( cons_P1582596238967110705on_nat @ X2 @ Xs ) )
      = ( ~ ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Xs ) )
        & ( distin8410642530753306484on_nat @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_745_distinct_Osimps_I2_J,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( distin2773651816175405460on_nat @ ( cons_S6797538523646374167on_nat @ X2 @ Xs ) )
      = ( ~ ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Xs ) )
        & ( distin2773651816175405460on_nat @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_746_distinct_Osimps_I2_J,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( distinct_nat @ ( cons_nat @ X2 @ Xs ) )
      = ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
        & ( distinct_nat @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_747_distinct_Osimps_I2_J,axiom,
    ! [X2: int,Xs: list_int] :
      ( ( distinct_int @ ( cons_int @ X2 @ Xs ) )
      = ( ~ ( member_int @ X2 @ ( set_int2 @ Xs ) )
        & ( distinct_int @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_748_list__update__code_I2_J,axiom,
    ! [X2: nat,Xs: list_nat,Y2: nat] :
      ( ( list_update_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat @ Y2 )
      = ( cons_nat @ Y2 @ Xs ) ) ).

% list_update_code(2)
thf(fact_749_list__update__code_I2_J,axiom,
    ! [X2: int,Xs: list_int,Y2: int] :
      ( ( list_update_int @ ( cons_int @ X2 @ Xs ) @ zero_zero_nat @ Y2 )
      = ( cons_int @ Y2 @ Xs ) ) ).

% list_update_code(2)
thf(fact_750_listrel1I2,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,X2: nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ X2 @ Ys ) ) @ ( listrel1_nat @ R ) ) ) ).

% listrel1I2
thf(fact_751_listrel1I2,axiom,
    ! [Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int,X2: int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) )
     => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X2 @ Xs ) @ ( cons_int @ X2 @ Ys ) ) @ ( listrel1_int @ R ) ) ) ).

% listrel1I2
thf(fact_752_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_753_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_nonneg
thf(fact_754_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z4 )
     => ~ ( ord_less_nat @ T @ X6 ) ) ).

% minf(7)
thf(fact_755_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z4 )
     => ~ ( ord_less_int @ T @ X6 ) ) ).

% minf(7)
thf(fact_756_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z4 )
     => ( ord_less_nat @ X6 @ T ) ) ).

% minf(5)
thf(fact_757_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z4 )
     => ( ord_less_int @ X6 @ T ) ) ).

% minf(5)
thf(fact_758_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z4 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_759_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z4 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_760_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z4 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_761_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z4 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_762_minf_I2_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X: nat] :
          ( ( ord_less_nat @ X @ Z5 )
         => ( ( P @ X )
            = ( P6 @ X ) ) )
     => ( ? [Z5: nat] :
          ! [X: nat] :
            ( ( ord_less_nat @ X @ Z5 )
           => ( ( Q @ X )
              = ( Q2 @ X ) ) )
       => ? [Z4: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ X6 @ Z4 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q2 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_763_minf_I2_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z5: int] :
        ! [X: int] :
          ( ( ord_less_int @ X @ Z5 )
         => ( ( P @ X )
            = ( P6 @ X ) ) )
     => ( ? [Z5: int] :
          ! [X: int] :
            ( ( ord_less_int @ X @ Z5 )
           => ( ( Q @ X )
              = ( Q2 @ X ) ) )
       => ? [Z4: int] :
          ! [X6: int] :
            ( ( ord_less_int @ X6 @ Z4 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q2 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_764_minf_I1_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X: nat] :
          ( ( ord_less_nat @ X @ Z5 )
         => ( ( P @ X )
            = ( P6 @ X ) ) )
     => ( ? [Z5: nat] :
          ! [X: nat] :
            ( ( ord_less_nat @ X @ Z5 )
           => ( ( Q @ X )
              = ( Q2 @ X ) ) )
       => ? [Z4: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ X6 @ Z4 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q2 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_765_minf_I1_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z5: int] :
        ! [X: int] :
          ( ( ord_less_int @ X @ Z5 )
         => ( ( P @ X )
            = ( P6 @ X ) ) )
     => ( ? [Z5: int] :
          ! [X: int] :
            ( ( ord_less_int @ X @ Z5 )
           => ( ( Q @ X )
              = ( Q2 @ X ) ) )
       => ? [Z4: int] :
          ! [X6: int] :
            ( ( ord_less_int @ X6 @ Z4 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q2 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_766_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z4 @ X6 )
     => ( ord_less_nat @ T @ X6 ) ) ).

% pinf(7)
thf(fact_767_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z4 @ X6 )
     => ( ord_less_int @ T @ X6 ) ) ).

% pinf(7)
thf(fact_768_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z4 @ X6 )
     => ~ ( ord_less_nat @ X6 @ T ) ) ).

% pinf(5)
thf(fact_769_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z4 @ X6 )
     => ~ ( ord_less_int @ X6 @ T ) ) ).

% pinf(5)
thf(fact_770_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z4 @ X6 )
     => ( X6 != T ) ) ).

% pinf(4)
thf(fact_771_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z4 @ X6 )
     => ( X6 != T ) ) ).

% pinf(4)
thf(fact_772_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z4 @ X6 )
     => ( X6 != T ) ) ).

% pinf(3)
thf(fact_773_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z4 @ X6 )
     => ( X6 != T ) ) ).

% pinf(3)
thf(fact_774_pinf_I2_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X: nat] :
          ( ( ord_less_nat @ Z5 @ X )
         => ( ( P @ X )
            = ( P6 @ X ) ) )
     => ( ? [Z5: nat] :
          ! [X: nat] :
            ( ( ord_less_nat @ Z5 @ X )
           => ( ( Q @ X )
              = ( Q2 @ X ) ) )
       => ? [Z4: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ Z4 @ X6 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q2 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_775_pinf_I2_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z5: int] :
        ! [X: int] :
          ( ( ord_less_int @ Z5 @ X )
         => ( ( P @ X )
            = ( P6 @ X ) ) )
     => ( ? [Z5: int] :
          ! [X: int] :
            ( ( ord_less_int @ Z5 @ X )
           => ( ( Q @ X )
              = ( Q2 @ X ) ) )
       => ? [Z4: int] :
          ! [X6: int] :
            ( ( ord_less_int @ Z4 @ X6 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q2 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_776_pinf_I1_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X: nat] :
          ( ( ord_less_nat @ Z5 @ X )
         => ( ( P @ X )
            = ( P6 @ X ) ) )
     => ( ? [Z5: nat] :
          ! [X: nat] :
            ( ( ord_less_nat @ Z5 @ X )
           => ( ( Q @ X )
              = ( Q2 @ X ) ) )
       => ? [Z4: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ Z4 @ X6 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q2 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_777_pinf_I1_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z5: int] :
        ! [X: int] :
          ( ( ord_less_int @ Z5 @ X )
         => ( ( P @ X )
            = ( P6 @ X ) ) )
     => ( ? [Z5: int] :
          ! [X: int] :
            ( ( ord_less_int @ Z5 @ X )
           => ( ( Q @ X )
              = ( Q2 @ X ) ) )
       => ? [Z4: int] :
          ! [X6: int] :
            ( ( ord_less_int @ Z4 @ X6 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q2 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_778_listrel1I1,axiom,
    ! [X2: nat,Y2: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Xs ) ) @ ( listrel1_nat @ R ) ) ) ).

% listrel1I1
thf(fact_779_listrel1I1,axiom,
    ! [X2: int,Y2: int,R: set_Pr958786334691620121nt_int,Xs: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y2 ) @ R )
     => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X2 @ Xs ) @ ( cons_int @ Y2 @ Xs ) ) @ ( listrel1_int @ R ) ) ) ).

% listrel1I1
thf(fact_780_listrel1I1,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Y2: sum_su4907927656612468301on_nat,R: set_Pr2535564044409765851on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X2 @ Y2 ) @ R )
     => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( cons_S6797538523646374167on_nat @ X2 @ Xs ) @ ( cons_S6797538523646374167on_nat @ Y2 @ Xs ) ) @ ( listre1270753918075970734on_nat @ R ) ) ) ).

% listrel1I1
thf(fact_781_Cons__listrel1E1,axiom,
    ! [X2: nat,Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ Ys ) @ ( listrel1_nat @ R ) )
     => ( ! [Y4: nat] :
            ( ( Ys
              = ( cons_nat @ Y4 @ Xs ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ R ) )
       => ~ ! [Zs2: list_nat] :
              ( ( Ys
                = ( cons_nat @ X2 @ Zs2 ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs2 ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_782_Cons__listrel1E1,axiom,
    ! [X2: int,Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X2 @ Xs ) @ Ys ) @ ( listrel1_int @ R ) )
     => ( ! [Y4: int] :
            ( ( Ys
              = ( cons_int @ Y4 @ Xs ) )
           => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y4 ) @ R ) )
       => ~ ! [Zs2: list_int] :
              ( ( Ys
                = ( cons_int @ X2 @ Zs2 ) )
             => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Zs2 ) @ ( listrel1_int @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_783_Cons__listrel1E1,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( cons_S6797538523646374167on_nat @ X2 @ Xs ) @ Ys ) @ ( listre1270753918075970734on_nat @ R ) )
     => ( ! [Y4: sum_su4907927656612468301on_nat] :
            ( ( Ys
              = ( cons_S6797538523646374167on_nat @ Y4 @ Xs ) )
           => ~ ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X2 @ Y4 ) @ R ) )
       => ~ ! [Zs2: list_S5253404628753938141on_nat] :
              ( ( Ys
                = ( cons_S6797538523646374167on_nat @ X2 @ Zs2 ) )
             => ~ ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ Zs2 ) @ ( listre1270753918075970734on_nat @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_784_Cons__listrel1E2,axiom,
    ! [Xs: list_nat,Y2: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y2 @ Ys ) ) @ ( listrel1_nat @ R ) )
     => ( ! [X: nat] :
            ( ( Xs
              = ( cons_nat @ X @ Ys ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ R ) )
       => ~ ! [Zs2: list_nat] :
              ( ( Xs
                = ( cons_nat @ Y2 @ Zs2 ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs2 @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_785_Cons__listrel1E2,axiom,
    ! [Xs: list_int,Y2: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ ( cons_int @ Y2 @ Ys ) ) @ ( listrel1_int @ R ) )
     => ( ! [X: int] :
            ( ( Xs
              = ( cons_int @ X @ Ys ) )
           => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y2 ) @ R ) )
       => ~ ! [Zs2: list_int] :
              ( ( Xs
                = ( cons_int @ Y2 @ Zs2 ) )
             => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Zs2 @ Ys ) @ ( listrel1_int @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_786_Cons__listrel1E2,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Y2: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ ( cons_S6797538523646374167on_nat @ Y2 @ Ys ) ) @ ( listre1270753918075970734on_nat @ R ) )
     => ( ! [X: sum_su4907927656612468301on_nat] :
            ( ( Xs
              = ( cons_S6797538523646374167on_nat @ X @ Ys ) )
           => ~ ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X @ Y2 ) @ R ) )
       => ~ ! [Zs2: list_S5253404628753938141on_nat] :
              ( ( Xs
                = ( cons_S6797538523646374167on_nat @ Y2 @ Zs2 ) )
             => ~ ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Zs2 @ Ys ) @ ( listre1270753918075970734on_nat @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_787_listrel_OCons,axiom,
    ! [X2: nat,Y2: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) @ ( listrel_nat_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_788_listrel_OCons,axiom,
    ! [X2: nat,Y2: int,R: set_Pr7995236796853374141at_int,Xs: list_nat,Ys: list_int] :
      ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ X2 @ Y2 ) @ R )
     => ( ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ Xs @ Ys ) @ ( listrel_nat_int @ R ) )
       => ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ ( cons_nat @ X2 @ Xs ) @ ( cons_int @ Y2 @ Ys ) ) @ ( listrel_nat_int @ R ) ) ) ) ).

% listrel.Cons
thf(fact_789_listrel_OCons,axiom,
    ! [X2: int,Y2: nat,R: set_Pr3448869479623346877nt_nat,Xs: list_int,Ys: list_nat] :
      ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ X2 @ Y2 ) @ R )
     => ( ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ Xs @ Ys ) @ ( listrel_int_nat @ R ) )
       => ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ ( cons_int @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) @ ( listrel_int_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_790_listrel_OCons,axiom,
    ! [X2: int,Y2: int,R: set_Pr958786334691620121nt_int,Xs: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y2 ) @ R )
     => ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel_int_int @ R ) )
       => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X2 @ Xs ) @ ( cons_int @ Y2 @ Ys ) ) @ ( listrel_int_int @ R ) ) ) ) ).

% listrel.Cons
thf(fact_791_listrel_OCons,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Y2: sum_su4907927656612468301on_nat,R: set_Pr2535564044409765851on_nat,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X2 @ Y2 ) @ R )
     => ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ Ys ) @ ( listre5398654966139722893on_nat @ R ) )
       => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( cons_S6797538523646374167on_nat @ X2 @ Xs ) @ ( cons_S6797538523646374167on_nat @ Y2 @ Ys ) ) @ ( listre5398654966139722893on_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_792_listrel_OCons,axiom,
    ! [X2: nat > nat,Y2: nat,R: set_Pr9093778441882193744at_nat,Xs: list_nat_nat,Ys: list_nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X2 @ Y2 ) @ R )
     => ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs @ Ys ) @ ( listrel_nat_nat_nat @ R ) )
       => ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ ( cons_nat_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) @ ( listrel_nat_nat_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_793_listrel__Cons1,axiom,
    ! [Y2: nat,Ys: list_nat,Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y2 @ Ys ) @ Xs ) @ ( listrel_nat_nat @ R ) )
     => ~ ! [Y4: nat,Ys4: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y4 @ Ys4 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Y4 ) @ R )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Ys4 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_794_listrel__Cons1,axiom,
    ! [Y2: nat,Ys: list_nat,Xs: list_int,R: set_Pr7995236796853374141at_int] :
      ( ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ ( cons_nat @ Y2 @ Ys ) @ Xs ) @ ( listrel_nat_int @ R ) )
     => ~ ! [Y4: int,Ys4: list_int] :
            ( ( Xs
              = ( cons_int @ Y4 @ Ys4 ) )
           => ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ Y2 @ Y4 ) @ R )
             => ~ ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ Ys @ Ys4 ) @ ( listrel_nat_int @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_795_listrel__Cons1,axiom,
    ! [Y2: int,Ys: list_int,Xs: list_nat,R: set_Pr3448869479623346877nt_nat] :
      ( ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ ( cons_int @ Y2 @ Ys ) @ Xs ) @ ( listrel_int_nat @ R ) )
     => ~ ! [Y4: nat,Ys4: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y4 @ Ys4 ) )
           => ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ Y2 @ Y4 ) @ R )
             => ~ ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ Ys @ Ys4 ) @ ( listrel_int_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_796_listrel__Cons1,axiom,
    ! [Y2: int,Ys: list_int,Xs: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ Y2 @ Ys ) @ Xs ) @ ( listrel_int_int @ R ) )
     => ~ ! [Y4: int,Ys4: list_int] :
            ( ( Xs
              = ( cons_int @ Y4 @ Ys4 ) )
           => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Y4 ) @ R )
             => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys @ Ys4 ) @ ( listrel_int_int @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_797_listrel__Cons1,axiom,
    ! [Y2: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat,Xs: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( cons_S6797538523646374167on_nat @ Y2 @ Ys ) @ Xs ) @ ( listre5398654966139722893on_nat @ R ) )
     => ~ ! [Y4: sum_su4907927656612468301on_nat,Ys4: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( cons_S6797538523646374167on_nat @ Y4 @ Ys4 ) )
           => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Y2 @ Y4 ) @ R )
             => ~ ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Ys @ Ys4 ) @ ( listre5398654966139722893on_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_798_listrel__Cons1,axiom,
    ! [Y2: nat > nat,Ys: list_nat_nat,Xs: list_nat,R: set_Pr9093778441882193744at_nat] :
      ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ ( cons_nat_nat @ Y2 @ Ys ) @ Xs ) @ ( listrel_nat_nat_nat @ R ) )
     => ~ ! [Y4: nat,Ys4: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y4 @ Ys4 ) )
           => ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ Y2 @ Y4 ) @ R )
             => ~ ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Ys @ Ys4 ) @ ( listrel_nat_nat_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_799_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y2: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y2 @ Ys ) ) @ ( listrel_nat_nat @ R ) )
     => ~ ! [X: nat,Xs3: list_nat] :
            ( ( Xs
              = ( cons_nat @ X @ Xs3 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ R )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_800_listrel__Cons2,axiom,
    ! [Xs: list_int,Y2: nat,Ys: list_nat,R: set_Pr3448869479623346877nt_nat] :
      ( ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ Xs @ ( cons_nat @ Y2 @ Ys ) ) @ ( listrel_int_nat @ R ) )
     => ~ ! [X: int,Xs3: list_int] :
            ( ( Xs
              = ( cons_int @ X @ Xs3 ) )
           => ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ X @ Y2 ) @ R )
             => ~ ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ Xs3 @ Ys ) @ ( listrel_int_nat @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_801_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y2: int,Ys: list_int,R: set_Pr7995236796853374141at_int] :
      ( ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ Xs @ ( cons_int @ Y2 @ Ys ) ) @ ( listrel_nat_int @ R ) )
     => ~ ! [X: nat,Xs3: list_nat] :
            ( ( Xs
              = ( cons_nat @ X @ Xs3 ) )
           => ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ X @ Y2 ) @ R )
             => ~ ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ Xs3 @ Ys ) @ ( listrel_nat_int @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_802_listrel__Cons2,axiom,
    ! [Xs: list_int,Y2: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ ( cons_int @ Y2 @ Ys ) ) @ ( listrel_int_int @ R ) )
     => ~ ! [X: int,Xs3: list_int] :
            ( ( Xs
              = ( cons_int @ X @ Xs3 ) )
           => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y2 ) @ R )
             => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs3 @ Ys ) @ ( listrel_int_int @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_803_listrel__Cons2,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Y2: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ ( cons_S6797538523646374167on_nat @ Y2 @ Ys ) ) @ ( listre5398654966139722893on_nat @ R ) )
     => ~ ! [X: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( cons_S6797538523646374167on_nat @ X @ Xs3 ) )
           => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X @ Y2 ) @ R )
             => ~ ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs3 @ Ys ) @ ( listre5398654966139722893on_nat @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_804_listrel__Cons2,axiom,
    ! [Xs: list_nat_nat,Y2: nat,Ys: list_nat,R: set_Pr9093778441882193744at_nat] :
      ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs @ ( cons_nat @ Y2 @ Ys ) ) @ ( listrel_nat_nat_nat @ R ) )
     => ~ ! [X: nat > nat,Xs3: list_nat_nat] :
            ( ( Xs
              = ( cons_nat_nat @ X @ Xs3 ) )
           => ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y2 ) @ R )
             => ~ ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs3 @ Ys ) @ ( listrel_nat_nat_nat @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_805_imp__le__cong,axiom,
    ! [X2: int,X7: int,P: $o,P6: $o] :
      ( ( X2 = X7 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
         => ( P = P6 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
           => P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
           => P6 ) ) ) ) ).

% imp_le_cong
thf(fact_806_conj__le__cong,axiom,
    ! [X2: int,X7: int,P: $o,P6: $o] :
      ( ( X2 = X7 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
         => ( P = P6 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
            & P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
            & P6 ) ) ) ) ).

% conj_le_cong
thf(fact_807_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_nat] :
      ( ( n_lists_nat @ zero_zero_nat @ Xs )
      = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% n_lists.simps(1)
thf(fact_808_of__int__pos,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_pos
thf(fact_809_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X2: int] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X2 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X2 ) ) ).

% of_nat_less_of_int_iff
thf(fact_810_listrel_Ocases,axiom,
    ! [A1: list_nat,A22: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R ) )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_nat ) )
       => ~ ! [X: nat,Y4: nat,Xs3: list_nat] :
              ( ( A1
                = ( cons_nat @ X @ Xs3 ) )
             => ! [Ys4: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y4 @ Ys4 ) )
                 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R )
                   => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys4 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_811_listrel_Ocases,axiom,
    ! [A1: list_nat,A22: list_int,R: set_Pr7995236796853374141at_int] :
      ( ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ A1 @ A22 ) @ ( listrel_nat_int @ R ) )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_int ) )
       => ~ ! [X: nat,Y4: int,Xs3: list_nat] :
              ( ( A1
                = ( cons_nat @ X @ Xs3 ) )
             => ! [Ys4: list_int] :
                  ( ( A22
                    = ( cons_int @ Y4 @ Ys4 ) )
                 => ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ X @ Y4 ) @ R )
                   => ~ ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ Xs3 @ Ys4 ) @ ( listrel_nat_int @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_812_listrel_Ocases,axiom,
    ! [A1: list_int,A22: list_nat,R: set_Pr3448869479623346877nt_nat] :
      ( ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ A1 @ A22 ) @ ( listrel_int_nat @ R ) )
     => ( ( ( A1 = nil_int )
         => ( A22 != nil_nat ) )
       => ~ ! [X: int,Y4: nat,Xs3: list_int] :
              ( ( A1
                = ( cons_int @ X @ Xs3 ) )
             => ! [Ys4: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y4 @ Ys4 ) )
                 => ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ X @ Y4 ) @ R )
                   => ~ ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ Xs3 @ Ys4 ) @ ( listrel_int_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_813_listrel_Ocases,axiom,
    ! [A1: list_int,A22: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ A1 @ A22 ) @ ( listrel_int_int @ R ) )
     => ( ( ( A1 = nil_int )
         => ( A22 != nil_int ) )
       => ~ ! [X: int,Y4: int,Xs3: list_int] :
              ( ( A1
                = ( cons_int @ X @ Xs3 ) )
             => ! [Ys4: list_int] :
                  ( ( A22
                    = ( cons_int @ Y4 @ Ys4 ) )
                 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y4 ) @ R )
                   => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs3 @ Ys4 ) @ ( listrel_int_int @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_814_listrel_Ocases,axiom,
    ! [A1: list_S5253404628753938141on_nat,A22: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ A1 @ A22 ) @ ( listre5398654966139722893on_nat @ R ) )
     => ( ( ( A1 = nil_Su3078906344068544711on_nat )
         => ( A22 != nil_Su3078906344068544711on_nat ) )
       => ~ ! [X: sum_su4907927656612468301on_nat,Y4: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat] :
              ( ( A1
                = ( cons_S6797538523646374167on_nat @ X @ Xs3 ) )
             => ! [Ys4: list_S5253404628753938141on_nat] :
                  ( ( A22
                    = ( cons_S6797538523646374167on_nat @ Y4 @ Ys4 ) )
                 => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X @ Y4 ) @ R )
                   => ~ ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs3 @ Ys4 ) @ ( listre5398654966139722893on_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_815_listrel_Ocases,axiom,
    ! [A1: list_nat_nat,A22: list_nat,R: set_Pr9093778441882193744at_nat] :
      ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ A1 @ A22 ) @ ( listrel_nat_nat_nat @ R ) )
     => ( ( ( A1 = nil_nat_nat )
         => ( A22 != nil_nat ) )
       => ~ ! [X: nat > nat,Y4: nat,Xs3: list_nat_nat] :
              ( ( A1
                = ( cons_nat_nat @ X @ Xs3 ) )
             => ! [Ys4: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y4 @ Ys4 ) )
                 => ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y4 ) @ R )
                   => ~ ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs3 @ Ys4 ) @ ( listrel_nat_nat_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_816_listrel_Osimps,axiom,
    ! [A1: list_nat,A22: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R ) )
      = ( ( ( A1 = nil_nat )
          & ( A22 = nil_nat ) )
        | ? [X3: nat,Y3: nat,Xs2: list_nat,Ys2: list_nat] :
            ( ( A1
              = ( cons_nat @ X3 @ Xs2 ) )
            & ( A22
              = ( cons_nat @ Y3 @ Ys2 ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
            & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_817_listrel_Osimps,axiom,
    ! [A1: list_nat,A22: list_int,R: set_Pr7995236796853374141at_int] :
      ( ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ A1 @ A22 ) @ ( listrel_nat_int @ R ) )
      = ( ( ( A1 = nil_nat )
          & ( A22 = nil_int ) )
        | ? [X3: nat,Y3: int,Xs2: list_nat,Ys2: list_int] :
            ( ( A1
              = ( cons_nat @ X3 @ Xs2 ) )
            & ( A22
              = ( cons_int @ Y3 @ Ys2 ) )
            & ( member4262671552274231302at_int @ ( product_Pair_nat_int @ X3 @ Y3 ) @ R )
            & ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ Xs2 @ Ys2 ) @ ( listrel_nat_int @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_818_listrel_Osimps,axiom,
    ! [A1: list_int,A22: list_nat,R: set_Pr3448869479623346877nt_nat] :
      ( ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ A1 @ A22 ) @ ( listrel_int_nat @ R ) )
      = ( ( ( A1 = nil_int )
          & ( A22 = nil_nat ) )
        | ? [X3: int,Y3: nat,Xs2: list_int,Ys2: list_nat] :
            ( ( A1
              = ( cons_int @ X3 @ Xs2 ) )
            & ( A22
              = ( cons_nat @ Y3 @ Ys2 ) )
            & ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ X3 @ Y3 ) @ R )
            & ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ Xs2 @ Ys2 ) @ ( listrel_int_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_819_listrel_Osimps,axiom,
    ! [A1: list_int,A22: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ A1 @ A22 ) @ ( listrel_int_int @ R ) )
      = ( ( ( A1 = nil_int )
          & ( A22 = nil_int ) )
        | ? [X3: int,Y3: int,Xs2: list_int,Ys2: list_int] :
            ( ( A1
              = ( cons_int @ X3 @ Xs2 ) )
            & ( A22
              = ( cons_int @ Y3 @ Ys2 ) )
            & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R )
            & ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys2 ) @ ( listrel_int_int @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_820_listrel_Osimps,axiom,
    ! [A1: list_S5253404628753938141on_nat,A22: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ A1 @ A22 ) @ ( listre5398654966139722893on_nat @ R ) )
      = ( ( ( A1 = nil_Su3078906344068544711on_nat )
          & ( A22 = nil_Su3078906344068544711on_nat ) )
        | ? [X3: sum_su4907927656612468301on_nat,Y3: sum_su4907927656612468301on_nat,Xs2: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat] :
            ( ( A1
              = ( cons_S6797538523646374167on_nat @ X3 @ Xs2 ) )
            & ( A22
              = ( cons_S6797538523646374167on_nat @ Y3 @ Ys2 ) )
            & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y3 ) @ R )
            & ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs2 @ Ys2 ) @ ( listre5398654966139722893on_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_821_listrel_Osimps,axiom,
    ! [A1: list_nat_nat,A22: list_nat,R: set_Pr9093778441882193744at_nat] :
      ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ A1 @ A22 ) @ ( listrel_nat_nat_nat @ R ) )
      = ( ( ( A1 = nil_nat_nat )
          & ( A22 = nil_nat ) )
        | ? [X3: nat > nat,Y3: nat,Xs2: list_nat_nat,Ys2: list_nat] :
            ( ( A1
              = ( cons_nat_nat @ X3 @ Xs2 ) )
            & ( A22
              = ( cons_nat @ Y3 @ Ys2 ) )
            & ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X3 @ Y3 ) @ R )
            & ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_822_mult2__ex__dom_Ocases,axiom,
    ! [X2: produc7312811325926348135st_nat] :
      ( ! [X: nat,Xs3: list_nat,Ys5: list_nat] :
          ( X2
         != ( produc921417020989057183st_nat @ X @ ( produc4487115339913071592st_nat @ Xs3 @ ( produc2694037385005941721st_nat @ nil_nat @ Ys5 ) ) ) )
     => ~ ! [X: nat,Xs3: list_nat,Y4: nat,Ys4: list_nat,Ys5: list_nat] :
            ( X2
           != ( produc921417020989057183st_nat @ X @ ( produc4487115339913071592st_nat @ Xs3 @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y4 @ Ys4 ) @ Ys5 ) ) ) ) ) ).

% mult2_ex_dom.cases
thf(fact_823_mult2__ex__dom_Ocases,axiom,
    ! [X2: produc6561234247366898463st_int] :
      ( ! [X: int,Xs3: list_int,Ys5: list_int] :
          ( X2
         != ( produc7930506808772752471st_int @ X @ ( produc920187109610959172st_int @ Xs3 @ ( produc364263696895485585st_int @ nil_int @ Ys5 ) ) ) )
     => ~ ! [X: int,Xs3: list_int,Y4: int,Ys4: list_int,Ys5: list_int] :
            ( X2
           != ( produc7930506808772752471st_int @ X @ ( produc920187109610959172st_int @ Xs3 @ ( produc364263696895485585st_int @ ( cons_int @ Y4 @ Ys4 ) @ Ys5 ) ) ) ) ) ).

% mult2_ex_dom.cases
thf(fact_824_mul__ex__dom_Ocases,axiom,
    ! [X2: produc5923727742716231015st_nat] :
      ( ! [Xs4: list_nat,Y4: nat,Ys4: list_nat] :
          ( X2
         != ( produc6629933339985824415st_nat @ nil_nat @ ( produc7518127839388293336st_nat @ Xs4 @ ( produc8282810413953273033st_nat @ Y4 @ Ys4 ) ) ) )
     => ~ ! [X: nat,Xs3: list_nat,Xs4: list_nat,Y4: nat,Ys4: list_nat] :
            ( X2
           != ( produc6629933339985824415st_nat @ ( cons_nat @ X @ Xs3 ) @ ( produc7518127839388293336st_nat @ Xs4 @ ( produc8282810413953273033st_nat @ Y4 @ Ys4 ) ) ) ) ) ).

% mul_ex_dom.cases
thf(fact_825_mul__ex__dom_Ocases,axiom,
    ! [X2: produc2984629891206486815st_int] :
      ( ! [Xs4: list_int,Y4: int,Ys4: list_int] :
          ( X2
         != ( produc5339354099187226199st_int @ nil_int @ ( produc1377355365024828980st_int @ Xs4 @ ( produc890397008584875393st_int @ Y4 @ Ys4 ) ) ) )
     => ~ ! [X: int,Xs3: list_int,Xs4: list_int,Y4: int,Ys4: list_int] :
            ( X2
           != ( produc5339354099187226199st_int @ ( cons_int @ X @ Xs3 ) @ ( produc1377355365024828980st_int @ Xs4 @ ( produc890397008584875393st_int @ Y4 @ Ys4 ) ) ) ) ) ).

% mul_ex_dom.cases
thf(fact_826_mul__ext__impl_Ocases,axiom,
    ! [X2: produc1828647624359046049st_nat] :
      ( ( X2
       != ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) )
     => ( ! [V2: nat,Va: list_nat] :
            ( X2
           != ( produc2694037385005941721st_nat @ nil_nat @ ( cons_nat @ V2 @ Va ) ) )
       => ( ! [V2: nat,Va: list_nat] :
              ( X2
             != ( produc2694037385005941721st_nat @ ( cons_nat @ V2 @ Va ) @ nil_nat ) )
         => ~ ! [V2: nat,Va: list_nat,Y4: nat,Ys4: list_nat] :
                ( X2
               != ( produc2694037385005941721st_nat @ ( cons_nat @ V2 @ Va ) @ ( cons_nat @ Y4 @ Ys4 ) ) ) ) ) ) ).

% mul_ext_impl.cases
thf(fact_827_mul__ext__impl_Ocases,axiom,
    ! [X2: produc1186641810826059865st_int] :
      ( ( X2
       != ( produc364263696895485585st_int @ nil_int @ nil_int ) )
     => ( ! [V2: int,Va: list_int] :
            ( X2
           != ( produc364263696895485585st_int @ nil_int @ ( cons_int @ V2 @ Va ) ) )
       => ( ! [V2: int,Va: list_int] :
              ( X2
             != ( produc364263696895485585st_int @ ( cons_int @ V2 @ Va ) @ nil_int ) )
         => ~ ! [V2: int,Va: list_int,Y4: int,Ys4: list_int] :
                ( X2
               != ( produc364263696895485585st_int @ ( cons_int @ V2 @ Va ) @ ( cons_int @ Y4 @ Ys4 ) ) ) ) ) ) ).

% mul_ext_impl.cases
thf(fact_828_mult2__impl_Ocases,axiom,
    ! [X2: produc1828647624359046049st_nat] :
      ( ( X2
       != ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) )
     => ( ! [V2: nat,Va: list_nat] :
            ( X2
           != ( produc2694037385005941721st_nat @ ( cons_nat @ V2 @ Va ) @ nil_nat ) )
       => ( ! [V2: nat,Va: list_nat] :
              ( X2
             != ( produc2694037385005941721st_nat @ nil_nat @ ( cons_nat @ V2 @ Va ) ) )
         => ~ ! [X: nat,Xs3: list_nat,V2: nat,Va: list_nat] :
                ( X2
               != ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ V2 @ Va ) ) ) ) ) ) ).

% mult2_impl.cases
thf(fact_829_mult2__impl_Ocases,axiom,
    ! [X2: produc1186641810826059865st_int] :
      ( ( X2
       != ( produc364263696895485585st_int @ nil_int @ nil_int ) )
     => ( ! [V2: int,Va: list_int] :
            ( X2
           != ( produc364263696895485585st_int @ ( cons_int @ V2 @ Va ) @ nil_int ) )
       => ( ! [V2: int,Va: list_int] :
              ( X2
             != ( produc364263696895485585st_int @ nil_int @ ( cons_int @ V2 @ Va ) ) )
         => ~ ! [X: int,Xs3: list_int,V2: int,Va: list_int] :
                ( X2
               != ( produc364263696895485585st_int @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ V2 @ Va ) ) ) ) ) ) ).

% mult2_impl.cases
thf(fact_830_the__elem__set,axiom,
    ! [X2: list_P3196869577824546119od_a_o] :
      ( ( the_el4730552915664895548od_a_o @ ( set_li2629784761229999580od_a_o @ ( cons_l9179691953475787517od_a_o @ X2 @ nil_li1372968520647675725od_a_o ) ) )
      = X2 ) ).

% the_elem_set
thf(fact_831_the__elem__set,axiom,
    ! [X2: produc1313295348606329723on_nat] :
      ( ( the_el8283737788879144688on_nat @ ( set_Pr937530902923597456on_nat @ ( cons_P1582596238967110705on_nat @ X2 @ nil_Pr8614093721167239297on_nat ) ) )
      = X2 ) ).

% the_elem_set
thf(fact_832_the__elem__set,axiom,
    ! [X2: sum_su4907927656612468301on_nat] :
      ( ( the_el7555745616574930200on_nat @ ( set_Su4848624856859743608on_nat @ ( cons_S6797538523646374167on_nat @ X2 @ nil_Su3078906344068544711on_nat ) ) )
      = X2 ) ).

% the_elem_set
thf(fact_833_the__elem__set,axiom,
    ! [X2: nat] :
      ( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X2 @ nil_nat ) ) )
      = X2 ) ).

% the_elem_set
thf(fact_834_the__elem__set,axiom,
    ! [X2: int] :
      ( ( the_elem_int @ ( set_int2 @ ( cons_int @ X2 @ nil_int ) ) )
      = X2 ) ).

% the_elem_set
thf(fact_835_Cons__lenlex__iff,axiom,
    ! [M2: int,Ms: list_int,N: int,Ns: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ M2 @ Ms ) @ ( cons_int @ N @ Ns ) ) @ ( lenlex_int @ R ) )
      = ( ( ord_less_nat @ ( size_size_list_int @ Ms ) @ ( size_size_list_int @ Ns ) )
        | ( ( ( size_size_list_int @ Ms )
            = ( size_size_list_int @ Ns ) )
          & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ M2 @ N ) @ R ) )
        | ( ( M2 = N )
          & ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ms @ Ns ) @ ( lenlex_int @ R ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_836_Cons__lenlex__iff,axiom,
    ! [M2: list_P3196869577824546119od_a_o,Ms: list_l8336264072933094989od_a_o,N: list_P3196869577824546119od_a_o,Ns: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ ( cons_l9179691953475787517od_a_o @ M2 @ Ms ) @ ( cons_l9179691953475787517od_a_o @ N @ Ns ) ) @ ( lenlex2563728411540087634od_a_o @ R ) )
      = ( ( ord_less_nat @ ( size_s3379185846251495609od_a_o @ Ms ) @ ( size_s3379185846251495609od_a_o @ Ns ) )
        | ( ( ( size_s3379185846251495609od_a_o @ Ms )
            = ( size_s3379185846251495609od_a_o @ Ns ) )
          & ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ M2 @ N ) @ R ) )
        | ( ( M2 = N )
          & ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Ms @ Ns ) @ ( lenlex2563728411540087634od_a_o @ R ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_837_Cons__lenlex__iff,axiom,
    ! [M2: sum_su4907927656612468301on_nat,Ms: list_S5253404628753938141on_nat,N: sum_su4907927656612468301on_nat,Ns: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( cons_S6797538523646374167on_nat @ M2 @ Ms ) @ ( cons_S6797538523646374167on_nat @ N @ Ns ) ) @ ( lenlex8545234769834248322on_nat @ R ) )
      = ( ( ord_less_nat @ ( size_s5686234912572693873on_nat @ Ms ) @ ( size_s5686234912572693873on_nat @ Ns ) )
        | ( ( ( size_s5686234912572693873on_nat @ Ms )
            = ( size_s5686234912572693873on_nat @ Ns ) )
          & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ M2 @ N ) @ R ) )
        | ( ( M2 = N )
          & ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Ms @ Ns ) @ ( lenlex8545234769834248322on_nat @ R ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_838_Cons__lenlex__iff,axiom,
    ! [M2: nat,Ms: list_nat,N: nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M2 @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R ) )
      = ( ( ord_less_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) )
        | ( ( ( size_size_list_nat @ Ms )
            = ( size_size_list_nat @ Ns ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N ) @ R ) )
        | ( ( M2 = N )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_839_Cons__lenlex__iff,axiom,
    ! [M2: produc1313295348606329723on_nat,Ms: list_P9127870862658512897on_nat,N: produc1313295348606329723on_nat,Ns: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ M2 @ Ms ) @ ( cons_P1582596238967110705on_nat @ N @ Ns ) ) @ ( lenlex6631009945862686470on_nat @ R ) )
      = ( ( ord_less_nat @ ( size_s7490768528479217773on_nat @ Ms ) @ ( size_s7490768528479217773on_nat @ Ns ) )
        | ( ( ( size_s7490768528479217773on_nat @ Ms )
            = ( size_s7490768528479217773on_nat @ Ns ) )
          & ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ M2 @ N ) @ R ) )
        | ( ( M2 = N )
          & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Ms @ Ns ) @ ( lenlex6631009945862686470on_nat @ R ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_840_Compl__iff,axiom,
    ! [C2: nat,A2: set_nat] :
      ( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A2 ) )
      = ( ~ ( member_nat @ C2 @ A2 ) ) ) ).

% Compl_iff
thf(fact_841_Compl__iff,axiom,
    ! [C2: list_P3196869577824546119od_a_o,A2: set_li5011860890973139239od_a_o] :
      ( ( member5261019209154752368od_a_o @ C2 @ ( uminus8391578155927437342od_a_o @ A2 ) )
      = ( ~ ( member5261019209154752368od_a_o @ C2 @ A2 ) ) ) ).

% Compl_iff
thf(fact_842_Compl__iff,axiom,
    ! [C2: produc1313295348606329723on_nat,A2: set_Pr2535564044409765851on_nat] :
      ( ( member5906418179908956708on_nat @ C2 @ ( uminus4127886064781087314on_nat @ A2 ) )
      = ( ~ ( member5906418179908956708on_nat @ C2 @ A2 ) ) ) ).

% Compl_iff
thf(fact_843_Compl__iff,axiom,
    ! [C2: sum_su4907927656612468301on_nat,A2: set_Su266273576555847555on_nat] :
      ( ( member8476201367316484836on_nat @ C2 @ ( uminus5421032341409327564on_nat @ A2 ) )
      = ( ~ ( member8476201367316484836on_nat @ C2 @ A2 ) ) ) ).

% Compl_iff
thf(fact_844_ComplI,axiom,
    ! [C2: nat,A2: set_nat] :
      ( ~ ( member_nat @ C2 @ A2 )
     => ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).

% ComplI
thf(fact_845_ComplI,axiom,
    ! [C2: list_P3196869577824546119od_a_o,A2: set_li5011860890973139239od_a_o] :
      ( ~ ( member5261019209154752368od_a_o @ C2 @ A2 )
     => ( member5261019209154752368od_a_o @ C2 @ ( uminus8391578155927437342od_a_o @ A2 ) ) ) ).

% ComplI
thf(fact_846_ComplI,axiom,
    ! [C2: produc1313295348606329723on_nat,A2: set_Pr2535564044409765851on_nat] :
      ( ~ ( member5906418179908956708on_nat @ C2 @ A2 )
     => ( member5906418179908956708on_nat @ C2 @ ( uminus4127886064781087314on_nat @ A2 ) ) ) ).

% ComplI
thf(fact_847_ComplI,axiom,
    ! [C2: sum_su4907927656612468301on_nat,A2: set_Su266273576555847555on_nat] :
      ( ~ ( member8476201367316484836on_nat @ C2 @ A2 )
     => ( member8476201367316484836on_nat @ C2 @ ( uminus5421032341409327564on_nat @ A2 ) ) ) ).

% ComplI
thf(fact_848_Nil__lenlex__iff1,axiom,
    ! [Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ns ) @ ( lenlex_nat @ R ) )
      = ( Ns != nil_nat ) ) ).

% Nil_lenlex_iff1
thf(fact_849_psubsetD,axiom,
    ! [A2: set_nat,B5: set_nat,C2: nat] :
      ( ( ord_less_set_nat @ A2 @ B5 )
     => ( ( member_nat @ C2 @ A2 )
       => ( member_nat @ C2 @ B5 ) ) ) ).

% psubsetD
thf(fact_850_psubsetD,axiom,
    ! [A2: set_li5011860890973139239od_a_o,B5: set_li5011860890973139239od_a_o,C2: list_P3196869577824546119od_a_o] :
      ( ( ord_le6036158148124213203od_a_o @ A2 @ B5 )
     => ( ( member5261019209154752368od_a_o @ C2 @ A2 )
       => ( member5261019209154752368od_a_o @ C2 @ B5 ) ) ) ).

% psubsetD
thf(fact_851_psubsetD,axiom,
    ! [A2: set_Pr2535564044409765851on_nat,B5: set_Pr2535564044409765851on_nat,C2: produc1313295348606329723on_nat] :
      ( ( ord_le188702667174282375on_nat @ A2 @ B5 )
     => ( ( member5906418179908956708on_nat @ C2 @ A2 )
       => ( member5906418179908956708on_nat @ C2 @ B5 ) ) ) ).

% psubsetD
thf(fact_852_psubsetD,axiom,
    ! [A2: set_Su266273576555847555on_nat,B5: set_Su266273576555847555on_nat,C2: sum_su4907927656612468301on_nat] :
      ( ( ord_le5614770093425365463on_nat @ A2 @ B5 )
     => ( ( member8476201367316484836on_nat @ C2 @ A2 )
       => ( member8476201367316484836on_nat @ C2 @ B5 ) ) ) ).

% psubsetD
thf(fact_853_ComplD,axiom,
    ! [C2: nat,A2: set_nat] :
      ( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A2 ) )
     => ~ ( member_nat @ C2 @ A2 ) ) ).

% ComplD
thf(fact_854_ComplD,axiom,
    ! [C2: list_P3196869577824546119od_a_o,A2: set_li5011860890973139239od_a_o] :
      ( ( member5261019209154752368od_a_o @ C2 @ ( uminus8391578155927437342od_a_o @ A2 ) )
     => ~ ( member5261019209154752368od_a_o @ C2 @ A2 ) ) ).

% ComplD
thf(fact_855_ComplD,axiom,
    ! [C2: produc1313295348606329723on_nat,A2: set_Pr2535564044409765851on_nat] :
      ( ( member5906418179908956708on_nat @ C2 @ ( uminus4127886064781087314on_nat @ A2 ) )
     => ~ ( member5906418179908956708on_nat @ C2 @ A2 ) ) ).

% ComplD
thf(fact_856_ComplD,axiom,
    ! [C2: sum_su4907927656612468301on_nat,A2: set_Su266273576555847555on_nat] :
      ( ( member8476201367316484836on_nat @ C2 @ ( uminus5421032341409327564on_nat @ A2 ) )
     => ~ ( member8476201367316484836on_nat @ C2 @ A2 ) ) ).

% ComplD
thf(fact_857_lenlex__irreflexive,axiom,
    ! [R: set_Pr2535564044409765851on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ! [X: sum_su4907927656612468301on_nat] :
          ~ ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X @ X ) @ R )
     => ~ ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ Xs ) @ ( lenlex8545234769834248322on_nat @ R ) ) ) ).

% lenlex_irreflexive
thf(fact_858_Nil__lenlex__iff2,axiom,
    ! [Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ns @ nil_nat ) @ ( lenlex_nat @ R ) ) ).

% Nil_lenlex_iff2
thf(fact_859_lenlex__length,axiom,
    ! [Ms: list_l8336264072933094989od_a_o,Ns: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Ms @ Ns ) @ ( lenlex2563728411540087634od_a_o @ R ) )
     => ( ord_less_eq_nat @ ( size_s3379185846251495609od_a_o @ Ms ) @ ( size_s3379185846251495609od_a_o @ Ns ) ) ) ).

% lenlex_length
thf(fact_860_lenlex__length,axiom,
    ! [Ms: list_S5253404628753938141on_nat,Ns: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Ms @ Ns ) @ ( lenlex8545234769834248322on_nat @ R ) )
     => ( ord_less_eq_nat @ ( size_s5686234912572693873on_nat @ Ms ) @ ( size_s5686234912572693873on_nat @ Ns ) ) ) ).

% lenlex_length
thf(fact_861_lenlex__length,axiom,
    ! [Ms: list_nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R ) )
     => ( ord_less_eq_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) ) ) ).

% lenlex_length
thf(fact_862_lenlex__length,axiom,
    ! [Ms: list_P9127870862658512897on_nat,Ns: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Ms @ Ns ) @ ( lenlex6631009945862686470on_nat @ R ) )
     => ( ord_less_eq_nat @ ( size_s7490768528479217773on_nat @ Ms ) @ ( size_s7490768528479217773on_nat @ Ns ) ) ) ).

% lenlex_length
thf(fact_863_mult2__ex__dom0_Ocases,axiom,
    ! [X2: produc3630873615015234996_nat_o] :
      ( ! [X: nat,Xs3: list_nat,Ys5: list_nat,B3: $o] :
          ( X2
         != ( produc6858984136714788332_nat_o @ X @ ( produc2959919240848031157_nat_o @ Xs3 @ ( produc5973304058656726566_nat_o @ nil_nat @ ( produc1747790020158526943_nat_o @ Ys5 @ B3 ) ) ) ) )
     => ~ ! [X: nat,Xs3: list_nat,Y4: nat,Ys4: list_nat,Ys5: list_nat,B3: $o] :
            ( X2
           != ( produc6858984136714788332_nat_o @ X @ ( produc2959919240848031157_nat_o @ Xs3 @ ( produc5973304058656726566_nat_o @ ( cons_nat @ Y4 @ Ys4 ) @ ( produc1747790020158526943_nat_o @ Ys5 @ B3 ) ) ) ) ) ) ).

% mult2_ex_dom0.cases
thf(fact_864_mult2__ex__dom0_Ocases,axiom,
    ! [X2: produc6353549441775477172_int_o] :
      ( ! [X: int,Xs3: list_int,Ys5: list_int,B3: $o] :
          ( X2
         != ( produc5683659465854387180_int_o @ X @ ( produc9173919776369438553_int_o @ Xs3 @ ( produc1348525947378492198_int_o @ nil_int @ ( produc8726017089342856835_int_o @ Ys5 @ B3 ) ) ) ) )
     => ~ ! [X: int,Xs3: list_int,Y4: int,Ys4: list_int,Ys5: list_int,B3: $o] :
            ( X2
           != ( produc5683659465854387180_int_o @ X @ ( produc9173919776369438553_int_o @ Xs3 @ ( produc1348525947378492198_int_o @ ( cons_int @ Y4 @ Ys4 ) @ ( produc8726017089342856835_int_o @ Ys5 @ B3 ) ) ) ) ) ) ).

% mult2_ex_dom0.cases
thf(fact_865_mult2__impl0_Ocases,axiom,
    ! [X2: produc430777597369845998_nat_o] :
      ( ! [B3: $o] :
          ( X2
         != ( produc5973304058656726566_nat_o @ nil_nat @ ( produc1747790020158526943_nat_o @ nil_nat @ B3 ) ) )
     => ( ! [V2: nat,Va: list_nat,B3: $o] :
            ( X2
           != ( produc5973304058656726566_nat_o @ ( cons_nat @ V2 @ Va ) @ ( produc1747790020158526943_nat_o @ nil_nat @ B3 ) ) )
       => ( ! [V2: nat,Va: list_nat,B3: $o] :
              ( X2
             != ( produc5973304058656726566_nat_o @ nil_nat @ ( produc1747790020158526943_nat_o @ ( cons_nat @ V2 @ Va ) @ B3 ) ) )
         => ~ ! [X: nat,Xs3: list_nat,V2: nat,Va: list_nat,B3: $o] :
                ( X2
               != ( produc5973304058656726566_nat_o @ ( cons_nat @ X @ Xs3 ) @ ( produc1747790020158526943_nat_o @ ( cons_nat @ V2 @ Va ) @ B3 ) ) ) ) ) ) ).

% mult2_impl0.cases
thf(fact_866_mult2__impl0_Ocases,axiom,
    ! [X2: produc7346466982839794670_int_o] :
      ( ! [B3: $o] :
          ( X2
         != ( produc1348525947378492198_int_o @ nil_int @ ( produc8726017089342856835_int_o @ nil_int @ B3 ) ) )
     => ( ! [V2: int,Va: list_int,B3: $o] :
            ( X2
           != ( produc1348525947378492198_int_o @ ( cons_int @ V2 @ Va ) @ ( produc8726017089342856835_int_o @ nil_int @ B3 ) ) )
       => ( ! [V2: int,Va: list_int,B3: $o] :
              ( X2
             != ( produc1348525947378492198_int_o @ nil_int @ ( produc8726017089342856835_int_o @ ( cons_int @ V2 @ Va ) @ B3 ) ) )
         => ~ ! [X: int,Xs3: list_int,V2: int,Va: list_int,B3: $o] :
                ( X2
               != ( produc1348525947378492198_int_o @ ( cons_int @ X @ Xs3 ) @ ( produc8726017089342856835_int_o @ ( cons_int @ V2 @ Va ) @ B3 ) ) ) ) ) ) ).

% mult2_impl0.cases
thf(fact_867_Cons__in__lex,axiom,
    ! [X2: int,Xs: list_int,Y2: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X2 @ Xs ) @ ( cons_int @ Y2 @ Ys ) ) @ ( lex_int @ R ) )
      = ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y2 ) @ R )
          & ( ( size_size_list_int @ Xs )
            = ( size_size_list_int @ Ys ) ) )
        | ( ( X2 = Y2 )
          & ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( lex_int @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_868_Cons__in__lex,axiom,
    ! [X2: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o,Y2: list_P3196869577824546119od_a_o,Ys: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ ( cons_l9179691953475787517od_a_o @ X2 @ Xs ) @ ( cons_l9179691953475787517od_a_o @ Y2 @ Ys ) ) @ ( lex_li6160749805527239089od_a_o @ R ) )
      = ( ( ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ X2 @ Y2 ) @ R )
          & ( ( size_s3379185846251495609od_a_o @ Xs )
            = ( size_s3379185846251495609od_a_o @ Ys ) ) )
        | ( ( X2 = Y2 )
          & ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Xs @ Ys ) @ ( lex_li6160749805527239089od_a_o @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_869_Cons__in__lex,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Y2: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( cons_S6797538523646374167on_nat @ X2 @ Xs ) @ ( cons_S6797538523646374167on_nat @ Y2 @ Ys ) ) @ ( lex_Su3183444693157923171on_nat @ R ) )
      = ( ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X2 @ Y2 ) @ R )
          & ( ( size_s5686234912572693873on_nat @ Xs )
            = ( size_s5686234912572693873on_nat @ Ys ) ) )
        | ( ( X2 = Y2 )
          & ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ Ys ) @ ( lex_Su3183444693157923171on_nat @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_870_Cons__in__lex,axiom,
    ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) @ ( lex_nat @ R ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R )
          & ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) ) )
        | ( ( X2 = Y2 )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_871_Cons__in__lex,axiom,
    ! [X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat,Y2: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ X2 @ Xs ) @ ( cons_P1582596238967110705on_nat @ Y2 @ Ys ) ) @ ( lex_Pr6276626268559582437on_nat @ R ) )
      = ( ( ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X2 @ Y2 ) @ R )
          & ( ( size_s7490768528479217773on_nat @ Xs )
            = ( size_s7490768528479217773on_nat @ Ys ) ) )
        | ( ( X2 = Y2 )
          & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( lex_Pr6276626268559582437on_nat @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_872_subrelI,axiom,
    ! [R: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ! [X: sum_su4907927656612468301on_nat,Y4: sum_su4907927656612468301on_nat] :
          ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X @ Y4 ) @ R )
         => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X @ Y4 ) @ S ) )
     => ( ord_le6961793561435735931on_nat @ R @ S ) ) ).

% subrelI
thf(fact_873_subrelI,axiom,
    ! [R: set_Pr9093778441882193744at_nat,S: set_Pr9093778441882193744at_nat] :
      ( ! [X: nat > nat,Y4: nat] :
          ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y4 ) @ R )
         => ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y4 ) @ S ) )
     => ( ord_le3678578370064672496at_nat @ R @ S ) ) ).

% subrelI
thf(fact_874_in__measures_I2_J,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Y2: sum_su4907927656612468301on_nat,F: sum_su4907927656612468301on_nat > nat,Fs: list_S8214525811314100556at_nat] :
      ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X2 @ Y2 ) @ ( measur7099086049050739021on_nat @ ( cons_S6883198266094849926at_nat @ F @ Fs ) ) )
      = ( ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) )
        | ( ( ( F @ X2 )
            = ( F @ Y2 ) )
          & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X2 @ Y2 ) @ ( measur7099086049050739021on_nat @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_875_subseqs_Osimps_I1_J,axiom,
    ( ( subseqs_nat @ nil_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% subseqs.simps(1)
thf(fact_876_in__measures_I1_J,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Y2: sum_su4907927656612468301on_nat] :
      ~ ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X2 @ Y2 ) @ ( measur7099086049050739021on_nat @ nil_Su430322706654367542at_nat ) ) ).

% in_measures(1)
thf(fact_877_subseqs__refl,axiom,
    ! [Xs: list_P3196869577824546119od_a_o] : ( member5261019209154752368od_a_o @ Xs @ ( set_li2629784761229999580od_a_o @ ( subseq3821054824087877782od_a_o @ Xs ) ) ) ).

% subseqs_refl
thf(fact_878_Nil__notin__lex,axiom,
    ! [Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys ) @ ( lex_nat @ R ) ) ).

% Nil_notin_lex
thf(fact_879_Nil2__notin__lex,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( lex_nat @ R ) ) ).

% Nil2_notin_lex
thf(fact_880_Cons__in__subseqsD,axiom,
    ! [Y2: product_prod_a_o,Ys: list_P3196869577824546119od_a_o,Xs: list_P3196869577824546119od_a_o] :
      ( ( member5261019209154752368od_a_o @ ( cons_P8471407633269037175od_a_o @ Y2 @ Ys ) @ ( set_li2629784761229999580od_a_o @ ( subseq3821054824087877782od_a_o @ Xs ) ) )
     => ( member5261019209154752368od_a_o @ Ys @ ( set_li2629784761229999580od_a_o @ ( subseq3821054824087877782od_a_o @ Xs ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_881_Cons__in__subseqsD,axiom,
    ! [Y2: nat,Ys: list_nat,Xs: list_nat] :
      ( ( member_list_nat @ ( cons_nat @ Y2 @ Ys ) @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
     => ( member_list_nat @ Ys @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_882_Cons__in__subseqsD,axiom,
    ! [Y2: int,Ys: list_int,Xs: list_int] :
      ( ( member_list_int @ ( cons_int @ Y2 @ Ys ) @ ( set_list_int2 @ ( subseqs_int @ Xs ) ) )
     => ( member_list_int @ Ys @ ( set_list_int2 @ ( subseqs_int @ Xs ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_883_subseqs__distinctD,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
     => ( ( distinct_nat @ Xs )
       => ( distinct_nat @ Ys ) ) ) ).

% subseqs_distinctD
thf(fact_884_subseqs__distinctD,axiom,
    ! [Ys: list_P3196869577824546119od_a_o,Xs: list_P3196869577824546119od_a_o] :
      ( ( member5261019209154752368od_a_o @ Ys @ ( set_li2629784761229999580od_a_o @ ( subseq3821054824087877782od_a_o @ Xs ) ) )
     => ( ( distin2380507709364613050od_a_o @ Xs )
       => ( distin2380507709364613050od_a_o @ Ys ) ) ) ).

% subseqs_distinctD
thf(fact_885_measures__less,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,X2: sum_su4907927656612468301on_nat,Y2: sum_su4907927656612468301on_nat,Fs: list_S8214525811314100556at_nat] :
      ( ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) )
     => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X2 @ Y2 ) @ ( measur7099086049050739021on_nat @ ( cons_S6883198266094849926at_nat @ F @ Fs ) ) ) ) ).

% measures_less
thf(fact_886_measures__lesseq,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,X2: sum_su4907927656612468301on_nat,Y2: sum_su4907927656612468301on_nat,Fs: list_S8214525811314100556at_nat] :
      ( ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) )
     => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X2 @ Y2 ) @ ( measur7099086049050739021on_nat @ Fs ) )
       => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X2 @ Y2 ) @ ( measur7099086049050739021on_nat @ ( cons_S6883198266094849926at_nat @ F @ Fs ) ) ) ) ) ).

% measures_lesseq
thf(fact_887_lex__take__index,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Xs @ Ys ) @ ( lex_li6160749805527239089od_a_o @ R ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3379185846251495609od_a_o @ Xs ) )
           => ( ( ord_less_nat @ I2 @ ( size_s3379185846251495609od_a_o @ Ys ) )
             => ( ( ( take_l5436743365630562101od_a_o @ I2 @ Xs )
                  = ( take_l5436743365630562101od_a_o @ I2 @ Ys ) )
               => ~ ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ Xs @ I2 ) @ ( nth_li3248255784598043854od_a_o @ Ys @ I2 ) ) @ R ) ) ) ) ) ).

% lex_take_index
thf(fact_888_lex__take__index,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ Ys ) @ ( lex_Su3183444693157923171on_nat @ R ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s5686234912572693873on_nat @ Xs ) )
           => ( ( ord_less_nat @ I2 @ ( size_s5686234912572693873on_nat @ Ys ) )
             => ( ( ( take_S1147720861222069983on_nat @ I2 @ Xs )
                  = ( take_S1147720861222069983on_nat @ I2 @ Ys ) )
               => ~ ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ Xs @ I2 ) @ ( nth_Su8471564416988646406on_nat @ Ys @ I2 ) ) @ R ) ) ) ) ) ).

% lex_take_index
thf(fact_889_lex__take__index,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
           => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
             => ( ( ( take_nat @ I2 @ Xs )
                  = ( take_nat @ I2 @ Ys ) )
               => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) @ R ) ) ) ) ) ).

% lex_take_index
thf(fact_890_lex__take__index,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( lex_Pr6276626268559582437on_nat @ R ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s7490768528479217773on_nat @ Xs ) )
           => ( ( ord_less_nat @ I2 @ ( size_s7490768528479217773on_nat @ Ys ) )
             => ( ( ( take_P4655611586102217833on_nat @ I2 @ Xs )
                  = ( take_P4655611586102217833on_nat @ I2 @ Ys ) )
               => ~ ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ ( nth_Pr3336893728888831618on_nat @ Xs @ I2 ) @ ( nth_Pr3336893728888831618on_nat @ Ys @ I2 ) ) @ R ) ) ) ) ) ).

% lex_take_index
thf(fact_891_sorted__list__of__multiset__singleton,axiom,
    ! [X2: nat] :
      ( ( linord3047872887403683810et_nat @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) )
      = ( cons_nat @ X2 @ nil_nat ) ) ).

% sorted_list_of_multiset_singleton
thf(fact_892_sorted__list__of__multiset__singleton,axiom,
    ! [X2: int] :
      ( ( linord3045382416894633534et_int @ ( add_mset_int @ X2 @ zero_z3170743180189231877et_int ) )
      = ( cons_int @ X2 @ nil_int ) ) ).

% sorted_list_of_multiset_singleton
thf(fact_893_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_nat,X2: nat,Ys: list_nat,Y2: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) ) @ ( listrel1_nat @ R ) )
      = ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
          & ( X2 = Y2 ) )
        | ( ( Xs = Ys )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_894_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_int,X2: int,Ys: list_int,Y2: int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Xs @ ( cons_int @ X2 @ nil_int ) ) @ ( append_int @ Ys @ ( cons_int @ Y2 @ nil_int ) ) ) @ ( listrel1_int @ R ) )
      = ( ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) )
          & ( X2 = Y2 ) )
        | ( ( Xs = Ys )
          & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y2 ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_895_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_S5253404628753938141on_nat,X2: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat,Y2: sum_su4907927656612468301on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( append3523812819975775986on_nat @ Xs @ ( cons_S6797538523646374167on_nat @ X2 @ nil_Su3078906344068544711on_nat ) ) @ ( append3523812819975775986on_nat @ Ys @ ( cons_S6797538523646374167on_nat @ Y2 @ nil_Su3078906344068544711on_nat ) ) ) @ ( listre1270753918075970734on_nat @ R ) )
      = ( ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ Ys ) @ ( listre1270753918075970734on_nat @ R ) )
          & ( X2 = Y2 ) )
        | ( ( Xs = Ys )
          & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X2 @ Y2 ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_896_subset__code_I3_J,axiom,
    ~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).

% subset_code(3)
thf(fact_897_subset__code_I3_J,axiom,
    ~ ( ord_le1271121679792220359od_a_o @ ( coset_2783394909775257274od_a_o @ nil_li1372968520647675725od_a_o ) @ ( set_li2629784761229999580od_a_o @ nil_li1372968520647675725od_a_o ) ) ).

% subset_code(3)
thf(fact_898_subset__code_I3_J,axiom,
    ~ ( ord_le6961793561435735931on_nat @ ( coset_1816015773736449646on_nat @ nil_Pr8614093721167239297on_nat ) @ ( set_Pr937530902923597456on_nat @ nil_Pr8614093721167239297on_nat ) ) ).

% subset_code(3)
thf(fact_899_subset__code_I3_J,axiom,
    ~ ( ord_le6595998496543023075on_nat @ ( coset_3487183839294692122on_nat @ nil_Su3078906344068544711on_nat ) @ ( set_Su4848624856859743608on_nat @ nil_Su3078906344068544711on_nat ) ) ).

% subset_code(3)
thf(fact_900_same__append__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_901_append__same__eq,axiom,
    ! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Ys @ Xs )
        = ( append_nat @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_902_append__assoc,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
      = ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_903_append_Oassoc,axiom,
    ! [A: list_nat,B: list_nat,C2: list_nat] :
      ( ( append_nat @ ( append_nat @ A @ B ) @ C2 )
      = ( append_nat @ A @ ( append_nat @ B @ C2 ) ) ) ).

% append.assoc
thf(fact_904_append_Oright__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ A @ nil_nat )
      = A ) ).

% append.right_neutral
thf(fact_905_append__Nil2,axiom,
    ! [Xs: list_nat] :
      ( ( append_nat @ Xs @ nil_nat )
      = Xs ) ).

% append_Nil2
thf(fact_906_append__self__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_nat ) ) ).

% append_self_conv
thf(fact_907_self__append__conv,axiom,
    ! [Y2: list_nat,Ys: list_nat] :
      ( ( Y2
        = ( append_nat @ Y2 @ Ys ) )
      = ( Ys = nil_nat ) ) ).

% self_append_conv
thf(fact_908_append__self__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_nat ) ) ).

% append_self_conv2
thf(fact_909_self__append__conv2,axiom,
    ! [Y2: list_nat,Xs: list_nat] :
      ( ( Y2
        = ( append_nat @ Xs @ Y2 ) )
      = ( Xs = nil_nat ) ) ).

% self_append_conv2
thf(fact_910_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_911_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_912_append__eq__append__conv,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,Us: list_l8336264072933094989od_a_o,Vs: list_l8336264072933094989od_a_o] :
      ( ( ( ( size_s3379185846251495609od_a_o @ Xs )
          = ( size_s3379185846251495609od_a_o @ Ys ) )
        | ( ( size_s3379185846251495609od_a_o @ Us )
          = ( size_s3379185846251495609od_a_o @ Vs ) ) )
     => ( ( ( append2159120467191820514od_a_o @ Xs @ Us )
          = ( append2159120467191820514od_a_o @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_913_append__eq__append__conv,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Us: list_S5253404628753938141on_nat,Vs: list_S5253404628753938141on_nat] :
      ( ( ( ( size_s5686234912572693873on_nat @ Xs )
          = ( size_s5686234912572693873on_nat @ Ys ) )
        | ( ( size_s5686234912572693873on_nat @ Us )
          = ( size_s5686234912572693873on_nat @ Vs ) ) )
     => ( ( ( append3523812819975775986on_nat @ Xs @ Us )
          = ( append3523812819975775986on_nat @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_914_append__eq__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
      ( ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        | ( ( size_size_list_nat @ Us )
          = ( size_size_list_nat @ Vs ) ) )
     => ( ( ( append_nat @ Xs @ Us )
          = ( append_nat @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_915_append__eq__append__conv,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Us: list_P9127870862658512897on_nat,Vs: list_P9127870862658512897on_nat] :
      ( ( ( ( size_s7490768528479217773on_nat @ Xs )
          = ( size_s7490768528479217773on_nat @ Ys ) )
        | ( ( size_s7490768528479217773on_nat @ Us )
          = ( size_s7490768528479217773on_nat @ Vs ) ) )
     => ( ( ( append2865821678206098582on_nat @ Xs @ Us )
          = ( append2865821678206098582on_nat @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_916_removeAll__append,axiom,
    ! [X2: nat,Xs: list_nat,Ys: list_nat] :
      ( ( removeAll_nat @ X2 @ ( append_nat @ Xs @ Ys ) )
      = ( append_nat @ ( removeAll_nat @ X2 @ Xs ) @ ( removeAll_nat @ X2 @ Ys ) ) ) ).

% removeAll_append
thf(fact_917_list__ex__append,axiom,
    ! [P: nat > $o,Xs: list_nat,Ys: list_nat] :
      ( ( list_ex_nat @ P @ ( append_nat @ Xs @ Ys ) )
      = ( ( list_ex_nat @ P @ Xs )
        | ( list_ex_nat @ P @ Ys ) ) ) ).

% list_ex_append
thf(fact_918_append1__eq__conv,axiom,
    ! [Xs: list_nat,X2: nat,Ys: list_nat,Y2: nat] :
      ( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) )
        = ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) )
      = ( ( Xs = Ys )
        & ( X2 = Y2 ) ) ) ).

% append1_eq_conv
thf(fact_919_append1__eq__conv,axiom,
    ! [Xs: list_int,X2: int,Ys: list_int,Y2: int] :
      ( ( ( append_int @ Xs @ ( cons_int @ X2 @ nil_int ) )
        = ( append_int @ Ys @ ( cons_int @ Y2 @ nil_int ) ) )
      = ( ( Xs = Ys )
        & ( X2 = Y2 ) ) ) ).

% append1_eq_conv
thf(fact_920_take0,axiom,
    ( ( take_nat @ zero_zero_nat )
    = ( ^ [Xs2: list_nat] : nil_nat ) ) ).

% take0
thf(fact_921_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_922_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_923_take__all__iff,axiom,
    ! [N: nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( ( take_l5436743365630562101od_a_o @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_s3379185846251495609od_a_o @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_924_take__all__iff,axiom,
    ! [N: nat,Xs: list_S5253404628753938141on_nat] :
      ( ( ( take_S1147720861222069983on_nat @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_s5686234912572693873on_nat @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_925_take__all__iff,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ( take_nat @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_926_take__all__iff,axiom,
    ! [N: nat,Xs: list_P9127870862658512897on_nat] :
      ( ( ( take_P4655611586102217833on_nat @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_s7490768528479217773on_nat @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_927_take__all,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s3379185846251495609od_a_o @ Xs ) @ N )
     => ( ( take_l5436743365630562101od_a_o @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_928_take__all,axiom,
    ! [Xs: list_S5253404628753938141on_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s5686234912572693873on_nat @ Xs ) @ N )
     => ( ( take_S1147720861222069983on_nat @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_929_take__all,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
     => ( ( take_nat @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_930_take__all,axiom,
    ! [Xs: list_P9127870862658512897on_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s7490768528479217773on_nat @ Xs ) @ N )
     => ( ( take_P4655611586102217833on_nat @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_931_nth__take,axiom,
    ! [I3: nat,N: nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( ord_less_nat @ I3 @ N )
     => ( ( nth_li3248255784598043854od_a_o @ ( take_l5436743365630562101od_a_o @ N @ Xs ) @ I3 )
        = ( nth_li3248255784598043854od_a_o @ Xs @ I3 ) ) ) ).

% nth_take
thf(fact_932_nth__take,axiom,
    ! [I3: nat,N: nat,Xs: list_S5253404628753938141on_nat] :
      ( ( ord_less_nat @ I3 @ N )
     => ( ( nth_Su8471564416988646406on_nat @ ( take_S1147720861222069983on_nat @ N @ Xs ) @ I3 )
        = ( nth_Su8471564416988646406on_nat @ Xs @ I3 ) ) ) ).

% nth_take
thf(fact_933_nth__take,axiom,
    ! [I3: nat,N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ I3 @ N )
     => ( ( nth_nat @ ( take_nat @ N @ Xs ) @ I3 )
        = ( nth_nat @ Xs @ I3 ) ) ) ).

% nth_take
thf(fact_934_nth__take,axiom,
    ! [I3: nat,N: nat,Xs: list_P9127870862658512897on_nat] :
      ( ( ord_less_nat @ I3 @ N )
     => ( ( nth_Pr3336893728888831618on_nat @ ( take_P4655611586102217833on_nat @ N @ Xs ) @ I3 )
        = ( nth_Pr3336893728888831618on_nat @ Xs @ I3 ) ) ) ).

% nth_take
thf(fact_935_nth__append__length,axiom,
    ! [Xs: list_int,X2: int,Ys: list_int] :
      ( ( nth_int @ ( append_int @ Xs @ ( cons_int @ X2 @ Ys ) ) @ ( size_size_list_int @ Xs ) )
      = X2 ) ).

% nth_append_length
thf(fact_936_nth__append__length,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X2: list_P3196869577824546119od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( nth_li3248255784598043854od_a_o @ ( append2159120467191820514od_a_o @ Xs @ ( cons_l9179691953475787517od_a_o @ X2 @ Ys ) ) @ ( size_s3379185846251495609od_a_o @ Xs ) )
      = X2 ) ).

% nth_append_length
thf(fact_937_nth__append__length,axiom,
    ! [Xs: list_S5253404628753938141on_nat,X2: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( nth_Su8471564416988646406on_nat @ ( append3523812819975775986on_nat @ Xs @ ( cons_S6797538523646374167on_nat @ X2 @ Ys ) ) @ ( size_s5686234912572693873on_nat @ Xs ) )
      = X2 ) ).

% nth_append_length
thf(fact_938_nth__append__length,axiom,
    ! [Xs: list_nat,X2: nat,Ys: list_nat] :
      ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) ) @ ( size_size_list_nat @ Xs ) )
      = X2 ) ).

% nth_append_length
thf(fact_939_nth__append__length,axiom,
    ! [Xs: list_P9127870862658512897on_nat,X2: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( nth_Pr3336893728888831618on_nat @ ( append2865821678206098582on_nat @ Xs @ ( cons_P1582596238967110705on_nat @ X2 @ Ys ) ) @ ( size_s7490768528479217773on_nat @ Xs ) )
      = X2 ) ).

% nth_append_length
thf(fact_940_list__update__length,axiom,
    ! [Xs: list_int,X2: int,Ys: list_int,Y2: int] :
      ( ( list_update_int @ ( append_int @ Xs @ ( cons_int @ X2 @ Ys ) ) @ ( size_size_list_int @ Xs ) @ Y2 )
      = ( append_int @ Xs @ ( cons_int @ Y2 @ Ys ) ) ) ).

% list_update_length
thf(fact_941_list__update__length,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X2: list_P3196869577824546119od_a_o,Ys: list_l8336264072933094989od_a_o,Y2: list_P3196869577824546119od_a_o] :
      ( ( list_u329291161401623989od_a_o @ ( append2159120467191820514od_a_o @ Xs @ ( cons_l9179691953475787517od_a_o @ X2 @ Ys ) ) @ ( size_s3379185846251495609od_a_o @ Xs ) @ Y2 )
      = ( append2159120467191820514od_a_o @ Xs @ ( cons_l9179691953475787517od_a_o @ Y2 @ Ys ) ) ) ).

% list_update_length
thf(fact_942_list__update__length,axiom,
    ! [Xs: list_S5253404628753938141on_nat,X2: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat,Y2: sum_su4907927656612468301on_nat] :
      ( ( list_u8577817230714425439on_nat @ ( append3523812819975775986on_nat @ Xs @ ( cons_S6797538523646374167on_nat @ X2 @ Ys ) ) @ ( size_s5686234912572693873on_nat @ Xs ) @ Y2 )
      = ( append3523812819975775986on_nat @ Xs @ ( cons_S6797538523646374167on_nat @ Y2 @ Ys ) ) ) ).

% list_update_length
thf(fact_943_list__update__length,axiom,
    ! [Xs: list_nat,X2: nat,Ys: list_nat,Y2: nat] :
      ( ( list_update_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) ) @ ( size_size_list_nat @ Xs ) @ Y2 )
      = ( append_nat @ Xs @ ( cons_nat @ Y2 @ Ys ) ) ) ).

% list_update_length
thf(fact_944_list__update__length,axiom,
    ! [Xs: list_P9127870862658512897on_nat,X2: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat,Y2: produc1313295348606329723on_nat] :
      ( ( list_u1840724836389618409on_nat @ ( append2865821678206098582on_nat @ Xs @ ( cons_P1582596238967110705on_nat @ X2 @ Ys ) ) @ ( size_s7490768528479217773on_nat @ Xs ) @ Y2 )
      = ( append2865821678206098582on_nat @ Xs @ ( cons_P1582596238967110705on_nat @ Y2 @ Ys ) ) ) ).

% list_update_length
thf(fact_945_Cons__eq__appendI,axiom,
    ! [X2: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs: list_nat] :
      ( ( ( cons_nat @ X2 @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_nat @ Xs1 @ Zs ) )
       => ( ( cons_nat @ X2 @ Xs )
          = ( append_nat @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_946_Cons__eq__appendI,axiom,
    ! [X2: int,Xs1: list_int,Ys: list_int,Xs: list_int,Zs: list_int] :
      ( ( ( cons_int @ X2 @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_int @ Xs1 @ Zs ) )
       => ( ( cons_int @ X2 @ Xs )
          = ( append_int @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_947_append__Cons,axiom,
    ! [X2: nat,Xs: list_nat,Ys: list_nat] :
      ( ( append_nat @ ( cons_nat @ X2 @ Xs ) @ Ys )
      = ( cons_nat @ X2 @ ( append_nat @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_948_append__Cons,axiom,
    ! [X2: int,Xs: list_int,Ys: list_int] :
      ( ( append_int @ ( cons_int @ X2 @ Xs ) @ Ys )
      = ( cons_int @ X2 @ ( append_int @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_949_append__eq__append__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Zs @ Ts ) )
      = ( ? [Us2: list_nat] :
            ( ( ( Xs
                = ( append_nat @ Zs @ Us2 ) )
              & ( ( append_nat @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append_nat @ Xs @ Us2 )
                = Zs )
              & ( Ys
                = ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_950_append__eq__appendI,axiom,
    ! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us: list_nat] :
      ( ( ( append_nat @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append_nat @ Xs1 @ Us ) )
       => ( ( append_nat @ Xs @ Ys )
          = ( append_nat @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_951_append__Nil,axiom,
    ! [Ys: list_nat] :
      ( ( append_nat @ nil_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_952_append_Oleft__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ nil_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_953_eq__Nil__appendI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_nat @ nil_nat @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_954_distinct__take,axiom,
    ! [Xs: list_nat,I3: nat] :
      ( ( distinct_nat @ Xs )
     => ( distinct_nat @ ( take_nat @ I3 @ Xs ) ) ) ).

% distinct_take
thf(fact_955_in__set__takeD,axiom,
    ! [X2: nat,N: nat,Xs: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) )
     => ( member_nat @ X2 @ ( set_nat2 @ Xs ) ) ) ).

% in_set_takeD
thf(fact_956_in__set__takeD,axiom,
    ! [X2: list_P3196869577824546119od_a_o,N: nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ ( take_l5436743365630562101od_a_o @ N @ Xs ) ) )
     => ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Xs ) ) ) ).

% in_set_takeD
thf(fact_957_in__set__takeD,axiom,
    ! [X2: produc1313295348606329723on_nat,N: nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ ( take_P4655611586102217833on_nat @ N @ Xs ) ) )
     => ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Xs ) ) ) ).

% in_set_takeD
thf(fact_958_in__set__takeD,axiom,
    ! [X2: sum_su4907927656612468301on_nat,N: nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ ( take_S1147720861222069983on_nat @ N @ Xs ) ) )
     => ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Xs ) ) ) ).

% in_set_takeD
thf(fact_959_take__Nil,axiom,
    ! [N: nat] :
      ( ( take_nat @ N @ nil_nat )
      = nil_nat ) ).

% take_Nil
thf(fact_960_take__0,axiom,
    ! [Xs: list_nat] :
      ( ( take_nat @ zero_zero_nat @ Xs )
      = nil_nat ) ).

% take_0
thf(fact_961_set__take__subset,axiom,
    ! [N: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).

% set_take_subset
thf(fact_962_set__take__subset,axiom,
    ! [N: nat,Xs: list_l8336264072933094989od_a_o] : ( ord_le1271121679792220359od_a_o @ ( set_li2629784761229999580od_a_o @ ( take_l5436743365630562101od_a_o @ N @ Xs ) ) @ ( set_li2629784761229999580od_a_o @ Xs ) ) ).

% set_take_subset
thf(fact_963_set__take__subset,axiom,
    ! [N: nat,Xs: list_P9127870862658512897on_nat] : ( ord_le6961793561435735931on_nat @ ( set_Pr937530902923597456on_nat @ ( take_P4655611586102217833on_nat @ N @ Xs ) ) @ ( set_Pr937530902923597456on_nat @ Xs ) ) ).

% set_take_subset
thf(fact_964_set__take__subset,axiom,
    ! [N: nat,Xs: list_S5253404628753938141on_nat] : ( ord_le6595998496543023075on_nat @ ( set_Su4848624856859743608on_nat @ ( take_S1147720861222069983on_nat @ N @ Xs ) ) @ ( set_Su4848624856859743608on_nat @ Xs ) ) ).

% set_take_subset
thf(fact_965_rev__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ( P @ nil_nat )
     => ( ! [X: nat,Xs3: list_nat] :
            ( ( P @ Xs3 )
           => ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X @ nil_nat ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_966_rev__induct,axiom,
    ! [P: list_int > $o,Xs: list_int] :
      ( ( P @ nil_int )
     => ( ! [X: int,Xs3: list_int] :
            ( ( P @ Xs3 )
           => ( P @ ( append_int @ Xs3 @ ( cons_int @ X @ nil_int ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_967_rev__exhaust,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ~ ! [Ys4: list_nat,Y4: nat] :
            ( Xs
           != ( append_nat @ Ys4 @ ( cons_nat @ Y4 @ nil_nat ) ) ) ) ).

% rev_exhaust
thf(fact_968_rev__exhaust,axiom,
    ! [Xs: list_int] :
      ( ( Xs != nil_int )
     => ~ ! [Ys4: list_int,Y4: int] :
            ( Xs
           != ( append_int @ Ys4 @ ( cons_int @ Y4 @ nil_int ) ) ) ) ).

% rev_exhaust
thf(fact_969_Cons__eq__append__conv,axiom,
    ! [X2: nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( cons_nat @ X2 @ Xs )
        = ( append_nat @ Ys @ Zs ) )
      = ( ( ( Ys = nil_nat )
          & ( ( cons_nat @ X2 @ Xs )
            = Zs ) )
        | ? [Ys6: list_nat] :
            ( ( ( cons_nat @ X2 @ Ys6 )
              = Ys )
            & ( Xs
              = ( append_nat @ Ys6 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_970_Cons__eq__append__conv,axiom,
    ! [X2: int,Xs: list_int,Ys: list_int,Zs: list_int] :
      ( ( ( cons_int @ X2 @ Xs )
        = ( append_int @ Ys @ Zs ) )
      = ( ( ( Ys = nil_int )
          & ( ( cons_int @ X2 @ Xs )
            = Zs ) )
        | ? [Ys6: list_int] :
            ( ( ( cons_int @ X2 @ Ys6 )
              = Ys )
            & ( Xs
              = ( append_int @ Ys6 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_971_append__eq__Cons__conv,axiom,
    ! [Ys: list_nat,Zs: list_nat,X2: nat,Xs: list_nat] :
      ( ( ( append_nat @ Ys @ Zs )
        = ( cons_nat @ X2 @ Xs ) )
      = ( ( ( Ys = nil_nat )
          & ( Zs
            = ( cons_nat @ X2 @ Xs ) ) )
        | ? [Ys6: list_nat] :
            ( ( Ys
              = ( cons_nat @ X2 @ Ys6 ) )
            & ( ( append_nat @ Ys6 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_972_append__eq__Cons__conv,axiom,
    ! [Ys: list_int,Zs: list_int,X2: int,Xs: list_int] :
      ( ( ( append_int @ Ys @ Zs )
        = ( cons_int @ X2 @ Xs ) )
      = ( ( ( Ys = nil_int )
          & ( Zs
            = ( cons_int @ X2 @ Xs ) ) )
        | ? [Ys6: list_int] :
            ( ( Ys
              = ( cons_int @ X2 @ Ys6 ) )
            & ( ( append_int @ Ys6 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_973_rev__nonempty__induct,axiom,
    ! [Xs: list_nat,P: list_nat > $o] :
      ( ( Xs != nil_nat )
     => ( ! [X: nat] : ( P @ ( cons_nat @ X @ nil_nat ) )
       => ( ! [X: nat,Xs3: list_nat] :
              ( ( Xs3 != nil_nat )
             => ( ( P @ Xs3 )
               => ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X @ nil_nat ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_974_rev__nonempty__induct,axiom,
    ! [Xs: list_int,P: list_int > $o] :
      ( ( Xs != nil_int )
     => ( ! [X: int] : ( P @ ( cons_int @ X @ nil_int ) )
       => ( ! [X: int,Xs3: list_int] :
              ( ( Xs3 != nil_int )
             => ( ( P @ Xs3 )
               => ( P @ ( append_int @ Xs3 @ ( cons_int @ X @ nil_int ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_975_split__list__first__prop__iff,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,P: list_P3196869577824546119od_a_o > $o] :
      ( ( ? [X3: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X3 @ ( set_li2629784761229999580od_a_o @ Xs ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys2: list_l8336264072933094989od_a_o,X3: list_P3196869577824546119od_a_o] :
            ( ? [Zs3: list_l8336264072933094989od_a_o] :
                ( Xs
                = ( append2159120467191820514od_a_o @ Ys2 @ ( cons_l9179691953475787517od_a_o @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y3: list_P3196869577824546119od_a_o] :
                ( ( member5261019209154752368od_a_o @ Y3 @ ( set_li2629784761229999580od_a_o @ Ys2 ) )
               => ~ ( P @ Y3 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_976_split__list__first__prop__iff,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P: produc1313295348606329723on_nat > $o] :
      ( ( ? [X3: produc1313295348606329723on_nat] :
            ( ( member5906418179908956708on_nat @ X3 @ ( set_Pr937530902923597456on_nat @ Xs ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys2: list_P9127870862658512897on_nat,X3: produc1313295348606329723on_nat] :
            ( ? [Zs3: list_P9127870862658512897on_nat] :
                ( Xs
                = ( append2865821678206098582on_nat @ Ys2 @ ( cons_P1582596238967110705on_nat @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y3: produc1313295348606329723on_nat] :
                ( ( member5906418179908956708on_nat @ Y3 @ ( set_Pr937530902923597456on_nat @ Ys2 ) )
               => ~ ( P @ Y3 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_977_split__list__first__prop__iff,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P: sum_su4907927656612468301on_nat > $o] :
      ( ( ? [X3: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X3 @ ( set_Su4848624856859743608on_nat @ Xs ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys2: list_S5253404628753938141on_nat,X3: sum_su4907927656612468301on_nat] :
            ( ? [Zs3: list_S5253404628753938141on_nat] :
                ( Xs
                = ( append3523812819975775986on_nat @ Ys2 @ ( cons_S6797538523646374167on_nat @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y3: sum_su4907927656612468301on_nat] :
                ( ( member8476201367316484836on_nat @ Y3 @ ( set_Su4848624856859743608on_nat @ Ys2 ) )
               => ~ ( P @ Y3 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_978_split__list__first__prop__iff,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ? [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys2: list_nat,X3: nat] :
            ( ? [Zs3: list_nat] :
                ( Xs
                = ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( set_nat2 @ Ys2 ) )
               => ~ ( P @ Y3 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_979_split__list__first__prop__iff,axiom,
    ! [Xs: list_int,P: int > $o] :
      ( ( ? [X3: int] :
            ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys2: list_int,X3: int] :
            ( ? [Zs3: list_int] :
                ( Xs
                = ( append_int @ Ys2 @ ( cons_int @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y3: int] :
                ( ( member_int @ Y3 @ ( set_int2 @ Ys2 ) )
               => ~ ( P @ Y3 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_980_split__list__last__prop__iff,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,P: list_P3196869577824546119od_a_o > $o] :
      ( ( ? [X3: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X3 @ ( set_li2629784761229999580od_a_o @ Xs ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys2: list_l8336264072933094989od_a_o,X3: list_P3196869577824546119od_a_o,Zs3: list_l8336264072933094989od_a_o] :
            ( ( Xs
              = ( append2159120467191820514od_a_o @ Ys2 @ ( cons_l9179691953475787517od_a_o @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y3: list_P3196869577824546119od_a_o] :
                ( ( member5261019209154752368od_a_o @ Y3 @ ( set_li2629784761229999580od_a_o @ Zs3 ) )
               => ~ ( P @ Y3 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_981_split__list__last__prop__iff,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P: produc1313295348606329723on_nat > $o] :
      ( ( ? [X3: produc1313295348606329723on_nat] :
            ( ( member5906418179908956708on_nat @ X3 @ ( set_Pr937530902923597456on_nat @ Xs ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys2: list_P9127870862658512897on_nat,X3: produc1313295348606329723on_nat,Zs3: list_P9127870862658512897on_nat] :
            ( ( Xs
              = ( append2865821678206098582on_nat @ Ys2 @ ( cons_P1582596238967110705on_nat @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y3: produc1313295348606329723on_nat] :
                ( ( member5906418179908956708on_nat @ Y3 @ ( set_Pr937530902923597456on_nat @ Zs3 ) )
               => ~ ( P @ Y3 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_982_split__list__last__prop__iff,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P: sum_su4907927656612468301on_nat > $o] :
      ( ( ? [X3: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X3 @ ( set_Su4848624856859743608on_nat @ Xs ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys2: list_S5253404628753938141on_nat,X3: sum_su4907927656612468301on_nat,Zs3: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( append3523812819975775986on_nat @ Ys2 @ ( cons_S6797538523646374167on_nat @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y3: sum_su4907927656612468301on_nat] :
                ( ( member8476201367316484836on_nat @ Y3 @ ( set_Su4848624856859743608on_nat @ Zs3 ) )
               => ~ ( P @ Y3 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_983_split__list__last__prop__iff,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ? [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys2: list_nat,X3: nat,Zs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( set_nat2 @ Zs3 ) )
               => ~ ( P @ Y3 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_984_split__list__last__prop__iff,axiom,
    ! [Xs: list_int,P: int > $o] :
      ( ( ? [X3: int] :
            ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys2: list_int,X3: int,Zs3: list_int] :
            ( ( Xs
              = ( append_int @ Ys2 @ ( cons_int @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y3: int] :
                ( ( member_int @ Y3 @ ( set_int2 @ Zs3 ) )
               => ~ ( P @ Y3 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_985_in__set__conv__decomp__first,axiom,
    ! [X2: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Xs ) )
      = ( ? [Ys2: list_l8336264072933094989od_a_o,Zs3: list_l8336264072933094989od_a_o] :
            ( ( Xs
              = ( append2159120467191820514od_a_o @ Ys2 @ ( cons_l9179691953475787517od_a_o @ X2 @ Zs3 ) ) )
            & ~ ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_986_in__set__conv__decomp__first,axiom,
    ! [X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Xs ) )
      = ( ? [Ys2: list_P9127870862658512897on_nat,Zs3: list_P9127870862658512897on_nat] :
            ( ( Xs
              = ( append2865821678206098582on_nat @ Ys2 @ ( cons_P1582596238967110705on_nat @ X2 @ Zs3 ) ) )
            & ~ ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_987_in__set__conv__decomp__first,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Xs ) )
      = ( ? [Ys2: list_S5253404628753938141on_nat,Zs3: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( append3523812819975775986on_nat @ Ys2 @ ( cons_S6797538523646374167on_nat @ X2 @ Zs3 ) ) )
            & ~ ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_988_in__set__conv__decomp__first,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
      = ( ? [Ys2: list_nat,Zs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs3 ) ) )
            & ~ ( member_nat @ X2 @ ( set_nat2 @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_989_in__set__conv__decomp__first,axiom,
    ! [X2: int,Xs: list_int] :
      ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
      = ( ? [Ys2: list_int,Zs3: list_int] :
            ( ( Xs
              = ( append_int @ Ys2 @ ( cons_int @ X2 @ Zs3 ) ) )
            & ~ ( member_int @ X2 @ ( set_int2 @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_990_in__set__conv__decomp__last,axiom,
    ! [X2: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Xs ) )
      = ( ? [Ys2: list_l8336264072933094989od_a_o,Zs3: list_l8336264072933094989od_a_o] :
            ( ( Xs
              = ( append2159120467191820514od_a_o @ Ys2 @ ( cons_l9179691953475787517od_a_o @ X2 @ Zs3 ) ) )
            & ~ ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_991_in__set__conv__decomp__last,axiom,
    ! [X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Xs ) )
      = ( ? [Ys2: list_P9127870862658512897on_nat,Zs3: list_P9127870862658512897on_nat] :
            ( ( Xs
              = ( append2865821678206098582on_nat @ Ys2 @ ( cons_P1582596238967110705on_nat @ X2 @ Zs3 ) ) )
            & ~ ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_992_in__set__conv__decomp__last,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Xs ) )
      = ( ? [Ys2: list_S5253404628753938141on_nat,Zs3: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( append3523812819975775986on_nat @ Ys2 @ ( cons_S6797538523646374167on_nat @ X2 @ Zs3 ) ) )
            & ~ ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_993_in__set__conv__decomp__last,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
      = ( ? [Ys2: list_nat,Zs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs3 ) ) )
            & ~ ( member_nat @ X2 @ ( set_nat2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_994_in__set__conv__decomp__last,axiom,
    ! [X2: int,Xs: list_int] :
      ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
      = ( ? [Ys2: list_int,Zs3: list_int] :
            ( ( Xs
              = ( append_int @ Ys2 @ ( cons_int @ X2 @ Zs3 ) ) )
            & ~ ( member_int @ X2 @ ( set_int2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_995_split__list__first__propE,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,P: list_P3196869577824546119od_a_o > $o] :
      ( ? [X6: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ X6 @ ( set_li2629784761229999580od_a_o @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys4: list_l8336264072933094989od_a_o,X: list_P3196869577824546119od_a_o] :
            ( ? [Zs2: list_l8336264072933094989od_a_o] :
                ( Xs
                = ( append2159120467191820514od_a_o @ Ys4 @ ( cons_l9179691953475787517od_a_o @ X @ Zs2 ) ) )
           => ( ( P @ X )
             => ~ ! [Xa: list_P3196869577824546119od_a_o] :
                    ( ( member5261019209154752368od_a_o @ Xa @ ( set_li2629784761229999580od_a_o @ Ys4 ) )
                   => ~ ( P @ Xa ) ) ) ) ) ).

% split_list_first_propE
thf(fact_996_split__list__first__propE,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P: produc1313295348606329723on_nat > $o] :
      ( ? [X6: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X6 @ ( set_Pr937530902923597456on_nat @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys4: list_P9127870862658512897on_nat,X: produc1313295348606329723on_nat] :
            ( ? [Zs2: list_P9127870862658512897on_nat] :
                ( Xs
                = ( append2865821678206098582on_nat @ Ys4 @ ( cons_P1582596238967110705on_nat @ X @ Zs2 ) ) )
           => ( ( P @ X )
             => ~ ! [Xa: produc1313295348606329723on_nat] :
                    ( ( member5906418179908956708on_nat @ Xa @ ( set_Pr937530902923597456on_nat @ Ys4 ) )
                   => ~ ( P @ Xa ) ) ) ) ) ).

% split_list_first_propE
thf(fact_997_split__list__first__propE,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P: sum_su4907927656612468301on_nat > $o] :
      ( ? [X6: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X6 @ ( set_Su4848624856859743608on_nat @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys4: list_S5253404628753938141on_nat,X: sum_su4907927656612468301on_nat] :
            ( ? [Zs2: list_S5253404628753938141on_nat] :
                ( Xs
                = ( append3523812819975775986on_nat @ Ys4 @ ( cons_S6797538523646374167on_nat @ X @ Zs2 ) ) )
           => ( ( P @ X )
             => ~ ! [Xa: sum_su4907927656612468301on_nat] :
                    ( ( member8476201367316484836on_nat @ Xa @ ( set_Su4848624856859743608on_nat @ Ys4 ) )
                   => ~ ( P @ Xa ) ) ) ) ) ).

% split_list_first_propE
thf(fact_998_split__list__first__propE,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X6: nat] :
          ( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys4: list_nat,X: nat] :
            ( ? [Zs2: list_nat] :
                ( Xs
                = ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs2 ) ) )
           => ( ( P @ X )
             => ~ ! [Xa: nat] :
                    ( ( member_nat @ Xa @ ( set_nat2 @ Ys4 ) )
                   => ~ ( P @ Xa ) ) ) ) ) ).

% split_list_first_propE
thf(fact_999_split__list__first__propE,axiom,
    ! [Xs: list_int,P: int > $o] :
      ( ? [X6: int] :
          ( ( member_int @ X6 @ ( set_int2 @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys4: list_int,X: int] :
            ( ? [Zs2: list_int] :
                ( Xs
                = ( append_int @ Ys4 @ ( cons_int @ X @ Zs2 ) ) )
           => ( ( P @ X )
             => ~ ! [Xa: int] :
                    ( ( member_int @ Xa @ ( set_int2 @ Ys4 ) )
                   => ~ ( P @ Xa ) ) ) ) ) ).

% split_list_first_propE
thf(fact_1000_split__list__last__propE,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,P: list_P3196869577824546119od_a_o > $o] :
      ( ? [X6: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ X6 @ ( set_li2629784761229999580od_a_o @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys4: list_l8336264072933094989od_a_o,X: list_P3196869577824546119od_a_o,Zs2: list_l8336264072933094989od_a_o] :
            ( ( Xs
              = ( append2159120467191820514od_a_o @ Ys4 @ ( cons_l9179691953475787517od_a_o @ X @ Zs2 ) ) )
           => ( ( P @ X )
             => ~ ! [Xa: list_P3196869577824546119od_a_o] :
                    ( ( member5261019209154752368od_a_o @ Xa @ ( set_li2629784761229999580od_a_o @ Zs2 ) )
                   => ~ ( P @ Xa ) ) ) ) ) ).

% split_list_last_propE
thf(fact_1001_split__list__last__propE,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P: produc1313295348606329723on_nat > $o] :
      ( ? [X6: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X6 @ ( set_Pr937530902923597456on_nat @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys4: list_P9127870862658512897on_nat,X: produc1313295348606329723on_nat,Zs2: list_P9127870862658512897on_nat] :
            ( ( Xs
              = ( append2865821678206098582on_nat @ Ys4 @ ( cons_P1582596238967110705on_nat @ X @ Zs2 ) ) )
           => ( ( P @ X )
             => ~ ! [Xa: produc1313295348606329723on_nat] :
                    ( ( member5906418179908956708on_nat @ Xa @ ( set_Pr937530902923597456on_nat @ Zs2 ) )
                   => ~ ( P @ Xa ) ) ) ) ) ).

% split_list_last_propE
thf(fact_1002_split__list__last__propE,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P: sum_su4907927656612468301on_nat > $o] :
      ( ? [X6: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X6 @ ( set_Su4848624856859743608on_nat @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys4: list_S5253404628753938141on_nat,X: sum_su4907927656612468301on_nat,Zs2: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( append3523812819975775986on_nat @ Ys4 @ ( cons_S6797538523646374167on_nat @ X @ Zs2 ) ) )
           => ( ( P @ X )
             => ~ ! [Xa: sum_su4907927656612468301on_nat] :
                    ( ( member8476201367316484836on_nat @ Xa @ ( set_Su4848624856859743608on_nat @ Zs2 ) )
                   => ~ ( P @ Xa ) ) ) ) ) ).

% split_list_last_propE
thf(fact_1003_split__list__last__propE,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X6: nat] :
          ( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys4: list_nat,X: nat,Zs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs2 ) ) )
           => ( ( P @ X )
             => ~ ! [Xa: nat] :
                    ( ( member_nat @ Xa @ ( set_nat2 @ Zs2 ) )
                   => ~ ( P @ Xa ) ) ) ) ) ).

% split_list_last_propE
thf(fact_1004_split__list__last__propE,axiom,
    ! [Xs: list_int,P: int > $o] :
      ( ? [X6: int] :
          ( ( member_int @ X6 @ ( set_int2 @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys4: list_int,X: int,Zs2: list_int] :
            ( ( Xs
              = ( append_int @ Ys4 @ ( cons_int @ X @ Zs2 ) ) )
           => ( ( P @ X )
             => ~ ! [Xa: int] :
                    ( ( member_int @ Xa @ ( set_int2 @ Zs2 ) )
                   => ~ ( P @ Xa ) ) ) ) ) ).

% split_list_last_propE
thf(fact_1005_split__list__first__prop,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,P: list_P3196869577824546119od_a_o > $o] :
      ( ? [X6: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ X6 @ ( set_li2629784761229999580od_a_o @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys4: list_l8336264072933094989od_a_o,X: list_P3196869577824546119od_a_o] :
          ( ? [Zs2: list_l8336264072933094989od_a_o] :
              ( Xs
              = ( append2159120467191820514od_a_o @ Ys4 @ ( cons_l9179691953475787517od_a_o @ X @ Zs2 ) ) )
          & ( P @ X )
          & ! [Xa: list_P3196869577824546119od_a_o] :
              ( ( member5261019209154752368od_a_o @ Xa @ ( set_li2629784761229999580od_a_o @ Ys4 ) )
             => ~ ( P @ Xa ) ) ) ) ).

% split_list_first_prop
thf(fact_1006_split__list__first__prop,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P: produc1313295348606329723on_nat > $o] :
      ( ? [X6: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X6 @ ( set_Pr937530902923597456on_nat @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys4: list_P9127870862658512897on_nat,X: produc1313295348606329723on_nat] :
          ( ? [Zs2: list_P9127870862658512897on_nat] :
              ( Xs
              = ( append2865821678206098582on_nat @ Ys4 @ ( cons_P1582596238967110705on_nat @ X @ Zs2 ) ) )
          & ( P @ X )
          & ! [Xa: produc1313295348606329723on_nat] :
              ( ( member5906418179908956708on_nat @ Xa @ ( set_Pr937530902923597456on_nat @ Ys4 ) )
             => ~ ( P @ Xa ) ) ) ) ).

% split_list_first_prop
thf(fact_1007_split__list__first__prop,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P: sum_su4907927656612468301on_nat > $o] :
      ( ? [X6: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X6 @ ( set_Su4848624856859743608on_nat @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys4: list_S5253404628753938141on_nat,X: sum_su4907927656612468301on_nat] :
          ( ? [Zs2: list_S5253404628753938141on_nat] :
              ( Xs
              = ( append3523812819975775986on_nat @ Ys4 @ ( cons_S6797538523646374167on_nat @ X @ Zs2 ) ) )
          & ( P @ X )
          & ! [Xa: sum_su4907927656612468301on_nat] :
              ( ( member8476201367316484836on_nat @ Xa @ ( set_Su4848624856859743608on_nat @ Ys4 ) )
             => ~ ( P @ Xa ) ) ) ) ).

% split_list_first_prop
thf(fact_1008_split__list__first__prop,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X6: nat] :
          ( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys4: list_nat,X: nat] :
          ( ? [Zs2: list_nat] :
              ( Xs
              = ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs2 ) ) )
          & ( P @ X )
          & ! [Xa: nat] :
              ( ( member_nat @ Xa @ ( set_nat2 @ Ys4 ) )
             => ~ ( P @ Xa ) ) ) ) ).

% split_list_first_prop
thf(fact_1009_split__list__first__prop,axiom,
    ! [Xs: list_int,P: int > $o] :
      ( ? [X6: int] :
          ( ( member_int @ X6 @ ( set_int2 @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys4: list_int,X: int] :
          ( ? [Zs2: list_int] :
              ( Xs
              = ( append_int @ Ys4 @ ( cons_int @ X @ Zs2 ) ) )
          & ( P @ X )
          & ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_int2 @ Ys4 ) )
             => ~ ( P @ Xa ) ) ) ) ).

% split_list_first_prop
thf(fact_1010_split__list__last__prop,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,P: list_P3196869577824546119od_a_o > $o] :
      ( ? [X6: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ X6 @ ( set_li2629784761229999580od_a_o @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys4: list_l8336264072933094989od_a_o,X: list_P3196869577824546119od_a_o,Zs2: list_l8336264072933094989od_a_o] :
          ( ( Xs
            = ( append2159120467191820514od_a_o @ Ys4 @ ( cons_l9179691953475787517od_a_o @ X @ Zs2 ) ) )
          & ( P @ X )
          & ! [Xa: list_P3196869577824546119od_a_o] :
              ( ( member5261019209154752368od_a_o @ Xa @ ( set_li2629784761229999580od_a_o @ Zs2 ) )
             => ~ ( P @ Xa ) ) ) ) ).

% split_list_last_prop
thf(fact_1011_split__list__last__prop,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P: produc1313295348606329723on_nat > $o] :
      ( ? [X6: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X6 @ ( set_Pr937530902923597456on_nat @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys4: list_P9127870862658512897on_nat,X: produc1313295348606329723on_nat,Zs2: list_P9127870862658512897on_nat] :
          ( ( Xs
            = ( append2865821678206098582on_nat @ Ys4 @ ( cons_P1582596238967110705on_nat @ X @ Zs2 ) ) )
          & ( P @ X )
          & ! [Xa: produc1313295348606329723on_nat] :
              ( ( member5906418179908956708on_nat @ Xa @ ( set_Pr937530902923597456on_nat @ Zs2 ) )
             => ~ ( P @ Xa ) ) ) ) ).

% split_list_last_prop
thf(fact_1012_split__list__last__prop,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P: sum_su4907927656612468301on_nat > $o] :
      ( ? [X6: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X6 @ ( set_Su4848624856859743608on_nat @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys4: list_S5253404628753938141on_nat,X: sum_su4907927656612468301on_nat,Zs2: list_S5253404628753938141on_nat] :
          ( ( Xs
            = ( append3523812819975775986on_nat @ Ys4 @ ( cons_S6797538523646374167on_nat @ X @ Zs2 ) ) )
          & ( P @ X )
          & ! [Xa: sum_su4907927656612468301on_nat] :
              ( ( member8476201367316484836on_nat @ Xa @ ( set_Su4848624856859743608on_nat @ Zs2 ) )
             => ~ ( P @ Xa ) ) ) ) ).

% split_list_last_prop
thf(fact_1013_split__list__last__prop,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X6: nat] :
          ( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys4: list_nat,X: nat,Zs2: list_nat] :
          ( ( Xs
            = ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs2 ) ) )
          & ( P @ X )
          & ! [Xa: nat] :
              ( ( member_nat @ Xa @ ( set_nat2 @ Zs2 ) )
             => ~ ( P @ Xa ) ) ) ) ).

% split_list_last_prop
thf(fact_1014_split__list__last__prop,axiom,
    ! [Xs: list_int,P: int > $o] :
      ( ? [X6: int] :
          ( ( member_int @ X6 @ ( set_int2 @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys4: list_int,X: int,Zs2: list_int] :
          ( ( Xs
            = ( append_int @ Ys4 @ ( cons_int @ X @ Zs2 ) ) )
          & ( P @ X )
          & ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_int2 @ Zs2 ) )
             => ~ ( P @ Xa ) ) ) ) ).

% split_list_last_prop
thf(fact_1015_in__set__conv__decomp,axiom,
    ! [X2: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Xs ) )
      = ( ? [Ys2: list_l8336264072933094989od_a_o,Zs3: list_l8336264072933094989od_a_o] :
            ( Xs
            = ( append2159120467191820514od_a_o @ Ys2 @ ( cons_l9179691953475787517od_a_o @ X2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_1016_in__set__conv__decomp,axiom,
    ! [X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Xs ) )
      = ( ? [Ys2: list_P9127870862658512897on_nat,Zs3: list_P9127870862658512897on_nat] :
            ( Xs
            = ( append2865821678206098582on_nat @ Ys2 @ ( cons_P1582596238967110705on_nat @ X2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_1017_in__set__conv__decomp,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Xs ) )
      = ( ? [Ys2: list_S5253404628753938141on_nat,Zs3: list_S5253404628753938141on_nat] :
            ( Xs
            = ( append3523812819975775986on_nat @ Ys2 @ ( cons_S6797538523646374167on_nat @ X2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_1018_in__set__conv__decomp,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
      = ( ? [Ys2: list_nat,Zs3: list_nat] :
            ( Xs
            = ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_1019_in__set__conv__decomp,axiom,
    ! [X2: int,Xs: list_int] :
      ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
      = ( ? [Ys2: list_int,Zs3: list_int] :
            ( Xs
            = ( append_int @ Ys2 @ ( cons_int @ X2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_1020_append__Cons__eq__iff,axiom,
    ! [X2: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,Xs5: list_l8336264072933094989od_a_o,Ys7: list_l8336264072933094989od_a_o] :
      ( ~ ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Xs ) )
     => ( ~ ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Ys ) )
       => ( ( ( append2159120467191820514od_a_o @ Xs @ ( cons_l9179691953475787517od_a_o @ X2 @ Ys ) )
            = ( append2159120467191820514od_a_o @ Xs5 @ ( cons_l9179691953475787517od_a_o @ X2 @ Ys7 ) ) )
          = ( ( Xs = Xs5 )
            & ( Ys = Ys7 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_1021_append__Cons__eq__iff,axiom,
    ! [X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Xs5: list_P9127870862658512897on_nat,Ys7: list_P9127870862658512897on_nat] :
      ( ~ ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Xs ) )
     => ( ~ ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Ys ) )
       => ( ( ( append2865821678206098582on_nat @ Xs @ ( cons_P1582596238967110705on_nat @ X2 @ Ys ) )
            = ( append2865821678206098582on_nat @ Xs5 @ ( cons_P1582596238967110705on_nat @ X2 @ Ys7 ) ) )
          = ( ( Xs = Xs5 )
            & ( Ys = Ys7 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_1022_append__Cons__eq__iff,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Xs5: list_S5253404628753938141on_nat,Ys7: list_S5253404628753938141on_nat] :
      ( ~ ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Xs ) )
     => ( ~ ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Ys ) )
       => ( ( ( append3523812819975775986on_nat @ Xs @ ( cons_S6797538523646374167on_nat @ X2 @ Ys ) )
            = ( append3523812819975775986on_nat @ Xs5 @ ( cons_S6797538523646374167on_nat @ X2 @ Ys7 ) ) )
          = ( ( Xs = Xs5 )
            & ( Ys = Ys7 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_1023_append__Cons__eq__iff,axiom,
    ! [X2: nat,Xs: list_nat,Ys: list_nat,Xs5: list_nat,Ys7: list_nat] :
      ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ( ~ ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
       => ( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) )
            = ( append_nat @ Xs5 @ ( cons_nat @ X2 @ Ys7 ) ) )
          = ( ( Xs = Xs5 )
            & ( Ys = Ys7 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_1024_append__Cons__eq__iff,axiom,
    ! [X2: int,Xs: list_int,Ys: list_int,Xs5: list_int,Ys7: list_int] :
      ( ~ ( member_int @ X2 @ ( set_int2 @ Xs ) )
     => ( ~ ( member_int @ X2 @ ( set_int2 @ Ys ) )
       => ( ( ( append_int @ Xs @ ( cons_int @ X2 @ Ys ) )
            = ( append_int @ Xs5 @ ( cons_int @ X2 @ Ys7 ) ) )
          = ( ( Xs = Xs5 )
            & ( Ys = Ys7 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_1025_split__list__propE,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,P: list_P3196869577824546119od_a_o > $o] :
      ( ? [X6: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ X6 @ ( set_li2629784761229999580od_a_o @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys4: list_l8336264072933094989od_a_o,X: list_P3196869577824546119od_a_o] :
            ( ? [Zs2: list_l8336264072933094989od_a_o] :
                ( Xs
                = ( append2159120467191820514od_a_o @ Ys4 @ ( cons_l9179691953475787517od_a_o @ X @ Zs2 ) ) )
           => ~ ( P @ X ) ) ) ).

% split_list_propE
thf(fact_1026_split__list__propE,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P: produc1313295348606329723on_nat > $o] :
      ( ? [X6: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X6 @ ( set_Pr937530902923597456on_nat @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys4: list_P9127870862658512897on_nat,X: produc1313295348606329723on_nat] :
            ( ? [Zs2: list_P9127870862658512897on_nat] :
                ( Xs
                = ( append2865821678206098582on_nat @ Ys4 @ ( cons_P1582596238967110705on_nat @ X @ Zs2 ) ) )
           => ~ ( P @ X ) ) ) ).

% split_list_propE
thf(fact_1027_split__list__propE,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P: sum_su4907927656612468301on_nat > $o] :
      ( ? [X6: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X6 @ ( set_Su4848624856859743608on_nat @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys4: list_S5253404628753938141on_nat,X: sum_su4907927656612468301on_nat] :
            ( ? [Zs2: list_S5253404628753938141on_nat] :
                ( Xs
                = ( append3523812819975775986on_nat @ Ys4 @ ( cons_S6797538523646374167on_nat @ X @ Zs2 ) ) )
           => ~ ( P @ X ) ) ) ).

% split_list_propE
thf(fact_1028_split__list__propE,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X6: nat] :
          ( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys4: list_nat,X: nat] :
            ( ? [Zs2: list_nat] :
                ( Xs
                = ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs2 ) ) )
           => ~ ( P @ X ) ) ) ).

% split_list_propE
thf(fact_1029_split__list__propE,axiom,
    ! [Xs: list_int,P: int > $o] :
      ( ? [X6: int] :
          ( ( member_int @ X6 @ ( set_int2 @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys4: list_int,X: int] :
            ( ? [Zs2: list_int] :
                ( Xs
                = ( append_int @ Ys4 @ ( cons_int @ X @ Zs2 ) ) )
           => ~ ( P @ X ) ) ) ).

% split_list_propE
thf(fact_1030_split__list__first,axiom,
    ! [X2: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Xs ) )
     => ? [Ys4: list_l8336264072933094989od_a_o,Zs2: list_l8336264072933094989od_a_o] :
          ( ( Xs
            = ( append2159120467191820514od_a_o @ Ys4 @ ( cons_l9179691953475787517od_a_o @ X2 @ Zs2 ) ) )
          & ~ ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Ys4 ) ) ) ) ).

% split_list_first
thf(fact_1031_split__list__first,axiom,
    ! [X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Xs ) )
     => ? [Ys4: list_P9127870862658512897on_nat,Zs2: list_P9127870862658512897on_nat] :
          ( ( Xs
            = ( append2865821678206098582on_nat @ Ys4 @ ( cons_P1582596238967110705on_nat @ X2 @ Zs2 ) ) )
          & ~ ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Ys4 ) ) ) ) ).

% split_list_first
thf(fact_1032_split__list__first,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Xs ) )
     => ? [Ys4: list_S5253404628753938141on_nat,Zs2: list_S5253404628753938141on_nat] :
          ( ( Xs
            = ( append3523812819975775986on_nat @ Ys4 @ ( cons_S6797538523646374167on_nat @ X2 @ Zs2 ) ) )
          & ~ ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Ys4 ) ) ) ) ).

% split_list_first
thf(fact_1033_split__list__first,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ? [Ys4: list_nat,Zs2: list_nat] :
          ( ( Xs
            = ( append_nat @ Ys4 @ ( cons_nat @ X2 @ Zs2 ) ) )
          & ~ ( member_nat @ X2 @ ( set_nat2 @ Ys4 ) ) ) ) ).

% split_list_first
thf(fact_1034_split__list__first,axiom,
    ! [X2: int,Xs: list_int] :
      ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
     => ? [Ys4: list_int,Zs2: list_int] :
          ( ( Xs
            = ( append_int @ Ys4 @ ( cons_int @ X2 @ Zs2 ) ) )
          & ~ ( member_int @ X2 @ ( set_int2 @ Ys4 ) ) ) ) ).

% split_list_first
thf(fact_1035_split__list__prop,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,P: list_P3196869577824546119od_a_o > $o] :
      ( ? [X6: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ X6 @ ( set_li2629784761229999580od_a_o @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys4: list_l8336264072933094989od_a_o,X: list_P3196869577824546119od_a_o] :
          ( ? [Zs2: list_l8336264072933094989od_a_o] :
              ( Xs
              = ( append2159120467191820514od_a_o @ Ys4 @ ( cons_l9179691953475787517od_a_o @ X @ Zs2 ) ) )
          & ( P @ X ) ) ) ).

% split_list_prop
thf(fact_1036_split__list__prop,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P: produc1313295348606329723on_nat > $o] :
      ( ? [X6: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X6 @ ( set_Pr937530902923597456on_nat @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys4: list_P9127870862658512897on_nat,X: produc1313295348606329723on_nat] :
          ( ? [Zs2: list_P9127870862658512897on_nat] :
              ( Xs
              = ( append2865821678206098582on_nat @ Ys4 @ ( cons_P1582596238967110705on_nat @ X @ Zs2 ) ) )
          & ( P @ X ) ) ) ).

% split_list_prop
thf(fact_1037_split__list__prop,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P: sum_su4907927656612468301on_nat > $o] :
      ( ? [X6: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X6 @ ( set_Su4848624856859743608on_nat @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys4: list_S5253404628753938141on_nat,X: sum_su4907927656612468301on_nat] :
          ( ? [Zs2: list_S5253404628753938141on_nat] :
              ( Xs
              = ( append3523812819975775986on_nat @ Ys4 @ ( cons_S6797538523646374167on_nat @ X @ Zs2 ) ) )
          & ( P @ X ) ) ) ).

% split_list_prop
thf(fact_1038_split__list__prop,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X6: nat] :
          ( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys4: list_nat,X: nat] :
          ( ? [Zs2: list_nat] :
              ( Xs
              = ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs2 ) ) )
          & ( P @ X ) ) ) ).

% split_list_prop
thf(fact_1039_split__list__prop,axiom,
    ! [Xs: list_int,P: int > $o] :
      ( ? [X6: int] :
          ( ( member_int @ X6 @ ( set_int2 @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys4: list_int,X: int] :
          ( ? [Zs2: list_int] :
              ( Xs
              = ( append_int @ Ys4 @ ( cons_int @ X @ Zs2 ) ) )
          & ( P @ X ) ) ) ).

% split_list_prop
thf(fact_1040_split__list__last,axiom,
    ! [X2: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Xs ) )
     => ? [Ys4: list_l8336264072933094989od_a_o,Zs2: list_l8336264072933094989od_a_o] :
          ( ( Xs
            = ( append2159120467191820514od_a_o @ Ys4 @ ( cons_l9179691953475787517od_a_o @ X2 @ Zs2 ) ) )
          & ~ ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_1041_split__list__last,axiom,
    ! [X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Xs ) )
     => ? [Ys4: list_P9127870862658512897on_nat,Zs2: list_P9127870862658512897on_nat] :
          ( ( Xs
            = ( append2865821678206098582on_nat @ Ys4 @ ( cons_P1582596238967110705on_nat @ X2 @ Zs2 ) ) )
          & ~ ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_1042_split__list__last,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Xs ) )
     => ? [Ys4: list_S5253404628753938141on_nat,Zs2: list_S5253404628753938141on_nat] :
          ( ( Xs
            = ( append3523812819975775986on_nat @ Ys4 @ ( cons_S6797538523646374167on_nat @ X2 @ Zs2 ) ) )
          & ~ ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_1043_split__list__last,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ? [Ys4: list_nat,Zs2: list_nat] :
          ( ( Xs
            = ( append_nat @ Ys4 @ ( cons_nat @ X2 @ Zs2 ) ) )
          & ~ ( member_nat @ X2 @ ( set_nat2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_1044_split__list__last,axiom,
    ! [X2: int,Xs: list_int] :
      ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
     => ? [Ys4: list_int,Zs2: list_int] :
          ( ( Xs
            = ( append_int @ Ys4 @ ( cons_int @ X2 @ Zs2 ) ) )
          & ~ ( member_int @ X2 @ ( set_int2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_1045_split__list,axiom,
    ! [X2: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Xs ) )
     => ? [Ys4: list_l8336264072933094989od_a_o,Zs2: list_l8336264072933094989od_a_o] :
          ( Xs
          = ( append2159120467191820514od_a_o @ Ys4 @ ( cons_l9179691953475787517od_a_o @ X2 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_1046_split__list,axiom,
    ! [X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Xs ) )
     => ? [Ys4: list_P9127870862658512897on_nat,Zs2: list_P9127870862658512897on_nat] :
          ( Xs
          = ( append2865821678206098582on_nat @ Ys4 @ ( cons_P1582596238967110705on_nat @ X2 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_1047_split__list,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Xs ) )
     => ? [Ys4: list_S5253404628753938141on_nat,Zs2: list_S5253404628753938141on_nat] :
          ( Xs
          = ( append3523812819975775986on_nat @ Ys4 @ ( cons_S6797538523646374167on_nat @ X2 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_1048_split__list,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ? [Ys4: list_nat,Zs2: list_nat] :
          ( Xs
          = ( append_nat @ Ys4 @ ( cons_nat @ X2 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_1049_split__list,axiom,
    ! [X2: int,Xs: list_int] :
      ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
     => ? [Ys4: list_int,Zs2: list_int] :
          ( Xs
          = ( append_int @ Ys4 @ ( cons_int @ X2 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_1050_append__listrel1I,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,Us: list_nat,Vs: list_nat] :
      ( ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
          & ( Us = Vs ) )
        | ( ( Xs = Ys )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us @ Vs ) @ ( listrel1_nat @ R ) ) ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Us ) @ ( append_nat @ Ys @ Vs ) ) @ ( listrel1_nat @ R ) ) ) ).

% append_listrel1I
thf(fact_1051_lex__append__leftI,axiom,
    ! [Ys: list_nat,Zs: list_nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R ) ) ) ).

% lex_append_leftI
thf(fact_1052_set__take__subset__set__take,axiom,
    ! [M2: nat,N: nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ M2 @ Xs ) ) @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_1053_set__take__subset__set__take,axiom,
    ! [M2: nat,N: nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_le1271121679792220359od_a_o @ ( set_li2629784761229999580od_a_o @ ( take_l5436743365630562101od_a_o @ M2 @ Xs ) ) @ ( set_li2629784761229999580od_a_o @ ( take_l5436743365630562101od_a_o @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_1054_set__take__subset__set__take,axiom,
    ! [M2: nat,N: nat,Xs: list_P9127870862658512897on_nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_le6961793561435735931on_nat @ ( set_Pr937530902923597456on_nat @ ( take_P4655611586102217833on_nat @ M2 @ Xs ) ) @ ( set_Pr937530902923597456on_nat @ ( take_P4655611586102217833on_nat @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_1055_set__take__subset__set__take,axiom,
    ! [M2: nat,N: nat,Xs: list_S5253404628753938141on_nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_le6595998496543023075on_nat @ ( set_Su4848624856859743608on_nat @ ( take_S1147720861222069983on_nat @ M2 @ Xs ) ) @ ( set_Su4848624856859743608on_nat @ ( take_S1147720861222069983on_nat @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_1056_same__length__different,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs != Ys )
     => ( ( ( size_size_list_int @ Xs )
          = ( size_size_list_int @ Ys ) )
       => ? [Pre: list_int,X: int,Xs4: list_int,Y4: int,Ys5: list_int] :
            ( ( X != Y4 )
            & ( Xs
              = ( append_int @ Pre @ ( append_int @ ( cons_int @ X @ nil_int ) @ Xs4 ) ) )
            & ( Ys
              = ( append_int @ Pre @ ( append_int @ ( cons_int @ Y4 @ nil_int ) @ Ys5 ) ) ) ) ) ) ).

% same_length_different
thf(fact_1057_same__length__different,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( Xs != Ys )
     => ( ( ( size_s3379185846251495609od_a_o @ Xs )
          = ( size_s3379185846251495609od_a_o @ Ys ) )
       => ? [Pre: list_l8336264072933094989od_a_o,X: list_P3196869577824546119od_a_o,Xs4: list_l8336264072933094989od_a_o,Y4: list_P3196869577824546119od_a_o,Ys5: list_l8336264072933094989od_a_o] :
            ( ( X != Y4 )
            & ( Xs
              = ( append2159120467191820514od_a_o @ Pre @ ( append2159120467191820514od_a_o @ ( cons_l9179691953475787517od_a_o @ X @ nil_li1372968520647675725od_a_o ) @ Xs4 ) ) )
            & ( Ys
              = ( append2159120467191820514od_a_o @ Pre @ ( append2159120467191820514od_a_o @ ( cons_l9179691953475787517od_a_o @ Y4 @ nil_li1372968520647675725od_a_o ) @ Ys5 ) ) ) ) ) ) ).

% same_length_different
thf(fact_1058_same__length__different,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( Xs != Ys )
     => ( ( ( size_s5686234912572693873on_nat @ Xs )
          = ( size_s5686234912572693873on_nat @ Ys ) )
       => ? [Pre: list_S5253404628753938141on_nat,X: sum_su4907927656612468301on_nat,Xs4: list_S5253404628753938141on_nat,Y4: sum_su4907927656612468301on_nat,Ys5: list_S5253404628753938141on_nat] :
            ( ( X != Y4 )
            & ( Xs
              = ( append3523812819975775986on_nat @ Pre @ ( append3523812819975775986on_nat @ ( cons_S6797538523646374167on_nat @ X @ nil_Su3078906344068544711on_nat ) @ Xs4 ) ) )
            & ( Ys
              = ( append3523812819975775986on_nat @ Pre @ ( append3523812819975775986on_nat @ ( cons_S6797538523646374167on_nat @ Y4 @ nil_Su3078906344068544711on_nat ) @ Ys5 ) ) ) ) ) ) ).

% same_length_different
thf(fact_1059_same__length__different,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs != Ys )
     => ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
       => ? [Pre: list_nat,X: nat,Xs4: list_nat,Y4: nat,Ys5: list_nat] :
            ( ( X != Y4 )
            & ( Xs
              = ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X @ nil_nat ) @ Xs4 ) ) )
            & ( Ys
              = ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y4 @ nil_nat ) @ Ys5 ) ) ) ) ) ) ).

% same_length_different
thf(fact_1060_same__length__different,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( Xs != Ys )
     => ( ( ( size_s7490768528479217773on_nat @ Xs )
          = ( size_s7490768528479217773on_nat @ Ys ) )
       => ? [Pre: list_P9127870862658512897on_nat,X: produc1313295348606329723on_nat,Xs4: list_P9127870862658512897on_nat,Y4: produc1313295348606329723on_nat,Ys5: list_P9127870862658512897on_nat] :
            ( ( X != Y4 )
            & ( Xs
              = ( append2865821678206098582on_nat @ Pre @ ( append2865821678206098582on_nat @ ( cons_P1582596238967110705on_nat @ X @ nil_Pr8614093721167239297on_nat ) @ Xs4 ) ) )
            & ( Ys
              = ( append2865821678206098582on_nat @ Pre @ ( append2865821678206098582on_nat @ ( cons_P1582596238967110705on_nat @ Y4 @ nil_Pr8614093721167239297on_nat ) @ Ys5 ) ) ) ) ) ) ).

% same_length_different
thf(fact_1061_not__distinct__decomp,axiom,
    ! [Ws: list_nat] :
      ( ~ ( distinct_nat @ Ws )
     => ? [Xs3: list_nat,Ys4: list_nat,Zs2: list_nat,Y4: nat] :
          ( Ws
          = ( append_nat @ Xs3 @ ( append_nat @ ( cons_nat @ Y4 @ nil_nat ) @ ( append_nat @ Ys4 @ ( append_nat @ ( cons_nat @ Y4 @ nil_nat ) @ Zs2 ) ) ) ) ) ) ).

% not_distinct_decomp
thf(fact_1062_not__distinct__decomp,axiom,
    ! [Ws: list_int] :
      ( ~ ( distinct_int @ Ws )
     => ? [Xs3: list_int,Ys4: list_int,Zs2: list_int,Y4: int] :
          ( Ws
          = ( append_int @ Xs3 @ ( append_int @ ( cons_int @ Y4 @ nil_int ) @ ( append_int @ Ys4 @ ( append_int @ ( cons_int @ Y4 @ nil_int ) @ Zs2 ) ) ) ) ) ) ).

% not_distinct_decomp
thf(fact_1063_not__distinct__conv__prefix,axiom,
    ! [As: list_l8336264072933094989od_a_o] :
      ( ( ~ ( distin690952206374581440od_a_o @ As ) )
      = ( ? [Xs2: list_l8336264072933094989od_a_o,Y3: list_P3196869577824546119od_a_o,Ys2: list_l8336264072933094989od_a_o] :
            ( ( member5261019209154752368od_a_o @ Y3 @ ( set_li2629784761229999580od_a_o @ Xs2 ) )
            & ( distin690952206374581440od_a_o @ Xs2 )
            & ( As
              = ( append2159120467191820514od_a_o @ Xs2 @ ( cons_l9179691953475787517od_a_o @ Y3 @ Ys2 ) ) ) ) ) ) ).

% not_distinct_conv_prefix
thf(fact_1064_not__distinct__conv__prefix,axiom,
    ! [As: list_P9127870862658512897on_nat] :
      ( ( ~ ( distin8410642530753306484on_nat @ As ) )
      = ( ? [Xs2: list_P9127870862658512897on_nat,Y3: produc1313295348606329723on_nat,Ys2: list_P9127870862658512897on_nat] :
            ( ( member5906418179908956708on_nat @ Y3 @ ( set_Pr937530902923597456on_nat @ Xs2 ) )
            & ( distin8410642530753306484on_nat @ Xs2 )
            & ( As
              = ( append2865821678206098582on_nat @ Xs2 @ ( cons_P1582596238967110705on_nat @ Y3 @ Ys2 ) ) ) ) ) ) ).

% not_distinct_conv_prefix
thf(fact_1065_not__distinct__conv__prefix,axiom,
    ! [As: list_S5253404628753938141on_nat] :
      ( ( ~ ( distin2773651816175405460on_nat @ As ) )
      = ( ? [Xs2: list_S5253404628753938141on_nat,Y3: sum_su4907927656612468301on_nat,Ys2: list_S5253404628753938141on_nat] :
            ( ( member8476201367316484836on_nat @ Y3 @ ( set_Su4848624856859743608on_nat @ Xs2 ) )
            & ( distin2773651816175405460on_nat @ Xs2 )
            & ( As
              = ( append3523812819975775986on_nat @ Xs2 @ ( cons_S6797538523646374167on_nat @ Y3 @ Ys2 ) ) ) ) ) ) ).

% not_distinct_conv_prefix
thf(fact_1066_not__distinct__conv__prefix,axiom,
    ! [As: list_nat] :
      ( ( ~ ( distinct_nat @ As ) )
      = ( ? [Xs2: list_nat,Y3: nat,Ys2: list_nat] :
            ( ( member_nat @ Y3 @ ( set_nat2 @ Xs2 ) )
            & ( distinct_nat @ Xs2 )
            & ( As
              = ( append_nat @ Xs2 @ ( cons_nat @ Y3 @ Ys2 ) ) ) ) ) ) ).

% not_distinct_conv_prefix
thf(fact_1067_not__distinct__conv__prefix,axiom,
    ! [As: list_int] :
      ( ( ~ ( distinct_int @ As ) )
      = ( ? [Xs2: list_int,Y3: int,Ys2: list_int] :
            ( ( member_int @ Y3 @ ( set_int2 @ Xs2 ) )
            & ( distinct_int @ Xs2 )
            & ( As
              = ( append_int @ Xs2 @ ( cons_int @ Y3 @ Ys2 ) ) ) ) ) ) ).

% not_distinct_conv_prefix
thf(fact_1068_list__update__append1,axiom,
    ! [I3: nat,Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,X2: list_P3196869577824546119od_a_o] :
      ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( list_u329291161401623989od_a_o @ ( append2159120467191820514od_a_o @ Xs @ Ys ) @ I3 @ X2 )
        = ( append2159120467191820514od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ I3 @ X2 ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_1069_list__update__append1,axiom,
    ! [I3: nat,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,X2: sum_su4907927656612468301on_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( list_u8577817230714425439on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) @ I3 @ X2 )
        = ( append3523812819975775986on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I3 @ X2 ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_1070_list__update__append1,axiom,
    ! [I3: nat,Xs: list_nat,Ys: list_nat,X2: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
     => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ I3 @ X2 )
        = ( append_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_1071_list__update__append1,axiom,
    ! [I3: nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,X2: produc1313295348606329723on_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s7490768528479217773on_nat @ Xs ) )
     => ( ( list_u1840724836389618409on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) @ I3 @ X2 )
        = ( append2865821678206098582on_nat @ ( list_u1840724836389618409on_nat @ Xs @ I3 @ X2 ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_1072_lex__append__leftD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ! [X: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_1073_lex__append__leftD,axiom,
    ! [R: set_Pr2535564044409765851on_nat,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat] :
      ( ! [X: sum_su4907927656612468301on_nat] :
          ~ ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X @ X ) @ R )
     => ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) @ ( append3523812819975775986on_nat @ Xs @ Zs ) ) @ ( lex_Su3183444693157923171on_nat @ R ) )
       => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Ys @ Zs ) @ ( lex_Su3183444693157923171on_nat @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_1074_lex__append__left__iff,axiom,
    ! [R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ! [X: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R ) )
        = ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_1075_lex__append__left__iff,axiom,
    ! [R: set_Pr2535564044409765851on_nat,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat] :
      ( ! [X: sum_su4907927656612468301on_nat] :
          ~ ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X @ X ) @ R )
     => ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) @ ( append3523812819975775986on_nat @ Xs @ Zs ) ) @ ( lex_Su3183444693157923171on_nat @ R ) )
        = ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Ys @ Zs ) @ ( lex_Su3183444693157923171on_nat @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_1076_lex__append__rightI,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o,Vs: list_l8336264072933094989od_a_o,Us: list_l8336264072933094989od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Xs @ Ys ) @ ( lex_li6160749805527239089od_a_o @ R ) )
     => ( ( ( size_s3379185846251495609od_a_o @ Vs )
          = ( size_s3379185846251495609od_a_o @ Us ) )
       => ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ ( append2159120467191820514od_a_o @ Xs @ Us ) @ ( append2159120467191820514od_a_o @ Ys @ Vs ) ) @ ( lex_li6160749805527239089od_a_o @ R ) ) ) ) ).

% lex_append_rightI
thf(fact_1077_lex__append__rightI,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat,Vs: list_S5253404628753938141on_nat,Us: list_S5253404628753938141on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ Ys ) @ ( lex_Su3183444693157923171on_nat @ R ) )
     => ( ( ( size_s5686234912572693873on_nat @ Vs )
          = ( size_s5686234912572693873on_nat @ Us ) )
       => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( append3523812819975775986on_nat @ Xs @ Us ) @ ( append3523812819975775986on_nat @ Ys @ Vs ) ) @ ( lex_Su3183444693157923171on_nat @ R ) ) ) ) ).

% lex_append_rightI
thf(fact_1078_lex__append__rightI,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,Vs: list_nat,Us: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R ) )
     => ( ( ( size_size_list_nat @ Vs )
          = ( size_size_list_nat @ Us ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Us ) @ ( append_nat @ Ys @ Vs ) ) @ ( lex_nat @ R ) ) ) ) ).

% lex_append_rightI
thf(fact_1079_lex__append__rightI,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat,Vs: list_P9127870862658512897on_nat,Us: list_P9127870862658512897on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( lex_Pr6276626268559582437on_nat @ R ) )
     => ( ( ( size_s7490768528479217773on_nat @ Vs )
          = ( size_s7490768528479217773on_nat @ Us ) )
       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( append2865821678206098582on_nat @ Xs @ Us ) @ ( append2865821678206098582on_nat @ Ys @ Vs ) ) @ ( lex_Pr6276626268559582437on_nat @ R ) ) ) ) ).

% lex_append_rightI
thf(fact_1080_lenlex__append1,axiom,
    ! [Us: list_l8336264072933094989od_a_o,Xs: list_l8336264072933094989od_a_o,R2: set_Pr7420778282784443591od_a_o,Vs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Us @ Xs ) @ ( lenlex2563728411540087634od_a_o @ R2 ) )
     => ( ( ( size_s3379185846251495609od_a_o @ Vs )
          = ( size_s3379185846251495609od_a_o @ Ys ) )
       => ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ ( append2159120467191820514od_a_o @ Us @ Vs ) @ ( append2159120467191820514od_a_o @ Xs @ Ys ) ) @ ( lenlex2563728411540087634od_a_o @ R2 ) ) ) ) ).

% lenlex_append1
thf(fact_1081_lenlex__append1,axiom,
    ! [Us: list_S5253404628753938141on_nat,Xs: list_S5253404628753938141on_nat,R2: set_Pr2535564044409765851on_nat,Vs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Us @ Xs ) @ ( lenlex8545234769834248322on_nat @ R2 ) )
     => ( ( ( size_s5686234912572693873on_nat @ Vs )
          = ( size_s5686234912572693873on_nat @ Ys ) )
       => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( append3523812819975775986on_nat @ Us @ Vs ) @ ( append3523812819975775986on_nat @ Xs @ Ys ) ) @ ( lenlex8545234769834248322on_nat @ R2 ) ) ) ) ).

% lenlex_append1
thf(fact_1082_lenlex__append1,axiom,
    ! [Us: list_nat,Xs: list_nat,R2: set_Pr1261947904930325089at_nat,Vs: list_nat,Ys: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us @ Xs ) @ ( lenlex_nat @ R2 ) )
     => ( ( ( size_size_list_nat @ Vs )
          = ( size_size_list_nat @ Ys ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Us @ Vs ) @ ( append_nat @ Xs @ Ys ) ) @ ( lenlex_nat @ R2 ) ) ) ) ).

% lenlex_append1
thf(fact_1083_lenlex__append1,axiom,
    ! [Us: list_P9127870862658512897on_nat,Xs: list_P9127870862658512897on_nat,R2: set_Pr8202701763255893447on_nat,Vs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Us @ Xs ) @ ( lenlex6631009945862686470on_nat @ R2 ) )
     => ( ( ( size_s7490768528479217773on_nat @ Vs )
          = ( size_s7490768528479217773on_nat @ Ys ) )
       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( append2865821678206098582on_nat @ Us @ Vs ) @ ( append2865821678206098582on_nat @ Xs @ Ys ) ) @ ( lenlex6631009945862686470on_nat @ R2 ) ) ) ) ).

% lenlex_append1
thf(fact_1084_subset__code_I2_J,axiom,
    ! [A2: set_nat,Ys: list_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( coset_nat @ Ys ) )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
           => ~ ( member_nat @ X3 @ A2 ) ) ) ) ).

% subset_code(2)
thf(fact_1085_subset__code_I2_J,axiom,
    ! [A2: set_li5011860890973139239od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( ord_le1271121679792220359od_a_o @ A2 @ ( coset_2783394909775257274od_a_o @ Ys ) )
      = ( ! [X3: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X3 @ ( set_li2629784761229999580od_a_o @ Ys ) )
           => ~ ( member5261019209154752368od_a_o @ X3 @ A2 ) ) ) ) ).

% subset_code(2)
thf(fact_1086_subset__code_I2_J,axiom,
    ! [A2: set_Pr2535564044409765851on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( ord_le6961793561435735931on_nat @ A2 @ ( coset_1816015773736449646on_nat @ Ys ) )
      = ( ! [X3: produc1313295348606329723on_nat] :
            ( ( member5906418179908956708on_nat @ X3 @ ( set_Pr937530902923597456on_nat @ Ys ) )
           => ~ ( member5906418179908956708on_nat @ X3 @ A2 ) ) ) ) ).

% subset_code(2)
thf(fact_1087_subset__code_I2_J,axiom,
    ! [A2: set_Su266273576555847555on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ord_le6595998496543023075on_nat @ A2 @ ( coset_3487183839294692122on_nat @ Ys ) )
      = ( ! [X3: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X3 @ ( set_Su4848624856859743608on_nat @ Ys ) )
           => ~ ( member8476201367316484836on_nat @ X3 @ A2 ) ) ) ) ).

% subset_code(2)
thf(fact_1088_coset__def,axiom,
    ( coset_nat
    = ( ^ [Xs2: list_nat] : ( uminus5710092332889474511et_nat @ ( set_nat2 @ Xs2 ) ) ) ) ).

% coset_def
thf(fact_1089_coset__def,axiom,
    ( coset_2783394909775257274od_a_o
    = ( ^ [Xs2: list_l8336264072933094989od_a_o] : ( uminus8391578155927437342od_a_o @ ( set_li2629784761229999580od_a_o @ Xs2 ) ) ) ) ).

% coset_def
thf(fact_1090_coset__def,axiom,
    ( coset_1816015773736449646on_nat
    = ( ^ [Xs2: list_P9127870862658512897on_nat] : ( uminus4127886064781087314on_nat @ ( set_Pr937530902923597456on_nat @ Xs2 ) ) ) ) ).

% coset_def
thf(fact_1091_coset__def,axiom,
    ( coset_3487183839294692122on_nat
    = ( ^ [Xs2: list_S5253404628753938141on_nat] : ( uminus5421032341409327564on_nat @ ( set_Su4848624856859743608on_nat @ Xs2 ) ) ) ) ).

% coset_def
thf(fact_1092_compl__coset,axiom,
    ! [Xs: list_nat] :
      ( ( uminus5710092332889474511et_nat @ ( coset_nat @ Xs ) )
      = ( set_nat2 @ Xs ) ) ).

% compl_coset
thf(fact_1093_compl__coset,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( uminus8391578155927437342od_a_o @ ( coset_2783394909775257274od_a_o @ Xs ) )
      = ( set_li2629784761229999580od_a_o @ Xs ) ) ).

% compl_coset
thf(fact_1094_compl__coset,axiom,
    ! [Xs: list_P9127870862658512897on_nat] :
      ( ( uminus4127886064781087314on_nat @ ( coset_1816015773736449646on_nat @ Xs ) )
      = ( set_Pr937530902923597456on_nat @ Xs ) ) ).

% compl_coset
thf(fact_1095_compl__coset,axiom,
    ! [Xs: list_S5253404628753938141on_nat] :
      ( ( uminus5421032341409327564on_nat @ ( coset_3487183839294692122on_nat @ Xs ) )
      = ( set_Su4848624856859743608on_nat @ Xs ) ) ).

% compl_coset
thf(fact_1096_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( ord_less_eq_nat @ K @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s3379185846251495609od_a_o @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_li3248255784598043854od_a_o @ Xs @ I2 )
                = ( nth_li3248255784598043854od_a_o @ Ys @ I2 ) ) )
         => ( ( take_l5436743365630562101od_a_o @ K @ Xs )
            = ( take_l5436743365630562101od_a_o @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_1097_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ord_less_eq_nat @ K @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s5686234912572693873on_nat @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_Su8471564416988646406on_nat @ Xs @ I2 )
                = ( nth_Su8471564416988646406on_nat @ Ys @ I2 ) ) )
         => ( ( take_S1147720861222069983on_nat @ K @ Xs )
            = ( take_S1147720861222069983on_nat @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_1098_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_nat @ Xs @ I2 )
                = ( nth_nat @ Ys @ I2 ) ) )
         => ( ( take_nat @ K @ Xs )
            = ( take_nat @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_1099_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( ord_less_eq_nat @ K @ ( size_s7490768528479217773on_nat @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s7490768528479217773on_nat @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_Pr3336893728888831618on_nat @ Xs @ I2 )
                = ( nth_Pr3336893728888831618on_nat @ Ys @ I2 ) ) )
         => ( ( take_P4655611586102217833on_nat @ K @ Xs )
            = ( take_P4655611586102217833on_nat @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_1100_listrel1E,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
     => ~ ! [X: nat,Y4: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R )
           => ! [Us3: list_nat,Vs2: list_nat] :
                ( ( Xs
                  = ( append_nat @ Us3 @ ( cons_nat @ X @ Vs2 ) ) )
               => ( Ys
                 != ( append_nat @ Us3 @ ( cons_nat @ Y4 @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_1101_listrel1E,axiom,
    ! [Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) )
     => ~ ! [X: int,Y4: int] :
            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y4 ) @ R )
           => ! [Us3: list_int,Vs2: list_int] :
                ( ( Xs
                  = ( append_int @ Us3 @ ( cons_int @ X @ Vs2 ) ) )
               => ( Ys
                 != ( append_int @ Us3 @ ( cons_int @ Y4 @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_1102_listrel1E,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ Ys ) @ ( listre1270753918075970734on_nat @ R ) )
     => ~ ! [X: sum_su4907927656612468301on_nat,Y4: sum_su4907927656612468301on_nat] :
            ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X @ Y4 ) @ R )
           => ! [Us3: list_S5253404628753938141on_nat,Vs2: list_S5253404628753938141on_nat] :
                ( ( Xs
                  = ( append3523812819975775986on_nat @ Us3 @ ( cons_S6797538523646374167on_nat @ X @ Vs2 ) ) )
               => ( Ys
                 != ( append3523812819975775986on_nat @ Us3 @ ( cons_S6797538523646374167on_nat @ Y4 @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_1103_listrel1I,axiom,
    ! [X2: nat,Y2: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat,Us: list_nat,Vs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R )
     => ( ( Xs
          = ( append_nat @ Us @ ( cons_nat @ X2 @ Vs ) ) )
       => ( ( Ys
            = ( append_nat @ Us @ ( cons_nat @ Y2 @ Vs ) ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% listrel1I
thf(fact_1104_listrel1I,axiom,
    ! [X2: int,Y2: int,R: set_Pr958786334691620121nt_int,Xs: list_int,Us: list_int,Vs: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y2 ) @ R )
     => ( ( Xs
          = ( append_int @ Us @ ( cons_int @ X2 @ Vs ) ) )
       => ( ( Ys
            = ( append_int @ Us @ ( cons_int @ Y2 @ Vs ) ) )
         => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) ) ) ) ) ).

% listrel1I
thf(fact_1105_listrel1I,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Y2: sum_su4907927656612468301on_nat,R: set_Pr2535564044409765851on_nat,Xs: list_S5253404628753938141on_nat,Us: list_S5253404628753938141on_nat,Vs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X2 @ Y2 ) @ R )
     => ( ( Xs
          = ( append3523812819975775986on_nat @ Us @ ( cons_S6797538523646374167on_nat @ X2 @ Vs ) ) )
       => ( ( Ys
            = ( append3523812819975775986on_nat @ Us @ ( cons_S6797538523646374167on_nat @ Y2 @ Vs ) ) )
         => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ Ys ) @ ( listre1270753918075970734on_nat @ R ) ) ) ) ) ).

% listrel1I
thf(fact_1106_take__Suc__conv__app__nth,axiom,
    ! [I3: nat,Xs: list_int] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
     => ( ( take_int @ ( suc @ I3 ) @ Xs )
        = ( append_int @ ( take_int @ I3 @ Xs ) @ ( cons_int @ ( nth_int @ Xs @ I3 ) @ nil_int ) ) ) ) ).

% take_Suc_conv_app_nth
thf(fact_1107_take__Suc__conv__app__nth,axiom,
    ! [I3: nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( take_l5436743365630562101od_a_o @ ( suc @ I3 ) @ Xs )
        = ( append2159120467191820514od_a_o @ ( take_l5436743365630562101od_a_o @ I3 @ Xs ) @ ( cons_l9179691953475787517od_a_o @ ( nth_li3248255784598043854od_a_o @ Xs @ I3 ) @ nil_li1372968520647675725od_a_o ) ) ) ) ).

% take_Suc_conv_app_nth
thf(fact_1108_take__Suc__conv__app__nth,axiom,
    ! [I3: nat,Xs: list_S5253404628753938141on_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( take_S1147720861222069983on_nat @ ( suc @ I3 ) @ Xs )
        = ( append3523812819975775986on_nat @ ( take_S1147720861222069983on_nat @ I3 @ Xs ) @ ( cons_S6797538523646374167on_nat @ ( nth_Su8471564416988646406on_nat @ Xs @ I3 ) @ nil_Su3078906344068544711on_nat ) ) ) ) ).

% take_Suc_conv_app_nth
thf(fact_1109_take__Suc__conv__app__nth,axiom,
    ! [I3: nat,Xs: list_nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
     => ( ( take_nat @ ( suc @ I3 ) @ Xs )
        = ( append_nat @ ( take_nat @ I3 @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I3 ) @ nil_nat ) ) ) ) ).

% take_Suc_conv_app_nth
thf(fact_1110_take__Suc__conv__app__nth,axiom,
    ! [I3: nat,Xs: list_P9127870862658512897on_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s7490768528479217773on_nat @ Xs ) )
     => ( ( take_P4655611586102217833on_nat @ ( suc @ I3 ) @ Xs )
        = ( append2865821678206098582on_nat @ ( take_P4655611586102217833on_nat @ I3 @ Xs ) @ ( cons_P1582596238967110705on_nat @ ( nth_Pr3336893728888831618on_nat @ Xs @ I3 ) @ nil_Pr8614093721167239297on_nat ) ) ) ) ).

% take_Suc_conv_app_nth
thf(fact_1111_bind__simps_I2_J,axiom,
    ! [X2: nat,Xs: list_nat,F: nat > list_nat] :
      ( ( bind_nat_nat @ ( cons_nat @ X2 @ Xs ) @ F )
      = ( append_nat @ ( F @ X2 ) @ ( bind_nat_nat @ Xs @ F ) ) ) ).

% bind_simps(2)
thf(fact_1112_bind__simps_I2_J,axiom,
    ! [X2: int,Xs: list_int,F: int > list_nat] :
      ( ( bind_int_nat @ ( cons_int @ X2 @ Xs ) @ F )
      = ( append_nat @ ( F @ X2 ) @ ( bind_int_nat @ Xs @ F ) ) ) ).

% bind_simps(2)
thf(fact_1113_maps__simps_I1_J,axiom,
    ! [F: int > list_nat,X2: int,Xs: list_int] :
      ( ( maps_int_nat @ F @ ( cons_int @ X2 @ Xs ) )
      = ( append_nat @ ( F @ X2 ) @ ( maps_int_nat @ F @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_1114_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% nat.inject
thf(fact_1115_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_1116_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_1117_Suc__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_1118_Suc__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% Suc_less_eq
thf(fact_1119_Suc__le__mono,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
      = ( ord_less_eq_nat @ N @ M2 ) ) ).

% Suc_le_mono
thf(fact_1120_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_1121_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_1122_negative__zless,axiom,
    ! [N: nat,M2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).

% negative_zless
thf(fact_1123_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_1124_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_1125_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_1126_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_1127_old_Onat_Oexhaust,axiom,
    ! [Y2: nat] :
      ( ( Y2 != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y2
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_1128_nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P @ N2 )
           => ( P @ ( suc @ N2 ) ) )
       => ( P @ N ) ) ) ).

% nat_induct
thf(fact_1129_diff__induct,axiom,
    ! [P: nat > nat > $o,M2: nat,N: nat] :
      ( ! [X: nat] : ( P @ X @ zero_zero_nat )
     => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
       => ( ! [X: nat,Y4: nat] :
              ( ( P @ X @ Y4 )
             => ( P @ ( suc @ X ) @ ( suc @ Y4 ) ) )
         => ( P @ M2 @ N ) ) ) ) ).

% diff_induct
thf(fact_1130_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_1131_Suc__neq__Zero,axiom,
    ! [M2: nat] :
      ( ( suc @ M2 )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_1132_Zero__neq__Suc,axiom,
    ! [M2: nat] :
      ( zero_zero_nat
     != ( suc @ M2 ) ) ).

% Zero_neq_Suc
thf(fact_1133_Zero__not__Suc,axiom,
    ! [M2: nat] :
      ( zero_zero_nat
     != ( suc @ M2 ) ) ).

% Zero_not_Suc
thf(fact_1134_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M4: nat] :
          ( N
          = ( suc @ M4 ) ) ) ).

% not0_implies_Suc
thf(fact_1135_not__less__less__Suc__eq,axiom,
    ! [N: nat,M2: nat] :
      ( ~ ( ord_less_nat @ N @ M2 )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
        = ( N = M2 ) ) ) ).

% not_less_less_Suc_eq
thf(fact_1136_strict__inc__induct,axiom,
    ! [I3: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I3 @ J )
     => ( ! [I2: nat] :
            ( ( J
              = ( suc @ I2 ) )
           => ( P @ I2 ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ J )
             => ( ( P @ ( suc @ I2 ) )
               => ( P @ I2 ) ) )
         => ( P @ I3 ) ) ) ) ).

% strict_inc_induct
thf(fact_1137_less__Suc__induct,axiom,
    ! [I3: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I3 @ J )
     => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
       => ( ! [I2: nat,J3: nat,K2: nat] :
              ( ( ord_less_nat @ I2 @ J3 )
             => ( ( ord_less_nat @ J3 @ K2 )
               => ( ( P @ I2 @ J3 )
                 => ( ( P @ J3 @ K2 )
                   => ( P @ I2 @ K2 ) ) ) ) )
         => ( P @ I3 @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_1138_less__trans__Suc,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I3 @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I3 ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_1139_Suc__less__SucD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_less_SucD
thf(fact_1140_less__antisym,axiom,
    ! [N: nat,M2: nat] :
      ( ~ ( ord_less_nat @ N @ M2 )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
       => ( M2 = N ) ) ) ).

% less_antisym
thf(fact_1141_Suc__less__eq2,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M2 )
      = ( ? [M5: nat] :
            ( ( M2
              = ( suc @ M5 ) )
            & ( ord_less_nat @ N @ M5 ) ) ) ) ).

% Suc_less_eq2
thf(fact_1142_All__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N ) )
           => ( P @ I ) ) )
      = ( ( P @ N )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ N )
           => ( P @ I ) ) ) ) ).

% All_less_Suc
thf(fact_1143_not__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M2 @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).

% not_less_eq
thf(fact_1144_less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ( ord_less_nat @ M2 @ N )
        | ( M2 = N ) ) ) ).

% less_Suc_eq
thf(fact_1145_Ex__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N ) )
            & ( P @ I ) ) )
      = ( ( P @ N )
        | ? [I: nat] :
            ( ( ord_less_nat @ I @ N )
            & ( P @ I ) ) ) ) ).

% Ex_less_Suc
thf(fact_1146_less__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_1147_less__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M2 @ N )
       => ( M2 = N ) ) ) ).

% less_SucE
thf(fact_1148_Suc__lessI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( ( suc @ M2 )
         != N )
       => ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).

% Suc_lessI
thf(fact_1149_Suc__lessE,axiom,
    ! [I3: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I3 ) @ K )
     => ~ ! [J3: nat] :
            ( ( ord_less_nat @ I3 @ J3 )
           => ( K
             != ( suc @ J3 ) ) ) ) ).

% Suc_lessE
thf(fact_1150_Suc__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_lessD
thf(fact_1151_Nat_OlessE,axiom,
    ! [I3: nat,K: nat] :
      ( ( ord_less_nat @ I3 @ K )
     => ( ( K
         != ( suc @ I3 ) )
       => ~ ! [J3: nat] :
              ( ( ord_less_nat @ I3 @ J3 )
             => ( K
               != ( suc @ J3 ) ) ) ) ) ).

% Nat.lessE
thf(fact_1152_transitive__stepwise__le,axiom,
    ! [M2: nat,N: nat,R2: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ! [X: nat] : ( R2 @ X @ X )
       => ( ! [X: nat,Y4: nat,Z4: nat] :
              ( ( R2 @ X @ Y4 )
             => ( ( R2 @ Y4 @ Z4 )
               => ( R2 @ X @ Z4 ) ) )
         => ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
           => ( R2 @ M2 @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_1153_nat__induct__at__least,axiom,
    ! [M2: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( P @ M2 )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M2 @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_1154_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
             => ( P @ M ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_1155_not__less__eq__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).

% not_less_eq_eq
thf(fact_1156_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_1157_le__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M2 @ N )
        | ( M2
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_1158_Suc__le__D,axiom,
    ! [N: nat,M6: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
     => ? [M4: nat] :
          ( M6
          = ( suc @ M4 ) ) ) ).

% Suc_le_D
thf(fact_1159_le__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_1160_le__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( M2
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_1161_Suc__leD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% Suc_leD
thf(fact_1162_Suc__inject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y2 ) )
     => ( X2 = Y2 ) ) ).

% Suc_inject
thf(fact_1163_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_1164_upt__conv__Cons__Cons,axiom,
    ! [M2: nat,N: nat,Ns: list_nat,Q3: nat] :
      ( ( ( cons_nat @ M2 @ ( cons_nat @ N @ Ns ) )
        = ( upt @ M2 @ Q3 ) )
      = ( ( cons_nat @ N @ Ns )
        = ( upt @ ( suc @ M2 ) @ Q3 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_1165_less__Suc__eq__0__disj,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ( M2 = zero_zero_nat )
        | ? [J2: nat] :
            ( ( M2
              = ( suc @ J2 ) )
            & ( ord_less_nat @ J2 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_1166_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M4: nat] :
          ( N
          = ( suc @ M4 ) ) ) ).

% gr0_implies_Suc
thf(fact_1167_All__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N ) )
           => ( P @ I ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ N )
           => ( P @ ( suc @ I ) ) ) ) ) ).

% All_less_Suc2
thf(fact_1168_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M3: nat] :
            ( N
            = ( suc @ M3 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_1169_Ex__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N ) )
            & ( P @ I ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I: nat] :
            ( ( ord_less_nat @ I @ N )
            & ( P @ ( suc @ I ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_1170_Suc__leI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).

% Suc_leI
thf(fact_1171_Suc__le__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
      = ( ord_less_nat @ M2 @ N ) ) ).

% Suc_le_eq
thf(fact_1172_dec__induct,axiom,
    ! [I3: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ( P @ I3 )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I3 @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ N2 )
                 => ( P @ ( suc @ N2 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_1173_inc__induct,axiom,
    ! [I3: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ( P @ J )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I3 @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ ( suc @ N2 ) )
                 => ( P @ N2 ) ) ) )
         => ( P @ I3 ) ) ) ) ).

% inc_induct
thf(fact_1174_Suc__le__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_le_lessD
thf(fact_1175_le__less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
        = ( N = M2 ) ) ) ).

% le_less_Suc_eq
thf(fact_1176_less__Suc__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_Suc_eq_le
thf(fact_1177_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1178_le__imp__less__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_1179_int__of__nat__induct,axiom,
    ! [P: int > $o,Z2: int] :
      ( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
     => ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
       => ( P @ Z2 ) ) ) ).

% int_of_nat_induct
thf(fact_1180_int__cases,axiom,
    ! [Z2: int] :
      ( ! [N2: nat] :
          ( Z2
         != ( semiri1314217659103216013at_int @ N2 ) )
     => ~ ! [N2: nat] :
            ( Z2
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).

% int_cases
thf(fact_1181_upt__conv__Cons,axiom,
    ! [I3: nat,J: nat] :
      ( ( ord_less_nat @ I3 @ J )
     => ( ( upt @ I3 @ J )
        = ( cons_nat @ I3 @ ( upt @ ( suc @ I3 ) @ J ) ) ) ) ).

% upt_conv_Cons
thf(fact_1182_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K2 )
               => ~ ( P @ I4 ) )
            & ( P @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_1183_upt__rec,axiom,
    ( upt
    = ( ^ [I: nat,J2: nat] : ( if_list_nat @ ( ord_less_nat @ I @ J2 ) @ ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J2 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_1184_upt__Suc,axiom,
    ! [I3: nat,J: nat] :
      ( ( ( ord_less_eq_nat @ I3 @ J )
       => ( ( upt @ I3 @ ( suc @ J ) )
          = ( append_nat @ ( upt @ I3 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I3 @ J )
       => ( ( upt @ I3 @ ( suc @ J ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_1185_upt__Suc__append,axiom,
    ! [I3: nat,J: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ( upt @ I3 @ ( suc @ J ) )
        = ( append_nat @ ( upt @ I3 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_1186_not__zle__0__negative,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).

% not_zle_0_negative
thf(fact_1187_negD,axiom,
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ zero_zero_int )
     => ? [N2: nat] :
          ( X2
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).

% negD
thf(fact_1188_negative__zless__0,axiom,
    ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_1189_inf__concat__simple_Ocases,axiom,
    ! [X2: produc8199716216217303280at_nat] :
      ( ! [F2: nat > nat] :
          ( X2
         != ( produc72220940542539688at_nat @ F2 @ zero_zero_nat ) )
     => ~ ! [F2: nat > nat,N2: nat] :
            ( X2
           != ( produc72220940542539688at_nat @ F2 @ ( suc @ N2 ) ) ) ) ).

% inf_concat_simple.cases
thf(fact_1190_upt__rec__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M2 ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_1191_hd__upt,axiom,
    ! [I3: nat,J: nat] :
      ( ( ord_less_nat @ I3 @ J )
     => ( ( hd_nat @ ( upt @ I3 @ J ) )
        = I3 ) ) ).

% hd_upt
thf(fact_1192_min__0R,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_1193_min__0L,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% min_0L
thf(fact_1194_min__Suc__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_min_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( suc @ ( ord_min_nat @ M2 @ N ) ) ) ).

% min_Suc_Suc
thf(fact_1195_int__eq__iff__numeral,axiom,
    ! [M2: nat,V: num] :
      ( ( ( semiri1314217659103216013at_int @ M2 )
        = ( numeral_numeral_int @ V ) )
      = ( M2
        = ( numeral_numeral_nat @ V ) ) ) ).

% int_eq_iff_numeral
thf(fact_1196_nat__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_numeral
thf(fact_1197_nat__neg__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = zero_zero_nat ) ).

% nat_neg_numeral
thf(fact_1198_int__ops_I3_J,axiom,
    ! [N: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% int_ops(3)
thf(fact_1199_nat__numeral__as__int,axiom,
    ( numeral_numeral_nat
    = ( ^ [I: num] : ( nat2 @ ( numeral_numeral_int @ I ) ) ) ) ).

% nat_numeral_as_int
thf(fact_1200_one__less__nat__eq,axiom,
    ! [Z2: int] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
      = ( ord_less_int @ one_one_int @ Z2 ) ) ).

% one_less_nat_eq
thf(fact_1201_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_1202_int__one__le__iff__zero__less,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ one_one_int @ Z2 )
      = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).

% int_one_le_iff_zero_less
thf(fact_1203_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_1204_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_1205_int__ops_I2_J,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% int_ops(2)
thf(fact_1206_nat__one__as__int,axiom,
    ( one_one_nat
    = ( nat2 @ one_one_int ) ) ).

% nat_one_as_int
thf(fact_1207_nat__induct__non__zero,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P @ one_one_nat )
       => ( ! [N2: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_1208_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_1209_diff__self__eq__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ M2 )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_1210_Suc__diff__diff,axiom,
    ! [M2: nat,N: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K ) ) ).

% Suc_diff_diff
thf(fact_1211_diff__Suc__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( minus_minus_nat @ M2 @ N ) ) ).

% diff_Suc_Suc
thf(fact_1212_diff__diff__cancel,axiom,
    ! [I3: nat,N: nat] :
      ( ( ord_less_eq_nat @ I3 @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
        = I3 ) ) ).

% diff_diff_cancel
thf(fact_1213_length__upt,axiom,
    ! [I3: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I3 @ J ) )
      = ( minus_minus_nat @ J @ I3 ) ) ).

% length_upt
thf(fact_1214_zero__less__diff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% zero_less_diff
thf(fact_1215_diff__is__0__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( minus_minus_nat @ M2 @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% diff_is_0_eq
thf(fact_1216_diff__is__0__eq_H,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( minus_minus_nat @ M2 @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_1217_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_1218_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_1219_Suc__diff__1,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
        = N ) ) ).

% Suc_diff_1
thf(fact_1220_diff__Suc__eq__diff__pred,axiom,
    ! [M2: nat,N: nat] :
      ( ( minus_minus_nat @ M2 @ ( suc @ N ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_1221_diff__less,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).

% diff_less
thf(fact_1222_diff__less__Suc,axiom,
    ! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).

% diff_less_Suc
thf(fact_1223_Suc__diff__Suc,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ N @ M2 )
     => ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M2 @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_1224_eq__diff__iff,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M2 @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M2 = N ) ) ) ) ).

% eq_diff_iff
thf(fact_1225_le__diff__iff,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% le_diff_iff
thf(fact_1226_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_1227_diff__le__mono,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_1228_diff__le__self,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).

% diff_le_self
thf(fact_1229_le__diff__iff_H,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C2 )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_1230_diff__le__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).

% diff_le_mono2
thf(fact_1231_diff__less__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( ord_less_nat @ M2 @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).

% diff_less_mono2
thf(fact_1232_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_1233_Suc__diff__le,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
        = ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_1234_diff__less__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).

% diff_less_mono
thf(fact_1235_less__diff__iff,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M2 @ N ) ) ) ) ).

% less_diff_iff
thf(fact_1236_minus__nat_Odiff__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% minus_nat.diff_0
thf(fact_1237_diffs0__imp__equal,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( minus_minus_nat @ M2 @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M2 )
          = zero_zero_nat )
       => ( M2 = N ) ) ) ).

% diffs0_imp_equal
thf(fact_1238_diff__commute,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K ) @ J ) ) ).

% diff_commute
thf(fact_1239_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I3: nat] :
      ( ( P @ K )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ ( minus_minus_nat @ K @ I3 ) ) ) ) ).

% zero_induct_lemma
thf(fact_1240_min__diff,axiom,
    ! [M2: nat,I3: nat,N: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M2 @ I3 ) @ ( minus_minus_nat @ N @ I3 ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M2 @ N ) @ I3 ) ) ).

% min_diff
thf(fact_1241_diff__Suc__less,axiom,
    ! [N: nat,I3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) @ N ) ) ).

% diff_Suc_less
thf(fact_1242_Suc__pred_H,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( N
        = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_1243_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
        = ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_1244_nat__numeral__diff__1,axiom,
    ! [V: num] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
      = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).

% nat_numeral_diff_1
thf(fact_1245_zle__diff1__eq,axiom,
    ! [W: int,Z2: int] :
      ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z2 @ one_one_int ) )
      = ( ord_less_int @ W @ Z2 ) ) ).

% zle_diff1_eq
thf(fact_1246_diff__nat__numeral,axiom,
    ! [V: num,V3: num] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V3 ) )
      = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V3 ) ) ) ) ).

% diff_nat_numeral
thf(fact_1247_int__le__induct,axiom,
    ! [I3: int,K: int,P: int > $o] :
      ( ( ord_less_eq_int @ I3 @ K )
     => ( ( P @ K )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ I2 @ K )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I3 ) ) ) ) ).

% int_le_induct
thf(fact_1248_int__less__induct,axiom,
    ! [I3: int,K: int,P: int > $o] :
      ( ( ord_less_int @ I3 @ K )
     => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
       => ( ! [I2: int] :
              ( ( ord_less_int @ I2 @ K )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I3 ) ) ) ) ).

% int_less_induct
thf(fact_1249_minus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( minus_minus_int @ zero_zero_int @ L )
      = ( uminus_uminus_int @ L ) ) ).

% minus_int_code(2)
thf(fact_1250_minus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( minus_minus_int @ K @ zero_zero_int )
      = K ) ).

% minus_int_code(1)
thf(fact_1251_int__diff__cases,axiom,
    ! [Z2: int] :
      ~ ! [M4: nat,N2: nat] :
          ( Z2
         != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% int_diff_cases
thf(fact_1252_int__minus,axiom,
    ! [N: nat,M2: nat] :
      ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M2 ) )
      = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ) ).

% int_minus
thf(fact_1253_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).

% int_ops(6)
thf(fact_1254_nat__diff__distrib,axiom,
    ! [Z3: int,Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ( ord_less_eq_int @ Z3 @ Z2 )
       => ( ( nat2 @ ( minus_minus_int @ Z2 @ Z3 ) )
          = ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z3 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_1255_nat__diff__distrib_H,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
       => ( ( nat2 @ ( minus_minus_int @ X2 @ Y2 ) )
          = ( minus_minus_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y2 ) ) ) ) ) ).

% nat_diff_distrib'
thf(fact_1256_zdiff__int__split,axiom,
    ! [P: int > $o,X2: nat,Y2: nat] :
      ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X2 @ Y2 ) ) )
      = ( ( ( ord_less_eq_nat @ Y2 @ X2 )
         => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y2 ) ) ) )
        & ( ( ord_less_nat @ X2 @ Y2 )
         => ( P @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_1257_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I: int,J2: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J2 @ I ) @ Js @ ( upto_aux @ I @ ( minus_minus_int @ J2 @ one_one_int ) @ ( cons_int @ J2 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_1258_nat__ivt__aux,axiom,
    ! [N: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N ) )
         => ? [I2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N )
              & ( ( F @ I2 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_1259_zabs__less__one__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
      = ( Z2 = zero_zero_int ) ) ).

% zabs_less_one_iff

% Helper facts (9)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X2: int,Y2: int] :
      ( ( if_int @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X2: int,Y2: int] :
      ( ( if_int @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( if_nat @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( if_nat @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X2: list_int,Y2: list_int] :
      ( ( if_list_int @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X2: list_int,Y2: list_int] :
      ( ( if_list_int @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_3_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X2: list_nat,Y2: list_nat] :
      ( ( if_list_nat @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X2: list_nat,Y2: list_nat] :
      ( ( if_list_nat @ $true @ X2 @ Y2 )
      = X2 ) ).

% Conjectures (2)
thf(conj_0,hypothesis,
    ! [I4: nat] :
      ( ( member_nat @ I4 @ ( set_nat2 @ cs ) )
     => ( ( c
          = ( nth_li3248255784598043854od_a_o @ cnf @ I4 ) )
       => ( ( ord_less_nat @ I4 @ ( size_s3379185846251495609od_a_o @ cnf ) )
         => thesis ) ) ) ).

thf(conj_1,conjecture,
    thesis ).

%------------------------------------------------------------------------------